:LETTERE AL NUOVO CIMENT0 VOL. 9, N. 14 6 Apr i l c 1974

R e n o r m a l i z a t i o n o f Yang-Mi l l s Theor ies .

1K. T o N ~

I s t i t u t o d i .Fisica dell' Un i ve r s i t h - P a d o v a

I s t i t u to .Yaz iona le d i F i s i c a N u c l e a t e - Sez ione di P a d o v a

( r i cevu to il 10 D i c c m b r e 1973)

The r e n o r m a l i z a t i o n of q u a n t i z e d Yang-Mil ls field t h e o r i e s is b y now a comple- t e l y so lved p r o b l e m (1-3). H o w e v e r t h e ex i s t i ng p roofs are r a t h e r invo lved .

An essen t ia l too l for t h i s r e n o r m a l i z a t i o n p r o g r a m is p r o v i d e d b y t h e S l avnov-Tay lo r i d e n t i t i e s (4.5) for c o n n e c t e d G r e e n func t ions . R e c e n t l y LEE (~) was able to de r ive f r o m t h e S l av~ov-Tay lo r i d e n t i t i e s new W a r d i d e n t i t i e s for one -pa r t i c l e i r r educ ib le G r e e n func t ions (ver t ices) . He a n d VELTMAN (7) h a v e s t r e s sed t h e fac t t h a t those i d e n t i t i t e s shou ld s impl i fy g r e a t l y t h e r e n o r m a l i z a t i o n p r o c e d u r e (*).

A n o t h e r i m p o r t a n t too l in t h i s r e s pec t is t h e r e c e n t l y p r o p o s e d d imens iona l r egu l a r i za t i on (s-lo).

In t h i s l e t t e r t h e Lee i den t i t i e s a n d t h e d i m ens iona l r egu l a r i z a t i on are exp lo i t ed to c a r r y ou t in a s imple w ay t h e r e n o r m a l i z a t i o n p r o g r a m for Yang-Mil ls field theor ies .

L e t us cons ider t h e L a g r a n g i a n

(1) L0 (o) 1 ~ = I ~ . ( ~ ) - - ~ C (q~)C (~) + � 89 ~ ,

w h e r e ~I~ is i n v a r i a n t u n d e r t h e in f in i t e s imal t r a n s f o r m a t i o n s

(2) ~o~0i = (z]~ Jr ti~q~j) ~ , i .e .

a n d

(3) C~(~) = _ F i ~oi,

~ t ~ ) (4) ~ ( ~ o ) = F~ (A~ + .

y(o) (~o~inv ~ 0

(1) G. T'HOOFT and M. VELT~AN: Nucl. Phys., $0 B, 318 (1972). (2) B. LEE and J. Z[~rN-JUSTI~r Phys. Rev. D, 5, 3121, 3137, 3155 (1972). (8) D. ROSS and J. TAYLOR: Nucl. Phys., 51B, 125 (1973). (~) A. SLAV~OV: Theor. and. Math. Phys. , 10, 99 (1972). (~) J. C. TAYLOR: NUCl. Phys., 33 B, 436 (1971). (~) B. LEE: Phys. Lett., 46B, 214 (1973). (~) M. VELTMX~r: CERN preprint TIt 1742 (1973). (*) Indeed, according to what announced by B. W. LEE (e) they have been used by him to generalize to any compact Lie groups the rcnormalization procedure along the line of ref. (2). (~) G. T'HOOFT and 5{. VELT~.~r: Nucl. Phys., 44 B, 189 (1972). (*) C. BOLLINI and J. GIAMBIA(~I: Nuovo Cimento, 12 B, 20 (1972). (10) j . ASHORE: Left. Nuovo Cimento, 4, 289 (1972); G. M. CICUTA and E. MONTALDI: Lett. Nuovo (~imento, 4, 329 (1972).

541

542 ~. TONIN

Here and in the following the compact notation of ref. (e) is used: Lat in indices have a threefold meaning, i.e. they stand for internal and Lorentz indices and space-time variables. Greek indices stand for both internal indices and space-time variables.

Repeated indices indicate summation over internal and Lorentz indices and inte- gration over space-time variables, u ~ are the Faddeev-Popov fields. ~i is the set of all the other fields (scalars, vectors and spinors). ~ are gauge parameters. With these notations L (~ is writ ten explicitly in the form - - inv

(5) L ( o ) 1 0 1 o 1 0 tnv -- ~ l"~ij q)i qJJ Jr- ~6 l~jk ~i ~J q)k Dr- ~ ll~ikh ~i ~i ~k q)h"

In order to assure the locality of the Lagrangian Lo, the quantit ies F~, A~ ,t~, F~, F ~ , F ~ a , must be ~-functions or derivatives of &functions in the space-time variables. We shall assume that A~ and / ~ contain at most first-order derivatives of &functions and t~ only &functions. Moreover we shall assume F~, F~ F~.~h such that all the terms in eq. (5) have canonical dimensions less than or equal to 4 (re- normalizable terms). C~(~) is chosen in such a way that the matrices (X~ -~ = F ~ --/7~/7~ and M ~ are invertible: their inverses are the zeroth-order propagators of the q-fields and FP particles respectively. A Lagrangian specified by a set of equations like eqs. (1)-(4) will be called quasi-invariant (G-quasi-invariant if the infinitesimal trans- formations under which L~=~ is invariant form a group G). The connected Green func- tions of a theory described by a C~-quasi-invariant Lagrangian obey the Slavnov iden- tities and therefore the corresponding vertices satisfy the Lee identities. Moreover the S-matrix elements are formally independent of the gauge C~(~) chosen to quantize the theory. Conversely a theorem of T+HOOFT and VELTMAN (1) asserts that if the quanti- ties A~ and/v~ ~ contain at most first-order derivatives of &functions and t~ only ~-func- tions and if the matrices M ~ and (Xi~ -1 have an inverse, the validity of the Slavnov identities implies that the transformations of eq. (2) form a group.

We note in passing that a simpler proof of the t 'Hooft-Veltman theorem can be obtained if their (~ group identities }> (Fig. B1, B 2 of ref. (1)) are exploited for vertices rather than Green functions. This proof however will not be given here. The renor- malizabflity of the Lagrangian Lo, if C~(r is a (~ renormalizable }) gauge, is evident by power counting. In the language of dimensional renormalization (n is the number of dimensions, ~ = n - - 4 ) it means what follows: it is possible to choose local and renor- malizable counterterms L (~), L (2), etc. in such a way that the Feynman graphs corresp- onding to the Lagrangian (1/~)L, where

L = L 0 § ~L (x) § ... § ~kL(k)§ . . . .

are regular in r = 0 and therefore remain finite in the limit r-+ 0. ~ is a loop expan- sion parameter in the sense that the graphs with l loops acquire the factor ~t-1. The true theory corresponds to ~ = 1. LCk} is the sum of the local counterterms one needs to renormalize k-loop graphs, once the graphs of lower loop order (up to k - - 1) have been renormalized. The coefficients of these eounterterms are poles in ~ = 0 of order less than or equal to k.

Assume that the transformations of eq. (2) form a compact group G o. The key probem is to show that, if L 0 is Go-quasi-invariant, also L is G-quasi-

invariant. (G o and G are, in general, isomorphic.) Then the renormalized matrix elements corresponding to the Lagrangian L are gauge

independent. Therefore if one can go continuously from a renormalizable gauge C ~ to the uni tary gauge (and this is generally possible) the theory is both uni tary and renorinalizable.

R E N O R M A L I Z A T I O N O F ] f A N G - M I L L S T H E O R I E S 5 ' ~

Some of the fields ~ (indeed some of the scalar ones) can have nonvanishing vacuum expectation values, let say e~ = ( 0 ~ 0 ) .

The v.e.v's ~ admit a loop expansion

o~ "K ~ k (k) (6) e ~ = ~ , ~ e~ .

k=O

ei are the v.e.v.'s of the renormalized theory and are finite numbers at v = 0. They are different from the v.e.v. 's ei of the theory described by the Lagrangian (1/~) ]5 0. In particular the lat ter have poles at v = 0.

The coefficients e~) in eq. (6) will be determined step by step in the course of the renormalization procedure.

Also the quantities F~ in eq. (3) can depend on the v.e.v. 's ei- For most of the gauges of interest

/v~ (~)ei = 0

holds identically. In the following we shall restrict ourselves to gauges which satisfy this condition, but this restriction is unessential. By a field transformation ~-~ ~ - one can reduce the v.e.v. 's of the shifted fields to vanish.

Of course if a Lagrangian is invariant under the infinitesimal transformation of eq. (2), the shifted Lagrangian is invariant under the transformation

After performing this shift, eqs. (1)-(4) maintain the same form in terms of the new fields, and eq. (5) acquires a new term linear in the field, i.e. F(~ moreover, also =P(~ =~(~ and A~ in the quoted equations depend on e. With this warning eq. (1) to (5) should be interpreted in the following as already expressed in terms of the shifted fields. The first coefficient e (~ in eq. (6) is fixed by the condition

F~(g ~ = 0,

which means that the Lagrangian --inv L(~ does not contain terms linear in ~ at zeroth order in ~.

Let us denote by (1/~)F(A, ~, (5) the vertex generating functional relative to the Lagrangian (1/~)Lo, where Ai are the classical fields associated with ~ ; and ~o ~, ~5 ~ are the classical fields associated with the FP particle fields u ~ , ~ .

Let us call - - X~j(A, w, ~) , - - X~c,(A, ~, 5~), - - X i~(A , oJ, (5) and - - G~,~(A, a~, Co) the inverses of the second-order derivatives of F(A, w, (5) with respect to Ai, A~.; A~, eo~; Ai, (5 ~ and w~, ~t~ respectively.

Consider the functionals

F(A) = F(A, 0, 0),

Ga~(A) = G~,~(A, O, O),

X~gA) = X i g A , O, O).

If Ai are equal to the v.e.v. 's of the fields qDi, X i j ( A ) and G ~ ( A ) are the propagators of the fieIds ~ and the FP particles, respectively.

544 ~. TO~'IN

Let us define the ver tex functions

(7)

They do not coincide with the ver tex functions (mult ipl ied by V) relat ive to the La- grangian (I/~/)L0, which are given by the functional derivat ives of F at A~= gs--e~; however, a t zeroth order in ~1, ~i = ~i.

The following definitions will be used

77(A) = - - t~ X~(A)G~r(A ) ~ G~(A)

and

Equat ion (4) implies

a'~y~ (A) ~=o" Y~;a...~,,...- 8A~,_. ~A~,,

(8) G~,~(A) = F~ (A~ + §

Performing a Legendre transformation (~) on th~ Sl~vnov-Taylor identi t ies in the form derived in ref, (~), one obtains

8F(A, (o ~5) 8I'(A, ~, ~) (9) [eo~(z~ -t- ti~A~) + t~X~-~(A, to, ~5)] + F~ A~ -- 0 .

~A ~ ~

Equat ion (9)cons idered at r (5 = 0, after a differentiation with respect to co ~, becomes

A [~r (A) ] (10) [AT + t~Ar247 ~)y~( ) ] [ ~ -t-F~F{A~ = O,

where eq. (8) has been used. The successive functional derivat ives of eq. (I0), at A = 0 are the Lee identi t ies

for the ver tex functions defined in eq. (7). For a few vertices of lowest order, t hey are

(11) (d~. + ~y~)Fi= O, a a ~x

(12) (A~ + ~y~ )y~j + (t~. + rites) F~ = 0 ,

(t~ + ~Y~;~)Yi~ (ti~ + ~Y~;~)Y~ + ~Yi;JkY~ =

] I ) e z m u t ~ t i o n s / L

x l ) e r m u t a t l o n s

�9 ~ ( t~+ Vy~;r)F~q+- 4 ~y~; , ,F~,+ -6~y~;~,y~q + 24 UY~;~'~F~ = 0,

(11) G. JONA L•SINIO: Nuovo Cimento, 34, 1790 (1964). (Is) L. QUAP.ANTA, A. ROUET, R. STORA aIld E. TIRAPEGUEI: in Proceedings o] the Marseille Con]erence, June 1972, edi ted b y C. P. KORTHALS-ALTES (1972).

REI~ORMALIZATION OF YAI~G-MILL8 THEORIES 5 4 5

where

y~ = F,r + F~'/~' .

Moreover from eq. (8) one gets

(16)

(17)

~ (,~, + v~),

These relations hold order by order in the pa ramete r V. Up to first order in ~ (0-loop and l- loop graphs) the ver tex functions appearing in eqs. (11)-(17) are

(18)

and

~o (1) ~ r,!o) V_p~, F~ = Vl~e~ + - - ~ + "t"

r, , = r,~ +- ' rJ? + ~ r . , T

..r

"K

Fijkht = ~1~ i j~h~ ,

(19)

~2~i = - g i + ~ ' i , T

r _r162

~ i ; j r , = ~ i ; j r ~ ,

In eqs. (18), (19) the poles are expl ici t ly exhibited. Notice tha t the residues of these poles are polynomials in the momenta (harmless poles in the language of ref. (s)).

The vertices F~ are defined in eq. (5). The coefficients s~ 1) in the loop expansion of e~, eq. (6), can now be fixed by the requirement tha t , at zeroth order in 7,

(20) ~ 0 (1) l~jei + P ~ = O .

At zeroth order in 7, eqs. (11)-(15) express s imply the fact t ha t ~i~,T(~ is" invariant under the var ia t ion ~0 given by eq. (2) and eqs. (16), {17) express the quasi-invariance of L o under ~o

Let us define the Lagrangian

1 1 (21) - Lt(~) = - ( L o + rjL(1)) ,

546 ~ . TONIN

where the counterterm 7].~(1)is given by

with

and

?~r(1)/~ = T(1) __ r (1 ) ~ i n v ~- ~A/u

_ _ T ( 1 ) 1 c r (1 ) f l _ _ . ( 1 ) f l ~ \ . , f l

- - u 2

y~l)a g i , (1)~ gii �9 = - - Y i i = - - �9 T T

The vertices relative to the Lagrangian (21) are free from poles in v = 0 up to l-loop graphs.

At first order in ~, the Lee identities eqs. (11)-(17) imply that LI(~) is quasi-invariant to first order in ~ under the transformations

(22) O l q 3 i = [ ( A s --7)~ i ) -~ ( t i j - - ~ i j ) ~ i ] ~ = [A~l)~('r/) -}- lij ( v } ) ~ j ] ( ~ .

In particular

(23) ~E(1)

(o) 7 ( Y i + y i ; j ~j) uJt .

In fact eqs. (11)-(15) give rise to equations for the residues of the poles appearing there, which amount to the relation

(24) ( b l A ~ E (~ A r . (1) - - 0

Recall that the variations ~o and (~1 and the quantit ies which appear in eqs. (23), (24) depend on s~: to be true, the Lee identities tell us tha t eqs. (23), (24) hold at the point ei = s~ ~ However by performing a shift on the fields, one can see immediately that they are valid also at the right values of the parameters si.

Since the variation of -inv L(~ + 'l~lnv'-r(~) under the transformations eqs. (22) is known, one can derive the Lee identities also for the theory described by the Lagrangian {21). They are analogous to eqs. (9), (10) with the addition of new terms which come from the noninvarianee at order ~2 of L (~ v(1) --i~v + ~i=v under the variation 61- The new terin that one must add to the Lee identi ty corresponding to eq. (9), is

(25)

where

L ~176 yij ~0i)]r

X , = A s + ~X~j(A, o, co) 8A~ + ~X~,(A, o, c5) ~ + ~ X ~ ( A , o, ~) ~ .

:RENORMA.LIZATION OF Y A N G - M I L L S T H E O R I E S 547

Here and in the following, the vertex generating functional, the vertex functions and M1 the order quantities previously defined in terms of L o, will be denoted by primed symbols when the starting Lagrangian is LI(V).

If one is interested in the Lee identities only to second order in ~/, one can replace in eq. (25) A~ by A~. With this approximation, the Lee identi ty corresponding to eq. (10) is

{26) ] (1,Ce ~ i ( A ) ] [ ~ - / ~-

(1) + ~3 ~Liav(A) " (1)cr _ _ ( 1 ) ~ - - .

$A~

The functional derivatives of eq. (26), at A = 0, are the Lee identities (to the order ~3) for the vertex functions F/~...~ . They are regular in ~ to first order in ~ and at order V2 they have harmless poles in z = 0. Moreover (1/~7)F~...i,, coincide with the vertices relative to the Lagrangian eq. (21) up to first order in ~.

Also the quantit ies A~I)~(V)+ VP~ and ~; t~ + VY~:r are regular at v 0 up to first order in ~/ and contain harmless poles at ~ = 0 of order ~3.

Let us call ~y~) and ~]3y~) the contributions of those poles and let us consider ~he variation

: (2)~ o~ 2 (2)c~ - - 1+(1)cr 2 (2)cr c~

Then from the harmless poles of order V2 which appear in the vertices Y~', F~j, etc., 2 r ( 2 ) one can construct a local counterterm ~ ~i~, in such a way that the Lagrangian

gives rise to Green functions (multiplied by ~) regular in z ~o the second order in ~. Moreover eq. (26) together with eq. (23) implies tha t L2(~) is quasi-invariant to the order ~3 under the variation ($3.

In fact eq. (26) gives rise to relations for the residues of the poles of the vertex func- tions, which amount to the condition

,~ i y ( 1 ) _ _ 0 "o "~ ~invr'(2) - - • (($3--($1)Li~ + ( ( $ 1 - - - o J ~ i n v -

The last term comes from the last term in eq. (26) when eq. (23) is taken into account. If eq. (24) is used, one finds tha t the variation (o) T(n ~T(2), ($~(Ll~-}-*/~i~ + ~ ~ l ~ is known and is of order ~3.

Clearly the same argument leading to L2(~) can be extended to higher orders and it allows one to define, for any k, a Lagrangian .Lk(71) which is quasi-invariant to the order ~ under the variation

(k)a a ($~, = ( A ~ ) ~ ( ~ ) + t . (~)~+) ($~ .

The quantities A~ k)~ and t~ )~ are polynomial in ~ of degree k and contain only harm- less poles in ~ = 0.

548 M. TONIX

Lk(O) is equal to /)o§ in such a way that the Green functions (multi- plied by ~) relative to the Lagrangian (1/~/)/)k(N) are regular in T up to the order ~*.

Looking at the dregree of divergence of the diagrams which contribute to A~ k)~ and t (k)~ -tr , one realizes tha t zJ~ ~)~ contains at most first-order derivatives of 6-function and i(k)~ only &functions. - i i

Therefore the t 'Hooft-Vel tman theorem can be applied and one can conclude tha t Lk(~) is quasi-invariant under a group G~ to the order ~k. Go and G k are in general isomorphic groups.

:k :tr r

I am grateful to Prof. G. COSTA for useful discussions.

Note added in proo]s.

After this le t ter was submitted for publication, a preprint by L ~ , (13) came out ia which a similar t rea tment of the renormalization of Yang-Mills field theories is presented.

(is) •. W. LEE: preprint NAL Pub. 73/71-THY (1973).