On gauge invariance of Yang-Mills theories with matter fields

IL NUOVO CIMENTO VOL. 109 A, N. 8 Agosto 1996

On gauge invariance of Yang-Mills theories with matter fields

M. DUTSCH

Institut fi~r Theoretische Physik der Universitdt Zi~rich Winterthurerstrasse 190, CH-8057 Zi~rich, Switzerland

(ricevuto il 9 Febbraio 1996; approvato il 22 Aprile 1996)

Summary. -- We continue the investigation of quantized Yang-Mills theories coupled to matter fields in the framework of causal perturbation theory. In this approach, which goes back to Epstein and Glaser, one works with free fields throughout, so that all expressions are mathematically well defined. The general proof of the Cg-identities (C-number identities expressing gauge invariance) is completed. We attach importance to the correct treatment of the degenerate terms and to the Cg-identities with external matter legs. Moreover, the compatibility of all Cg-identities with P-, T-, C-invariance and pseudo-unitarity is shown.

PACS 11.10 - Field theory. PACS 12.38 - General properties of QCD (dynamics, confinement, etc.).

1. - I n t r o d u c t i o n

In this paper we complete the study in ref. [1-4] of gauge invariance for Yang-MilIs theories coupled to matter fields in the framework of causal perturbation theory. This approach, which goes back to Epstein and Glaser [5], has the merit that only well-defined quantities, namely free fields, appear, in contrast to the ill-defined interacting fields in the usual Lagrangian formalism. The central objects in this approach are the n-point distributions Tn which appear in the formal power series of the S-matrix

(i.i) S ( g ) = l + ~ 1 I n=l --~. d4Xl ""d4xnTn(Xl' "'" Xn)g(Xl)""g(Xn)'

where g(x) is a tempered switching function. The T~'s may be viewed as mathematically well-defined time-ordered products. They are constructed inductively from the given first order

(1.2) T 1 (x) = T A (x) + T~ (x) + T~ (x),

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1146

with

(1.3)

(1.4)

(1.5)

def ig A~b (x) F[ ~ (x): , T A (x) = ~-f~b~ :Az~ (x)

T? (x) ~ - igfab~ :A~ (X) ub (x) 3z Uc (X): ,

T~ (x) de=f ijna (x)AZa (x),

where the mat ter current jz~ is defined by

f , de g (1.6) jza(x) = -~- : ~ ( x ) y z ( i [ ~ ) ~ F ~ ( x ) : .

M. DOTSCH

Herein, g, g ' are coupling constants, which are independent for the time being, f~b~ are the structure constants of the group SU(N) and - ( i / 2 ) ; t~ , a = 1, ..., N 2 - 1 denote the generators of the fundamental representation of SU(N). The gauge potentials A~,

F~'~ f3ZA[- 8~A~ and the ghost fields u~, ~a are massless and fulfil the wave

equation. The mat ter fields F and ~ ~ F + 7o may be massless or massive and satisfies the free Dirac equation

i y , ~ F ~ (x) = M~Z y3z (x), (1.7)

with a diagonal mass matrix Ma~ = m~ 5 az, m~ I> 0. We want the mat ter c u r r e n t Jtta to be conserved

(1.8) c~j~,a (x) = 0,

which requires s = M + ) ~ ( = M;t~), Va = 1, ..., N ~ - 1. By means of Schurs lemma, we conclude that j,~ is conserved, if and only if the mass is colour independent

(1.9) Ma~ = mS,z , m I> 0.

Therefore, we only consider mat ter fields fulfilling (1.9). The most important property of the S-matrix (1.1) to be proven is gauge

invariance, which means roughly speaking that the commutator of the Tn-distributions with the gauge charge

(1.10) QdeS f dax(~A~oUa) t = const

is a (sum of) divergence(s). In first order this holds true

(1.11) [Q, TA(x) + T~(x)] = igv((TA~(x) + Tl~/~(x)),

where

def .

(1.12) TA/~ (X) = igfabc :A~La (X) U b (X) F[f' (x): ,

ON GAUGE INVARIANCE OF YANG-MILLS THEORIES ETC.

(1.13) T~/~I (x) def ig = - 2f~br : U a ( X ) U b ( X ) ~ V U c (X) : ,

1147

and, by means of the current conservation (1.8),

(1.14) [Q, T~ (x)] = i3, T~/~ (x)

with

(1.15) def .

T~I (x) = ij~ (x) u~ (x).

Note that [Q, T~] is not a divergence. In order to have gauge invariance in first order, we are forced to introduce the ghost coupling T~ (1.4). We define gauge invariance in arbitrary order by

(1.16) [Q, T~(Xl, . . . , X n ) ] = i ~ ~T~/t(Xl . . . . , x~). / = 1

The divergences on the r.h.s, of (1.16) are given by n-th order T-distributions from a different theory which contains, in addition to the usual Yang-Mills couplings (1.2) a so-called Q-vertex, defined by T{/~ = TA~ + T~I + T ~ (1.12), (1.13), (1.15). (See ref. [2] for more details.) The operator gauge invariance (1.16) can be expressed by the Cg-identities, the C-number identities for gauge invariance [2,3]. It is a remarkable fact that the gauge invariance of quantized non-Abelian theories can be formulated by the simple condition (1.16) involving only the well-defined asymptotic fields (which are free fields) and that this condition does not connect different orders of the perturbation series. To clarify the latter point, note that the 4-gluon interaction is not a fwst-order term, it is of second order, namely the free normalization term

(1.17) Cig 2 :A~a (xl) A,b (xl) A~ (x2) A[ (xz): fab~fd~c 5(Xa -- X2)

of T2(xl, x2). Gauge invariance (1.16) in second order fLXeS the value of the normalization constant C uniquely and (1.17) agrees with the usual 4-gluon interaction (see [1] or subsect. 3"2).

Gauge invariance in first order requires that the coupling constants in T A (1.3), T~ (1.4), T~I (1.12) and T~/1 (1.13) agree (= g) and that the same must be true for the coupling constants in T1 ~ (1.5) and T~I (1.15) (= g'), while g' ~ g is possible. However, by considering again the tree diagrams in second order we find that gauge invariance (1.16) is only fulfilled if

(1.17a) g' = g.

This will be demonstrated in subsect. 3"2, case 5). In all other calculations we replace g' by g. (1.17a) is the universality of charge in non-Abelian gauge theories.

Our gauge invariance (1.16) implies the invariance of the S-matrix (1.1) (in the formal adiabatic limit g ~ 1) with respect to the following gauge transformations of the free fields:

z ~tY (1.18) ~(x)----)r exp[-i;LQ']O(x)exp[i;~Q'], ~ At', Fa , ua, ~a, ~f,~,

1148 M. DUTSCH

where

def. f (1.19) Q, dej( _ 1)Qg Q, Qg = z d3x : ?~a (X) ~0 U a (X): .

t = const

Qg is the ghost charge operator [6]: [Qg, u~] = - u ~ , [Qg, ~a] = Ua. Obviously, these transformations (1.18) are linear and the transformed fields are still free fields, as it must be in causal perturbation theory. Note that the latter fact excludes a local transformation ~ ~ (x) --* exp [iA(x)] y~ ~ (x). The infinitesimal versions 5~ = ~ I~ = 0 q~ of (1.18) are proportional to the (anti)commutators

(1.20)

5A~ = - i( - 1)Qg [ Q , A~] = ( - 1)Qg a" ua ,

5F~ v = - i( - 1)Qg[Q, r ~ ~] -- 0,

5ua = - i( - 1) Qg {Q, u~} = 0,

5a"~a = - i ( - 1)qg {Q, 3 " ~ } = - ( - 1)Qg 3"F~ v ,

bFa = - i( - 1)qg[Q, YJa] = 0,

5 ~ = - i ( - 1)Qg[Q,~] = 0.

These transformations are the free-field version of the famous BRS-transformations [7]. Remark: the transformations (1.18) do not completely agree with the transformations in ref.[1]: r However, due to [Q', T~(X)] = ( - 1)Qg[Q, T~(X)], our gauge invariance (1.16) implies the invariance of the S-matrix with respect to both kinds of transformations.

Because we consider transformations of the free fields only and, especially, since we cannot t ransform ~p, ~, the reader may think that (1.16) is only a restricted gauge invariance. But the main purpose of gauge invariance is to eliminate the unphysical degrees of freedom in the proof of unitarity on the physical subspace. However, there are no unphysical particles in the mat ter sector. Therefore, it causes no troubles tha t we cannot t ransform ~p, ~. In fact, we succeded in proving the physical unitari ty by means of (1.16) (sect�9 5 of ref. [4]). Moreover, the gauge invariance (1.16) yields the usual Ward identities in QED (ref. [2, 8, 9]). For Yang-Mills theories we have proven [10] that our Cg-identities (expressing (1.16)) imply the usual Slavnov- Taylor identities[11-13]. We emphazise that the Cg-identities contain more information than the Slavnov-Taylor identities, because in the latter the inner vertices are integrated out with g(x) - 1. In order to derive the Slavnov-Taylor identities from our Cg-identities, the latter integration must be carried out and we have to eliminate the distributions with one Q-vertex [10]. However, this elimination cannot be done completely in the case with a pair (~, ~ ) among the external legs. But in this case also Taylor [11] is forced to introduce the Q-vertex (1.15) to formulate the Slavnov-Taylor

�9 �9 �9 - 1 - 2 identities. He needs the C-number distributions t-~(A), t-~(A) (see subsect. 42), which are precisely the distributions with Q-vertex which we cannot eliminate [10].

We do not consider it as a weakness that we study the simple gauge transformations (1�9 only. By contrast, it is a virtue that the invariance with respect to these simple transformations implies the full content of the usual gauge invariance. However, we have not yet studied the independence from the gauge fixing�9

The proof of (1.16) follows the inductive construction of the Tn, T~/z. The crucial

ON GAUGE INVARIANCE OF YANG-MILLS THEORIES ETC. 1149

step is the causal distribution splitting dn = rn - an. The problem is that we do not have a general formula for a covariant splitting solution rn at our disposal [2]. In QED the non-vanishing mass of the fermions ~, ~ seems to guarantee the existence of the central (or symmetrical) splitting solution, which is obtained by a dispersion integral from dn (ref. [5, 8]). However, in the case of non-Abelian gauge theories, a mass m > 0 (1.9) of the matter fields does not help us in this respect because of the self-interaction of the gauge bosons.

The general proof of gauge invariance (1.16) in ref. [1-4] is not complete concerning the degenerate terms and the coupling to matter fields. We shall close these gaps and prove the compatibility of gauge invariance with discrete symmetries and pseudo- unitarity.

2. - Out l ine o f the p r o o f o f gauge invar iance

The proof follows the inductive construction of the T n , Tn/t. Therefore, it is by induction on n, too, like most proofs in causal perturbation theory. The operator gauge invariance (corresponding to (1.16)) of A,[, R~ and Dn = A~ - R ~ ,

(2.1) [Q, Dn(x l , ..., Xn)] = i ~ ~D~n/l(Xl, ..., Xn), l = l

has been proven in subsect. 3"1 of ref. [2] in a straightforward way. This proof is very instructive because it shows that our definition (1.16) of gauge invariance is adapted to the inductive construction of the Tn's. However, the distribution splitting Dn = -= R n - An is done in terms of the numerical distributions dn = rn - an, because they depend on the relative coordinates only and, therefore, are responsible for the support properties. We see that we have to express the operator gauge invariance (2.1) by the Cg-identities for D n , the C-number identities for gauge invariance, which imply the operator gauge invariance (2.1) of Dn.

However, there is a serious problem: consider the 1 = 1 term on the r.h.s, of (2.1)

(2.2) ~ D~n/1 (Xl, ...) =

= (3 ld~V)(x l , . . . )U(Xl)Av(x2) + d~V(Xl, . . . )~zu (x l )Av (x2 ) + . . . .

If d ~ ~ contains a contribution with a factor 6(xl - x3), then the terms with different field operators may compensate, due to the identity

(2.3) [U(Xl) - u(xs)] 3~15(x1 - xs) + 5(Xl - xs) ~/~ u(xl ) = O.

(Note that the contribution with u(x3) comes from the term with x~ ~ x~ exchanged, which belongs to the l = 3 term in (2.1).) Even the definition of the C-number distributions in R ' , A" (and therefore in Dn = R " - A ' ) has a certain ambiguity because terms N (36):A...: can mix up with terms N (~ :~A... : - (~ :F . . . : . To get rid of these ambiguities, we choose the convention of only applying Wick's theorem (doing nothing else) to

(2.4) A ' ( X l , ...; xn )= ~ T k ( Y ) T n - k ( Z , xn), Y, Z

where the (already constructed) operator decompositions of Tk, Tn- k are inserted. In

1150 M. DUTSCH

this way we obtain the natural operator decomposition of A"

r (2.4a) A~ = ~ aS: ~ : , 6~

where the sum runs over all combinations ~ of free field operators. We similarly proceed for A~/t, R/~ and R(~/l. We do not change this operator decomposition in constructing D~, Dn/1, Rn, Rnfl , An, An~l, T~, T~/l, Tn, Tn/1 and Tn, Tn/l. More

l de~ , 1 precisely, we first define d~ ) = r ~) - a '~ ) . Then we split the numerical distributions

d~ ) with respect to their supports d~ ) = r~ ) - a~ ) . Next we define ~§ ~fr(1)_ o - r '~ ) and symmetrize it, which yields t~ ) . The definition

d e f ~ (1) (2.4b) T~(/1) = ~ t~ : ~:

t~

gives T~(/~) in the_ natural operator decomposition. In order to obtain this decomposition of T~(/t) which is the n-point distribution of the inverse S-matrix we insert the natural operator decomposition of T~(/l), R'(/t) and of R~(/t) into T~(/1)= = - T,(/1) - R'(/t) - R~(/t). Thereby, R~[(/t) is an object similar to R~(/l) (see (5.23)) and its natural operator decomposition is defined analogously to (2.4), (2.4a). Note that this procedure fixes the numerical distributions uniquely, up to the normalization in the causal splitting d~ ) = r~ ) - a~ ) . The latter can only be done if ~ z (t) has causal support.

_ def . ,

In sect. 3 of [5] it is proven that the operator-valued/)~(/t) = A~(/~) - R~(/z) = ~ ~(l) : dr:

has causal support. However, due to compensations similar to (2.3), we cannot directly conclude that this holds true for the numerical distributions d~ ) . A solution of this problem is given in sect. 4 of[5](*). Let us sketch it. To simplify the notations, we

def consider a free scalar field q~(x) with the self-interaction T l ( x ) = ~:q~(x)~: in first

def y . ,

order. Moreover, we introduce the vertices T ; ( x ) = ~ ( m . / ( m - r)!): ~(x) ~ - ' ' 0 <<. ~< r ~< m, which are sub-Wick monomials of T1 (x) - T o (x). Then, F,~ ~ ~ (Xl . . . . , x~)

denotes the n-th order F-distribution (F = A ' , R ' , D, A, R, T ' , T, T) with a vertex T~9(xy) at xj for all j = 1 . . . . , n. The operator decomposition

(2.4c) Fn~ . rn ( X l , " " , Xn ) =

= E ( ~ , F r l + s l " r~ t+S~(X l , . . . , X n ) ' Q ) : ~ ( X l ) S l " " ~ ( X n ) S ~ : ,

s, . . . . s~, O<st <~m- r~ 81! . . . 8 n !

F = A ' , R ' , D, A, R, T ' , T, T, is trivially fulfilled in first order n = 1. Epstein and Glaser prove that (2.4c) holds true in all orders if the inductive construction of all T ~ ' ~ , 0 ~< ri ~< m, is strictly done in the way (2.4), (2.4a), (2Ab) described above. In other words, (2.4c) agrees with the natural operator decomposition of F~ 1 ~ Obviously the numerical distributions (t~, D r l + sl .r , + s~ ( X l , . . . , Xn ) if2) o f

D ~ ~ "rn(x 1, ... , X~) have causal support, since this holds true for ~nDr~+s~" ~ + ~ . The translation of this formalism to our Yang-Mills model is straightforward. The operators A(x ) and F(x), respectively ~p(x) and ~(x), must be considered as independent fields. Our T~(/t)-distributions are the 0 .0 T~(/t). The normalization of the

(*) I am grateful to Prof. R. Stora for discussions about this point.


corresponding numerical distributions (tg, 81.. ~ .., T</l) ( x l , . x~)f2) is not restricted in any way. This will be important for our proof of gauge invariance.

Let us return to our operator gauge invariance (2.1) (respectively, (1.16)). Starting with the natural operator decomposition of D~ (T~, respectively) and D~/L (T~/~), we commute with Q or take the divergence 8~ ~ and obtain the natural operator decomposition of (2.1) ((1.16), respectively).

However, due to (2.2), (2.3), the Cg-identities for D~ cannot be proven directly by decomposing (2.1). We must go another way: instead of proving the operator gauge invariance (1.16), we prove the corresponding Cg-identities (by induction on n ), which are a stronger statement. In this framework the Cg-identities for D~ can be proven by means of the Cg-identities for Tk, Tk in lower orders 1 ~< k ~< n - 1.

The Cg-identities for D~ (or T~, respectively) are obtained by collecting all terms in the natural operator decomposition of (2.1) (or (1.16), respectively) which belong to a particular combination : 6: of external field operators. By doing this, one has to take care of the following two specifications A) and B).

A) There is a speciality concerning matter fields. A divergence 8~ ~ on the r.h.s. of (2.1) ((1.16), respectively) acting on %0 or ~, appears always in the form xvSVF or 8v~X, (see subsect. 4"2 below). Due to the Dirac equation

(2.5) y ~ 8 v ~ ~ = - im~ ~, 8 ~ ~ X ~ = im ~ ,

we do not obtain a new field operator, as this is the case in pure Yang-Mitls theories (e.g., 8vu(x) is not proportional to u(x)). The terms with the divergence acting on ~f or

and those with the divergence acting on the numerical distribution, belong to the same Cg-identity. Therefo_re, we always apply the Dirac equation (2.5), i f the divergence acts on ~ or ~.

B) The arguments of some field operators must be changed by using &distributions, i.e. by applying the simple identity

(2.6) :B(xi) ~(X): ~(G - xk)... = :B(xk) ~(X): 5(xi - xk). . . ,

where X ~ ( x ~ , x2, ..., x~o) and ~(X) means the external field operators besides B. Soon, this will be explained further.

By means of A) and B) we are now able to give a precise definition of our assertion that the Cg-identities hold: we start with the natural operator decomposition of (1.16), applying always the Dirac equation according to A ). Using several times the identity (2.6), we can obtain an operator decomposition

(2.7) [Q, T~ (X)] - i 2 8z T~fl (X) : ~ v; (X): G (X): / = 1 )

(where vj(X) is a numerical distribution and : ~ ( X ) : a normally ordered combination of external field operators) which fulfils

(2.8) vj (X) = 0, Vj.

The decomposition (2.7) must be invariant with respect to permutations. The latter means that the numerical distribution belonging to :G(;rX): 0 r e S t ,

def. ;rX = (X~l, ..., x~)) is obtained from the numerical distribution belonging to :G-(X): (which is r;(X)) by permuting the arguments with ;r (i.e. it is given by vj(zX)).

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Otherwise, we would get contradictions in the Cg-identities (2.8). We call (2.7) the Cg-operator decomposition of (1.16).

A Cg-identity is uniquely characterized by its operator combination :6z:. The terms in a Cg-identity are singular of order [9]

(2.9) I ~ l + 1

at x = o, where

3 (2 .10 ) 16~1 = 4 - b - g~ - g a - ~ ( g v + g ~ ) - d .

Here, b, g~, ga, gv, g~ are the number of gluon, u, ~7, to, ~-operators, respectively, in ~ , and d is the number of derivatives on these field operators. This was shown in ref. [2].

In the various diagrams contributing to T~ and Tn~/l, we have the basic external field operators A, F, u, 3~, ~ and ~. Going over to [Q, T j , we get one external field operator ~u or 3F in each non-vanishing term. In ~ ~ T~/l, the derivative may act on the numerical distribution or on an external field operator. In the first case all external field operators are basic ones (i.e. A, F, u, ~ , ~f and ~). In the second case we always apply the Dirac equation (2.5) and decompose

1 F (2.10a) ~ A , = ~ ~ + (~,A~)~,

def where (~,A~)~ =(1/2)(3~Az + 3~A~). We see that in this second case terms with one non-basic field operator au, 33~, ~F, (~A)~ and terms with only basic ones appear. (The latter are the (1/2)F-terms in (2.10a) and the 8F, ~ - t e r m s . ) With that we are able to define the various types of Cg-identities (see sect. 2 of ref. [3]):

Type Ia: : 6z: contains one non-basic field operator ~u or aF. All terms of [Q, Tn] belong to type Ia.

Type Ib: :6z: contains one non-basic field operator 83~7 or (SA)~.

Type II: :6z: consists of basic field operators only.

The following statement is a part of our assertion that the Cg-identities hold: the natural and the Cg-operator decomposition agree for the terms of type Ia or Ib (i.e. the terms with one non-basic field operator). The 5-identity (2.6) and the Dirac equation (2.5) are applied for terms of type II only.

Now we classify the various terms in (1.16), respectively (2.7), in another way. There are no pure vacuum diagrams, i.e. terms with no external legs. This is obvious for [Q, T~]. Concerning the divergences, note that the diagrams of Tn/1 fulfil g ~ = g a + 1.

The distributions Dn, D~/1, An, An~t, Rn, R~/~ contain connected diagrams only, due to their causal supports. However, disconnected diagrams appear in A' , A'/l, R~, R'/z and therefore also in T~, T~/z. They fulfil the Cg-identities (on Tn-level) separately. This can be proven easily by means of the Cg-identities for their connected subdiagrams, which hold by the induction hypothesis.

Let us consider a connected diagram in the natural operator decomposition of (1.16). We call it degenerate, if it has at least one vertex with two external legs; otherwise the connected diagram is called non-degenerate. Let xi be the degenerate


vertex with two external fields, say BI, B2. Such a degenerate term (i.e. a term belonging to such a degenerate diagram) has the following form:

(2.11) :Bl (x i )B2(x i )Bs(x j~) ... Br (xjr_2):

A(xi - xk) t~_ 1 (xl - Xn , . . . , x i - Xn, . . . , X n - 1 -- Xn) ,

where k ~ i, Jt ~ i (Vl = 1, ..., r - 2) and the coordinate with bar in tn_ 1 must be omitted. In general, there is a sum of such terms (2.11) belonging to the fLxed (degenerate) operator combination : ~: ---- :B1 (xi) B2 (xi) B8 (Xjl)... Br(xj~_2):. For A ( x i - xk) the following possibilities appear:

a) A = D F , ~ D F , ~ v D F (lu ;~ v ) , ~ Q ~ u ~ v D F (/z ;~ v ;~ o ;~ /~) , S F ,

b) A = 5(4), 35(4)

If A = 3SF, we apply the Dirac equation y v ~SF = -- i5 (4) _ imSF. The 36 (4)-terms in b) cancel (see remark 3) below). If a degenerate term (2.11) with A = 5 (4) (type b)) can be transformed in a non-degenerate one by applying (possibly several times) the identity (2.6) only, we call it 5-degenerate; if this is not possible we call it t ru ly degenerate. All other degenerate terms (i.e. the terms of type a)) are called truly degenerate, too.

Remarks: 1) By considering the possible diagrams for a 5-degenerate term, we shall see (at the beginning of sect. 4) that all 5-degenerate terms can be transformed in non-degenerate form by applying

(2.12) : B l ( x i ) B 2 ( x i ) . . . : 5 ( x i - X k ) . . . + ( X i ~ =

---- :B1 ( x i ) B2 (Xk) . . . : 5 ( X i -- Xk) . . . + :B1 ( xk ) B e ( x i ) . . . : 5 ( x i - Xk) . . . ,

only, where the term with x~, xk exchanged is taken into account. This identity (2.12) is a special case of (2.6). However, (2.12) does not suffice to transform all truly degenerate terms from the natural into the Cg-operator decomposition; (2.6) is needed for this purpose. We recall that we apply the transformation (2.12) to the 5-degenerate terms of type II only (i.e. the 5-degenerate terms without non-basic field operator). For the 5-degenerate terms of type Ia, b the natural and the Cg-operator decomposition agree.

2) In the process of distribution splitting 5-distributions are produced (e.g., [3D(x) = 0, W l D r e t ' a v ( x ) = (~(x)). Therefore, one might think that the distinction in 6-degenerate and truly degenerate terms is not the same for Ag, R ' , D~ and A~, R~, T~. For second-order degenerate diagrams there are really more 5's in A2, R2, T2 than in A~, R~, De. However, 5-degenerate terms do not appear at all in this order (see sect. 3). In higher orders n I> 3 we take the natural splitting (see subsect. 4"2 of ref.[14]) for the degenerate terms (2.11). This splitting does not generate new 6-distributions.

3) There are terms in the natural operator decomposition of (1.16) which have a propagator A(x~ - xk) = 35(x~ - xk) in (2.11). They are generated by the divergence 3~ ~ (term l = i in ~ ~l T~/l) acting on

l

( 1 ) (2.13) g~a c~v8"~DF(x~ -- xk) -- - ~ g ~ 5 ( x i -- xk) + antisymmetrizations.

1154 M. DUTSCH

The second term is the 4-gluon interaction (1.17), which propagates in the inductive construction from second order (see subsect. 3"2) to higher orders. Doing the necessary antisymmetrizations in Lorentz indices, the ~6(xi - xk)-terms coming from the first term in (2.13) cancel with the ones coming from the 4-gluon interaction term (see sect. 2 of ref. [3]). Therefore, we need not to care about such ~5-terms.

The truly degenerate terms fulfil the Cg-indentities (on Tn-level) separately, by means of the Cg-identitiesfor their subdiagrams (subsect. 3"1). The latter hold by the induction hypothesis. The exception are some tree diagrams in second and third order, which need an explicit calculation (subsect. 3"2).

The disconnected and the truly degenerate terms cancel separately in (1.16). There remain the non-degenerate and 5-degenerate ones, which are linearly dependent. Therefore, the 5-degenerate terms of type II must be transformed in non-degenerate form by using (2.12). In this way we obtain completely new Cg-identities, in contrast to the disconnected and the truly degenerate Cg-identities, which rely on Cg-identities in lower orders.

Therefore, it is not astonishing that the difficult part of the proof of the Cg-identities concerns the non-degenerate :~: of type II (including 5-degenerate terms). In sect. 4 we prove the Cg-identities for the non-degenerate and 5-degenerate terms. In part a) we prove them for An, R" (and therefore also for Dn) by means of the Cg-identities in lower orders. In the process of distribution splitting the Cg-identities can be violated by local terms only which are singular of order ] ~1 + 1 (see ref. [2, 9]), i.e. the possible anomaly has the form

I~l +1 (2.14) a(xl, . . . , Xn) = E C b D b ( ~ 4 ( n - 1 ) ( X l -- Xn . . . . ) .

Ibl =0

We see that we only have to consider Cg-identities with

(2.15) ] ~z] ~> - 1.

This is only possible for Cg-identities with 2-, 3-, 4-legs and one Cg-identity with 5-legs ( :0: = :uAAAA:). For the latter, the eolour structure excludes an anomaly (2.14) (sect. 4 of ref. [4]). The Cg-identities with 2-, 3- and 4-legs without external operators ~p or ~ are deduced and proven in ref. [2, 3] and [4]. In subsect. 4"2 we list the Cg-identities with : 6z: containing ~ or ~. Their proof is given in subsect. 4"3: first we restrict the constants Cb in the ansatz (2.14) for the possible anomaly by means of eovarianee, the eolour structure (see appendix A) and invarianee with respect to permutations of the inner vertices and C-invarianee. Then, we remove the possible anomaly by finite renormalizations of the t-distributions in the Cg-identity. If a certain distribution t appears in several Cg-identities, the different normalizations of t must be compatible. For certain Cg-identities (:~z: = :uAA:, :uAAA:, :uuO~A:) the removal of the anomaly is only possible, if one has some additional information about the infrared behaviour of the divergences with respect to inner vertices (sect. 2 and 3 of ref. [4]). Luckily, this additional input is not needed for the Cg-identities with : 6z: containing ~), ~. C-invarianee does mainly the job of restricting the constants Cb (2.14) in the latter case.

To complete the inductive step, one has to prove the Cg-identities for T~ which is the n-point distribution of the inverse S-matrix. This was proven in a simple and short way at the end of subsect. 3"4 in ref. [2].

The latter proof is implicitly repeated in sect. 5, where we prove the compatibility


of the various normalization conditions, which are covariance, P-, T-, C-invariance, pseudo-unitarity on the whole Fock-space [4] and gauge invariance.

The reader may wonder why we shall spend so much words about changing the arguments of some field operators by means of the simple 6-identity (2.6). However, the linear independence of different field operator combinations :O(X): is lost, if 6-distributions are present (see (2.6) and (2.3)). Therefore, by means of (2.6), one must combine terms, having different :~(X): in the natural operator decomposition of [Q, T n ] - i ~ a l T ~ / t . If these combinations are not done in the right way, the

Cg-identities (2.8) do not hold.

3. - Truly degenera te t e r m s

The prototype of a truly degenerate term in the natural operator decomposition of (1.16) reads

(3.1) :B1 (Xl) B2 (xl) B3... Br: A ( X l - x2) . . . , zl ;a~ 6 ,

see fig. 1. The subdiagram with vertices {x2, ...x~} is an arbitrary connected diagram. For example, it is possible that x2 is an external vertex, "i.e. that we have an external field operator B~ (x2), 3 ~< s ~< r.

Replacing in (3.1) z l ( x l - xz) by 6 ( x l - x2), the resulting term needs not to be 6-degenerate. The counter examples are the two local tree terms in fig. 2 and 3 and their combinations with (3.1) which are given in fig. 4. All terms belonging to fig. 2-4 are truly degenerate, although they have a factor 6(x~- x2) (or even 6 ( x l - x2)" �9 6(x2 - x3)). A tree diagram in order n I> 4 has r i> 6 external legs. By means of (2.9), (2.10) the corresponding numerical distribution is singular of order [2, 9]

(3.2) I~l + 1 ~ < 5 - r ~ < - 1 .

(Remember that we are considering terms in the (natural) operator decomposition of (1.16), therefore, the singular order is ] ~] + 1 and not I g~ ].) We conclude from (3.2) that a tree diagram in order n I> 4 has non-local support, i.e. one propagator is A(Xj -- Xl) ;~ (~(Xj -- Xl).

These results lead to the presumption that all truly degenerate terms are exactly the terms belonging to the diagrams in fig. 1-4, where the vertices may be permuted. We do not try to prove this statement because we only need the following weaker

B 1

B2

A (xl-x2)

B 3

x2 ~ Br

Fig. 1. - The two field operators B1 (xl)Be (xl) cannot be partitioned on two different vertices by means of (2.6). Therefore, terms of (1.16) having the structure of fig. 1, are truly degenerate.

1156 M. DUTSCH

5 (Xl--X2) X 1

Fig. 2. - Four external legs can be partitioned on two vertices by means of (2.6). Therefore, such a term in the natural operator decomposition of (1.16) is truly degenerate.

version (which holds obviously): all truly degenerate terms are either the local tree terms symbolized in fig. 2 and 3, or they have the form given by the diagram in fig. 5,

with X1 ~f(xl . . . . , Xr), )[2 ~(Xr + 1, ..., Xn), A(Xl -- Xr § 1) ~ 5 (Xl -- X~ + 1). Again, the vertices may be permuted. The subdiagrams with vertices X1 (X2, respectively) are ar- b i t rary connected diagrams. Especially, for X1 = (Xl)(r = 1), fig. 5 agrees with fig. 1. Note that there are also non-degenerate and 5-degenerate te rms belonging to fig. 5.

In subsect. 3'2 we prove the Cg-identities for the local t ree terms of fig. 2 and 3, in subsect. 3"1 we prove them for all te rms of the type given by fig. 5.

3"1. Tree-like diagrams with non-local propagator. - There is a heuristic argument for the gauge invariance of the terms symbolized by fig. 5: in the process of distribution splitting the gauge invariance (2.1) of Dn can be violated by local t e rms only. However, the terms of fig. 5 have a non-local propagator A(Xl -Xr+I) . Therefore, they are non-local and cannot spoil gauge invariance.

The problem with this argument is tha t it relies on a unique splitting of a (numerical) distribution into a local and a non-local part , which does not exist. The re is an explicit counter example: first note that instead of summing up all diagrams

1 X3 /

Fig. 3. - Five external legs and three vertices--a truly degenerate term in the natural operator decomposition of (1.16).


X 1

5 ( x l - x 2 )

X,

Fig. 4. - Combinations of fig. 2 and 3 with fig. 1. Not all external legs can be parti t ioned on d i f f e r e n t vertices. Therefore, such terms in the natural operator decomposition of (1.16) are t r u l y degenerate.

wi th p e r m u t e d ve r t i ces , we m a y s m e a r ou t one d i a g r a m w i t h a s y m m e t r i c a l t e s t function. Then , b y m e a n s of t he i d e n t i t y (we a s s u m e y and y ' to be i n n e r ver t i ces )

(3.3a) ( a q D r ~ t ( x - y ) ( ~ - 3 ~ , ) [ 5 ( y - y ' ) t ( z - x n , . . . , y ' - x ~ , . . . ) ] .

�9 : B l ( x ) B 2 ( x ) B s ( z ) . . . : , (p(x , y , z , . . . , y ' , . . . , x ~ ) ) ,

(3.3b) = - ( ~ ( x - y ) 5 ( y - y ' ) t (z - x~, . . . , y ' - xn . . . . )-

�9 : B l ( x ) B 2 ( x ) B s ( z ) . . . : , ~ ( x , y , z . . . . , y ' , . . . , x . ) ) -

--(~eDret(X - y ) ~ ( y - y ' ) t ( z - x n , . . . , y ' - Xn , . . . ) "

" : B I ( x ) B 2 ( x ) . . . : , a ~ q ~ ( x , y , z . . . . . y ' , . . . , x~ )} +

+ (~oDret (X - y ) ( ~ ( y - y ' ) t ( z - x ~ , . . . , y ' - x ~ , . . . ) "

�9 : B l ( x ) B 2 ( x ) . . . : , a ~ , q ~ ( x , y , z . . . . , y ' , . . . , Xn)} =

= - ( 5 ( x - y ) 5 ( y - y ' ) t ( z - x ~ , . . . , y ' - x ~ , . . . ) "

�9 : B I ( x ) B 2 ( x ) B 3 ( z ) . . . : , ~ ( x , y , z, . . . , y ' , . . . , x~)},

1158 M. DUTSCH

x 1 Xr+l

A (x l -Xr+l)

Fig. 5. - In the natural operator decomposition of (1.16) some non-degenerate, some 5-degenerate and all truly degenerate terms (apart from the terms of fig. 2 and 3) have the structure of fig. 5, where A ~ ~(4). The Cg-identities for these terms are proven in subsect. 3"1 by means of the Cg-identities for the two (arbitrary) subdiagrams with vertices X1, respectively X~.

which holds on symmetric test functions ~(x, y, z, ..., y ' , ..., x~), the special t ruly degenerate term (3.3a) (see (3.1), fig. 1) can be t ransformed into a ~-degenerate te rm (3.3b) (see (2.12)). Moreover, if t is local, the resulting term (3.3b) is local (in all variables) and, therefore, could spoil gauge invariance. Note that such transformations (3.3) are forbidden in obtaining the Cg-operator decomposition from the natural one (see (2.6), (2.7), (2.8)). (Especially, they are excluded in the definition of 5-degenerate and truly degenerate terms, which assures the uniqueness of this definition.)

In sect. 5 of ref. [3] we gave a proof for the Cg-identities of the terms (3.1) (fig. 1), for pure Yang-Mills theories. The proof there is an explicit calculation of all cases and relies on the Cg-identities for the subdiagram with vertices {x2, ...Xn}. In three steps A), B), C) we now reformulate that proof and generalize it to the terms in fig. 5. Moreover, mat ter fields are included.

A) First we introduce some notations

-- def B 1 ~ A def -- def def _ def B 6 d e f ~ . (3.4) B0 = 0, , B2 = F , Bs = u , B4 = au , B5 = ~ ,

Furthermore, let A~t(x - y ) be the (,contraction, of B~(x) with B t ( y ) , but with the Feynman propagators D F ( x - y ) instead of D + ( x - y), - D - ( x - y); and with S 7 (x - y) instead of - S § (x - y), S - (x - y) , e.g.,

(3.5) { A~[~b(x) = i g Z V ~ b D F ( x ) , A65az(x) = - iSazSF( - X),

A~4ab(X) = iJab~DF (x) , Ala = O, Aot = O, V t .

Note that Ll~t ~ 5, Vs, t. (We omit the 4-gluon interaction terms (1.17), (2.13).) For simplicity, we omit colour and Lorentz indices. We define the index Q(s) by the equation

(3.6) [Q, B~]~ = i~BQ(s)

(commutator [. . .]_ for s = 0, 1, 2, 5, 6, anticommutator [.. .]§ for s = 3, 4), e.g., Q(1) = 3 , Q ( 2 ) = 0 . For the subdiagram with vertices X1 =(Xl , . . . , x 0 (X2--

ON GAUGE INVARIANCE OF YANG-MILLS THEORIES ETC. 1 1 5 9

= (x~ § t, ..., x~), respectively), we write

(3.7) f . 8 ~ ... Tr(/z)(X1) = ~ . T r ( / t ) ( X 1 ) B s ( x l ) . + (1 = 1, . . . r) ,

Tn - ~(/z') (X2) = ~ :Bt (x~ + 1) T t - ~(/l') (X2): + ... ( l ' = 1 - r, 1 = r + 1 . . . . n) , t

where the dots mean terms without external leg at xl (x~ + l, respectively). There may be a second external leg at Xl (x~ +1, respectively), but Bs (xl) (Bt (x~+ 1), respectively) is the one which is contracted in fig. 5. Then we have

(3.8) f __ . S t .

T~(X1, X~) - E . T ~ (X1)Ast (x l - Xr + l) Tn-r (X2) . + . . . , st

_ _ . S t ,

T n / l ( X l , X 2 ) - ~ . T r / l ( X 1 ) z ~ s t ( X l - X r + l ) T n _ r ( X 2 ) . + .. . for 1 ~< r , st

~st . T r ( X l ) A s t ( x l - x r + l ) T n _ r / 1 , ( X 2 ) . . . . T n / l ( X 1 , X2 ) _ . s t . +

for l~ {r + 1, . . .n} .

The terms symbolized by fig. 5 are given by the natural operator decomposition of

�9 A st (xl - x~ + 1) - iT~./1 (X1) ~A st (xl - x~ + 1)} Tn t - r (X2): +

+ :T~ (X1) {Ast(Xl - Xr+ 1)(----- [Q, Tt-~(X2)]~ -

_ i t ~ l ~ t , T t_r / l , (X2 + i~Ast (Xl _ Xr + l) Tn_r/ l (X2 ) + . . . .

B) In order to obtain the Cg-operator decomposition of (3.8a), we consider the Cg-operator decomposition of the two subdiagrams with vertices X1, respectively )(2. By the induction assumption, the Cg-identities are true for (T~(X1), Tr/l(X1)). Considering only operator combinations : ~(X1): = : ~ (X1)B(j)(xl): (no sum over j ) which contain an external field operator B(j)(xl) , B ( j ) = B s , ~Bs, i3Bq(s), this means

(3.9a) [Q, T~ (X1)]~ Bs (xl): +- :T~ (X1) i ~BQ(~) (xl): -

- i ~ :~lTrS/l(Xl)Bs(xl): - i:TrS/l(X1)~Bs(xl): I=1

(3.9b) = ~ . v ~ ( X 1 ) : ~ j ( X O B ( 2 ( x l ) : 3

1160 M. DOTSCH

with

(3.10) ~- (X1) = 0, Vj.

More precisely: start ing with the natural operator decomposition of (3.9a), one obtains (3.9b) by applying the Dirac equation (2.5) and the 5-identity (2.6) only. Note that v~j(X1) is a sum of contributions coming from the various terms in (3.9a). (Remark: a term v,~(X1):~.(X1)B(~)(xl): (no sum over j ) in (3.9b) could have a contribution from a term tl (Xl, ..., xk ...) - 5 ( x k - Xl)B(3)(xk) (k ~ 1) in [Q, T~ (X1)] - - i ~ z T ~ / l ( X 1 ) , which has not the form of the terms in (3.9a). However, there is a

t second term t2 with Xl, xk exchanged: t2(x l , . . . , x k . . . ) = t l (xk , . . . , x l . . .) ~ 5(xk - - x l ) B ( j ) ( x l ) . The sum tl + t2 must be partitioned in a symmetrical way (see the comment to (2.7), (2.8)) on the two operator combinations :~.(X1)B~.)(xl ) : and :~. (X{)B(j)(xk): , where XI' is obtained from X1 by exchanging Xl, xk. Taking t2 with :gZj (X1)B(j)(Xl): and tl with :Oj (X~)B(i)(xk): , we really obtain (3.9).)

Now we replace in the natural operator decomposition of (3.9a) (with the Dirac equation (2.5) applied) and in (3.9b) the external leg B(j)(xl) by the ~(big external leg>>

A (j)t (xl - xr + 1) T t - r (X2), where Tn t _ ~ (X2) is in the natural operator decomposition t

Tt~ - r (X~) = ~ t~n - ~)i (X2): gZ (X2):. In detail this replacement reads i

(3.11) B~--~ E A ~ t T t _ ~ , aBs---> E 3 A ~ t T t _ r , iaBQ(~)----> E i 3 A Q ( ~ ) t T t _ ~ . t t t

Note that 3B5 always appears in the form ~ ~.B5 = - i m B s , which we replace by - imA 5~ T~_ The latter is meant by writing 3A r- 56 T~ _ ~ in the following. Analogously we proceed with ~B~. For example, for s = 2(B2 = F) it happens that ~A~t(x 1 - xr+ 1) contains terms - - 5 ( x l - x ~ + ~ ) . These 5-terms are new terms, produced by the replacement (3.11). We omit them on both sides of the resulting equation, since we are interested in terms - A ( x ~ - x r + l ) , A ~ 5 only. In this way eq. (3.9) remains true

(3.12)

1 - - _ ~ - } T ~ _ ~ ( X 2 ) . iTS/l(X1)~Ast(xl X r + l ) + Tr(Xl)i~AQ(s)t(X 1 X r + l ) t . ___

J nod~

= ~ , v v ( X 1 ) A ( j ) t ( X l - X ~ + l ) t ~ n - ~ ) i ( X 2 ) : ~ j ( X 1 ) f f t i ( X 2 ) : I , i j t ~ r)

where ,nod- means the natural operator decomposition and ~,~-5, that the terms 5(xl - xr+ 1) are omitted. The numerical distributions on the r.h.s, vanish by means

of (3.10). The only difference between the 1.h.s. and the r.h.s, of (3.12) is that the arguments of some field operators B(xk), k e {Xl, ..., Xr} are changed by using the 5-identity (2.6). Analogously, by means of the Cg-identities for (Tn- r(X2), Tn- ~/t (X2))


and with Tr ~ (X1) = ~. t~ (X1): ~" (X1): (natural op. dec.), one proves 3

(3.13) { ( n r )

:Trs(Xl) Ast(Xl--Xr+ 1) +-[Q, T t - ~ ( X 2 ) ] ~ - i ~ ~l,T~-~/1,(X2) + l '=1

T i ~ Z J s t ( X l - X r + l ) T t - r / l ( X 2 ) - i ~ z J s Q ( t ) ( X l - X r + l ) T t - r ( X 2 ) ] : I nod, +~

= E t~ (X1)A ~(~)(xl - x~ + 1) T(n - r)i (X2): ~ (Xl) ~ i (X2): I s~3 §

with

(3.14) v(~ - r)i (X2) = 0,

C) The non-trivial step in the proof is the cancellation of the terms

�9 s ( X 1 ) i ~./~ 1 ) T ~ t - ~ ( X ~ ) : +-. T~ Q(s)t (Xl -- Xv +

V/.

and

- :T~r (X1) i 3/1 sQ(t)(Xl - Xr + 1) Ttn r (Z2):

in the sum of the 1.h.sides of (3.12) and (3.13). Since the (anti)commutators of Q with F , u, ~f and ~ vanish (1.18), the following cases appear only:

1) s = 4, t = 1: ~BQ(~)(xl) = ~B2(xl) = ~ F ' Z (xl), Bt(xr+l) = Ae(xr+l ) for (3.12); B~ (Xl) ~- ~ t~ ?~(Xl) , ~BQ(t) (Xr + 1) -= ~B3 (xr + 1) -- ~Q U(Xr + 1) for (3.13). Omitting the terms - 5(Xl - xr+l), one easily verifies ~A21(xl - xr+l) + 3~43(xl - X~+l) = 0.

2) s = 4, t = 2: ~BQ(s)(Xl) = 9vFVZ(xl) , B t ( X r + l ) = F e ~ ( X r + l ) for (3.12); BQ(t) (xr+ 1) = 0 and therefore /1 ~q(t) (xl -- xr+ 1) = 0 for (3.13). Omitting the terms - ~ ( x l - X~+l), one obtains ~A22(x1 - x~+l) = 0.

The further cases s = 1, t = 4 and s = 2, t = 4 are completely analogous to 1), 2). Taking this cancellation into account, the sum of the 1.h.sides of (3.12) and (3.13) is exactly the natural operator decomposition (with the Dirac equation applied) of the terms in (3.8a) (symbolized by fig. 5).

Adding up the r.h.sides of (3.12) and (3.13), the above cancellation 1), 2) (of all terms with the commutated leg contracted) happens, too. This can be seen in the following way: :T~(X1)i~BQ(8)(xl): in (3.9a) is the term with Q commutated with B~ (xl). Therefore, it belongs to a Cg-identity of type Ia and, by means of our induction assumption, is unchanged by going over from the natural operator decomposition to the Cg-operator decomposition (see sect. 2). This remains true for the terms ++- :T~r (X1) i 3zJ Q(s)t (Xl -- Xr + 1) Tt - ~(X2): in (3.12). The analogous s tatement holds true for the terms - :T3(X1)i3A~Q(t)(Xl - x~+ 1) Ttn-r(X2): in (3.13). In other words, the terms cancelling above agree identically in the 1.h.s. and r.h.s, of (3.12), respectively (3.13). Therefore, they also cancel in the sum of the r.h.sides of (3.12) and (3.13), without use of any (5-distribution. We conclude that the Cg-operator decomposition of the terms in (3.8a) is obtained by the sum of the r.h.sides of (3.12) and (3.13). Since the numerical distributions of all terms in the latter sum vanish (see (3.10),

1162 M. D~STSCH

(3.14)), the Cg-identities hold. The reasoning given six, seven sentences above for . S t . +-.Tr(X1)i~AQ(s)t(x I - x r+l )Tn_r(X2) . holds true for all terms in (3.8a) (fig. 5) of

type Ia, b: the natural and the Cg-operator decomposition agree for the lat ter terms. "

3"2. Tree diagrams in second and third order. - In ref. [1] we considered gauge invariance in the form

[Q, T~] = sum of divergences.

We proved it there for pure Yang-Mills theories in second order and for the tree diagrams in third order. Here, we return to these proofs by using the Cg-identity technique. Note that the Cg-identities imply gauge invariance in the stronger form (1.16), which contains the Q-vertices. Moreover, mat ter fields are included here.

Let us consider gauge invariance (1.16) in second order

2 (3.15) [Q, Te (Xl, xe)] - i E ~ T~/t (x~, xe).

/=1

In this sect. 3 we only consider the tree diagrams in (3.15). (The loops are non-degenerate.) We collect all terms in the natural operator decomposition of (3.15) belonging to a fixed tree operator combination :t~:: the non-local terms (i.e. the terms -- DF (x, -- x2), ~DF(xl- -X2) , ~t t~vDF(x l - -X 2) (no contraction of it, v), ~ ~ ~ D F (xl - x2) (no contraction of/~uv), S F (xl - x2) or S ~ (x2 - xl)) cancel. This was proven in subsect. 3"1. There remain the local terms ( - 5 (x~- xe), - 3 5 ( x ~ - x~)). In order to get them in the Cg-operator decomposition (2.7), we need to apply the identity (2.6). Since local terms can only appear for I~l I> - 1 (see (2.9)), we merely have to consider the cases:

(3.16)

1) : ~1 (Xl, x2): : :n/~a (Xl) ~vUb (Xl)AQd (x2)A~e (X2): ,

2) : ~2 (Xl, x2): = :Az~ (Xl) U b (xl)Aed (x2)A~ (x2): ,

3) : ~ (xl, x2): -- :Az~ (xl) ub (x l )A~ (x2) FQ~ (x2): ,

4) :5Z4(x 1, x2): = :A~a(Xl)Ub(Xl)Ud(X2)~eUe(X2):,

5) : t~(Xl , X2): = :Ua(Xl)A~b(Xl) ~(X2).. . ~0(X2): �9

Additionally, we have the cases with Xl, x2 exchanged in (3.16). The partition of the local terms on the two operator combinations : ~. (Xl, x2): and : ~- (x2, xl): must be done in a symmetrical way (see the comment to (2.7), (2.8)). Note that the local terms, which we shall compute, do not cancel in the sum v j ( x l , x2 ) :~ j (X l ,X2) : + -~- T,j(X2, X l ) : ~ ( X 2 , Xl): , in all cases (3.16).

Le t us consider the three tree terms

(3.17) T2 (xl, x2) =

_/g2 - - :Aza (xl)A,b (xl)AQd (x2)niie (X2): fabcfdec {[g~ ( ~ 3e D F (Xl -- X2) +

+ CagV~(5(xl - x2)) - g ~ ( ~ D ~ ( x l - x2) + Cag~QS(xl - X2))] - [)~o~]} + . . . .


(3.18) T~/l(X~, x2) =

- ig ~

2 - - :n#a (Xl) u b (Xl)Aed ( x2 )A~ (xe): f a ~ f ~ {[gZ~ (~" 3 eD~ (Xl - - X2) +

+ Cbg 'e5(x l - x2)) - g ~ ( ~ o Q D F ( x I -- X2) + Cbg~eS(Xl -- X2))] -- [)[ <--) ~)]} § ...

and the term of T2/2 obtained from (3.18) by exchanging xl and x2. (Note T2/2(x1, x2) = T2/1 (x2, xl).) These tree diagrams have singular order [9] (o = 0 and, therefore, a free normalization term ~ Ca,, b S ( X l - x2) has been added.

Case 1) This case is of type Ia, whereas (2)-(5) are of type II. Due to [Q, A,a (x)] = i $ , u , (x), we have two equal terms from Q commuted with (3.17) which

�9 261 V contribute. There is also a contribution from ~3. T2/l(X~, x2), generated by the divergence 3~ acting on ub (Xl) in (3.18). Considering the local terms only, we see that this Cg-identity is fulfilled iff

(3.19) Ca = Q .

Case 2) There is only a contribution from the divergence ~1 acting on the numerical distribution in the term (3.18) of T~/1 (x l , x2). Because of [3D F = 5, local terms appear in this Cg-identity only. One easily obtains that the latter is equivalent to

1 (3.20) Cb -

2

Gauge invariance fLXeS the values of Ca, Cb uniquely. The Ca-normalization term (in (3.17)) is the 4-gluon interact ion. It propagates into higher orders in the inductive construction of the T~'s (see subsect. 4"2 of ref. [14]).

Case 3) The divergence 9~ ~ acting on the numerical distribution in

(3.21) T[/I (x~, x2) = - i g 2 : A z a ( x i ) u b ( x l ) A 3 ( x 2 ) F e ~ ( x 2 ) : �9

" f a b c f d e c [ g ~ D F ( x l -- X2) -- g ' ~ D F ( x l -- X2)] + ... ,

produces a non-local and a local term. We need to consider the latter only

(3.22) ~1T[ /1 (xl , x2) - /g2 - - :Az~ (xl)ub (x l )A~ (x2)Fete (x2): "

"g~ (fabcfgec -- fdbcf~c) 5 (Xl -- X~) + . . . ,

where we have antisymmetrized f~bcfd~c in a, d, because the other terms have this antisymmetry. There is another local contribution, coming from the Q-term in (3.18), with the divergence acting on Aza(Xl). The resulting $ , A , a ( x l ) must be antisymmetrized in v/~, since the numerical distribution has this antisymmetry. We

1164 M. DUTSCH

end up with

(3.23) ~ T[/1 (x~ , xe) =

ig2 : A ~ a ( X 2 ) U b ( X l ) n ~ ( x 2 ) F ~ e ( x l ) : ~ = - - . g fadcfebc2Cb(~(Xl -- X2) -t- . . . 2

for this second local contribution. Due to (3.20) and the Jacobi identity, the sum of (3.22) and (3.23) vanishes.

Case 4) 3~ acting on the numerical distribution in

(3.24) T~/1 (Xl, x2) = ig 2 :A~a (Xl) ub (xl) Ud (X~) ~ ~ (Xe): "

"f~fd~ [g~e ~'DF (Xl -- X~) -- g~e 3ZD F (Xl - xe)] + ...

produces a non-local and a local term. The latter is

(3.25) ~ T ~ / ~ ( x l , x~) = 2 :Az~(Xl)U~(X~)Ud(X~)~e~(x~):"

"g#Q (fabcfdec --fadcfbec) (~(Xl - - X2) -~- . . . .

Another local contribution is generated by the divergence ~ acting on the numerical distribution in

(3.26) T ~/1 (xl , x2) ig2 --~ : U b ( X l ) U d ( X l ) A ~ a ( X 2 ) ~ U e ( X 2 ) : fbdc faec~VDF (Xl -- X2) -~- . . . . 2

This second local contribution cancels with (3.25) by means of the Jacobi identity.

Case 5) Now we assume that the coupling constants g ' (1.6) (of the mat te r fields) and g (1.3) (of pure Yang-Mills theory) are independent. Replacing in (3.24) the open ghost line - gfd~c :Us (X2) a~ ~ (X~): by the open mat ter line g ' ( 1 / 2 ) : ~ ( x e ) y e , ~ p ( x 2 ) : , we obtain a first local contribution analogously to (3.25):

igg ' ~ ~ :u~ (xl)Az~ (Xl) ~(x2) r~)~c y3(x2): f~br ( ~ ( X l - - X2) "~- . . . . (3.27) ~ T2/l(Xl, x2) - 2

Let us consider the two C-conjugated Compton diagrams

(3.28) T~/1 ( X l , X2) - - - - [:uo(xl)A,b(x2) ~(x2)r" ,~bS~ (x2 - x ~ ) ; t ~ ( x ~ ) : +

Using

(3.29)

+ :ua(xl)A,b(x2) ~ ( x l ) r ' ~ a S ~ ( x l - x2),~bT~'~f(x2):] + . . . .

{ ~1 (~ (x l ) y~S F (xl - x2)...) = - i~(x2) (~(X 1 - - X 2 ) . . * ,

~Xl (... S F (x 2 _ Xl ) ~]v~(xl)) ~_ ... i(~(x2 _ Xl ) ~(x2) ,


we obtain two further local contributions in 3~ T[/1 (x~, x2). These three local terms cancel by means of

(3.30) 2if~b~c -{- ~ b / ~ a - - ~ a ~ b -'~ 0 ,

if and only if g ' = g (see (1.17a))..In QED the ghost fields couple to the matter fields only. Therefore, the term (3.27) is absent. The sum (3.28) of the two C-conjugated Compton diagrams is already gauge invariant there.

We turn to the tree diagrams in third order, which have five external legs. In the natural operator decomposition of

(3.31) [q, T~ (x,, x2, x3)] - i ~ c~ Ti/z(x,, x2, xa) l = 1 5-legs

the non-local terms - A (~) (x~l - x~) A (2) (x~ - x~) or ~ A (3) (x~l - x~) 8(x~ - x~) (for an arbitrary z e S s ; /I (~), A (2), A(8 )~5) cancel (see subsect. 3"1). For the A (3) 5-terms the cancellation works only, if x~ is replaced by x~ (or vice versa) in the arguments of some field operators, using 5(x=e- x~). These replacements are determined by the Cg-identities for the subdiagrams with vertices {x~, x~} (see (2.6), the tree diagrams in second order and subsect. 3"1). There remain the local terms -58(xl-x3,x2-x3). Due to (2.9), (2.10), they must have the field operators :uAAAA:. (In all other cases one has a derivative on the external field operators and, therefore, I~[ + 1 is smaller than zero.) We write the local terms in the following symmetrical form:

C ~ ~s (~ _ x~, x~ - x~)[: ~ ~ (xl): + : ~ , ~ (xe): + : ~ , ~ (x~): ] bcd~ t~ kwl (3.32)

with

(3.33) : t~ abcdezv~ (X): = :Ua (X) A~b (X) Avc (x) A~d (X) A~ (x): .

~ must be inva~ant with respect to permutations of (/~, b), (v, c), The constant ~ab~de C ~ which is Lorentz- and SU(N)-invariant (see (;t, d), (v, e). There is no such tensor ~b~d~

sect. 4 of ref. [4]). This finishes the proof of the Cg-identities for third-order tree diagrams.

Remark: the Cg-operator decomposition of (3.31) which we constructed, has the following property: all local terms belong to a :gz 6 (x~):, i = 1, 2, 3. For example, the

dg operator combination : 07: = :u(xl)A(xI)A(x2)A(x3)A(x3): contains non-local terms only. This is not necessary: one can construct another Cg-operator decomposition of (3.31) in which all local terms are partitioned in a symmetrical way on the operator combinations of the non-local terms (e.g., :07: ). Then :0-6: would not appear. We followed this latter procedure in our Cg-operator decomposition (3.16) of the second-order tree terms, with the exception of : ~ : . (Only local terms exist with operators :uAAA: .)

4. - Non-degenerate and 5-degenerate terms

In this section we prove the Cg-identities belonging to non-degenerate operator combinations : ~: in (2.7) (including 5-degenerate terms if : ~: is of type II) and the Cg-identities with 5-degenerate :~: in (2.7) which are of type Ia or Ib.

1166 M. DUTSCH

First we characterize the diagrams of the 5-degenerate terms in (1.16). In (3.2) we realized that the tree diagrams can have local support in second and third order only. Obviously, this result holds true even for arbitrary subdiagrams: a term in (1.16) has at most two 5(4)-distributions (as propagators) in direct neighbourhood. Neither a chain of three (or more) ~(4), nor three ~(4) joining the same vertex, do appear. Consequently, all diagrams of the 5-degenerate terms have the following form:

1) fig. 1 with A ( x ~ - x2) replaced by 5 ( X l - x2), or

2) the first diagram of fig. 4 with zl(x2 - x3) replaced by 5(x2 - xs).

Every 5-degenerate term can be transformed in non-degenerate form by applying (2.12) once in 1) and twice in 2). Remark: In (3.32) we proved that the tree terms with local support cancel in third order. We argued by means of the permutation symmetry of the external field operators. However, this symmetry is lost if this tree diagram is a subdiagram. Therefore, case 2) must be considered, too.

4"1. Proof of the Cg-identities for A ' , R" and D~. - By means of the Cg-identities for Tk and Tk in lower orders 1 ~< k ~< n - 1, we are going to prove the Cg-identities for A~. The proof for R~ is completely analogous. Together, we shall obtain the Cg-identities for D~ = R~ - A~. This proof was given already in subsect. 3"4 of ref. [2]. However, we did not care about the degenerate terms and the coupling to matter fields was not included there.

A) In the natural operator decomposition of [Q, A ~ ] - i~OzA' / l , we omit all t

disconnected terms and apply the Dirac equation (2.5) to all terms of type II and obtain

(4.1) [Q, A" (X)] - i Y~ 31A "/t (X) = ~, ~XnJ--'(1) ~.-'~) -tV'l" ~(1 ) (X):, 1 3

where a~31)(X) does not vanish in general. By means of (2.12), we transform the 5-degenerate terms of type II in non-degenerate ones. Furthermore, we omit all truly degenerate terms. In this way we obtain the Cg-operator decomposition (see (2.7), (2.8)) of the non-degenerate and 5-degenerate terms

(4.2) [Q, A d (X)] - i • atA'/t (X) I = ~" a~) (X): 9 (2) (X):. 1 non-deg j

All :~(2)(X): of type II are non-degenerate. Our aim is to prove

(4.3) a~j 2) (X) = 0, Vj.

For this purpose we construct in B) another operator decomposition of [Q, A ' ] - - i Y~ 3tA~/z, which fulfils (4.3) by construction. In part C), we shall prove the agreement

l of the two operator decompositions constructed in A) and B).

B) We insert in (see (2.4))

l , l, xt e


(4.4b) Y, Z l, xl e {Z , x . }

the Cg-operator decompositions in lower orders k, n - k (see (2.7), (2.8))

(4.5) [Q, Tk(Y)] - i ~ 3~Tk/t(Y) = ~ k r ( Y ) : ~ ( Y ) : , I, xl e Y "r

(4.6) ~k~(Y) = 0, Vr,

(4.7) [Q, T~_k(Z, x~)] - i ~ 3~T~_k/~(Z, x~) = ~ v ( . _ k ) ~ ( Z ) : ~ ( Z ) : , l, xl ~ {Z , x~ } s

(4.8) ~(n - k)8 (Z) = 0, Vs.

For Tn_k(Z, x~) in (4.4a) and Tk(Y) in (4.4b) we insert the natural operator decomposition. Afterwards, we only apply Wick's theorem to (4.4a), (4.4b) and omit all disconnected terms. The resulting operator decomposition

(4.9) (4.4) = (4.4a) + (4.4b) = ~. a~j ~) (X): 9 (3) (X): 3

fulfils

(4.10) a~} ) (X) = 0, Vj,

due to (4.6), (4.8). Again, we omit all truly degenerate terms. (In part C) we shall see that (4.1) and (4.9) can be obtained from each other by applying the 5-identity (2.6) only. Since the distinction in truly degenerate and 5-degenerate/non-degenerate terms does not depend on the application of (2.6), we omit the same terms here, as we did in part A).) Moreover, we tranform the 6-degenerate terms of type II in non-degenerate form by means of (2.6). (Note that (2.6) is needed here, instead of the more special (2.12) in (4.1), (4.2).) There are several possibilities to do this transformation for a single term. It does not matter which one we choose. But the permuted terms must be transformed in the same way. Then, the sum over permutations is a totally symmetrical partition of the external legs on the vertices. For example a 5-degenerate term 5(S)(xt- x3, x2 - xs):Bl(xs)B2(x3)...: goes over in ~ 5(S)(xl-x3, x2-x~):BI(X~I)B2(x~2)...:. We end up with the operator

z c S ~

decomposition

(4.11) (4.4) ] non-deg = E a~ 4) (X): Gj (4) (X): , 3

which consists of 5-degenerate Ia, b and of non-degenerate : 9 (4) (X): of all types, and fulfils

: 9 (4) (X): of type

(4.12) a~9 ) (X) = 0, Vj,

by means of (4.10).

C) We are going to prove that (4.2) and (4.11) agree

(4.13) : 9 (2) (X): = : 9 (4) (X):, Vj, a~j 2) (X) = a~ 4) (X) = 0, Vj,

after relabeling the index j in a suitable way.

1168 M. DU~TSCH

In A) we first applied Wick's theorem to A~(X)= ~ Tk(Y) Tn-k(Z, x~), where Y,Z

Tk(Y) and T~_k(Z, Xn) are in the natural operator decomposition, and similar for A~/t(X). Afterwards, we commuted with Q, respectively took the divergence 8~ and applied the Dirac equation (2.5) to obtain (4.1).

Note that the operation of taking the divergence 8t (with application of the Dirac equation) commutes with contracting, i.e. applying Wick's theorem. This relies on the fact that S (-+), which is produced by contracting ~ and ~, fulfils the Dirac equation (2.5), too. Moreover, the operation [Q, .] commutes with contracting. The latter was shown in detail in subsect. 3"4 of ref. [2]. (The coupling to matter fields causes no complications, since Q commutes with ~p, ~.) The non-trivial step in that proof there is the cancellation of the terms, arising by contracting the commutated leg. This is exactely the same cancellation as demonstrated at the end of the proof in subsect. 3"1.

Reversing now in A) the order of these operations, we obtain the following statement: inserting the natural operator decomposition (with the Dirac equation applied), instead of the Cg-operator decomposition, for

(4.14) [Q, Tk(Y)] - i ~, 8x, Tk/l(Y) l, Xl �9 Y

and for

(4.15) [Q, T~_k(Z, x~)] - i ~, 8x~Tn-k/~(Z, x~) l, xl~{Z,x~}

in (4.4a), respectively in (4.4b), and doing the other steps in exactly the same way as in B), we obtain (4.1) instead of (4.9). We know by the induction hypothesis (see (2.7)) that the natural operator decomposition (with the Dirac equation applied) and the Cg-operator decomposition can be obtained from each other by applying (2.6) only. The latter property survives contracting between Y and (Z, x~) in (4.4a), (4.4b). Therefore, (4.1) and (4.9) can be obtained from each other by applying (2.6) only.

The following argument is shown for (4.4b). The procedure for (4.4a) is completely analogous. Let us consider a term of (4.4b), in which we inserted for (4.15) different operator decompositions in A) and B). The corresponding subdiagram with vertices {Z, x~}, belonging to (4.15), must be truly degenerate or 5-degenerate in order ( n - k) and it must be of type II, because the natural and the Cg-operator decomposition agree for non-degenerate terms and for terms of type Ia, b. Therefore, the (whole) considered term of (4.4b) is of type II, too. (The proof is finished for the terms in (4.4b) of type Ia, b.) At the beginning of this sect. 4, we realized that there are at most two 5(4)-distributions in direct neighbourhood. Consequently, we only must consider two cases which read (before contracting in (4Ab))

Case 1)

(4.16a) tkt(Y): ~t (Y): :BI(yl)...Br(yr)Br+l(xt)...: 5(Xl - x2) t n - k ( Z , Xn) =

(4.16b) = t-kt(Y):dtt (Y): :BI(Zl).. .Bv(zr)Br+l(xl). . . : 5(xl - x2)t~-k(Z, x~),

with Yl, .-., Y~, Zl, ..., z~ e {xl, x2} r {Z, x~}, xt e [{Z, x~}\{Xl, x2}], 2 ~< r ~< 4;


Case 2)

(4.17a) tkt (Y): g~t (Y): :B1 ( y l ) . . . Br (Yr) B~+ 1 (Xt)...: 5 (s) (xl - X~, x2 - X~) t~_~ (Z, Xn) =

(4.17b) =~t(Y):~t(Y): : B I ( Z l ) . . . B r ( z r ) B r + l ( X t ) . . . : ~ ( S ) ( X l - X s , X2--x3)tn-k(Z, Xn),

with Yl, ..-,Y~, zl, ..., z~e {Xl, x2, x3} r {Z, x~}, xl e [{Z, Xn} \ {X l , X2, X3}], 3 ~-~ r ~-~ ~< 5, where t~ (Y): ~t (Y): (no sum over t) is a term of the natural operator decomposition of Tk(Y). The other factors (:B1...B~B~+I ...: 5(4/s)t~_k) are a term of the natural operator decomposition of (4.15) (for (4.16a), (4.17a)), or they are the corresponding term in the Cg-operator decomposition of (4.15) (for (4.16b), (4.17b)). These terms belonging to (4.15) are 5-degenerate for r = 2 in case 1) (r = 3 in case 2)) and truly degenerate for r = 3, 4 in case 1) (r = 4, 5 in case 2)).

Now we consider the contractions of : t~t (Y): with B~, ..., B~. If there remain s = 3 or 4 field operators B1, ..., Br uncontracted in case 1) (s = 4 or 5 in case 2)), the resulting diagram is truly degenerate and, therefore, we omit it.

If all operators B~ . . . . , B~ are contracted (s = 0), the resulting operator combinations in (4.16a) and (4.16b) ((4.17a) and (4.17b), respectively) agree and the numerical distributions are equal (e.g., D (+) ( x - Xl ) ~ ( Xl - x2) = D (§ ( x - x~) ~ ( xl - x2), x e Y ) .

Let us consider case 1) with s = 1 or 2 field operators uncontracted, e.g., B 1 or BIB2 . I f such a term is 5-degenerate (this is possible for s = 2 only), it must be transformed in non-degenerate form by applying (2.12) for (4.16a), respectively (2.6) for (4.16b). Now we consider in each case the sum with the term with Xl, xe exchanged. This sum is the same for (4.16a) and (4.16b), namely for s = 1

(4.18) t k t ( Y ) : O t ( Y ) B l ( x l ) B 2 ( x l ) . . .B~ (x l )B~+I (X l ) . . . : 5(Xl - x 2 ) t n - k ( Z , 2r n) ~-

+ ~ t ( Y ) : O t ( Y ) B I ( x 2 ) B 2 ( x l ) . . . B ~ ( x l ) B ~ + I ( X t ) . . . : ~ ( x I - x 2 ) t n - k ( Z , X~),

where all operators B2(Xl) . . . . , B~(xl ) are contracted with ~t(Y) in both terms; for s = 2 this sum reads

(4.19) tkt (Y) : Ot (Y)B1 (Xl) B2 (x2) B8 (Xl) ... B~(Xl)B~+I (xl)...: (~(Xl - x2) t~-k (Z, xn) +

+ tkt (Y): ~t (Y)B1 (x2)B2(xl)B~ (Xl) .. . Br(Xl)Br+ 1 (Xl) . .-: ~(Xl - x2) tn -k(Z, Xn),

where all operators B~(xl), ..., B~(Xl) are contracted with gzt (Y) in both terms. We turn to case 2) with the operators B1, ..., B~, s = 1, 2 or 3, uncontracted. Again,

the 5-degenerate terms must be transformed in non-degenerate form by using (2.12) for (4.17a), respectively (2.6) for (4.17b). Here we consider the sum of all 6 te rms generated by permutations of xl , x2, xs. These sums agree for (4.17a) and (4.17b): in both cases we obtain

(4.20) : . . . B1 ( X ~ l ) . . . B~ (x~)Br+ 1 ( x l ) - . . : 5 (s) ( X l - x3, x2 - x 3) . . . . ~eS3

There occur combinations of the various cases which we have discussed. These combinations are no problem, because they concern disjoint groups of vertices, each group having two (case 1)) or three (case 2)) vertices. �9

1 1 7 0 M . D U T S C H

4"2. The C g - i d e n t i t i e s w i t h m a t t e r f i e ld s . - Going over from n - 1 to n, the really n e w Cg-identities belong to non-degenerate :~ : . Therefore, only such Cg-identities are considered in this subsection. We recall our convention of denoting numerical distributions of non-degenerate terms which we have introduced in ref. [2]:

(4.21) t~... ab..~ (X2, X3, --. ) :Aa (x2) Bb (X~)... :

means the sum of the connected diagrams with external field operators (legs) A~ (x2), Bb (x~)..., a and b are colour indices. (We only write the index ~7 instead of ~ for an external field operator 3 , u ( x i ) . ) The superscripts a2 show that this term belongs to Tn~/~(x~, x2, x3, . . . ) with Q-vertex at the second argument (= x3) of the numerical distribution t. An immediate consequence of this notation is the relation

(4.22) ~1 t ~ . ~b. (x~, x2, ...) = - - t ~ . .~. (x2, x~, .. .),

where we have a minus sign, if A, B are both ghost or both matter operators and a plus sign in all other cases. The Lorentz indices of the two operators A, B must also be reversed in (4.22). Note that (4.22) particularly holds, if A and B are the same field operators. Moreover, the sum over permutations of the vertices is present in T~(/~). For example,

(4.23) Tn (xt, ..., x~) =

= E l ~ i ~ 2 ~ k < l ~ j , ~ k

�9 ~p(x~)A ,~ (xk )A~b(x l ) : + .. . + (disconnected terms),

where the coordinates with bar must be omitted. The matrix multiplication : ~ t ( . . . ) F : concerns the spinor space and the space of the fundamental representation.

The Lorentz structure T~/~ - y" (1.15) propagates to higher orders in the inductive construction of the T , / l . We conclude

u l u - 1 . t_~2 . - 2 .. (4.24) t ~ . . . = ~ t~v u . . , ~ . . = t ~ v u . ~ ,

which implies by means of the Dirac equation (2.5)

(4.25) 3~ ~ T~/1 (x l , x~) = : ~ ( x l ) ( 7 ~ ~ + i ra) -1 ..., t ~ . . (xl, x2, ...) ~2(x2) ...: + . . . .

In appendix A the following colour structures are proven:

(4.26) t ~ Z = V~pc~ a~ , t~Baa~ : T ~ p B ( 2 a ) a f l ,

where B = A, F, u and a, fl are the indices of the fundamental representation. The Cg-identities for pure Yang-Mills theories with 2-, 3-, and 4-legs were given in

ref. [2,3]. Adding the coupling to matter fields, the form of these Cg-identities remains the same, as far as we do not admit e x t e r n a l ~p or ~. Moreover, the presence of i n n e r matter lines does not change the symmetry properties of the numerical distributions. Therefore, the forms of the possible anomalies and of the normalization polynomials are unchanged. We see that the Cg-identities without external F or ~ can be proven as in the case of pure Yang-Mills theories (see ref. [2,4]).

: F(x~) t~AA~b (X~, Xj, Xk, Xl, Xl, ...,X~,Xj,Xk,Xl, ..., Xn)"

ON GAUGE INVARIANCE OF YANG-MILLS THEORIES ETC. 1 1 7 1

However, there are additional Cg-identities with external legs ~f or ~. An opera tor combination : ~ : , characterizing a Cg-identity, must fulfil g~ = g~ + 1 (see (2.10) for the notations). Consequently, only the following new Cg-identities appear (with at most 4-legs), where we use the classification given in sect. 2:

Type Ia: for the external legs :~ : = :-~(x~). . .~f(x2)3~u~(x3): we get

(4.27) ~ ~ V ~ f A = V ~ u ,

where the factor )~ is omitted (see (4.26)). Fo r : 6z: = : ~ ( x l ) . . . ~(x~)2~u~ (xs)A~,b (x4): we obtain

(4.28) ,z , ~s, t ~ ~AAab -= v~ ~uAab ,

for : 0-: = : ~ ( x ~ ) . . . F(xe) ~ u ~ ( x ~ ) F ~ (x4): we obtain

(4.29) t-~';AF~ = t~?(~F~

and for : gz: = : ~(Xl) ... F (xe) Ua (x~) 3 , F ~ (x4): we get

(4.30) ~ F ~ = 2 TM ~ V ~ - u ~ u ~ .

Type Ib: for :~ : = : ~(x~). . . y~(xe)u~(x~)(1/2)($,Az~(x~) + 3,A,~(x4)): we get

(4.31) t~4~A~ + (v , - ~ ) = 0.

Type II: for :t~: = :~(x~) . . .y~(x~)u~(xs): we obtain by means of (4.25) and (4.26)

(4.32a)

(4.32b)

(4.32c)

0 = r ' X2 ' X3 ' X4 ' . . . ) + / = 3

--2 + ( y v 8 ~ + i m ) -~r~u (x l , x2, xa, x4, . . . ) + V ~ u ( X l , x~, xs, x4, . . . ) ( ~ y - i m ) ~

+ i g [d(x8 - x l ) v ~ v ( x l , x2, x4, . . .) - T ~ y ~ ( X l , x2, x4, . . . ) d ( x 2 - x3)] § 2

g (4.32d) + : - ~ z ~ ( x l - x 2 ) t ~ ( x 3 , x2, x4, . . . ) .

2

In QED the corresponding Cg-identity has only three terms: the 1 = 3-term of (4.32a) (vertex) and the ~-degenerate terms (4.32c) (self-energy). The ~-distributions in the ~-degenerate te rms (4.32c), (4.32d) are produced by the divergence $~ of ~ 3l Tn/t,

l

acting on -~(xt) y ~ S F (Xl -- "), S F (. _ Xl) y v yj (Xt) (4.32C) or on a" D F (Xt -- ") (4.32d). The 2-legs Cg-identity (1 /2)[ t (~g az -t~u~g ~] = t ~ ~ (see ref. [2]) is inserted in (4.32d).

1172 M. DUTSCH

For :0.: = :~(x~) . . . Vd(x~)u~(xs)A~(x4): we get similarly to (4.32)

(4.33a) 0 ~ ~ z~ = ~ t ~ a ~ ( x ~ , x~, x~, x4, x~, ...) + / = 3

(4.33b) + ( ~ ~ + ira) t ~ a ~ (Xl, X~, XS, X4, X~, ...) +

-2t t ~-- + t ~ A ~ ( x ~ , X~, x~, X4, 3?5, . . . ) (3x2~2 u -- ira) +

(4.33c) + ig [2~5(xa ~' __ -- Xl)~bT~flA(Xl, X2, X4, 3? 5 , . . . ) - -

2

- ~ w ~ ( x ~ , x~, x~, x~ . . . . ))~5(x~ - x~);t~] -

(4.33d) g y x f ~ 2 c S ( X l - x ~ ) t ~ ( x 4 , xa, xe, x~, ...) - 2

(4.33e) ig [yziib ~(x4 -1 -- X l )~a~V;u(Xl , X2, X3, XS, . . . ) -- 2

-- ~ f l u ( X l , X2, X3, X5, . . . ) ~ a ~ ( X 2 -- X4)~b~] ~] q-

(4.33f) + gf~b~ ~ ~ v~ ~A (xl , x2, xa, xs, ... ) ~ (x8 - x4).

Again, all &distributions in (4.33c)-(4.33f) are generated by the divergence ~ , acting on ~(Xl) r ~ S F ( x l - "), sF(" --XI)~'y~(Xt) (4.33C), (4.33e) or on 3 " D F ( x ~ - ") (4.33d), (4.33f). The 3-legs Cg-identity t~X '~= (1 /2 ) [g~" t (~A- g a~t~u~K4] (see ref. [3]) has been used in (4.33d).

For :~ : = : ~(x~).. . ~ ( x e ) u ~ ( x s ) F ~ ( x 4 ) : we proceed in a similar way and obtain by means of (4.25), (4.26)

~ ~x~t~lttv (4.34a) O = ~ ~r Xe, X~, X~, X~, ...) + l=3

(4.34b) + (y , $x~ + ira) ~ " ~puFab(Xl, X2, X3, X4~ X5, - . . ) -Jr

+ ~ b ~ l , X2, Xa, X4, Xs, . . . ) ( ~ ira) +

(4.34C) 1 [4~4 ~

+ - L ~ A ~ b ( X ~ , X2, X~, X4, Xs, ...) -- (/~ o r ) ] + 4

(4.34d) /,/1" + ig [ ; t~(x3 - Xl)~bV~vF(Xl, X2, X4, Xs, ...) -- 2

--T~,pF(Xl, X2, X4, X5, . . . ) .~ .bS(X2 -- X3)~a ] - -

(4.34e) - g ~ x f ~ c ~ ( x l - x2)t$:i~(x4, xs, x2, xs, ...) +

~tT (4.34f) q-gfabc~cT~yjF(Xl, X2, X3, X5, . . . ) 5 ( X 3 -- X4).


The 5-distributions in (4.34d), (4.34e) are produced by the divergence 3~t, acting on ~(xt) ~ ~ S F (xt - "), S F (" - xl) ~" ~fl(xz) (4.34d) or on ~ 'DF (xt -- .) (4.34e). The 3-legs Cg-identity t ~ ' ~ = (1/2)[g~'tu~F--g~t~u~F] (see ref.[3]) is inserted in (4.34e). The (~-distribution in (4.34f) comes from a 4-gluon interaction term (see (3.17)-(3.20), (2.13)). In (4.34c), (4.34f) the derivative 3z of ~ 3t T~/~ acts on a field operator A, which

l

gives ( 1 / 2 ) F (remember (4.31)). In the other terms (4.34a), (4.34b), (4.34d), (4.34e) 3t acts on the numerical distribution.

4"3. C-invariance and conservation of the Cg-identities in the splitting. - The distribution splitting is the critical step in the proof of gauge invariance. In fact, the axial anomaly appears in this step in QED with pseudovector coupling: the normalization conditions of vector and axial current conservation are not compatible (see ref. [15]). However, these are two different types of gauge invariance, whereas all our Cg-identities express the same gauge invariance (1.16). Therefore, it is not astonishing that all Cg-identities can be fulfilled simultaneously, as we are going to prove.

In the process of distribution splitting, only the Cg-identities with I ~ l I> - 1 can be violated (see (2.15)). These are the identities (4.27), (4.32) and (4.33). The first one is easily preserved in the splitting by defining ~ T~v~ by means of (4.27). This is allowed since the corresponding d-distributions fulfil (4.27).

To prove (4.32), (4.33) we make the ansatz (2.14) for the possible anomaly. Since C-invariance essentially restricts these anomalies, we first study non-Abel ian C-invariance[4]: the Dirac equation and its adjoint (2.5) are invariant under C-conjugation, which transforms ~p, ~ similarly to QED [9]

(4.35) Uc~f~(x) Uc I = C ~ ( x ) T , U c ~ ( x ) Ud 1 = ~f~(x)rC,

where a phase factor has been chosen suitably and the matrix C satisfies

(4.36) ~2 t t T : - C - l y t t C , /A = 0, 1, 2, 3,

(4.37) C-- - C T= - C - 1 = - C §

Then the fermion current transforms as follows:

(4.38) Ucj~ UC 1 = g Uc: ~/tt,~a~/): U C 1 m g 2 -2 : ( C - l ~ ' ) i Y / [ ~ C k l ~ l : =

g .~TCT~,,uTc-I~fl: = g : ~ , a * ~ , Z ~ f l : g a a , j g , , 2 2

where the definition

def T (4.39) U ~ , ~ , = -~a* = - ) ~

of U~, has been used. The ;ta can be chosen to have the following property under complex conjugation:

(4.40) ;t* = Va)ia, V~ = --+ 1,

1174 M. DIJTSCH

where no summation over a is done. Then, U~, is a simple diagonal matrix

(4.41) U~,, = - - 5 ~ ' V a , a = 1, . . . , N 2 - 1 ,

which fulfils U -1 = U T = U and one easily obtains

(4.42) fabc = U~a, Ubb' U~c, f~'b'~' , - d~b~ = U ~ , Ubb, Ucc, da,b,~,

by means of (4.39), (3.30) and {)~, ~b} = (4 /N)5~bl + 2d~b~it~ (see appendix A of ref. [4]). In order that T~ (1.5), TI~I (1.15) and T~ (1.4) are C-invariant, UcTI ( /1 )UC 1 -~ = T~(/1), the gauge boson and ghost fields must transform according to

(4.43) Ua~,A~, = U c A a U ~ 1 , U ~ , u ~ , = U c u ~ U ~ 1 , Ua~,g~, = U c ~ a U j 1 �9

Obviously, these C-transformed fields fulfil the wave equation, too. The restriction of the unitary operator Uc to the gluon-ghost sector, which implements U~,, is explicitly constructed in appendix A of ref. [4]. One easily verifies by means of (4.42), (4.43) that T~ (1.3), T~I (1.12) and T~/1 (1.13) are C-invariant, too. Moreover, the transformed fields U c ~ U ~ ~ , U c ~ U ~ 1 (4.35)and U c A U d 1 , U c u U c 1 , U c ~ U ~ 1 (4.43) fulfil the same (anti)commutation relations as the original ones [9]; otherwise we would have a contradiction to the unitary implementability of these transformations.

We turn to the inductive step. One easily finds that A~(/z) (2.4) and R'(/z) are C-invariant and, therefore, also D n ( / l ) = R ~ ( / l ) - A n ( / l ) . Assuming R n ( / t ) to be a splitting solution of D.(/1), we conclude that UGRn(/1) Uc 1 is a splitting solution of Uc D,~(/l) U ~ 1 = D~(/t) . Consequently,

(4.44) c def 1 R~(/~) = ~ (R~(/~) + UcR~(/~) Uc ~)

is a C-invariant splitting solution of Dn( /1 ) , and the corresponding Tn(/1) is C-invariant, too

(4.45) Uc Tn U c 1 = Tn ' Uc Tn/l V c i = Tn/l ' l <~ n .

Let us consider numerical distributions with one external pair (~, F). By means of (4.35), (4.37), (4.43) and (4.45) we obtain

(4.46a) T ~ ( x l , . . . , Xn) = ~ {: ~ ( x ~ ) t ~ , B a.. (X~, XN, Xk, ...)~f(Xj): + k ~ i < j ~ k

+ : ~ ( x j ) t ~ , 8 .~... (x j , x~, xk, . . . ) ~ (xi): } :Ba (xk)...: + ...

(4.46b) = U c T ~ ( X l . . . . . x~) U ~ 1 =

= E Uab{: ~ (x j ) Ct~oB .b ..(Xi, X 2, Xk, "") TC I~)(X~): ~- k ~ < j ~ k

+ : ~(x~) C t ~ 8 . . b . (x~, x i , xk , . . . ) ~ C -~ ~(x~): } :B~ (xk)...: + . . . ,

where B~, ... is a gluon or a ghost operator and the transposition ~,T- in (4.46b) concerns the spinor space and the space of the fundamental representation. Considering the numerical distributions belonging to the same operator combination in (4.46a) and (4.46b), we may not d i rec t l y conclude that they are equal, because


different operator combinations are not linearly independent if 5-distributions are present. However, the ~t-distributions in (4.46) are C-invariant, i.e. the mentioned equality holds true, but a somewhat complicated argument is needed to prove this. We avoid it by the simple symmetrization

(4.47) ~ B . . ~.. (Xi, X~, Xk, ...) =

1 2 [t-~B .a. (Xi, Xj, Xk . . . . ) + U~bCt~B .~b...(xj, x~, xk, . . . )Tc - i ] .

Due to (4.45), these t ' are numerical distributions of the original Tn. For the t-distributions without external (~, ~), we see from (4.43) that C-invariance concerns the colour structure only; the symmetrizations analogous to (4.47) are symmetrizations of the colour tensors. For Tn/l we proceed in the same way. Due to the distinction of the Q-vertex, it happens that C-invariance connects different numerical distributions. For example, we obtain for the C-symmetrized t ~ u . , -2 t~,~u.-distributions (see (4.24))

- r l . (4.48) t ~ u . a (xl x2 xs, . . . ) = - U a b C t ' '2 ' ' ~ p u . . b.. (X2 , X l , X3, " " ) T C - 1

Again the transposition ,T , concerns the spinor space and the space of the fundamental representation.

We return to the ansatz for the possible anomalies in (4.32) and (4.33). Note that the latter Cg-identity appears one step later in the inductive proof. Taking (2.14), Lorentz covariance, the colour structures (A.11), (A.12) and invariance with respect to permutations of the inner vertices into account, we obtain

(4.49) a~ ~ u ~ X l , x2, xs, x4, ...) + ... = 1=3

= (Ko + K l ~ / V ~ x v 1 -b K 2 y ' ~ ~ + K 3 ~ / v ~ x v ~ ) ( ~ 4 ( n - 1 ) ( X l - x n , . . . , X n _ 1 -- X n ) ,

n + l

(4.50) ~ ~§ ~ ~uAab(Xl, X2, X3, X4, X5, ..., X~+I) + ... = 1=3

= 7 f' (g4 (5 ab l g • Y -~ K 5 d~b~ 2~ + K6 f~b~ 2~) 5 4n ( x I _ Xn + 1, . . . , Xn -- Xn + 1)"

In these two equations we go over to the C-symmetrized numerical distributions t.'., v' (4.47). Since this symmetrization is linear and commutes with taking the divergence, we obtain the C-symmetrized anomalies on the r.h.s.. The latter read by means of (4.35), (4.36), (4.37), (4.39) and (4.42)

~xt ~.~l (4.51) ~ ~ ( x l , x2, x~, x4, ...) + ... = l = 3

(4.51) = { I [ K I + K217"(9,a + c~xe)+ K3~/v~xs}(~4(n-1)(Xl-Xn, . . . , X n _ l - - X n ) ,

n + l

(4.52) ~ ~x~ § t~ ~ v ~-~wuAab~.~l, X 2 , X3 , X4 , . . . , X n + l ) ~ . . . : I = 3

= 7" K6f~b~2c (~4n (X 1 _ X n + 1 , " " , Xn - - X n + 1)"

1176 M. D~TSCH

In contrast to the : ~ f u A : - C g - i d e n t i t y in QED (see ref. [16]), the anomaly (4.50) is not completely removed by the C-symmetrization (4.47). The non-Abelian Cg-identity (4.33) contains completely new terms. We already have met this phenomenom in second order: the term (3.27), which has the C-invariant colour structure f~A~ (4.52), is absent in QED.

Now we are going to remove the possible anomalies (4.51) and (4.52) by means of f in i te renormalizations of the t'.., v' -distributions. For simplicity we omit the primes. Taking the singular order of the numerical distributions [9], Lorentz covariance, the colour structures (A.8), (A.11) and invariance with respect to permutations of the

�9 �9 �9 ~ ~ ' 3 inner vertices into account, the normahzatmn polynommls of V ~ A = V ~ (see (4.27)), v - ~ (1 = 4, ...n), -~ -e v ~ , V~r and v~v must have the following forms:

v : N ~ f l u ( X l ' . . . , X n ) -_ C 3 ~ ] v ( ~ 4 ( n _ l ) ( x l _ X n X n _ l , (4.53) N ~ A (x~, ..., x~) ~s . . . . , -- X~)

7,I _ X n ) , (4.54) N ~ w ( x l , ..., Xn) = C07~4(~- 1)(Xl - x~, ..., xn-1 l = 4, . . . n ,

(4.55)

(4.56)

- -1 . . . , C 154(n N~vu (xl, X n ) . : - 1)(X 1 - - X n ' " " , X n _ 1 - - X n ) ,

- - 2 N ~ u ( X l . . . . , X n ) C 2 5 4 ( n - 1)(Xl X n . . . . , X n - 1 - - X n ) ,

N ~ (Xl , . . . , X n _ 1) -----

= [C4 -~- C 5 ~ ] u ~ x l -I- C 6 ~ 2 r ~ x 2 ] ( ~ 4 ( n - 2 ) ( X l - X n - 1 , "'*, X n - 2 - - X n - 1 ) -

. t~V~A and t ~ A have singular order [9] The 4-legs distributions t~v~A (1 = 3, 4, . .n), -~ -2 ~) = -- 1 and, therefore, have no freedom of normalization. Note that t ~ and tA~ could be renormalized, too. However, this causes a chain reaction of other renormalizations in order to maintain the Cg-identities already proven (see ref. [2, 3]). We shall see that the removal of the anomalies (4.51), (4.52) is possible without renormalizing tu~, tA~ .

- -1 - - 2 C-invariance restricts the normalization constants in N~v~, N~v~ (4.55) and Nvv (4.56)

(4.57) C1 = C2, C5 = - C6.

In the step from n - 1 to n we must find constants Co, C1 = Ce, Cs, C4, C5 = - C6 fulfilling (see (4.32))

(4.58) {1 }

__ ~ X 1 -~[K~ + K~17 (a~ + 8~2) + K 3 y ~ a p d4(~-i)(x~ - x~, . . . , x~_~ - x~) =

~ ~ . ~ ( x ~ , x2, x3, x~, ...) + (~,~ c~' + z m ) N ~ ( x ~ , x2, x~, x4, ...) + / = 3

( - -

+ N ~ ( x l , x2, x3, x t , . . . ) ( ~ 2 7 v - im) +

-~- ~ [ 6 ( X 3 -- X l ) N ~ f l ( x l , x 2 , x4, . . . ) - N ~ f l (X l , x2 , x4 . . . . ) 6 ( x 2 - x3)] ,

for arbitrarily given K1, K2, Ks, in order to remove the anomaly (4.51). This eq. (4.58)

ON GAUGE INVARIANCE OF YANG-MILLS T H E O R I E S ETC. 1177

is equivalent to the linear system

(4.59)

K3 = C3 - Co + igCs ~

which has a 2-dimensional set of solutions for Co, C1, C3, C5. Having performed a renormalization fulfilling (4.59), the Cg-identity (4.32) holds true and there remains a C-invariant and gauge-invariant freedom of normalization: (4.53), (4.54), (4.55) and (4.56) with Ck replaced by C~ (k = 0 , 1, ...6), the latter fulfilling (see (4.57), (4.59))

(4.60) I C~' = C~, C~ = - C~, [ ' ~ ' C ' . - C d + C l ' = 0 , ~3 - o + igC~ = O

Note that C2 is restricted neither by C-invariance nor by gauge invariance. The freedom (4.60) is used in the inductive step from n to n + 1, where the

anomaly in (4.52) must be removed. We have to solve the equation (see (4.33))

(4.61) - 7 ~ K 6 f a b c , ~ c ( ~ 4 n ( x l - - X n + l , . . . , X n - - X n + l ) :

= [~aS(X~ - X~),~bN~A(Xl, X~, X4, X 5 . . . . ) - -

- - N ~ A ( X l , X2, X4, X5, ...)~.b6(X2 -- X~)s --

ig [7~b(~(x4 _ x l ) ,~a~, 1 2 ~ ( x l , x2, x3, xs, ...) -

- - , 2 - N ~ u ( x l , x2, x3, xs, ...)~taS(x2 - X 4 ) ~ b ~ ] # ] §

+ gfabc~cN~A (xi, X2, X~, XS, ...)5(X3 -- x4),

for an arbitrarily given K6. Inserting (4.53), (4.55) we obtain

ig (4.62) -K6f~bc,~c = -~(C~ + C~')[)~, ;tb] + g C ~ f ~ b ~ = -gClf~b~,~c,

where (3.30) has been used. The linear system (4.60), (4.62) has a one-dimensional set of solutions for C~, C1' C' ' . , 3, C~ Therefore, the anomaly (4.52) can be removed.

For n = 3 the story is simpler, since ~35(x l - x3, x2 - x~) = - ( ~ ' + ~2) . �9 5 ( x l - x~, x 2 - x3) and N ~ u (1 = 4, . . .n) does not appear, but the results are the same.

For the Cg-identities without external ~, ~, we found the following rule in t rying to remove the possible anomaly of a certain Cg-identity: every renormalization of the 5-degenerate terms which preserves all other Cg-identities, can be absorbed in a renormalization of the non-degenerate terms. In other words: we do not gain an additional freedom of normalization from the 5-degenerate terms to remove the anomaly (see ref. [3, 4]). The above calculation shows that this rule holds true also for

1178 M. D(~TSCH

(4.32c), and one easily verifies it for renormalizations of t ~ in (4.32d). However, this rule is broken in (4.33): there is no freedom of normalization of the non-degenerate terms (4.33a), (4.33b), but the possible anomaly (4.52) can be removed by renormalizing the (5-degenerate terms (4.33c)-(4.33f). This is a new feature, which can be understood by the fact that Q commutes with ~ and F (1.20).

In order to complete the proof of the Cg-identities we still have to show that the Cg-identities of type Ia, b belonging to (5-degenerate :~: can be maintained in the process of distribution splitting. This holds generally true for all Cg-identities of type Ia, b, as we realized in sect. 2 of ref. [3]: the Cg-identities of type Ia are iden t i f i ca t ions of numerical distributions of the theory with one Q-vertex with numerical distributions of the normal theory, both in the natural operator decomposition (see, e.g., (4.27)-(4.30)). They can easily be preserved in the process of distribution splitting, because we are free to normalize the extended theory properly. The Cg-identities of type Ib concern the L o r e n t z s t ruc ture only (see, e.g., (4.31)). They hold true in the natural operator decomposition, if the Lorentz structure is preserved in the process of distribution splitting, which is always assumed.

Remark: The Cg-identities of type Ia,b belonging to ~-degenerate :~: can be proven directly, without using the Cg-identities for A ' , R ' proven in subsect. 4"1. The corresponding terms have one non-basic field operator. Therefore, the (Ldistribution which is responsible for the J-degeneration originates from a 4-gluon interaction term (see (1.17), (2.13), (3.19), (3.20)). The corresponding diagram is obtained by replacing in fig. 1 A(x~- x2) by (~(xl- x2). The non-basic field operator is an element of {B3, ..., B~}. Analogously to subsect. 3"1 the considered Cg-identities can be reduced to the corresponding Cg-identities for the subdiagram with vertices x2, ..., x~.

5. - Discrete symmetr ies and pseudo-unitari ty

5"1. Compat ib i l i t y o f P-, T-, C- invar iance and pseudo-un i tar i t y . - First we study the behaviour of the free-field operators with respect to parity P and time reversal T (see ref. [9, 17, 18]). Since the free Dirac equation (2.5) is diagonal in the colour space, we can adopt the transformations of the matter fields from the Abelian case

(5.1) U p F a ( x ) Up 1 = i T ~ U e ~ , ( x ) U p s = - i ~ ( X p ) 7 ~ ,

(5.2) V T ~ a ( X ) VT 1 : ~]SC~2a(XT), V T~a(X) VT 1 : ~a(XT) C-I~] 5 ,

def~ o x r ~ f ( where Xp = ( x , -x ) , - x ~ x) and some phase factors have been chosen in a suitable way (see sect. 4.4 of ref. [9]). Up is unitary, whereas VT is antiunitary, otherwise the (anti)commutation relations of the free-field operators would not be invariant with respect to these transformations [9, 17,18]. Then, the matter current j~ (1.6) transforms according to (remember (4.39))

(5.3) Upffa(X) Up 1 = j , ~ ( X p ) , V T j ~ a ( X ) V T 1 = - - Ua~j#b(XT).

Note that it is not a printing mistake that/~ is once an upper and once a lower index. We want T1 to be P- and T-invariant

(5.4) U p T I ( X ) U p I = T l ( x p ) , V T T I ( x ) V ~ 1 = TI(xT) = - -T I (XT) ,

where TI(/,) is the first-order T-distribution of the inverse S-matrix. Then, due to

ON GAUGE IN'VARIANCE OF YANG-MILLS THEORIES ETC. 1179

U Q U -1 = Q for U = Up, VT, gauge invariance implies

i ~ r T1/ lv (XT) = [Q, T1 (XT)] = VT[Q, TI(X)] V T 1 =

= VviO~ T{/1 (x) V~ 1 = - i( - 9xr) VT TI~/1 (x) V~ 1

and a similar statement for parity. Therefore, P- and T-invariance means for T1/1

(5.5) UpT{/I (X) U f 1 = T1/1 v(Xp), VTT~ / I (X)VT 1 = T1/lv(XT) = -- T1/I,(XT).

Inserting T1 ~ (1.5), T~I (1.15) and T~ (1.4) in (5.4), (5.5), we see that the gluon and ghost fields must transform according to

(5.6) UpA~(x ) Up 1 = A ~ ( X p ) , U p u a ( x ) U p 1 = Ua(Xp) , UpUa(X) U p 1 = u a ( x p ) ,

{ VTA~ (x) VT 1 = - U~bAzb (XT),

(5.7) VTU a (X) V T 1 = -- Vab Ub (XT) ,

VTUa (X) V T 1 = -- Uab Ub (XT).

One easily verifies that T A (1.3), TI~I (1.12) and T~/1 (1.13) are P- and T-invariant, too. We leave it to the reader (or refer to [9,17,18]) to check that the Dirac equation, respectively, wave equation and the (anti)commutation relations of the free-finld operators are invariant with respect to the transformations (5.1), (5.2), (5.6) and (5.7).

Next we consider pseudo-unitarity. The conjugation , ,K, was introduced in sect. 5 of ref. [4] and in [6]. It is related to taking the adjoint ,, + , and transforms the free- field operators in the following way:

{ A I' (x) g = A t' (x), u(x) g = u(x) , ~(x) g = - ~(x), (5.8) ~ ( x ) K = ~ ( x ) + , ~ ( x ) K = ~ ( x ) + .

Obviously -K, agrees with taking the adjoint on the physical subspace, which is defined by excluding scalar and longitudinal gluons and the ghost fields u, ~ (see sect. 5 of ref. [4]). In contrast to the C-, P- and T-transformations, the conjugation -K- cannot be unitarily or antiunitarily implemented in Fock space, since

(5.9) (B~ B2) K = B K B g ,

where B1, B~ are arbitrary free-field operators. It is easy to check that T~ and T~/~ are pseudo-unitary

(5.10) T1 (x) K = T1 ( x ) -- - T 1 ( x ) , T1/1 (X) K ---- T1/1 ( x ) -- - Tl l 1 (X) .

We turn to the inductive step in the construction of the Tn(/t)- By means of the fact that A'(/1), R~(/1) are sums of tensor products of lower Tk(/~), Tk(/~), k < n (see (2.4)), one obtains the P-, T-invariance and pseudo-unitarity of A'(/~), R'(/~) and D~(/l)= R~(/1)- - A ' ( / t ) (see ref. [9]). The discrete symmetries P, T, C and pseudo-unitarity can be violated (by local terms) in the causal splitting D~(/~) = R~(/~) - A~(/t) only. Analogously to (4.44), one obtains a P-, respectively, T-invariant or pseudo-unitary splitting

1180 M. DUTSCH

solution by symmetrizing an arbitrary splitting solution (see ref. [9]). For example, the symmetrization

(5.11) RU def 1 K ~(/t) = -~ (R~(/1) - R~(/l))

yields a pseudo-unitary splitting solution. Note that these symmetrizations only change the normalization of R~(/z). The question is whether these symmetrizations are compatible. Since

(5.12) Up e = Uv, ~r = Uv, VT Up = Uv Up VT,

where Uv is a so-called valency operator, U~ = 1, the group generated by Uc, Up, VT has more than 8 elements. Due to U~ = 1, [Uc, Up] = O, [Ue, VT] = 0, it has 16 elements. By symmetrizing an arbitrary splitting solution with respect to this group, we obtain a C- and P- and T-invariant splitting solution (see sect. 4.4 of ref. [9]).

Now we add the symmetrization (5.11) with respect to pseudo-unitarity. To prove

(5.13) (: 6z: )K = : 6zK:,

for an arbitrary operator combination 6t, one needs (5.9) and that ,,K- transforms a creation operator in an annihilation operator and vice versa. Because B -~ UBU -1 (for U = Uc, Up, VT) transforms a creation operator (annihilation operator, respectively) in a creation (annihilation) operator, we obtain

(5.14) U:5~: U -1 = :UdzU-I: , U = Uc, Up, VT.

By means of (4.35), (4.43), (5.1), (5.2), (5.6), (5.7) and (5.8) one easily verifies

(5.15) U B g U - I = ( U B U - 1 ) K, U = U c , Up, VT, B = A , F , u , ~ , ~ , ~ f .

Together with (5.13) and (5.14) we conclude

(5.16) U(: ~ : )g V - 1 ___ : u ~ t K u - l : : :( U ~ U - 1 ) K : _ (V : ~t. U - l ) g "

This implies for U = Uc (with R,(/t) = ~. rj : ~ : ) 3

(5.17) t~c'r R K~(/l) U.-1c = E r ? U c ( : ~ j : ) K U c ----E r~* (Uc: ~ j : . Uc1) K = (UcRn(fl) Uc1) K , 3 J

where * is the complex conjugation. This means that the C-transformation R~(/t)--~ --~ UcR~(/l)U~ 1 (see (4.44)) commutes with R~(/z) -o-R~/ t ) (see (5.11)). Therefore, through

~Cu def 1 K K -1 - - U c R ~ ( / I ) U ~ ], (5.18) /~n(/l) = ~- [R~(/~) R~(/l) + UcR~(/t) U j 1

Cu we obtain a C-invariant and pseudo-unitary splitting solution Rn(/l ) . Similarly one concludes from (5.16) that the P- and T-transformations (on the space of the retarded

K (see sect. 4.4 of ref. [9] for the distributions R~(/~)) commute with R n ( / l ) ---->- Rn(/l ) detailed form of the P- and T-transformations). Moreover note that R~(/~) --* - R~/z) is an involution. K Consequently, the group G, generated by R~(/t) ---) - R,(/t) and by these


P-, T-, C-transformations (on the space of the R~(/t)'s), has 32 elements. By symmetrizing an arbitrary splitting solution with respect to this group

l g~G (5.19) R~(/l) = - ~ R~(/~)

and constructing

(5.20)

T~ ~ R~ ' - R ' ,

def s p T~lz = R~lt - R~lt,

T ! . . , T ~ ( X l , . . . , Xn) ----- - ~ . ~ n (X~l , �9 Xz~n),

1 T~/t(xl, . . . , x~ n) -- -~. ~z T(~/=-,l(x=l, ..., Xzn),

we obtain a T~(/1)-distribution, which fulfils all these symmetries

U C T~(/t ) (X) U c I = TSn(/1) (X),

U p T ~ ( X p ) U p 1 ~- T~(X),

s - 1 T sv iv~ Ue T~/ tv(Xe) U~ = n / ~ l ,

(5.21) VT T~ (XT) V~ 1 = T~ (X),

- - s 8V Vr T~/zv (XT) W 1 = T~/~(X),

--s K s T~(/~) (X) = T~(/z) (X),

where X ~f(xl , ..., x~), Xp ~f(Xlp, ..., X~p), XT ~f(XlT, ...X~T) and T~(/t) is the n-th order T-distribution of s - l ( g ) . The latter means

(5.22) T~(X) = - T~(X) - ~, T~_~(Y) Tk(Z) = YUZ=X, YgO, ZgO

= - T~ (X) - R" (X) - R~ (X)

and analogously for -~ T~/~. Thereby, R~ is the usual R~-distribution of the causal construction and R~ is a similar object

(5.23) R" (X) d=ef ~ Tn - k (U) Tk (V, Xn) . U U V = { x l , . . . , X n - 1} , U ~ 0

Note that R" and R~ are given by the induction hypothesis and that their sum R" (X) + R~ (X) = ~, Tn - k (Y) Tk (Z) is symmetrical with respect to permutations of Xl, ..., x~. Therefore, the latter holds true also for T~ (X).

Moreover, ff we start with a Lorentz-covariant and SU(N)-invariant Rn(/L) in (5.19), the resulting T~(/z) is Lorentz covariant and SU(N) invariant, too.

We turn to the numerical distributions t ~ in the natural operator decomposition of T~(/l). We assume them to be Lorentz covariant, SU(N) invariant and to respect the permutation symmetry (5.20). (Since we have chosen the natural operator decomposition, these properties could be violated by local terms only. For example, a certain numerical distribution could have a non-covariant normalization term which drops out in the sum over all diagrams.) Similarly to C-invariance (see (4AS)-(4A7)), it requires some work to prove the P-, T-, C-invariance and the pseudo-unitarity

1182 M. DUTSCH

of the t~-distributions belonging to T~(/t). We avoid it by symmetrizing them with respect to G

(5.24) t~,dej 1 g~a 32 t~,

analogously to (4.47). Thereby the transformation t--* tg, g e G is defined in the following way: it is induced from the operator transformation

(5.25) T ( X ) - ~ , g ( T ) ( X ) = U c T ( X ) U ~ 1 , UpT(Xp)Up ~, VTT(XT)VT ~, T(X) K,

(see (5.21)). Applying these transformations to T ( X ) = ~ . t j (X ) :~ . (X ) : , we define 3

g(T)(X) = ~ tjg (X) : Oj (X):, e.g., for T~/t (X) = ~. tj ~t (X): ~. (X): and the P-transformation 3 J

g(T~/t)(X) = E t~, (Xp) Up : O~ (Xp): U p 1 de f E tffg l ( X ) : ~ j (X): , i 3

more precisely

d e f l (5.26) tfgt(X) = ti~(Xp) for :~-(X): = Up:ffti(Xp): Ufi I .

Note { Up: ~i (Xp): Up 1 l i} = {: ~ (Z): ]j} and that tj (X) and tjg (X) belong to the same operator combination :~j(X): in T(X), respectively g(T)(X). For g being the C-conjugation, we refer the reader to (4.46)-(4.48). Let us illustrate this transformation by another example: by identifying the numerical distributions in T~/z(X) and in VT Tn/~ (XT) VT1 belonging to the operator combination : ~ ( X l ) . . . F(x2)ua(xs)Azb(x4): we get

~vltt = - - Ub~) ~ C t ~ A c ~ (Xr) , ~f luAabg(X ) ( Uac) ( 5 7~1 * ~]5C-1

where ~(1) is a numerical distribution of the natural operator decomposition of T,(/l). According to the construction after (2.4b), this means that ~(z) is defined by

(5.27) t ~f - t - r ' - r" ,

with t, r ' , r" being the numerical distributions in the natural operator decomposition of Tn(/Z), R~(/l), Rn'(/t). (Remark: in subsect. 4"1 we have proven the Cg-identities for r ' . Analogously one proves the Cg-identities for r". With (5.27) we conclude that the Cg-identities for t imply the Cg-identities for t.) Due to (5.21), the t~'-distributions (5.24) are still numerical distributions of T~(/1).

5"2. Compatibil i ty wi th the Cg-identities. - The last step is to prove the compatibil i ty of the discrete symmetries and pseudo-uni tari ty wi th the Cg-identities. For this purpose we consider finite renormalizations T~(/t)-o T~(/t)+ N~(/~), N~(/t)= = ~. n] ~) : ~.: and a possible anomaly ~ (X) = ~ aj : ~j: violating the operator gauge

3 3 invariance (1.16)

(5.28) ~ 3t T~/t (X) + i[Q, T~ (X)] = ~ (X). l

ON GAUGE INVAR[ANCE OF YANG-MILLS THEORIES ETC. 1 1 8 3

Due to (5.22), T~(/~) must be renormalized, too

(5.29) T,(/l) ""-> Tn( / l ) -]- Nn( /1 ) -'~ Tn(/1) - Nn(/l),

and taking additionally the gauge invariance of R'(/z) and R~(/l) into account we conclude

(5.30) • a~ Tn/t (X) + i[Q, Tn (X)] = ~ ( X ) = - . s (X). l

Analogously to (5.25), (5.26), the operator transformations N---)g(N), ~ - ~ g ( _ ~ ) , g e G, induce transformations n~ (t)---) n~ ) , aj---) ajg of the corresponding numerical distributions.

By means of

(5.31) Q K = Q, UcQUCi = Q, U p Q U f i l = Q, V T Q V T 1 = Q,

we obtain from (5.28)

(5.32) ~ atg(T~/~) + i[Q, g(T~)] = ( - 1)r(g)g(~r g �9 G , l

where T(g) is the number of time reversals in g. Next we consider a Cg-identity belonging to (5.28)

(5.33) ~, 31 tt + t = a, t

with t being a sum of terms, possibly containing 5-degenerate terms. According to (5.25), (5.26), the numerical distributions tl, t, a are replaced by tlg, tg, ag in the step from (5.28) to (5.32). Therefore, performing the operator decomposition in (5.32), the Cg-identity (5.33) changes into

(5.34) ~ ~itlg + tg = (--1)T(g)ag. 1

(We have pointed out several times that such an operator decomposition is not unique. (5.34) can be deduced rigorously from (5.33). But for this purpose some properties of the transformations t(~)--> t(t)g, a - o ag must be worked out. Note tha t these transformations are not linear or antflinear in general.) Similarly to (5.33), (5.34), one obtains the following statement: if nl, n remove the anomaly in (5.33), i.e. they fulfil

(5.35) ~ ~t nl + n = - a ,

the transformed normalization polynomials remove ( - 1)r(~)ag

(5.36) ~ $ln~ + n g = - ( - 1)T(g)ag. Z

NOW we make the ansatz for the possible anomalies with the symmetrical t ~'-distributions (5.24)

(5.37) ~ $1tS + t *' = a s- 1

1184 M. D'UTSCH

By means of (5.34) we conclude

(5.38) a ~ = ( - 1)T(g)a~.

In the case of C-invariance this has given us a restriction of the possible anomalies (4.51), (4.52). We have proven that there exist normalization polynomials n{, n ~ of t ( , t ~' which remove the anomaly a ~, i.e. they fulfil (5.35). The question is whether t{ '+ n~, t ~' + n ~ are still invariant with respect to G: (t~tl + n(~))g = t~ + n(~) ? Generally, this is not true. But we can symmetrize the normalization polynomials with respect to G

(5.39) n{,def 1 ~ n~, n~, def 1 - - - - n ; .

32 g~G 32 g~e

We conclude by means of (5.36), (5.38) that n ( , n ~' remove the anomaly a ~, too. Moreover, t ( + n ( , t ~' + n ~' are still invariant with respect to G.

Summing up, we have proven that the numerical distributions in the natural operator decomposition of Tn, Tn/~ can be normalized in such a way that, simultaneously, they are Lorentz covariant, SU(N)-invariant, P-, T-, C-invariant, pseudo-unitary and fulfil the Cg-identities.

I would like to thank F. Krahe and Prof. G. Scharf for stimulating discussions. I am grateful to A. Aste and again Prof. G. Scharf for reading the manuscript. Finally, I thank my fiancee A. Schneider for bearing with me during working at this paper.

A P P E N D I X A

Colour structure in the fundamental representation

In this appendix we study the structure of the colour tensor of an arbitrary diagram with external field operators

(A.1) case 1) :5z1: = :~(xl)~f~(x2): ,

(A.2) case 2) : ~ : = : ~, (x l )~(x2)ua(x3) : ,

(&3) case 3) : ~ : = : ~ a ( X l ) ~ f l ~ ( X 2 ) u a ( x 3 ) A ~ ( x 4 ) : .

These tensors are SU(N)-scalars (A.1), -vectors (A.2), -tensors of second rank (A.3) in the fundamental representation. They are used in subsect. 4"2, 4"3 for the determination of

the colour structure of the numerical distributions t~v, t~v~ (4.26), l the normalization polynomials of t-~r t~vA, t~vu (4.53)-(4.56),

the ansatz for the possible anomalies of the Cg-identities characterized by : t~: and : ~ : (4.49), (4.50).


All diagrams with external legs :6z1:, 6~: or :6~: have an open mat ter line going through the diagram from ~ to ~. For every vertex on this line we have a matrix 2r ( i = 1,2, ...l) (see (1.5), (1.15)). These matrices g e t multiplied: : ~ a ( X l ) ( ~ a~ ~ ae"" j~ at)aft ~) fl (X2) : , since the propagators S~(Z), s r ~ t/av , s~F~ are -- (~ ~ . By applying several times

(A.4) 2 a ~ b ~-- 2 (~ ab 1N • N + dab~ ~ ~ + ifabc ~ ~,

this matrix product can be reduced to

(A.5) Ella2 ...a t 1N • N + F~la2 ...atc~c ,

where F I ~ at and 2 F~. . . a t~ are invariant SU(N)-tensors of rank r = l, 1 + 1 respectively. (A basis for the latter tensors (up to rank r = 5) was given in appendix A of ref. [4]. We shall repeatedly use these bases.) The other vertices and the other inner lines of the diagram (which may include mat ter loops) contribute another invariant SU(N)-tensor Fb~b~. b,. The total colour structure of the diagram is

1 2 (A.6) Fb, b~...b~2a~2~ ""2at = Fb~b2..bsF~ia2. a t l N • + r b l b 2 .b~F~iaz .a tc~c �9

There are contractions between the indices bI, b~, ..., bs and a l , a2, . . . , at. In case 1) all these indices must be contracted (s = l)

(~-7) .atFala2" . a t l N x N + F a l a e 2 Fa ~ a~ 1

Since 1 = Fal~...af~l~...at C � 9 and since there exists no SU(N)-veetor 2 Fa~ ~...at F~ ~... ate, we obtain

(A.8) (A.7) ---- C 1 N • N .

In case 2) one index (= a) is not contracted. There are two possibilities: 1) a � 9 . . . . ,bs} ( s = l + l )

(A.9) Faa~a2.. azF~la21 ...az 1N x N -[- Faala2 ..aiFala22 . . .alc~c "

2) a �9 {al, ..., at}, (s = l - 1)

(A.10) I 1 + F a l ~ z a l - i F ' ~ , a 2 . . . a . at 1 C ~ ff Faloa...al_lF~la2...a...az_l N x N . . . . "

In both cases the term ( F F 1 ) a l N • vanishes because there exists no SU(N)-vector (FF1)a . All SU(N)-tensors of second rank are multiples of 5a~. Therefore, (FFe)a~ must have this form for 1) and 2). We obtain

(A.11) (A.9), (A.10) = C;ta, C �9 C .

In case 3) two indices (= a, b) are not contracted in (A.6). Then, (FF1),, is an SU(N)-tensor of second rank: (FF1)ab = C 15ab, C 1 e C (see above), whereas ( F F 2 ) ~ is an SU(N)4ensor of rank 3. Therefore, the latter must have the form (FF~)ab~ = = Ceda6c + CSfab~, C 2, C8 �9 C. Consequently, we obtain the total colour structure

(A. 12) C1 5 ab 1N • N + C2 dabc ;t ~ + CSf~b~)~ ~.

1186

R E F E R E N C E S

M. DUTSCH

[1] DUTSCH M., HURTH T., KRAHE F. and SCHARF G., Nuovo Cimento A, 106 (1993) 1029. [2] DUTSCH M., HURTH T., KRAHE F. and SC~ARF G., Nuovo Cimento A, 107 (1994) 375. [3] DUTSCH M., HURTH W. and SCHARF G., Nuovo Cimento A, 108 (1995) 679. [4] DUTSCH M., HURTH T. and SCHARF G., Nuovo Cimento A, 108 (1995) 737. [5] EPSTEIN H. and GLASER V., Ann. Inst. Poincarg A, 19 (1973) 211. [6] KRAHE F., preprint DIAS-STP-95-02. [7] BECCHI C., ROUET A. and STORA R., Commun. Math. Phys., 42 (1975) 127; Ann. Phys.

(N.Y.), 98 (1976) 287. [8] DUTSCH M., KRAHE F. and SCHARF G., Nuovo Cimento A, 103 (1990) 903. [9] SCHARF G., Finite Quantum Electrodynamics, 2nd edition (Springer-Verlag) 1995.

[10] DUTSCH M., preprint ZU-TH 30/95, hep4h/9606105. [11] TAYLOR Y. C., Nucl. Phys. B, 33 (1971) 436. [12] SLAVNOV A. A., Theor. Math. Phys., 10 (1972) 99. [13] 'W HOOFT G., Nucl. Phys. B, 33 (1971) 173. [14] D~TSCH M., KRAHE F. and SCHARF G., Nuovo Cimento A, 106 (1993) 277. [15] DUTSCH M., KRAHE F. and SCHARF G., Phys. Lett. B, 258 (1991) 457; Nuovo Cimento A, 105

(1992) 399. [16] DUTSCH M., HURTH T. and SCHARF G., Phys. Lett. B, 327 (1994) 166. [17] HURTH T., Ann. Phys. (N.Y.), 244 (1995) 340. [18] BOGOLUBOV N. N., LOGUNOV A. A. and TODOROV I. T., Introduction to Axiomatic Quantum

Field Theory (W. A. Benjamin, Inc.) 1975.

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On gauge invariance of Yang-Mills theories with matter fields