Part I

N o n - l i n e a r Fie ld Trans format ions in 4 D i m e n s i o n s

Transforming fields non-linearly causes problems in quantum field theory: products

of fields at one and the same space-time point are singular and hence have to be

made well-defined prior to any application. The ambiguities inherent in any such

r~ormalization have to be understood and to be taken care of.

These remarks constitute the program for the present part: First it is recalled

what renormalization is about; then those examples are presented where non-linear

field transformations have been mastered (in 4-dimensional space-time).

R e n o r m a l i z a t i o n Theory~

a Shor t A c c o u n t of Resu l t s and P rob l ems*

DIETER MAISON

Max-P1anck-Institut fi/r Physik und Astrophysik

- Werner-Heisenberg-Institut flit Physik -

P.O.Box 40 12 12, Munich (Fed. Rep. Germany)

1. H i s to ry

Historically Quantum Field theory arose from the attempt to quantize charged

particles coupled to the electromagnetic radiation field. Already the first calcula-

tions by Dirac, Heisenberg and Pauli treating the interaction between the particles

and the radiation field as a small perturbation were plagued by infinities for energies,

polarizabilities e tc . . Not all of these came as a surprise since infinite self-energies

resp. -stresses were already known from the classical theory of point particles cou-

pled to the electromagnetic field. Although it was remarked that from a pragmatic

point of view the parameters of non-interacting (bare) particles or fields are unob-

servable and can therefore be made suitably infinite in order to cancel the infinities

arising from the interaction, this position is quite unsatisfactory as it renders the

starting point of the calculations, the Lagrangean, ill-defined. A more satisfactory

attitude is to take the divergencies as an indication that the theory is incomplete and

should be embedded in a theory behaving more decently at short distances resp. at

large momenta. A divergent but renormalizable theory could then be considered as

an 'effective' low energy approximation which is made self-consistent by the renor-

malization of a finite number of parameters diverging with the high energy cut-off.

*Chapters 2 and 3 have been added for the convenience of readers less familiar with the formalism of perturbation theory.

In fact we may even learn some interesting things studying the cut-off dependence

of the theory considering it as an 'effective' low-energy theory. For instance the

question of 'naturalness' of super-renormalizable couplings resp. mass terms arises

precisely from the self-consistency of the 'effective' theory.

However, quite independently of the particular 'philosophy' favoured to cope

with the undesirable presence of the divergencies, it turns out to be possible to

develop calculational procedures avoiding the infinities and reducing them to an

ambiguity which can be removed through a fit to the observed values of the param-

eters ('Renormalization Theory').

In the early days of perturbative quarttization the main emphasis was put on

finding simple calculational schemes mitigating the unwanted divergencies. How-

ever it was soon recognized that subtracting infinity from infinity was not a terribly

unique recipe. 'Hence there was a definite need for a structural investigation of

the divergencies of QED and its consistent removal' (Dyson). In addition, beyond

the one-loop approximation one was faced with a principal problem in form of the

so-called overlapping divergencies. The consistent removal of these turned out to be

a rather tricky entertainment leading to a satisfactory answer only after a number

of erroneous steps about which A. Wightman commented: 'Renormalization Theory

has a history of egregious errors by distinguished savants. It has a justified reputa-

tion of perversity; a method that works up to 13 th order in the perturbation series

fails in the 14 th order.' Here Wightman refers to a method of Ward to renormalize

QED that works perfectly well until one meets graphs of the type

X X

when things go wrong.

The difficulties pertaining to the proper treatment of overlapping divergencies

were finally resolved by a systematic approach based on general postulates like

locality, unitarity and Poincar~ invariance. This 'axiomatic' approach emerging

from ideas of Stueckelberg was fully formalized by Bogoliubov and resulted in a

rigorous construction of the renormalized perturbation expansion to all orders due

to the penetrating work of Hepp. A particularly powerful formulation was given

by Zimmermann, who succeeded to resolve the result of the recursive addition of

counter-terms to the Lagrangean resp. subtractions of vertex functions on Feynman

amplitudes into a closed expression called the 'forest formula'. Many of the further

developments of renormalization theory used this particularly lucid formalism.

Characterizing the renormalized theory by abstract principles instead of defining

it through a particular subtraction scheme has the advantage that one can study

its properties in a scheme independent way. It only remains to show that there

exists some method leading to the desired result, whatever method is used in any

particular case turns into a matter of convenience. Some renormalization schemes~

as e.g. Zimmermann's have simple formal properties making them ideally suited for

general considerations, whereas others like dimensional renormalization are more

suitable for actual calculations.

An approach staying as closely as possible to the 'axioms' of renormalization

theory was given by Epstein and Glaser [1]. Using the x-space support properties

of advanced and retarded Green functions they can avoid undefined quantities al-

together. The recursive construction of the perturbation series is reduced to the

problem of 'cutting' distributions. At this point the usual ambiguities of the result

emerge, which can be removed as usual by suitable normalization conditions.

The 'axiomatic' approach also turns out to be a fiducial guide on the treach-

erous field of theories with local invariance groups. As in the early days of renor-

malization theory also in this case the situation was and still is characterized by

misinterpreted calculational results: and inconsistent assumptions leading often to

paradoxical conclusions. What we can, however, learn from an excursion into the

history of perturbative renormalization is the fact that - as frequently in science -

progress is stimulated by these paradoxical results which can only be resolved by

clarifying the basic physical requirements masked by complicated calculational pro-

cedures erroneously taken to be a substitute for the latter. Clearly that does not

mean that we should underestimate the value of intelligent calculational methods

which after all make the renormalized perturbation expansion more than an exercise

in mathematics. The overwhelming success of perturbative QED in cases like the

higher order corrections to the anomalous magnetic moment of the electron or the

muon is an impressive example. In fact, for calculations beyond one loop in the

Weinberg-Salam theory it may be vital to find a renormalization scheme minimizing

the calculational effort exhausting easily the capacities of even the biggest existing

computers.

The renormalized perturbation expansion has also been a powerful guide for

non-perturbative considerations. Much of the work of LSZ on quantum field theory

has been abstracted from the perturbative series. Of particular importance is the

development of 'Constructive QFT' emerging from the attempt to use renormaliza-

tions as suggested by perturbation theory, but otherwise proceed non-perturbatively.

Its recent development is strongly influenced by the close connection between 'eu-

clideanized' relativistic quantum field theories and and the theory of phase transi-

tions in statistical mechanics. The essential conceptual tool is the 'renormalization

group' of Wilson, which also provides a new understanding of the concepts of renor-

malization theory. Renormalizable theories turn out to be related to the fixed points

of the renormalization group transformation. This viewpoint supersedes the con-

ventional perturbative classification and may also allow consistent theories which

are perturbatively non-renormalizable.

2. The Free Field

The n-dimensional scalar free field ~(x) of mass m >_ 0 is a Wightman field [2]

acting on a Hilbert space of free particles, the Fock space 9 v. ~" has the structure of

a direct sum ~" = ( ~ = 0 "T'N of N-particle spaces ~'N which are symmetric tensor

products of the one-particle space ~'1 = L2(d#) with dp = 8(p2 _ m2)O(pO)dnp. ~'0 = C~ is the (no-particle) vacuum sector. ~'0 and ~'1 carry irreducible unitary

representations of the Poincar~ group through

U(h,a)f~ = ~2

U(A, a)¢(p) = eipa¢(A-lp)

inducing a unitary representation on ~'.

The free field ~(x) may be defined through its truncated Wightman functions

[2]

1

w T = 0 for k # 2

= iA+(xl -- x2, m 2)

It obeys the field equation

(a 2 + ~2)~,(x) = o

dgp(p) = f 6(p 2 - ~2)e (p° )dp(~2)

with some (signed) measure dp(x 2) leading to the two-point function

(a, ~p(x)~.(o)~) = i X A+(x, ~2)dp(~2)

If the moments Kj = f x2Jdp(a2) vanish for 0 < j < J (for J sufficiently big) the

two-point function of ~p(X) becomes differentiable. Hence generalized free fields

£pp(x) may be used as regularized versions of ~(x).

derived from the Lagrangean

= / +

Similarly one may define a generalized free field £pp(x) replacing the measure

d#(p) by a superposition

3. W i c k P r o d u c t s

The Wick products :qo(X): = :T(Xl). . . qO(Xk): can again be conveniently defined

through their vacuum expectation values

(fl, :~(Xl):... :~(xk):a)= ~ 1-IwT(zi) ~ z~=~ x ~

where the sum runs over all possible ordered pairs Zi = (xa 1, xa2) where the xa i

are elements of different Xj's.

Example:

( ~ , :~( Xl)~P( X2 )::~P(x3)~( x4):~ ) =

-- A + ( X l -- x 3 ) A + ( x 2 - - x4 ) - - A + ( X l - - x 4 ) A ÷ ( x 2 - - x3)

The Wick product :qo(X): remains well-defined even if all the elements of X =

{Xl , . . . , xr} coincide, leading to the Wick power ~ .

The definition of Wick products can be generalized to derivatives of qo(x) intro-

ducing a suitable multi-index notation [1]

: ~ r : ( x )

r! -- : 1-I l_~( Oa ~( x) ~r(a).

r ( . ) ! ' r , . •

where only a finite number of r(a) are different from zero.

The vacuum expectation values (~, ~ . . . :grk ' -~f l ) can be evaluated with r l l ~'k! /

the formula given above. To each term A+(Z1) . . . A+(Zk) corresponds a graph G

whose vertices are the x i and whose lines connect the vertices given by the Zj's.

Example:

A + ( x 1 - - x 2 ) A + ( x 1 - - x 3 ) A + ( x 2 - - x 3 ) A + ( x 2 -- x3 ) contributing to

(a, :~2:(~):~3:(~2 ) :~3:(~3)a ) 2! 3! 3~

gives the graph

2

3

Wick's theorem allows to expand multiple Wick products into simple ones:

:T(X1):... :¢P(Xk): ----

(a, :~(X 1 \ Y1):..- :~(Xk \ Yk) :~) :v(Y1) .. . ~(Yk): l~cx~

From this formula one easily derives the corresponding expansion for products of

Wick powers

rl ! "'" rk ! :~(rl--Sl):(Xl) :~p(rk--Sk):(Xk)O,~ ~Sl (Xl)

' - s k ) ! - ' : Sl ! "'" sk !

Analogous formulae hold for generalized free fields. Sufficiently regularized free fields

y)p(X) allow for the definition of the time-ordered functions resp. products

(~,T:~p(X1):. . . :~p(Xk):~) obtained by replacing the A+ functions by (regular-

ized) Feynman propagators

1 [ e ipz AF, p (2~r)n J p2 _ g2 + io ampdp(~2)"

The corresponding graphs are called Feynman graphs.

The generalization of Wick's theorem to time-ordered products (well-defined

only for regularized fields) is

T:~(X1): . . . :~(Xk): =

(a, T:T(X 1 \ Y1):.-. :T(Xk \ Vk):fl):T(Yl)..- ~(Yk): ~CX~

10

4. The Sca t te r ing ope ra to r

The scattering operator S (S-matrix) providing a unitary map between the Fock

spaces of in- and outgoing asymptotic particles can be characterized by 'axioms'

derived from its physical interpretation. Following Bogoliubov [3] one considers the

scattering operator S(g) in the presence of 'external' classical fields g(x) assumed

to be smooth and localized (e.g. of compact support) which are coupled to suitable

quantum fields. The corresponding interacting quantum fields can then be defined

by

O(x) = S(g,O)-ii6h(x) S(g,h)[h: 0

where we have distinguished the particular field O by its external field h(x).

In order to avoid problems with interactions of infinite duration resp. spatial

extension it is convenient to replace also the coupling constants by such localized

functions. The adiabatic limit g(x) -+ const, can then be studied separately. Hence

we shall for the moment not distinguish between external fields and coupling con-

stants.

The required properties of S(g) are:

i) S(O)= 1 (Normalization)

ii) U(A,a)S(g)U(A,a) -1 = S(D(A)g(A-I (x -a) ) ) (Poincax~ invariance) where

D(A) is the finite dimensional representation of the homogeneous Lorentz

group corresponding to the covaxiance of g(x)

iii) S(g)S+(g) = S+(g)S(g) = 1 (Unitarity)

iv) If the support of g lies outside the causal past of the support of h,

i.e. supp g VI (supp h + l / - ) = ~), then

S(g)- lS(g + h) = S(h) (Causality)

The perturbation expansion of S(g) is a power series in the coupling constants

resp. external fields g(x)

c¢ i k / S(g) = 1 + ~ k.' T k ( X l , ' " , x k / g ( x l ) ' " g ( x k / I I d x i

k=l

In order to avoid questions of convergence of the series it is usuMly interpreted

as a formal power series in g. Since the individual terms of the expansion axe in

11

general unbounded operators some care has to be taken to find a suitable common

invariant domain in .~ [1]. As long as one studies only Green functions (compare

next paragraph) this problem is avoided.

Given the Lagrangean E. = £0 + E.int one can use the canonical formalism to

derive a formal expression for S(g), the so-called Gell-Mann Low formula

1 i k [ = E -~.(-h) a Tf~int(Xl) "• "f~int(Xk) 1-I dxi

where the symbol T denotes time ordering.

From this expression for S(g) one reads off Tl(x)g(x ) = ~£int(X).

For the definition of the interacting field tending asymptotically to ~(x) it is

necessary to include in f~int a term f~o(x) j (x)dnx linear in the free field ~fl. This

implies [4] that Lint must be a finite linear superposition of Wick monomials ~ .

(For dimension n = 2 the free field ~(x) has vanishing canonical dimension; in this

case also infinite sums are possible.)

In contrast to the first non-trivial term in the Gell-Mann Low formula all the

higher ones are in general ill-defined due to the T-products involved. A procedure

constructing well-defined Tn's to a given T 1 resp. £~int in accordance with the 'ax-

ioms' i)-iv) is called a renormalization. Such renormalizations involve in general

some arbitrariness. This arbitrariness is however severely restricted by the validity

of the axioms i)-iv) as expressed by the

THEOREM A [3]:

Two renormalized perturbation expansions for S(g) fulfilling the axioms i)-iv)

which coincide up to the (k - 1) th order (k > 2) may differ at the k th order

at most by a completely local term of the form

r "

Ak(Xl, . . . ,Xk) = ~ Pr(O)5(Xl - x2).. 5(Xk_ 1 - - ~. (*)

where the P 's are differential operators further restricted by conditions ii) and

iii).

Such a difference is called a finite renormalization. Another central result of renor-

malization theory is that such finite renormalizations can be absorbed into a redef-

inition of T1 i.e. Lint.

12

THEOREM B [3]:

Given two renormalized expansions S(1)(g) and S(2)(g) for the same f-.int one

can find a set of A£k ' s of the form (*) such that after the replacement

1 fl~int(x) -+ f~int(X) + E -~. / AEk(X, Xl,"" ,xk-X) YI dxi

k>2

in the construction of say S(1)(g) they coincide.

The local operators A E k ( x l , . . . , xk) can be constructed recursively from the An's:

2i A£1 = A1 AE2 = A 2 - -~T£intA 1 etc.

In order to make the construction of S(g) independent of the renormalization scheme

it is necessary to specify a set of normalization conditions (compare below).

An important corollary to Theorem B is the relation between two different renor-

malizations of the same (composite) interacting quantum field, distinguished by its

source h(x), i.e. O(x) = S(g,0) -1 ~-]~S(g,h)lh= 0

COROLLARY:

The interacting quan tum fields Oi(x) (i = 1, 2) referring to two different renor-

malized S-matrices S(i) for the same Eint such that S(1)(g,0) = S(2)(g,0) are

related by the Zimmermann identity [5]

0 (I) _ 0 (2) = ~ . rj(g)O~l)(x) 3

where the sum runs over a suitable basis of interacting quan tum fields Oj and

the coefficients rj(g) are at least of order g.

Theorem B is also exploited in the renormalization through qnfinite ' counterterms

added to if.inf. Introducing regularized (cut-off) fields ~Op one may take the naive

Gell-Mann Low formula for S(g)p replacing £int by £int + A£p where A/2p is

chosen such that S(g)p has a well-defined limit with the properties i)-iv) when the

regularization p is removed.

If the original Lagrangean contains all the terms required as counterterms the

replacement E -+ E + AE amounts to a replacement g --+ gren + Agp and a renor-

realization of £0. In order to decide which terms are required as counterterms or,

]3

to pose it differently, what is the minimal ambiguity introduced through the renor-

malization process one has to control the singularity of Tn(x l , . . . , Xn) for coinciding

arguments. This is achieved through the so-called Power Counting Rules based on

the assignment of a canonical dimension d to Wick powers

= E ( - - - y - + I l) n - 2

Ot

From this one arrives at the UV power counting degree w (degree of UV singularity)

{ ~ : ~ ° r l : Xl ~ " ~ ] considered as distributions [1]. of the Green functions n,T • . . r k : J

• d :~ri: w ( r l , . . . ,rk) = y~.( (--~-il. ) - n) Jr n

:~oTI: Xl :~Tk:/xk ) one assigns the degree of its To the t ime ordered product T ~ • ' ' r k :

vacuum expectation value.

As observed above £int is (for n ~ 2) a finite sum of Wick monomials. In this

case this power counting formula allows to restrict the ambiguity in the construc-

tion of the Tk's resp. to characterize the type of counter terms required in A £ k .

Theories for which the degree w of the counterterms A £ k does not increase with

k are called renorraalizable (by power counting). For this classification one takes

into account only those terms of Lint referring to the genuine coupling constants.

Among the renormalizable theories one distinguishes super-renormalizable theories

requiring only a finite number of counterterms, i.e. w >__ 0 only for finitely many

Tk's , and strictly renormalizable theories for which the degree of the Wick polyno-

mial A~: k is independent of k. Theories for which the degree of A~: k grows with k

and which therefore involve necessarily infinitely many coupling constants are called

non-renormalizable. For n -- 2 or more generally for fields of vanishing canonical

dimension some other concept of renormalizability is required.

Obviously power counting also restricts the coefficients in the Zimmermann iden-

tity.

14

5. Genera t ing Funct ions

The generating functional

ikf Z(j) = ~ ~ T(k)(xl, . . . , xk ) j (x l ) . . . j ( xk )dx l . . , dx k k=0

of the time-ordered Green functions is given in terms of the S-matrix by Z(j) = (f~, S(g, j)~2), where we have distinguished the external field coupling to qp(x) by the

letter j. From the knowledge of Z(j) one can completely reconstruct the operator

S(g, j). The perturbation expansion of the Green functions yields terms of the type

, , :qPrl:(Yl) xPrt:(Yl) f~) . . .

which can be represented by Feynman graphs.

The generating functional for connected Green functions is given by Zc(j) =

In Z(j). This terminology refers to the fact that connected Green functions corre-

spond to connected Feynman graphs.

For the renormalization the generalized vertices or one-particle irreducible (1PI)

Green functions are most important, because their response to a change of f-'in~ is

most transparent and in addition their renormalization is sufficient to make Z(j) well-defined [3]. Their generating functional P(¢) is obtained from Zc(j) through a

Legendre transformation. Setting

¢(x) = ~j(x)Zc(j) - ¢0

with ¢0 = ~ Z c ( j ) I j = O " and resolving this equation (recursively) with respect to

j(x) as a functional of ¢(x) one puts

F(¢) = Zc( j (¢) ) - / j ( x ) ( ¢ ( x ) + ¢0)dx

To the order g the vertex functional r(¢) = f-,iat where on the r.h.s, the quantum

field qo(x) is replaced by ¢(x).

Frequently F(¢) is considered as a formal power series in h whose powers count

the number of loops of the corresponding Feynman graphs.

15

The power counting rules allow to assign a degree w to each 1PI Green function

F(k) in the expansion

ikf r(¢)= Z r(k)(xl,'",xk)¢(Xl)"'¢(xk) dxl''dxk k=0

The ambiguity in the construction of S(g) can be removed in a transparent way

posing w + 1 normalization conditions for all I'(k)'s with co _> 0.

Example: Taking f£int(x)d4x = g f ~ d4x and m 2 > 0 one may pose the

normalization conditions in momentum space

~(2) (p, _p)Ip2=rn2 = 0

O~-F (2) (p, -p)[p2=m2 = i OpZ

~,(4) (Pl, P2, P3, P4)Ip2=m2,(pi+pj)2= 4m2(iej ) = - ig

where we have suppressed 5-functions expressing momen tum conservation and as-

sumed that the symmetry ~o --+ - T is preserved.

6. I n f r a r e d P r o b l e m s

Infrared problems may arise in theories involving massless fields in the 'adiabatic

limit ' , i.e. when the space-time cut-offs from the genuine interaction Lagrangean

are removed. In the case m ~ 0 it can be proven [6] that the strong adiabatic

limit for S(g) exists in 9 v if suitable normalization conditions are properly taken

into account. On the other hand, if massless fields are involved the strong adia-

batic limit will in general not exist, because the asymptotic fields are not really

free i.e. non-interacting. In order to avoid this 'dynamical ' IR problem one may

however study the weak adiabatic limit of the off-shell Green functions T(k) resp.

F(k). At this level incurable (perturbative) IR deseases occur, if massless fields have

super-renormalizable couplings. In order to control the situation suitable IR power

counting rules were developed [7, 8].

But even when the result of IR power counting is admissible it is still neces-

sary to guarantee that massless fields do not develop super-renormalizable couplings

through radiative corrections. This has to be insured through proper normalization

conditions for certain vertex functions F(k) of these fields. In certain cases this may

]6

be impossible due to an IR-instability of the interaction Lagrangean. If this happens

it is necessary to take these radiative super-renormalizable couplings into account

already in the tree approximation thus changing completely the perturbation ex-

pansion [9].

Another form of such off-shell IR problems arises with fields of canonical dimen-

sion zero. A typical case are canonical scalars in two dimensions, e.g. the coordinate

fields of non-linear ,,-models. A further case are sub-canonical scalars as e.g. the

lowest component of vector superfields in four dimensions [10]. In these cases al-

ready the free propagator needs an IR regulator. Only very special Green functions

have a chance to have a decent adiabatic limit independent of the IR regulator in

this case. In addition this perturbative limit may not give the correct physical result

due to an illegal interchange of limits (compare discussion session l).

7. The Action Principle

Schwinger's action principle [11] first studied by Lowenstein [12] and Lam [13] in

its renormalized form describes the change of the Green functions under infinitesimal

variations of the Lagrangean. It is particularly important in the study of symmetry

transformations acting on the quantized fields.

In its naive form the action principle has two parts:

i) Infinitesimal variations of external fields, coupling constants (treated on the

same footing in this setting) resp. parameters in/30 result in

~g(x) Z(g) = ( a , T f~(x)dnxe~f £1~dx)d~xa) .

The insertion of ~ f ~(x)dnx inside the time-ordered product is called an

operator insertion [12] and also denoted by ~ f f~(x) dnx Z(g).

ii) Infinitesimal variations of the quantized field ~ yield

(fl, T~,iV(x ) e~ f z'"(z) d'~xfl) = O . ~Z(g)

For the generating functional of the renormalized Green functions Z(g) the insertions

calculated naively pick up radiative corrections. So for example ii) is changed into

where A is a local operator insertion which is at least of the order h.

17

8. S y m m e t r i e s

Up to now we have discussed the renormalization of theories for which the La-

grangean contains a complete set of Lorentz covariant Wick monomials of a scalar

field compatible with the power counting rules. In practice one is, however, inter-

ested in theories implementifig special global or local symmetry transformations. At

the classical level (tree approximation) this requires the introduction of multiplets

of fields with possibly different Lorentz covariance properties and the construction

of invariant Lagrangeans. Upon quantization it is important to give precise con-

ditions how such symmetries are to be implemented since the Lagrangean as well

as possibly non-linear field transformations are no more well defined. This remark

is also to be understood as a further warning that it is not enough to take some

classical Lagrangean, write down Feynman rules and subtract infinities. In view of

the~ambiguities inherent in the renormalization procedure it is important to give

conditions on the renormalized generating functional Z(g) resp. F(~) expressing the

symmetry of the theory and making the result renormalization scheme independent.

In the case of continuous symmetries it is standard to consider the variation

of Z(g) under infinitesimal transformations of the fields. Suppose we have a mul-

tiplet of fields ¢(x) (elementary or composite) closed under the local infinitesimal

transformations

with some constant matrices t i and that the (classical) Lagrangean has the structure

£~(x) = £inv(X) + JT(x)q~(x)

then the classical action f f..dx satisfies

+ wi) f Zdx = 0

where the Wi's are differential operators

The naive action principle ii) implies the Ward identities

WiZ(g, J) = 0

18

We may now require the same Ward identity for the renormalized generating func-

tional as a substitute for the invariance of the quantized Lagrangean.

There are essentially two different strategies to construct renormalized Green

functions obeying the Ward identities:

i) take a manifestly invariant renormalization scheme, e.g. using some invariant

regularization;

ii) take an arbitrary renormalization scheme and exploit the freedom to perform

finite renormalizations (compatible with power counting) to enforce the valid-

ity of the Ward identities.

It turns out that there are cases, where even strategy ii) fails (e.g. Adler-Bardeen

anomaly). Using the Lie algebra structure of the differential operators Wi it is

possible to characterize possible obstructions (called anomalies) to the construction

of a Z(g, J) fulfilling the Ward identities [14]. The study of these anomalies gives

rise to interesting problems in cohomology theory which triggered a fruitful dialog

between mathematicians and physicists.

9. Non- l inear field t rans format ions

Particular problems arise with non-linear field transformations typical for fields

taking their value on general manifolds (e.g. non-linear a-models). In view of the

geometrical nature of such theories one has to require that also the quantized theories

should be invariant under transformations compatible with the geometric structure

of the manifold, a.e. diffeomorhisms (general coord, transfs.), affine transfs. (gauge

transfs.), isometries (rigid motions) etc.. In order to linearize them as required for

Ward identities one is in general forced to introduce an infinite string of composite

fields. A simple example is

~ = ~2 g~2 = 2~3 , , ° ' '

A possible way out of this dilemma seems to be the introduction of anti-commuting

parameters or ghost fields ~ la BRS, such that 5 2 = 0 (compare the lectures by Stora

and Blasi).

]9

10. Specific Regularization Schemes

Obviously it is very convenient to use an invariant regularization resp. renor-

malization scheme, if some symmetry is to be implemented, since the number of

terms in £'int to be taken into account is in general much larger for a non-invariant

procedure. This is a particular problem for local gauge invariances which are highly

restrictive.

There are essentially two schemes that have been invented to deal with gauge

theories:

i) dimensional regularization;

ii) higher covariant derivative regularization.

Dimensional regularization works well for vector gauge theories like QCD, but fails

for chiral gauge theories like the Weinberg-Salam theory. Even when the axial

anomaly cancels algebraically there are troubles with ")'5 in the dimensional scheme.

Also for supersymmetric theories dimensional regularization is not suitable (in the

sense of an invariant regularization). The modification proposed by Siegel [15] known

as 'regularization by dimensional reduction' is plagued by inconsistencies whose

effects beyond the l-loop approximation are not under control.

The higher covariant derivative method advertized in [16] does not seem to yield

a consistent renormalization procedure (compare the lecture by S~n~or).

In view of this situation it would seem highly desirable to invent some invariant

regularization scheme for chiral gauge theories with algebraic anomaly cancellation.

This could perhaps terminate the state of confusion about the applicability of the

dimensional scheme prevalent in the present literature [17].

REFERENCES

[1] H. Epstein and V. Glaser, Ann. last. Henri Poincard 29 (1973) 211.

[2] R. Jost, The General Theory of Quantized Fields, Amer. Math. Soc., Prov-

idence R.I., 1965.

20

[3] N.N. Bogoli'ubov and D.V. Shirkov, Introduction to the Theory of Quan-

tized Fields, Wiley-Intersience, New York, 1959.

[4] H. Epstein, Nuov. t im. 27 (1963) SS6.

[5] W. Zimmermann, Ann. Phys. (N. Y.) 77 (1973) 536.

[6] H. Epstein, in Renormalization Theory, G. Velo and A. Wightman eds.,

Dordrecht, 1976.

[7] J.H. Lowenstein and W. Zimmermann, Nud. Phys. B 86 (1975) 77.

[8] P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977) 55.

[9] G. Bandelloni, C. Becchi, A. Blasi and R. Collina, Commun. Math. Phys.

67 (1978) 147.

[10] O. eiguet and K. Sibold, Nud. Phys. B 247 (1984) 484, Nucl. Phys. B 248

(1984) 336 and Nucl. Phys. B 249 (1984) 396.

[11] J. Schwinger, Phys. Rev. 82 (1951) 914, Phys. Rev. 91 (1953) 713.

[12] J. Lowenstein, Commun. Math. Phys. 24 (1971) 1.

[13] Vuk-Ming P. Lain, Phys. Rev. D 8 (1973) 2943.

[14] C. Becchi, A. Rouet and R. Stora, Ann. Phys. (N.Y.) 98 (1976) 287.

[15] W. Siegel, Phys. Lett. 84 B (1979) 193.

[16] L.D. Faddeev and A.A. Slavnov, Gauge Fields, Introduction to Quantum

theory, Benjamin, Reading, 1980.

[17] a. van Damme, Nucl. Phys. B 227 (1983) 317;

M.E. Machacek and M.T. Vaughn, Nucl. Phys. B 222 (1983) 83;

I. Jack and H. Osborn, preprint DAMTP 84/2.

21

SOME GENERAL REFERENCES ON RENORMALIZATION THEORY

N.N. Bogoliubov and D.V. Shirkov, Quantum Fields, Benjamin, Cummings, Reading

Mass., 1983.

E.R. Speer, Generalized Feynman Amplitudes, Princeton University Press, Prince-

ton, 1969.

K. Hepp, Th~orie de la renormalisation, Lecture Notes in Physics Vol. 2, Springer,

Berlin, 1969.

C. de Witt and R. Stora eds., Renormalisation Theory in Statistical Mechanics and

Quantum Field Theory, Gordon and Breach, New York, 1970.

G. Velo and A.S. Wightman eds., Renormalization Theory, Reidel, Dordrecht, 1976.

O. Piguet and A. Rouet, Symmetries in perturbative quantum t~eld theory, Phys.

Rep. 76 (1981) 1.

C. Becchi, The renormalization of gauge theories, Les Houches 1983, B.S. deWitt

and R. Stora eds., Elsevier, 1984.

O. Piguet and K. Sibold, Renormalized Supersymmetry, Birkh£user, Boston, 1986.

SOME REMARKS FOR THE CONSTRUCTION OF YANG-NILLS FIELD THEORIES

Roland S~n~or

C e n t r e de P h y s i q u e T h ~ o r t q u e , Eco le P o l y t e c h n l q u e

91128 P a l a t s e a u Cedex, F r a n c e

One o f t h e most c h a l l e n g i n g p rob lem t h a t p e o p l e which a r e fond o f r i g o r o u s r e s u l t s

i n p h y s i c s would l i k e t o s o l v e i s t o p r o v e t h e e x i s t e n c e o f t h e n o n - p e r t u r b a t i v e

Yang-Mills model .

I w i l l p r e s e n t h e r e t h e f i r s t s t e p o f an a p p r o a c h t o t h i s q u e s t i o n worked o u t

in c o l l a b o r a t i o n (1) w i t h J . Feldman from U.B.C. (Canada) and J . Magnen and V.

R i v a s s e a u f rom Eco le P o l y t e c h n i q u e ( F r a n c e ) . To p r e c i s e t h e g o a l I w i l l s a y t h a t o u r

a m b l t l o n i s to c o n s t r u c t a f i n i t e volume pu re Y a n g - N i l l s E u c l i d e a n f i e l d t h e o r y in

t h e s m a l l c o u p l i n g r e g i m e . We a l s o f i x t h e gauge g r o u p t o be SU(2) , t h i s l a s t

r e s t r i c t i o n b e i n g o n l y f o r n o t a t i o n a l r e a s o n . Adding some m a t t e r f i e l d s w i l l

p r o b a b l y n o t make t h e p r o b l e m much more h a r d e r . The o n l y s t r o n g h y p o t h e s i s a r e t h e

one c o n c e r n i n g t h e f i n i t n e s s o f t h e volume s i n c e t h e removal of t h i s c o n d i t i o n w i l l

mean t h a t we know how t o d e a l w i t h t h e i n f r a r e d p rob lem i n gauge t h e o r y and t h e one

c o n c e r n i n g t h e s m a l l n e s s o f t h e c o u p l i n g c o n s t a n t . T h i s w i l l be an a l t e r n a t i v e to

t h e l a t t i c e a p p r o a c h o f T. Ba laban ( 2 ) .

In f a c t i n t h e l a s t few y e a r s , we (3) and o t h e r s (4) were a b l e t o c o n s t r u c t

a s y m p t o t i c a l l y f r e e E u c l i d e a n f i e l d t h e o r i e s l i k e t h e 8 r o s s - N e v e u i n 2 d i m e n s i o n s

and t h e i n f r a r e d ~4 i n 4 d i m e n s i o n s . The o n l y d i f f e r e n c e be tween Y a n g - N i l l s and

t h e s e t h e o r i e s i s t h e gauge i n v a r i a n c e . Up t o now t h e a p p r o a c h e s t o t h i s p r ob l em t r y

n o t to b r e a k t h e s y m m e t r i e s due t o t h i s t n v a r l a n c e . E i t h e r t h e y were a t t e m p t s to

b u i l d f i e l d s on some i n v a r i a n t m a n i f o l d s a s t h e s p a c e o f o r b i t s (5) o r to

a p p r o x i m a t e them I n gauge i n v a r i a n t way (6) o r , t h e y p r e s e r v e i n v a r i a n c e by l o o k i n g

a t t h e Wi l son a c t i o n on a l a t t i c e .

To c o n s t r u c t s o m e t h i n g as c o m p l i c a t e d a s Y a n g - N l l l s f i e l d s one needs t o s t a r t

f rom some s i m p l e r o b j e c t s t h a t we know f o r s u r e and t o g e t t h e whole t h e o r y a s a

l i m i t i n g p r o c e s s d e f i n e d w i t h t h e i r h e l p . In t h e l a t t i c e c a s e , t h e l a t t i c e p l a y s t h e

r o l e o f an u l t r a v i o l e t c u t o f f and f o r each l a t t i c e s i z e one has t h r o u g h t h e Wi l son

a c t i o n a w e l l d e f i n e d model (7) t h e main prob3em i s t h e r e f o r e t o go t o t h e

c o n t i n u o u s l i m i t i . e . t o l e t t h e l a t t i c e s i z e go to z e r o . In t h i s c a s e one work w i t h

t h e g r o u p , t h e f i e l d s which a r e a g e b r a e l e m e n t s a r e r e c o v e r e d a t t h e l i m i t . The

F e d e r b u s h a p p r o a c h u s e t h e c o n t i n u u m f i e l d s a r e smeared in o r d e r to d e f i n e l a t t i c e

e l e m e n t s . The p u r e l y c o n t i n u o u s v e r s i o n t r i e s t o d e f i n e a r e g u l a r i z e d i n v a r i a n t

23

diffusion process on the invariant manifold approching close enough the Yang-Mills

action.

The purpose of this talk is to show that it is possible to start wlth an

approximation which does not preserve gauge invarlance, the gauge Invarianoe being

recovered when we remove the approximation. In other word the Yang-Mills action is

stable with respect to some perturbations which break gauge invarlanee. It seems

surprisingly that this stability was never questioned before.

In a first section we recall what are the main ingredients which are needed to

prove the existence of an asymptotically free Euclidean field theory. We then

discuss a possible choice of covariant regularization and show that it does not

work. Finally we explain what is this stable non covariant way of regularizing the

theory.

I. Survey of the methods used in constructive field theory

In the constructive approach to asymptotically free models (such as Oross-Neveu in 2

dimensions or the infrared • 4 in 4 dimensions) when dealing with expressions of the

form

r - s ( ® ) I e W dO(x) J X

one g e n e r a l l y s p l i t s t h e a c t i o n S i n t o 2 p a r t s L I and L o and d e f i n e a r e f e r e n c e

measure with the free part Lo: it is a Oaussian measure which is perfectly well

defined in the Euclidean framework. Then integral above is replaced by

[ e -LI(O) dp(O)

For the simplicity of notations we will take as reference model an ultraviolet

asymptotically free bosonic scalar field theory.

The basic tools to control the theory are

a) truncated perturbation expansions for the fields which are small

b) the positivity of the interaction to dominate the fields which are large

c) the asymptotic freedom for the convergence purpose

24

the notion of small and large fields being explained later. By the positivity of the

interaction one means the boundedness from below of the Eucliean action.

How do we use a) b) and c)? The answer is by doing a phase space analysis. One

defines slices of momenta (Mi} i=1,2 ..... M>l. Assoclated,to these slices one

introduces:

I) a field decomposition at each point of space

~.~@i

related to the reference measure dp(~). In fact dp(@) is a Gaussian measure of mean

0 and covariance C (generally C = (-a + m2) -1) and one writes

~(p) = Z ~i(p)

i

where ~i(p) . ~(p)~i(p), with ~i(p) a function which localizes p to be roughly in

the slice i, i.e. of order M i, and

~i(p) = 1 i

Thus dp - ~ dp i from which follows the field decomposition.

2) a space c e l l decomposition

For each scale i one writes R d as an union ~i of disjoint cubes a of volume

N -dl and each space integration related to this scale is decomposed according to ~i"

3) a spllting of the fields into high and low momentum ones relatlvely to the

scale i

@h,i " j ~ i @j and @l , i j < i

At first approximation the high momentum fields wlll be the small fields and the low

momentum fields will be the large fields.

Then one pe r fo rms an e x p a n s i o n made of 2 p a r t s :

an h o r i z o n t a l one (a c l u s t e r e x p a n s i o n ) to t e s t t h e s p a t i a l c o u p l i n g

between distant cubes of a lattice ~i" If the theory is massive this gives an

exponentlal clustering

a vertical one to test the coupling between momentum slices. If the

25

renormalization has been performed this also gives an exponential decrease.

The theory is then expanded in terms of graphs whose dominant contribution

comes from the lowest order ones. These graphs have vertices localized in the cubes

of the lattices {~i} and lines given the Ci's. The coupling constant renormalization

leads at each vertex to replace the initial coupling by a running one whose index is

related to highest momentum line hooked to it. To control the flow of this running

coupling constant one needs to know the 4-point function to the one and two loops

order. More precisely in the case of the second order loops the divergent

subdiagrams have only to be renormalized "usefully".

2. How to regularize the Yang-Nills functional

To d e f i n e a Y a n g - M i l l s f u n c t i o n a l one t h e r e f o r e needs t o s t a r t f rom some

r e g u l a r i z a t i o n . We a l r e a d y d i s c u s s some o f them in t h e i n t r o d u c t i o n . Ano the r

p o s s i b l e one which i s e x t e n s i v e l y u sed in p e r t u r b a t i o n t h e o r y i s t h e d i m e n s i o n a l

r e g u l a r i z a t l o n . U n f o r t u n a t e l y we c a n n o t used i t In a f ramework o f f u n c t i o n a l

i n t e g r a t i o n s i n c e we d o n ' t know how t o d i m e n s i o n a l l y i n t e r p o l a t e f u n c t i o n a l s p a c e s .

I t r e m a i n s a method a d v o c a t e d f i r s t by B. Lee and J . Z i n n - J u s t i n (8) and t h e n by L.

Faddeev and A. S l avnov (9 ) : t h e r e g u l a r i z a t i o n by h i g h e r c o v a r i a n t d e r i v a t i v e s . ~e t

us d i s c u s s i t .

The idea is to add to the Yang-Mills Lagrangian Ly.M ' m I/4 F~vFpv, written for

example in the Landau gauge, a regularizing term L R = 1/4 A -~ D=FpvD2F v where

F a = ~pA~ v a abCAbAC ~ v - ~ A p - gC p v

ab = 6 a b a _ gcaCbAC D = - D~Dp and Dp P

The indices a,h.., are related to the gauge group and summation over them is

implicit. Since we choose SU{2) as gauge group, C ahc is the completely antlsymmetric

tensor with 3 indices.

One then extract from the whole Lagrangian the quadratic part to define in the

usual way a Gaussian measure. The corresponding propagator has for Fourier

t r a n s f o r m :

26

D~v(k)ab ffi 8 a b I ( a ~ v - kkPkV ) 1 + X k p k v 1 q s k s + kG/A4 k s k s + IJ

The i n t e r a c t i o n t e r m s a r e sums o f monomia l o f t h e form aPA a w i t h a + p - 4, a>2 when

t h e y come from F s and a + ~ ffi 8, a>2, when t h e y come f rom DSFD2F. As i t i s w e l l

known a l l t h e g r a p h s a r e r e g u l a r i z e d e x c e p t t h e 1 -1oop o n e s w h i c h n e e d an e x t r a

r e g u l a r i z a t i o n . I t was shown (F-S) t h a t a P a u l i - V i l l a r s r e g u l a r i z a t i o n d o e s n o t

b r e a k g a u g e i n v a r i a n c e a t t h e 1 - l o o p l e v e l and t h e r e f o r e can be u s e d t o c o m p l e t e t h e

r e g u l a r i z a t i o n o f t h e t h e o r y .

We c l a i m t h a t r e n o r m a l i z a t i o n b r e a k s t h e g a u g e i n v a r i a n c e o r c o n v e r s e l y t h a t i f

we w a n t t o m a i n t a i n g a u g e t n v a r i a n c e t h e n t h e r e a r e i n f i n i t i e s .

At t h e l - l o o p o r d e r , a s s a i d b e f o r e , t h e P a u l i - V i l l a r s t r i c k r e g u l a r i z e s t h e

d i a g r a m s b u i l t w i t h t h e v e r t i c e s o f F s l e a d i n g t o a new v a l u e o f g : gR " g z * / z s w i t h

t h e c o n v e n t i o n a l d e f i n i t i o n o f t h e c o u n t e r t e r m s z , . . . . S i m i l a r l y t h e r e a r e g h o s t s

c o u n t e r t e r m s z~ and z s and t h e Ward i d e n t i t i e s i m p l y z~z 2 - z s z ~. T h i s i m p l i e s t h a t

ab w h i c h was a f u n c t i o n o f g becomes a f u n c t i o n o f gR" t h e c o v a r i a n t d e r i v a t i v e Dp

Gauge i n v a r i a n c e w i l l mean t h a t t h e h i g h e r c o v a r t a n t t e r m h a s t o be r e p l a c e d by

z 3 D s ( g R ) F p v ( g R ) D S ( g R ) F p v ( g R )

But t h e r e a r e no new d i v e r g e n c e s i n t r o d u c e d by t h e v e r t i c e s o f L R, t h u s t h e

8 - p o i n t f u n c t i o n n e e d n o t t o be r e n o r m a l i z e d . On t h e o t h e r hand t h e r e a r e no

c o r r e c t i o n s o f t h e form ( - a ) S A a . T h i s i m p l i e s t h a t z 3 - 1 and g i s u n c h a n g e d i n t h e

h i g h e r c o v a r l a n t d e r i v a t i v e t e r m , t h u s l e a d i n g t o a b r e a k d o w n o f g a u g e t n v a r i a n c e .

A n o t h e r a r g u m e n t l e a d i n g t o t h e same c o n c l u s i o n h a s b e e n o b t a i n e d by

P. B r e i t e n l o h n e r and D. N a i s o n .

We t h e r e f o r e c h o o s e t o r e g u l a r i z e t h e Y a n g - N i l l s f u n c t i o n a l by u s l o g an u s u a l

E u c l i d e a n t n v a r t a n t c u t o f f , t h u s b r e a k i n g g a u g e i n v a r i a n c e .

3 . The non c o v a r i a n t r e g u l a r i z a t t o n

In t h i s s e c t i o n we w i l l d i s c u s s wha t a r e some o f t h e c o n s e q u e n c e s o f i n t r o d u c i n g a

non c o v a r t a n t r e g u l a t o r . A l l we w i l l s a y c o n c e r n ~ t h e l - l o o p o r d e r . Comments w i l l be

g i v e n a t t h e end f o r t h e n e x t o r d e r .

The r e s u l t s a r e t h e f o l l o w i n g ones :

27

I) because of the non covariance of the regulator apart from the usual

gauge Invarlant counterterms there are 2 purely Euclidean invariant contributions: a

mass correction and a term proportional to (ApAp) 2

2) the non gauge invariant logarithmic divergences are in fact finite

3) there is a large class of cutoff functions which have the property of

not spoiling the posltlvity of the interaction.

We choose for simplicity to work in the Feynman gauge with A-field propagator

ab I Dpv(k ) - 8ab SPY ~ ~(k) - 8ab 8pc D(k)

w i t h cu to f f ~. The i n f r a r e d behaviour can be taken account e i t h e r by i n t roduc ing a

fictitious mass in the propagator or by the introduction of periodic boundary

conditions. The ghost propagator has the same form.

Let us compute the term proportional to (ApAp) =- ([A[) =. The diagrams which

contribute to it are (see Fig. 1)

G z G= G 3 G .

Flg.l Contributions to A 4

One f i n d s t h a t t h e z e r o momentum c o n t r i b u t i o n i s g i v e n by

36[ D(k) 'd'k - 9 0 [ k ' ( D ( k ) 1 3 d ' k ÷ - J J J J

AS can be s e e n e a s i l y t h e l e a d i n g c o n t r i b u t i o n of e a c h i n t e g r a l i s C s t LnA, i f ^ i s

t h e u l t r a v i o l e t c u t o f f and t h e sum o f a l l t h e c o e f f i c i e n t v a n i s h e s ; t h u s t h e l - l o o p

c o n t r i b u t i o n t o A 4 i s f i n i t e .

ab S i m i l a r l y one can compute t h e 2 - p o i n t f u n c t i o n r p v ( p ) . I t i s p r o p o r t i o n a l t o

8 ab and one g e t s

28

I D ( k ) D ( p - k ) [ S p v ( l O ( p - k ) = + 8 k . ( p - k ) - 4 ( p - k ) p ( p - k ) v - 4 k p ( p - k ) v - l O k p ( p - k ) v ] d ~ k

- 6SpvJD(k)d+k - 2JD(k )D(p -k ) [ kv (k -p )p ]d4k

which correspond respectively to the diagrams Z=, Z~ and ~3 of Fig.2.

0 0

Fig.2 Contributions to A 2

One can expand the 2-point function around p~O and one gets

ab ~ab{spvbt + + (p=8 v_p~pv ) + rpv(p) = P~Pvb= - . .

The c o e f f i c i e n t h t i s negat ive and behaves as A =, hut b=

logarithmically divergent is finitel

instead of being

We have thus encounter twice the same mechanism: although we break the gauge

invariance the leading logarithmic divergences have gauge invariant coefficients.

At t h i s l e v e l , t h e one l o o p l e v e l , one can u n d e r s t a n d t h i s i n two d i f f e r e n t

ways.

One i s by u s i n g t h e p r o o f g i v e n i n t h e book o f F a d d e e v and S l a v n o v t h a t a t t h i s

o r d e r t h e P a u l i - V i l l a r s r e g u l a r i z a t i o n i s a g a u g e i n v a r i a n t r e g u l a r i z a t i o n .

C o m p a r i n g t h e e f f e c t o f t h e two r e g u l a r i z a t i o n s one can s e e e a s i l y t h e a b o v e

a s s e r t i o n b e c a u s e o f t h e a d d i t i v i t y o f t h e l o g a r i t h m s . The o t h e r way, w h i c h i s t h e

one w h i c h can be g e n e r a l i z e d , i s by w r i t i n g S l a v n o v - W a r d i d e n t i t i e s .

Let us write these identities. One starts wlth an action of the form:

29

T2(x) - a P ~ ( x ) D p ~ ( x ) ] d 4 x - [ J P ( X ) A p ( X ) d ' x L = - ¼1[F'(x' - 2

X - ~ I j ' ( x ) ( ~ - ' - ' ) ( x - Y ) ' ( Y ) d ~ x d ~ + I a P ~ ( x ) ( ~ - t - a ) ( x - y ) a p ~ ( y ) d ' x d ' y

1 - a v - ; ",

The f i r s t two i n t e g r a l s a r e t h e c l a s s i c a l Y a n g - M i l l s a c t i o n and s o u r c e t e r m s . The

o t h e r t h r e e t e r m s a r e t h e c o r r e c t i o n s due to t h e i n t r o d u c t i o n o f a c u t o f f . With

= aPAp, t h e q u a d r a t i c t e r m s g i v e a p r o p a g a t o r f o r t h e A - f i e l d

kPkV 1 ~ab [ ~ v _ (1 -x - i )k2-~m2]g(k)

Lk~+m 2

C o n s i d e r i n g _ t h e g e n e r a t i n g f u n c t i o n a l

G(J) - e U dA d~d~

we w i l l o b t a i n some i d e n t i t y by e x p r e s s i n g t h a t i t i s i n d e p e n d a n t o f any change o f

t h e i n t e g r a t i o n v a r i a b l e s . S i n c e a p a r t o f L i s i n v a r i a n t by t h e B .R .S .

t r a n s f o r m a t i o n me c h o o s e t o p e r f o r m t h e change

Ap ) Ap + 8A p w i t h ahpa . D~b(x)~b(X)8~

w i t h 8~a(X) = k Ya(X)8~

g w i t h 8~a(X) ffi 2 Cabc~h(X)~c(X)a~

o n

i~ L I -- a ( X ) e ~ dA d-~d~ ~ 0

and look a t t h e f i r s t o r d e r c o n t r i b u t i o n i n 8~. We g e t a f t e r an i n t e g r a t i o n by p a r t

w i t h r e s p e c t t o t h e q u a d r a t i c p a r t and c h o o s l n g from now on k - 1

I n t r o d u c i n g

a s

• 'IJCbcdCbof(1 o L - o

t h e g h o s t 2 - p o i n t f u n c t i o n G b a ( A ; y , x ) - < ~ a ( X ) ~ ( y ) > one can r e w r i t e i t

30

I[~pA~a(x) + I J~(u)D~cGca(A;u,x)d4ul e L =

= [ [ - g [ [ C b c d ( 1 - ~ ) ( V , U ) A P ( v ) 8-----=- v , x ) d ' u d % j j c SA~(u) Gda(A;

-;=JJCbcdCbef(1-~)(v,u)A;(u){%e(A;v,)~.eda(A;v,x)-Gde(A;v,u)~pGca(A;v,x)}]e L

Expanding Gba as a f u n c t i o n o f A and g

G b a ( A ; y , x ) = -Sab~(Y-X) - g I C a b c ~ ( Y - u ) ~ p A ~ ( u ) ~ ( u - x ) d ' u

, g = J l C a c d C b c e ~ ( Y - v ) ~ , ~ ( v - u ) ~ ( u - x ) A ~ ( u ) A : ( v ) d ' u d ' v , O(g')

one now compu te t o g i v e an e x a m p l e t h e one l o o p c o n t r i b u t i o n t o t h e 2 = p o i n t

function. One gets

pp r ~ ; ( p ) = -2Sab(PVA(p) - p=BV(p)) -4SabCV(p )

with

A(p) = [p.k D(k)D(p-k)d4k J J

If we now use the small p expansion of rPa;(p) one sees that

ppr pv = -4pvC v

thus expanding CV(p) as ap v + bp=p v + . . . one f i n d s t h a t b= = b and

3 A -~" P

b - , ~ [ ~ " ( k ) ~ ( k ) d4k 2 J

i s f i n i t e ( t a k e ~ o f t h e fo rm ~ ( k A - i ) ) .

The g e n e r a l f e a t u r e w h i c h makes t h e

i n v a r l a n t c o r r e c t i o n s ( w h i c h s h o u l d v a n i s h

c o n t r i b u t i o n i n 1 - ~. A f t e r a c h a n g e o f

d i v e r g e n c e w h i c h may a p p e a r i s c a n c e l l e d by t h i s d i f f e r e n c e .

B~(p) = [k p D ( k ) D ( p - k ) d 4 k CP(p) = [ ( 1 - ~ ( p - k ) ) k p D(k) d4k J

result finite is that the non gauge

if ~ --) 1) have in the Integrand a

variable k--)k^ -i the logarithmic

I t r e m a i n s t o show t h a t we c a n c h o o s e t h e c u t o f f i n s u c h a way t h a t t h e f i n i t e

c o n t r i b u t i o n t o ( [ A I ) ~ d o e s n o t s p o i l t h e p o s i t t v i t y o f t h e I n t e r a c t i o n i . e . h a s a

positive sign.

T h i s c a n be a c h i e v e d i n t h e f o l l o w i n g way: t a k e t h e c u t o f f f u n c t i o n t o be o f

31

the form ~(k) - I/2[~(akA -I) + ~(a-~kA-i)] and use the fact that for any reasonable

function

I~(akA_t) p d4k -- ~ -Ln(ah -I) + finite terms if p~l k 2

and for a large enough

d~k - L n ( a A -~ ) + f i n i t e t e r m i f p~ l

I[ " [ + . Then the contribution looks like

36x2 90x6 54×14 1 r Lna I - + j + finite terms = 2.55 Lna + finite terms

L 4 8 16

whe re t h e f i n i t e t e r m s a r e u n i f o r m l y bounded when ^ and a go t o +w.

F i n a l l y i t r e m a i n s t o s t u d y t h e 2 - l o o p s c o n t r i b u t i o n s . The same t y p e o f

m e c h a n i s m a p p l i e s . A g a i n , t h e ma i n f e a t u r e i s t h e f i n i t n e s s o f non g a u g e i n v a r t a n t

l o g a r i t h m i c a l l y d i v e r g e n t t e r m s . U s u a l l y 2 - l o o p t e r m s w h i c h a r e l o g a r i t h m i c a l l y

d i v e r g e n t do n o t b e h a v e a s a l o g a r i t h m b u t a s a s q u a r e o f l o g a r i t h m . However b e c a u s e

we a r e w o r k i n g i n an e f f e c t i v e c o u p l i n g c o n s t a n t s cheme i t c an be shown t h a t t h e

d i v e r g e n c e i s t h a t o f a s i n g l e l o g a r i t h m ( s e e ( 3 ) ) h e n c e l e a d i n g t o t h e same

a n a l y s i s .

REFERENCES 1 F e l d m a n , J . , Magnen, J . , R i v a s s e a u , V . , S 6n~or , R . , t o be p u b l i s h e d 2 B a l a b a n , T . , E e n o r m a l i z a t i o n Group Approach t o L a t t i c e F i e l d T h e o r i e s , Commun. Math . P h y s . 109, 249 (1987) 3 F e l d m a n , J . , Magnen, J . , R i v a s s e a u , V . , S~n~or , R . , A R e n o r m a l i z a b l e F i e l d T h e o r y The M a s s i v e 6 r o s s - N e v e u Model i n Two D i m e n s i o n s , Commun. Math . P h y s , 103, 67 (1986) 4 Gawedzk i , K . , K u p i a i n e n , A . , G r o s s - N e v e u Model t h r o u g h c o n v e r g e n t p e r t u r b a t i o n e x p a n s i o n s . Commun. Math . P h y s . 102, 1 (1986) 5 A s o r e y , M., M i t t e r , P . K . , R e g u l a r i z e d C o n t i n u u m Y a n g - M i l l s P r o c e s s and Feynman-Kac F u n c t i o n a l I n t e g r a l , Commun. Math. P h y s . 80, 43 (1981) 6 F e d e r b u s h , P . , A P h a s e C e l l Approach t o Y a n g - M i l l s T h e o r y , Commun. Math . P h y s . 107 319 (1986) 7 S e i l e r , E . , L e c t u r e N o t e s i n P h y s i c s , V o l . 1 5 9 , B e r l i n , H e i d e l b e r g , New-York: S p r i n g e r (1982) 8 Lee , B. W., Z i n n - J u s t i n , J . , P h y s . Rev. D, 5 , 3137 (1972) 9 F a d d e e v , L. D . , S l a v n o v , A. A . , Gauge F i e l d s , B e n j a m i n , R e a d i n g (1980)

NON-LINEAR FIELD TRANSFORMATIONS

Simple Examples and General Remarks

K. Sibold Max-Planck-Inst i tut fCir Physik und Astrophysik~

- Werner-Heisenberg-Inst i tu t fLir Physik - Post fach #01212, D-8000 Munich #0, Fed. Rep. of Germany

Table of Contents

I. Linear Symmetry Transformations

2. BRS-Transformations, Slavnov Identi ty

3. Wess-Zumino Model without ' Auxiliary F ie lds

t~. General Formalism for Non-linear Symmetry Transformations

Appendix: Notations, the Action Principle

I. Linear Symmetry Transiormations

In order to famil iar ize ourselves with the problems to come let us f i rs t consider a

simple model and perform therein all s teps in detail which for more complicated cases will

perhaps only be sketched.

Consider an isovector field (') qO -_ ¢~ - ~,

for which an invariant Lagrangian reads

(1)

\ t / (2)

The invariance of the classical act ion

under the f ield transformations

~ -- ~% ~ ~ (~1

can be expressed by the Ward-identity (WI)

~P

We note in addition that the algebra of the symmetry transformations (0) is translated to

33

the algebra of the WI-operators-

i.e. the W's sat isfy commuta t ion rela t ions like the angular momentum. The problem to be

solved is now easily formulated: ex tend the classical ac t ion to the genera t ing funct ional for

ve r tex funct ions

r + 4 r (7)

(formal power series), such tha t the WI (5) holds for[". The l a t t e r then expresses the sym-

met ry con ten t of the theory.

If we were to use a symmet r ic regular iza t ion and subsequent renormal iza t ion we could

very simply ensure the ex is tence of (7) and the val idi ty of (5) by giving Feynman rules

where ~ , m, g are replaced by ~ r e n ' mren ' gren" Since in general such symmet r i c regular i -

za t ions do not exis t we t r e a t already the above simple example by a more general method

(see R. S:ei 'a in [1]).

We invoke the ac t ion principle (see Appendix)

which tel ls us t ha t opera t ing with W k on ~ leads to a local inser t ion of specif ied dimension

(here #). This inser t ion is of order h a t leas t and or ig inates from the fac t tha t our t echnique

of making f in i te the one-par t i c le - i r reduc ib le diagrams was in conf l ic t with the symmet ry ;

(in the c lass ical theory the re are no subt rac t ions to be performed, P,/. was symmet r i c ,

whence the order Tn fOrAk). Using (d . Appendix)

= & k * (9)

we ar r ive at

Up to now the i n se r t i on /k k was r e s t r i c t ed only by locali ty and power counting, but the alge-

bra (6) implies addi t ional cons t ra in ts . Act ing with W l on (10), sub t rac t ing it from the cor-

responding re la t ion with l,k in te rchanged and using (6) we derive the so-cal led Wess-Zumino

consis tency condit ions

Hence to lowest order of Tl we have reduced the quantum problem of es tabl ishing (5) to

solving the classical problem

34

where 1~ k consis ts of sums of polynomials in ~ of dimension less than or equal to four.

Multiplying (12) by Ejkl and summing over k,l we have the condi t ion

In order to solve (13) for ~ 1 we use (13) again

We have produced the Casimir opera tor for our group[

(14)

(15)

(We have used the a lgebra and again (13)). Or

(16)

Since the Casimir opera tor has an inverse and commutes with W. we may solve for /~. J l

!

(17)

i.e. ~ j is the var ia t ion of a local inser t ion:

(18)

and is i tse l f local. This f ac t is suf f ic ien t for repair ing the WI (8) which was damaged by the

sub t rac t ion procedure . Eq. (10) reads now

~ = W ~ . ~ ( f - A m ) (19}

hence

and the viola t ion of the symmet ry has been pushed to the order ~ k ( instead of A k). Since

~, is local and has dimension -~ 4 it can be absorbed as a c o u n t e r t e r m into the in t e rac t ion

with which we ca lcu la te diagrams. Recurs ively we can by this procedure es tabl ish the WI to

all orders. The remain ing f ree p a r a m e t e r s a re those of the symmet r i c c o u n t e r t e r m s and are

to be fixed by normal iza t ion condit ions. The symmet ry r equ i r emen t and the normal iza t ion

condi t ions fixed uniquely the theory.

2. BRS Trans io rmat ions , Slavnov Ident i ty

As the f i r s t example of a non- l inear field t r ans fo rma t ion we discuss the Becch i -Roue t -

S tora (BRS) t r ans fo rma t ions in a pure Yang-Mills theory with s imple gauge group. They read [2]

:35

sAr = ~rc. + ~[.c,>k r]

$ c e = ~, C~ c+ /

s c _ : 1

s Z : O

=_ ~ ~ ~ A,~ c_+ (21)

(The T~generate the fundamental representation; c +

fields) B is a Lagrange mul t ip l ier f ield).

a r e t h e a n t i - c o m m u t i n g F a d d e e v - P o p o v

We observe i i rs t t h a t

- - ! (22)

i .e . t h e t r a n s f o r m a t i o n s s a r e n i l p o t e n t on t h e f ie lds . C o n v e r s e l y we c o n v i n c e o u r s e l v e s t h a t

wi th t h e a n s a t z

S C~ 'L C i " C~

the postulate of ni lpotency requires x"" to be the structure constants of a Lie algebra and

~'- ~ , 6 ~ adj. by a redef in i t ion of the fields. Hence the ni lpotency of the transformat ions

embodies tho algebraic structure on the level of the fields. The hope is that on the level of

functionals i t w i l l also determine the theory.

Since on t h e

t i ons an i n v a r i a n t

g a u g e f ie ld A t h e B R S - t r a n s f o r m a t i o n s look like local g a u g e t r a n s f o r m a -

in t e r m s of A is jus t t h e Yang-Mi l l s L a g r a n g e a n

~ ~ 1:~.~'~ ~ ,

In o rde r to ob t a in a p r o p a g a t o r for t h e f ie ld A g a u g e f ix ing is r equ i r ed

wh ich can be i n c o r p o r a t e d in a B R S - i n v a r i a n t f ash ion :

The main problem in quantizing the BRS-symn~etry consists in giving a meaning to those

field variations which are composite operators. This can be done by coupling them to ex-

ternal fields and taking care of them as of any other interaction vertex in the theory: they

contribute to Feynman rules) power counting and subtractions) admit counterterms) etc.

C o n s i d e r i n g t h e e x t e r n a l i i e lds as i n v a r i a n t unde r s we c a n e x p r e s s t h e na ive B R S - i n v a r i a n c e

o f

36

by writ ing

(29)

Now, it is suggest ive - and can indeed be rigorously just if ied (cf. b. Sect. t~) _ to rewr i te

this as a ~ . non- l inear equat ion

. - t t 'd '~ "i~t . + % ) =0 (30)

the Slavnov- ident i ty . Our task is to solve (30) w h e n ~ has been replaced by

t he ,genera t ing func t iona l f o r ve r t ex funct ions.

To begin with we observe t ha t any funct ional ~ can be split into

- - t. * (3,)

(where ~ does not depend on B) by imposing the gauge condit ion

For, this condit ion is l inear in the quant ized fields and can then be sat isf ied naively in many

renormat iza t ion schemes. Using (3I) we rewr i te the Slavnov-ident iy as

u ~ r ~ ~ ~ + ~ ) ~p ~ ~ ~ (33)

I f we want to ach ieve ~ ) - ' 0 we have a necessary condition-" d i f f e r e n t i a t i o n wi th respect to

B of (33) yields

(34)

i.e., the funct ional P depends on c and ~ only through the special combina t ion

= ft. + ~ c_ (35)

(N.B.; ( 3 ~ ) i s just the equat ion o¢ motion of the ghost c ). Again, l inear field equat ions, like

(34), can be naively implemented in many renormal i za t ion schemes, hence we adopt one of

this type and rewr i te now the Slavnov- ident i ty as

37

(37)

That this seemingly t r iv ia l l inearization of the Slavnov-identity is, in fact, of true import-

ance is revealed by the following properties of the ~,'s

It will be immedia te ly clear t ha t these re la t ions express the ni lpotency of the BRS-t ransfor -

mat ions on the funct ional level. Indeed, the ac t ion principle (App.) tells us

act) -_ A . T ' (,0)

Using on the l.h.s. (36) and on the r.h.s. (A.3) we find

I

Acting with ~ o n this equat ion and employing (38) the re follows

hence to lowest order in Tl

(,3)

With the notation b ~ ~,~dhis consistency condition for the BRS-transformation is wri t ten as

9o A : o (.,)

We note that b is completely known (in particular that the external fields transform under

b) and that it is nilpotent

~ t =- 0 (,5)

due to (39) and (29). Solving (,#) is not easy and the actual solution represents a mile-stone

in the history of Yang-Mills theory [2]. The solution of (#4) reads

" T ) (,6)

where the second term is the celebrated ehiral anomaly. Due to parity we have r = 0 in our

model and BRS-invariance can thus be restored: e.g. in one loop we define

f' = P - A (,7)

we know

38

.. ~ ~)

Recursively we can thus establish BRS-invariance to all orders.

(~8)

(~9)

3. Wess-Zumino Model without Auxiliary Fields

N = l supersymmetry t ransformat ions are linear and close off-shel l (i.e. without use of

equations of motion) due to the exis tence of so-called auxiliary fields. The simplest example

is provided by the Wess-Zumino model [3]. It has a classical act ion

~ I [ ~ , ~ ,v~ , ~ - ~ . ~ . ~ (~0)

+ ~¢.~ ~ , - ~ - F ~ e . ¢ . 1)

invariant under the supersymmetry t ransformat ions

(We use Weyl spinors, cf. [4] for notat ion and conventions). F is the auxiliary field because

it can be algebraically expressed in te rms of the other fields (on shell)

- ~ = ~ - ~ - ~ i ~ , = ~ + } ' A ~ (~2)

Inserting this into (50), (51) we obtain

which is naively invariant under

~ i n v is, of course, a per fec t ly leg i t imate action but the t ransformat ions (54) are now non-

linear and their an t i - commuta to r s close only if one uses the equation of motion for the

spinor field, i.e. on-shell. Whereas the renormalizat ion of (50), (51) is by now textbook wis-

dom [4], tha t of (53), (5#) is not quite s t ra ightforward.

In analogy to the previous case we introduce [5] an external source u for the non-linear

variation

39

(~)

and calcula te the variation of ~ in the bilinear fashion known ~rom BRS

(~6)

which is again composite. A t this point one could introduce a new source, calculate the va-

r iat ion, f ind a composi te object, etc. - potent ia l ly an in f in i te sequence. But we observe that

the spinor equation of mot ion reads

which is just what occurs in (56). E l iminat ing the breaking term by (_57) we end up wi th

Due to the non-l ineari ty of the field t ransformat ions we have a ~ ' -b i l inear Wl; but due to

the conspiracy of (56) and (54) we need only one source and have just a harmless inhomo-

geneous te rm in (58); (harmless because it is linear in the quantized field).

If we wish to set up consistency condit ions for

(59)

we have to know how an insertion t ransforms under supersymmetry and what the algebra of

those t ransformat ions with ~u.(. I~) is. The l inearization of the Wl-operator, which was so

suggestive in the BRS-case (cf. transit ion from (36) to (37)), should in fac t be viewed as

answering the question: how does an insertion t ransform? In order to find that out we recall

that insert ions are di f ferent ia l ver tex operations [6] and consider an infinitesimal variation

of the act ion

z (6o)

(61)

(Of. with BRS: ~ is the analog of ~ (37)). One can check by s t ra ightforward calculation P,

that the ~ s and L~(r)satisfy the following algebra

I~ (P) ~ o , (62)

r' ~.~,. ~2 Txi. crl * gb u,(r~ ~ g .,e ue. l~

40

Here Wp denotes the WI-operator for the translations(which yields zero if /~ is the act ion or

the ver tex functional because those are t ranslat ion invariant).

The higher order machinery is now prepared: we use (A.3) for (59), act suitably with ~fg

resp. kia on (59), and use (62). We obtain for the insertions A , ~ k the following consis tency

conditions

.txg , = 0

(~.~ ~-W~t'~'and we calcula ted to lowest order in h). It is now a ma t t e r of some algebraic work [5]

to show that as a consequence of (63) one has

i.e. the cohomology of the supersymmetry defined by (58) is still trivial (as it was in the

linear real izat ion [7]). As usual we can absorb ~k into P and proceed by induction to show

that the WI (58) holds to all orders; i.e. supersymmetry in the version of (58) has again

defined the model and by studying e.g. the Callan-Symanzik equation one comes to the

conclusion tha t physicswise the linear version [3] and the non-linear version (58) are equiva-

lent[S] .

4. General Formalism for Non-linear Symmetry Transformations [8]

Up to now we based our discussion on P . The reason for this is twofold:

(I) the lowest t e rm of ~ in its h-expansion can be identif ied with the

classical act ion on which all manipulations are most familiar;

(2) ~ governs the renormalizat ion: connected or general Green 's func-

tions are divergent only if their one-par t ic le- i r reducible parts are

so. But the action principle, the equations of motion, e tc . , are

generally proved with the help of Z (or Z c) - the generat ing func-

tional for (connected) Green 's functions. In fac t , non-linear field

t ransformat ions are eas ies t formulated on Z (or Z c) and not on P .

So let us study, again f irs t in the linear case, the t ransi t ion from ~ to Z C

" (%)

be the linear t ransformat ion law for the e lementary fields i~ m.

W~P

and to Z. Le t

(65)

(66)

41

is t he WI-opera tor . Z c is de f ined via t he L e g e n d r e t r a n s f o r m a t i o n

(67)

where

(68)

The inver se t r a n s f o r m a t i o n is given by

(69)

On Z the WI-operator reads c

On ~ = e. (71)

i t has the same form. Before proceeding let us note that for composite operators ~ wi th

source Jc the Legendre transformation is given by

= ~ ) t'_ (72)

in contrast to (68), (69) since the Legendre transformation is perlormed only wi th respect to

elementary fields, i.e. those operators producing one-part icle poles, wi th respect to which

one can speak of one-particle reducibi l i ty or i rreducibi l i ty. Eq.(72) wi l l soon be seen to

cause the di f ferent form a WI-operator has on Z c and on ~ .

Let now ~ denote al__l] elementary fields of the theory [8]. Suppose the polynomials ~kt-d~J

occur as non-linear variations

Then we in t roduce sources Jc for t he ~ , need possibly sou rces for t he double va r i a t ions , e t c .

H e n c e we c o l l e c t all f ie lds and sources in

a s sume t h a t t he f ie ld t r a n s f o r m a t i o n law is l inear in

(75)

and admits a certain algebra

-- (76)

We f o r m u l a t e t h e WI-opera to r for t h e s e t r a n s f o r m a t i o n s on Z c in ana logy to (70) as

42

(77)

The Legendre t r ans fo rma t ion with respec t to the e l e m e n t a r y fields

: ¢+, = 4 , . ~ , 6 ~ . ~ 'P (+, ~c. l -= -7_+.+- I~.4,~ (78)

inverse: ~ ~-,~

leads to the WI-operator

~ . L t ' ) --" .: I;. ( - ~'P++

° - t ,+ , G , : )

'~e.. ) 1%+ '+'++

(79)

which is non- l inear in P . For consis tency condi t ions we have to know how an inser t ion

t ransforms (cf. a. Sect. 3)

f' -', f' * ~ . 6 t

~A

,, (?) (-) + ~ . ( O

(80)

(8l) The a lgebra of the t r ans fo rma t ions (76) leads to the a lgebra of the WI-operators

(82)

From the examples above i t should by now be clear that this algebra leads to the

consistency condit ions

for the inser t ions appear ing in

P,( ~k " ~'~ i. ) • It should also be clear that the desired Wl

can be proved if and only if the solution of (83) is the t r iv ia l one ¢~

(85)

(86)

This f inishes the p re sen ta t ion of the general case of non- l inear symmet ry t r ans fo rmat ions .

43

APPENDIX: Notat ions, the Act ion Principle

Z c

Z

is the genera t ing funct ional for ver tex ~unctions,

is the genera t ing funct ional for connec ted Green ' s funct ions,

is the genera t ing funct ional for general Green ' s funct ions.

e l emen ta ry fields

Jc sources for composi te opera tors

m,g pa rame te r s

(A.1)

The ac t ion principle [9] s t a t e s

(A.2)

i.e. opera t ing with a d i f fe ren t ia l opera tor on ~ , Z c or Z one obtains a local inser t ion of

specif ied power counting. If inser t ions a re defined as Zimrnermann-Lowens te in normal pro-

ducts [10] one has the following examples

Another simple but very powerful re lat ion is the fo l lowing

& . £ = & * ~ ( & ~ ) (A.3)

Here & ' ~ denotes the generating funct ional for al l ver tex functions housing the special

ver tex A . The Eq.(A.3) means that in the loop expansion the f i rst term is the t r i v ia l one:

the ver tex & i tself . Example; Ia ¢ t

~- . genuine loop diagrams A

i.e. only the 2-point funct ion (/\ " ~ h a s a t r ee contr ibut ion, all o thers s t a r t with genuine

loops.

REFERENCES

[ i ]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[ i0]

44

G. Velo, A.S. Wightman (eds.), Renormalization Theory, D.):Reidel Publ. Co.

Dordrecht Holland 1976

C. 13ecchi, A. Rouet, R. Stora, Ann. of Phys. 98(1976) 287

3. Wess, B. Zumino, Phys. Lett. #9B (197#) 52

O. Piguet, K. Sibold, RenormaJized Supersymmetry, Birkh~iuser Boston 1986

O. Piguet, K. Sibold, Nucl. Phys. 13253 (1985) 269

3.H. Lowenstein, Comm. Math. Phys. 2__~ (1971) l

O. Piguet, M. Schweda, K. Sibold, Nucl. Phys. 1317# (1980) 183

P. 13reitenlohner, D. Maison in Supersymmetry and its Applications= Super-

strings~ Anomalies and Supergravity (eds. G.W. Gibbons, S.W. Hawking,

P.K. Townsend) Cambridge 1986

Y.M.P. Lain, Phys. Rev. D6 (1972) 2145, 2161

T.E. Clark, 3.H. Lowenstein, Nucl. Phys. BI13 (1976) 109

3.H. Lowenstein, W. Zimmermann~ Comm. Math. Phys. #__~ (1975) 73

3.H. Lowenstein, Comm. Math. Phys. #7 (1976) 53

SUPERSPACE RENORMALIZATION OF N = i, d = 4 SUPERSYMMETRIC GAUGE THEORIES

Olivier Piguet

Theory Division, CERN, CH-1211 Geneva 23, Switzerland

Contents: I. Non-Linear Field Renormalization

II. Conformal Invariance

These two somewhat unrelated talks deal with the renormalization of N = 1 supersym-

metric gauge theories in four-dimensional space-time. We are working in the

superfield formalism, i.e., in a linear realization of supersymmetry, with a

supersymmetry- invariant gauge-fixing condition. This is to be contrasted with the

Wess-Zumino gauge approach, where the non-linear realization of supersymmetry causes

some difficulties which are still awaiting a complete solution I). However, although i 2),3)

renormalization is made simpler by the superfield approach , a substantial price

has to be paid, due to the fact that the gauge superfield is dimensionless and

massless.

Indeed, the consequences of this fact are, first, the occurrence of a non-linear

renormalization of the gauge superfield 3)'4), a phenomenon which was also met later

on in the study of two-dimensional ~-models 5)-7), and, second, off-shell infra- red

singularities due to a propagator of the form i/k 4 for this same gauge superfield.

This is the subject of the first talk, where we show that the infinite set of

arbitrary parameters describing the non-linear field renormalization *) are gauge

parameters, and thus do not contribute to physical quantities like Green functions

of gauge-invariant operators. The method of the proof consists of allowing these

parameters to transform under BRS and of proving the corresponding Slavnov identity.

This procedure is explained in Re f. 9) for the case of gauge parameters in ordinary

Yang-Mills theories, and was in fact already advocated in Re f. I0). The application

to the supersymmetric case we discuss here was given in Re fs. 3) and 4). We also

briefly describe in this first talk the use of this procedure for curing the infra-

red singularity, by introducing an infra-red cut-off mass and showing that it is a

gauge parameter 11).

The second talk deals with the problem of finite theories. More precisely, we

consider theories with vanishing Callan-Symanzik Z-functions, namely conformal

invariant theories, which can be interpreted as finite "on the mass-shell". For

these, in particular, the Green functions of gauge-invariant operators without

*) This phenomenon was also discovered, independently, by explicit one-loop graph computations 8) .

46

anomalous dimensions, e.g., conserved flavour currents, have no ultra-violet diver-

gences. We shall show that N = 1 super-Yang-Mills theories coupled with matter

indeed have vanishing fl functions, if they satisfy three conditions which can be 12),13)

checked by simple one-loop computations These criteria may be expressed in

the following way.

(I) The gauge coupling B-function vanishes in the one-loop approximation.

(2)

(3)

The anomalies of the axial currents associated with the set of chiral invarian-

ces of the superpotential, i.e., of the action describing the self-interaction

of the matter fields, vanish.

The coupling constants are completely reduced14..1 In other words, all matter

self-interaction coupling constants k I can be chosen in a consistent way as

functions of the gauge coupling constant g , so that the theory depends only

on on_~e coupling constant.

These criteria will be shown to be sufficient for the vanishing of the B-functions.

On the other hand, condition (I) is clearly necessary. Condition (3) is also neces-

sary in view of the lower-order calculations of Re f. 15). Let us mention that in

the latter reference, as well as in the remaining literature, the vanishing of the

anomalous dimensions of the matter fields is required. This is indeed sufficient for

the matter self-interaction B-functions to vanish, but in general not necessary.

Our three criteria can be seen to be fulfilled 12) by the extended N = 4 super-Yang-

Mills theory, as well as by a class of N = 2 theories, all written in terms of N = 1

superfields. This confirms the known results 16)'17) The criteria are also satis-

fied by N = 1 theories with complex representations for the matter fields 13).

I. - NON-LINEAR FIELD RENORMALIZATION 3)'4)

I.i Classical Theory

The field content of the theory is given by a set of real gauge superfields

~i(x,e,8) (dimension 0), a set of Lagrange multiplier chiral superfields BX(x, 0) (dimension i) and a set of anticommuting ghost and antighost chiral superfields

i c~(x,8) and c_(x,e) (dimensions 0 and i). No coupling with matter fields will be

*) We consider a simple gauge group; thus there is only one gauge coupling constant.

considered in this section.

use the matrix notation

47

The superscript i is the Yang-Mills index, and we shall

i i i = # zi' B = B ~i' c± = c+z i (i.i)

where the matrices z. are the generators of the gauge group in the fundamental i

representation. Notations and conventions are those of Re f. 3). The gauge group is

chosen to be simple.

The BRS transformations may be written as

se ~ = e¢c+ - g+e ~

s 0 = c + - c + + ½ [ ¢ , c + + c + ] + . . . E Q S ( ¢ , c + ) ( 1 . 2 )

SC+ = --C~

SC_ = B, sB = 0

and are nilpotent:

s 2 = 0. (1.3)

i i Introducing external superfields O

we can write an action invariant under (1.2) as: i

c+ respectively,

(1.1) is used]:

1 Tr fdSFaF Fs(~,c±,B,p,o) = _ 128gZ

and i coupled to the BRS variations of # and

[matrix notation

1 + Tr fdV{- ~ (DDc_ + DDc_)Qs(O,c +) + OQs(¢,c+) }

- Tr fdSoc 2 - Tr fdSoc 2

+ Tr fdV{(DDB + DDB)~ + =BB}

(1.4)

where

48

F(x = D[~ , e~ = e-([Dc( e¢ (1.5)

dV = d4xDD DD, dS = d4xDD, dS = d4xDD (1.6)

D , D~ are the superspace covariant derivatives.

The BRS invariance of (1.4) may be expressed by the Slavnov identity 18)

OF 6F S(F) = : Tr fdV Op 6¢

6F OF + 1"r fdS [ -~ "~-+ OF + B Oc "-':-'_] - c . c . = 0

(1.7)

and the gauge fixing [last terms in (1.4)] by the (linear) gauge-fixing condition

6F 1 . . . . . 0--B = ~ DD DD~ + = DD B. (1.8)

The theory is further specified by supersymmetry and rigid invariance

i i 0rig ~ = i [¢,m], ~ = 00 ~i' co = const. (1.9)

= ¢, c±, B, p,

18) expressed through the Ward identities

6F w C = -i ~ f6=+-~= 0

6r Wrigr = -i ~ ~6rig +~-~ = 0

(1.10)

From now on, we define the theory through the functional identities (1.7), (1.8) and

(I.I0). This is the appropriate way for the extension to the quantized theory.

Whereas the requirements (1.8) and (i. I0) are straightforward, we shall see that the

action (1.4) is not the most general classical solution of the Slavnov identity

(1.7). In order to investigate this, let us consider the following stability prob-

lem: the special solution F (1.4) being given, find the most general form of the s

perturbed action (¢ small)

49

r = r + ~A(¢, c+, c_, B, p, ~) (1.11) s

fulfilling all of our requirements, and having its dimension bounded by four in

order to preserve power-counting renormalizability. From supersymmetry and rigid

invariance (1.10), we know that A is a linear combination of superspace integrals of

rigid-invariant superfield monomials - there is an infinite set of them, since @ is

dimensionless.

The gauge-fixing condition (1.8) implies

r(~, c+, c_, B, p, o) = r(¢, c+, ~, o)

+ F~I~--~DDB + DDB)@ + BB}

(i.12)

with

= p - ~(DDc_ + DD~_). (1.13)

That the dependence on c_ and p occurs through the combination D (1.13) is a conse-

quence of the ghost equation

6 1 ~ DD 6 Gr = : [ -~-f_+ ~ -~p ] r = 0 (1.14)

which in turn follows from the gauge-fixing condition and the Slavnov identity.

The ansatz (1.12) allows us to write the Slavnov identity (1.7) in the form

S(r) = ½ B~F = 0 (I.15)

with

(1.16)

(We drop all trace and integration measure symbols.) The functional-dependent linear

operator (1.16) obeys the identities

B B y = O, V Y (1 .17) Y Y

50

B 2 = 0, if B y = 0. (1.18) Y Y

F and F s being each decomposed according to (1.12), (1,11) reads

= + ~ A(~, c+, ~], ~). (1.19) ~s

Substituting this in the Slavnov identity (1.15) and retaining the terms of first

order in E, we obtain for A the equation

bA = 0 ( 1 . 2 0 )

with b = : Brs-, b 2 = 0 (1.21)

[The nilpotency of b follows from (1.18) since F is a solution of the Slavnov iden- S

tity.] Note that b, when acting on ~ and c+, coincides with the BRS operator s

(1.2). But it acts non-trivially on the external fields:

6~s ~s b D = ~-, bo = 6c+ (1.22)

To solve (1.20) is a cohomology problem, with the coboundary operator given by

(1.21). The most general solution A having ghost number 0 *) and dimension 4 has the

form 4)

A

A = z FSYM(#) + b A(<~, c+, ~, o) (1.23)

where

= _ __I_I ~dSF~F FSYM 128 Tr (I 24)

is the super-Yang-Mills gauge-invariant action occurring in (1.4), and ~ is an

arbitrary local functional of dimension 4 and ghost number -i:

A

A = Tr fdVf(~)D - [x Tr fdSc+o + c.c.] (1.25)

with

f(~) = k~ I Xk(~) k (1.26)

....................................................................................

*) The ghost numbers of ~, c+, c_, p, ~ are O, i, -i, -i, -2 respectively.

51

or, more precisely:

Qk t m . ¢i I .ik fi (~) = k~l ~i Xk,m i(il...1 k) "" (1.27)

• are the ~k invariant tensors of rank k+l, symmetric in their k where t~(il...ik)

last indices (rigid invariance is taken into account), z, x and Xk, ~ are arbitrary

parameters.

Computing b~ according to the definitions (1.21) and (1.16) we find (integration

measures and trace symbols omitted)

#s 6~s x(c÷ 6~s 6Ps~ bA : f { f i T i - ~i~J f i 6-~j ÷ ~ - ° T~-~} ( l .2g)

5.f. = (5/8#J)fi(~). with 3z

Substituting (1.23) into (1.19) yields, at the first order in g,

?(~, c+, D, ~) = Fs(~, c+, ^ D,̂ ~)Jg2+g2_ez (1.29)

with

¢i = ¢i + efi(~)' Ni = Di - e~jSifj(#)

J% =

c+ (l+~x)c+, ~ : (l-~x)~.

(i.3o)

This means that the general solution in the neighbourhood of the special solution s

is obtained by a coupling constant renormalization g2 + g2 _ ez and the field

substitutions (1.30). For c+ this is just a usual field amplitude renormalization,

but for # we have a generalize&, non-linear field amplitude renormalization.

We notice that the sources ~ and ~ for the BRS transformations of # and c+ are

redefined, too: this amounts to a redefinition of the BRS transformation laws (1.2)

s ÷ s, such that

e <~= e ¢ ~ + - c+ e ~, ~c+ :-cA2+ (1.3l)

This is in fact 3) the most general change of s keeping its nilpotency - which is

implicitly contained in the definition of the Slavnov operator [see (1.17) and

(I.i8)].

52

The relevance of studying the general solution in the infinitesimal form (i. II) lies

in the fact that it yields the general structure of the counterterms of the

quantized theory in the perturbative framework.The occurrence of the non-linear

renormalization (1.30) was indeed confirmed by explicit one-loop computations 8)

which showed the presence of infinities, absorbable only through a non-linear

redefinition of ~.

One has, however, to look for the general classical solution of the Slavnov identity

in finite form, since this is the starting point for the perturbative construction

of the quantum theory. It turns out 4) that this general solution is again obtained

by a substitution exactly as in (1.29), but now with the from a special solution Fs

finite field redefinitions

^ ^

#i = Fi(~)' ~i = Dj[ Fj-I(%)]$ = F(~)

A Zc+ ~ = Z- 1 (~ c÷ = C+~ C+

(1.32)

where [in the short-hand notation (1.26) instead of (1.27)]

F(#) = Z~ # + k~2 ak(~)k (1.33)

is an a r b i t r a r y i n v e r t i b l e , d i m e n s i o n O, f u n c t i o n of ~. Zc+ , Z a k are a r b i t r a r y

constants.

We may conclude that the theory depends on infinitely numerous parameters and hence

is non-renormalizable! The following formal argument suggests that the parameters

akare in fact gauge parameters, hence non-physical. (A rigorous proof will be given

in the next subsection for the quantized theory.) We first observe that the A

substitutions (1.32) for ~ and ~ (we now take c+ = c+, ~ = o) take place in F as

defined by (1.12), and not in the whole action F . But if, after this, we perform s

the inverse transformation ~ ÷ F-I(~), and similarly for ~, in the whole action F

(this defines a canonical transformation) we arrive at the equivalent action

F'(~,c+,c_,B,p,(~) = Fs(#,c+,~],c~) + f{l (DDB + DD B) F-I(~) + ~BB} (1.34)

We see in this new formulation that the n~mbers a k now parametrize the non-linear

gauge fixing condition

6F ' I -- 6B = ~ DD DD F-I(Q) + cc D'D B (i.35)

which replaces the linear condition (1.8): they are indeed gauge parameters.

53

1.2 Renormalization

The quantum theory is described by the generating functional Z(J¢,Jc+,Jc_,JB,p,~ ) of

the Green functions, or by

Z c = T log Z (1.36)

which generates the connected Green functions, or by the vertex functional

F(~,c+,c_,B,p,~) = Zc(J¢,Jc+,Jc_,JB,p,~) - f{J¢~ + Jc+C+ + Jc_C_ + JB B} (1.37)

which generates the one-particle irreducible graphs and coincides with the classical

action at ~ = O, ~ being taken as the perturbation expansion parameter (loop

expansion). The theory is defined by requiring the supersymmetry and rigid

invariance Ward identities (i. I0), the linear gauge fixing condition (1.8) and the

Slavnov identity (1.7). The latter reads, for the Green functional Z, in short-hand • *)

notatlon :

6 6 6 SZ =: f{-J# -~p + Jc+ ~ + Jc_ 6--~B} Z - 0 (1.38)

with S 2 = 0.

As we Nave seen, the theory depends on the infinite set of parameters a k describing

the general non-linear ~-field renormalization. We will now show that this depen-

dence has the peculiar form

Da k Z ~ S(Ak.Z) (1.39)

where A k are some insertions with ghost number -1, or, equivalently for F:

Dak p N B~ (Ak.P) (1.40)

with B~ defined by (1.16). The consequence of (1.39) is that the S-matrix - if it

can be defined - is independent of the ak's. More generally one can define the Green

*) In order to avoid here any infra-red problem caused by the presence of dimension- less fields we introduce masses which preserve supersymmetry but break BRS invariance. Hence the Slavnov identity can only hold up to soft terms: this is expressed by the sign ~ in all subsequent identities.

54

functions of gauge invariant operators Qa through the introduction of BRS invariant

external fields qa" Their generating functional

Zinv(q) = Z(J,P,~,q)Ij=p=o= 0 (1.41)

is then ak-independent:

~a k Zinv(q) = [S(Ak.Z)]j=p=o= 0 = 0. (1.42)

The validity of (1.39) or (1.40) is easily checked in the classical approximation.

[In the infinitesimal form (I.ii) this directly follows from the fact that the a k-

(or Xk-) dependent part of the perturbation A is of the form B~f(~)k~ as can be seen

from Eqs. (1.23)-(1.26).]

In order to extend the property (1.40) to the quantum theory, we require the latter

to obey the new Slavnov identity

~r S(r) = Sold(r) + ~ Xk~k ~ 0 (1.43)

where we have introduced an infinite set of anticommuting parameters x k. This

amounts to considering the ak's as transforming under BRS (in a way respecting the

nilpotency):

s a k = Xk, sx k = 0 (1.44)

It is checked that this works by differentiating (1.43) with respect to Xk, which

yields

~r B~ ~r ~ 0 (1.45) Da k ~x k

with

B~ = B~ Id + ~ x k ~ (1.46) ~a k

Equation (1.45) indeed reproduces (1.40) with the identification

~r (1.47) Ak.r = Dx k "

55

The construction of an Xk-dependent classical action fulfilling the new Slavnov

identity is straightforward. By standard arguments 18) we know that the construction

is then feasible at all orders of perturbation theory if the cohomology equation

bA = 0 (1.48)

admits only trivial solutions

A

A = bA. (1.49)

Here, A being a local functional A(~,c+,D,o,ak,x k) of dimension four and ghost

number one, the solution A must be local, too, with dimension four and ghost

number zero. The coboundary operator is

b = B - , b 2 = 0. (1.50) Fclassical

In particular

ba k = Xk, bx k = 0. (1.51)

4) Concerning the x k and a k dependence of A, the cohomology is that of polynomials ,

and is thus trivial:

A

A = bA 1 + A 2 (1.52)

with

DA 2 ~A 2

bA 2 = 0, Da k ~x k 0. (1.53)

The remaining cohomology problem (1,53) is well known3)'19): the only non-trivial

solution is the chiral anomaly, which we assume to be absent.

We have proved in this way the possibility of constructing a theory obeying the new

Slavnov identity (1.43). In other words there always exists a set of insertions A k

such that the physical a k independence condition (1.39) holds.

*) The ak's play the role of coupling constants. Since they couple with increasing powers of ~ as k increases, any term of a given order in4~ (i.e., given number of loops) and of a given degree in ~ can only depend on a finite number of a k.

56

1.3 The off-shell infra-red problem 3)'II)

Since the e = 0 component of the gauge superfield ¢ is of dimension zero, and

massless in the case of strict gauge invariance, it has a propagator of the form

I/k 4, which causes infra-red (IR) divergent Green functions.

In order to cure this disease we take advantage of the freedom of doing an arbitrary

field redefinition (1.32). We choose a e-dependent redefinition

¢ + F(¢) = (1 + ½ #2~)2~2)¢ (1.54)

where ~ has the dimension of a mass (~ must be changed accordingly). The

substitution of (1.54) in r [according to (1.29)] has the effect of changing in

particular the above IR-singular propagator into i/(k2-~2) 2 . Thus ~2 is an

IR-cut-off.

Moreover, this IR-cut-off appearing as a parameter of the field redefinition, is a

gauge parameter, in the same sense as the parameters a k previously discussed. The

proof is the same, too, although the presence of fields staying massless (but with

non-singular propagators) complicates considerably the technical task of solving.the

BRS-cohomology.

Thus the physical quantities do not depend on the IR-cut-off. In other words, the

IR-singularities cancel when computing these quantities.

Of course, the ansatz (1.54) breaks supersymmetry explicitly. This soft breaking is,

however, controllable and can be shown not to affect the physical quantities.

II. CONFORMAL INVARIANCE 12)'13)

Let us consider the super-Yang-Mills model of Section I, with a simple gauge group

G, and couple it with chiral matter fields A R. Here R labels both the field and the

irreducible representation of G where it lives. Its BRS transformation is

sAR = -c+i TiR AR

(2.1)

s~ = AR Ti - R c+,

i representing the generators of G in the representation R. the Hermitian matrices T R

The Slavnov identity (1.7) has a corresponding piece:

57

S ( F } . . . . + E f d S 5F 6P _ c . c . ~ 0 ( 2 . 2 )

% SY R 6A R

where YR is the external field coupled to the BRS transformation of A R. The matter

field contribution to the action (1.4) is

i

~i T R A R fdS ~ k I WI(A) + YR sAR] + c.c Fmatter = 1/16 fdV ~ AN e + (2.3)

where WI(A) is a basis of invariant cubic polynomials of A, and the kl'S are the

"Yukawa" coupling constants. Mass terms, not necessarily gauge invariant but

supersymmetric, are supposed to be present in order to avoid the off-shell infra-red

problem discussed in Section 1.3. The Slavnov identity is thus softly broken, as

well as all other following equations [this is expressed by the sign ~ in (2.2)].

Our strategy for studying the properties of the Callan-Symanzik ~ functions and for

finding conditions under which they vanish is based on the existence of a

BRS-invariant supercurrent 3) V obeying the Ward identity

Da Va& ~ -2w r - 4/3 D (S+S 0) (2.4)

with

v=. = ~.== v, 5&s = 5~ so = 0.

w is a functional differential operator expressing the different symmetries " . . 1 8 )

(superconformal group) involved. The letters V, S and S O stand for insertions . V

and S are BRS-invariant, i.e.,

B~ V N 0, B~ S ~ 0 (2.5)

[see (1.16) and Ref. 18) for the definition of B~].

Let us forget SO, an effect of the gauge fixing, irrelevant for the present • 2 0 )

discussion. The supercurrent V contalns among its components an axial current

(e = 0 component) associated with R-invariance 21~'3~ and the conserved symmetric

energy-momentum tensor (eOO-component). The BRS-invariant chiral insertion S, of

dimension 3, and of order ~, describes in particular the anomalies of the axial

current and of the trace of the energy-momentum tensor, the latter being related to

the dilatation anomaly, hence to the Callan-Symanzik equation. The precise relation

is the following. We expand S in a basis of BRS-invariant insertions L of dimension n

58

3 defined through the action principle 18) by

Vnr ~ fdSL n + fdSL n

where the V's are a basis of BRS-invariant differential operators: n

Vg = bg, V k = V)~I = b~l' ak'

V¢ = N¢ =: N¢ - N - N - N- - N B - N~ + 2~ p c_ c_

VRs = NR S =: NAR S - NyRs

V+ = N+ =: N - N + c.c. c+ (~

the N's being the counting operators

N~ = fdV ~ , NRs = fdSA R 6 6A S , etc.

One sees in particular that

LRs = (AR 6-----_ YS 7yR ) F 6A s

+•c+ 5 L+ = Tr(c - ~ ~) I ~

and one can show that

(2 .6)

(2.7)

(2 .8)

(2 .9)

L~ = DD %~, L k = DD %k

where %~ and ~ are BRS-invariant.

We thus write

+ - %~ L~ - R~S ySR LRs - Y+ L+ - ~ ~k Lk' S = ~g Lg ~ ~XI LXI

The connection with the Callan-Symanzik equation

(2.1o) I

(2.11)

59

CF =: ~ ~

+ - -- -

[Zama~ma + ~g~g + ~kl~kl ~kl~kl y~N~ •SRNR S y+N+ -

- YkDak]F ~ 0

(2.12)

3) follows from the identity

g a ma ~m r + fdSS + c.c = 0 (2.13) a

(summation is over all mass parameters of the theory).

of basis for the insertions LRs , the new basis {L0a,LIA } Let U S perform a change

being defined according to (2.6) through the counting operators

~ S ~R

N0a = RZ, S ea R N S

~ ~R NIA = R~S fASR S

(2.14)

where the operators N0a form a basis of counting operators annihilating the

superpotential terms W I of the action (2.3):

N0a WI(A) = 0, Y I (2.15)

N

The NIA complete the basis of matter field counting operators.

The expansion of S in this new basis reads [with (2.10) taken into account]

- - m

+ - y~ DD %~ - E a Y0a L0a - ~ YIA LIA - y+L+ - ~ Yk DD %k" S = ~gLg ~ ~k I Lkl

(2.16)

A key remark, now, is that the conditions (2.15) defining the insertion L0a express

the invariance of the theory (at the classical approximation) under the chiral

transformations

RsAS ' S 6 aA R = i ea 6aYR = -i ea RYs (2.17)

(which obviously leave invariant the rest of the action). This invariance can be

extended to the quantum theory, but the associated axial currents become anomalous.

A relation between the coefficients 6, y of (2.16) and the coefficients of these

80

anomalies, as well as with the coefficient of the anomaly for the axial R-current

will follow from the fact 12) " that the BRS-invariant chiral insertion T = S,Lg,...,

can be written in the form

T N D--D [rK ° + jinv] + T c (2.18)

where T c is genuinely chiral [i.e., it cannot locally be written as D-~(...)] and

jinv is BRS-invariant. The insertion K ° is not invariant, although D-DK ° is, and is

defined *) through the supersymmetric descent equations

• • ,

B~ K ° N~& K I~ B~ K lu ~ ODD + 2DD) ~ K 2

B~ K 2~ ~ Du K 3 B~ K 3 ~ O, D.~ K 3 = 0 (2.19)

(the superscript denotes the ghost number).

The dimensionless insertion K 3 is proportional to c3+ and can he shown to be finite,

hence uniquely defined up to a numerical factor, chosen to be 1/3 by convention. It

turns out that K ° is then uniquely defined up to an invariant. The coefficient r in

(2.18) is thus defined unambiguously and is moreover gauge independent.

The genuinely chiral insertion T c being expanded in terms of the chiral insertions

LIA , L+, we can write (2.18), for T = S,Lg,..., as

-- 1 jinv]j S N-~[rK ° + jinv] + S c Lg N DD[(.̂ ~---2r----r + Izo g- r )K ° + + L c

g g g (2.20)

-~[rxIKO inv c ~ D--~[roaKO + inv L c LX I ~ + Jl ] + LI Loa J0a ] + 0a"

The corresponding expressions for L~, Lk, LIA, L+ have a coefficient r = 0 due to

our choice of basis. The coefficients r, r , r and are of order • at least. kI g roa The zeroth order coefficient in L comes from the fact that

g

D-DK ° = Tr F=F + 0(~) (2.21)

is the integrand of the Yang-Mills action (1.4).

*) Up to an invariant.

61

Moreover, the coefficients r and r in (2.20), which can be interpreted as the 0a

anomalies of the axial currents associated with R-invariance and the chiral *)

invariances (2.17), respectively, can be proved to be non-renormalized: they have

only one-loop contributions, which may be computed, with the result

1 512(4~)z (-3C2(G) + ~ T(R))

1 T(R) roa = _ 256(4~)-'/~ eaRR

(2.22)

T(R) is defined for the irreducible representation R by

"" i " (2.23) T(R)6 lj = Tr T R TJR

and

C2(G) = T(ad) (2.24)

R is the quadratic Casimir operator of the group. The numbers e

a R (2.14) and (2.15).

were defined by

The substitution of the expressions (2.20) in (2.16) and the identification of the

coefficients of K ° yield the equation

( 1 r = ~g ~ + rg) + ~ ~k I rkl - Z a r0a Y0a" (2.25)

This is the announced relation between the Callan-Symanzik functions and the anomaly

coefficients r, r0a. One sees in particular that r is proportional to the one-loop

~g function.

If the representations R of the laatter fields are chosen such that the coefficients

(2.22) vanish,

r = r0a = 0

then Eq. (2.25) becomes homogeneous in ~g, ~k I.

(2.26)

*) 12) nes 3 3 ii The proof is based on: (i) the finite s of K = i/3 Tr c +; ('') the Callan- Symanzik equation, obeyed by the superspace integrals of the insertion S and L0a without any of their own anomalous dimensions.

62

If, moreover, the theory can be completely reduced 14), i.e., that all! Yukawa

coupling constants ~I can be chosen as power series in the gauge coupling ~onstant g, • 14)

then these functions ~l(g) must be solutions of the reduction equatlons

~I ~g ~g ~I (2.27)

and Eq. (2.25) becomes

(I 0 = ~g l--~g + rg + r~l ~g ~i ) (2.28)

whose solution is, in perturbation theory,

~g = 0. (2.2'9)

This also implies the vanishing of all ~I' due to (2.27).

We have thus proved that the model is asymptotically scale invariant, as announced

in the Introduction, if, first, the representations are such that the vanishing of

the quantities (2.22) holds and, second, that the reduction equations (2.27) admit

non-trivia~ solutions. The latter is, as a rule, true if it is verified in the

one-loop approximation.

A closer look at the superconformal group reveals that the physical quantities are

also superconformally covariant.

83

REFERENCES

1) P. Breitenlohner and D. Maison - These Proceedings.

2) S.J. Gates, M.T. Grisaru, M. Ro~ek and W. Siegel - "Superspace", Benjamin/Cummings (1983).

3) O. Piguet and K. Sibold - "Renormalized Supersymmetry", BirkhaHser, Boston (1986).

4) O. Piguet and K. Sibold - Nucl.Phys. B197 (1982) 257; B248 (1984) 301.

5) G. Bonneau - These Proceedings.

6) K.S. Stelle - These Proceedings.

7) A. Blasi and R. Collina - These Proceedings.

8) J.W. Juer and D. Storey - Phys.Lett. BII9 (1982) 125; Nucl.Phys. B216 (3'983) 185.

9) O. Piguet and K. Sibold - Nucl.Phys. B253 (1985) 517.

I0) H. Kluberg-Stern and J.B. Zuber - Phys.Rev. DI2 (1975) 467, 482, 3159.

11) O. Piguet and K. Sibold - Nucl.Phys. B248 (1984) 336; B249 (1984) 396.

12) O. Piguet and K. Sibold - Phys.Lett. B177 (1986) 373; Int.J.Mod. Phys. A1 (1986) 913.

13) O. Piguet and K. Sibold - Conference "Renormalization Group-86", JINR, Dubna (1986).

14) W. Zimmermann - Commun.Math.Phys. 97 (1985) 211; R. Oehme, K. Sibold and W. Zimmermann - Phys.Lett. 153B (1985) 142.

15) A.J. Parkes and P.C. West - Phys.Lett. B138 (1984) 99; Nucl.Phys. B256 (1985) 340.

16) S. Mandelstam - Nucl.Phys. B213 (1983) 149; L. Brink, O. Lindgren and B. Nilsson - Nucl.Phys. B212 (1983) 401.

17) P.S. Howe, K.S. Stelle and P. West - Phys.Lett. 124B (1983) 55; P.S. Howe, K.S. Stelle and P.K. Townsend - Nucl.Phys. B236 (1984) 125.

18) K. Sibold - These Proceedings.

19) O. Piguet and K. Sibold - Nucl.Phys. B247 (1984) 484.

20) S. Ferrara and B. Zumino - Nucl.Phys. B87 (1975) 207.

21) P. Fayet - Nucl.Phys. B90 (1975) 104.

N = 2 S u p e r s y m m e t r i c Yang-Mil l s Th e o r i e s

in t h e W e s s - Z u m i n o G a u g e

PETER BREITENLOHNER

Max-Planck-Institut ffir Physik und Astrophysik

- Werner-Heisenberg-Institut ffir Physik -

P.O.Box 40 12 12, Munich (Fed. Rep. Germany)

1. INTRODUCTION

We consider the renormalization of the N = 2 Yang-Mills multiplet coupled to

matter and of the N = 4 Yang-Mills multiplet in the Wess-Zumino gauge. Since there

is no formulation of the N = 4 theory with auxiliary fields we have to describe it

as a N = 2 theory with one matter multiplet in the adjoint representation and have

to impose additional constraints later on in order to guarantee the N = 4 supersym-

metry. In view of the fact that all known consistent renormalization schemes violate

either supersymmetry or gauge invariance, we study the possible anomalous radiative

corrections to both the BRS and SUSY Ward identities.

The analogous program for the N = 1 theory has been performed in the very

elegant superfield formulation, i.e. with unconstrained multiplets. This approach has

the problem that there are massless scalar fields of canonical dimension zero. These not

only require an IR-regulator destroying explicit BRS (Slavnov) or SUSY invariance, but

also open the door for an infinite parameter group of field redefinitions [1]. Furthermore

for extended supersymmetry there does not seem to exist an acceptable supermultiplet

to put the Faddeev-Popov ghosts in. Last not least there is no superfield version of

the N = 4 theory available. All these problems are avoided using the so-called Wess-

Zumino gauge [2]. Yet, there is a price to pay:

i) the supersymmetry variations are non-linear;

ii) the commutator of two supersymmetry transformations contains a covariant

translation instead of an ordinary, field independent one;

iii) the gauge fixing term violates supersymmetry explicitly.

We find, however, that this price is low compared to the trouble one avoids. Hence we

shall use the Wess-Zumino gauge.

In this article we will concentrate on the non-linearity of the supersymmetry trans-

formations in the Wess-Zumino gauge. This non-linearity originates from the non-

65

linearity of the 'field dependent gauge transformations' and as a first step we will

analyze the complications due to this non-linearity.

2. INVARIANT AND NON-INVARIANT REGULARIZATION SCHEMES

In this section we want to discuss in quite general terms the situation if some sym-

metry of the classical theory is or is not explicitly preserved by the regularization and

renormalization procedure. In addition we want to recall that a renormalization proce-

dure which deserves that name is more than a prescription to obtain finite results from

divergent expressions. A renormalization procedure must in addition satisfy Hepp's

axioms [3] which are equivalent to those locality and causality requirements which are

the starting point to construct the perturbation expansion.

2.1. INVARIANT REGULARIZATION SCHEMES

If an invariance of the classical theory is explicitly preserved by the regu]arization

and renormalization procedure the resulting renormalized theory will certainly be in-

variant. This very simple fact has motivated the invention of various regularization

and renormalization procedures which preserve one or the other type of symmetry.

There are renormalization procedures which explicitly preserve supersymmetry, but

none of them preserves gauge invariance. Conventional dimensional renormalization

[4, 5] clearly violates supersymmetry because the structure of supersymmetry multi-

plets is different in different space-time dimensions. The so-called 'regularization by

dimensional reduction' [6] method was soon found to be inherently inconsistent by its

very inventor [7]. In spite of this it seems still to be quite popular [8]. The method of

'higher covariant derivatives' [91 either breaks gauge invariance [10] or does not regular-

ize one-loop diagrams and has to be supplemented by another regularization breaking

one of the desired invariances. A systematic study of such hybrid regularizations seems

to be missing. Moreover it is not clear whether this method can be extended to N = 2

supersymmetry.

Given the fact that there is no acceptable renormalization scheme which explicitly

preserves gauge (or rather BRS) invariance and supersymmetry we have to study the

consequences of violations of supersymmetry (and possibly other symmetries) by the

process of renormalization.

66

2.2. NON-INVARIANT REGULARIZATION SCHEMES

It is a fact, although not a widely recognized one, that the existence of an invariant

regularization with respect to some desired symmetry is only of marginal interest from

a more general standpoint. In fact, since the pioneering work of Becchi, Rouet and

Stora [11] on the renormalization of gauge theories it is understood that the problem

of genuine anomalies in Ward-Takahashi identities is a purely algebraic one. It can

be reduced to the cohomology theory of Lie algebras. Looking at the problem from

this more general standpoint one has freed oneself from the necessity to refer to any

particular renormalization scheme. We will only assume that Lorentz invariance and

invariance under global compact groups (in our case SU(2) x SU(2)) are preserved.

This is no loss of generality because it is known that these symmetries can always be

restored (absence of anomalies for these symmetries).

The classical Lagrangean is highly restricted (relations between coefficients and

absence of certain terms) by symmetry requirements. If a symmetry is destroyed

by the regulari~tion procedure, there is no more reason for these restrictions in the

regularized Lagrangean. Such a restriction has in fact no meaning independent of

a particular renormalization scheme. We must, therefore, start from a more general

'effective' Lagrangean containing all possible terms with arbitrary coefficients with no

other restrictions than those imposed by power counting and by symmetries which are

respected by the renormalization procedure.

The invariance or non-invariance of the renormalized theory under the desired

symmetries is most easily expressed by the absence or presence of anomalous terms

in the corresponding Ward-Takahashi identities. One must try to adjust the many

additional parameters in the effective action in such a way that all anomalies are

removed. The resulting symmetric renormalized theory should then have as many free

parameters as the original classical theory. They can be fixed by suitable symmetric

normalization conditions, such that the resulting theory is completely determined by

symmetry requirements and normalizations conditions independent of any particular

scheme.

3. WARD IDENTITIES AND WESS-ZUMINO CONSISTENCY CONDITIONS

Invariances of the Lagrangean are reflected at the level of generating functionMs

by Ward identities. In this section we want to derive these Ward identities and in

particular point out the difference between linear and non-linear transformations of

the elementary fields ~ = ( ~1) .

6"7

3.1. LINEAR TRANSFORMATIONS OF THE FIELDS

Let us consider a set of (infinitesimal) transformations ~i of the fields ¢ which leave

the Lagrangean invariant

~i~inv = 0 (3.1)

and satisfy the commutation relations of some Lie algebra

[6i, 5j] = f i jk6k . (3.2)

In order to simplify the following discussion we ignore (for the moment) all sign factors

due to Fermi fields or supersymmetry transformations.

If all the fields transform linearly we have

6i¢ = tic (3.3)

with some matrix representation t i of the Lie algebra. In order to generate Green's

functions we have to use the 'classical' Lagrangean obtained by adding source terms

f-'cl = f-.inv + jT ¢ . (3.4)

This classical Lagrangean satisfies the identities

(6i + Wi)£cl =- 0 (3.5)

where the Wi's are differential operators in the sources

W i = - - j T t i ~ T (3.6)

which satisfy by construction the commutation relations

[Wi, Wj] = f i jkWk . (3.7)

The naive action principle (valid for the tree approximation) implies that the gen-

erating functional Z(j) of the connected Green's functions satisfies the naive Ward

identities

WiZ(j ) = 0. (3.8)

These relations will, in general, not be true for the renormalized theory. The renor-

maiized action principle [12, 13, 14, 5], which is a consequence of general results from

renormalization theory, yields

Wi Z = A i Z (3.9)

68

where the anomalies A i are integrated local operator insertions of appropriate dimen-

sion and covariance which are at least of order O(h), i.e. vanish in the tree approxima-

tion. In order to study how the Ai's change if we perform finite renormalizations (i.e.

exploit the freedom inherent to any renormalization scheme) we have to introduce the

generating functional F(¢) of 1PI Green's functions (vertex functions) obtained from

Z by a Legendre transformation with respect to the sources j . We first define

solve for j(¢) and define

with the consequence

5Z cT(j) = ~ - (3.10)

F(¢) = Z - j T ¢ (3.11)

j T ( ¢ ) = . 5F 5¢" (3.12)

Note that the tree approximation Fcl of F coincides with l~in v

r = r e / + O(h) , rc~ = Z:in~ (3.13)

Upon Legendre transformation the Ward identities take the form

Wi(F) = AiF (3.14)

where the differential operator

is linear in F.

satisfy the Wess-Zumino consistency conditions [15]

T T 5F wi(r) - ¢ ti ~ (3.15)

As a consequence of the commutation relations (7) the anomalies A i

wr(Air)- wr(A~r) /~k~r (3.16)

where

does not depend on F.

• T T 6 X w r x = ¢ t~ ~ = w~(x) (3.17)

69

3.2. NON-LINEAR TRANSFORMATIONS OF THE FIELDS

Let us now consider the case where the variations of the elementary fields are some

non-linear expressions Pi in the fields

and consequently

6i¢ = Pi(¢) (3.18)

6i(£inv + jT ¢) = jT pi(¢) . (3.19)

In order to express these changes through differential operators Wi acting on sources

we have to introduce additional sources for all non-linear expressions (composite fields)

appear ingin thePi ' saswel las fora l l the i r i te ra tedvar ia t ions . L e t ~ = ( ~ ) bealt

these fields, ¢ are the elementary ones as before and ¢c are all the composite fields

(possibly infinitely many), such that the infinitesimal variations are again linear (in O)

6 i~ = t i ~ • (3.3')

J ) and before: the differential Similarly we introduce sources J = jc proceed a s oper- /

ators

(3.¢)

act on the generating functional Z(J) =- Z(j, jc).

The vertex functional F is obtained from Z by a Legendre transformation with

respect to the sources j for elementary fields but not the sources jc for the composite

fields. We define

solve for j(¢,jc) and define

with the consequence

6Z cT (j, jc) = -~f (3.10')

r(¢,jc) = z - j T ¢ (3.11')

jT(¢,jc) = 6F 6¢"

In the tree approximation we find

Fd = £inv +JT¢c.

The differential operator ~F

\ --3c /

(3.12')

(3.13')

W/(I ~) ~_ (¢T (3.15')

70

is now non-lineax in F and

does depend on I'.

3.3. THE COItOMOLOGY PROBLEM

We can now use the fact

assume

6X , ,T ~ . ¢ Wi(X) (3.17') + ( 0 -~ )~i -.7c /

AiF = A i + O(liAi) , (3.20)

A i = hmAl m) + o(hm), (3.21)

and use the consistency conditions (16) to the order h m to obtain the cohomology

equations

~t (m) ct (m) - W j A i W~ Aj = f i jkA~ m) (3.22)

where W {l is given by eq. (17) resp. (17') with the replacement F ~ Feb The difference

between the two cases of linear and non-lineax transformations is that the structure of

W cl is much simpler in the former case.

If we change the effective action (which is in a certain sense the renormalized version

of f-'inv or rather Fcl )

Feff ~ Feff + li mA(m) (3.23)

the corresponding change in the anomalies is

W, cl A(m) (3.24) zxl m ) - - * A l m ) + i L •

This poses the following cohomology problem: Given some anomalies Aim) which

necessarily satisfy the consistency conditions (22). If there exists some integrated local

expression X such that

w/ x (3.251 then the change Feff ~ Feff - hmX will remove all anomalies in this order in h,

otherwise there is a genuine anomaly.

The procedure described above is the standard procedure to establish BRS in-

variance. The main difficulty arises from the fact that the differential operators W el

mix terms with a different number of sources. The situation is, however, not too bad

for BRS transformations because they axe nilpotent and therefore they require only a

finite number of je's. For SUSY in the Wess-Zumino gauge the situation is more com-

plicated because there is an infinite number of jc's and moreover almost all of them

have negative (power counting) dimension.

71

4. N = 2 SUPER YANG-MILLS WITII MATTER

We use a notation with SU(2) covariant N = 2 Majorana spinors and assume

that the gauge group is simple and non-chiral and that all matter fields are massless.

These theories have a global SU(2) × SU(2) invaxiance but only the diagonal SU(2)

subgroup is made explicit by our notation. The fields of these theories are contained

in one N = 2 Yang-Mills multiplet [16]

(Aa, S, P; A; 5 ) (4.1)

in the adjoint representation of the Lie algebra (or equivalently with values in the Lie

algebra) as well as one N = 2 matter multiplet (hypermultiplet) [16]

(A, A; a; F, F) (4.2)

transforming under an arbitrary representation p of the Lie algebra. This repcesenta-

tion need not be irreducible but we assume it to be real.

The Lagrangean has the form £inv = £YM + ~M where

l ( - 1 F a b 1DaS DaS ~Dap DaP - i - ~'YM = g2 \ 4 " Fab + • + • -~A . "TaDa)t (4.3)

+ -~D. 5 - -~)~, (S - i'~hP) x )t - (S × P). (S × P))

and

1 ,CM = 2DaA" DaA + ~Da.4" Da.4- ~ , T a D a a + ~F. F

1-. + 1~ . ~ + A. p(DIA+ ~A.p(D) x d - ip (X) (A+ iCA~. a

i 1A" (P(SlP(S) + p(P)p(PI)A + -~ . p(S + i'~hP)a + 2 1-.

+ ~A. (p(S)p(S) + p(P)p(P))A.

(4.4)

The supersymmetry variations 5(i~Q)¢ generate the N = 2 SUSY algebra [17]

2[5(ielQ),5(ie2Q)l=-i~le2(5(S)÷e(Z)) -~175c25(P)-i~lTae25(Ta) (4.5)

where 6(Z) is the central charge transformation (acting on the matter fields only),

6(Ta) = Oa - 6(Aa) is the gauge covariant translation, and 5(Aa), 6(S), ... are field

dependent gauge transformations. These field dependent gauge transformations are

the source of all non-linearities in the transformation laws and force us to introduce an

infinite number of composite fields with increasing dimensions.

72

If we define a spinor derivative 7:) by

5(i~Q)¢ = i¢-~¢

we can rewrite the SUSY algebra in the form

v v = -8~(~(s) + 5(z))

~75:D = - 8 5 ( P )

~)3~aD .-= -8i~(Ta)

~3,a"/5"71:) = 0

~7abr-*D = 0

and find in addition

(4.6)

(4.7)

and on the ghost fields c, e and B

1 sc = B, sB = O, s ~ = x ~ x ~ (4.11)

z

such that

The gauge fixing term

~,~II = - ~ s ( c . OaAa - l c . B)

= ( B . OaA a - ~ B . B - c. OaDae)

{~, ~} = 0 . (4.12)

(4.13)

s A a = D a G s S = ~ x S , . . . s - P = p ( ~ ) F (4.10)

[V, 5(Z)] = 0 . (4.8)

These commutation relations (7) generate an infinite dimensional algebra through the

identities

[:D, 6(Ta)] = -[~D, 6(Aa)] = -5(T)Aa)

IV, ~(x) ] = ~(vx) (4.9)

[6(Ta), 5(Tb) ] = -5 ( Fab )

[5(Ta), 6(X)] = ~(DaX)

where X can be any of the covariant fields S, P , . . . .

The BRS transformation acts in the usual way on the physical fields

73

added to the Lagrangean is BRS invariant. Finally the ghost fields are invariant under

SUSY transformations

with the consequence

:Pc = :P~ = :pb = 0 (4.14)

{:P, s} = 0. (4.15)

With our definition (14) the gauge fixing term (13) is not SUSY invariant

1 :P£q)II = "~g2S( c" 7aOa A) • O . (4.16)

This leads to a slight complication because supersymmetry is already broken at the

tree level. Due to the form s(. . . ) the breaking term (16) does, however, not affect

the physical sector (gauge invariant amplitudes). The following table collects all the

relevant transformations together with some notation

7)

~(Z)

~(Ta)

~(X)

~(Ta)

~(X)

AI m)

A S

AW

AZ

Aa

AX

ha

;X x

t

S

W

Wz

Wa

Wx

(4.17)

where again X can be any of the covariant fields S, P, i . . . Since all insertions are

integrated ones we have actually ~(Ta) = -,5(Aa). The transformations $(Ta) and g(X) are not yet defined but will turn out to be useful later on.

Assume that all anomalies have been removed up to the order h m-1 for some

m > 1, i.e. they all start at the order h m. Our aim is to show that the consistency

conditions (3.22) imply that they can all be removed by a suitable change in Feg (in

this order hm). This will then define appropriately renormalized composite operators

such that BRS and SUSY invariance are restored.

In the following sections this program will be performed for some of the symmetries

under consideration.

74

5. BRS INVARIANCE

As a first step we want to study the BRS anomaly A S which must satisfy, due to

the commutation relation (4.12), the usual consistency condition

= o. (5.1)

In spite of the infinitely many sources and the complicated structure of the differential

operator $ it is relatively easy to show that any such anomaly has the structure

A s = S X s " (5.2)

with some X$ and can therefore be removed by a suitable finite renormalization.

Since there are infinitely many composite fields with increasing power counting

dimension the dimension of their sources will soon get negative. Consequently the usual

power counting arguments cannot be used to control the occurenee of such sources, e.g.,

in A S. We can, howevor, rearrange the sources J : (je, Jc) as J = (ja, ka, j a , K a)

in the following way: The elementary fields ¢2a = (Aa, S, P, t , D, A, .4, a, F~ _F, c,

~) are coupled to j a and their BRS variations *~os to k s. Note that the elementary

field B = sc is contained among sc, os. The 'physical' composite fields 62s = (Fab,.. .),

all with vanishing BRS charge, are coupled to j a and their BRS variations to s e a to

K a .

We can decompose Fcl and S according to their degree in the sources j s and K s

and find Fcl = F 0 + F 1, ,5' = $0 + '5'1 where (up to signs due to anti-commuting fields)

r0 : rc b=K=0 (5.an)

Pl : JS(I)s + K%(I)a (5.3b)

6 js___5__5 6F 0 6 (5.4a) SO : s~°a 6~os q- 6K s + 6~o---a 6k ---~

5 + (5.4b)

The first two terms in $0 are nice and simple, the third term involving field equations

in the presence of sources k s is somewhat messy and $1 contains the really unpleasant

contribution of the infinite sequence:of sources J~ and K s to the field equations. The

nilpotency of ,5 implies

, 5 0 2 {,5,0, $1 } =`5~2 = 0 . (5.5)

Similarly we express A 8 in the form

n----O

75

where each term An is homogeneous of degree n in the J ' a and K's and obtain the

consistency conditions

SOA0 = 0 (5.1')

S O A n T S I A n - l = 0 for n _ > l .

It can be shown that the first of these equations implies that there exists an X 0 such

that A 0 = SoX O. Here we can use the original argument of [11] with minor modifica-

tions due to the presencoof the elementary fields L3, F and ff of dimension two. and

we have to use the fact that the gauge group is non-chiral and therefore there can be

no Adler-Bardeen anomaly.

We can now use the following recursive argument to show that the anomaly has

indeed the form (2): Assume the first non-vanishing term in the sum (1') has the form

An_ 1 = SOXn_ 1 (true for n = 1). Subtracting this Xn_ 1 from Feff removes this term

and modifies the next one An such that the consistency condition (1 I) implies

SoAn = 0 . (5.7)

Using the fact that the anti-commutator

{u" So} = J + u° u--a (5 .8 ) j '

yields the counting operator for the degree of homogeneity in the J ' s and K's, we find

from eq. (7)

An = SoXn with Xn 1K~ 5 A = . . ( 5 . 9 )

Thus we have shown that the anomaly A S can indeed be written in the form (2).

Next we must study the remaining finite renormalizations compatible with BRS

invariance, i.e. the most general solution of the equation

SA c = 0 (5.10)

where A£ is an operator insertion of dimension four with vanishing ghost charge

whereas A S had dimension five and ghost charge one. Repeating the arguments used

to analyze A S we find

A£ = SX£ + A~ "i" (5.11)

where A~ "i" is a gauge invariant expression (without sources and ghost fields). Unfor-

tunately the decomposition (11) is not unique, i.e. there exist some X£ such that SX£

is a (non-vanishing) gauge invariant expression.

76

As a next step X£ and A~ "i" could be chosen in such a way that (if possible) A W

is removed. All higher anomalies AZ, A S , . . . would then automatically vanish. This

should then fix all the coefficients in Feff except a few, i.e. the requirement of BRS and

SUSY invariance should uniquely determine the theory up to a redefinition of'the gauge

coupling constant g and three (physically irrelevant) wave function renormalizations.

6. FIELD DEPENDENT GAUGE TRANSFORMATIONS

6.1. SUSY WARD IDENTITIES

Due to the commutation relations (4.7) we have the consistency conditions

~ A W = - 8 i ( A S + Az)

]/V75Aw = - S A p

) /VTaAw = - 8 i A a

)TVTa75~A W = 0

~Tab"TAW = 0 .

(6.1a)

(6.1~)

(6.1c)

(6.1d)

(6.1e)

, they

additional consistency conditions.

Yang-Mills multiplet

~ A a = 7aA (6.2a)

z~s = A (6.2b)

z~P = - i ~ s a (0.2c)

ab " ~ = - i T a D a ( S + i'~5P) - -~7 Fab + 75 S × P - ~D (6.2d)

V D = i 'Y ( -TaDaA - ( S - i'y5P ) x ~) (6.2e)

yield, together with the identities (4.8, 4.9),

W ( A z + AS) -- (Wz + WS)zXW = Z~ (6.3a)

14)Ap - W p A W = -i75A~ (6.3b)

W A a - W a A W = - ' laA~ (6.3c)

]/V~A + W A £ W =- - i ' Ta (Wa/ k S - W S A a ) + . . . - "~/k D (6.3d)

W A D -- W D A W = - i f a ? ( W a A A - W A A a + . . . ) . (6.3e)

The supersymmetry transformations laws for the

If there were only the field dependent gauge transformations ~(S), 5(P) , . . .

would generate an infinite dimensional and essentially free Lie algebra which would

yield no useful consistency conditions. In the present case, where all field dependent

gauge transformations are generated from supersymmetry transformations, we have

77

Each of the anomalies A w, A S, . . . is BRS invariant and has a decomposition into a

piece (depending on sources and/or ghost fields) which is itself a BRS variation and

a gauge invariant piece (compare eq. (5.11)) but again these decompositions are not

unique.

6.2. A NEW SET OF TRANSFORMATIONS

The form of the consistency conditions (1, 3) would simplify considerably if we could

first remove the anomalies Aa, AS, . . . for the field dependent gauge transformations.

In spite of the fact that all these transformations essentially form a free Lie-algebra

this can indeed be done. Note that all these transformations commute with s (again

we ignore a sign change due to fermion fields e.g. in g(A)). It should, therefore, not be

too much of a surprise that each of them can be expressed as the anti-commutator of

a new transformation with s. We thus introduce the new transformations ~(Ta) and

~(X) which act on the fields as follows (all physical fields are annihilated)

~v

c

B

Ab

Y

0

OaC

Aa

0

0

0

0

X

0

0

Oa - {$(Ta),

0

0

0

Fob DaY

{$(x), 0

0

0

DbX

~(X)Y

(6.4)

an X and Y can be any of the covariant fields S, P, . . . .

These new transformations are linear and the corresponding Ward identity opera-

tors are rather simple. Moreover they are all nilpotent and mutually anti-commute. As

a consequence the corresponding consistency conditions are extremely powerful. Once

we have removed their anomalies/~a a n d / ~ x the anomalies Aa and A X for the field

dependent gauge transformations vanish automatically.

The Lagrangean £inv = £ YM -4- ~,M is obviously invariant under these new trans-

formations but the gauge fixing term EVl] is not (compare eq. (4.17)). This explicit

breaking which is already present at the tree level can be taken into account by addi-

tional terms in the Ward identity operators. In order to do so we have to add some

more sources. The constant (space-time independent) sources j a and /'~'a couple to

integrated composite fields f ~a and f s~a which contain the variations of £¢1I under

iterated application of all the transformations under consideration. Since we can treat

78

these new sources J and/ '( in almost the same way as the J 's and K's all results about 0

BRS invariance remain valid. Among the new (integrated) composite fields there are

in particular f ~a = ~(Ta)/:~II and f ~ x = ~(X)Z~II (for X = S, P , . . . ). The non-

invariance of/:~II yields the inhomogeneous terms - ~ j ~ and -0J--~ in l/Va and l/~rx

respectively. For each such transformation, e.g. for ~(S), we can use the identity

{ I~S, - ' IS } = 1 (6.5)

and find indeed that

Zx S = 17VsAf. " with AZ: = - J S A s . (6.6)

Subtracting this term from Feg removes this anomaly (to the order in h under con-

sideration). The fact that this can be done should not be very surprising, in a sense

we are just redefining the composite field ~(S)/:¢II which is the tree level value of this

anomaly. Having done this we are left with modified values for all the other anomalies.

In the next step we can remove one of the remaining anomalies, say/~p in exactly the

same way. Moreover since j S anti-commutes with l~Zp the/~S removed in the previous

step stays absent. Repeating this process we can construct a AZ: such that

ha = ¢¢oZXL, (6.7) 2X X = I;VxA£ for all X .

This AZ: is not yet BRS-invariant and will, therefore, introduce a new BRS-anomaly

A S . In the process described above we have, however, introduced at least one explicit

power of J, i.e., increased the number of J ' s and K's which played a crucial r61e in the

removal of the BRS-anomaly A$. In order to remove all the anomalies AS, /~a and

2x X simultaneously we have to use the process described above after each step of the

inductive procedure used to remove A s (compare eq. (5.9)).

The anomalies A(Ta) and A(X) for the field dependent gauge transformations will

now all vanish automatically. At the same time the vast majority of the parameters in

the effective action has been determined. All the remaining freedom should now suffice

to remove the SUSY-anomalies.

79

7. CONCLUSIONS

We have shown that one can study the renormalization of supersymmetric Yang-

Mills theories in a way which does not depend on any particular renormalization

scheme. Since there is no known reliable renormalization procedure which respects

both BRS invariance and supersymmetry we have to assume that BRS and/or SUSY

invariance are destroyed in the renormalized theory. This forces us and allows us to

start from a more general Lagrangean having only those symmetries which are respected

by the renormalization procedure (i.e. Lorentz invariance and a global SU(2) x SU(2)).

The anomalies automatically satisfy Wess-Zumino consistency conditions which do or

do not guarantee that the parameters in the effective action can be chosen in such a

way that the renormalized theory is BR~ and SUSY invariant (cohomology problem).

In spite of all the complications due to the non-linearities of the SUSY transfor-

mations in the Wess-Zumino gauge it is possible to analyze this cohomology problem

(although this analysis is not yet entirely completed). At present we are able to show

that the anomalies for BRS transformations and field dependent gauge transformations

can be removed by suitable finite renormalizations. We are confident that the same can

be done for the SUSY anomaly, i.e. that there is no genuine anomaly. Once this has

been achieved, all parameters of the theory are determined by symmetry requirements

and by a few (gauge invariant and supersymmetric) normalization conditions.

REFERENCES

[1] O. Piguet and K. Sibold, Nucl. Phys. B 247 (1984) 484, Nud. Phys. B 248

(1984) 301 and Nuc/. Phys. B 249 (1984) 396;

O. Piguet in this volume.

[2] J. Wess and B. Zumino, Nucl. Phys. B 78 (1974) 1.

[3] K. Hepp, in Renormalisation Theory in Statistical Mechanics and Quantum

Field Theory, C. deWitt and R. Stora eds.

[4] G. 't Hooft and M. Veltman, Nuc/. Phys. B 44 (1972) 189;

C.G. Bollini and J.J. Giambiagi, Phys. Lett. 40 B (1972) 566;

G.M. Cicuta and E. Montaldi, Nuovo Cimento Left. 4 (1972) 329.

[5] P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977) 11.

[6] W. Siegel, Phys. Left. 84 B (1979) 193.

80

[7] W. Siegel, Phys. Lett. 94 B (1980) 37.

[8] G. Altarelli, M. Curci, G. Martinelli and S. Petrarca, Nucl. Phys. B 187

(1981) 461;

N. Marcus and A. Sagnotti, Caltech Preprint CALT-68-1128 (1984).

[9] A.A. Slavnov, Teor. Mat. Fiz. 13 (1972) 174.

[10] R. S6n6or, in this volume.

[11] C. Becchi, A. Rouet and R. Stor~t, Ann. Phys. 98 (1976) 287.

[12] J. Schwinger, Phys. Rev. 82 (1951) 914, Phys. Rev. 91 (1953) 713.

[13] Yuk-Ming P. Lam, Phys. Rev. D 8 (1973) 2943.

[14] J. Lowenstein, Commun. Math. Phys. 24 (1971) 1.

[15] J. Wess and B. Zumino, Phys. Left. 37 B (1971) 95.

[16] S. Ferrara and S. Zumino, Nucl. Phys. B 79 (1974) 413;

A. Salam and J. Strathdee, Nud. Phys. B 80 (1974) 499;

P. Fayet, Nucl. Phys. B 113 (1976) 135.

[17] P. Breitenlohner and M.F. Sohnius, Nucl. Phys. B 165 (1980) 483.

IIIIITIIE MISS 6ENEIITIIH IN SCILE IHIIIIINT SYSTEMS WITR

SPINTINEIUS SYMMETRY BIEIKIIIN

R. Collina Dipartimento di fisica dell' Universita; Genova

Istituto Nazionale di Fisica Nucleare; Sezione di Genova

INTRODUCTION

The mass generation induced by radiative corrections present a marked

phenomenological interest [I]; it has been suggested in the literature that the mechanisms

may explain the nature of electron's and/or plan's mass [1,2] or a partial justification of

the different mass scale which are present in the unified theoMes. For these reasons

several mechanisms of radiative mass generation have been described in perturbative

quantum field theory. The phenomenon for example arises in some H. K. gauge models [3]

when the classical potential energy of the scalar fields is invariant with respect to a

symmetry group which contains the gauge group as a proper subgroup, this larger

symmetry being violated by the other terms of the Lagrangian. It follows that there are

"accidentally" more Goldstone bosons than those implied by the symmetry of the model

which are not reabsorbed by the H. K. mechanism (systems with pseudo Goldstone bosons)

[11. Radiative masses are also present in particular supersymmetric models (O'Raifertaigh

model) where their appearance is tMggered by the presence of an Infra Red (I.R.) anomaly

I4]. Here we shall be concerned with another, relevant, class of models, namely those with

spontaneous symmetry breakdown of the classical scale invariance. The model where this

mechanism first appeared was proposed by S. Coleman and E. J. Weinberg in 1973 [5];

Subsequently many papers discussed the subject and in particular the possibility of

extending this class of models to an arbitrary order in covariant perturbation theory.

A first difficulty is due to the presence of two different scale parameters: the strength

of a field vacuum expectation value and the renormalization scale, which could be mixed

by the radiative corrections. A regulaMzation independent solution of this problem is

obtained by the identification of a local Ward Identity (W.I.) which, at the classical level,

enforces the recursive condition that the trace anomaly be given by a scale invariant

operator [6].

The second main difficulty in obtaining a complete quantum extension of theories with

spontaneously broken dilatation symmetry is related to the fact that they are necessarily

accompanied by an I.R. instability brought about by radiative mass generation; this

instability is controlled by a suitable modification of the perturbative development [7].

In this note we shall provide a rigorous description of these models, in a covariant

82

perturbative context, f i rst using the W.I. instrument, and successively identifying, starting from a suitable regularization, an effective scheme of computing Feynman amplitudes [8].

We focus the attention on the main features of the problem, omitting the numerous technical details which are treated in the literature.

2-THE LOCAL SCALE INVARIANCE

To illustrate the general framework let us adopt a simple reference model built with a two component scalar field

_~ = ( ~ , % ) ( i )

whose classical action is invanant under the scale transformation of ~t and ~z + v and

under the inversion ~ ~- ~1-

One first and obvious remark is that a rigid scale transformation produces, in the

shifted fields, non integrable vertices; therefore we need a local description of

spontaneously broken scale invaMance. Then we must consider the following local

transformations

x I~ -~ x I~ - ~(x)

~ ( x ) = ~ ( x ) ~ ( x ) i : i, 2 (28)

which are general coordinate transformations. The natural next step is to introduce a

metric field belonging to the Einstein-Riemann representation i. e.

(2b)

Unfortunately this procedure is of no help in identifying scale invariant theories. Indeed

the classical terms

f~x(-det~J~2*12 f~x(-detbj/2(~, v)2 (3)

are invariant under the transformation in Eq.s(2) and in the flat limit (gpv ~ ~pv) they

correspond to mass terms.

A correct description, which solves our problem, is obtained in terms of tensorial

densities, with suitable Weyl weights which exclude the presence of couplings such as in

Eq.(3), given by

~1(x) = (-det~v~Ita~1(x), (4a)

83

((~z(X) + v) = (-det~i~) I/s (cl~ z + v),

~I:~(x) : (-det~i:~)I/~gI:~(x)

(4b)

(4~)

which transform according to

G~t (x) = x ~ ( x ) ~ (x) + 1/4 ~ x~(x)~(x), 8(l~(x) = X,~(x)~c~(x) + ! / 4 ~X.P'(x)(cI~(x) +

8~(x) = ~(x)~(x) + l/2 ~ ~(x)~(x)

- ~x~(x)~(x)- ~x~(x)~(x).

v),

(so)

(5b)

(5c)

The use of densities also requires the introduction of a covaMant derivative

where

and

~,(x) = -I/e :In(-detb~,,)

~(x) = ~P(x)a~(x) + ~ xP(x)~p(x) - l/4 ~ap xP(x).

(6)

(7)

(8)

Observe that the choice (4c)is equivalent to

- dell :~(×)I -- I. (9)

The scale transformation ( spontaneously broken ) in Eq.(5) and (8) can be summarized in a

local functional differential operator Wi~(x) which in the flat limit ~I~v -~ ~,, ~I~ -~ 0 is

given by

( Wi~(x))F1. = 2~o(818E~o _ :l~14818~x) - 114 ~818~p

+ ~ 818(~ - 114 ~[ ((~(x) + vi)818( ~ ], v i = v 8i2 (io)

The classical theory is then completely identified by the general solution of the W.I.

(W~(X)rCL-)~I = 0 (I I)

and by the stability condition of the classical vacuum, i. e.

Fct.= _ j'd4x{112 ~I~(x)(~ + c~i~(x))(~(x) + vi)( ~ + oJ1,(x))(~(x) + v i)

+ a((~1(x)) 4 + b212 (~l~(X)((~z(x) + v) z } (12)

and

84

~I8¢,f~CUl~=o = o. (13)

3 - RADIATIVE MASS AND THE PERTURBATIVE EXPANSION

ReferMng to the action in Eq.(12) we see that the classical theory is defined only if the

vacuum is an indifferent equilibMum state for any field translation. But the quantum

fluctuations select a stable equilibrium state by introducing a linear restoring force

(radiative mass term). This can intuitively be seen by looking at the level curves of the

classical potential (V = ad~14 + b212~iz(dpz+v) 2) of the model in fig. I.

. . . . . - = . . . . . . . . . - ~ . , . ~-=-...-.-.-.,,-~- ....

fig.1 In other words at the classical level, from Eq.(12), we have

%

ml 2 = b2v 2 and rn2 z = 0 ; (14)

but the last condition is not preserved, to the one loop level, in the ordinary perturbative expansion. Indeed at f i rs t order we have the diagrams in fig.2, where the continous and dotted lines stand for the qb I and ~ propagator respectively. Now the only counterterm containing a linear contMbution in the ~P2 field compatible with the symmetry is

c ( ½ , v) 4 (! 5)

which is necessary for the vacuum definition to the one loop level; i.e. the coefficient c

-0": ::0:: : :0 :0 : :0 : : 0 --0-- 0

!

fig.2

85

must be chosen to cancel the contribution of the last diagram in fig.2. This choice

produces finite nonvanishing contributions for the mass and vertex corrections related to

the ~ field.*

Numerically we have

m2 2 = (41T)-ZT) 2b 2 m 2 + O(Tl 2)

= ~ m22(1) + O(~z2), where m2= b2v 2 (16)

This radiative mass term induces an infrared sickness of the usual perturbation theory

and the cure is a modified perturbative expansion [7]. Indeed the mass generation due to

radiative correction can be phIIsically interpreted only in view of a summation of the

perturbative series generating a mass term in the propagator of the (I)2 massless particle.

The first important difference between such a perturbative approach and the usual one

arises in the construction of the Fock space as a consequence of the finite mass

renormalization of the d72 field, which in a formal non covariant language, could be carried

out by a singular BogoIiubov transformation [9]. From a covariant point of view, this is

equivalent to adding to the free lagrangian the mass term generated by the radiative

corrections at lowest order; i. e. we perform the substitution in the naive massless

propagator

p-2 -~ (p2 + TimzZ(1))-l. (17)

But in this new perturbative approach the equivalence between T) and loop ordering is not a pMoM guaranteed. Indeed in a completely massive model, when the propagator mass vanishes, a given amplitude depending from Pl ..... I~ external momenta, behaves like:

Z I ~n Cr~ (Pl,---,l~)(lnt ~)m- (18a) (~-nD

Hence in our case the generic N-loop graph wi l l depend on #z as:

T z ~ Cam (Pl ..... Pk)( ~ln~ )m, (18b) (~-~D

and the loop factor h N may be completely hidden by Tz ~n . For example the dia~am in

fig.3 has the leading power T~ 2 independently of the number of loops. We have proved [7] that i t is sufficient to exclude, with a suitable extra subtraction, at the one loop level the

propagator correction, at zero external momentum, to obtain a new consistent

perturbative expansion. The resulting perturbative development is consistent in the sense

that i t is a formal power series in 6~n with Ir~ corrections, and the ordering of the

* The result ls scheme independent; for instance In the B.P.H.Z. renormallzatlon, where the tadpole

contributions ere absent, one meg flx the c parameter tomelntaIn the vanishing of the ~>2 mass contribution, but

then efinfts trlllneer(l~ 2 vertex correction survives, whlch Is Incompatible with the Infrared pawercoumtlno~

86

.... - , ,

fig.3

diagrams is compatible with the loop ordering. In particular a generic proper Green

function G is a formal power series in ~ of the type

G = Go + Z ~(,+0 G(e)(Ir~) (19)

n --I

where G(,) has contributions from a finite number of diagrams with at most n loops.

Moreover the term G(,)(Ir~) does not depend upon the vertices of the effective lagrangian

~n.)(Ird~) for n'>n and the only contributions from ~n)!In~) are tree approximation terms.

The proof, given in Ref.[7], is according to the B.P.H.Z. method extended to include

massless particles [I0].

4-RENORMALIZATION OF THE WARD IDENTITY

We return to the W.l. in Eq.(l I) in order to discuss its renormalizability. The operators

WI~(X) satisfy local commutation relation i.e.

I w~(x), Wv(y) ] =- al~ 8(x-y) Wv(x) - a/~cv 8(x-y) %(y). (20)

Let assume that the vertex functional of the renormalized theory satisfy

W~Q(~,x)F = 0

where WI~Q(~,x) are a quantum extension of the operators W~(x), in particular

(21)

[WI~Q(~,x) , WvQ(~,y)] = - 8/axi~ 8(x-y) WJ:I(~,x) - a /~v 8(x-y) Wi~I~(~,y) (22a) and

lim Wl~Ci(~,x) = Wg(x). (22b)

~-~0

It is easily seen that the operators W~Q(li,x) are not unique, indeed they are not

invarlant under a local redefinition of the sources such as

87

Ji(~) -~ Ji(×) + c z(~(×))Ji(×), (23a)

(23b)

(23c)

where Ji(x) = - ~18~i(x)7 is the variable conjugated to ~(x) in the Legendre

transformation. The substitutions in Eq.s(23) performed in the Z[J] functional (or Y[~])

amount to a new choice of the time-ordered products and have no effect on the physical

interpretation of the model. The question is then, if all the possible quantized versions of

the theory correspond to Wi~(~,x) operators which are related by source transformations

as in Eq.s(23).

In fact we prove that the anomalies of the dilatation W.I. in Eq.(1 I) are related to an

instability of Weyl's representation for the metMc field, which can be reabsorbed by a

• source redefinition [~]. In particular for the anomalies we have

(w~(×)r)~- ~l(x), (24)

where the subscMpt F.L. is for flat limit and l(x) is a local operator obeying the

minimaIity conditions of being of canonical dimension 4 and which cannot be wMtten as a

divergence (e.i. I(x) ~ ~KI~(x)). !(x) is the well known Callan-Symanzik or trace anomaly

[11]. This anomaly can be .reabsorbed by a source redefinition. Indeed adding to the

Lagrangian in Eq.(12) the coupling (~I~V(x) - TII~V)I(x) and considering the altered

representation of the metMc field

8~I~V(x) = ~P(X)Sp~I~V(x), ½(I÷ ~)~p~,P(x)~V(x) - ~p;~(x)~PV(x) - ap~,V(x)~(x) (25)

we obtain, at the lowest order in %1 and in the flat limit, the new anomaly free W.I.

(26)

But the description of the anomalg by the term - -~8/8~x~F does not guarantee i ts

mintmalitg and we are forced to analyze the W.I. outside of the f lat l imit. An alternative wag, which is sufficient to solve the problem without going out of the f lat l imit, consists in the characterization of the scalar operators by introducing a source for them; i.e. a classical scalar field Z(x) with vanishing canonical dimension [6].

The strategy is that of trying to reabso~ the trace anomalies bg transferring the instability, of the metric field representation to the Z(×) field. Thus we put

Sz(x) = xP(x)opz(x) - ~ a ~ P and analyze, in the flat limit, the new W.I.

(27)

88

( w~(x), ~.(x);/s~:(x), • ~/~;;~(x))F IF~- o. (20)

This identity is resolved, order by order, by a Lagrangian containing arbitrary powers of Z, but depending upon a finite number of parameters. The iterative procedure analyzed in

Ref.[6] shows that the order h ~ is completely identified by the invariant term at the

order ~N+1~l-t; the free parameters are those of the ecale-invariant original theory, plus a finite number of the physically irrelevant terms containing derivative couplings for the Z field.

5-DIMENSIONAL RENORMALIZATION

In the previous section we have illustrated the steps leading to a complete formal

proof of renormalizability for a model with spontaneous symmetry breaking of scale

invariance, but the problem of identifying, starting from a suitable regularization, an

effective scheme of computing Feynman amplitudes remains open. To this end, and also in

view of analyzing gauge models, a dimensional renormalization scheme appears as o

natural choice. In this context a clear definition of scale invariance for geneMc space-

time dimension d is required. According to the lines proposed before the behavior of a

field under infinitesimal local scale transformation is specified by assigning the Weyl

weight. Then a regulaMzed version of the local scale transformation is identified by a

Weyl weight depending on the space time dimension d. For example for a scalar field (~(x)

8~i(x) = ~ ( X ) ~ i ( x ) + (d-2)/2d ~ ;~t~(x)(~(x) + vi). (29)

This choice assure that, in the f lat l imit, the operator ~ l ~ i ~ i is scale invariant, but.the

operator 414 has weight (2d-4)/d instead of I, as necessary for scale invariance. The way out of this diff iculty is to introduce, already at the classical level, a spurion field O(x) with non vanishing vacuum expectation value and caning the Weyl weight necessary to

compensate that of the operator 414. i.e. the new invariant operator is 414 (Q + 1~(4-d)/z) z where t~ is the scale parameter. It is convenient to define the dimensionless external field

~(4-d)/2T.(X) -- O(X). (30)

Referring to our simple model we have the new local scale transformations (with spontaneous breakdown)

8~(x) = ;~l~(x)~(x) * (d-2)/2d 81~ ~,l~(x)(~(x) + ~(~'4)/2v~), (31a)

~l~V(x) = ~P(x)Sp~llV(x)+ 2/d 8p ;~P(x)~tiV(x) - 8p~x)~P~(x)- 8p~.V(x)~Pt~[x), (31b)

89

8o~I~(X) : ~P(x)Sp~i~(x) + ~ ~P(x)~p(X) -(d-2)/2d ~81~ ~P(x),

8~.(x) = XP(x)Sp~.(x) + (4-d)12d 8p },I~(x)(}:(x) + I).

(31c)

(3id)

In Eq.(31a) the mass scale parameter I ~ has been introduced to mantain the mass

dimension of v independent of the space-time dimensionality. Hence the new W.I. in the

flat limit is

2~(8/8~I~F eL- TI~/dS/8~X]~ZL) - (d-2)/2 d ~8/Scup] "eL

+ ~ 8/8~]~L_ (d-2)/2 d ~[ (d~(X) + I~(~-4)12Vi)8/8<~Y a"" ]

+ ~7.(X)8/8~.(x)~'" - (4-d)/2d ~[(~.(x) + I)8/8~.(x)F eL] =0 (32)

and Eq.(32) can be thought to hold for any complex value of the dimension d. Setting

d = 4- 2v (33)

the general solution of Eq.(32), constrained by the classical vacuum condition, appears as

the obvious extension, to include the spurion field, of the classical action in Eq.(12)

and

r~. = _ fd4-2Vx{ 112~V(x)(& + w~(x))((~i(x) + ~-Vvi)(~ + wv(x))((~i(x) + ~Vvi)

+ai~Zv(1 + ~.(x))2(~i (x))4 + b2/2 i~2v(1 + ~.(x))Z~IZ(x)(~(x) + p-Vv)Z }

+ a finite number physically irrelevant terms depending on ~Z, (34a)

818(~rc~-I~ o = o. (34b)

The classical vacuum condition in Eq.(34b) must be maintained at the higher orders as a

necessary constraint for the proper vertex functional. On the other hand the only

lagrangian counterterm containing a linear term in the relevant scalar field which can be

introduced without affecting the W.I. is

~2~(I + E(x))2(d~z(x) + ~-Vv)4. (35)

And is the choice of this term that discriminates among the possible quantum extensions

of the classical model.

Then the complete bare lagrangian obeing the W.I. in Eq.(32), is, up to a multiplicative

field redefinition

= f~1.+ ~2Vb2A[v,a(~/v~V,b(~Iv) v] (I + ~.(x))2(c)2(x) + ~Vv)4. (36)

where A[v,a(~/v~V,b(~Iv) v] is a metamorphic function of v, computed order by order in terms of tadpole Feynman diagrams.

9o

We observe that the proposed dimensional extension acts as a regularization only for

the U.V. divergencies; the !.R. problems, related to the radiative mass generation, must be

taken care of, already at the bare level. According to the philosophy discussed in Section 3

the new Feynman rules include the 1-1oop radiative mass hmzZo) in the d~z propagator and

the extra subtraction concerning the one loop mass correction. It is in terms of these new

rules that the A coefficient must be evaluated. For example the diagram in fig.4 has a

contribution ~In~.

s S

[ "\ Ac' l

fig.4

Finally let us remark that the meaning of the theory is, a priori, not evident. Indeed the

W.I. in Eq.(32) is altered by any renormaltzation procedure which, order by order, removes

at least the v poles. It is then relevant to discuss the class of W.I. which are obtained

from the regularized one through a multiplicative renormalization procedure. In Ref.[8] it

is shown that the theory can be made finite with a multiplicative ";-dependent

renormalization of the fields

hI~V(x) -~ H(:~)hPV(x), (where hI~'(x) = ~I~V(x) - TI'L~v),

~p(X) -~ L(Z)Wp(X),

T. _~ M(Z) = Zf(?-), with f(O) ~ 0

(37a)

(37b) (37c)

(37d)

which produces a renormalized W.I. equivalent, up to a finite multiplicative 7.-dependent

renormalization, to that discussed in Sect.4 in a regularization independent scheme. Thus

we have a precise regularization procedure to analyze the phenomenon of radiative mass

generation. If we are not interested in Green's functions with ~. external legs, we can set

T. = 0 and the bare lagrangian, which in the flat limit becomes

+ t j .2Vb2A[v,a(~/v)2V,b( l~ /v)~ ' ] (<~z(x) + p.'Vv)4, (3B)

is the only information needed for computations.

We observe also that the main interest of utilizing a dimensional approach is in the

minimal subtraction scheme. In particular P. Braitenlohner and D. Maison have shown that,

in a completely massive theory which has no I.R. problems, when the minimal subtraction

scheme corresponds to a multiplicative renormalization process, the related constants

are mass independent[ 121.

91

In our scheme this minimality condition is appareni~ly violated for two main reasons i.e.

the presence of tadpoles, whose compensation is required already in the bare theory, and

the radiative mass which requires a new propagator accompanied by an (I.R.) extra

subtraction for the one loop mass correction.

Concerning the tadpole subtraction it can be automatically implemented by a suitable

alteration of the Feynman rules, which consists in neglecting the tadpole contMbutions to

Feynman graphs. For example in fig.5 are represented the one loop and two loops mass

corrections of <~I and (~2 fields respectively. |..-.. .... .....

, © I ! ,

,, ; ~...e I. ........... J

8.. @ - - © ..... © ..... @ .... O - -

I- . . . . . . . . . . . . " 0 . . . . . . . . ~ " " ' "

.... 0 - - .Q., I= J

fig.5

The new propagator and the extra subtraction required by the presence of radiative

mass induce, from three loops on, in the divergent part of the diagrams, terms

proportional to the Into 2. But the IogaMthmic terms vanish in the sum of all the diagrams

contMbuting to a given Green's function at a fixed perturbative order. The cancellation

mechanism is the same as suggested by P. Breitenlohner and D. Maison for the I.R.

subtraction[13]. For example in fig.6 the diagram (a) contains a divergent contMbution

proportional to ~In(mZli~ z) which is cancelled

! l I | '~ s s

(a) (b) (c)

fig.6

by analogous terms from diagrams (b) and (c). Thus the considered renormalization

procedure is minimal and the corresponding constants are also mass independent.

6-CONCLUSIONS

In this note we have analyzed in a simple model the mechanism of radiative mass

generation brought about by the spontaneous breaking of dilatation symmetry in a

perturbative context. The main points are the use of a local W.I. which formalizes, by

means of a spurion field, the minimality requirement of the trace anomaly which appears

92

as an instability of the classical representation of the general coordinate transformation

in Weyl scheme. The consistent inclusion of the radiative mass in the propagator of the

related field requires a new perturbative expansion which is compatible, although not

coinciding, with the loop ordeMng. The modified perturbation expansion resolves the I.R.

problems connected with the radiative mass.

We have also discussed how an extension to arbitrary space time dimensions d of the

local dilatation W.I. provides an effective scheme for computations, which yields a

quantum extension of the theory equivalent to that obtainable in a regularization

independent way. An attractive feature of the procedure is that it maintains the

renormalization constants mass independent.

Finally we observe that the scheme here illustrated on a simple model with only scalar

fields can be directly employed to characterize to all orders the quantum extension of

gauge models with spontaneously broken dilatation invariance, since the B.R.S. and W.I.

operators commute. One interesting case in this class is the model proposed by S. Coleman

and E. Weinberg in 1973.

Acknowledgement. I'am indebted to A. Blasi for a critical revision of the manuscript.

REFERENCES

[I] S. Weinberg - Phys. Rev. Lett.29(1972),1698; Phys. Rev.D13(1976),974; Phys. Rev.

D7(1973),2887.

[2] H. Georgi, S. Glashow - Phys. Rev.DT(1973),2457.

[3] P.W. Higgs - Phys. Lett. 12(1964), 132.

T. Kibble - Phys. Rev. 155(1967), 1544.

[4] T.E. Clark, O. Piguet, K. Sibold - Nucl. Phys.B99(1977),292.

W.A. Bardeen, O. Piguet, K. Sibold - Phys. Lett.72B(Ig77),231.

[5] S. Coleman, J.E. Weinberg - Phys. Ray.D7(1973), 1888.

[6] G. Bandelloni, C. Becchi, A. Blasi, R. CollI~a - Nucl. Phys.B197(1982),347.

[7] G. Bandelloni, C. Becchi, A. Blasi, R. Collina - Commun. Math. Phys.67(1978), 147.

[8] C. Becchi, A. Blasi, R. Collina - Nucl. Phys.B274(1986), 121.

[9] N.N. Bogoliubov - Exp. Theor. Phys. (USSR)34(1958),58; Nuovo CimentoX(1958),794.

[ 10] J.H. Lowenstein, W. Zimmerman - Commun. Math. Phys.44( 1975),73.

J.H. Lowenstein - Commun. Math. Phys.47(1976),53.

[I I] C.G. Callan, J.S. Coleman, R. Jackiw - Ann. of Phys.59(1970),42.

C.G. Callan - Phys. Rev.D2(1970), 1541.

K. Symanzik - Commun. Math. Phys. 18(1970),227.

[I 2] P. Breitenlohner, D. Maison - Commun. Math. Phys.52(1977), I I.

G. Bonneau - Nucl. Phys.B 167(I 980),261; B 171 (I 980),447.

[I 3] P. Breitenlohner, D. Maison - Commun. Math. Phys.52(1977),55.

Discussion session on par t I:

Non- l inear field t rans format ions in 4 dimensions

To Seminar of Olivier Piguet:

In the N = 1 SUSY case with superfields there is an off-shell infrared problem.

Writing e.g. $(F) ,,o 0 can then not be understood in the sense of scaling. I.e. the

insertion Arm in

s(r)=A'.r

does not necessarily die out for large Euclidean (non-exeptional) momenta, because

that limit does not exist.

Question: What is the precise meaning of ",,~ 0"?

Answer: In a context where the scaling limit cannot be performed it just means that

At m has UV-power counting degree 3 instead of 4. But, in an IlZ-regularized

theory where the scaling limit can be performed it is indeed distinguished from a

hard insertion by being soft. The real goal in pure SYM would be the construction

of gauge independent operators where one of the gauge parameters is the infrared

regulator #2. There the supercurrent is of primary interest. In general SYM one

would in addition expect that gauge independent matter mass insertions A* exist

which have power countig degree 3 and permit the scaling limit i.e. are truly soft.

To Seminar of Peter Breitenlohner:

There is an alternative approach to using directly the infinitely many non-linear

field transformations and sources, namely to employ a differential algebra. This for-

mulation has an infrared problem due to the presence of a constant anti-commuting

parameter of positive dimension: it leads to the insertion of superrenormalizable

vertices.

94

Question: Does this method really avoid the need for defining all the mentioned

non-linear transformations or does it only produce the illusion that it does?

Answer: Like in the case of gauge transformation - versus Bt{S-transformations it

really makes superfluous to define those composite operators.

Question: Where is the information referring to the SUSY content?

Answer: It is contained in the respective Slavnov identity.

Question: Should one take for serious the infrared problem and solve it?

Answer: Yes, these parameters are decisive ingredients of the complete theory. Sug-

gestion: Make these parameters x-dependent. E,g. try the structure of ex-

ternal supergravity and perform the adiabatic limit (constant external fields). The

study of the infrared problems is then at the same time a useful preparation for

supergravity itself.

P a r t I I

N o n - l i n e a r a - M o d e l s

Non-linear a-models considered as classical field theories have a geometrical struc-

ture. The Lagrangian describes harmonic maps from space-time to a Riemannian

target space with a prescribed metric. The main problem of general quantized

non-linear a-models is their non-renormalizability. In more than two space-time di-

mensions they are non-renormalizable by power counting. Even in two dimensions

renorma/ization requires infinitely many parameters describing not only arbitrary

changes of the coordinates (fields) but also arbitrary deformations of the metric. In

order to specify a particular a-model one has to characterize its metric within the

general class of all metrics. One such possibility is provided by spaces like spheres

which can be characterized by their isometry group.

THE NON--LINEAR SIGMA MODEL..

C. B e c c h i .

D i p a r t i m e n t o d i E i s i c a - U n i v e r s i t a d i G e n o v a a n d

I s t i t u t o N a z i o n a l e d i F i s i c a N u c l e a t e , S e z i o n e d i G e n o v a , I t a l y .

Introduction

The non-linear sigma models have been introduced more than 15 years

ago (1,2) to describe the infrared properties in d>2 space-time

dimensions of systems with symmetry spontaneously broken according to

the Golstone-Nambu mechanism.

The first step in their construction consists in the choice of a

non-linear representation [2] of the spontaneously broken symmetry

group. This leads to the study of models based on coset (homogeneous)

spaces (3). That is, the field carrying the Goldstone degrees of

freedom belongs to the quotient space of the broken symmetry group G

with respect to the stability group H of the classical vacuum

configuration.

An important example of tills kind is that of the chiral models in 4

space-time dimensions, where the group G is identified with the chiral

e. g. SU(3)*SU(3) group and the stabi|ity group H is the diagonal

(vector> SU(3). Another interestig example is the Heisenberg model in

d<3 space dimensions [&). Here G=O(n), n being the number of components

of the unit spin vectors, while H=O(n-l~.

More recently non linear sigma models have been discussed in d=2

dimensions to understand the physical space-time Structure of string

theories [5]. In this case coset spaces do not play any particular

role; rather people study models where the field belongs to more

general RJeman~ia~ manifolds.

From the point of view of perturbation theory, space-time dimension 2

is particularly relevant, since for d=2 the non-linear sigma models are

power-counting renormalizable. As it is well known, this means that,

developing the lagrangian density in powers of the field, one does not

find any coefficient with negative mass dimension; not even any with

98

p o s i t i v e d i m e n s i o n e x c e p t f o r t h e i n f r a r e d r e g u l a t o r t e r m w h o s e r o l e

w i l l b e d i s c u s s e d i n t h e f o l l o w i n g .

Some p a r t i c u l a r models based on c o s e t spaces f as e. g. the Heisenberg

model, in 2 d imensions have been proved to be " r e a l l y " r e n o r m a l i z a b l e

( 6 , 7 , 8 ) . By t h i s we mean t h a t the p e r t u r b a t i v e expans ions of Green

( c o r r e l a t i o n ) f u n c t i o n s of p h y s i c a l l y meaningful] o p e r a t o r s , those

independent of the p a r t i c u l a r c o o r d i n a t e s chosen to i d e n t i f y the f i e l d

c o n f i g u r a t i o n s , a re un ique ly i d e n t i f i e d in terms of a f i n i t e number of

symmetry and n o r m a l i z a t i o n c o n d i t i o n s and hence depend on a f i n i t e

:lumber of pa rame te r s .

I t t u r n s a l s o out a t d=2 t h a t , i f the manifold M has " n e g a t i v e "

c u r v a t u r e ( t h a t o f t h e s p h e r e ! ) , t h e c o r r e s p o n d i n g m o d e l i s

a s y m p t o t i c a l l y f r e e . I n t h i s s i t u a t i o n W i l s o n r e n o r m a l i z a t i o n g r o u p

a n a l y s i s ( 9 ) g i v e s s o m e i n t e r e s t i n g s u g g e s t i o n s . F i r s t ( l O l , a b o v e 2

d i m e n s i o n s , t h e u l t r a v i o l e t p r o p e r t i e s o f t h e m o d e l a r e d e t e r m i n e d b y a

n o n - t r i v i a l , u l t r a v i o l e t s t a b l e f i x e d p o i n t a n d b e y o n d p e r t u r b a t i o n

t h e o r v t h e m o d e l s e e m t o b e r e n o r m a l i z a b l e , p e r h a p s u p t o 4 d i m e n s i o n s

( 4 1 . S e c o n d l y , a t d = 2 , t h e l o n g d i s t a n c e p r o p e r t i e s o f t h e t h e o r y a r e

e x p e c t e d t o b e f i x e d b y t h e p o s s i b l e p r e s e n c e o f a n i n f r a r e d u n s t a b l e

f i x e d p o i n t . T h e n a t u r e o f t h i s i n s t a b i l i t y h a s b e e n s t u d i e d

~ 4 , 6 , 1 1 , 1 2 ) i n t h e c a s e o f t h e H e i s e n b e r g m o d e l i n t h e l i m i t w h e r e t h e

number of f i e l d components tend to i n f i n i t y . The most r e l e v a n t

sugges t ion , emerging from t h e s e s t u d i e s i s the p resence of a

d imens iona l t r a n s m u t a t i o n mechanism g e n e r a t i n g a mass gap.

Of course the r e n o r m a l i z a t i o n group r e s u l t s and s u g g e s t i o n s r e f e r to

very p a r t i c u l a r models on c o s e t spaces . I t i s by no means obvious t h a t

they could be extended to a g e n e r i c compact mani fo ld . Indeed W i l s o n ' s

a n a l y s i s (9) i s based on the h y p o t h e s i s t h a t the r e l e v a n t t heo ry ( t h a t

co r r e spond ing to a f i x e d p o i n t ) c o n t a i n only a f i n i t e number of

pa rame te r s , t h i s i s not the case of a Riemannian mani fo ld (3,13) whose

g e o m e t r i c a l p r o p e r t i e s a re c h a r a c t e r i z e d by a g e n e r i c m e t r i c t e n s o r .

There fore one should expec t to have meaningful quantum t h e o r i e s only

foF some s p e c i a l c l a s s of m a n i f o l d s which r ema ins to be d i s c o v e r e d .

Th i s problem i s ana logous , and perhaps s t r i c t l y connected, to t h a t of

e x t e n d i n g the c l a s s of models which a re proven to be " r e a l l y "

r e n o r m a l i z a b l e in the sense d i s c u s s e d above. I t looks r e a s o n a b l e t h a t

t h i s c l a s s would c o n t a i n a t l e a s t a l l t he models based on c o s e t spaces ,

s i n c e the a c t i o n of t h e s e models i s i d e n t i f i e d , up to a f i e l d

r e d e f i n i t i o n co r r e spond ing to a c o o r d i n a t e t r a n s f o r m a t i o n , by the

99

invariance of the action under the isometries of the manifold. To the

infinitesimal generators of the isometry group there correspond at the

quantum level Ward identities {6] constaining the Green functions of

the theory. The stability of these identities under quantum corrections

guarantee the "real" renormalizability of the coset space models.

Taking as a guide the renormalization group point of view we should

not be satisfied with the coset spaces. Indeed, if we assume that the

renormalization group action be smoot:h enough to deform the field

manifold without violating its global topological properties, we expect

that r-euormalization group fixed points should exist not corresponding

to coset spaces. This would happen e. g. in the case of two dimensional

manifolds of genus larger than one. Referring to this situation one

should wonder if the sigma models on complex algebraic curves are

"really" renormalizable.

Let us also mention that a possible way to build new "really"

renormalizable models could be based on the "reduction mechanism"

proposed by R. Oehme and W. Zimmermann (]A] and successfully applied to

a vast class of theories [15).

We have given a typical example of the questions which remain open

even after remarkable Friedan's thesis {161 on the renormalization of

non-linear sigma models. In this thesis Friedan gives a complete set of

rules, based on dimensional regularization S to characterize the

possible divergences appearing in the perturbative construction of the

theory. The analysis starts from the choice of a sultable coordinate

system, the geodesic normal coordinates [13,17) corresponding to the

Dare metric of'the model.

I n t h e r e s t o f t h i s p a p e r we s h a l l r e c a l l a n d d i s c u s s t h e g e n e r a l

a s p e c t s o f t h e a n a l y s i s o f r e n o r m a l i z a t i o n o f n o n - l i n e a r s i g m a m o d e l s ,

e~idencing the r e s u l t s which are independent of the f i e l d

Paramet r iza t ion . Our aim i s not to p resen t new r e s u l t s , but to e xh i b i t

the s t a t u s of the problem in i t s s imples t p o s s i b l e form.

We sha l l begin our a n a l y s i s r e c a l l i n g the main formal s t e p s of the

c o n s t r u c t i o n ~ a quantum theory and the d i f f i c u l t i e s o f the

p e r t u r b a t i v e approach connected with the presence of divergences.

Following Friedan, we sha l l d i s c u s s the c o n s t r a i n t s connected with the

geometrical p r o p e r t i e s o f the models. We sha l l then analyse the

u l t r a v i o l e t d ivergences appearing order by order in the p e r t u r b a t i v e

expansion, comparing the case of a generic Riemannian manifold with

t h a t of a cose t space.

100

Formal construction of the quantized theory.

The classical field is a function on R ~ takin9 value on a C~Riemannian manifold M with metric tensor g~j. The classical action is defined:

A ----- -~flXJ" :}ii{~} ~+i ~ +~ (1)

A lattice regu!arized version (A ) of the action is built [181

replacin9 R with the lattice Z . Labelling by p the points of the

lattice and bY p- the links, we write:

wl~ere D(# , #') is the distance between the corresponding points on the

man1 fold.

Formally the quantization of the model is based on the measure:

_A

or in i t s l a t t i c e r e g u l a r i z e d v e r s i o n : (9)

-A.

where at*_. isd. t h e c o , , a r i a n t l y cons t an t , measu re {1~1 on t h e m a n i f o l d (ck~.-~_ = ~ det,q dx ).

For the models of relevant interest in statistical mechanics the loop

orderin9 parameter ~ is replaced by the "bare" temperature t {4] (which

has to be renormalized as any other physical quantity).

Of course the quanEum measure has to be written explicitly in .terms of

coordinates on the manifold ; in 9eneral this requires more than a

single coordinate system which is limited to a local chart not covering

M.

To avoid this difficulty we assume t.hat the quantum fluctuations at

the point, x be damped when the point 9oes to infinity. That is: lira

~x '} where is point on the manifold M. We also assume that m m any J

101

the quantum fluctuations never exceed the border of the coordinate

chart centered in m. This hypothesis looks perfectly reasonable for

space-time dimension d>2. If d=2, large fluctuations at infinity could

be responsible of the expected infrared instabilities. In any case an

assumption of this kind seems to be technically unavoidable (163. [t

implies that the quantum measure is decomposed in disjoint

contributions corresponding to the different asymptotic values m of the

f i e l d .

We c a l l m t h e " c o n s t a n t b a c k g r o u n d f i e l d " n o t t o b e c o n f u s e d w i t h

o r d i n a r y v a r i a b l e " b a c k g r o u n d f i e l d " u s e d a s a v a l i d t e c h n i c a l t o o l f o r

c a l c u l a t i o n p u r p o s e s [ 1 , 1 7 , 1 9 ) .

F o r e v e r y c o n s t a n t b a c k g r o u n d v a l u e m we c h o o s e a c o o r d i n a t e s y s t e m

. E ~ ) m a p p i n g a n e i g h b o r h o o d o f t h e o r i g i n o f R ~ , i f n i s t h e d i m e n s i o n

o f t h e m a n i f o l d , i n t o a n e i g h b o r h o o d o f m.

I n t h i s c o o r d i n a t e s y s t e m t h e m e t r i c t e n s o r i s w r i t t e n :

and hence the quantum measure is:

N o t i c e t h a t . i n g e n e r a l i t i s n o t p o s s i b l e t o a s s i g n a c o o r d i n a t e

s y s t e m c e n t e r e d a t e v e r y p o i n t o f a m a n i f o l d w i t h c o n t i n u o u s t r a n s i t i o n

f u n c t i o n s b e t w e e n e v e r y p a i r o f o v e r l a p p i n g s y s t e m s . H o w e v e r t h i s i s

n o t a n o b s t a c l e t o o u r c o n t r u c t i o n i £ t h e q u a n t u m m e a s u r e i s i n v a r i a n t

u n d e r c o o r d i n a t e t r a n s f o r m a t i o n s . I n t h e f o l l o w i n g we s h a l l a s s u m e o l l l y

l o c a l l y t h e s m o o t h n e s s o f t h e t r a n s i t i o n f u n c t i o n s .

We h a v e t h u s i n t r o d u c e d i n t h e m e t r i c t e n s o r , a n d h e n c e i n t h e a c t i o n

a n d i n t h e q u a n t u m f u n c t i o n a l m e a s u r e , t h e d o u b l e d e p e n d e n c e o n t h e

f i e l d ~ a n d o n t h e c o n s t a n t b a c k g r o u n d m . I n o t h e r w o r d s we h a v e

i n d e p e n d e n t l y a s s i g n e d t h e m e t r i c t e n s o r i n e v e r y c h a r t . I t r e m a i n s t o

a s s u r e t h a t t h e d i f f e r e n t l o c a l a s s i g n e m e n t s o f t h e m e t r i c t e n s o r

c o r r e s p o n d t o a u n i q u e , g l o b a l l y d e f i n e d t e n s o r o n t h e m a n i f o l d M.

We f i n d i n F r i e d a n ' s t h e s i s [ 1 6 3 how t h i s c o n d i t i o n c a n b e w r i t t e n i n

t e r m s o f a " n o n - l i n e a r c o n n e c t i o n " O. G i v e n a t t h e p o i n t m a t a n g e n t

v e c t o r v t o M, Q d e f i n e s a c o r r e s p o n d i n g d e r i v a t i v e a c t i n g o n f u n c t i o n s

o f t h e d o u b l e v a r i a b l e m a n d ff :

c ,DI - ( v ,

102

Hese ( v , ~ ) means the o r d i n a r y p a r t i a l d e r i v a t i v e wi th r e s p e c t to the

background [] induced by the v e c t o r v, whi le the secon ter[] d e f i n e s a

p a r t i a l d e r i v a t i v e wi th r e s p e c t to the quantum f i e l d

The r equ i r emen t t h a t the f u n c t i o n f([], ff ) i d e n t i f i e s a un iqu~

, g l o b a l l y de f i ned , s c a l a r f u n c t i o n on H, i s w r i t t e n :

Th i s i s c a l l e d the " c o m p a t i b i l i t y c o n d i t i o n " f o r the c o o r d i n a t e choice

E ( 4 ) .

I f we a s s i g n f o r every p o i n t [] in an open s e t U a b a s i s (m)

(i=1 . . . . . n) of T (H), the t angen t space to the [ ]an i fo ld a t the p o i n t m,

we have c o r r e s p o n d i n g l y a system of d e r i v a t i v e s ~ = ( v i , D ) , and i f the

Lie p roduc t r u l e s a r e g iven:

[ v?-) c~) v (-) (8)

we have the i n t e g r a b i l i t y c o n d i t i o n s (commutation r u l e s ) :

;i

I t i s a p p a r e n t t h a t t h e c o m p a t i b i l i t y c o n d i t i o n l e a d s d i r e c t l y t o a

f u n c t i o n a l c o n s t r a i n t f o r t h e a c t i o n a n d f o r t h e q u a n t u m m e a s u r e .

I n t e r m s o f t h e q u a n t u m m e a s u r e o n e d e f i n e s t h e c o r r e l a t i o n f u n c t i o n s

o f s o m e p h y s i c a l l y m e a n i n g f u l e . g . s c a l a r f u n c t i o n h ( m , ~ ) :

w h e r e we h a v e a l s o i n t e g r a t e d o n t h e c o n s t a n t b a c k g r o u n d m.

I n t h e p e r t u r b a t i v e f r a m e w o r k t h e c o r r e l a t i o n f u n c t i o n s a r e c o m p u t e d

i n t e r m s o f F e y n m a n a m p l i t u d e s w h i c h c o i n c i d e w i t h t h e c o r r e l a t i o n

( G r e e n ) f u n c t i o n o f t h e f i e l d v a r i a b l e s . T h e i r f u n c t i o n a l g e n e r a t o r i s

the F o u r i e r transformed quantum measure:

This functional generator has only local meaning and it is not

103

independent of the choice of local coordinates, nevertheless it is a

necessary tool of perturbative renormalization.

The perturbative development of the functional generator Z is affected

with two kind of divergences.

There are ultraviolet divergences wh~th are intrinsically related to

the definition of the quantum measure and have to be controlled by a

suitable regularization and cured by the renormalization procedure.

We shall discuss the renorma]ization in the following always

understanding dimensional regularization. This is indeed particularly

suitable to preserve the constraints defining the quantized version of

the model.

There are also infrared divergences, since the quantized field is

massless and even the propagator is ill defined in two dimensions. To

avoid this difficulty we shall introduce a mass term for the quantum

field {61 preventing too large long wavelenght field fluctuations. This

mass term ruins the compatibility condition for the action since it

introduces an attractive force toward the background. In the fol]owing

we shall forget this problem since the effect~ of the mass term are

"soft" i. e. negligible at short wavelenght.

However one should remember that at the end of every computation a

zero-mass limit has to be performed to recover the original geometrical

structure of the theory. This limit, which has been only studied in the

case of coset space models {18,20}, in general does not exist, and, in

the most favourable situation, it is meaningful only' for some special

class of correlation functions.

Renormalization.

To discuss the renormalization of our theory we have, first of all, co

write the "compatibility" condition for the Feynman functional Z,

For every vector field v i (m) we introduce a constant (m and x

independent) Grassmann (anticommuting) variable C A and an anticommuting

source: ~(x). Then we add to the action A the term:

104

t h u s m o d i f y i n g t h e f u n c t i o n a l :

F o r t h i s new g e n e r a t o r we h a v e a " S l a v n o v - l i k e c o m p a t i b i l i t y

condition" [8,21):

T h e m i s s i n g t e r m s i n d i c a t e d by d o t s i n t h e r i g h t - - h a n d s i d e h a v e t o b e

s u i t a b l y c h o s e n t o m a k e t h e S o p e r a t o r n i h i l p o t e n t . T h i s i s a l w a y s

p o s s i b l e a n d t h e n u m b e r o f n e e d e d t e r m s d e p e n d s on t h e p a r t i c u l a r

o f t h e b a s i s { v i ~ ( 2 2 ) . choice

In order to make as plain as possible the formalism, in complete

generality we shall choose locally vector fields generating independent ;i

translations; i. e. such that the structure constants F and hence the

further terms vanish. The reader should keep in mind that this has only

the consequence of simplifying the formulae.

Our Slavnov identity, which is now reduced to the first two terms of

Eq(14), is equivalent to the prescription of Eq(9) for the metric

tensor and of the integrability condition in EQ(II).

AS usual, the consequences of Eq(14) for the quantum extension of our

theory are analysed introducing the vertex generator (effective action)

of the theory, which is defined in terms of the connected functional: W~

= In Z, by the Legendre transformation:

T h e S l a v n o v i d e n t i t y i s w r i t t e n i n t e r m s o f ~ a s f o l l o w s :

; gr qC,,L)F+ :o , , o ,

T h e a n a l y s i s o f t h i s e q u a t i o n f o l l o w s a , b y now s t a n d a r d , i t e r a t i v e

p r o c e d u r e . F i r s t o n e n o t i c e s t h a t E q ( 1 6 ) i s a u t o m a t i c a l l y s a t i s f i e d b y

t h e " b a r e " , d i m e n s i o n a l l y r e g u l a r i z e d , p r o p e r a m p l i t u d e s . R e m e m b e r i n g

t h a t ~ i s a f o r m a l p o w e r s e r i e s i n ~ ( t ) w i t h z e r o t h o r d e r v a l u e e q u a l

to the complete action A~ = A + A S , and assuming that EQ(16) has not

been broken by the renormalization procedure up to the (n-l)-th order,

o n e g e t s f o r t h e n - t h o r d e r s i n g u l a r t e r m s . S :

r , i )i C'

105

Since S i s n i h i l p o t e n t , t h i s equat ion has the general s o l u t i o n :

The f i r s t term in the r i gh t -hand s ide i s the first__t~order v a r i a t i o n of

the action under the singular transformation: ~A~4~,~, and hence it

cq]lects the terms which are trivially compensated by a field

redefinition. Notice that the validity of our assumption on the

independence of our theory of the particular choice of coordinates is

confirmed by the fact that every possible field redefinition is

automatically reabsorbed into our scheme.

The second term contains the non-trivial divergences, those affecting

the coordinate independent properties of the metric tensor and of the

[~on-linear connection. While the non-linear connection is uniquely

identified by the metric tensor(14), this one can be freely deformed in

al, infinite number of independent ways. This in general means an

infinite number of independent divergent contributions requiring each a

different normalization condition.

In this situation the model is not "really" renormalizable.

It remains to discuss how the infinite number of independent

normalization conditions can be replaced, in some special cases, by a

finite number of constraints whose implementation makes the theory

"really" renormalizable. Our discussion will be brief and necessarily

limited to the, up to now, only known case, that of the coset space

mudels.

We have already recalled that in a coset space the metric tensor is

constrained and identified up to coordinate transformations by an

Jsometry group G. To the infinitesimal generators of this group there

correspondia system of Killing vector fields (13), which in terms of

our coordinates will be defined by the system of differential

operators:

n o t to be mistaken with the non-linear connection.

The "compatibility condition", prescribing the global definltness the

106

KiJ. l i n g vec to r f i e l d s , r educes to the equa t ion : (x ,D )=0.

The i somet ry c o n d i t i o n s associa{ted with the k i l l i n g f i e l d s X e can be

t r a n s l a t e d in to a system of f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s for the

a c t i o n :

e.:o C:~

In t u r n t l l i s sys tem g e n e r a t e s a sys tem of W a r d i d e n t i t i e s commuting

with the S]avnov i d e n t i t y or , even b e t t e r , a f t e r the i n t r o d u c t i o n o f a

s u i t a b l e new sys tem of c o n s t a n t ant icommuting c o e f f i c i e n t . s , C~, i t can

be i n s e r t e d i n t o the Slavnov i d e n t i t y s imply adding a new term~othe

a c t i o n :

m o d i f y i n g t h e S l a v n o v o p e r a t o r :

s - sz r, i ndependen t ly of the p a r t i c u l a r formal a t t i t u d e , the Sl~stant i a l

consequence of the e x i s t e n c e of i s o m e t r i e s a t the quantum leve l i s t h a t

t h e n o n t r i v i a l s i n g u l a r t e r m s i n ~ L a r e , o r d e r b y o r d e r . c o s t r a i n e d b y

t h e s a m e c o n d i t i o n a s t h e a c t i o n ( E q ( 2 0 ) ) . T h i s i n g e n e r a l m a k e s f i n i t e

t h e n u m b e r o f i n d e p e n d e n t , n o n - t r i v i a l , d i v e r g e n c e s a n d h e n c e " r e a l l y "

r e n o r m a l i z a b l e t h e t h e o r , ~ - .

Conclus ion .

We n o w u n d e r s t a n d t h e d i f f e r e n t r o l e s o f " c o m p a t i b i l i t y c o n d i t i o n g "

and i somet ry Ward i d e n t i t i e s . While the f i r s t c o n d i t i o n s s imply ensu re

t h a t tile m a t r i x g~(m,f) appea r ing in the a c t i o n i s r e l a t e d to the

g l o b a l l y d e f i n e d m e t r i c t e n s o r of the manifo]d wi thout any p a r t i c u l a r

c o n s t r a i n t fo r the manifo]d i t s e l f , the Ward i d e n t i t i e s co r respond ing

to t he K i l l i n g vec to r f i e l d s i d e n t i f y the Riemannian mani fo ld a s a

c o s e t space depending on a f i n i t e number of pa r a me te r s . I t i s t h i s

second s t e p which makes " r e a l l y " r e n o r m a l i z a b l e the theory producing an

107

essential reduction of tile number of free parameters.

References.

(I) K. Meetz, J. Math. Phys. I0 (1969), 65.

J. Honerkamp, Nucl. Phys. B 36 (1972), 130.

G. Ecker and J. Honerkamp, Nucl. Phys. B 35 (1971), 481.

(2) C. G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177

(1969), 2247.

S. Weinberg, Phys. Rev. 166 (1968),1568.

S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1968), 2239.

(Bl S. Helgason, Differential Geometry and Symmetric Spaces,

Academic Press, New York (1962)

(4) E. Brezin and J. Zinn-Justin, Phys, Rev. B l& (1976), BllO.

(5) E. S. Fradkin and A. A. Tseytlin, Nucl. Phys. B 261 (1985), I.

C. G. Callan, D. Friedan, E. J. Martinec and M. J. Perry,

Nucl. Phys. B 262 (1985), 593.

C. Lovelace, Phys. Letters 135 B (1984), 75.

f6) E. Brezin, J. Zinn ,Justin, Phys. Rev. Letters 36 (1976), 691.

E. BrezJn, J. C. Guillou and J. Zinn-Justin, Phys. Rev. D 14

(1976), 2615.

[7) G. Bonnaeu and F. Delduc, Nucl. Phys. B 266 (1986), 536.

F. Delduc and G. Valent, Nucl. Phys. B 253 (1985), 494.

G. Valent, Nucl. Phys. B 238 (1984), 142.

(8) A. Blasi and R. Collina, Nucl. Phys. B (1987) in publication.

[9) K. G. Wilson and J. Kogut, Phys. Reports C 12 (1974), 75.

[I0) A. M. Polyakov, Phys. Letters 59 B (1975), 79.

[11) G. Parisi, Phys. Letters 76 B (1978), 65 and Nucl. Phys. B 150

(1979), 163.

[12] F. David, Nucl. Phys. B 293 (1982), 433 and Nucl. Phys. B294

(1984), 2 3 7 .

[13] A. Lichnerowicz, Theorie globale des connections et des

groupes d'holonomie, Edizioni Cremonese, Roma (1962).

[14] R. Oehme. W. Zimmermann, Commun. Math. Phys. 97 (1985),569.

[15) o. Piguet and K. Sibold, this hook.

[16J D. Friedan, Phys. Rev. Letters 45 (1980) 1057 and Ann. Phys.

108

(N. Y . ) 163 ( 1 9 8 5 ) , 3 1 8 .

{17 ] L. A l v a r e z - G a u m e , D. F r i e d a n a n d S . M u k h i , Ann . P h y s . (N. Y . )

134 ( 1 9 8 1 ) , 8 5 .

{18 ] S . E l i t z u r , I n s t i t u t e f o r A d v a n c e d S t u d i e s ( 1 9 7 9 ) a n d

N u c l . P h y s . B 2 1 2 ( 1 9 8 3 ) , 5 3 6 .

( 1 9 } P . S . H o w e , G. P a p a d o p o u l o s a n d K. S . S t e l l e , N u c l , P h y s .

[20) F. David, Phys. Letters 96 B (1980), 371 and Commun. Math.

P h y s . 81 ( 1 9 8 1 ) , 1 4 9 .

( 2 1 } A. A. S l a v n o v , T . M. O. 10 ( 1 9 7 2 ) , 1 5 3 .

C. Becchi, A. Rouet, R. Stora, Phis. Letters 52 B (1974), 344.

( 2 2 ) M. H e n n e a u x , P h y s . R e p o r t s 126 ( 1 9 8 5 ) 2 .

B.R.S. renormalization of O(n+l) non linear o-model

A.Blasi

Dtplrttmnto dl Fllica dell'Ualver~iti"

I.N.F.N. 8eztene dl Genova

Introduction

We propose a new method of analyzing the perturbative renormalizability of the O(N+I)

non linear o-model in two space -time dimensions. Historically the model is built with a

field vector lying on the surface of the N+I sphere of unit radius and the action is the

free one[I]. The resolution of the unit length constraint yields the interaction and a

particular parametrization of the action itself in terms of N independent fields. The

original O(N÷I) symmetry is no longer linearly realized but only a O(N) subgroup

maintains a linear representation, while the remaining N generators are non linear. The

model provides a simple example of non linear symmetry [2] which, in general, poses the

problem of introducing a denumerable set of sources in the action in order to control the

behavior under renormalization of the non linear transformations [3]. In a conventional

approach this problem does not appear only in the parametMzation corresponding to the

erthogonal projection [4].

This we shall refer to as the algebraic problem. We shall show that replacing the

commuting parameters of the transformations with anticommuting ones, allows the

definition of a nihilpotent B.R.S. operator which can be used to control the algebraic

properties of the model with a finite set of external fields [5].

Now the algebraic problem is not the only one connected with this model; in two

dimensions the I/k2 propagator is ill defined and we need a way of regularizing it. The

commonly adopted procedure is to introduce a mass term with precise transformation

properties under the group . The choice, natural in the orthogonal projection [4], is

somewhat freer in other parametrizations [3]. Here we shall show that there is a

preferred mass term and that the B.R.S. algebraic operator can be extended, still

remaining nihilpotent, so that the mass term can be included in the invariant action.lt

will also turn out that this choice is the convenient one to discuss the zero mass limit of

the theory [6].

According to the above lines, we first consider the algebraic aspect and then the

inclusion of the mass to arrive at the complete B.R.S. operator

110

The renormalizabiIitg of the model can now be analyzed by standard methods which, in the regularization independent approach we adopt, amount to a stability check for the

classical action and to the computation of the first cohomology class.

We shall report only the results of the necessary algebraic computations whose details can be found in [7].

Finally let us remark that our method can be applied to solve the algebraic problem (modulo group theoretical dif f icult ies) for non linear o-models built on any symmetric or homogeneous space [8].

The algebra

The group structure of the symmetric 0(N+ I)/O(N) space is the following: denote by Wij= -

W~ the infinitesimal generators of the linearly realized 0(N) subgroup and by W i the

remaining ones with commutation relations

[Wij,Wkl] = 8~Wil + 811W~ - 8~WjI - 8jIW~ (2Ja)

[Wij,W k] : 8kj W i - 8 kiwi l;j;k;l=1...N (2.1b) [Wi,Wj] = - Wij (2.1c)

The fields of the carrier space are ~(x) i=1....N and the infinitesimal generators are

realized as

wij =j'd2x(~i(x)S/S{j(x)-~j(x)S/S,:h(x)) = .rd2xpij,~(x)S/SC~(x) (2.2)

so that dp2(x)= ~(x) ~(x) is invaMant under Wij ,and

Wi = .~d2xlX(¢ z) 8~+a(Cz )q~q~l~] SlS~(x)= j'dZxP~8/8~(x) (2.3)

The algebraic commutation relations in (1.11 impose the condition

2[x(¢2) -,-o(~ )Ix £ x(~) o(~ ) /~=- I (2.4)

where xe=d;~IddpZ and there is only one arbitrary function (say X) in the model. Any choice

of ~ corresponds to a particular projection of the original N+I sphere: for example ~k=(1- q~z)1/2 or

111

~,=I/2(I- ~) identify the orthogonal and stereographic projections respectively.

'The classical action invariant under (2.2-2.3) is given by

A- I/gSeXt j(x) (x) j(x)1 with

(x2+ ¢ r'[sij +

(2.5)

(2.6)

In order to unify notation, introduce a single greek index c~={i:ij} so that the commutators

in (2.1) are written as

[Wa, W~]=fap~/W~/ (2.7)

These relations can be embedded in a "rigid" B.R.S. scheme by means of anticommuting

parameters C e, where C U is coupled to WIj and C i to W i and it is convenient to assign to

the C e a negative unit of Faddev-Popov charge.

We now define the B.R.S. operator

S = CaWe - !I 2 C a C a f~/818C~' (2.8)

whose nihilpotency is a direct consequence of (2.7) and the Jacobi identity for the

structure constants f~/.

The classical action in (2.5) is the general Faddev-Popov neutral, local functional

invariant under s.

Due to the nonlinear kernel of W~, the s operator cannot be directly employed to

discuss the renormalizability of the model, but it has to be translated in a suitable

functional form with the help of a set of auxiliary external fields ~/i(x),i=1....N,carrying a

positive unit of Faddev Popov charge and assigned a canonical dimension equal to two. The

new classical action becomes

to1= A + C a j'd2xp~(x) ~/i(x) (2.9)

and the B.R.S. symmetry is descMbed by the functional relation

(s to1)= j'd2x( 8 ro1/87i(x) 8 rcl/8~(x ) ÷ i/2 ca ca fa~ar¢|l~c~ (2.1o)

TO check the nihilpotency of the operator in (2.10) we have to rewMte it for the

connected functional Z¢I[J] = 7ci + fd2xji(x)~(x) evaluated atJi(x)= -8 Y¢118~(x) , where

112

it reads

S ZolIJ]= [ fdZx(-Ji(x) ~/8~/i(X))+ 1/2 C~ C 13 f~l],fc3/OC~,]Zo][j] (2.11 )

We have now a functional identity which could be used as the basis to analyze the quantum corrections to the classical action in (2.9); as mentioned in the Introduction we sti l l to define the propagator with the addition of an appropriate mass term to the action.

The mass term

Let us first notice that the nihilpotency of the BRS operator can be easily exploited to

introduce in the action an arbitrary mass term n~f((1)z) provided we also introduce its

variation coupled to an external field. This is the approach considered in [7]; here we

shall illustrate an alternative procedure which identifies a preferred mass term [9].

There is one projection, namely the orthogonal [4],where the natural choice is to select

the mass along the projection axis; here the mass is given by (I- ~z)I/z and its

transformations are

Wi(1- (~)I/2=-~ Wi ~= (I- (~)ll2 8~ (3.1)

In order to reproduce (3.1) in an arbitrary parametrization, we introduce composite

operators(interpolating fields)

~(X)=~(X) G(c~(x)) (3.2)

and impose at the classical level the analog of (3.1),i.e.

W i M(¢~)--- ~ Wi~4~= M(¢~-)~ (3.3)

The solution to (3.3) is easily found to be

G--M/~ and MI/M--- I/2~(~ + o) (3.4)

which identifies both the mass term and the interpolating field up to a multiplicative

constant.

113

To implement (3.4) beyond the classical level we need sources Ki(x) for the composite

operators £4(x) and co(x) for the mass term which are assigned a canonical dimension

equal to two and are Faddev Popov neutral.

The complete classical action is now

rm= .l'd2xKi(x)-~(x)+ fd2xo~(x)M((~2) + rc1 (3.5)

with Ycl given in (2.9). The behavior of the first two terms on the r.h.s, of (3.5) under

infinitesimal transformations is computed by (2.2;2.3;3.3);explicitely we have

Wij j'dZxKi(x)~(x)= j'd2x(Ki(x)=-j(x)-Kj(x)~(x)) =.rd2x(Kia/sKj-Kjs/SKi)I'm

W i j'd2xKlix)5(x)= j'd2xKi(x)M(~) = j'dZxa/awY m

(3.6a)

(3.6b)

Wij SdZxco(x)M(q~2)=O (3.7a)

(3.7b)

According to the above expressions ,we modify the BRS operator in(2.1 I) to the new form

S= j'd2x(-Ji(x)8/Syi(x))+ 1/2C~C~f@~,81~Cy+2CiJ J'd2xKja/~Ki+C~ j'd2xKia/aw

c~ j'~x(~.,rrC-)S/aK~ (3.8)

where we have shifted w(x) to co(x)+m 2 in order to have an explicit mass parameter.A

direct check shows that S 2 still vanishes.

The proof of the renormalizability of the model is now based on the possibility of

extending to all perturbative orders the identity

SZ[Ji;yi;Kj;co] =0 for the connected functional Z,or

(3.9)

(sr)=j'd2x(SrlsT~(x)ar/aC~(x) ÷ I/2c~ c~ f~var/ac ~ +2CiJJ'd2xKjaf'laKi ÷dJ'exNSr/sw - dfex(~÷~)sr/s~ (3.1o)

for the vertex functional £ whose classical approximation is given in (3.5).

The procedure we shall follow is the one adopted when no regularization is assumed

and it requires two independent checks; first we must control that the classical model is

stable and then that the first cohomology class of the S operator is empty [10].

114

In our case the computation is completely algebraic i.e. the group structure is sufficient to guarantee the above conditions.

Stability

We shall analyze this property by means of the identity (3.10) for the vertex functional;

notice that the expression for (SY) is invariant under the field, source and parameter redefinition

~(x) 4 (~(x)Z(~(x)) (4.18) 7i(x) -) ~j(x) 7j(x) with ~j(x)=Z(~(x)) 8ij+Z~(~(x)) (~i(x) (~j(x) (4.1b)

Ki(X) ") a Ki(x) (4. I c)

~o(x) -> a ~o(x) (4. I d)

m 2 "~ a rn2 (4.1e)

g -) zgg (4.1f)

Suppose we now perturb the classical in (3.5) wi th a term

or!-- fex j(x) (x) &¢j(x), c fexR=(x) .fexK,(x) (x) + fdZxm(x)E(~), m 2 fdZxF(~) (4.2)

which to first order in £ satisfies

(S[Fm+EF ! ])=0 (4.3)

where Au(x) , R~(x), Bt(x),E(x), F(x) are functions of the fields ~(x) without space-time

derivatives.

The model is said to be stable i f such perturbation can be reabsorbed by performing on the classical action a transformation as in (4.1).

It is well known that s tabi l i ty is a necessary condition for renormalizabil i ty and that i t becomes also suff icient i f there is a regulaMzation prescription which preserves the BRS identity.

To proceed with the stabi l i ty check, we select in (4.3) the f i rs t order in E and obtain

115

SLq:{ fex( Js x)s/s- x), ,i/2c% f %c +2dJfex jS/s +C i j'dZxKiSlS~o - C i.rex(~o+m2)8/81 ~ }F, (4.4)

where the linearized operator S L is still nihilpotent.

We discuss the solution of (4.4) by setting to zero the coefficients of the independent

expressions in the external fields and anticommuting parameters. First select the

coefficient of ~/i(x)C~Ci3 which yields the system of equations

W~ R~(x) - W~R~(x) -fd2yR~j(y)S/8(~j(y) PI~(X)+ J'd2yRi~j(Y)S/Sd~(y) Pa~(x)

+ f~I3TR~i(X) = 0 (4.5)

whose solution, analyzed in detail in [7],is

R~j~--O P~j(x)=A((~(x))Sij + Z(~(x))~(x)(~j(x) with

2[A'(a+;K)+},e(Z+A)] _ (ZX+~A)/~ =0

(4.6a)

(4.6b)

Notice that the constraint (4.6b) corresponds to the first order perturbation X -> X+cA

o -> a+EY. ofthe algebraic closure condition (2.4). These terms can be reabsorbed by the choice

,' (~.~(xj)--1 b(x) with (~-2~t~2)b=A (4.7)

Suppose we perform the substitution (4.7) in the classical action so that the

contribution C~fd2xR~(x)oYi(x) disappears from £I and Z(~Z(x)) is now fixed; we then

select the coefficients of C~Ki(x) and C~(x) which yield the equations

WijBl(X) - 2811Bj(X) "0 (4.8a) WjB~(x)- SijE(x)=O (4.ab) B~(x) + WiE(x) --0 (4.8c)

with solutions :

Bi(x) =(~i(x)B(q~Z(x)) (4.9a)

E(x) =XB(~(x)) (4.9b) provided

E~/E = _ 112X(X + a) (4. I0)

Comparing (4.9:4.10) with (3.4)we have E=kM and B=kM/~,.

116

Finally the external field independent contributions to (4.4) satisfy

W fd2x j(x) C (x) :0 (4.1 la) fO~Xl~(X) *wifo~xF(x) :0 (4.1 l b)

which imply F(x) = E(x) (4.12a) Aij(x) - rgu(x) (4.12b)

All these terms can be generated from the classical action through the transformations in (4.1) with a= I+ck zg=l+¢r.

The stabil i ty check is now completed and we shall consider the problem of the presence of anomalies in the next section.

CohomoloH

In a regularization independent approach we still have to show that the BRS invaMance

can be maintained at all perturbattve orders, i.e. that there are no anomalies.

The steps through which this control is performed are well known and are based on an

inductive procedure. The vertex functional r' is a formal power series in the loop ordering

parameter fl with the zeroth order given by the BRS invariant classical action in (3.5).

Suppose that (3.10) holds up to the order n- I, then at the next order we find

(SY)(")=A (5.1)

where, by the Quantum Action PMnciple [11], we know that ~ is a local functional of the fields with canonical dimension less than or equal to two and with a negative unit of Faddev Popov charge.

The nihilpotency of the BRS operator insures that ~ obeys the consistency condition [12]

SLL~ = 0 (5.2)

where SLis the linearized operator in(4.4). The solution of (5.2) can be wMtten as

Z~ = SLY+ ~o (5.3)

117

and A, is not a BRS variation. The number of independent terms (if any) in ~# is the

number of anomalies and coincides with the dimensionality of the first cohomology space.

Therefore we seek the general solution of (5.1); if this implies that Z~# vanishes than

we say that the breaking A is compensable and the needed counterterm is exactly L~

We parametrize A and ~, as:

~=Ca j'd2x ~i(x)'yi(x)+ fd2xKi(x)Di(x)+ fd2x~(x)A(x)+m2 .j'dZxB(x)* fd2XL~o(X) (5.41))

where L~z(x), Ao(X) depend upon the fields ~(x) and two space time derivatives, all other

coefficients being functions of ~(x) alone.

Insert now (5.4a) into (5.2) and isolate the independent terms beginning with the

coefficient of CaCI~/.~x) which yields

8c Jc" 2c'cJ j'E (5.5)

The expression in the r.h.s, is compensable by the choice of the corresponding term in given by

c

with

R(x)=(~E)/dpz -2E)t(h+o) - HA -2~#(8+H(~ 2)

(5.6a)

(5.6b)

The apparent undetermination in (5.6b) is easily explained if we observe that the

compensability relation A = SEa determines 4, once ~ is given by the consistency condition (5.2), only up to terms SL~. Wecan use this degree of freedom to eliminate the

undetermination in A ; indeed the only candidate with the correct quantum numbers is

with variation

SL~=~fO2x[L(2~I~+~,)~/i +(2L~l-2Ll(h+a)-L~)(~k~/k]+(~/kindependent terms) (5.7b)

and by a suitable choice of L we can eliminate either H or ® in (5.6a).

Let us analyze now the remaining contributions which do not contain space time

derivatives; the consistency condition (5.2) reduces them to the form

118

(5.8)

which is compensable by

j.o~×~(x)A(×), reSd2xa(x)+ 2 i'd ~<i(x)~(x)o(x) provided

D(x)+2Al(x)(o+~)(x) =-A(x)

D(x)+2B'(x)(o+X)(x) : -B(x)

A(x) - D(x) X(x) = D(x)

(5.9)

(5. I Oa)

(5.10b)

(5.1Oc)

In particular, the compensability of the last term on the r.h.s, of (5.8) is insured by the choice in (5.10) through the use of the algebraic closure relation (2.4).

Finally the external fields independent part of A containing two space time derivatives when inserted in (5.2) becomes

dw~ fo2x [%(x)&~(x) ~%(x)l which is trivially compensable by

(5.1 la)

(5.11b)

We conclude therefore that the breaking A is compensable and there are no anomalies.

The proof of the renormalizability of the model is now completed if renormalizability

is intended in the broader sense specified in [13]; indeed the fields ~(x) and the external

fie]ds ~/~) have renormalization "constants" which are functions of d~2(x).

However, looking at the list in (4.1) we see that all other external fields and

parameters are renormalized with true constants which are formal power seMes in fl.

This suggests that the interesting objects to look at are the connected Green functions

which have external legs of interpolating fields ~(x) i.e. the ones obtained from the

connected functional by deriving w.r.t, the external fields Ki(x). In this case we can set to

zero all the uninteresting sources and anticommuting parameters to obtain from (3.8;3.9)

(5.12)

which can be used to characterize the mass insertions in these Green functions in a

119

manner identical to that proposed in [6] and thus (5.12) is the starting point to analyze

the zero mass limit of the theory.

References

[ 1 ] M.Gell-Mann, M.Levy -Nuovo Cimento 16(1960) 705

[2] S.Coleman, J.Wess, B.Zumino -Phys.Rev. 177 (1969) 2239

M.Bando, T.Kuramoto, T.Maskawa, S.Uehara -Prog.Theor.Phys. 72(1984) 313;

7__2(I 984) 1207

[3] G.Bonneau, F.Delduc -Nucl.Phys. B266 (1986) 536

[4] E.Brezin, J.Zinn-Justin, J.C. LeGuillou -Phys.Rev. D ! 4 (1976) 2615

[5] C.Becchi -These proceedings

[6] F.Davi~I -C.M.P. 81 ( 1981 ) 149

[7] A.Blasi, R.Collina -"Renormalization a la BRS of the non linear ~ model" to appear in

Nuclear Physics B

[8] D.H.Friedan -Annals of Phys. 163 (1985) 318

6.Valent -Nucl.Phys. B238 (1984) 142

F.Delduc, G.Valent -Nucl.Phys. B253 (1985) 494

[9] C.Becchi, A.Blasi, R.Collina - in preparation

[I0] C.Becchi, A.Rouet, R.Stora -Annals of Phys. 98 (1976) 287 and "Gauge field models" in

Renormalization Theory edited by G.Velo, A.S.Wightman-Reidel Publ.Co. 1976 [ 11 ] Y.M.P.Lam -Phys.Rev. D6 (1972) 2145 ; D7(I 973) 2943

J.Lowenstein -C.M.P. 24 ( 197 I) I

[ 12] O.Piguet, A.Rouet -Phys.Reports 76C ( 198 I) I

[ 13] G.Bonneau -Nucl.Phys. B221 (1983) 178

RENORHALI ZATION OF BOSONI C NON-LINEAR o--HODELS BUILT ON COMPACT HOHOGENEOUS HANIFOLDS

Guu BONNEAU

Laboratoire de Phystque Th~orique et Hautes Energies Unlver'slt~ Par,is VII, Tour 24, 2 place dussleu,75251 PARIS CEDEX 05, FP, ANC.,E

Abstract. We review the quantum status of the non-linear bosonic o'-models built on compact homogeneous spaces. The subclass of K~hler manifolds can be parametrized in such a way that multiplicatlve renormalizablllty holds, to all-order or perturbation theory. The essential ingredients are the homogeneity of the space and the existence of a charge Y that separates the fields in ~ and $ : for these K~ihler manifolds, a family of coordinate frames exists such that the non-linear isometries are holomorphlc. The method is exemplified on the special case SU(3)/(U( 1 )xU( 1 )).

The material presented here results from work done in Part8 with Francois DELDUC and Galliarlo VALIANT

L INTRODUCTION AND GENERAL SURVEY

Non-linear o'-models on homogeneous (coset) spaces G/H play an essential role in the analysis of symmetry breaking : whenever the symmetries of a model are broken down from a compact group G to a closed subgroup H, the associated Goidstone bosons are described by a non-linear ~-model on the coset G/H [i]. In a 2-dimensional space time, the renormalization program can be undertaken : the aim of this talk is to enumerate what is known on that subject and give some new results, depending on the geometry of the manifold G/H. The n fields ~ 's being understood as coordinates on the n-dimensional real Riemannlan manifold G/H whose metric is gjj [~], the lnvariant action ls wri t ten as

I [~] = ½J'd2x gij [~] ap~ i ap~J (1)

If ~1" (resp. ~ ) is the Lie algebra of G (resp. H) and its generators are separated in Hi c ,S~ and Xa c ~ - ~ . the commutation relations are

[Hi, Hj] = fij k Hk (2.a) [Hi, Xa] = fja b Xb (2.b) [Xa, X b]= fad °X o+ fab iH i (2,C)

In the symmetric space case, the lab ° vanish. Symmetric spaces are Ilsted in [2] and the corresponding o--models are known to be tntegrable [3]. These spaces are Elnstein : P~j = c g jj with c > 0 in the compact case, which is responsible for (one-loop) asymptotic freedom, There is no general proof of renormalizability for these models in the real case

121

but for S N = SO(N*I)/SO(N) [4], the Grassmannlan SO(N)/($O(p)x$O(N-p)) [5] and SU(N)/SO(N), SU(2N)/Sp(N) recently analysed [6]. Moreover, these proofs use a definite choice of parametrlzation of the coset space (plus dimensional regularization, that is l ic i t for non-linear bosonic o--models). A promising approach a IaBRS. allowed A. Blasi and R. Collina to give a renormallzation proof for SN case that does not rely upon any regularlzation and is independent of the choice of a parametrizatlon of the sphere - as soon as the sugroup H is linearly realized [7].

On the contrary, in the hermltian (~) ~yrnmetric case. a complete analysis and

renormalizabillty proof was given by F. Delduc and G. Valent [9] using a special parametrization of the coset space where the isometries are holomorphic in ~)~and

$~(the complex coordinates adapted to the hermltian structure that exists in that case

[8] ). Moreover, these hermltian symmetric spaces are known to be K~ihler (~') manifolds

(ref.[2] page 372). In our way towards an unified treatment of non-linear o--models built on a

coset space G/H, we first looked at homogeneous K~hler manifolds. For a systematic study of complex homogeneous spaces, the following mathematical results are in order: Theorem I

Let G be a compact connected semi-simple Lie group and H a proper closed connected subgroup. Then three equivalent propositions are: (i) G/H is a complex homogeneous space and G and H have the same rank, (i i) G/H is KQhler,

( i i i ) H is the centralizer of a torus (~) in G. The equivalence (i) ¢~ ( l id is proved in ref.[lO.a], the equivalence (i i i ) ¢~(ii) in ref.[I O.b]. Notice that hermitian symmetric spaces [9] correspond to a one-dimensional torus [1 i].

Homogeneous K~ihler spaces are widely discussed in the mathematical l i t terature but have been studied only recently in theoretical physics for supersymmetrlc model building [11]. Here we want to emphasize the pioneering work of M. Bando, T. Kuramoto, T. Maskawa and S. Uehara [ 12], hereafter referred to as B.K.M.U. The classical analysis of bosonic homogeneous KQhler o--models relies on their method which was also cleared up In K. Itoh, T. Kugo and H. Kunltomo's work [13]. Indeed, the B.K.M.U. method offers a parametrlzation of the manifold adapted to the hermltlan structure and where, as in the symmetric case [9], the isometries are holomorDhic.

In the next section, we shall exemplify the general method with the $U(3)/(U(I)xU(1)) case (which Is K~hler, see theorem I). Using B.K.M.U. parametrizatlon of a coset space, we obtaln the classical action depending on :3 parameters and the expression of the lsometries - which are of the deslred holomorphlc form : 8Q = f(~), 8Q = f [$). A detailed analysis of this model wi l l be presented elsewhere [14].

e'J An introduction to complex manifolds and K~thler geometry can be found in these proceedings ([8] and references therein).

(~) A torus means a direct product of any U(I) subgroups of G, (U(I)) < with k( r = rank G. Its centralizer means the subgroup of all the elements of G that commute with these U(I)'s. It has cJeaHy the same rank as the group b.

122

In section III, we then explain the B.K.M.U. construction of the general homogeneous K~hler o'-model. The isometries are holornorphlc :

8~o~= ¢o~ + F%~ { ~ } _ ~(~" + ~G~ {4,} ~ ~)~" (3)

where F~[~} and %~,{~} are finite order polynomials in ¢, and ¢ and ~ are the

parameters of the non-linear transformations. In the hermltlan symmetric case, F~,{¢}

~R~,a,~_ ~a where l~,a~- is the Riemann vanishes and ~G~,{~} is linear In ~ : ~G°~(¢} = i

curvature of the manifold at ~ = $ = o [9]. We also explain the origin of the Ktihler

potential. In section IV, using dimensional regularization and the holomorphic expression

(3) of the isometries in B.K.M.U. parametrization (also valid in any holomorphically related parametrization that keeps the linearly realized H transformations), we are able to prove the all-order multiolicative renormalizabilitv, with no field renorrnalization at all. This new result comes essentially from two facts: (i) the existence of a charge that separates ~ and $, that is to say the complex character

of the manlfold, (li) the existence of a (class of) coordinate system(s) In which the non-linear Isometries are holomorphic, a property that results from the K~lhler character of the manifold.

II. CLASSICAL ANALYSIS OF THE HOHOGENEOUS SPACE SU(3)/(U(i)xU(1))

This six-dimensional real manifold is a complex three-dimensional K~hler one (theorem I). It is then natural to choose complex coordinates ~$oz adapted to the

hermitian structure [8] : in such coordinates, if the isometries are holornorphic, the hermitian structure of the metric will be stable.

The su(3) algebra is the commutator algebra of 3 x 3 antihermitian traceless matrices. We take for ~ the subalgebra of diagonal matrices spanned by IH i and iI-I 2

l! °, !1 I!° H 1= - H2= 1 (4) 0 0

and choose ('9 to parametrlze the orthogonal complement ( 9 " ~ ) I of ,~ in ~ by :

E o - = E~x~ • ~ x ~ , x ~ - - I x j (5) ~1 ~2

Notlce that the X , X _ do not belono to the aloebra of su(3) but to Its complexlfled O~ O~

algebra. However, ~ combination (5) does belong to Lle(SU(3) ).

P) This choice corresponds to the choice of a comDleX structure in 6/H [8]. AS known from [12,13], others choices are possible : this wil l be explained in detail elsewhere [14].

123

II. I. Standard ~arametrization of the coset soace G/H. - The Coleman, Wess and Zumino parametrization [15] of the coset space G/H

(hereafter referred to as C.W.Z.) will then be given, in these complex coordinates, by a unitary matrix U[~,~ ] :

U[~,$ ] = exp ( ¢~X + $~- X _ )

The left action of an element g ~ G on the representative U is

(6)

g U[~,$ ] = U[~',$ '] hi ~,$ ; g] (7) U i - - i where h ~ H is the transformation depending on g needed to obtain a [~,~ ] of the

form (6). The transformed field ~' depends on ~ and $ and the isometries wi l l not be holomorphic ones.

This can be traced back to equ. (6) where only a definite combination of X ,

X_ belongs to ~ . If in (6) one could forget the relation $~ = [~ }~ one would "divide"

the representative matrix U[~,$] by exp( $~ X _) and parametrize the coset space by

fields ~ only. This remark then leads us to consider the complex extension of the algebras ~ and ~ ( ~ and ~ respectively). As wi l l be shown in the next subsection, this use of complex algebras solves the problem of findlng a complex parametrlzation of G/H in which the isometries are holomorphic.

II. 2 B.K.M.U. oarametrizatlon of the coset soace 5U(3)/((U(1)xU(1)) In this example, one goes from su(3) algebra to sl(3,C) :

[o~ i T1 ~ ~ll 1 go ~ £3 ~2 1-12 ~i ~i, Tll are complex numbers (8) ~l ~2 ~3 with O~ 1 + O~ 2 * ~ 3 = 0

We divide the generators of go _ No into two sets : the upper triangular ~I=X+~ and the lower triangular ~ X_~. One easily verifies that the X+,~ are positively charged generators, with respect to the U( 1 ) charge Y = Hi + 1-12:

[Y, X+~ ] = Cl~ X+,= q2 = q3 = 2, ql = 4 (their hermitian conjugates X_~ being negatively charged). ,,,, The set ( ~,o, X+ ) of non negatively charged generators span a subalgebra ~ of go Let A" be the corresponding subgroup (H" ~ HC). The coset space G°/~r can be described ,~/a C. WZ by three complex fields ~ :

~(~) = exp(~i X_.i )exp(~2 X_,2 ) exp(~3 X_.3 ) = ~ I (9)

where ~,i = ~i+ ~2~3

and will shortly be proved to be homeorphlc to G/H [13].

124

The action of an element g ~ G c G c on ~'[~] is

g ~'{~} = ~'[~'] h[~;g] ( 1 O) where the transformation h ~ H is now indeDendent of $ and is uniouelv fixed by the form (9) of ~¢{(I)] and ~'{(l)'}. The infinitesimal version of (I O) writes'

One obtains'

~(4>÷a4~1= ~z 1 ~(~1 o~ 2 ,q~' IB I e 2 O o~ 3

g E-l[(I);g]

o( i = I + ~i ~,I + }3~3 , 0(2= I + ~2 (~2-~:3 (~3-~i (l)2 (I) 3 , ~,I(~,2o~;3 = I

I-lI=~I , i-~2=~2_~I~5;3 T]3= ~3+~I~2

and :

~,(I)2 = 62+ ~:3 ~1 + ~ 2{(I)2]2 + ~1 (~1(I)2

(11)

(12)

(13)

II.3 G°I~ homeomorDhic to GIH

In some neighbourhood of the origin, any element of G ° can be written as'

g = ~'[(~] exp{#~{~,$ } X+,,J exp[ci{(I),$ ] ~} (14) Restricting to g ~ G, we can express a representative of each class of G/H as

U[(~,$ ] = ~'[~} exp{a~[~,$ ] X+,~} exp{cJ[~,$ } ~} (15) where now d , i = 1,2 are real. Moreover d and 8 ~ are fixed by the unitarity of U. Equation (15) then proves the homeomorphism between G/H and G°/~ [13]. Here one obtains :

U =

, / ; J;

, I

w th: v= +l '12+lel 2, W= +1 "12+1 312,

c 1" -½1,o9w *¼Logv, c 2 - -¼Logv.

(16)

11.4. Invariant action and vlelbeins The Lie algebra valued Maurer-Cartan one form T = u-ldU can be decomposed

as' T =iuJ jH i + e ax +T~ ~X _ (17) -,8 +,8

Due to equation (15), e a = ea~d~ ~ (no d$ ~) and from equations (16) one obtains

125

e i - d~l+ ~3d~ 2

e j - e' v J;

, e3= + _ _ e I

w J ;

(18)

Under a group transformation U' = 9 U h -i (7) and then T' = hTh 4 + hdh -i. So e a

transforms homogeneously and the eae a (a = 1,2,3, no summation) are invariant. As a

consequence, the general SU(3)I(U(1)xU(1)) invariant action depends on 3 positive parameters ga

= e a eb _ a , b = 1 ,2 ,5 ( 1 9 ) g ~ Tla6 ~ 13 where "q~ = ga #ab iS the (fiat) tangent space metric. Here one gets :

9 4 a .$~a.$~ = (91492 ÷ 93]]1 a.~ l ÷ ~s a .~212/ (vw) +

8o~8~[g2 Lo9 v + 93 Log w ] 8 ~ ° ~ $ ~ (20) For 9i = 92 +93 the metric is explicit ly a regular Ktihler one, the K~hler potentials being given by c I and c 2 of last subsection ' thls ls general [1:;] as i t wll l appear In the next section.

I I i. CLASSICAL ANALYSIS OF ANY HOMOGENEOUS K~,HLER COSET SPACE G/H

As explained in the introduction, G is compact, connected and semi-simple and

the subgroup H is the centralizer of a torus. Let IYj (j = 1,2.. ,k ,< r) be the generators of

that torus and S a the other generators of H. By construction [Yj, S a] = O. Whatever a

Y-charge e~ T (Y = ~. i=i,..,k ciYi ) be, the antihermitian generators of ~ - ~ (in a

suitable unitary representation of G) have not a definite charge. But, if one goes to the

complex extensions ~o and ,~6 c, one can separate the generators of ~o _ ~c into two

parts ' the X+,~ and the X_,~ = -(X+,~) ¢ having positive and negative Y-charge,

respectively. Then, with ~ - {iYj, Sa, X+,~} the coset space (*) G°IH " is parametrized by

the complex fields (I)~:

~¢(6J = exp ~ X_~ (21 )

The special form of H (the centralizer of a torus) ensures that there is no generator in

(~) H" is th~complex subgroup of G ° whose Lie algebra generators are the non-negatively

charged {~e, }.

126

~ _ ~o with a vanishing %)-charge, which would make the B.K.N.U. trick Impossible. Indeed, In such a case the homeomorphlsm between 6e/H " and G/H would be lost. We recall that the choice of %) is equivalent to a choice of a complex structure in G/H, whlch is not unique [ 12-14].

As previously explained (equ. 1 O) the isometrics are holomorphlc |

~{~,] -- g ~{~} 5-~[$;g] Moreover the inrlnitesimal non-]inear (~) 9 transformations being written

(22)

9 = exp {~= X_,~+ ~X+,~ } (23) one can use the Hausdorf formula and get the following expression for the non- linear infinitesimal isometrles '

85~ = ~+~.p) l F~,r..a, p ~4~ >"....4~ ~'p+ ~'1~1%~','"'~'p ¢#~'""" ~'P (24) Both sums are finite, since all components of the field ~) carry positive V-charge.

In the symmetric coset space case [9], the F%;rl....;r p valllsh and the ~ ~G~'l'"';rn

reduce to the bilinear !R~2 6;r~ ~,~& ( R ~ is the Riemann curvature at the origin (I)--~; 0).

The correspondance between C.W.Z. and B.K.I'I.U. parametrizations (equ. (15)) is st i l l valld and we now explain why the real functions ci{$,$ } are the K~hler potentials of the model (i = 1,..k) [1:3]. From (7)(22) and (15) one gets (the %)~'s are hermltlan matrices):

U[~',$ '] = g ~'{~} exp{a~{$,$ } X,,~) exp{ba{$,$ } Sa} exp{d{$,$ } V i] h-1($,$ ;9] = ~¢{$') ~[$;g] exp(a~{$,$ ) X%J exp[ba{~,$ ] S a} exp[d{$,$ ] Yi) h-1{$,75 ;9} - ~'{$'} exp(a'~[$,$ ] X+~) exp[b'a[~,$ ) Sa) exp{c'i{$,$ } YJ

With h'[~;g] = exp[a'~[~;g} X+~} expt~a{~;9} S,} exp[di{4~;9} Y~} and hq[~,$ ;9} = exp{~Sa] exp{i;k ~ Yi } where b'aand ;k i are real functions, one gets :

c" [~ ,$ ] = cJ[~,$} + ½[ cf{~;9} + cf[~;9} } (25) as the c ~ and c' are real functions. Equation (25) means that the c~{~,$ ] are K~hler potentials. The general expression of the K~hler potential K[~,$] is then

K{(I),$ } = _Z~=I,..~ k 9i c'{4),$ ] (26)

(let us recall that k is the dimension of the torus whose centralizer is H). Notice that the general invariant metric depends on m parameters (m ,> k) where m is the number of

(~] As usual, the subgroup H is supposed to be linearly realized, and only the non-linear isometries are delicate.

127

Irreduclble representations of H contained in GIH ( 3 in the special case of sectlon II). One can show that the metric g~ = a~ K[¢,~] is a regular one if all the g~'s are

different from O. As a consequence, GIH is a K~hler manifold, even if the more general inVariant metric on the manifold is not K~hler : as indicated in refs. [12,13,16], the supersymmetric extension of such bosonic models will enforce relations among the m coupling constants in order that the metric be explicitly K~hler as required [I 7,8].

I V. ALL ORDER RENORMALISABILITY OF HOMOGENEOUS KAHLER BOSONIC o'-MO[)~L~

We have obtained - explicitly in the SU(3)I(U(1)xU(1)) case -, the most general invariant metric on the homogeneous K~hler manifold GIH

9~: ~ g, "rbiBie ~e i~ (27)

where the sum runs over all the irreducible representations I~ of H contained in G/H, and

~B~ is the matrix of the H-invariant sesquilinear form in the representation I~. The

expression for the non-linear Isometries has been found to be

6@~ = ~+~'p)1F°~ ~@ ~'l....@~'p ~'ml ~'l'"'i'p p~,i....~, p + G ~ ~l~ (~'i.... *~'P (28)

AS the transformation law does not close, we introduce an infinite number of sources (refs. [4b,9] ):

rO=r~a~.(~) ~d2 x .. _ i i ~'n=2, ~[ LIn ¢I''''¢n + -i. _i n . + Lln~ I..,,(~ j , In = {il,...in}

The Ward identities for the non-linear symmetry (28) may be written (on the generating functional F' for IPl Green functions) w r--w_r=o

E

p 8 r 8 r

,f--,p

-L~i~2[# ~@ + p)1 ~ F=I __ + {0Ci~-*(~'2} _ ] 8L~',...~ z

(29)

- ~ L~..~ [~ =----------+~ F =' ,~3 , ° a~...~ r~ ~","'~', a ~ . . . ~ , ~ . . . ~

+ [ ~I-~2-~'""'~n} ] + [¢' ~L-~" )a¢ }

Write the general solution as r = r ° + r i (it gives the stability of the classical action

as well as, due to the use of symmetry preserving dimensional regularization, the structure of one-loop divergences). Due to power counting, F 'i has the general structure

1 2 8

- I n - _ I n -

r i = ~ d2×{A{i~,*, ~1~*,,,*) + ~.2,.,~{ L,n T { * , * }+ LInT { * , * ) }1

Extracting terms in [ in from the linearized Ward identity satisfied by r i, i

following equations for the tensors T n{~,$}

(30)

we get the

{ii2 • . i I .... • .... 6~T = ~o~ [F'loc~,T'2~' + ~'p~2 F o~,l....~,p{ T ~'I ~'pi2 _ ~'2 T~'I ~'p}

; [q i2]] i i ' J n ~oc I ¢ i i i 2 . . . . i n ' . - - i n 6~T = [6 o~T + FIlo~,T ~'12

i I .... i n - . + ~'m2 F o~,l....~rp[ T ~'! .... ~'Pi2 _ ~'2...~'n T~'I .... ~'p] ]

+ [ , , - , i2- " - , 'n 11 , n>.3

(3 l.a)

(31 .b)

In the symmetric case, the right hand side of (31.a)vanishes. One can then use the fact that there is no dimension zero monomial of the fields (no derivatives), lnvariant under the non linear symmetry to get T hi2 = O. Then, by recurrence, all tensors T h'Jn vanish (ref.[9]).

In this homogeneous case, one uses the Y-charge .operator. The

parametrlzatlon has been choosen such that any tensor component T fi'Jn has a well

defined Y-charge. These tensor components can then be ordered according to their charge,

regardless to the number of indices they carry. ~ has a positive charge, and thus one

sees from (3 l.a-b) that the variation of a component with a definite charge contains only components with lower charges. Then, starting from the lowest charged component, one obtains its invariance and then its vanishing. One can then prove by induction on the Y-charge that all tensors T ~i'~n vanish. The symmetry is not renormalized in that

coordinates. With regard to Afa~, ~,~,$] it should be invariant under the non linear

symmetry (28) ' the solution is known and allows only ¢0uDling constants Tla B

renormalization, but no field renormalization.

This proves the multlplicative renormallzabllity of homogeneous K~hler bosonlc non linear o--models, with no fleld renormalizatlon in this nice B.K.M.U. parametrization of the coset space G/H. ( Of course, a soft mass term has to be added for quantization, but this does not change the result.). The essential ingredients were homogeneity and the existence of a U(1) charge Y that separates ¢°~and ¢o~.

V. CONCLUDING REMARKS

Using the B.K.M.U. parametrlzation of homogeneous K~hler coset spaces G/H whlch leads to holomorphlc isometrles, we have been able to prove the all-order

129

renormalizability of the corresponding bosonic non linear o--models. Moreover, in that coordinates the fields are unrenormallzed. A regulator free treatment ~ laB.R.S, could be done along the lines of ref.[7].

Supersymmetric extensions of these models are possible and this restricts the number of arbitrary coupling constants. Of course, dimensional regularization is no longer usable for renormalizability proofs.

We emphasize that, although homogeneous spaces have, for a given dimension,

less stringent geometries than symmetric spaces (for instance, when going from CP 3 = SU(4)/U(3) to SU(3)/(U(1)xU(t)), both describing three complex f~elds, the isometry

group get restricted from SU(4) to SU(3)), it was corresponding o--models.

stil l possible to renormalize the

References

[ I] J. Goldstone, Nuov. Cim. 19 (1961 ) 165

[2] S. Helgason "Differential geometry, Lie groups, and symmetric spaces" (Academic Press, 1978)

[3] H. Eichenherr and M. Forger,Comm. Math. Phys. 82 (l g81) 227 and references therein

[4] a) E. Br~zin, J. Zinn Justin and J.-C. Le Guillou, Phys. Rev. D14(1976)2615

b) G. Bonneau and F. Delduc,, Nucl. Phys. B266 (1986) 536

[5] G. Valent, Phys. Rev. D30 (1984) 774

[6] A.V. Bratchikov and I.V. Tyutin, Theor. Math. Phys. 66 (1986) 238

[7] A. Blasi,"B.R.S. renormalization of O(n÷ 1) non-linear o'-model ", these proceedings

[8] 0.. Bonneau, "K~hler geometry and supersymmetric non-linear o--models " an introduction", these proceedings

[9] F. Delduc and G. Valent, Nucl. Phys. B253 (1985) 494

[10] a) H.C. Wang, Amer. Jour. Math. 76 (1954) I,A. Borel and F. Hirzebruch, Amer. Jour. Math. 80 (1958) 458, especially page 501 and chapter IV

b) A. Borel, Proc. Nat. Acad. Sci. USA 40 (1954) 1147

[ 11 ] M. Bordemann, M. Forger and H. R6mer, Comm. Math. Phys. 102 ( 1 g86) 605

[12] M. Bando, T. Kuramoto, T. Maskawa and S. Uehara, Phys. /ett. 138B (1984) g4 and Progr. Theor. Phys. 72 (Ig84) 313

[I 3] K. I toh, T. Kugo and H. Kunitomo,Nucl. Phys. B263 (1986) 2g5

[ 14] G. Bonneau, F. Delduc and G. Valent, in preparation

[15] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (lg6g) 2239

[ 16] C.L. Ong, Phys. Rev. D31 (1985) 327 I

[17] B. Zumino, Phys. Lett. 87B (197g) 203

Nonlinear Field Renormalizations in the Background Field Method

K.S. Stelle*

TH Division

CERN

CH-1211 Geneva 23

Switzerland

A B S T R A C T

The use of the background field method in intrinsically nonlinear theo-

ries such as a-models requires nonlinear field renormalizations of the quantum

fields that cannot be deduced from divergent graphs with external background

lines only. We show how these necessary renormMizations are to be derived,

thus allowing computation to arbitrary loop order.

1. I n t r o d u c t i o n

The background field method is a useful computational tool in quantum field theories

that allows one to compute radiative corrections while maintaining manifestly the symme-

tries of the theory under consideration [1]. In this article, which is based upon work done

together with with P.S. Howe and G. Papadopoulos [2],we explain how the method must be

applied in the case of intrinsically nonlinear theories such as a-models. Nonlinear a-models

give rise to new problems in their quantization because maintaining covariance in the pertur-

bative quantum theory requires a nonlinear split between the background and the quantum

fields. Although a-models are renormalizable in two space-time dimensions [3], they are not

simply multiplicatively renormalizable. In particular, since the scalal fields of the model

have dimension zero, there is nothing to stop them from acquiring arbitrary functional field

renormalizations. Because of this, the renormalization of a a-model in the background field

method highlights features of the method that were not brought to the fore in the case of

four dimensional gauge theories.

It should be noted that maintaining background symmetries manifest in the quantum

formalism is not a substitute for BRS invariance. In Yang-Mills theories, BRS invariance

* On leave of absence fxom the Blaekett Laboratory, Imperial College, London SW7, England.

131

must be used to establish multiplicative renormalizability, whether one is performing back-

ground field quantization [4] or following the classical covariant quantization procedure. BRS

techniques also turn out to be necessary in the case of models without internal symmetries,

such as a-models taking their values in target manifolds without isometries. In this case, the

background field method has the task of maintaining manifest the invariance under reparame-

terization of the background scalar fields. BRS methods are needed in this case to control the

relation between the background, quantum and total fields. Indeed, similar BRS techniques

were used in the original proof of renormalizability of the a-models in quantization about a

constant background [3].

It may be useful to illustrate the issues involved in the context of four-dimensional

¢4 theory. The action is

S= / d4x(l (ot~¢) 2 1--2~2 - - ( 1 . 1 )

and the generating functional for connected Green's functions W[J] is defined by

= / T ~ ¢ exp i(I[¢]+fJ¢) (1.2)

The generating functional for 1PI graphs is related to W by a Legendre transforma-

Evidently

w[~, J] = w[J] - ] J~. (1.6)

Further, we define a new F-functional by taking the Legendre transformation with respect to

J only,

F[~, r] : W[~, J] - f J r . (1.7)

It is not difficult to show using the trivial shift symmetry 5~(x) = ~(x), 5r(x) = -q(x) that

depends only on ~ + r , i.e. ~F[~, r] _ ~F[~, r] (1.8) ~(x) ~r(x)

(1.5)

tion

W[J] = F[¢] + / J¢ (1.3)

where the argument of F is the vacuum expectation value of the quantum field ¢ in the

presence of the source J. We now split the total field ¢ into a background field ~(x) and a

quantum field r(x)

¢(x) = ~(x) + r(x) (1.4)

and define a new functional [5] l/~-[~, J] by

132

and that

F[~, 0] = r[~,]. (1.9)

(1.9) is the key equation; it states that the standard 1 P I functional can be computed by

calculating the 1 P I background functional with no external quantum ¢r lines.

So far, all our considerations have been formal and we have not taken into account

the effects of renormalization. Expanding out the interaction term in the action, one finds

A 4 A 4 = + 4 3. + 2 + + (1.10)

and in principle these various vertices could be renormalized differently. Now of course this

doesn't happen because of the linear-splitting Ward Identity (1.8); furthermore (1.8) also

tells us that the wave-function renormalizations for ~ and ~r are actually the same,

Z~ = Zr. (1.11)

Thus, the Ward identity (1.8) has the Consequence that the counterterms are func-

tionals of the total field ¢, and may be deduced from graphs with no external quantum

lines. In particular, the renormalization of the various vertices involving the quantum field

is performed by renormalizing A as deduced from diagrams with only external background

lines and then substituting the corresponding bare A into the right hand side of (1.10). The

renormalization of the quantum field ~r can also be deduced from graphs with only external

background lines, as expressed in (1.11), but in fact these multiplicative renormalizations can-

cel out in graphs with no external quantum lines. This can clearly be seen diagramatically:

the factors of Z~r cancel between the propagators and vertices.

In the case of nonlinear theories such as a-models, the split between background and

quantum fields needs to be nonlinear in order to maintain reparameterization covariance.

This will give a more complicated Ward identity than (1.8), requiring BRS techniques for its

proof. The result will be that the counterterms a reno t simply functionals of the total field,

and additional quantum field renormalizations will be necessary.

2. T h e N o n l i n e a r a - M o d e l

Let ~ be two-dimensional spacetime and M be a Riemannian manifold with metric

g. Then the a-model field is a map ¢ : E --~ M represented in local coordinates by ¢i(x).

The Lagrangian for the model is

1 L = "~gij(¢) Ol~qJiol~q jj. (2.1)

133

The quantization of general a-models in two space-time dimensions based on the

Lagrangian (2.1) has been discussed from a number of different points of view. Friedan

[3] considered fluctuations of the ~-model fields about a constant background, and then

proceeded with the standard approach to quantization. Within this framework, he proved

the renormalizability of the theory. An important element in this approach was the proof that

the counterterms for a given constant background are geometrical expressions related to those

for nearby constant backgrounds in a natural way. The other approaches to the quantization

of ~-models have made use of various forms of the background field method [6,7]. This is the

most convenient way to carry out renormalization calculations, and counterterms have been

derived using the method in a number of theories at low loop orders. The implications of the

background field method at higher loop orders in c~-models have not so far been investigated,

however. This is the issue to which we now turn.

The action I = f d2x L is not invariant under any symmetries for a general target

manifold M, but it is reparameterization invariant in the sense that L has the same form in

any coordinate chart. This can be rephrased slightly: let f be a diffeomorphism of M onto

itself; then it induces a new map, Ct = f o ¢, and we have

I[/,~, ¢'1 = z[9, ¢] (2.2)

or, infinitesimally,

I[g - £~g, ¢ + v] = I[g, ¢] (2.3)

where v is the vector field generating the diffeomorphism and £v denotes the Lie derivative.

Evidently, a diffeomorphism only induces a symmetry of I , I[¢'] = I[¢], if it is an isometry,

£vg = 0. The standard generating functional of connected Green's functions is defined as

before

e i }~,.~[,[] = f v¢ ~xp ~ {~[¢] + I ~x J~(x) ¢~(x)} (2.4) but the source term clearly spoils reparameterization invariance.

Formally, the effect of a diffeomorphism is given by

eil4~-£vg[ J] = f 9¢ ~ p i {zig, ¢1 + I J- (¢ + v)}. (2.5)

Since coupling the source to a function of ¢ leads to the same S-matrix, one concludes that

the latter is reparameterization invariant and that models with metrics which are related by

a diffeomorphism are physically equivalent.

It is possible to avoid spoiling reparameterization invariance in the Green's functions

by the source term if one uses an unconventional source term [3] f d2x h(x; ¢(x)). The

functional eiW[h] = f 9¢ ~xp i {I[g, ¢] + I h(x; ¢)} (2.6)

134

will be reparameterization invariant provided that h(x; ¢(x)) is defined to transform as a

scalar. Note that the explicit functional dependence of h on xg and ¢i(x~) means that

f h(x; ¢) is equivalent to a sum of an infinite number of source terms coupling to all powers

of the quantum field. Diffeomorphic metrics yield completely equivalent functionals of the

form (2.6) since

Wg-r.~g[h - £vh] = Wg[h]. (2.7)

Thus, the nonlinear sigma model is really a theory of an equivalence class of models defined

by diffeomorphic metrics. In practice, renormalization calculations will be performed for

the conventional functional (2.4) with a single source coupled to the quantum field, but the

counterterms can then easily be transformed into forms appropriate to (2.6). For the time

being, we will concentrate on renormalizing the functional (2 4), but will return to (2.6) later

on.

In proceding to quantize the theory using the background field method, we must now

split the total field into background and quantum parts. However, a straightforward linear

background-quantum split, ¢i = ~oi + nil does not lead to a manifestly covariant formalism,

since 7r i cannot be interpreted as a vector. In order to achieve manifest covariance, it is

therefore necessary to use a nonlinear split and this can be based on geodesics [7]. Let Ok(s)

be an interpolating field with

• = ¢ • ' d s s=0

that satisfies the geodesic equation

d2~i i d¢j d~k ds 2 "4- Fjk ds ds

= (i and ~i(1) = ¢ i (2.8)

- - - 0 . ( 2 . 9 )

Then we can solve (2.9) with the initial conditions of (2.8) to get

; , (2.10)

where

and

x = - E L r ( . 71=2 13,[ 31...3n (2.11)

F}I...j, " ... - F i = V ( j 1 V j n - 2 jn_l,jn_2)(Cfl) ( 2 . 1 2 )

In (2.12) ~7 indicates that the covariant derivative is to be taken with respect to the

lower indices only. In this way of splitting, the quantum field is taken to be ~i the tangent

vector to the geodesic ~i(s) at s = 0. Now, ~ has a geometrical interpretation: it is a cross-

section of the bundle over ~ obtained by pulling back the tangent bundle of M with the

135

background field ~o, { e F[~*(TM)].

expand the Lagrangian one sets [8]

so that

This fact ensures the covariance of the expansion. To

(2.13)

1 d " L ( s ) ~=o L(¢) = L(1) = y ~ n! (ds )"

7~=0

n=O

where Vs is the covariant derivative along the curve ¢i(s). The series (2.14) is easily evaluated

using the formulae

dOi 0 d o i d¢J V ~ o " ¢ i = V~'"~'s - " ds + Ot '¢krikJ(¢) ds , V~gij = O,

v d¢ i i s-'~- 0 [Vs, V~]X k d¢ r'-Q ,T~jr~k "v'g = , = --~-sv,.,. l~eij.~ , (2.15)

where in the last equation X i is an arbitrary vector. Hence, all the vertices derived in this

expansion involve tensorial functionals of the background metric glj(qo). If we introduce the

split (2.10) into W [ J ] and drop the term f J . ~ , we obtain

e il~.'[~,,l] = f :DTr exp i(1[¢] + f J~,r i)

= / exp i + f + xi ) ) . (2.16)

Now (2.16) is not quite what we want since the source is still coupled to the (non-covariant)

function X i, so we define a new functional

e iW[~']] = / :D~ exp i (1"[¢] + f J i l l ) . (2.17)

The Feynman Rules for this functional will be manifestly covariant, but as we shall

shortly see, it is not enough to compute 1 P I graphs with no external lines if one is to

determine the counterterms necessary for higher loop calculations from lower loop graphs.

136

3. T h e N o n l i n e a r S p l i t t i n g W a r d I d e n t i t y

The action

I[¢] -= I [% ~] = I [~ + ~r] (3.1)

is invariant under the obvious symmet ry

= ( 3 . 2 )

and this leads to a simple Ward Identi ty for the linear split t ing F[~o, ~r] functional:

6P 5F 5--- 7 = ~i (3.3)

with the obvious solution

P[% ~r] = P[~p + 7r]. (3.4)

Now, we can reformulate (3.2) in terms of t ransformations # of ~ and ~:

5,j~i = qi

5,,~i : F ~ ( ~ , ~)~J (3.5)

where the functional F/j is determined by the requirement tha t ¢i be invariant, i.e.

= " ~ ,~ i r , J ~k (3.6) 5,1¢ i qJSJ¢ i + ' y ~ k'~ = 0

where 5 5

5i = - - 5~ - (3.7)

(note tha t since the ¢ ~ (% ~) relation is local in x#, we could have used ordinary partial

derivatives and ordinary summat ion here). It is easy to derive the identi ty

5[jFik] q- F [jSgF k] = 0 (3.8)

with the aid of which it can be verified that the t ransformat ions (3.5) are Abelian. However,

they are nevertheless nonlinear and this fact requires tha t they be handled with care at the

q ua n t um level, since the t ransformations themselves will require renormalization. To s tudy

# From now on, we use DeWitt notation, so that an index "i" does double duty as a tensor index and a spacetime point x t~. Summation over indices includes integration over spacetime.

137

the associated Ward Identities it is therefore convenient to consider the associated B.R.S.

transformations. We introduce a ghost field ci(x) and define

S¢fl z ~ C i

s i= F cJ s e ~ ~ O. (3 .9 )

Then s 2 = 0 by virtue of (3.8). To obtain the Ward Identi ty it is necessary to modify

the action by including s~ i coupled to an anticommuting source Li. However, since Fijc j has

power counting weight zero it will mix with all other possible dimension zero operators. To

allow for this operator mixing we therefore define

• (3 .10) E = I + La~N~

where

Ni~=sA~ a = 0 ,1 , . . . oc

A~o - s~ ~ ; Loi - Li. (3.11)

The set {A~} are all possible dimension zero vectorial functions of ~ and ~, this set being

sufficient as we shall see. If we assign dimension zero and ghost number ng = 1 to c then the

L's have dimension 2 and ghost number - 1 . It is clear tha t

s E = 0 ,

since s¢ = 0 and ~2 = 0. This can be rewritten as

i 5E 5E 5E sE = c ~ + SL~ 5~ - 0

since

(3.12)

(3.13)

5~i_ 5E 5L~

At the quantum level, the structure of the 1PI functional F[~o, ~, L, c] is determined

by the background reparameterization invariance and by the shift Ward identity

i 5F ~F 5F c ~ + SLi 5~ i - O . (3.14)

Reparameterizat ion invariance is realized linearly on the quantum fields, yeilding the repa-

rameterization Ward identity

£ v g i j ~ - i Sr~ o¢'5F v - Ojvi~J777, ~ = 0 . (3.15)

138

Assuming the existence of an ultraviolet regularization scheme tha t preserves reparameteri-

zation invariance #, the linearity of (3.15) in F implies tha t the same relation will be satisfied

by the divergent par t .of F, F D. In the present case of a purely bosonic a-model , dimensional

regularization can be used as an invariant ultraviolet regulator. Thus, the divergences will

be manifest ly invariant under reparameter izat ions of the target manifold.

The shift Ward identi ty (3.14) is nonlinear in F and consequently requires some care

in its analysis. We wish to subtract the divergences so tha t the renonnalized F, F ~, continues

to satisfy (3.14). As in the Yang-Mills case, we proceed order by order in h. For example,

the one-loop divergences satisfy

__D 5F~)~S hE 5F~) = c + + - - = 0

The solution to (3.16) is given by

F~) = G[¢] + D~X[~, ~, L, c], (3.17)

where G[¢] is a reparameter izat ion invariant functional of the to ta l field ¢, and X is an

a rb i t ra ry functional with ghost number - 1 and dimension zero ( remembering tha t it is an

integrated functional). Tha t (3.17) solves (3.16) follows from the fact tha t sG[¢] = 0 and by

vir tue of the fact tha t the opera tor DE is nilpotent,

DsD E = 0 (3.18)

as may be verified directly. Tha t it is the most general solution can be proved using similar

arguments to the Yang-Mills case [9]. X has the general form

x = zo L. A ( 3 . 1 9 )

for some (infinite) constants Za,o. Expanding out (3.17) yields

• 5E = c [ ¢ ] - . ' Z,~,sLa~No + Zo,~A,~ ~'~ • (3.20)

I t follows from the reparameter iza t ion Ward identi ty (3.15) tha t r ~ ) [ % ~, c = L = 0] is co-

variant, so tha t the A~ are indeed vectors as claimed, and in addit ion G[¢] must be of the

form -½TijOt,¢'Og¢ J. To summarize, the one loop divergences comprise metric divergences

# Two dimensional massless theories also require infrared regularization. This can be done in a repa- rameterization and shift (3.5) invariant way by including a potential m2V(¢) , where V is a scalar function of the total field• The function V(¢) will also have to be renormalized, e.g. at the one-loop order by a term proportional to DiOiV. Since these renormalizations are proportional to rn 2, they are clearly distinguishable lYom the other ultraviolet divergences with which we are chiefly concerned.

139

(from G), i bE nonlinear ~ renormalizations ( Z 0 ~ A ~ ) and multiplicative renormalizations of the

infinite set of sources {L~i}. The sources {L,~i) are needed only in the proof of renormaliz-

ability and will ultimately be set to zero. Thus, the important renormalizations are those of

the metric and of the quantum field ~.

One can iterate the above procedure loop by loop. In the end, the renormalized action

has the form 0 i s(r)[~, ~, L, ~1 = ±°Iv, ~0] + L,~N~(~, ¢) (3.2~)

where I ° includes the metric counterterms, i.e. gij "-'+ gOj = gij + ~ T i j and

L ° i = L.,3iZoc,

Z0aA~(~2, ~0) = ~i. (3.22)

The proof that (3.21) satisfies the splitting Ward identity

ci6E(r)~ + 5E(") ~E(") bop ~ 5Li 5~------- ~ - 0 (3.23)

follows from the observation that

where

8°E (r) = 0 (3.24)

sOqo i = c i

5E (~) s°~°i= F~(qo~°)c j - 5LOi

s°c i = 0 (3.25)

because

Ni(% ~) = sA/(% ~) =~ Ni(% ~0) = sOA i ,c,(~o, ¢,o,) . (3.26)

Using (3.25), (3.24) is

5E(r) 6E(r) c'6~*)~, + 6L0 ~o, - 0 . (3.27)

If one then changes variables from (~o, ~0) to (% ~) and uses (3.22), one readily sees that (3.27)

implies (3.25). Strictly speaking, we should show that the solution (3.21) is unique; this can

be done along the same lines as the Yang-Mills case as discussed, for example, in Ref. [9].

To summarize then, the consequences of the nonlinear splitting Ward Identity are

given in equations (3.21) and (3.22); in addition to the metric renormalizations, there are

nonlinear renormalizations of the quantum field ~i that are not derivable from expanding out

the metric counterterms. Furthermore, they do not cancel out in higher loop graphs and

140

therefore must be taken into account. In the next section we give an algorithm for computing

them.

4. C o m p u t a t i o n a l A l g o r i t h m

In this section we show how to renormalize the functional

eiW[~,J] = f 73~ei(Z[~41+ f .7~) (4.1)

From the preceding section we know that the renormalized functional is given by

ei~~[~J] = f D~e i(I°[~'~°]+f'l~) , (4.2)

which follows from (3.21) and (3.22) upon setting the Lai to zero. The task is to compute

the metric contributions and the (nonlinear) quantum wave function renormalizations. This

could, of course, be done straightforwardly by computing all 1PI graphs with both external

quantum and background lines. To do this would be to violate the spirit of the background

field method, however, in which the effective action Fifo] is computed from graphs with no

external quantum lines. Nonetheless, as we have shown, the nonlinear renormalizations of the

quantum field must be taken into account in order to correctly subtract the theory. Moreover,

these renormalizations cannot be deduced from the divergent parts of F[~] alone.

In order to deduce the necessary renormalizations of gij and ~i without computing

graphs with external ~ lines, we consider instead of (4.1) the generalized functional (2,6),

eiW[h] = f D~ei{I[~4]+f /,(x,¢)} (4.3)

The functional (4.3) can be renormalized in the same way as (4.1) since the modified source

is a functional of the total field, but the wave-function renormalizations cancel out since W[h] is essentially the vacuum functional for the modified action I + f h. So (4.3) is renormalized

by

h ~ h0= S ( h + H(g,h)) (4.4) ,

where the source counterterms H are linear in h because h has dimension two. As with

the metric counterterms, the h counterterms can be classified according to their conformal

weights under the scalings gij "+ A-lgij, h ~ A-th. For example, the one-loop counterterms

have conformal weight zero, so there are only two possibilities, ~TiXTih and Rh. The latter

does not occur since the Feynman rules involve only derivatives of h.

141

In terms of the nonlinear ~ - ( split, h has the expansion

1 • ' h(x, ¢(x)) = h(x, + (4.5)

If we choose the condition

V(h...Vi,)h ¢=~ = 0 n ¢ 1 (4.6)

then the source term f h reduces to a conventional source term as in (4.1). When this is

done after renormalization, it is necessary to perform a quantum field redefinition in order

to recast the renormalized integrand into the form of (4.2).

To see how this works, consider the bare source h ° which occurs in the renormalized

functional e i ~ V ( r ) [hi = J :D~ exp i(I°[gG ~] + f h°(x; ¢)) ; (4.7)

one finds

= h + 4@eViVih + g > 2 loop terms . (4.8) #-~h 0

Expanding now h ° using (4.5) and (4.6), one finds

#-~h°=hi(~i+t=l f i X~t)(% ~ ) ) , h i = V i h ¢= , (4.9)

where the X~t ) are covariant expressions corresponding to g loops involving all powers in the

quantum field ~. These arise from the renormalization of h upon the imposition of (4.6).

Note that Xit ) is determined entirely by the g-loop contribution to h ° in (4.8); Xit ) contains

terms of arbitrary order in the quantum field ~ and is a power series in the regulator e -1 up

to order e ~t. For example, from the one-loop contribution to h ° given in (4.8) one obtains

• 1 2 i 2VJRik~J~k_~2V~Rjk~J~k+O(~3)) " X~I)- 4~re (3/~j ~j + ' " " (4.10)

We can regain the form (4.2) by changing variables in the functional integral so that the

source hi couples to ~i, with the resulting bare quantum field ~0i in the action given by

• ,

{0i = ~i _ Xil)(% {) _ Xi2)(% ~) + X{1) X~I ) + g > 2 loop terms. (4.11)

Again, we emphasize that the expression for the bare quantum field ~0i at a given loop order

g is derived from the renormalization of h at loop orders up to g given in (4.8).

In order to compute the renormalization of I and h in (4.7), it is convenient in practice

to consider a more general functional

e IW[~'h;J] = f D~ exp i (I[v, ~] + / h + f Ji~ i) . (4.12)

142

This functional obviously reduces on the one hand to (4.3) for Ji = 0 and on the other hand to

(4.1) for h = 0. In order to renormalize (4.12), we require metric, h and ~ renormalizations.

Since h has dimension two, its presence does not affect the renormalization of ~i which

has dimension zero. Thus, by our previous discussion, the renormalization of ~i is given in

terms of the renormalization of h by (4.11). We also know from the results' of section 3

that the renormalization of h in (4.12) is given by functionals of the total field ¢(~, ~), so

it may be deduced from diagrams with no external ~ lines. Hence, by calculating with the

general functional (4.12), we may deduce the renormalization of the quantum field ~i via the

renormalization of h from diagrams with no external quantum lines, and then set h = 0 to

obtain the renormalized (4.2).

5. C o n c l u s i o n

In this article, we have analyzed the general structure of the counterterms for a

nonlinear a-model defined on an arbitrary Riemannian manifold. In addition to the expected

counterterms which are functionals of the total field ¢i, there are additional nonlinear field

renormalizations of the quantum field ~i which must be performed even if one wishes to

calculate only diagrams without external quantum lines. These nonlinear renormalizations

of ~i can be calculated from the renormalization of a generalized source h(x; ¢i(x)), which is

treated as if it were a potential for the a-model.

The renormalizations of the quantum field that we have discussed above are required

for the correct subtraction of subdivergences in all higher loop orders. Since these renor-

malizations are nonlinear, they do not simply cancel out as do the multiplicative quantum

field renormalizations in four-dimensional Yang-Mills theories [4,5]. At the two-loop level, the

subdivergences that are removed by these renormalizations are proportional to the classical

equations of motion for the background. At higher loop orders, one will encounter second and

higher order variations of the action with respect to the background fields. Properly taking

account of these subdivergences is necessary for the separation of ultraviolet from infrared di-

vergences. For example, if one uses dimensional regularization plus a potential incorporating

a mass m as an infrared regulator, at the two loop order there are simultaneous ultraviolet

and infrared subdivergences proportional to ~gn (m--~2"~ that are cancelled by the quantum field

renormalization [2].

For simplicity, we have been concerned in this paper only with a-models without

fermionic fields and without torsion. All of the above considerations are of equal importance

in more general cases. Recent work on the renormalization of nonlinear a-models with torsion

[10] has confirmed the general structure described here.

143

References

[1] B.S. de Witt, in Quantum Gravity 2, eds. C. J. Isham, R. Penrose and D.W. Sciama

(Clarendon Press), 449; G. 't Hooft in Proc. 12th Winter School in ~heoretical Physics

in Karpacz, Acta Univ. Wratisl. no. 38, (1975); D.G. Boulware, Phys. Rev. D23

(1981) 389.

[2] P.S. Howe, G. Papadopoulos and K.S. Stelle, Institute for Advanced Study preprint,

Dec. 1986.

[3] D. Friedan, Phys. Rev. Left. 45 (1980) 1057; Ann Phys. 163 (1985) 318.

[4] H. Kluberg-Stern and J.B. Zuber, Phys. Rev. D12 (1975) 482, 3159.

[5] L. Abbot, Nucl. Phys. B185 (1981) 189.

[6] J. Honerkamp, NucI. Phys. B36 (1972) 130.

[7] L. Alvarez-Gaum~, D.Z. Freedman and S. Mukhi, Ann. Phys. 134 (1981) 85.

[8] S. Mukhi, Nucl. Phys. B264 (1986) 640.

[9] S. Joglekar and B.W. Lee, Ann. Phys. (N.Y.) 97 (1976) 160.

[10] C.M. Hull and P.K. Townsend, University of Cambridge D.A.M.T.P. preprint, Mar.

1987.

144

KA'HLER GEOMETRY AND SUPERSYMMETRIC NON-LINEAR or-MODELS : AN INTRODUCTION

Guy BONNEAO

Laboratoire de Physique Th~orique et Hautes Energies, Univerait~ Paris VII, Tour 24, 2 place Juesieu 75251 PARIS CEDEX 05, FRANCE

Abstract. The necessary and sufficent conditions for a supersymmetric extension of a bosonic non-linear o'-model to exist are reviewed. The framework for the perturbative

analysis of such models is sketched with emphasis on some delicate points. These are

exemplified on the "proof" of all-orders finiteness of hyper-Kahler supersymmetric non- linear o'-models.

I.INTRODUCTION

The importance of K~hler geometry for supersymmetric theories was stressed in 1979 by B. Zumino [1]. He studied the SUSY extensions of a bosonic non-linear o'-rnodel

whose fields take values in a complex Kahler manifold and showed that if the

supersymmetry is N=I for four-dimensional space-time, i t turns to N=2 in two space-

time dimensions. Necessary and suff icient conditions for extended supersymmetry in two

space-time dimensions where later on given by L.A]varez Gaum6 and D.Z.Freedman [2] and

we shah review them in the f i rs t part of this talk. Perturbation theory for supersymmetMc non-linear o'-rnode]s wi l l then be sketched (background field method

with normal coordinates, D.Friedan [3] renormalizabil i ty in the space of metrics, one loop

calculations and all-orders finiteness conjectures) wi th emphasis on some delicate - and,

to our mind, controversial - points.

Let us also mention another application of K~hler geometry • in general

relat iv i ty, i t is well known that the geometry of self-dual gravitational instantons is

that of an hyper-K~h]er manifold (see G. Va]ent contribution to this worshop [4]).

Our interest for Ktihler geometry comes from our attempts at defining a physical

f ield theory through characterizations other than the usual Ward identit ies linked to

isometries (symmetric spaces or homogeneous manifolds [5]).Such characterizations

could be S matrix properties such as non production [6 ] , scale invariance or geometric

properties such as K~hler or hyper-K~hler structure.

145

II. SUPERSYMMETRIC EXTENSIONS OF BOSONIC NON-LINEAR c-MODELS

This section closely fol lows the discussion made in ref [2].

Start ing wi th the bosonic action

I[~b] = ~- d2x gij[~b] cbp~b bp ( 1 )

where the n f ields ~bi's are understood as coordinates on an n-dimensional real Riemannian manifold J~, whose metric is ~j[~b], an N= 1 supersymmetric extension wi l l be:

I ' D~ is the covariant derivat ive: D ~ i= 8p~JVj~ i= 8~dji+ ~jk 8~#~J~ k • ~jk and Ri~ the

Christoffel connection and Riemann curvature corresponding to the metric c3j. The

supersymmetry transformations

commute wi th coordinate reparametrizations of J~L In ref [2], i t is shown that there is a second supersymmetry leaving the action (2) invariant, and satisfying SUSY algebra (*)

wi thout central charges { da),~(b)} = 2 ~abp (4)

If and only if a tensor ldj [~b] exists wi th the properties :

flj fJk = - (Yk (5.a) fij (~k fkt = gJ (5.b)

Vk tij = 0 (5.c) This second supersymmetry is completely fixed in function of f~a {~}:

6(2)~ i = ~ [fij~bj}, 6 (2) [ fij~bJ}- -i ~ ~b i ¢ - rijk {~ fJm@m]fknkb n

and i has the same expression as (3) wi th ~i changed to fijd~J.

We now interpret equations (5) from a geometric point of view [7] ' - Equation (5.a) <=> f~, is an almost complex structure. Complex coordinates Z ~,

2&can be defined locally that'diagonalize the almost complex s t ruc tu re f%~= i f~-- = - i

(o~,~, = t ...... m, where n=2m ).

- Equations (5.a+b) ~ g ~j is hermitian wi th respect to f'j <=> the Riemannian manifold (Jl~,,ojj) is an almost hermit.ian manifold. In the complex coordinates Z°~,Z ~ , g..~: g ~ :o.

(* ) I f the manifold is irreducible, i t is shown in [2] that any new fermionic invariance of the action is necessarily a supersymmetry satisfying the algebra (4).

146

An antisymmetric tensor is then defined f~a = ~kfka = -f j , and a 2-form, called the Kahler

form ' Q = 1 flj d~ ~^ d~J = ig,~ dZ ~^ dZ ~ (6)

All these are local properties whose global extension is possible if, and only if, fEjsatisfies the integrability condition that the Nijenhuis tensor vanishes

~ = f~i( ~ ~ j - 8j fk l ) - ( i <--> j ) = O. (7 )

- Equations (5.a+7) <=> ~ is a complex manifold. Then, in each open set one can choose

complex coordinates such that, in the intersection of 2 charts, the coordinate systems Z ~, Z '= are holomorphically related'

z '~ = r= ( z ) , 2 '~ = ~ ( 2 ) - Equations (5.a+b +7) ¢~ (a~.~j) is an Hermitian manifold. The property g=13 = g~,~ = 0 is

preserved by an analytic change of coordinates.

- Equations (5.a*b~c) ¢~ JIrL. is a Kahler manifold <=> d£~ = O. This is a global notion ((5.c)@

Nij k= O) which is a strong restriction on E¢. i t means that, in a coordinate frame adapted

to the hermitian structure fl j, there exists a K~hler potential K(Z,Z ) such that

g,~ = 82 K(Z,2 )/(SZ ~ 82~). (8) I

K(Z,Z) is defined up to a K~hler transformation •

K[Z,Z ] --* K[Z,Z ] + f[Z] + f [Z ] (9)

If two covariantly constant complex structures exist, satisfying the Clifford algebra that

results from supersymmetry (equ.4) • f(a} f(b} + f(b) f(a) = _26ab a,b = 1,2

then f(3) = f(i) f(2) iS also a covariantly constant complex stucture and we get N=4

supersymmetry. We then have three covariantly constant complex structures f(a)

a= 1,2,3, satisfying the SU(2) (quaternionic) relations : ~a]i k f~'b}ka = _(~ab ~ij + ~-~ab¢ f(c)ij ( l O)

and ~ is called an hvDer-K~hler manifold.

The following table summarizes the known results on supersymmetric extensions of bosonic non-linear o--models defined on a Riemannian manifold (,,~,, i~j). (A

4-dimensional N-supersymmetric theory gives through dimensional reduction to di2 a 2N-

supersymmetric one : this just i f ies the d = 4 column of the table). We emphasize that this table shows an equivalence between a geometric property of a manifold and some physical symmetry of a (supersymmetric non-linear o" -) model built on that manifold. For general

bosonic models, such equivalence does not seem to exist, and, in view of quantization one

might hope to take such geometrical property as the definition of the theory.

147

i

mension

extended ,SUSY

N=i

N=2

N = 4

d=2

no restriction on JI¢c

~ . is K~hler

~ . is hyper-K~hler

d=4

is K~hler

~, is hyper-KQhler

No extension exists

(bosonic sector ne~ds spinl )

With this in mind, we now discuss the perturbative approach to these supersymmetric non-linear or- models.

II I.PERTURBATION THEORY FOR SUPERSYMMETRIC NON-LINEAR ~-MODELS

The background field method is inherent in the perturbative study of generalized non-linear or- models 6 . ~ ~ ' ~ o J ~ [3].

We also add a mass term to the action by hand, since here we are not concerned with the infra-red behaviour.

II I. 1 BackGround field method with normal coordinates The usual background field splitting ~i = lick + C~ leads to a non covariant

formalism as Q i is not a vector field under reparametrizations. So, in i972, J.Honerkamp

introduced normal coordinates [B] (1)i - lii~l~ ÷ ~i ÷ Xi ( ~i~ick, ~ ) (il)

where the quantum field ~i is the tangent vector to the geodesic (I) i (t) at t=O :

~(0) = ~i~o k , ¢I)i(I)= d# , d(:l)i(t)/dt It=o = t~ i (12)

The resulting Feynman rules will be manisfestly covariant and, due to power

counting, the divergences of the theory could be compensated for by covarlant metric counterterms TO)ij [3]"

~ j [~ ] - ' ~ r e u [ Q l = Qj[Q] + ( lh/¢)T(~)u[gl + ...... (13)

(dimensional regularization, compatible with reparametrization invariance is used here). Finiteness proofs for supersymmetric non-linear or-models wil l be based upon an analysis

of the T(l)ij allowed in perturbation theory.

148

A few comments are in order:

C.1. the polynomial character of the divergences, used to write equ. (13), supposes

substraction of subdivergences : these ones, being necessarily ambiguous, have to be

precisely fixed through normalisations conditions. Here, in the absence of isometries for the general Riemannian metric !~j, an infinite number of such normalisation conditions -

i.e of ~ - ~ physical parameters - is necessary. This non renormatisability is usually

circumvented by invoking minimal schemes : we emphasize that this relies heavily on a

definite regularization and hides the difficulty. A correct approach would be to prove, as

was done in other examples with an infinite number of parameters [9], that only a f inite

number of them are physical ones.

C.2. As shown by B. De Wit and M.T.Grisaru [10], when non linearly realized symmetries

are present, the "on-shell counterterms" are not necessarily symmetric, even i f a

symmetry preserving regulator existed. As a consequence, when studying extended

supersymmetry, one should devise a background field splitt ing such that the

transformations of the quantum field under reparametrizations and extended

supersymmetry are linear: this is not so simple [11].

C.3. Moreover, in d : 2 non-linear o--models where the canonical dimension of the field

vanishes, non-linear field renormalizations are involved[g]. The usual argument -that

quantum field renormalizations are unnecessary in the background field method -seems

dif f icult to maintain, and indeed, i t has been proved in a recent calculation (see K.S. Stelle contribution to this worshop [t2]) that background field counterterms are n..ot sufficient to compensate for non local higher loop divergences ({1/~}log m2/p 2 terms in a

two-loop calculation).

C.4. Quantization of such supersymmetric theories relies heavily on the existence of a

supersymmetry preserving regulator. The common practice is to use dimensional

reduction [13a)] :al l the spinor -or supersymmetric covariant derivatives - algebra is

done in d = 2 dimensions and, after, momenta are analytically continued in the complex d- plane. Unfortunately, this method suffers from mathematical inconsistencies [13b),c)]

and one cannot rely upon it. In superfield calculations, convergence improvements occur

only after such manipulations are down on divergent integrals, and thus they suffer from the very ambiguities that make normalisation conditions necessary.

Ill. 2. One-loop calculations and all-orders finiteness conjectures in the supersymmetric

case. One-loop on-shell divergences are unambiguous and proportionnal to the Ricci

tensor I~j [g] (refs.[g,3]). Two- and three-loop calculations, with minimal substraction

149

and dimensional reduction, indicate (but for comments C.1 and C.4) that no new divergent

contribution appears after the one-loop order. This led to the conjectures - and claimed proofs - that N = 1 supersymmetric non-linear o-models have only a one loop divergence

or, less ambitiously, that N= 2 Ricci f lat supersymmetric cr- models are all-orders

f inite [14]. All these "proofs" suffer from the diff icult ies previously mentioned, and a

recent four-loop calculation[15], done under the same hypothesis, shows that they are

uncorrect. As N = 2 Ricci f lat supersymmetric non-linear ~r-models are discussed in C.N

Pope contribution to this workshop [I 6], we now present and comment the less ambitiOUS conjecture of all-order finiteness for supersymmetMc non-linear o'-models built on

hyper-K~hler manifolds[17](hyper-K~hler implies Ricci flatness [2]).

The equivalence discussed in section II between N : 2 (resp. N : 4)

supersymmetry and the Kah]er (resp. hyper-K~h]er) character of the metric, plus the

hypothesis of a supersymmetry preserving regulator, imply that the cOunterterms should

be Kahler ( resp. hyper-K~hler). The K~hler character of the cou~terterms T E means that they are of the form

T E = 8 8 ~ S[2,Z ] where, ~ - ~ w ~ , S{Z,2 ] is not a globally defined function. However, in

refs.[17a),l I], arguments are given which indicate that, except at the one loop order where IR~ = 8 ~ L o g det Igl, S[Z,Z ] is a globally defined function. N = 2 supersymmetry,

plus Ricci-flatness of the metric to get rid of the one loop contribution, then insures that the K~h]er form £~ (equ. (6)) stays in the same cohomology class in HIA(JIq,).

On the other hand, the hyper-K~h]er character of the counterterms means that the Ricci flatness of the metric is preserved in higher orders

N=4 SUSY => R ~[g]= R [ g , T ] = O (14)

Then the Ricci form d = i R E dZ~^dZ is the same for metrics g and g+T whose K~hler

forms f~ and ~3' are in the same cohomology class. The manifold being supposed c.ompact

(**) and connected, the uniqueness theorem of Calabi[18] asserts that these K~hler forms are the same and, as a consequence, T ~ vanishes, to all-orders of perturbation theory.

Q,E.D. We hope to have clearly pointed out the delicate points of such a proof (see

comments C._J_I and C._~4). Another "proof" of finiteness of N=4 supersymmetric ~r-models

exists, based upon quantization in harmonic superspace [19]. It of course suffers from the same difficulties.

(**)Here, as a consequence of Ricci flatness, the f i rs t Chern class vanishes. Concerning compactness, we do not expect that perturbative results sould depend on this mathematical necessary hypothesis (see also [17b]).

150

IV. CONCLUDING REMARK

If we expressed doubts on some "proofs" in the litterature on supersymmetric non-linear o--models, however we think that something should be true in these up to five-

loop order calculations [15]. As in the early days of O.E.D. recalled by D. Maison in his

introductory talk [20], when there was no satisfactory way to get rid of infinities

(renormallzatlon theory being not yet at hand), new methods are probably necessary to explain these "experimental" results.

[~] [2] [3]

[4] [5]

[o] [7]

[8] [9]

[10] [11] [12] [13]

[14]

[15]

[16] [17]

[18] [i9] [20]

REFERENCES B. Zumino, Phys.Lett.87B(1979) 203 L. Alvarez-Gaum~ and D.Z. Freedman, Com.Math.Phys.80 ( 1981 ) 443 DiH.Friedan, Phys.Rev.Lett. 45 (1980) 1057 and Ph.D. thesis, August 1980, published in Ann.Phys. 163 (1985) 318 G. Valent, Methods in hyper-K~h]er o'-mode]s building,these proceedings G. Bonneau, Renorma]ization of bosonic non-linear o'-models built on compact homogeneous manifold, these proceedings G. Bonneau and F. De]duc, Nuc].Phys.8250 (1985) 551 a) A detailed introduction to K~hler geometry for physicists can be found in L.Alvarez-Gaum~ and D.Z.Freedman lecture at Erice 1980, "Unification of the fundamental particu]e interactions", eds. 5. Ferrara et alL, Plenum New York 1980, page 41, b) for a more mathematical point of view, see 5.Ga]lot contribution in "premiere classe de Chem et courbure de Ricci : preuve de laconjecture de Calabl" 5oci~t~ math~matique de France, AsteMsque n°58 (1978) J.Honerkamp, NucI.Phys.B36 (1972) 130 a) O. Piguet and K. 5ibold, these proceedings b) G.Bonneau and F. De]duc, Nuc].Phys. B266 (1986) 536 B. de Wit and M.T. GMsaru, Phys. Rev. D20 (1979) 2082 P.S. Howe, G. Papadopoulos and K.S. Stelle, Phys. Lett. 174B (1986) 405 K.S. 5telle, these proceedings a) W. Siege], Phys.Lett.84B ( 1979) 193 b) W. Siege], Phys.Lett.94B (1980) 37 c) L.V. Adveev et alL, Phys.Lett. 1058 (1981) 272 Unpublished prepMnts by L. Alvarez-Gaum~ and P. Ginsparg, by C.M. Hull ; L.Alvarez-Gaum~, 5. Coleman and P. Ginsparg, Comm.Math.Phys. 103 (!986) 423 M.T. GMsaru, A.E.M. van de Ven and D. Zanon, Phys.Lett. t 73 B (1986) 423, M.T. GMsaru, D.I.Kazakov and D. Zanon, HUTP prepMnt 1987 C.M. Pope, these proceedings a) L. Alvarez-Gaum~ and P. Ginsparg, Comm. Hath. Phys. 102(1985) 311 b) C.M. Hull, Nucl.Phys. 8260 (1985) 182 For a pedagogical review on Ca]abi-Yau theorems, see the book refered under[7b)] A. Galperin et all., Class. Quant. Gravity 2 (1985) 617 D. Maison, these proceedings.

METHODS IN HYPERKKHLER ~ MODELS BUILDING

D. OLIVIER,G. VALENT

LPTHE, Universit~ Paris VII, T. 24, 5°~tage

2, place Jussieu 75251 PARIS CEDEX 05

FRANCE

1 ° INTRODUCTION

The study of supersymmetric extensions of bosonic non linear O"

models has received increasing attention these last years. The main

reason being that they are deeply relatedlalready at the classical le-

vel, with the theory of G structures which is by itself a field of in-

terest. Indeed for a four dimensional base manifold the metric which

defines the ~ model should be K~hler (K) (rasp. Hyper-K~hler (HK)) to

accomodate for N = 1 (resp N=2) supersymmetries [I] .

At the quantum level, the increase in the number of complex struc-

tures linked to extended supersymmetries is commonly believed to give

milder ultraviolet divergences [27 . This indicates that if something

can "stabilize" at all the G structures this should be supersymmetry.

However the increase of complexity in constructing explicit metrics

is significant : for K. ones it is completely solved through the exis-

tence of a potential (in holomorphic coordinates), for the H.K. ones no

such a general characterization has yet been found.

In view of these remarks the use of N=2 susy in HK building seems

therefore an attractive approach, which already led to interesting

general results [3] .

Recently an unconstrained N=2, D=4 superfield formalism was cons-

tructed [43 : the so called Harmonic Superspace (HSS). In this frame-

work, starting from a given superfield lagrangian one is in principle

able to extract out a bosonic sector which must be HK. This raises the

hope of a Systematic approach to the construction of HK metrics.

152

Indeed the superfield lagrangians corresponding to Taub-NUT ~

and the higher dimensional metrics of Calabi [6J are Eguchi-Hanson

known.

It is the aim of this talk to review the results obtained for

Taub-NUT in [5 7 and to present some generalizations of them [7,8,14J.

2 ° HK METRICS AND GRAVITATIONAL INSTANTONS

Let us first recall some basic results in four dimensional HK me-

tries, using the notations of [9].

The euclidean distance writes in terms of the vierbein:

A=o and the connexion 60ASand curvature RAB result from :

~A%= d&3A5 ~ 60AcA~DO%

60& 5 = - 60~A

The curvature (as well as the connexion) can be splitted using self-

duality :

E+ ÷ R +_

This splitting can be interpreted in terms of the holonomy algebra

whose generators are precisely the RAB. For a general metric the holo-

nomy algebra is so(4) and its generators can be splitted into two sub-

sets R~B and RAB which generate respectively su(2~ and su(2)_ •

It follows that a metric with self dual curvature (RAB = 0) has

holonomy su(2~ /i) sp(1) and is therefore HK, as first obtained in[l].

However such an argument does not tell anything on the complex struc-

tures. Here we will present a proof of this result which gives in ad-

dition their explicit form.

For any 4 dimensional metric we begin by defining a triplet of

2-forms :

153

b

It is readily checked that thequaternionic multiplication law holds :

<F4 ~ (%~': - a~ ; ? + ~ ~ ~ c~>?

and therefore any 4 dimensional metric exhibits a triplet of (almost)

complex structures and it is (almost) hermitian with respect to them.

be further shown ~] that they satisfy : It can

~ = -~7 i ̂ J~

Now if the metric has self dual curvature, its connexion is a pure

gauge. There exists a matrix M G SU(2)_ such that :

Then if we rotate the vierbein :

A

we get 60~% O.

It follows that the triplet of complex structures (Fi)~ extracted

from~(~) are then covariantly constant establishing the HK character

of the metric.

It is interesting to notice that such an HK structure implies Ricci

flatness and ensures that any euclidean HK metric is a gravitational

instanton.

Let us now turn to the HSS derivation of Taub-NUT.

154

3 ° Taub-NUT INSTANTON AND HSS

In this formalism the basic object is the unconstrained superfield

+ [4] superspace { ~,~ ~ ÷ --_ ~ } q --- which lives in the analytic N=2 ~ j ~+.~

and is defined by :

+ F "~ ~ e * ~ ' ~ A'~ . O*e ~" M - .~ e ~ O"~N - q = +

The price to pay for unconstrained superfield is the existence of

infinitely many auxiliary fields which are displayed in the harmonic

expansions of F +, A- , etc... For instance F + writes :

In this expansion the spinor coordinates fi(i=l,2) and their charge

conjugate partners ~i are the physical bosonic fields. They correspond

to a super isospin 1/2. The infinite tail of higher isospins must be

expressed in terms of the coordinates using the field equations.

For Taub-NUT the action is [5] :

(see [4,5] for the notations -the involution ~ of these references is

simplified here to ~).

The first attractive feature of this formalism is that the isome-

tries of the bosonic sector can be read off from the superfield lagran-

gian. Here we have the supersymmetric SU(2) for which fi and ~i are

doublets and the so-called Pauli-Gursey U(1) :

u , 0G *

The second nice feature is that, given the lagrangian, the remaining

steps to obtain the metric are all "deductive". Let us describe the HSS

algorithm :

155

i) solve for the field equations :

and get F + A- in terms of the coordinates fi ~i ; l •

# - - 2) integrate over , e÷ in (I) and get rid of the auxiliary fields

M,N,P using their equations of motion• One remains with :

3) extract out the singlet part from this expression• This gives

you an HK metric !

In order to express the final result it is convenient to define :

i) bispinor coordinates : ~ u/'~ --_. --~ ~:~

where fi,l, fi,2 are spinors under SU(2) s and fl,a, f2,a are spinets

under SU(2)pG. These coordinates are constrained by pseudo-reality

2) the vierbein E ia :

with the same pseudo-reality constraint as the coordinates. For Taub-NUT :

The relevant information on Taub-NUT can then be summarized in a

triple :

• isometries ~ ~U(~3~ ~ ~.~.

• distance ~ ~----" £ ~ E i ~ ~ } • ( 3 )

• triplet of closed 2 forms : ~ ~b)= ~4)~ E i~ ~

Let us observe that the structure displayed by equations (3) was

by Sierra and Townsend ~0] for any N=2 susy O- model first obtained

using constrained superfields. However the detailed form of the vierbein

(2) was found in [9 using HSS.

156

These relations have such a nice structure, that one may address

the following question: to what extent do these relations define Taub-

NUT and this, independently of the use of HSS.

Let us try to construct the most general metric satisfying (3) with

isometrics SU(2) S x U(1)pG Its vierbein should write

+

because of the isometries. The closedness of the triplet~ ij imposes :

,:la

x : ( a - , ~ , ) ~

d x = - , XVT -×@--r'3 k-a % T= %-o

A-~5

(4)

The function T(s), which remains free, reflects the arbitrariness

in the definition of s.

The system (4) with the choice T = 0 leads to the vierbein (2) (pro-

vided that we look for a metric which is flat at the origin).

This proves that, at least at the classical level, Taub-NUT is "uni-

quely" defined by requiring (3) and the isometries SU(2)s x U(1)pG.

"Uniquely" means here up to a reparametrization of s :

4 ° SYMMETRY BREAKING AROUND Taub-NUT

We have generalized in [8] and [14] respectively both of the pre-

vious analyses.

Let us present our results.

i) the HSS analysis was applied to the lagrangian :

where

= ~ .

~_d _ -7--

157

induces a symmetry breaking of SU(2) s down to U(1) s ; therefore we

have U(1) s x U(1)pG as isometries.

E .L~I _

It leads to the vierbein

where :

and to a triplet of closed 2-forms ~ij as in (3).

At that stage one has to face the problem of the identification of

this HK metric.

In fact GIBBONS [ii~ pointed out that this metric should be related

to the multicenter ones [12]. Indeed using polar coordinates :

d

and rotating the vector o the form :

~--= ~ ~

and defining :

Vx ~ I ~ ra l

the distance derived from E ia becomes

÷ ~ = V'~G~+~, ~ ' ) ~ +

which is the standard form of the multicenter metrics. The lagrangian

we started from corresponds to a potential V with one center (cor-

158

responding to Taub-NUT) plus a symmetry breaking term with dipolar

structure.

2) the "relations (3)" analysis was applied to obtain the explicit

form of a metric which generalizes Taub-NUT by lowering its isometries

to SU(2)_. The detailed computations are too hairy to be presented

here [143 but lead ultimately to the known gravitational instanton of

Atiyah and Hitchin [13~.

It is interesting to note that this metric has not yet been re-

covered in the HSS approach in spite of the fact that the generic form

of its lagrangian is well known [5] . It appears that the breaking of

U(1)pG introduces in the field equations non linearities which are hard

to deal with.

In our opinion this second approach may give some help to the HSS

analysis.

REFERENCES

[~ ALVAREZ-GAUME L., FREEDMAN D.Z., Commun. Math. Phys. 8_O0, 443 (1981)

2] For a critical discussion, see the contributions of C. BECCHI and G. BONNEAU to this workshop.

[3] HITCHIN N.J., KARLHEDE A°, LINDSTROM U., ROCEK M., Commun. Math. Phys. 108, 535 (1987).

[4] GALPERIN A., IVANOV E., KALITZIN S., OGIEVETSKY V., SOKATCHEV E., Class. Quantum Grav. l, 469 (1984).

5] GALPERIN A., IVANOV E., OGIEVETSKY V., SOKATCHEV E., Commun. Math. Phys. 103, 515 (1986).

~6] GALPERIN A. IVANOV E., OGIEVETSKY V., TOWNSEND P.K., Class. Quantum Gray. ~, 625 (1986).

[7] OLIVIER D., VALENT G., preprint PAR-LPTHE 86/22, unpublished.

8] OLIVIER D., VALENT G., preprint PAR-LPTHE 86/49 to appear in Physics Letters B.

[9] EGUCHI T., GILKEY P., HANSON A., Phys. Rep. 66, 213 (1980).

~ SIERRA C., TOWNSEND P., Phys. Lett. 124B, 497 (1983).

b yGIBBONS G.W., Private Communication.

~HAWKING S.W., Phys. Lett. 60A, 81 (1977). GIBBONS G.W.,HAWKING S.W., Phys. Lett. 78___BB, 430 (1978).

t59

[13]ATIYAH M.F., HITCHIN N.J., Phys. Lett. 107A, 21 (1985). GIBBONS G.W., MANTON N.S., Nucl. Phys. B274, 183 (1986).

[14]OLIVIER D., VALENT G., in preparation.

SIGMA MODEL B-FUNCTIONS AT ALL LOOP ORDERS

C.N. Pope CERN, CH-1211 Geneva 23

Switzerland

i. Introduction

One of the most remarkable claims, indeed perhaps the remarkable claim, of

string theory is that it constitutes a consistent quantum theory, possibly finite,

that incorporates general relativity as a low-energy effective limit. This low-energy

theory corresponds to Einstein's theory of gravity described by the Lagrangian ~-gR,

together with higher-order terms involving more derivatives, such as /-g (Riem) 2,

etc. One way to investigate these higher-order terms in the effective Lagrangian is

to calculate string scattering amplitudes, in which the external lines are chosen to

be on-shell gravitons. Such amplitudes have been calculated, for the type lib and

heterotic string theories, with up to four external gravitons I), and give rise to

terms in the effective action up to quartic in Riemann tensors.

A different approach to determining the low-energy effective action is from

the a-model point of view. The two-dimensional worldsheet action for a closed string

coupled to a non-trivial curved background metric gij is classically invariant under

conformal rescalings of the worldsheet metric. Consistency requires that this

conformal invariance be preserved at the quantum level, in other words that the

worldsheet stress tensor should not develop a trace-anomaly. This is equivalent to

the condition that the B-function ~ij that describes the renormalization of the

target -space metric gij should vanish. The resulting equation,

is believed to be equivalent to that obtained from the low-energy effective action

for the string.

In this paper we review some results concerning the form of the contributions

to B-functions for a-models in non-trivial backgrounds. Section 2 is concerned with

supersymmetric a-models with curved target-space metrics gij" The forms of the metric

counterterms that can arise are tightly constrained by supersymmetry considerations.

A sequence of candidate counterterms at all loop-orders can be constructed2); the

four-loop term had already been found by direct calculation 3), and shown to be 2) ,4) ,5)

equivalent to (Riem) 4 term in the type liB string effective action . It is not

161

known whether the higher-loop terms in the sequence actually occur with non-zero

coefficients in general, nor whether other independent counterterms can arise.

In section 3, we consider a rather simpler o-model corresponding to an open

bosonic string in flat 26-dimensional space, in the background of a non-zero

electromagnetic field F . The string couples to the potential A only via a boundary

term, and because the problem is therefore essentially one-dimensional, it turns out

that the Z-function _B~ describing the renormalization of A~ can be calculated

exactly. Although perhaps of limited physical relevance, this provides an interesting

toy example in which one could, in principle, study the non-perturbative structure of

the quantum theory.

2. Supersymmetric o-models

The action for the supers~metric o~odel is

J 5 " (2) - °al

where gij(X) is the metric on the target manifold M, B± ~ / ~ + B/~O, X i = _ are the a

co-ordinates on the target space, and , h R are left and right moving fermions. It

is invariant under the supers~metry transfomations

: F~ ~ XK X ~

~ ' X" X ~ S k~, = - F,i, ~ /k~ ~ + i £ ~ 9 + ,

(3)

where ha = hieS1 and e@(X)z satisfies eaeai j = gij" This N = 1 supersymmetry is in fact

comprised of two independent supersymmetries, of type (i,0) and (0,i), corresponding

to the independent parameters e L and eR, and is often referred to as (i,I)

supersymmetry.

If the target manifold M is KNhler, then there is a second, independent,

supersymmetry with parameters (eL, eR) , in which (~, k R) in Eq. (3) are replaced by

(Ji#hJL, jijkj = k. ), where jij is the complex structure on M, with Jij -Jji = gik J j

162

the K~hler form. This N = 2 supersyrametry is further enlarged to N = 4 if M is

hyperk~hler, the analogous additional supersymmetries occurring for each of the three

complex structures j(1)ij, j(2)ij and j(3)ij.

The action (2) may be written in terms of N = 1 superfields as

(4)

i 8Rk~ + where ~i = X i + 8LkL + 8LSR Fi and D L = ~SL + iSLe_, D R = DSR + iSRD +. The

fermionic co-ordinates @L and 8 R are left and right-handed Majorana spinors. In the

N = 2 case, where M is a K~hler manifold with K~hler metric gij' the action may be

written in terms of N = 2 superfields as

(5)

where K(e) is the Kghler potential.

The N = 4 (hyperk~bler) models are finite 6" ~. The reason for this is

essentially that the order-by-order renormalization of gij must preserve the

hyperk~hler condition. In particular, this implies that the metric is Ricci-flat and

K~hler, and hence (for a given complex structure) unique. Thus no counterterms can

arise.

For the N = 2 (K~hler) models, one can integrate out two of the e's in

Eq. (5), to obtain Eq. (4) with gab = ~a~b K in complex co-ordinate notation.

Likewise, counterterms Al will take the form fd2~d4eS, and can be integrated out to

give Aga~ = ~a~S. General arguments based on considering the conformal weights of

the counterterms show that, beyond one loop, the functions S must be globally-defined

scalars constructed from powers of the Riemann tensor on M and its covariant

derivatives 7) .

Since all hyperk~hler spaces are K~hler, and all K~hler spaces are Riemannian

spaces, it follows that the N = 4, N = 2 and N = 1 supersymmetric a-models can all be

viewed as N = i models. It therefore follows from the above discussion that all

metric counterterms Agi~ J must be Riemannian expressions which vanish if gij is

hyperk~hler and must take the form &gab = ~a~b S if gij is K~hler. The problem of

constructing candidate counterterms can thus be reduced to finding symmetric tensors

Ti~ J built from Rijkl'S and their covariant derivatives that satisfy these

requirements.

1 6 3

In order to construct such tensors, it *is convenient to employ real rather

than complex co-ordinates in the case of K~hler spaces. To do this, we define

projectors ~±.J: i

i Z - •

+ •

These project onto holomorphic and antiholomorphic indices, so that Va++ ~ iJVj, V a

<-~-iJv''3 Thus counterterms must take the form Tij~ +(ik -o_i)IVkViS in K~hler spaces,

i.e.

where we use the hat notation 8), defined by V^ = J.Jv. for all vector V.. A candidate i i j i

counterterm is acceptable if S vanishes when gij is hyperk~hler, and if Eq. (7) can

be written without complex structures, i.e. without hats. This last requirement is

highly non-trivial.

The first example of an acceptable counterterm that does not vanish in

Ricci-flat spaces was found in Re f. 3) for N = 2 o-models, and it was shown that it

in fact occurs with non-zero coefficient at the four-loop order. The scalar S is

given by

(8)

It is straightforward to check that S vanishes if gijis hyperk~hler. This follows

immediately from the form of the Riemann tensor on hyperk~hler spaces,

Rijkl +÷ e ~y6~ABCD , where ~, ~ are Sp(1) indices, A, B are Sp(n) indices

(dim M = 4n), and QABCD = Q(ABCD)" To prove that Tij in Eq. (7) can be written

without complex structures, one must show that VZ V~ S can be written without hats.

It is a tedious and not entirely straightforward matter to do this from Eq. (8) by

repeated use of cyclic and Bianchi identities 8). One also needs to use the fact that,

• it follows that V^U I = -Vi UI, V~ = -Vi, and from the properties of Jij' l

Rijk~ = -Rij~l. This last property follows from the covariant-constancy of Jij' which

also implies that hats can pass freely through covariant derivatives.

There is a much easier way to derive the result that riVeS can be written

without complex structures, which also lends itself immediately to a generalization

to candidate counterterms at arbitrary loop order. By using a cyclic identity, one

can easily see that Eq. (8) may be written as

164

This admits a natural generalization to an expression of n'th power in Riemann 2)

tensors ,

c~ ~ r~ ~ R ~, (i0)

This vanishes for hyperk~hler metrics for all n. It is now almost trivial to show

that V~V^S (n) is Riemannian, and hence T! n) is a candidate (n + l)-loop counterterm, _ ( ~ n ) J" . ~ 2) tj

with 1.. glven oy 11

---- +

v , - * ' . +

. - a

+ ~'r~ ~" ~ ~r~ ~ ~

The constraints on a function S if it is to have the property that VzV@S is

independent of J.. are very restrictive, and it is tempting to conjecture that the 13

examples of Eq. (10) are essentially exhaustive. Unfortunately there seems to be no

obvious way of constructing the most general such function, and so in the light of

past experience with o-models, such speculations are probably ill-advised. What can

be said is that beyond one loop, the counterterms must certainly take the

cohomologically trivial form 5a5~S on a Kghler space, where S is globally defined.

Thus for a K~hler metric to yield a zero of the B-function, it must satisfy

~.~ -" ~o. ~ S (12)

for some globally defined scalar formed from Riemann tensors and covariant

derivatives. It follows therefore that M must have vanishing first Chern class.

Whether Eq. (12) would admit non-trivial (i.e. non-hyperk~hler) solutions is not

clear. One can invoke Yau's theorem 9) to construct an infinite sequence of metrics (r)

gij ' defined by

165

R:..c { 1. <<+'' ) =- ~<. "~-~ S t ~.<'-'), ~l~ i

(o) with gij Ricci-flat, but making any rigorous statements concerning the convergence

of the sequence would appear to be problematical [see, for example, Ref.lO)].

3. Open strings in background ~au~e fields

We now turn to an example of a ~-model in which it appears to be possible to

compute a B-function to all loop orders. It corresponds to the open bosonic string,

propagating in a flat (26-dimensional) background, but in the presence of a non-zero

gauge potential A . This couples to the string only at its ends, and the action is

I (14) J J ~

Expanding around a classical solution X ~, we write X ~ : X~ + H~(~), where as

usual, we employ a normal co-ordinate expansion in the quantum field ~ around the

point X~. Dropping the bar, we have

+ t v,v~ ~ . t~'~:,~:~.~+ ~ ~. ~ i~-f ~ f ~ + (15)

To compute the Z-function, we must consider all one-particle irreducible diagrams

that contribute counterterms to SA X ~ with i/e poles in dimensional regularization. Ii)

This has been discussed at the one-loop level , and extended to all loops in Re f.

1 2 ) •

At one-loop, counterterms come from diagrams with one vertex VF X ~ together

with any number of insertions of the vertex F~. It follows from symmetry arguments

that only terms with an even number of such insertions contribute, and so

schematically one has

166

i• + +F F VF k ~F X w~

~- . . .

(16)

where the dots indicate quantum lines corresponding to ~. On the boundary DE, the

propagator G v takes the form loglz-$'[, or in momentum space, G~v ~ 6~v/lP]" Thus

for all lines with dots, the time derivative on ~ brings down a p which cancels the

propagator, yielding a 6-function 6($-~'). The result is that Eq. (16) reduces to

F F

+ • . .

where the closed loop yields a factor /_~dp/lPl common to all terms.

(17)

Introducing an infra-red mass regulator m, and dimensionally regularizing,

this gives a factor

oo

i

(i8)

The infra-red divergence is irrelevant as far as the B-function is concerned, and so

we obtain, taking account of combinatoric factors II),

(19)

ii) which can be summed to give

(20)

Here F 2, F 4, etc. denote matrices, with components F FPv, F PFp6F kFkv, etc. It was ~P

shown in Ref. Ii) that although ~ itself cannot be obtained from the variation of

= any action, the equation ~ 0 is equivalent to that obtained from the Born-lnfeld

action

Thi s y i e l d s t h e f i e l d equa t ion W = 0, where 6S z JLW'GA and P P

~ h u s , from Eq. (201,

The ma t r ix F2 i s nega t ive s emi -de f in i t e , and so ( l - ~ ~ ) i s a non-degenerate ma t r ix ,

i m p l y i n g t h a t @ v a n i s h e s i f and on ly i f F i s a s o l u t i o n o f t h e Born-Infeld f i e l d 11) I-r P"

e q u a t i o n (22) . The e x t e n s i o n o f t h e @ - f u n c t i o n c a l c u l a t i o n t o a l l loop o rde r s12 ) i s most

e a s i l y d i scussed by cons ide r ing a t y p i c a l diagram. From Eq. (151, one of t h e

c o n t r i b u t i o n s a t two loops w i l l be g iven by

V F i As b e f o r e , t h e p ropaga to r s on l i n e s wi th d o t s a r e c a n c e l l e d , g i v i n g

where each loop g i v e s a f a c t o r JdpIp . We aga in r e g u l a r i z e us ing Eq. (181, but now we

must t ake c a r e t o s u b t r a c t ou t t h e one-loop subdivergent diagrams, by in t roduc ing

a p p r o p r i a t e counter terms. The n e t e f f e c t i s t h a t t h e remaining c o n t r i b u t i o n from Eq.

(25) i s a pu re ly 1 / c2 d ivergence , and thus does not a f f e c t t he p-function, which i s

determined e n t i r e l y by t h e 1 / ~ pole . By s i m i l a r arguments one can show t h a t a t

n - l o o p s t h e c o u n t e r t e r m s c o n t r i b u t e pure 11," po le s , and aga in do no t c o n t r i b u t e t o

B P . There a r e two s p e c i a l c a s e s t h a t must be c o n s i d e r e d , i n which d o t s become

"trapped" i n c lo sed loops . This can happen wi th e i t h e r one o r two d o t s , g i v i n g

f a c t o r s o f Idp o r Jpdp r e s p e c t i v e l y . Both o f t hese van i sh i n dimensional

r e g u l a r i z a t i o n .

168

We see therefore that the all orders expression for ~ is precisely the same

as the one-loop expression Eq. (20) 12). Of course this result is highly dependent on

the renormalization scheme used. It is consistent with the results of a three-loop

calculation in Re f. 13).

In view of the conjectured equivalence of demanding conformal invariance of

the ~-model on the one hand, and solutions of the string equations of motion on the

other, the above result appears somewhat surprising at first sight. String scattering

calculations yield amplitudes with non-polynomial dependence on the external momenta,

and thus the string effective action will involve arbitrarily many derivatives on

F whereas Eq. (21) involves none at all. However, there need not necessarily be

any conflict between these results, since it is solutions of the two systems of

equations of motion that are supposed to be in correspondence, which does not

necessarily mean that the equations themselves must have the same form.

Finally, we remark that it was also shown in Re f. 12) that the same

Born-lnfeld action, Eq. (21), arises in the case of the ten-dimensional superstring

in an arbitrary purely bosonic gauge field background. This result 12), obtained for

the Neveu-Schwarz-Ramond action, is different from that obtained by Tseytlin 14), who

used the Green-Schwarz action. It is not clear what the origin of this discrepancy

is; it may perhaps be an indication that in a non-trivial background, the two

formalisms are not equivalent. This may be similar to the situation in the case of • 15),10)

closed superstrings in curved background spacetlmes

ACKNOWLEDGEMENTS

I am grateful to E. Bergshoeff, M. Freeman, E. Sezgin, M. Sohnius, K.

Stelle and P. Townsend, with whom the work described in this paper was carried out,

for extensive discussions.

169

REFERENCES

i) D.J. Gross and E. Witten, Nucl. Phys. B277 (1986), i.

2) M.D. Freeman, C.N. Pope, M.F. Sohnius and K.S. Stelle, Phys. Lett. 178B (1986), 199.

3) M.T. Grisaru, A.E.M. Van de Ven and D. Zanon, Phys. Lett. 173B (1986), 423; Nucl. Phys. B277 (1986), 388 and 409.

4) M.D. Freeman and C.N. Pope, Phys. Lett. 174B (1986), 48.

5) M.T. Grisaru and D. Zanon, Phys. Lett. 177B (1986), 347.

6) C°M. Hull, Nucl. Phys. B260 (1985), 182; L. Alvarez-Gaum~ and P. Ginsparg, Comm. Math. Phys. 102 (1985), 311.

7) P.S. Howe, G. Papadopoulos and K.S. Stelle, Phys. Lett. 174B (1986), 405

8) C.N. Pope, M.F. Sohnius and K.S. Stelle, Nucl. Phys. B283 (1987), 192.

9) S.T. Yau, Proc. Natl. Acad. Sci. 74 (1977), 1798.

10) M.D. Freeman, C.N. Pope, M.F. Sohnius and K.S. Stelle, CERN preprint TH.4632/87, to appear in the Proceedings of the Colloquium on Strings and Gravity, Meudon, Paris, September 1986.

Ii) A. Abouelsaood, C.G. Callan, C.R. Nappi and S.A. Yost, Princeton preprint (1986).

12) E. Bergshoeff, C.N.Pope, E. Sezgin and P.K. Townsend, Trieste preprint (1987), to appear in Phys. Lett. B.

13) H. Dorn and H.J. Otto, Z. Phys. C 32 (1986), 599.

14) A.A. Tseytlin, Nucl. Phys. B273 (1986), 391.

15) M.D. Freeman, C.M. Hull, C.N. Pope and K.S. Stelle, Phys. lett. 185B (1987), 351.

THE d=2 CONFORMALLY INVARIANT SU(2) o-Model

WITH WESS-ZUMINO TEPJM AND RELATED CRITICAL

THEORIES +)

R. Flume

Physikalisches I n s t i t u t , Univers i t~ t Bonn

Nussallee 12, D-5300 Bonn I

Abstract: The st ructure of the four point cor re la t ion funct ions of the conformally

invar ian t SU(2) o-model is presented. Relations of the SU(2) model to other c r i t i c a l

theories are pointed out.

I . ) A f ter the poineering work of Belavin, Polyakov and Zamolodchikov / I / much pro-

gress has been made during the last two years in the analysis of two-dimensional c r i -

t i ca l systems in s t a t i s t i c a l mechanics and f i e l d theory /2 / . The l i s t of f i e l d theo-

re t i ca l models which so fa r have been solved f l ) and are p a r t i a l l y i den t i f i ed wi th

c r i t i c a l systems of s t a t i s t i c a l mechanics contains

i ) a l l un i tary models wi th central Virasoro charge c < I / 3 / - / 6 /

i i ) un i tary theories wi th N=I and N=2 supersymmetry (c < ~ (N=I), c < 3 (N=2))

171 - I101

i i i ) models rea l i s ing parafermion algebras I111

i v ) a-models on group manifolds wi th Wess-Zumino term /12/ - /17/ .

I want to report in th is cont r ibut ion on recent work spec ia l l y devoted to the

SU(2) model /14/ - /16/ and to point out l inks of th is model wi th the other theories

( i ) - ( i i i ) in the above given l i s t . (~echnical - not conceptual - complications have

t i l l now impeded the e x p l i c i t so lu t ion - that i s , the construct ion of cor re la t ion

funct ions - of o-models on groups of higher rank).

I I ) The observation that o-models on group manifolds aquire through the addi t ion of

a Wess-Zumino term a conformally invar ian t f ixed point at a non-zero value of the

coupling constant is due to Witten /12/ . I w i l l not discuss the Lagrangian version

of the "Wess-Zumino-Witten (WZW) model". Instead I assume fo l lowing Knizhnik and

+)Talk presented at the Ringberg Workshop "Renormalization of Quantum Field Theories wi th non- l inear Field Transformations", Feb. 16 - 20, 1987.

f i ) At least in the sense that the s t ructure of the operator product algebra is known.

171

Zamolodchikov /13/ - the exi:stence of two Kac-Moody algebras of currents as const i -

tu t i ve s tar t ing point.Let G be a simple compact group. We suppose that in the theory

under consideration two sets of currents { j r } and { j~} occur generating the symme-

t ry group GL~G R (G R = G L = G) (a=1, . . . }d im G labels a basis of the Lie-algebra

of G)o The general frame work is two-dimensional Euclidean f i e l d theory. Let z (with

complex conjugate z) be a complex coordinate for the Euclidean plane. The currents

.a and "B JL JR are supposed to sa t i s fy the conservation equation

.a .a ~ JL = 0 = ~z JL '

and to obey the commutati.on relat ions - conveniently quoted in terms of a Laurent

decomposition ~ L,~ =~--T ) ,

I (Ib)

fabc denote here t o t a l l y antisymmetric structure constants of the Lie algebra of G.

The normalization is assumed to be chosen such that the long roots of the Lie alge-

bra have unit length. The parameter k on the r .h .s , of ( la ) and ( Ib ) , the so called

Kac-Moody central charge, is taken to be a posi t ive integernumber.(Thisisthenecess-

ary and suff ic i :ent cQndition to ensure the existence of unitary highest weight re-

presentations of the Kac-Moody algebra ( la) and ( Ib ) , c f . /18/ , /19/) . The energy

momentum tensor in the f i e l d theoret ical sector generated by the currents is known

to be of the Sugawara form. I ts two independent components taken into the z-z and

z-z direct ions read as (/20/)

7-_ (2a)

172

where the double points denote normal ordering with respect to current frequencies.

C v is the eigenvalue of the quadratic Casimir operator in the adjoint representation.

Energy momentum conservationis expressed through

~ Tzz = 0 = az T~

The Laurent components of Tzz and T~

: .t-~-~ ~rZ

generate the Virasoro algebras

(3a)

. = C. 1,1.3 _ ~,~÷ v,~, o ( 3 b )

E L . , L ~ ] = o (~c)

~, ct~ C a~ : de., G" ) c . . ~ -c - - -7

(3d)

Eqs. (3a) - (3d) are straightforward consequences of the Kac-Moody algebra relations

( la) - ( Ic) .

The Sugawara form of the energy momentum tensor supplies the key to the solution

of the WZW models. The observation is due to Knizhnik and Zamolodchikov /13/(cf . also

Dashen and Frishman, ref . /20/).To explain the i r observation we have to introduce

the notion of primary f ie lds / I / : An operator ~ is called primary with respect to the

Kac-Moody and Virasoro algebras (I) and (3) i f the following highest weight relations

are sat is f ied,

L + ~ p = - c + ~ = j : , ~ ( t -~ J~,~i ~ : o V + , o ,

~and ~y are here the scaling dimensions of ~ with respect to dilations in z

and z resp., t a t a L,~ and R,~ denote representation matrices of the Lie algebra of

G R and G L resp. (~ is supposed to stand for an operator multiplet carrying some irre-

173

a ducible representations of the two groups). JL ,n~ ' Ln~° etc. are shorthands for

All non-primary operators of the theory are determined through the primary ones via representation theory of the two algebras (I) and (3) f2) . The solution of the current

and energy-momentum Ward ident i t ies of correlat ion functions

< ~ c=~,z , j . So~ ( z . . z . ) >

of primary operators (~i (with scaling dimensions Ai ' 3i and carrying representa-

tions {tL, i } and {tR, i } resp.) are par t icu lar ly simple. One finds

• ( - 1

o,

(5)

m

~},. I- z-z~ ; (6)

(7)

Knizhnik and Zamolodchikov compare the Ward ident i t ies (6)and (7) with the fol low- ing relat ions derived from the operator product expansion.

f2) I t is assumed here tac i te ly that the ident i ty is the unique operator in the theory which commutes with a l l currents JR and JR "

174

% . (z) @~ ~ z ~ . ) = v , , c , , . .

~- regular terms

4"- regular terms

(9)

where CL, i (CR, i ) denotes the eigenvalue of the quadratic, Casimir operator, in re-

presentation carried by cPi. Ident i fy ing the singular terms in (z-z i ) of Eqs. (6)

and (8) ((7) and (9)) oneobtains f i r s t for the scaling dimensions the relations

~#,,L

and second, af ter use of the current Ward ident i t ies (5) ((6)) a system of d i f fe -

rent ial equations - the "Knizhnik-Zamolodchikov equations" - for the correlat ion

functions of primary f ie lds

O z ~ < ~ c z , ) . . ~p.~z,,) >

I - K j ~

Jtz ~ : ~

® L,,:I 7- L- Z<j

(lO)

and an analogous system of d i f ferent ia l equations in the complex conjugate variables

zi" The general solution of the Knizhnik-Zamolodchikov equations for the four point

functions of the SU(2) ~-model has been found in references /14/ and /15/. Let

{~)m~Z{} = -- ~-J _~Oj(z)be a primary operator mul t ip let carrying the isospin j represen-

tat ion of G L = SU(2) f3) . I choose isospins Ji " ' " J4 sat isfy ing the relations

3

,I, " i - "< J, 4 J,, .o , ,

f3) I ignore for the moment the z-dependence of the operators and do also not spe- c i f y under what representation -cPj transforms with respect to the iHght handed SU(2).

175

Global SU(2) invariance implies that in a cor re la t ion

the isospins Ji " ' " J4 are combined in to SU(2) s ing le ts . With the special choice

((11)) of isospins one can form ( I+ I ) d i f f e ren t s inglets and to th is corresponds a

decomposition of in to ( I+ I ) components: F j l " ' " J4

( F~I~ ) 4''J #i+., /

Ths bas i s of i nva r i an t s I choose i s const ructed by p r o j e c t i o n of the tensor product

of r ep re sen t a t i ons of i sosp in Jl and J2 in to the r ep re sen t a t i on with i sosp in

Jl + J2 - ( k - l ) , k = 1 . . . . . l + l a n d c o n t r a c t i o n with the same r e p r e s e n t a t i o n combin-

ing J3and J4" The Knizhnik-Zamolodchikov equation becomes an ( l+ l ) -d imens iona l ma-

t r i x d i f f e r e n t i a l equation fo r the vec to r funct ion F. I t s ge ne ra l so lu t ion can be

represented in terms of a 1-dimensional contour i n t eg ra l of the Euler type ( / 14 / ,

/I 5/) j__,

# 4

- ~ - 1 ~-1

~=~ (12) # J,

L--&

Several comments and remarks are to be made.

i ) The sum of exponents attached to any of the in tegrat ion var iables t I . . . t I is

equal to -2. The solut ion (12) could so be interpreted as integral over vertex ope-

rators of scal ing dimension I (with respect to scal ing in z) i f the in terac t ion of

these "ver t ices" wi th the external points z I . . . z 4 were not s l i g h t l y unsymmetric.

i i ) The matr ix d i f f e r e n t i a l equation has in the case under consideration a funda-

mental system of ( I+ I ) independent vector so lut ions. Those are found by making d i f f e -

rent choices of contours in (12). In order to construct a proper four point funct ion

one has to consider in addi t ion the general so lut ion of the Knizhnik-Zamolodchikov

equations in the complex conjugate variables zi and then to superimpose products of

special solut ions of the z- and z-equations so that the resu l t ing expression is a

one-valued funct ion in the Euclidean plane. This program is known as the conformal

176

bootstrap. I t fixes in particular up to an overall normalization the expansion co-

eff ic ients of the opera~or algebra. For details of the conformal bootstrap I refer

to /14/ and /15/.

i i i ) The functions given by Eq. (12) are identical in structure with the four point

functions of certain closed subalgebras of s tat is t ica l mechanics systems with central

charge c < I . (cf. /1 / ) . The ties of the SU(2) model with these and other systems

w i l l be discussed in the following section.

I I I ) A general recipe to construct the cor re la t ion funct ions of c r i t i c a l systems wi th

c < I has been given by Dotsenko and Fateev /6 / . The four point funct ions found by

these authors have the general form

= IT

m~ I . . I W E ~ - F F a . E , " '

t : t J=l o .-T]-. (.E~ - E~j ).z .

A

• (E4,..,E_, z,,..,~ V) (t~,.., E,, z1,..,z~) (13)

"" J ~ : 1 ~---1 ('(c)

A A

TF IT C:I $-'~ tz.j

where the exponents a~(aIX&), b (~ ) are functions of c and m (n). Comparing with

Eq. (12) one sees that for the case m = 0 or n = 0 in Eq. (13) one encounters the

same type of contour integrals as in the SU(2) model. Those special cases corres-

pond ~o closed operator subalgebras of the c < I models.

The functions F represent the general solution of a l inear d i f ferent ia l equa- n ,m

tion of (n+1). (m+1)-th order which can be derived from degeneracy relations of the

Virasoro algebra. Special solutions are again given through specificchoices of con-

tours. To be able to execute the conformal bootstrap program, as i t has been sketched

177

in the previous section, one has to know how d i f fe ren t special solutions are related

to each other under analyt ic continuation. One needs in other words information on

the monodromy group of the system of functions represented by the contour integral

(13). The problem is fac i la ted through the observation of Dotsenko and Fateev (6)

that the monodromy group factor izes on the pieces fm and fn in the integrand on the

r .h .s , of Eq. (13). I t means, that themonodromygroups of Fo, n and Fm, o determine

completely the monodromy of Fm, n. One is so in th is respect lead to the considera-

t ion of two seperate SU(2)-l ike s i tuat ions. The analysis of closed operator algebras

in the c < I systems can in fact be deduced step by step from the corresponding

analysis in the SU(2) model ( /16/ ,~/21/) .

Another in terest ing connection between s ta t i s t i ca l systems with c < I and the

SU(2) model has been observed by Gepner /22/. He notes that the characters of the

highest weight SU(2) Kac-Moody representations transform under the modular group very

s i m i l i a r l y as the characters of the highest weight Virasoro representations in the

c < I systems. Gepner uses the s i m i l i a r i t y to relate modular invar iant combinations

of SU(2) Kac-Moody characters to modular invar iant par t i t ion functions of the sta-

t i s t i c a l systems. The subject has fur ther been persued by Cappell i , Itzykson and Zuber

/23/.

S t i l l another aspect re la t ing the representation theory of the SU(2) Kac-Moody

algebra and the Virasoro algebra with c < I has been pointed out by Goddard, Kent

and Olive /24/: Start with a theory real is ing the tensor product of two SU(2) Kac-

Moody algebras (that i s , G L = G R = SU(2) ®SU(2)) with central charges k I and k 2 for

the two factors. Suppose that one can construct a sector in which the diagonal sub-

group of SU(2) ® SU(2) is annihi lated. In th is sector are real ised, i f k I and k 2 are

properly chosen, the highest weight representations of some c < I Virasoro algebra.

Goddard, Kent and Olive show that in th is way a l l representations of a l l c < I uni-

tary models can be reached. There should be a conceptual l ink between the algebraic

Goddard-Kent-Olive pmcedure and the above mentioned f i e l d theoret ical relat ions which

has to be found.

The Dotsenko-Fateev Coulomb gas construction is easi ly supersymmetrized by adding

a free Majorana f i e l d . The unitary theories with N=I supersymmetry have been worked

out along these l ines in re f . /8 / . The previous remark on monodromy groups applies

also in th is case.

I f i n a l l y mention the works of Fateev and Zamolodchikov /11/ and of Qiu /10/. I t

was shown in re f . /11/ that theories with an underlying parafermion algebra can be

mapped isomorphically on the SU(2) WZW model. Qiu /10/ found recently the operator

solution of the degenerate c r i t i c a l systems with N=2 supersymmetry in terms of ope-

ratorsof parafermion theories. The N=2 theories are therefore also by the isomorphism

of Fateev and Zamolodchikov related to the SU(2) WZW model.

178

References

/ I / A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl.Phys. B241 (1984) 333.

/ 2/ J.L. Cardy, in phase Transitions and Cri t ical Phenomena vol. 11, ed. C.Domb and

J.L. Lebowitz (Academic Press: New York, 1986), and references therein.

/ 3/ D. Friedan, Z. Qiu and S.H. Shenker, in Vertex Operators in Mathematics and

Physics, eds. J. Lepowsky et al . (Springer 1984); Phys. Rev. Lett. 52 (1984)

1575.

/ 4/ G.F. Andrews, R.J. Baxter and P.I. Forrester, J. Stat. Phys. 35 (1984) 193.

/ 5/ D.A. Huse, Phys. Rev. B 30 (1984) 3908.

/ 6/ V.S. Dotsenko and F.A. Fateev, Nucl. Phys. B240 (1984) 312.

/ 7/ D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575.

/ 8/ M. Bershadsky, V. Knizhnik and M Teitelman, Phys. Lett.151b (1985) 31.

/ 9/ W. Boucher, D. Friedan and A. Kent, Phys. Lett. B172(1986) 316;

P. di Vecchia, J.L. Petersen and H.B. Zheng, Phys. Lett. 174B (1986) 280;

S. Nam, Phys. Lett. B172 (1986) 323.

/10/ Z. Qiu, Inst i tu te for Advanced Study, Princeton preprint (October, 1986).

/11/ A.B. Zamolodchikov and V.A. Fateev, Zh. Eksp. Teor. Fiz 89 (1985) 380, (Sov.

Phys. JETP 62, 215).

/12/ E. Witten, Comm. Math. Phys. 92 (1984) 455.

/13/ V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83.

/14/ V.A. Fateev and A.B. Zamolodchikov, Yad. Fiz. 43 (1986) 75.

/15/ P. Christe and R. Flume, Nucl. Phys. B282 (1987) 466.

/16/ P. Christe and R. Flume, Phys. Lett . B (1987) in press.

/17/ D. Gepner and E. Witten, Princeton University preprint (April 1986).

/18/ V.G. Ka¢, In f in i te - dimensional Lie algebra - an Introduction (Birkh~user,

1983),

/19/ P. Goddard and D. Olive, in Vertex Operators in Mathematics and Physics,

eds. J. Lepowsky et a l . (Springer 1984).

/20/ R. Dashen and Y. Frishman, Phys.Rev. D11 (1975) 2781.

/21/ P. Christe, work in preparation.

/22/ D. Gepner, Princeton University preprint (1986).

/23/ A. Cappelli, C. Itzykson and J.-B. Zuber, Saclay preprint PhT 86/122 (1986).

/24/ P. Goddard, A. Kent and D. Olive, Comm. Math. Phys. 103 (1986) 195.

THE TWO-DIMENSIONAL O(n) NONLINEAR o'-MODEL FROM A WILSON RENORMALIZATION GROUP VIEWPOINT

P.K. MITTER and T.R. RFIMADAS Laboratoire de Physique Th~orique et Hautes Energies

Universit6 Pierre et Marie Curie, Paris

* Laboratoira as~ci~ au CNR$ UA 280

and

School of Mathematics, T~a Institute of Fundamental Research Homi Bhabha Road, Bombay 400 005

1 ° INTRODUCTION

Consider the O(n) nonlinear o'-model in two (euclidean) dimensions. With a

latt ice cut off -on a latt ice of spacing a- the theory is defined by Rn-valued, spins ?on a latt ice with the constraint ~ l ~ = f Z'(~) and the bare action

= ,

It is believed that (Polyakov [1], Brezin et al [2,3])

i) The theory has a unique phase for all valness of Z,g. This phase has exponentially decreasing correlation functions.

i i ) A scaling l imi t of the theory exists as a continuum field theory wi th a mass-gap, Provided that as a -~ 0 we let g(a) ~ 0 and Z(a) ~ 0 at rates computable in

@

perturbation theory.

It is clearly important, to see what Wilson [4] renormalization group techniques or phase cell expansions (Glimm-Jaffe [5]) so successfully applied to renormalizable asymptotically free 4-fermionic theories [15,16] in 2-d (Gawedski-Kupiainen [6], Feldman et al [7]) can tell us about the 2-d non-linear ~r-model. But here we immediately face the problem that the simplest and most

appealing renormalization group transformations destroy the ~-function constraint

in the model. Here we adopt the prescription : drop the constraint I ~ I z I and take as bare action ~('~)

180

L

(1.2)

Then for large 9, we are approximating the or-model. The idea is to let ~k go to inf inity as a -~ O. Of course ~. has to go to inf ini ty at some minimal rate, and this wi l l be a crucial concern below.

We have three sets of results to report. The f i rs t (section 2) is a perturbative computation implementing the above idea. After applying a magnetic field to break the vacuum degeneracy and to get perturbation theory started, we show that up to one loop we can recover the usual perturbative results, including the renorma]ization constants provided ~ goes to inf inity at a certain minimal rate. In fact this rate is given by :

X- - ;L

(1.3)

where ~. is an arbitrary positive dimensionless constant and ~(a) is given by the 2-loop asymptotic freedom formula.

In Section 3 we try to understand the success of this perturbative computation by studying UV cutoff removal using Wilson RG transformations. We use Pauli-Villars regularization and a continuous Wilson RG which we study in a "local approximation" (this is the f i rst step of a more complete study). We show that under RG flow (and up to O(g2)) the parameter ; t is driven to a fixed point. We show how asymptotic freedom is recovered, and show how to compute the renormalized trajectory in weak coupling in g2(M) the running coupling constant. Then g2 turns out to be the only "marginal" variable, all other parameters contracting to fixed point values (there are no "relevant" parameters). This is completely consistent with the previous perturbative computation except that we miss out the wave function renormalization due to the local approximation.

Our third set of results (section 4) is from a study of the Wilson approximate recursion [ 14"~,#~ r l~] ] for the model. This can also be interpreted as the renormalization group transformation for a certain hierarchical version of the model. This yields a fixed point in the variable ~ but not in ~L. The theory can be renormalized around this fixed point, and would be asymptotically free. At the present however, we believe that this fixed point is an artif ice of the hierarchical recursion we have chosen, since its existence in the full model would cause problems.

181

This work was begun in January 1986. In the meanwhile Gawedzki-Kupiainen have published (Gawedzki-Kupiainen [13]) a construction of the o'-model in a

different hierarchical version. The picture we have presented in Section 3 is in qualitative agreement with theirs.

We thank Sourendu Gupta for help with numerical computations and Giovanni Felder for a discussion of his work on the local approximation to the Renormalization Group. P.K. Mitter thanks the Tata Institute of Fundamental Research and the National Board of Higher Mathematics (India) for hospitality. T.R. Ramadas thanks the LPTHE, Universit@ Paris Vt for hospitality.

2" THE PERTURBATION THEORY N

Consider the ac t ion (w i th b o = e r e~ ~- Tr ~ ,'1". e~ = o )

S A = +1 o 191 - t

-e L 2 I z I 2.

(2.1)

2 and go are Here A is a momentum cut-off and the parameters Z, t~ 2, m~. 2 ~ ~ (for fixed A ) this A -dependent in a way to be determined. If we let m o-

becomes the o'-model with a wave-function renormalization Z = 2g~ and the addition of a magnetic field (the mass term #~ I~1 z becomes in this l imi t a constant). We are, however, going to let rn2~ ~ ~ along with the cut-off. Note that,

in comparing (1.2) and (2.1)

, v Z Z

(In particular, m 2 = 4 A ~" 2,. U"

Define e = o - + ~o~,/=.

in terms of (t.3)). we have then (with m¢--2 _ m2+ p#)

~- (2.3)

We can now study the perturbation theory in go We summarise the (one-loop)

computations in figures 1,2 and 3. (The computations were done with ~=I and the

182

Green's functions then rescaled. Thus in the figures Z=I).

2 / A 2 )) 1. We also assume The computations are done under the assumption m~. ~t 2 / A 2 << 1 (and this is consistent with the way i1 o gets renormalized). The effect

of the f i rs t assumption is to "contract the o'-lines" so that, for example

6 -

we obtain then (upto 0 ), for the Fourier transform of the "pion" two point function <'-rrC,). -~-(~)~> ~- , ,,E ~- ~,-me,-,4"~,~

>,(o)-- , -{ :L- :c^. r J

where ~ 2 k

° I (_ A k ~-ro

and for the proper l-pion irreducible four point function at O-momentum

~:~ (°b = -po (-.-,)~+,)[~* ~-

(2.3)

(2.4)

Note that the quadratically divergent diagrams have cancelled, leaving only logarithmic divergences.

If we take

, r o = E'/,

~,= ~_ - &-2) g' _T.̂ ,~, , v 2.

7= ~_g-r

% /-

(2.5)

we get finiteness of both (0) = Z21~4~'=i(0) and~{2)(o). This agrees with the computations o:r Br~zin-Zinn Justin [2].

In fact the conditon ~ z >>:t is not necessary. As long as ~ is a

positive constant we obtain the same physical results, with only f inite changes in the renormalization constants.

183

3 ° THE WILSON RENORMALIZATION GROUP 'AND CUT OFF REMOVAL

Our starting point is the bare action-

(3.1)

(3.2)

Here A N = L N (in mass units) is the U.V. cut off,'i.e. A N -* 0% L > t. We have

H - - b ~

introduced a Pauli-Vi l lars cut off. ~ o is a free dimensionless parameter (the

dimensionless o'-mass). In Section 2 we have seen that by adding a magnetic field

(to avoid perturbation theory I.R. divergences) and choosing Z(N), ~(N) in the

standard way [2] , we recover i- loop non-linear or-model results. The fact that the cutoff removal is achieved by lett ing the dimensional mass 2 ~ - A 2 ~o with d'- ~1

~-o arbitrary immediately suggests (by virtue of our knowledge of cr i t ical

phenomena due to Wilson [4,8]) that under the RG the dimensionless parameter :;L is driven upto O(g 2) to a fixed point ("infra-red stable"). In this sction we wi l l

veri fy this point as well as how asymptotic freedom is obtained by making some approximations to the ful l Wilson RG.

Remark We shall work with the symmetric form of the theory i.e. start from (3.t), (3.2). To avoid I.R. divergences we should put a space-time volume cutoff. However as we shall see (and as is well-known) in the Wilson RG transformation no I.R. divergences are met. Hence the volume cutoff is ignored, although i t is not d i f f icu l t to keep track of i t (F.J. Wegner [9]). All parameters in (3.2) are dimensionless. Hence by resca]ing

0

where

I ~.. Z ~, 2. I ~2. .

The free part now has a "unit cutoff" and Z, go carry the cutoff dependence.

(3.4)

184

Hold the points of (3.3) non-coincident ' Ix i - xjl >> L -M (M fixed) L M will be our

reference momentum scale. In the Wilson method [4] at each RG step we lower the

cutoff from I to L-11 and rescale back to I. After (N-M) steps. We get '

. . . . = < . . . . . . (3.5) N - P l

"-V-d. eN) "V" ¢~) (3.6) --~ %8. ~-I

In (3.5) we have ignored a slowly varying f in i te (N-independent) momentum dependent factor. The idea is now to choose Z(N), go(N) such that

-/;-rn ~/-M -- ¢ ,,'~ ~-s N--> ~o (3.7)

in which case V M is the renormalized trajectory (L M is the renormalization scale).

For our purposes here we adopt a continuous form of the Wilson RG (Wilson-Kogut

[4], Wegner [9]). We lower the cutoff continuously from I to e-tl and scale back to I. Starting from t = 0 we integrate out to tN_M=(N-M) log L and the renormalized

t rajectory is

Under the continuous Wilson RG, Vo(N) --* vt(N) given by •

_ , _ % ¢ - ) .

(3.8)

(3.9)

where ~c~ is a Gaussian measure wi th covariance •

C h [ d-2~ e

k ~ (3. I O)

Note that the hard propagator C~ has n_£ I.R. divergences, so that in (3.9) no I.R.

divergences occur. From (3.9-3.10) we easily drive the non-linear functional di f ferential equation, (Wilson [4], Wegner [9], see also Po]chlnski [10])

185

9k

3/" (~) } "n ( z _ ~ (N )

(3.11)

K (*-~) = el_~. ~ e e (3.12)

Because of the non-local kernel K(x-y), the third term in (3.11) generates non-local contributions to the effective potential vt(N). This is also the source of

the wave function renormalization. As a f i rs t step in the ful l analysis we make the local approximation (for a recent study, see Felder [11]) to the RG equation (3.11 ). This consists in replacing K(x-y) where i t s i ts in the third term of (3. I 1 ) by

(2) (x-y). This has the disavantage that we miss out the wave function renormalization which wi l l be recovered by going beyond the local approximation in the second step of the analysis. However i t has the great advantage of giving a purely local evolution under the RG which can be throughly analyzed and gives a very interesting picture of what is essentiall ly going on.

In the above local approximation the evolved potential V t remains local (we

drop the superscript "N" which refers to the ini t ial condition)

and calling I ~ (=) = q"

I /I " I " ) +

(3.14)

where v't, v" t are f i rs t and second derivatives with respect to "r' . This is a 2

-dimensional non-linear partial differential equation. In the local approximation, no wave functional renormalization is generated and hence we can set Z = 1, so that (from (3.4)) our in i t ia l local potentia! i s

186

I Here _i~. is the minimum of Vo( ,r ) (its dependence on the UV cutoff N wil be

fixed later) and ~og 2 is the curvature at the minimum which defines ;L o In

the following :~ o is an N-independent free positive parameter. We now define ~.~

as the minimum of the evolved potential vt(~'). Define

^ J ) (3.16)

Then ~t has minimum at ?' = O. The differential equation (3.14) gives

A ~-_.-vk = 2"u-~ +

/:t D ,__ ^, (3.17)

Taking another derivative of (3.i7) w.r. to O/t(O) = 0 gives us-

"r" , and demanding that

d J e = - , _ (3.18)

Let us expand out v t as a power series in '7', upto O( "r ).

= t + ) ~m 2 5}

(3.19)

The linear term is always absent because of (3.18). We need to go upto the ~' ~ term to get the asymptotic freedom formula upto order (g2)3 (i.e. 2 loops) correctly. We have nevertheless kept the @,6 term to show that i t and (also beyond) does not affect the renormalization analysis,(3.18) can now be writ ten :

1 8 7

(3.20)

From (3.17), (3.19) and using (3.20) we get the following flow equations for the parameters in (3.19) '

~t Zq ~ (3.21)

(3.22)

5! ,,___~ ~ - ~ o ~ t ~ ~ ~ o(a~ ) 3

(3.24)

Finally,

(3.25)

3 To obtain the flow (3.20) for (g2 t) exactly to order (g't) , (2-loop information), we

need Wt, ~ exactly to order g2 t. From (3.21-3.22) we see that we need ~ t,

"~t to 0(1), and Pt (the "r' 6 term) plays no role for this part of the calculation.

Let ~. t, Wt, t , "~ t , ~t denote the solutions of the approximate flow

equations obtained from (3.2i-3.25) by putting g2 = 0. As t --~ ~, these parameters tend to fixed point values ( 2,÷, W~, ~ .,, -} ~, 9~).

-• = - I . --" - ~ i _ _ In particular ~, = I ~ ~ g ~ ,~ q ^

This for large t we can replace (3.21-3.22) by,

188

Z 2.

(3.26)

where a,, b, contain the above fixed point information. From (3.26) i t is easy to see

tha for large t

A 7-

Plugging in (3.27) in (3.20) for large t we obtain.

(3.27)

z. 2 3

b 2~ (3.28)

The discrepancy in the leading term from its true value (n-2)/(211) may be attributed to the local approximation.

The init ial value - g2 (N) is now fixed by taking the two loop formula

(3.29)

and demanding that for tN_ M = (N-M) log L g2N_ M is held fixed as g2M_

(normalization condition at L M, M fixed). With this choice of g2(N) one obtains that the solution g2N_ M of (3.28) (with three loop and higher terms) stabilize as N i-b ~,

to g2 M < ~. This phenomenon is well known (P.K. Mitter learnt this in 1978 from the

late K. Symanzik in the context of Creutz' simulations). The init ial g%(N) being thus fixed, we go back to (3.26) and see that for large t ( - (N-M) log L), (3.26) settles down to

_..1:2,

i f 2 Z

d b (3.3oi

189

Letting now t = tN_ M = (N-H) log L ~ ~, we see that ~ t, wt approach "fixed

N --~ c~

points" A

A similar discussion holds for (3.23-3.25)

(3.31 )

As a consequence we obtain as N -b~ , the renormalized t ra jectory

(3 32)

where

A

-z z ~.3

~)s~, 6) (3.33) &

where •.(M) =,~. + O(g2) etc.

We note that g is the only margina} variable, and otherwise all other parameters contract to fixed point values under RG.

The above discussion is clearly deficient since it is based on the "local approximation" to the RG : thus the wave function renormalization is missed out, and the numerical coefficients in the asymptotic freedom formula are inaccurate. Moreover (3.33) is just a weak coupling expansion (in gM ) presumably valid for

small fields. Nevertheless it explains (we hope clearly) the mechanisms at work. A more detailed study wi l l be published elsewhere [12]. The above picture that we have derived is similar to (but not all details are the same as) that obtained by Gawedski and Kupiainen [13] in a study of a hierarchichal model.

4 ° THE WILSON APPROXIMATE RECURSION FORMULA

This recursion is co -~ co' where

r - e = C.~sb ~ e J

(4.1t

190

and L > 1. This can be regarded either as an approximation to the full renormalization group transformation as in [4] or as the renormalization group transformation for a certain hierarchical model. Note that i t differs from the recursion used by Gawedzki-Kupainen[13]. While comparing with our earlier discussion the identification that is made is

(i) We choose Z-1 (in the hieararchical approximation there is no true wave-function renorm al i zation)

(ii) --> % U¢ V, where by ~ oWe denote the initial

point for the recursion.

We shall consider O(n) invariant co. This ondition is clearly preserved by the

recursion. We write

A

where O )0, v(O) = 0 (this defines g if v has a unique minimum which is taken at the

origin -this wil l be the case in our approximate considerations. In general a more rigorous definition of g wil l have to be given). Then the recursion (4. I) become ~

( ) e : a s E "E e T 5 T e TO~ •

I~, 2 Z

where E - ~ (~1)~ ,and (og,]~ is the minimum of co'

We can put g = 0 in the above recursion keeping 6 fixed, to obtain

- "n. %" ~@)L r~.,~ f ~ - _ ..~_, e = ~ s e z 5'z '. J J

(4.4)

191

We have proved (see the Appendix) that for n=3, there exists an ~=6 ~>0 such

that this recursion possesses a fixed point - ~3" - This proof, however, gives very

l i t t le information. A saddle point computation shows that in the n = ~ l imi t the above transformation has a fixed point v ~ , with E = 1. This is done closely

following Ma [14] and yields ~__ v~ ~- ~-~ where f is the function

2 ~ = o L z('~+O-l-Zu. (4.5)

A ful l analysis of the scaling fields is possible and shows that this fixed point is at t ract ive.

i~ I

^~ Figure 4 - A sketch of v=~

We have also studied (4.4) (for n=3) numerically. This confirms the n =

picture.

We have studied the recursions (4.3) and (4.4) to order 1 In. We get

~ J(4 .6) I'/" ^ , (z:~

+ o

^ ~ (,'-) ith ^ where v~ denotes the derivative of v:~. Continuing this way one gets what

we believe is an asymptotic series for £~. Note, howeve that we do not expect an exact formula for ~ ~

192

One might attempt to take a scaling l imit of the hierarchical theory by letting

.i~l ~- A i- 2

_ ,

(4.7)

(4.8)

where 2~*is the coefficient of the expansion of v n around O, i.e., A~ ~-~ (~') = ~ "~ + 0 C"l"~ In fact this programme can be carried

out to order 1/n. However, note that the coefficients of the asymptotic freedom formula refer to the fixed point v n and are hence not exactly known.

Finally we remark that in the variable 2,_ (cf., [1.3]) we do not find a fixed point. This is in contrast to sections 2 and 3 and points to the fact that in two dimensions hierarchical models cannot be wholly trusted.

REFERENCES

[1 ] A.M. Polyakov, Phys. Lett. 59 B, 79 (1975).

[2] E. Br~.zin and J. Zinn-Justin, Phys. Rev. B ~ 3110 (1976).

[3] E. Br~zin, J.C. Le Guil]ou and J. Zinn-Justin, Phys. Rev. D ill, 2615 (1976).

[4] K.G. Wilson and J. Kogut, Phys. Rep. 12 C, 75 (1974).

[5] J. Glimm and A. Jaffe, Fortschritte der Physik 21. 327 (1973).

[6] K. Gawedzki and A. Kupiainen, Comm. Math. Phys. 102, 1 (1985).

[7] J. Feldman, J. Magnen, V. Rivasseau, R. 5~n~or, Comm. Math. Phys. 103, 67(1986).

[8] K.G. Wilson, Phys. Rev. ~ 291 i (i973).

[9] F.J. Wegner, in "Phase Transitions and Critical Phenomena, vol. 6. Eds. C. Domb and M.5. Green, Acad. Press, London, New-York i 976.

[ 1 O] J. Polchinski, Nucl. Phys. B231,269 (1984).

[11] G. Felder, IHES, Bures-sur-Yvette, preprint 1986.

193

[12] P.K. Mi t ter and T.R. Ramadas, in preparation.

[i 3] K. Gawedzki and A. Kupiainen, Comm. Math. Phys. i06, 533 (1986).

[i 4] S. Ma, "Modern Theory of Critical Phenomena, New-York, Benjamin 1976.

[i5] P.K. Mitter and P.H. Weisz, Phys. Rev. D8, 44i0 (1973).

[ 16] D. Gross and A. Neveu, Phys. Rev. D i O, 3235 (i 974).

I I

~ , 7. q7 . ,-" •

LI a"d'o I .7,,~Z ~- .*" _L % ~" %

/ / '~.

FiqUFe I - The. basic pFopagator and wTrtice.s

,'3 I ~- + ( ~ _ , ) _ _ _ - -

2. 2 (~-,~ ___E~.--

_'_3"____

~ ~C~- " ) - - - ~ - - -

F~qure 2 - The combinatories for the "pion"2pt. function

194

F/oure 3 - The I - ~ irreducible proper 4point function. (external legs deleted everywhere, but so indicated only in the firs diaqram.

= - '','I.<> - , )+ ,>&- . ,,, + _ ~.~,, & , , ] - +~&_ , :,,

+ ~ , , z (2 ~ - O + ( z + z ) ( . , , - , ) ~ ,,, , - . , , ,, , a. '° (.,.,.,) } , , × . . . . x , . . "+

/

<,

. - " . . II, ~L I

"m ~4 i4 % - - \ /

4

- ~ ~° ~"~,-,~'+~'E. ~ "~ ", ,' -i ",,..~-, / _ 3 ~ 4

/

- ~ - - - c-<~" '~+~ ~ ~ . , , . . . ~ . . . - " ' ~ ° (~'c.-,) '+£~-.-,)) ' ' I G 1 Z ' l {>'#.. 2 / ' / ,,

\

, ~ - ~0 ,<£, (.,,-,),,-z,,3 ~- , 16 x 5~ . . " " , : i(o,cL I - . ,...

$ i'l ( ix ~" / ! 4 -i (. z . ,~- , , .., / +..,:j_.~: ~_,~'+~.,<..-o) ]:::-.{ + ~ , - ~ o ( . ~ _ j + ~ . ~ . - , ~ ) . , . > _ ~ - . ~ . 1 6 x4 ~,

/

, , , ~ (2~,~c,.o.2,~Co-,) ., . . . , . ~ ) . - .4. '+ - ' ' ' 1 ° ' ,- , (~,<~#-,.)+ . ". .. 16~ 14/ ~" " " ~ 16~llt - - e 2q~, ~ ? , . . ~ _ n ) ) / , / ,

/

+ ~ , z ' & - , ) . , 3 , . 2 ~,-, ", - " ~ o ( ~ c-,,-o+ ,, " ' I~,, '~. ~ , " ~ , ~>,,z ' , .~ ~ ? ( ' , , - 0 ) / ',,,

_ ,.~ , t I_4 °

,,=,<----~ . " " . ,~. - + ~.< ~ , - o ) . . " "..

195

APPENDIX The recursion (4.4) for n = 3, L =2.

A

§1. We denote g(~r) = e LI . Then (4.4) becomes

~I. I~ _~Is (A-I)

If we define s = e = e 2 we get [c = "varying constant"]

~5 ~ L4 (A-2)

Where ~ = e . We shall now change the def ini t ion of c and ~ in an e

inessential way. Note that ft is a decreasing function, f~ -~ 0 as x ~ oo, and

_r'/,,=_cfl'-ac'(")./~). Wedef inec s.t. f '(O)= 1 and ~ s.t. fl(1) = a

for some 0 < a < 1 f ixed once and for all.

Let K >> 1, and define

~--( :f~ C ~o,K] ,{(o)=.l , :r('~L)= o , 3C non increasing)

This is a closed convex set in c[0,1 ]

t

§2. We define a transformation f ~ Tf = f on ~ . This is done by regarding f

as a function on [0 ,~ , performing the t ransformat ion f --* f~ as in S 1, and defining f

= ?(K~'~ where 2(K is the characterist ic function of [O,K].

~3. The Leray-Schauder Fixed Point Theorem applies to the transformation T, and we obtain a f ixed point f~

K

get a priori estimates on ~K ~ and 3C~ $4. We

196

(A-3)

~';, c~) ~_ ~-'/~

We can now get a fixed point of (A-2) by taking a weak limit-of the fK

§5. We have very l i t t le information about the fixed point g*. We can, however show lira inf g.*(x) = 0

X ---~ o o

NONLINEAR ~ -MODELS WITH BOUNDARY AND OPEN STRINGS

H. Dorn, H. J. Otto

Sektion Physik, Humboldt-Universit~t Berlin, DDR

InvalidenstraBe 42, Berlin 10#0

Abstract:

We present a short review of some psrturbative results concerning

the renormalization of nonlinear ~ -models with boundary relevant to

the description of open bosonic strings. Special emphasis is given

to the so-called ~ -function approach to derive effective actions for

the massless excitations of the string. We comment also on the correct

definition of the 2 dimensional stress tensor in the presence of anti-

symmetric tensor field background.

I. Introduction

Relativistic strings describe as its excitations an infinite tower

of usual point particles. The corresponding mass scale is the Planck

mass ~/ i . Usual physics appears to be that of the massless exci-

tations. Therefore, one is interested in an effective action for the

massless fields derived via an o<' expansion from string theory. In

this sense string theory can provide a guiding principle to reach

consistent anomaly free unified field theories. Typical string induced

effects in such theories are e. g. the Chern-Simons completion of the

field strength of the antisymmetric tensor field which is crucial for

anomaly concellation / I /, higher curvature terms for gravity and its

supersymmetric extensions / 2 /, and structures of nonlinear electro-

dynamics and its non-Abelian generalization / 3 /. There are several

methods to derive the effective action, and within the inherent ambi-

guities due to renormalization scheme dependence and/or the freedom to

redefine the fields all of them yield the same result / 4, 9 /.

In this talk we restrict ourselves to the # -function approach which

is based on the requirement of conformal invariance of the generalized

nonlinear [ -model describing the motion of the string in background

fields corresponding to its own massless excitations. This procedure

we demonstrate for open bosonic strings / 6, 7 /. The limitation to

198

bosonic strings is made for simplicity. Although closed strings

appear to have better phenomenological prospects open strings still

are not ruled out and in particular due to its simple coupling mech-

anism to background gauge fields allow to some extent statements in

arbitrary order of ~' / 7, 8 /. They seem to be also a suitable

testing ground for ideas concerning an interference of usual ultra-

violet divergencies and divergencies emerging in limiting cases of

the topology of the string world surfaces / 9, qO /.

Our g-model describes an open string x ~ ( z ) in the background

fields corresponding to the massless excitations of open and closed

strings: gauge field A~ (x), gravitational field G~v (x), anti-

symmetric tensor field B~ (x) and dilaton ~ (x).

g = SM + g~

,L6 ~7~v I ~ x'~ ~ + o< s ~ (~

(1)

~ is the 2 dimensional metric, R (2) the corresponding curvature

scalar and k (s) the external curvature of the boundary ~M . (We

consider ~ positive definite. To reach the pseudo-Riemannian

(Minkowski) case by analytic continuation B~ , G~v and A~ have to

be real.) In the Abelian case S~M describes the coupling of an ex-

ternal gauge field to opposite charges sitting at the ends of the

string. The generalization to Yang-Mills fields naturally leads to the Wilson loop. The term including k (s) ensures the correct coup-

ling of the dilaton ~ (x) to the Eulsr characteristics / 10 /. From

the pure ~ -model point of view it is quite natural to couple x ~ (z)

to both open (A~) and closed (G, B, ~ ) string excitations. This

coupling is also justified within the string picture / qO /.

The rest of the talk is organized as follows. In section 2 we

199

scetch the relation between conformal invariance and renormalization

group ~ -functions / 11 - 13 /. Section 3 reviews our own calculation

of ~ -functions in lowest orders / 6 /. Section 4 discusses the prog- ress made in summing the ~' series / 7, 8 / and the attempt to in-

clude effects of higher topologies of the parameter domain M (string

loops) / 9, 10 /. Finally, section 5 gives a preliminary account of

some results concerning the definition of the 2 dimensional stress

tensor in the presence of B~ (~).

2. ~ -functions and conformal invariance

We treat the renormalization of the nonlinear g -model given by

(I) in the generalized sense of Friedan / 14 /, i. e. all counter-

terms are classified according to their two and D-dimensional

structure and e. g.

M ~ ~

is interpreted as a renorm~lization of the D-dimensional metric G~

and the gauge field A~ , respectively. Then one defines (/~ renormali-

zation scale)

A ~

with corresponding eqs. for ~8£a, ~ " and #£ . Due to two and D-

dimensional covariance the trace of the 2 dimensional stress tensor

has the structure

+ gauge field contribution +

+ terms vanishing on shell (3)

+ ... + nonlocal terms

~4

with coefficients ~ introduced independent of ~ at this moment. Be-

sides these two kinds of operator ~ -functions ~ (x), ~ (x) it is

sometimes useful to consider ~ (x) depending on a background con-

figuration ~ (z) / 15 /.

2OO

The most detailed discussion of the relation between ~, ~ , ~ is

given in refs. / 11, 13 /

with the field strength for B~v given by

The terms containing the quantities W and L are due to operator

mixing with total derivative terms and vanish up to 2 loop order

/ 11, 13 /. The remaining terms in ~-~ involve the dilaton field

and are due to the explicit breaking of the Weyl invariance

~_~(~)S~(~) by the dilaton coupling in (I).

Consistency of the string theory requires Tmm = 0 as an operator

statement. The resulting conditions ~=~+= ~ B = 0 can be inter-

preted as equations of motions for the background fields.

The whole procedure up to now has been performed for closed

strings only. We don't know how to include the gauge field contribu-

tion into eq. (3). The main obstacle is the independence of the gauge

field coupling in (I) of the 2-dimensional metric. Therefore, for the

time being we only can assume that there is a boundary contribution

to Tram parametrized by some #~ = ~ ~ ~ . An obvious source

for a contribution to ~ ~ is a boundary term due to the dilaton

part of the action. To some extent the situation resembles that of

the B~v part in the stress tensor. The classical stress tensor does

not depend on B~v , but there is a B~v -part in the trace anomaly

(3), (5). The derivation in refs. / 11 - 13 / is based on dimensional

regularization. As we shall see in section ~ there are problems con-

cerning the application of dimensional regularization in the pre-

sence of an ~ ~v term. Hence further study is required to per-

form a regularization scheme independent analysis.

201

~, Lowest order calculations

The # -functions for the fields of the excitations of the closed

string are / q6, I~ =, 13 /

~ ~'( R~.,, - .s?.?,~ g,,~'~ ) + 0(~ '~)

= - ~ .sx/,,,, + o ¢ ~ ' ~ ) ( 7 )

Calculations to the next order have been done in refs. / 17, q3 / and

in the supersymmetric case up to # and 5 loops / 18 /.

We illustrate the method of calculation by a short review of ref.

/ 6 / where the standard treatment of ~ -models without boundary has

been extended to the boundary case. The main ingredients are the

background field method, the use of Riemann normal coordinates and a

careful treatment of boundary terms' The calculation has been per-

formed neglecting @ for G, B, A background at I loops, pure A back-

ground at 2 loop and Abelian A at 3 loop level.

Making a background-quantum split x ~ ~- ~ x ~ + y~ and introduc-

ing the normal coordinates :~

we get

H

+ term linear in ~ .

(9)

D~ , D~ are the covariant derivatives corresponding to the con-

nections ~ ~ ~ +~ J~. ~ respectively. D~ , D~ are their 2

202

The expansion of the Wilson loop is based on

- ' F A x /~

I 'c=~

AS propagator for the quantum field ~ A = V A ~ where

D-dim. vielbein we choose

(lo)

is the

| I ~A(~) ,~ ~ , j -__ ;'n-<~' ~AB ~ & , £ ) (11)

with N denoting the Neumann function for M. Then we get for the ef-

fective action at I loop order

- r ~ = ~t J a ~ A/c~,~ L ( ~ - - W ~ % / . ~ ) s ~" ~-~ D~S/.~ ~ " ] ~ ~ t,f

D (A' P ) is the covariant derivative with respect to gravitational and

gauge field background. The ultraviolet divergence of the integral

over M will be cancelled by G~v and B~ renormalization leading to

the first two lines of eq. (7). The boundary terms should be related

to A/~ renormalization. However, first a comment on the relation

between B~ and abelian F~ is in order. If B M = 0 the model is

invariant with respect to B ---~B + dA . This invariance is lost

in the presence of a boundary. The action S M now is sensible to pure

gauge contributions to B and these cannot be distinguished from

S~M with abelian A. With respect to A,~ renormalization the terms

proportional to the normal derivative pose some problems. They are

absent if S = 0, then we get for nonabelian A

D ~ ° - 4 + #4 =-~' o e'~)~ F~x+ O(~'~) (13)

203

and for abelian A

If S ~ 0 one can try to eliminate the normal derivative term by the

use of the boundary condition for the background configuration. This

is straigthforward for abelian A only

leading to

, -~-~ 8/~x ) -

,< ~ S~e " ~ ~ .~ ) * . . . - ~ . (F.~+~.~, ~.~) ( & ~ + ~ , (16)

For the case of pure gauge field background fields (G = B = ~ = O)

we find at 2 loop order

' o I - ~ - ~ ( 1 7 )

independent on the renormalization scheme and for abelian A at 3 loop

order

• a ,,g" F x - <,3 ('~'~r~,x 0(: < ' ' (18)

The equation ~ = 0 agrees at this order of ~' with the equation of

motion for the generalized Born-Infold action / 3 /

~ ¢ £ ( 4 4 2 [~ 'F ) J

4. Gauge field ~ -function to all orders in

loop corrections

!

and dual

Using in (11) instead of the usual Neumann function with constant

normal derivative a function which satisfies the boundary condition

(15) for constant F one finds at lowest order / 7 /

=-~ F'~'~ ( 4 - ~'~'~ ) "~" " (19)

204

This is equivalent to summing up all diagrams based on the old N and

selecting contributions to the first order in the derivative of F.

There are claims that (19) is correct for arbitrary abelian ? / 8 /.

The extension of any kind of these considerations to nonabelian A is

nontrivial since the noncommutativity introduces new sources for

ultraviolet divergencies which complicate the combinatorics in higher

orders.

The trick of ref. / 7 / also works for constant G, B, A, ~ back-

ground and yields for this more general case / 10 /

,2. Iro~ I , A ~ . V @

/3 _F.L ] (20)

Assuming as announced in section 2 that the difference between ~A k

and the corresponding Weyl anomaly coefficient ~ is only due to

explicit symmetry breaking by the dilaton coupling one gets / 10 /

(2q)

The conformal invariance requirement ~ x = 0 is equivalent to the A~

equation of motion for the action

On the other hand ~ @ ~ ~ ~s = = = 0 can be deduced as eqs. of motion

for

e [ ¢] (23)

Now the authors of ref. / 10 / make the crucial step to postulate the

total effective action as

= sclosed open (24) Seff eff + K'Seff

with an up to this point free parameter K. While leaving the equation

205

of motion for the gauge field A~ unchanged the change to Sef f mod-

ifies the eqs. for the other fields. In particular the gauge field

now can act as a source for gravity. This seems to be the most essen-

tial achievement of this approach since in the standard picture the

gauge field sitting at the boundary cannot influence the closed

string field ~ -functions. Further analysis reveals / 10 / this pure

-model recipe as the consequence of coupling usual ultraviolet sin-

gularities and singularities for a vanishing hole on the string world

sheet in the sense of ref. / 9 /.

~. A comment on the 2 dimensional stress tensor in the

presence of B~

Although one expects a boundary contribution containing ~A in the

trace anomaly (3) there is still no proof available. As already men-

tioned the main obstacle is the absence of 2 dimensional gab in the

corresponding coupling term in the action. Just the same situation we,

find for the coupling to B~v • Since the abelian gauge field via

Stokes theorem can be included as a pure gauge part into B~ it is

sufficient to study the B~ case. To test our technical means we do

this first for the closed string ~ M = O. We will find that even in

this situation the up to now used treatment via dimensional regular-

ization poses some serious problems.

To have a handle for a B~v contribution to the stress tensor we

use the canonical definition

(25)

We have dropped ~ and use flat gab" Eq. (25) agrees with the defini--

tion via ~/~ and the B~v dependence vanishes in 2 dimensions since

the tensor

pmnab e 6 ~ . . . . (26)

is annihilated by any factor antisymmetric in m, n. We can however

use the dimensional continuation of

206

in the evaluation of corresponding Feynman diagrams.

Since T ab of eq. (25) is not derived via a parametric differenti-

ation with respect to gab one is not sure whether there is a non-re-

normalization theorem. Of course it would be possible to construct a

pure formal recipe along this line of arguments by defining ~---~

a way to ensure the product rule in (27).

But to be on safer ground we decided to study the one loop re-

normalization of T ab explicitely. For the general dimension two op-

erator H~.~ (x) ~x~×v we found in arbitrary regularization

(28)

_ ~_ [ ~x ~- ~-~ (Y~ H~"o ~ R ~,~,~ v ~ -~ 31~ H ~'~ s ~ sLe)

where x is a background configuration. Special cases of this formula

with summed m, n are contained in ref. /13 /,too.

Applying this formula to T ab and using dimensional regularization

I I supplemented by relations derived from (27) like N(O) : ~ a-~

207

(~.,~p~pX~,~x v- ~-,,p~px~Om,~ ~) =

= ~ - ~ ) ( ~ s ~ * ~ ~-~ ~o~ % ~ )

and the convention

pmnab = 0 Amn

(29)

(30)

for any antisymmetric A~n not containing e tensors we find that the

expectation value of T a is made ultraviolet finite by G and B re-

normalization only.

< r ( ~ ° , ~ ° ) ~ > ~ = T 6 c ~ , ~ ) ( ~ ) -

- - ~ × a~, D~'&,r,, 4

(31)

0(~ ~) + nonlocal terms.

However, there is a finite renormalization of all components of Tab.

For the trace it is welcome ~ince

<rc~°, ~ ° ) ~ ~ : g [ ~ o ~ v (~v- ~ s ~ )

+ i ~ 9 ~ 6 ~ ~ ~ (32)

- ~ C ~ ~ ~ ~ &~" ) ]

fits into (3). The finite renormalization of the off diagonal ele-

ments of T ab in the case S $ 0 with its related interpretation as a

violation of 2 dimensional general covariance should be an artifact

of dimensional regularization. Just the same conclusion with an S • S

term but without the total derivative part has been reached in ref.

208

/ 19 / by an analysis of the conformal operator product expansion for two T's. Obviously, further investigations of the antisymmetric tensor

coupling in a renormalization scheme independent way are necessary

to exclude an unwanted anomaly in the off diagonal elements of T ab .

References

/ I / M. Green and J. H. Schwarz, Phys. Lett. 149 B (1984) 117

/ 2 / B. Zwiebach, Phys. Lett. 156 B (198~) 315

/ 3 / Eo S. Pradkin and A. A° Tseytlin, Phys. Lett. 163 B (198~) 123

/ 4 / A. A° Tseytlin, Phys. Lett. 176 B (1986) 92

/ 5 / H. Dorn and H.-J. Otto, Proc. Symp. Ahrenshoop 86, Berlin - Zeuthen, IfH PHE 87-

/ 6 / H. Dorn and H.-J. Otto, Z. Phys. C 32 (1986) 599

/ 7 / A. Abouelsaood, C. G. Callan, C. R. Nappi and S. A. Yost, Princeton prepr. (1986)

/ 8 / E. Bergshoeff, E. Sezgin, C. N. Pope and P. K. Townsend, Trieste prepr. (1986) C. N, Pope, talk at this workshop

/ 9 / W. Pischler and L. Susskind, Phys. Left. 173 B (1986) 262

/10 / C. G. Callan, C, Lovelace, C. R. Nappi and S. A. Yost, Princeton prepr. (1986)

/11 / A. A. Tseytlin, Phys. Lett. 178 B (1986) 34

/12 / G, Curci and G. Paffuti, CERN prepr. TH 4558/86 G. M. Shore, Bern prepr. BUTP-86/16

/13 / A. A. Tseytlin, Lebedev prepr. 342 (1986)

/14 / D. Priedan, Phys. Rev. Lett. 45 (1980) 1057 and Ann. Phys. 163 (1985) 318

/q5 / C. G. Callan. E. Hartinec, M. J. Perry and D. ~riedan, Nucl. Phys. B 262 (1985) 593

/16 / E. Braaten, T. L. Curtright and C. K. Zachos, Nucl. Phys. B 260 (q985) 630

/17 / B. Fridling and A. E. M. van de Ven, Nucl. Phys. B 268 (1986) 719

/18 / ~. T. Grisaru, A. E. M. van de Ven and D. Zanon, Nucl. Phys. B 277 (1986) 388, 409 M. T. Grisaru, D. I. Kazakov and D. Zanon, Harvard Univ. pre- pr. HUTP-86 A 076

/19 / T. Banks, D. Nemeschansky and A. Sen, Nucl. Phys. B 277 (1986) 67

Discuss ion session on p a r t II:

Non- l inea r a - m o d e l s

A number of questions asked repeatedly during the individual lectures were

collected and presented at the discussion session. They concerned the topics

i) Off-shell IR-problem;

ii) Covariance properties of Green's functions and observable quantities;

iii) Renormalizability;

iv) Conformal invariance.

The written answers to these questions are not those of any individual speaker, but

represent the general opinion emerging during the discussion.

To i): The free part of the action of the non-linear a-model describes a massless

scalar field. In space-time dimensions d _< 2 a given background field configuration

is unstable against long-wavelength fluctuations of this massless field in view of

d2k_ikx Therefore a mass term has the IR-singularity of its 2-point function f--~-e .

to be introduced which stabilizes a given classical background configuration. This

automatically breaks the Ward (Slavnov) identity furnishing the independence of

the Green's functions of the background configuration (compare Becchi's lecture).

This breaking is considered to be soft (Stelle's lecture).

Question: Is the mass term really soft or is it potentially harmful?

Answer: Concerning the renormalization procedure the effect of the mass term in

the UV can be separated by power-counting. This, however, does not mean

that the theory has a finite zero-mass limit and if so that the latter is correct.

Experience with the O(N)-model (David) suggests the following situation. If the

target manifold is a coset-space and if the mass term has a well-defined covariance

with respect to the isometries of the space it should be possible to extend the result

of David saying that Green's functions which are invariant under the isometries

have a finite zero-mass limit. This ought to be connected with the fact that the

breakdown of the isometry artificially introduced by the mass term is averaged to

210

zero in the case of invariant Green's functions. Nothing is known in the general case

of non-standard models (no coset-space). Even if the IR-limit exists in perturbation

theory the result may be incorrect as the case d = 1 shows. It may be that the

perturbative approach does not adequately reproduce the large field fluctuations at

long wavelength.

To ii): It is possible in the construction of non-linear e-models to use covariant

(normal geodesic) coordinates for which the covariance of the action and the counter

terms under coordinate transformations is preserved using dimensional regulariza-

tion with minimal subtractions.

Question: What are the covariance properties of connected Green's functions?

Answer: The generating functional for connected Green's functions (free energy)

depending on linear field sources has no simple covariance property under

coordinate transformations. However, in principle, covariance could be rescued for

the ~effective action' defined as the Legendre transform of the connected functional.

One should notice that in the case of coset-spaces the connected functional is con-

strained by the Ward identity representing the action of the isometry group on the

coset-space and that this is sufficient to define the renormalized theory.

Question: How can one understand the covariance of the 'effective action' using

dimensional renormalization?

Answer: A simple argument in favour of the covariance of dimensional renormaliza-

tion with minimal subtraction could be based on the fact that the covariant

construction of the dimensionally regularized theory ill terms of the normal geodesic

coordinates with respect to the bare metric tensor is valid in any integer space-time

dimension. This ceases to be true if a Wess-Zumino term is added to the e-model

action.

211

Question: Do Green's functions for coordinates in the target manifold make any

sense or should one restrict oneself to the discussion of observables and

what are relevant observables ?

Answer: Everybody agrees that there is no observable meaning to coordinates of

a manifold. On the other hand the Green's functions for coordinates may

contain relevant information about observable quantities such as the particle spec-

trum (masses etc.). The situation may be viewed in analogy to gauge theories where

there seems to be no doubt about the usefulness of the Green's functions of gauge

potentials having a similar status as coordinates. On the other hand it would be

definitely desirable to construct Green's functions for observables like the energy-

momentum tensor. This would in particular allow to extract information about the

current algebra of the energy-momentum tensor relating to the conformal properties

(Virasoro algebra) of the theory.

To iii): There is ample literature about the renormalization of non-linear a-models

in two space-time dimensions in particular for special types of manifolds (K~hler

etc.). Of special interest are supersymmetric extensions.

Question: Are non-linear a-models in two space-time dimensions renormalizable

theories?

Answer: Everybody agrees that the general non-linear a-model for an arbitrary Rie-

mannian manifold is not renormalizable in the conventional sense, because

the action is unstable against arbitrary deformations of the metric. This instability

leads to infinitely many parameters of the renormalized theory. On the other hand

there is also agreement that renormalizability could be proved for coset spaces except

for the cases where there are antisymmetric couplings. This could be done either

using an invariant regularization (e.g. dimensional) or by discussing in detail the

cohomology of the isometry group. It is suggested that some special G-structures

- that is Riemannian manifolds whose holonomy group is G, a subgroup of the or-

thogonal group - could perhaps lead to renormalizable models with a finite number

of parameters. This however does not seem to be true for K~hler manifolds where

212

G is a unitary group. Another possible such case is that of hyper-K£hler models

having the scalar content of N = 4 supersymmetric models. In any case, nothing is

known for supersymmetric models, because dimensional renormalization is not an

invariant scheme for such theories.

Assuming that the Wilson renormalization group preserves global topological con-

straints and that fixed points correspond to renormalizable models one might con-

sider 2-dimensional target manifolds. In this case the only topological invariant is

the genus of the manifold. It is suggestive to speculate that the fixed points yield

algebraic surfaces, even if some arguments (Lott) seem to indicate an instability of

the constant curvature configurations (e.g. spheres) against the formation of corners.

To iv): In connection with string theories a particular emphasis is put on the con-

struction of non-linear cr-models with vanishing fl-function (conformal invariance).

Question: Is the vanishing of the/~-function ('finiteness') a possible handle on the

renormalization problem for manifolds without isometries, e.g. hyper-K£hler

models with a Wess-Zumino term?

Answer: The requirement of conformal invariance seems to restrict rather drastically

the ambiguities involved in defining these theories (Flume), to the extent

that explicit formulae can be given for the 4-point functions of primary fields. How-

ever, the construction of the Virasoro algebra for the energy-momentum tensor has

not yet been achieved in renormalized perturbation theory.

Part III

Cohomological and Geometrical Methods~

Relation to String Theory

A free string moving in a classical background manifold can be described by the

equations of motion for a quantized non-linear a-model in two-dimensional space-

time with this manifold as target space.

Cohomological and geometrical methods play an important rble in renormaliza-

tion theory, in non-linear a-models and in string theory. It is tried to point out

possible links between these subjects.

R emarks on Slavnov Symmet r i e s

R. STORA

LAPP, Annecy le Vieux, France

and

Theory Division, CERN

Geneva, Switzerland

Slavnov symmetries have been discovered some ten years ago [1] within the

framework of perturbative Yang-Mills type gauge theories. They were found as non-

linear local field transformations, depending on an anticommuting parameter, which

leave the Paddeev Popov-gauge fixed Yang-Mills action invariant. They turned out

to be the ideal tool to perform the perturbative renormalization of Yang-Mills the-

ories, taking full advantage of locality and power counting.

It was soon realized [1] [2] that the Slavnov symmetry could be cast into the

form of a cohomological statement related to a differential algebra which is a trivial

extension of the cohomology algebra of the gauge Lie algebra.

This class of algorithms was extended to the analysis of a number of similar

situations:

- other gauge Lie algebras including diffeomorphisms, and their local anomalies;

- graded Lie algebras occurring e.g. in globally as well as locally supersymmetric

theories;

- more general differential algebras [3];

- constrained systems in Hamiltonian dynamics [4].

Similar constructions have also proved useful in statistical mechanics, and, in

particular in the method of stochastic quantization [5].

216

The subject has been actively revived recently outside the small group of adepts

in connection with string theory, mostly, however, in the form close to the Hamilto-

nian version analyzed by the Lebedev group [4], in which both locality and geometry

fail to serve as leading principles. (See however [6]).

Let us finally mention the applicability of Slavnov type algorithms to express

coordinate independence of a-models [7] one of the main topics covered during this

meeting.

Given this list of examples, one may wonder what features they share in common,

besides being cohomology theories.

One answer is the following:

One starts from a (local) classical action S(_¢) whose arguments are fields, some

of which will be quantized, whereas some others will remain classical sources. Con-

sider the situation where S(¢) admits infinitesimal zero modes

5¢(x) = Pi(¢)5)ti(x) (1)

where the Pi's are ¢ dependent linear operators and the 5)~i's infinitesimal parame-

ters, i.e.

5S a f dx~--r(gS¢ (x) =- 0 (2)

for 5¢ given by (1). In gauge situations the 5Ai's are fields and the Pils are differ-

ential operators which depend locally on ¢.

If these zero modes are integrable into " leaves" in field space, one may consider

the ~i's as parameters along these submanifolds, and 5 as differentiation along these

submanifolds: locally in field space, one has a parametrization

¢ ( x ) =

=

where Qi are (not necessarily local) linear operators. So, we may define

(3)

217

s = 5 - - * s 2 = 0 ( 4 )

so that S(¢) is s-invariant, with

s¢. = Qi(¢)w i (5) sw* = O;

of course s 2 = O.

Conversely given (5) such that s 2 = 0 Frobenius formally guarantees that the

infinitesimal zero modes are integrable. The locality question is not under control

and is sensitive to the choice of parametrization (3). For instance, in the Yang-Mills

case let us parametrize the Yang-Mills field a according to

a = g - l ~ g + g - l d g

0

with g in the gauge group, a subject to a gauge condition. We have

(6)

6ga = Da(g-15g) = - d ( g - 1 5 g ) - [a,g-15g] (7)

(assuming 6gd + d6g = 0). 5g plays here the role of 5~. Trying to eliminate g in

terms of a, in (7) would in general result into non locality. However, we know that,

defining

w = g-15g

we have

= _

1

s 2 = O

( 8 )

218

which are the well known local formulae for the cohomology of the gauge Lie algebra.

When zero modes affect quantized fields, the Faddeev-Popov procedure allows

to extend the initial differential algebra by introducing the antighost field ~ and the

Lagrange multiplier b in such a way that

leaves the gauge fixed action

s~ = - b

b=0 (9)

s g / = (b,g) + (i0)

invariant, whatever gauge functions g have been chosen. It is to be noted that the

geometrical natures of ff~, b essentially depend on the choice of the gauge functions

g.

When the zero modes do not give rise to an integrable situation no general

theory is known at the moment: this is the problem of auxiliary fields, namely,

given a non-integrable situation, can it be viewed as a restriction of an integrable

o n e . 7

It may happen that, due to a "bad" choice of a gauge function and or imposing

locality, zero modes appear in the ghost sectors [6] [8]. This poses a slightly more

general problem namely, given s with 8 2 = 0 acting on some set of fields ¢, and

given C-dependent zero modes @i(¢):

6¢ = c i¢ i (¢) (ii)

can one find ~? ci such that together with

:~¢ =- s ¢ - ~ _ , c i ¢ i ( ¢ ) (12)

one achieves ~2 = 0.

It is fair to say that no general scheme has been found so far which is able to

deal with the quantization of degenerate actions as stated above; only classes of

examples have been collected, of which we have quoted representatives.

219

ACKNOWLEDGEMENTS

These remarks constitute by no means a fair review of the work which has been accomplished and contributes to the understanding of Slavnov symmetries. They rather try to extrapolate from the experience which has been gained by collecting and working out examples towards the formulation of some open problems. The author wishes to thank many colleagues for sharing their views concerning this intriguing area, and, in particular, C. Becchi, L. Baulieu, L. Bonora, P. Cotta- Ramusino, O. Piguet, K. Sibold. Also, it is a pleasure to thank the organizers of this meeting for the opportunity they gave us to meet our colleagues and hear about recent developments.

REFERENCES

[1] C. Becchi, A. Rouet, R. Stora, Ann.Phys. 98 (1976) 287.

[2] C. Becchi, A. Rouet, R. Stora, in: Renormalization Theory Erice 1975; G. Velo, A.S. Wightman, Ed. NATO ASI Series Vol.C23, Reidel 1976; C. Becchi, A. Rouet, R. Stora in: Field Theory Quantization and Statistical Mechanics Tirabegin ed. Reidel 1981.

[3] S. Boukraa: The BRS Algebra of a free differential algebra, Nucl. Phys. B (to appear).

[4] M. Henneaux: Phys.aep. 126 (1985) 1.

[5] Zinn-Justin, Nucl. Phys. B 275 (1986) 135[FS17]

[6] L. Baulieu, C. Becchi, R. Stora, Phys. Left. 180 B (1986) 55 C. Becchi: On the covariant quantization of the free string: the confor- real structure Nucl. Phys. B (to appear); L. Baulieu, M. Bellon, Beltrami parametrization and string theory, LPTHE 87-39; L. Baulieu, M. Bellon, R. Grimm: Beltrami parametrization for super- strings, LPTHE 87-43; L. Baulieu, B. Grossmann, R. Stora Phys. Left. 180 B (1986) 95; J.P. Ader, J.C. Wallet Phys. Left. 192 B (1987) 103.

[7] See e.g.C. Becchi's talk at this meeting.

[8] L. Baulieu, J. Thierry-Mieg Nuc/. Phys. B 228 (1983) 259.

SUPERSYMMETRIC PROPERTIES OF FIELD THEORIES IN IO-D

L. Bonora (z)<2), M. Bregola <4), K. Lechner (s),

P. Pasti (2)<3), M. Tonin <2)(3)

(I) CERN, TH Division

(2) Dipartimento di Fisica dell'Universit~ di Padova,

Via Marzolo 8, Padova.

(3) INFN, Sezione di Padova

(4) Dipartimento di Fisica dell'Universit~ di Ferrara,

Via Paradiso 12, e INFN, Sezione di Bologna.

A field theory which represents the low energy limit of a string

theory is perhaps the most important piece of information that we

can extract from that string theory. The effective field

theory can be obtained by direct string amplitude calculations; or

else we can start from the field theory content, which is

unambiguosly defined by the zero mass excitations of the string,

and follow at present two methods. The first consists in writing

down a sigma model of the string interacting with background fields

corresponding exactly to the zero mass spectrum of the string: the

requirement of conformal invariance, through the vanishing of the

beta functions, will provide us the equations of motions of the

background fields. In the second approach one simply takes from the

string theory the field content and with these fields tries to

construct a field theory. If this field theory turns out to be

consistent and, especially, if it enjoys some kind of uniqueness

it, hopefully, will represent the low energy limit of the string

theory.

In this talk we will be concerned with the second approach. It

221

is well known that the zero modes of the heterotic string and type

I superstring correspond to the field content of a supersymmetric

Yang-Mills (SYM) theory coupled to supergravity (SUGRA) in ten di-

mensions. In other words, beside the graviton, the dilaton, the

"two-index photon" and the gauge potentials, we have the gravitino,

the dilatino and the gauge superpartners.

Of course constructing a field theory with such a field content

is interesting in itself, even without reference to string theory.

However the number of theories one could construct would be

enormous, due to the different choices of the gauge group , were

it not for the requirement of chiral anomaly cancellation. As is

well known Ill chirai anomalies cancel in 10-D SYM+SUGRA theories

provided that the gauge group is EsxEs or SO(32), and provided a

two form field B and a three form field H (its "curvature") are

defined in such a way as to satisfy the equation

H = clB , q w~y - C = ~ L (i)

w h e r e W3y ( b 2 ~ L ) i s t h e g a u g e ( L o r e n t z ) C h e r n - S i m o n s f o r m i . e .

b e i n g t h e g a u g e ( L o r e n t z ) c o n n e c t i o n ; cz and c2 a r e c o n s t a n t s ,

w h i c h m u s t a s s u m e p a r t i c u l a r v a l u e s f o r t h e a n o m a l y t o c a n c e l ( f o r

e x a m p l e , f o r S 0 ( 3 2 ) , c 1 = c 2 = 1 ) . In e q . (1 ) B i s i d e n t i f i e d w i t h t h e

t w o - i n d e x p h o t o n . T h i s i d e n t i f i c a t i o n r a i s e s s e v e r a l p r o b l e m s w h i c h

a r e i n p a r t a d d r e s s e d h e r e and i n p a r t i n a p a r a l l e l t a l k [ 2 ] . H e r e

we s t u d y t h e p r o b l e m o f i m p l e m e n t i n g s u p e r s y m m e t r y i n t h e p r e s e n c e

o f c o n d i t i o n (1 ) . T h i s p r o b l e m a r o s e h i s t o r i c a l l y i n t h e f o l l o w i n g

way . C h a p l i n e and M a n t o n [3] s u c c e e d e d a f e w y e a r s ago i n w r i t i n g

down a l a g r a n g i a n o f a SYM t h e o r y c o u p l e d t o SUGRA i n IO-D. T h e y

assumed a condition similar to (I) except that c2=0. As a

consequence their theory is plagued by chiral anomalies. Green and

Schwarz, through eq. (i) were able to cancel anomalies, however in

this way they destroyed manifest supersymmetry in the Chapline-

222

-Manton lagrangian.

There have been a few attempts [4] to restore supersymmetry by

suitably modifying the Chapline-Manton lagrangian, but this

approach appears rather unwieldy. Another, more promising approach

is based on supermanifold techniques [s,6,7] Supermanifold

techniques have the disadvantage that, at least in ten dimensions,

they cannot give us a Lagrangian but only the equations of motion,

but of course they have the advantage of involving superfields

instead of component fields and of resorting to a "supergeometry"

which is very powerful (although probably not still understood).The

supermanifold approach is the one we have used to solve the problem

of implementing supersymmetry in the presence of condition (I).

Let us consider a supermanifold with coordinates z,= {x u , @ ~},

m=l,...,10, ~=i,...,16. We will use throu&~hout mostly flat-indexed

objects. F l a t indices are

alphabet: ex. A={a,=}, a=l,...,10,

from world indices to

supervielbeins e~ (z) and

e~(z). Next we

Lorentz superconnection W~ B

superconnection A together with the

and F given by

denoted by the first letters of the

==i,...,16. In order to pass

flat ones we introduce the

the corresponding superforms eA=dz M

mimic usual differential geometry and introduce a

and a Lie algebra-valued

torsion T A , the curvatures RA s

T A = B cA

F = d A + A A (2)

Here d = ~ N ~ is the total differential in the superspace, d~ r,

while ~= £A~A is the covariant differential w.r,t, both

gauge and Lorentz indices. Moreover we introduce a 3-superform H

and a 2-superform B and write the analogue of eq. (i) where the

fields are replaced by the corresponding superfields. Henceforth

eq. (i) will be regarded as a superfield equation. Corresponding to

223

e q s . (1 ) and (2 ) we h a v e t h e i d e n t i t i e s

dH = c,T~ (FF)- ¢~D(RR) (3)

AT A = e s I~B' (4)

AKA~= 0 (5)

AF-0 (6)

We use the basis of supervielbeins {e A } to define the components of

a given superform. For instance TA=½ eSeCTcB A.

In order to extract, from this very general geometric setting,

the equations of motion, the procedure consists in reducing the

number of degrees of freedom by imposing constraints on the

components of T A, F and H. The Bianchi identities (3)(4) and (6)

will produce the equations of motion for the surviving degrees of

freedom. Eq. (5) does not give additional information.

Following [7] we postulate the following constraints

T$~ - #. F¢~ ;

F¢~ = 0

H¢~/-- 0 ;

T¢/_- 0 = T,#;

(8)

(9)

The c o n s t r a i n t s

p r o p o r t i o n a l t o c2 ( s e e b e l o w ) .

Here we use two sets of F-matrices, with

indices, respectively. They satisfy the condition

r ÷ ¢i

( 9 ) w i l l be m o d i f i e d i n t h e f o l l o w i n g by t e r m s

u p p e r a n d l o w e r

224

Moreover F a~'''aK stands for the antisymmetrized product of k F-

matrices normalized to unit weight.

With the above constraints it is possible to solve completely

the T-Bianchi identities, eq. (4). One obtains for example,

where L~.b =- F~. ,T{6 The constraints also imply

= T cb'~ R.~ ~'b' o~ (P~'~'~.~.),~T~...~..Br..~

; ~/'~-_.LT..(r'") "~ ; ~'5-¢" ( lO)

nuoTab¢. = T_ (P } h_T_ q/'r',_ I~',~ I,

( l l )

and similar expressions for the other components of the curvature.

The F-Bianchi identities, eq. (6), give

n~ = P~,~ ~' , ~b = ~ (~b)2 ~ X ~ (12)

which leads to the field equations in the pure YM sectors. The H-

-Bianchi identities, eq. (3) have been solved for the particular

case c2=0. In this way the equations of motion of the Chapline-

-Manton model have been recovered. Explicit formulas can be found

for example in ref. [7].

It is worth recalling at this point that the identification with

the physical fields is obtained by associating the dilaton to the

field ~ of eq. (9) at 0 =0, the dilatino to D= • for 0=0, while

graviton and gravitino are associated with the space-time-like

e~ and spinor-like vielbein e~ respectively. The vielbein

225

remaining identifications are the obvious ones.

Before extending the previous analysis to the case c2#0 we have

to introduce some notational conventions. First we decompose any n-

-superform ~] i n t o Z ~p,q w h e r e ~p,, is an n - s u p e r f o r m P+~:~

homogeneous of degree p in the space-time-like vielbeins e a and of

degree q in the spinor-]ike vielbeins e a. We do the same with the

exterior differential operator d (when applied to scalar forms). To

this end it is convenient to regard it as an operator s(~nding

(p,q)-superforms into a sum of (p~ ,ql )-superforms and split it into

homogeneous pieces according to the degree (r,s)=(p P -p,q# -q).

Therefore

(13)

w h e r e d , D , T , r h a v e d e g r e e ( 1 , 0 ) , ( 0 , 1 ) , ( - 1 , 2 ) and ( 2 , - 1 )

r e s p e c t i v e l y . More p r e c i s e l y d=eaDa + T ( 1 , o ) , D=eaDa + T ( 0 , 1 ) ,

T = T ( - 1 , e ) p T = T ( 2 , - 1 ) : Da and Da a r e t h e c o v a r i a n t f l a t s p a c e - t i m e

and s p i n o r d e r i v a t i v e s a c t i n g o n l y on t h e c o m p o n e n t s , w h i l e T < r , s >

a c t s o n l y on t h e v i e l b e i n b a s i s . In p a r t i c u l a r T e a = F a a ~ e a e $ ,

T e a = O .

From t h e n i l p o t e n c y o f d, d2=O, a s e t o f u s e f u l r e l a t i o n s among

~ , D , T and r f o l l o w . In p a r t i c u l a r :

T' 0 , DT+TD 0 dT+T + 0

Now we a r e r e a d y t o t u r n t o t h e d i s c u s s i o n o f t h e g e n e r a l c a s e ,

c 2 ~ 0 . The d i f f i c u l t y i n t h i s c a s e l i e s on t h e f a c t t h a t , i f we s e t

Q = T r ( R , R ) = ~ Q(,~ , Qo,4 and Q1,3 a r e n o n - v a n i s h i n g u n l i k e t h e

c o r r e s p o n d i n g s u p e r f o r m s i n T r ( F , F ) ( w h i c h v a n i s h due t o t h e

c o n s t r a i n t s 8 ) . As a c o n s e q u e n c e t h e c o n s t r a i n t s (9 ) a r e u n t e n a b l e .

T h e r e e x i s t s h o w e v e r t h e o p p o r t u n i t y t o u s e t h e r e s u l t s o f t h e c a s e

c2=0 t o s o l v e t h e c a s e c l # O , by r e l y i n g on t h e f o l l o w i n g r e m a r k a b l e

p r o p e r t y o f T r ( R R ) . t e l

226

Lemma

Tr(RR) = c 4 X , K (14)

where X is a gauge and Lorentz invariant 3-superform and K is a

gauge and Lorentz invariant 4-superform such that Kz,s=O=Ko,4 and,

moreover K2,2 has the same structure as Tr(F,F)2,s.

The proof goes as follows. From the identity dTr(R,R)=O we write

down the equations

TQ~.,. + DQ,.~ * d Oo, L, = 0 (15a)

T Q,,~ + D Qo,~, = 0 (15b)

T Qo,~ = 0 (lSe)

In other words, eq. (15c) means that 00,4 is a

coboundary operator T. It is easy to show that

cohoundary

cocycle of the

~t is indeed a

Qo, = T (16)

where (Y1,z)a~o = Ra~ c4 c* Tac I c L . Inserting eq. (16) into (15b) we

obtain

T ( ( ~ , ~ - DY,,~. ) = 0 (17 )

227

The c o c y c l e Q I , s - D Y I , z i s n o t a c o b o u n d a r y b e c a u s e o f t h e p r e s e n c e

o f a 1440 i r r e d u c i b l e r e p r e s e n t a t i o n o f S 0 ( 1 , 9 ) . H o w e v e r o n e c a n

p r o v e t h a t , i f we i n t r o d u c e t h e t h r e e - s u p e r f o r m Z 1 , 2 s u c h t h a t

G (18)

then TZl,2=O and

Q,, , - D %,~ : DZ,,, + T X~,, (19)

Then setting XI,2=Yz,2+ZI,s we h a v e

Q~,~ = DX,,z + T X,,, (20)

Oo,~ = TX,,, (21)

Now inserting eq. (20) into eq. (iSa) one finds

T(¢ , , , - OX,,, - ~- X,,~ ) = 0 (22)

Again this T-cocycle is non-trivial, however one can write in

complete generality:

Oi,i = b X~,, ÷ d X4,z + T X~,o * K,,~ (23)

w h e r e

228

Since

we see that K2,2 has the same group theoretical structure.

Defining X=XI,2 + X2,1 + X3,o and setting K=Q-dX, we see

lemma is proved.

Now set

(Tr(FF):,.~)as,r,~ : (~ ' ) ' : l ' Tr("XlrX;)(r",,l).~/~

t h a t t h e

A

H = H,c,X (25)

From eq. (3) one gets the identity

dH = qD(FF)- qK (26)

which is formally identical to the case cs=O since K and Tr(F,F)

have the same structure. Therefore by imposing the new constraints

which are formally the same constraints as in the case c2=0, we can

solve all the Bianchi identities. Moreover we can use the equations

already found for the case c2#0 and obtain the corresponding ^

equations for c2=0, by simply replacing everywhere H with H and

Tr(FF) with ciTr(FF)-csK. All the quantities Xs,1, Xs,0, Ks,s,

K3,1, K3,I, K4,o as well as the equations of motion can be

explicitly calculated. The complete results are reported elsewhere

[io]. Here we would like to make a few remarks.

One of the equations obtained is the following:

A (28)

Tabc appears in all the equations as an auxiliary superfield. In

229

p a r t i c u l a r i n e q . ( 2 8 ) i t a p p e a r s a l s o i n a h i g h l y n o n l i n e a r w a y

i n t h e e x p l i c i t e x p r e s s i o n o f W a ~ . I f we w a n t t o e l i m i n a t e t h e

d e p e n d e n c e on T a b c we m ay s o l v e r e c u r s i v e l y e q . ( 2 8 ) , o b t a i n i n g a n

infinite series in H,L and H. This is what the approximate

solutions already existing in the literature suggested.

It is clear that we are helped to think the field theory we have

found has something to do with string theory, if it is

characterized by some kind of uniqueness, or, at least, it is not a

member of an infinite set of inequivalent field theories

(equivalence here is meant up to field redefinition). A critical

point in this regard is the choice of the constraints (7) (8) and

(9). Other constraints have been proposed [5], hut they are shown

to be equivalent to these.

So one might hope that proving uniqueness, after all, is not

such a forbidding task. However, regarded in complete generality,

the problem is daunting, the number of irreducible components of

the fields involved being too high. We need some restrictions.

Fortunately a few restrictions are provided by the anomaly problem,

by which we mean finding non trivial solutions of the Wess-Zumino

consistency conditions which reduce to the usual (non

supersymmetric) ones when we take e =0 in the superconnections.

When in the usual case a non trivial solution exists, while in the

superfield formulation it does not, we assume this as an

inconsistency of the theory. We transform this into a selective

criterion for consistent superfield formulations. It is rather

easy to show with the method of ref. [ii], that in the pure gauge

sector the constraints (8) is a necessary and sufficient condition

for such non trivial solution to exist. It is also easy to realize

tha~ the existence of a solution depends on constraints on the

relevant curvature. In the pure SUGRA sector the situation is more

complicated than in the pure SYM sector because ~ b contains

many irreducible components w.r.t. S0(1,9) [Eq. (ii) is a very

particular representation of ~ b , which is due to the

230

constraints (7)]. Nevertheless the above criterion seems to impose

extremely sharp restrictions. Work in this direction is in

progress.

The last remark concerns ghosts. The equations of motion we find

[i0], unambiguosly contain ghosts. For example the equation for the

Ricci tensor takes the form

R~b --- [ ] R~b + " ' "

which unveils the presence of a ghost which propagates at the

Planck mass. Such ghosts are likely to be eliminated by non-local

field redefinitions [12]. However one can argue whether such a

procedure is a correct one. Perhaps these ghosts are a consequence

of the genuine stringy nature of the field theory we are

considering. From this point of view they may be far from

unappealing.

R e f e r e n c e s

231

[ I] M. Green and J. Schwarz, Phys. Lett. 149B, I17 (1984).

[ 2] L. Bonora, P. Cotta-Ramusino, and M. Rinaldi these

proceedings.

[ 3] G.F. Chapline and N.S. Manton, Phys. Lett. 120B, 105 (1983).

[ 4] L. Romans and N. Warner, Caltech preprint CAL-68-1291, 1985.

S.K. Han, J. Kim, I. Koh and Y. Tanii, Phys. Rev. D34 (1986),

5 3 3 .

[ 5] B.E.W. Nilsson, Nucl. Phys. 188 (1981), 176.

B.E.W. Nilsson and A.K. Tollsten, Phys. Lett. 169B (1986),

369; Phys. Lett. 171B (1986), 212.

[ 6] E. Witten, Nucl. Phys. B266 (1986), 245.

[ 7] J. Atik, A. Dhar and B. Ratra, Phys. Rev. D33 (1986) 2824.

[ 8] S. McDowell and M. Rakowski, Yale preprint YTP, 86-01.

[ 9] L. Bonora, P. Pasti and M. Tonin, DFPD 21/86, to appear in

Phys. Lett. B.

[i0] L. Bonora, M. Bregola, K. Lechner, P. Pasti and M. Tonin, in

preparation.

[ii] L. Bonora, P. Pasti and M. Tonin, Nucl. Phys. B286 (1987),

150.

[12] S. Veser and A.N. R e d l i c h , Phys. L e t t . 176B, 350 (1986 ) .

G E N E R A L I Z E D W E S S - Z U M I N O T E R M S

L. Bonora (*)

Theory division, CERN, CH-1211 Geneve 23

P. Cotta-Ramusino (**)

Dipartimento di Fisiea dell'Universith di Milano

and Istituto Nazionale di Fisiea Nucleate, Sezione di Milano

Via Celoria 16, 20133 Milano

M. Rinaldi

International School for Advanced Studies

Strada Costiera 11, 3~100 Trieste

A possible way to s tudy string theories is to write down sigma-models which

describe the string propagating in a background represented by fields corresponding

to the zero mode excitations of the string itself [1] . Typically the pure bosonic par t

of the sigma-model action will contain the te rm

(1) g°b(h*r) b + eab(h*S)ab

integrated over a given Riemann surface S which represents the Euclidean world-

sheet of the string: gab is the inverse of the intrinsic metric over S; h : S ~ M is an

embedding of S into the target space M (the space-time in the string approach), eab is

the two-dimensional ant isymmetric tensor, F is a metric on the target space and B is

a two form on M. It is well known that requiring conformal invariance in the form of

vanishing beta functions gives us information about the dynamics of the background

fields. Unfor tunately information concerning the geometry and the topology of the

(*) On leave of absence from Dipartimento di Fisica dell'Universit~ di Padova and Istituto Nazionale di Fisica Nucleare, Sezione di Padova (**) Work supported in part by: Ministero Pubblica Istruzione (research project on "Geometry and Physics ~)

233

target space is hardly obtainable in this way. However pieces of information about the

geometry of the target space can be gotten from anomaly cancellation. Of course in

order to have anomalies involving background fields, we have to have chiral fermions

interacting with background fields. This is what happens for example in the heterotic

string sigma model. Again typically we will have in mind a Lagrangian which beside

the term (1) contains also:

(2) < > + < >

where A and w are, respectively, connections on a principal gauge bundle with struc-

ture group G and on the orthonormal bundle O r M over M, while ~h; A(~h; ~) denotes

the Dirac operator coupled to the pulled-back connections h~ A (h~w) as explained by

the following diagrams [2]

(3)

(4)

ho h~P ~ P

S ho M.

D

ho h~OrM ~ O r M

S ho M.

where h0 (~0) is a bundle map. The field A is assumed to be a section of the bundle

S ± @ h*TM, where S ± is the spinor bundle over S with positive (negative) chirality,

and ¢ is a section of the vector bundle S ~: @ h*V with V being a vector bundle

associated to P. Notice that ¢ and A have opposite chirality.

The action given by (1) + (2) gives rise to chiral anomalies which, contrary to

what occur in field theory [3] , can cancel if certain geometrical restrictions over the

target space are fulfilled. This talk is devoted to explaining this qualitatively new

phenomenon.

For the sake of simplicity, unless otherwise stated, henceforth we will refer to the

diagram (3) , that is a sigma-model in which the world-sheet fermions are coupled to a

pulled back gauge connection. In other words, in the fermionic part of the Lagrangian

234

we consider only (3) . The chiral anomaly corresponding to such a model can be

constructed in the following way. Referring to the diagram (3) we consider the maps

~)ho ho (5) h~P × ~ho , h~P - > P

where .6ho is the group Autv(h~ P), that is the group of vertical automorphisms of

h~ P, namely the group of gauge transformations of h~P, and eVho is the evaluation

map defined by eVho (u, ¢) ---- ¢(u), for any u E h; P and ¢ E .~ho. [We remark that

the construction we are illustrating is valid for a generic sigma-model, in which case

one considers a generic map f0 : S ~ M, not a particular map like an imbedding.

We limit ourselves to imbeddings having in mind the string case].

Now in the Lie algebra of G, we consider the polynomial K with two entries which

is invariant under the adjoint action of G, and a generic connection ~ on P. We can

construct the connection eV~o h~ ~ on h~ P x ~ho, and, given a fixed connection A0

on h~ P , we can consider the 3-form [3]

(6) Wg((h0 o eVho)* ~, Ao)

on S × ~ho"

[Here WK(A, Ao) for two generic connections A, Ao is a basic 3-form such that

dWK(A, Ao) = K(F ,F) - K(Fo,Fo), F and Fo being the curvatures of A and Ao,

respectively [3]].

The form (6) contains all the information concerning the properties of an anomaly.

If we differentiate it in S × ~ho we obtain

(7) (d + ~)WK((ho o eVho)* ~,Ao) = 0

where d and 6 are the exterior differentials in S and ~ho respectively. The iden-

tity (7) splits into several equations according to the order of the form in M and ~ho.

In particular at the identity of .~h0, we have

(8) 6i(.) Wi~(h~ ~,Ao) + di (.) i (.) WK(h~ ~,Ao) = 0

which is the consistency condition for the non integrated anomaly i(.)WK (h~ ~, Ao).

Her e i(.) denotes the map Z ~ iz , where Z is any vector field in Lie~ho, and

i z denotes the interior product, which does not act, however, on Ao. The form

235

i(.)WK(f~ ~,A0) is the anomaly relevant to the model we are considering. Its co-

efficient can be calculated through a perturbative expansion or by using the index

theorem, but it will not play any r61e in the following.

Contrary to what happens in field theory, in the present context the above

anomaly can be cancelled in some cases. A condition for this to occur is K(F~, F~)

being in the kernel of the Well homomorphism, that is

(9) K(F~, F~) = dH

where H is a 3-form on M, and F~ is the curvature of ~.

One can see immediately that in field theory the condition (9) cannot be satisfied.

The reason is that in gauge field theory the diagram analogous to ('3) is /

P , EG

M f~ BG.

where P is the relevant gauge principal bundle BG and EG the classifying space and

universal bundle, respectively, f is the classifying map and ] any bundle map which

covers f . In this case the rSle of the connection ~ is played by a universal connection

on EG and the analog of eq. (9) is false. In other words the fundamental difference

between sigma-models and gauge field theories is that the latter can be regarded as

limiting cases of the former when the target space coincides with the classifying space.

In turn this property of field theory of being characterized by ~universality" properties

is a consequence of the requirement of locality [3]. Eq. (9) which would represent a

violation of locality in field theory, is allowed in sigma-model and represents simply a

(mild) geometrical condition on the target space. We could phrase this by saying that

in sigma-models the locality requirement is less restrictive than in field theory.

We are going to show now that, thanks to condition (9), it is possible to construct

a (generalized Wess-Zumino) counterterm which reproduces the anomaly (6), so that

by subtracting it from the quantum action we obtain an anomaly free theory.

To this end let us consider the space of paths over Imb(S,M), the space of

imbeddings of S into M, with initial point a fixed imbedding h0, and denote it by

Pho Imb(S,M). Next we define the projection ~rl : Pao Irnb(S, M) , Irab(S,M)

236

defined by sending every pa th into its endpoint. 7rl is a principal fibration with fiber

represented by the space ~ho Imb(S, M) of loops passing through h0.

First of all we remark that , by using the parallelism induced by the connection ~,

we can construct a homomorphism [4]

(10) r : {'lho Imb(S, M) , Aut~ h 3 P .

Next consider the diagram

(:1)

h~P x Pho Imb(S, M) ~ 7rl* eV*P

S x Pho Imb(S, M) ,r,

ev*P , P

S x Imb(S,M)ho ~% M,

where Imb(S, M)ho means the component of Irnb(S, M) connected to h0, ~1 and

e---~ are the canonical bundle maps corresponding to Irt and ev, respectively. The

isomorphism between Irl*eV*P and

h~P x Pho Imb(S, M)

is induced by the connection ~, analogously to the homomorphism (10) . Now

consider again the connection ~ on P and construct the connection ~ : ~ * ~ on

h~P × Pho Imb(S, M). Then, if A0 is a fixed connection on h~ P , it is easy to show

that

02 ) dCW~(~l~*~, A0) - ~ e v * H ) = 0

due to eq. (9) .

Now we are going to use the contractibili ty of the space Pho Imb(S, M) to prove

that there exists a two-form 3 on S × Pho Imb(S, M) such that

(13) wK( I * A0) - =

The key point is tha t any pa th space is contractible, so that we can construct a

retraction

r : S x Pho Irnb(S, M) ~ S

which togheter with the inclusion i : S , S x Pho Imb(S, M) forms a map i o r :

S × Pho Irnb(S, M) , S x Pho Irnb(S, M), which is homotopic to the identity. So

237

there exists a homotopy operator ~ such that

(14) 1" - (i or )* = +

Using eq. (14) one can easily show that

(15) /3 -- ~ (WK ('~1-~* ~, Ao) - ~ev*H)

fulfills eq. (13) .

The generalized Wess-Zumino term is by definition

(16) /M BW Z

where B Wz is the (2,0)-component of ft. By construction, if we take the variation

of fM BWZ along the fiber ~ho Imb(S, M), we obtain the (integrated) anomaly of

eq. (8) . Of course this identification is permitted if we can identify the variation

along the loop space with the variation in the gauge group ,~ho. This is accomplished

through the homomorphism (10). Finally, by subtracting the term (16) with a suitable

coefficient we can rid the model of the anomaly in question.

Now let us return to the full Lagrangian (1) + (2), which can be suitably

described in terms of the bundle P + O r M over M. We can consider the di-

agram (11) with P substi tuted by P + OrM. The anomaly is generated by

WK(~I~-~*~,Ao) -- WK(~l-~*~hWo), where A0, w0 are fixed connections on P and

OrM. If eq. (9) and an analogous equation for the connection ~/ is satisfied we can

eliminate the two anomalies separately. However there exists also a weaker condition,

that is:

(17) K(F~,F~) - K(F~,F~) = dH.

If eq. (17) is satisfied, in exactly the same way we have proceeded above, we can

construct a generalized Wess-Zumino term fM B~o Wz which cancels the total anomaly.

This is a geometrical description of the Green-Schwarz cancellation mechanism in

sigma-models. Needless to say, the term Bto Wz does not have anything to do with the

B ~o~ represents B of eq. (1), except that they are both 2-forms. We can say that w z

a very complicated interaction of the bosonic part of the heterotic string with the

gauge and Lorentz connections, which annihilate exactly the anomaly generated by

the fermionic part. However this is not yet a satisfactory cancellation scheme. Indeed

238

both f M BW Z and f M W Z Bto t are functionals on Pho Imb(S, M). We have to make sure

that the exponential of the product of 2~i times our (generalized) Wess-Zumino term,

descends to a functional over the appropriate degrees of freedom.

A similar situation is met in Witten's bosonized sigma-model [5] where a Wess-

Zumino term is added in order to reproduce the features of the original fermionic

model. The construction turn out to be a particular case of that described above:

the target space M is replaced by a Lie group G, the relevant maps are maps from

M to G, Map(M, G) and instead of a generic connection we have the Maurer-

Cartan form of G. The construction is simplified and leads to a Wess-Zumino

term of the form fMxIEv*TK(O), where Ev is the double evaluation map Ev :

M × I x Map.(M, PeG) ~ G and P~G is the path space over G wit h initial point

the identity. It is easy to realize that

exp(2ri f Ev*TK(O)) M × I

which is a functional from the space of paths over Map(M, G) to ~ , descends to a

functional over Map(M, G) only ff for any loop I : M × S 1 ~ G passing through the

identity, the integral fMxs~ I*TK(O) is an integer. This requirement gives rise to the

coupling quantization condition.

From the previous example it is clear that we have to study for instance the

behaviour of the generalized Wess-Zumino term B Wz under gauge transformations

which imply changing the homotopy class within ~ho (whenever rO(~ho) # 0). As a

consequence global anomalies become involved. In other words we have to make sure

that also global gauge anomalies are eliminated. If this happens, the generalized Wess-

Zumino term, when restricted to a functional over the the loop space ilho Imb(S, M),

descends to a functional over the subgroup of the group of gauge transformations,

given by the image of the homomorphism (10) .

239

I. References

[1] E.Fradkin and A.Tseytlin:Effeetive field theory from quantized string; Phys.Let.

158B; 316 (1985);

[2] G.Moore, P.Nelson: The aetiology of sigma-model anomalies; Comm.Math.Phys.

100, 83 (1985);

[3] L.Bonora, P.Cotta-Ramusino, M.Rinaldi and J.Stasheff:The evaluation map in

field theory, sigma-models and strings-I; CERN-TH 4647/87; to be published in

Comm.Math.Phys.;

[4] L.Bonora, P.Cotta-Ramusino, M.Rinaldi and J.Stasheff: The evaluation map in

field theory, sigma-models and strings-II; in preparation;

[5] E.Witten:Non abelian bosonization in two dimensions; Comm.Math.Phys. 92;

455 (1984).