Z. Phys. C - Particles and Fields 30, 501-505 (1986) Ze,sc.,. P a r t i e s f~r Physik C

and Fk k:ts �9 Springer-Verlag 1986

One-Loop N-Point Functions in the Light-Cone Gauge

George Leibbrandt ~ and Su-Long Nyeo 2

1 Guelph Waterloo Program for Graduate Work in Physics and Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, N1G2Wl, Canada

2 Guelph Waterloo Program for Graduate Work in Physics, Department of Physics, University of Guelph, Guelph, Ontario, N1G2W1, Canada

Received 15 September 1985

Abstract. We consider, in the light-cone gauge, the possible structure of the counterterms arising in the solution to the renormalization equation. Using the structure of these counterterms and the nonco- variant formalism of the integrals as guidelines, we also examine to one-loop order the divergence and nonlocality of the N-point gluon vertex functions for N > 5 .

1. Introduction

The renormalization of Yang-Mills theory in the light-cone gauge has been investigated by several authors I-1-4]. One of the major problems is how to renormalize the divergent, but nonlocal, terms arising already at the one-loop level. While opinions differ on the precise technical tools to be employed, there is reasonable agreement that the final proof of the renormalizability of the theory (provided it exists in the first place), would somehow have to involve the BRS invariance of the Lagrangian. As is well known, this invariance leads to the Slavnov-Taylor identities which have to be satisfied to all orders in perturbation theory. We note that in this BRS ap- proach, ghost fields are definitely required [53, even though they are known to decouple from physical S- matrix elements.

Tricky as it may be, we believe that the non- locality problem mentioned above is not untractable, as recent studies have indicated. It was demonstrat- ed, for example, that BRS-invariant counterterms for the one-loop gluon self-energy may indeed be de- rived in a consistent fashion. Unfortunately these counterterms are insufficient to account for the qua- dratically nonlocal terms in the one-loop three-gluon vertex [4].

The purpose of this paper is to investigate the possible structure of the counterterm solution to the renormalization equation and to discuss the non- finiteness of the one-loop N-point functions in the case where N > 5.

2. The Renormal izat ion Equation

The Yang-Mills Lagrangian density in the light-cone gauge* n ,A~=O, n2=nunu.=O, where A~ are gauge (or gluon) fields and n~ is a constant null-vector, reads

1 U, = L - - . Aq2, c~ gauge parameter,

2~

1 (12'a ~2 a a ab b +ju)Du co ~ ~ b . . . . b c - - g g J Ix. co c o ,

ab ab abe c D, =0~(~ - g f AI,,

F a ~ A a c~ A a 4 - f a b c A b A c " (2.1)

g is the gauge coupling constant and fabr are the group structure constants; r/a and co" denote ghost fields, and J ~ , K a external BRS sources. The action S ' = j d x E , dx=_d4x, is invariant under the following BRS transformation [6] :

a ab b cSA, = 2D u co ,

2 ~'abc o,)b o c (~coa= _ ~ gj

2 a &l ~ = ~ n . A . ,

(2.2)

)~ being an anticommuting constant.

* We use the metric g,,.=(1, - 1 , - 1 , -1). Greek letters are Lo- rentz indices, Latin letters gauge indices. No distinction is made between upper and lower indices

502 G. Leibbrandt and S. Nyeo: One-Loop N-Point Functions in the Light-Cone Gauge

Consider the generating functionals for Green functions:

exp {i W [j~, ~-", ~a; j~, K.]} =Z[j~, "(~, ~a; jau, K"]

= ~DAauD~I~Dof exp{iS' + ~dx[j~A~ + ~co" + lfl ~]},

(2.3)

�9 a a where j , , ~-a and ~" are external sources for A,, o" and ~/~, respectively. The functional Z generates the complete Green functions, while W generates the connected Green functions. Invariance of the integ- ral (2.3) under the transformation (2.2) and under any local change of the variable q~ leads to the following identities for W:

j 6W 6W tax ;(x)

and

nu 8W ] "a 6j.(x) ~(x) =0

(2.4)

bW _ _ a X = 0 . ( 2 . 5 )

"" aJ~(x) + ~ ( )

The generating functional for one-particle-irreduc- ible Green functions is defined by

. . . . . W [ j # , ~ , ~ , # , 3 F[A,,co ,t 1 ,Ju,K"]= .~ 7a ~_~.j. Koq �9 a a ~a a - ~dx[j.(x)Au(x) + ~ (x) c9 (x) + ~(x) ~"(x)], (2.6)

with the dual relations

3W bW ~W ajau(x)-A~ (x), c~ ~-,(x) -- c~ (~a(x ~ -- ~a(x);

(2.7a)

6F 8F 6F 5A~u(x)- J~u(x)' a~0"(x)-~"(x)' 6~/"(x)- {"(x);

(2.7b) and

bW 3F bW 8F a a ' 5Ju(x ) 6J,,(x) 6g"(x) c~ga(x)

(2.7c)

To denote the expectation values of the fields in (2.6)-(2.7) we use the same symbols as for the fields in (2.1). In terms of the functional F, (2.4)-(2.5) be- come, respectively,

~dx[ 8F 8F 8F 8F 6A~(~) aS~(x) ~ 6o~ 6K~

6F (2.8a)

and

6F (SF =0. (2.8b) g / # a

6J.(x) 6~~

Equations (2.8) may be simplified by introducing

F - F + ~--~ ~ dx(nuA"u)2:

~dx[ 6f 6F 6f af ] " - - ~ O aA".(x) aJ;(~) + ~ aU(x)

and

(2.9a)

6/~ 6/~ =0. (2.9b) n. 6J~(x) 6q~

These are the Slavnov-Taylor identities, which are to be satisfied in every order of perturbation theory:

f = f(o) + f(1) + f(2) + . . .

with f (o )=S =~dxL. From (2.9a), the divergent part ^(i) ~(i) ^(i) ' �9 of F =Ffi.ite +Fdi v satisfies the renormalization equa-

tion (to nth order)

A/~(ilv ) = 0 , (2.10)

where

A_~dx F [ 5S (5 6S 3

~s a ~s a 1 + 6oO(x) 6KO(x) 4 -~s ~ &oa(x) I (2.11)

is nilpotent: A2=0. The general solution to (2.10) is

fa~=aa~v + AX, (2.12)

where Gdi v is an arbitrary, gauge-invariant func- a tional of A, and its derivatives, and X an arbi-

a trary local functional of Au, co a, ~7 a, J , and K". The identity (2.9b) implies that :7" and J~ appear in the combination (J~4-n,~l"). It is also convenient to in- troduce the conserved quantity Ng, called the ghost number. If Ng[LJ - 0 , a reasonable assignment is

Ng[A".]--O, Ng[co~] = - 1 , Xg[lla]=- +1, (2.13)

Ng[J~]= +1, Ng[K"]= +2,

so that Ng[F~ and N g [ A J = - I . The structure of X must, accordingly, satisfy Ng[X] = + 1.

3. Solution to the Renormalization Equation (2.10)

In a previous article [3] we suggested the following forms for the functionals Gdi v and X (the integration

G. Leibbrandt and S. Nyeo: One-Loop N-Point Functions in the Light-Cone Gauge 503

symbol for x is suppressed):

Gdiv 1 a 2 1 a 2 = - ~ a l ( F ~ ) (3.1) -~a2(n,F~) ,

X - A a a a , a a a --a3 u(Ju+n~,rl )+aan~Aon,(Ju+nurl )

+ a 5 [n* ~(n~O~)- 1 nxA~ ] nu(j ~ + n, rff)

+ a 6 co~K ~. (3.2)

The null vector n * = ( n o , - n ) is conjugate to n, =(n o, n), and arises from (dimensionally regularized) Feynman integrals such as [7-1

d2~ 2p. n* - ~ ( p _ k ) 2 k . n - n . n ~ I , f = iTr2(2- co) -1. (3.3)

The divergent constants a~, i= 1 .. . . ,6, may in prin- ciple be found by comparing the terms obtained from the expressions (3.1) and (3.2) with the terms calculated from the corresponding Feynman dia- grams.

It was shown that the a~ can indeed be deter- mined from the YM one-loop self-energy and from the local terms in the three-gluon vertex. On the other hand, it became also clear that the structure of (3.1) and (3.2) is insufficient to account for the non- local terms in the 3-gluon vertex, such as [4]

H{u} = n,, n,= n,~ {(/)2 n,. p 3 _ p2 n*. p2)/n, n* P2 .np 3" n} (s),

P l + P z + P3 = 0 , (3.4)

where { }cs) denotes cyclic symmetrization over the indices 1, 2 and 3:

{guzu3(P2 -- P 3 ) u ~ } (s) = g ~ 2 u 3 ( P z - - P3)u~

+gu~,~(P3-P~)~= +gu~,=(P~-P2)u~, etc.

A possible remedy is to add extra terms to the expressions in (3.1) and (3.2)�9

Let us first consider the allowed counterterms that may be included in Gd~ v. As far as locality is concerned in (3.1), there is no a priori reason to exclude terms of the type

�89 (n*F a~2 and _ ! h n*- ~-a ~-~ (3.5) 1~ # #v] 2 ~ 2 ~ t ' ~ v * # X * v ~ "

Actually, it turns out that these terms are not needed to one-loop order.

As far as (3.2) goes, the number of possible can- didates, with the proper dimensions of mass, is much larger. Here are a few examples (Ju + n~/7 ~ = J~):

. - . b4[(noa~)-~a~A~]n Y~, b3n~A~nuJ~,' u u

bs[n~O~(n~t?)-~ ~ , -1 -, (3.6) n~A;.]n~3~(n~O~) nuJu, a b c * b * c 1 ~a b6g f n~ A~n,,A~(n c3 ) n~Ju,

bsgf~bc[n~ Or i b -a (3.6) n z A ~] n~A~(n o a ~)- 1 n,J ~,

b9 gfOb~ [n~ ar Z n~A~] [n* a~(n ~a o)- i n~A~] �9 % a , ) - ' ~.Y~.

Some of the expressions are irrelevant at the one- loop level. We also observe that most of the terms in (3.6) are proportional to f~b~ and hence to various triple combinations of n and n* such as: n,n v n~, from b6, and n~n~n~, from bT, b 8 and b 9. The set {n'n'no,, n,n~n~} arises from two A-fields and one J- source�9 Note that only those n's and n*'s contracted

a either with A n or J~ are considered. Needless to say, there is another category of pos-

sible candidates which are proportional to f~b~f~d~ and containing both n's and n*'s. For the sake of simplicity we shall consider terms having n's only. The relevant factors are then proportional to four n's (like n~nun~n,~ ) or, equivalently, to three A's and one J, and account for the four-gluon vertex. The last two paragraphs would seem to suggest, therefore, that the number of A fields in the func- tional X is one less than the number of light-cone vectors n,. In fact we shall see in the next section that the general structure of X contains (N-1)A ' s , where N is the number of noncovariant light-cone vectors n: nu,, ..., n,~,.

First, however, let us mention that the four-gluon a b c d vertex F~,p~ and the three-gluon vertex FA"~ are re-

lated by the Slavnov-Taylor identity

ir, F2~;~(r, p, q, k)= g[f~b~ FSe(q , k, r + p)

+ f"~eF~bf~(p,k,r +q)

+ f~e~Ff~(p,q,r+k)] (3.7)

with incoming momenta r, p, q and k: r + p + q + k ~0 .

4. Non-Finiteness of General N-Point Vertex Functions

Consider the solution for the counterterms (2.12), /~(1)=Gdiv-t-zlX. Gdi v is explicitly gauge-invariant iv

and hence local in space-time. Its structure can be obtained from the tensor Fly, the covariant deriv- ative Dab. , and from the constant vectors n, and n* [8]. The second term A X, involving the functional X, is BRS-invariant by construction. But apart from being restricted by (2.9b) and carrying the proper mass dimension and ghost number, the structure of X is arbitrary. In particular, X may not only contain local, but also nontocal terms. (Clearly, X is the only place where nonlocality can be introduced, since Gdi v

504 G. Leibbrandt and S. Nyeo: One-Loop N-Point Functions in the Light-Cone Gauge

A/~ 1

A,u N ] ! \ A,, 4

A~k_ I Apj

Fig. t. A general one-loop N-point gluon vertex diagram. This diagram, with N>5, is finite in covariant gauges and in the axial (or planar) gauge, but divergent in the light-cone gauge

composition formula

k .n (p -k ) .n p.n k ( p - k ) . n + , pu.#O, (4.4)

which may be considered as part of the light-gauge formalism. The crucial point is that formula (4.4) raises the degree of divergence of all those Feynman integrals which contain double or more factors, such as [k.n(p-k) .n] -1 or [(k.n)2(p-k).n] -1. Hence, the application of formula (4.4) complicates matters considerably. Suppose we set, as a special case, one of the external incoming momenta p~, P2 and P3 equal to zero, say p~=0, so that p 2 = - p 3 - p , and consider the following term arising from a vertex graph:

d2~ p~ n~ n~2n~ S (p + k)~(k, n) ~(p + k). n" (4.5)

For this and similar integrals naive power counting is not allowed. Double application of formula (4.4) leads to [3]

has to be local.) This explains the introduction of nonlocal terms in (3.2) and (3.6), and also of other nonlocal terms that might be needed for the nonlo- cal N-point functions, specifically for N > 5.

Let us briefly discuss why nonlocal terms are generally required in the light-cone gauge. The sim- ple reason is that the N-point functions, N=>5, (Fig. 1) in the light-cone gauge are divergent. To see this, we first consider the one-loop three-gluon ver- tex, which involves typically three bare propagators and three bare three-gluon vertices. The bare gluon propagator reads [9]

- i [gu~ (n,k~+_n_~k,)], k2+ie k.n J e>0, (4.1)

while the bare three-gluon vertex is given by

gf,bc [(p _ q)~ gx, + (q _ k)zg, ~ + (k - P)ug~ ~.3 (4.2)

with incoming momenta p, q and k: p + q + k = 0 . The unphysical singularites of (k-n) -1 in (4.1) are treated with the prescription I-7, 10]

1 k.n* - - - § e > 0 (4.3) k.n ~o k .nk.n* +ie'

leading to integrals such as (3.3). Explicit evaluation of a typical Feynman graph also involves the de-

2p 2 p. n* - (4.6) n .n*(p. n) 2 nulG2G3I"

This expression, a special case of (3.4), is propor- tional to 3n's - one n, from each propagator - and requires X to contain term(s) proportional to two A's and one ]. Hence X and 6X/6J~ are quadratic in A~,a so that the counterterms D = _~r(1)dlv will be of order five in A~. This implies that several nonlocal counterterms (with divergent constants determined by the three-gluon vertex) are also introduced for the four- and five-point functions.

Similarly, the evaluation of the four-point func- tion would give factors proportional to 4n's and would require X to contain terms cubic in A~. These would generate several counterterms for the five- and six-point functions. This seems to suggest the following hierarchical pattern: the evaluation of an N-point function produces divergent factors propor- tional to Nn's and requires X to contain terms of

a order ( N - 1 ) in A,. These terms will, in turn, gener- ate partial counterterms for the (N +I ) - and (N+2)- point functions.

The above structure seems to be exhibited only in the light-cone gauge and is certainly not shared by other known gauges, such as the covariant gauges or the axial (or planar) gauge. In the latter gauges,

a the counterterms are at most quartic in Au, since the associated Feynman integrals do not "generate" ex-

* (cf. (3.3)). Hence, N-point tra parameters, such as nu functions, with N->_ 5, are finite in these gauges.

G. Leibbrandt and S. Nyeo: One-Loop N-Point Functions in the Light-Cone Gauge 505

5. Summary

The renormalizability of Yang-Mills theory in the light-cone gauge in the framework of BRS in- variance remains an intriguing question. In this paper we examined the possible structure of the counter- terms arising from the solution of the renormalization equation. This renormalization equation is sufficiently arbitrary to admit also nonlocal counterterms which are needed to "absorb" the nonlocality of the three- gluon vertex function.

Using the structure of the counterterms required for the three-gluon vertex, we found that the five- point function is likewise nonlocal and divergent. Moreover, the nonlocality (and divergence) of ad- missible terms in the functional X seems to be in- timately connected with the number of noncovariant light-cone vectors nu. It turns out that the number N of light-cone vectors, n~ln~2...nuN , where N > 5 is ex- actly one more than the number of gauge fields A, . . .Au~ ~ in X. In other words, the N-point vertex function contains divergent, nonlocal expressions proportional to n~l n~,2...nu~,.

We should mention that Lee and Milgram have discussed renormalization in the light-cone gauge using an entirely different approach, but also ob-

tained a set of counterterms. We refer to their work for further details [1].

Acknowledgement. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant No. A8063.

References

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3. A. Andra~i, G. Leibbrandt, S.-L. Nyeo: Guelph preprint Mathematical Series 1985-100,

4. M. Dalbosco: Phys. Lett. 163B, 181 (1985) 5. See, for example, A. Andra~i, J.C. Taylor: Nucl. Phys. B192,

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(1976) 7. G. Leibbrandt,: Phys. Rev. D29, 1699 (1984) 8. G. Leibbrandt, T. Matsuki, M. Okawa: Nucl. Phys. B206, 380

(1982) 9. Here we let the gauge parameter c~-,0. The renormalization

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