Local operator products in gauge theories. I

ANNALS

OF PHYSICS

108,

233-287 (1977)

Local Operator

Products

in Gauge

Theories.

I

SATISH D. JOGLEKAR* The Institute jbr Advanced Study, Princeton, New Jersey 08540, and Stanford Linear Acreierator Center, Stanford Uniwrsity, Stanford, California 94305 Received October 16, 1976

In this and a subsequent paper, we discuss the renormalization and gauge invariance of products of local gauge-invariant operators and their operator product expansions in gauge theories in detail. In this paper, we study in detail the following problem which when solved leads to the complete understanding of the operator product expansion of a product of two gauge-invariant operators. The problem is to consider the complete set of color singlet local functionals of gauge, matter, and ghost fields @‘[A, c, F, Q,]) in a non-Abelian unbroken gauge theory of strong interactions containing fermions and find what subset of these local operators have the property that all of their “physical matrix elements” are q-independent to all orders in the perturbation theory. (Note: All the phrases in quotes and the above statement are to be defined in the text by certain limiting procedures whereupon they become precise mathematical statements.) Here, 7 is a gauge parameter. We believe that the discussion given here can be extended to spontaneously broken gauge theories in a straightforward manner, where of course, the physical matrix elements exist in perturbation theory. The conclusion is that the only nontrivial operators that have this property to all orders are all the gauge-invariant operators and certain gauge variant operators which are a subset of the renormalization counterterms a gauge-invariant operator at zero momentum needs. However, these gauge-variant operators are not new physical entities in the sense that their “renormalized physical matrix elements” are not independent to the “S-matrix elements.” These results will be useful in understanding which operators appear in the operator product expansion of, say two currents, letting one use these methods with full understanding in gauge theories. As a by-product, certain technical discussions in this paper can be used to considerably shorten the proof of a theorem in the renormalization of gauge-invariant operators previously given.

1. INTRODUCTION Operator product expansion [l] has been used extensively in obtaining results in asymptotically free gauge theories of strong interactions such as those on electroproduction, deepinelastic scattering [2], weak decays of hadrons [3]. Operator product expansion in gauge theories has not, however, been fully understood and understanding of someof the questions related to operator product expansion is necessary before one can confidently use these methods in gauge theories. The difficulties in understanding operator product expansion has been at two levels, * Research sponsored by the Energy Research and Development E(ll-l)-2220.

Administration,

Grant No.

233 Copyright All rights

6 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSN

0003-4916

234

SATISH D. JOGLEKAR

they both have been caused by the so-called ghost mixing in covariant guageswhich means that the renormalization counterterms needed for the renormalization of a gauge-invariant operator are not gauge invariant in form and further cont.& &ost fields. At the practical level this raises several questions. Previous extensive works [4-71 on this subject have answered several questions as to what the structure of the other operators that arise as renormalization counterterms is and what the structure of the renormalization matrix is. Knowing the answer to the latter question, one knows how to conveniently compute some of the eigenvaluesl of the anomalous dimension matrix. The second question of immediate practical importance is whether these new operator? are of any physical signiflcance.3 Contained in this are several subquestions such as: (i) Are their matrix elements nonzero? (ii) If so, can they be measured at least in principle.. (i.e., are they “physical observables?“). (iii) Do they appear in the operator product expansion at all? (iv) If so, do they appear at a stage so that their effect can be measurable in practical applications insofar as the effect of the gauge invariant observables is? At another level, one faces the question of understanding the gauge invariance of the operator product expansion in gauge theories. Indeed, unless one understands this, one cannot understand the question about the physical significance of these new operators. In a rather vague language the problem is the following. What one measures in the processes mentioned above is the physical matrix elements* of a product of current at short distances5 which ought to be gauge independent.6 This quantity is expressed as a series of matrix elements of local field operators possibly involving gauge-variant’and ghost operators which certain coefficient functions. Our object here, is to use the fact that the sum of these terms is gauge independent to infer what sort of local field operators enter the sum by studying the gauge dependence of the matrix elements of local field operators in gauge theories. At this point, we must explain what sort of matrix elements we are talking about. In application of an asymptotically free gauge theory it is an inherent assumptions that the physical hadrons are bound states of quarks and grluons and what one 1 That is, eigenvalues corresponding to the eigenstates of the renormalization matrix containing gauge invariant operators of class I. See, for details, Refs. [5-71 or Section 2. 2 One should not, at this stage, even assume that only the gauge invariant operators and their counterterms appear in the operator product expansion. (WT identities for
OPERATOR PRODUCTS IN GAUGE THEORY, I

235

measures(and what ought to exist and be gauge independent) are the matrix elements of these operators in physical hadron states; which we do not, however, know how to write explicitly.g What we shall do is to consider the matrix elements of operators (local operators or their products) between elementary fields of the theory, hereafter ccferred to as the “quark-gluon matrix elements.” As is well known, these matrix elements may not exist in higher orders in perturbation serieson account of infrared divergencesbut that the off-mass-shellGreen’s functions from which these are derived exist [8]. What we are interested in here are not the matrix elements themselves but their variation with the gauge parameter 7. We shall therefore devise a suitable onshell-limiting process for this variation. Thus we shall show, for a product of two gauge-invariant currents suitably renormalized, has the property that its ph\,sical truncated Green’s functions satisfy,lO [9] (1.1) (barring a certain set of exceptional momenta of the external lines.) In a loose language ( i ~(j,(~).h@)l~u~,n is a truncated Green’s function whose renormalization parts have been renormalized off-mass shell (pz = --p2 for gluons), and is evaluated off the mass shell at pL” -: -pz for gluons. (The precise definitions will be explained in Section 2.) Note that the above is a rigorous mathematical statement even though the matrix element lim,~+o(l 27j,(x) j,CO))l),2,, may not exist. We shall, for brevity of the language, refer to a statement such as Eq. (I .I) as the gauge independence (here, q-Independence) of the matrix element under consideration, their existence is neither implied nor necessary.We shall also refer to lim pz + 0 as the on-shell limit. In this paper. we shall consider one specific question. The above discussion was intended to explain its place in the general discussion of operator product renormalization and expansion. We shall restrict ourselvesto the linear covariant renormalizable gaugeswith a variable real gauge parameter 7. We shall consider the complete set of local field operators {F[A. c, c; ~o]~ and’l find out what operators in this set have the property that their physical truncated Green’s functions, suitably renormalized off-mass shell, satisfy

iie., has “gauge-independent” matrix elements to all orders. In a subsequent paper. 9One may considergaugeinvariant local field operatorswith the quantumGreen’sfunctionsof the operatorunderconsideration(suchas ~~(~~rs~(x)O~(y)u~~(~)> for :?r I 0 ! 7;). We shall consider such quantitiesin particular,in the succeeding paper.However,it isnot easyto, if it ispossible, to get the results of this paper by considering only these Green’s functions. lo It may be worth reminding the reader that the definition of a statement lim,. .“A A(x) : B makes reference to A(x) for x in some open domain (a, a + 6), 6 > x; it makes no reference to A(a). I1 Here, q0 is the unrenormalized gauge parameter and 17 is the renormalized gauge parameter.

236

SATISH

D.

JOGLEKAR

we shall discuss the general question of renormalization of operator products in gauge theories with a particular emphasis on application to product of two currents. The plan of this paper is as follows. In Section 2, we shall explain all our notations which are almost identical to those in Ref. [5], note down several definitions and discuss the results on the WT identities and on the renormalization of gauge-invariant, operators. We shall also explain more precisely how we do the off mass-shell renormalization. In Section 3, we shall discuss some of the isolated topics which are essential, to the future discussion. In particular, we shall consider variation with 7, of a certain1 kind of Green’s functions, and of Green’s function of operators in some more details, following the discussion of Lee and Zinn-Justin [IO]. In Section 4, we shall take up the main problem as explained in the next to then preceding paragraph. Here we shall briefly outline the method. It is easy to show that the matrix elements of any local field operator not depending on 7 explicitly, are, T-independent in the tree approximation. We therefore consider the y-variation of the physical Green’s functions of any such functional in one loop approximation. A part of this variation can be canceled by suitable changes in the renormalization counter-1 terms for the operator. The remaining part is, in general, a nonlocal quantity with a specific structure and may contain divergences. Now one has, of course, the option of modifying the operator under consideration in a local manner without altering its tree level matrix elements. We enumerate such possible alterations and show that they cannot, in general, cancel (in the on-shell limit) this remaining piece entirely. We then formulate a necessary condition that this piece can be removed by such modifications (modulo additional renormalization counterterms, of course). This restricts greatly the possible functions in the subset satisfying Eq. (1.2). The problem, then, becomes that of finding local functions all of whose matrix elements vanish up to n-loops and have the possibility of having gauge-independent matrix elements in the (n + 1) loop approximation; this is to be done for each y1starting from zero. Then, in Section 5, we shall take up these operators and show that they can be renormalized to all orders to yield gauge-independent matrix elements. The conclusion of these sections is that the local field operators whose (renormalized) matrix elements are gauge independent to all orders (and have at least one nonvanishing matrix element) are (i) all the gauge invariant operators (of class I and class II in the classification scheme of Section 2E), (ii) a certain subset of other operators of class II which we shall state explicitly in Section 4. The latter set of operators are not,-however, separate physical entities in that their matrix elements are trivially related to S-matrix elements (i.e., determined given the S-matrix elements). It should be remarked that there are other gauge-variant operators in class II that have nonvanishing matrix elements not trivially related to the S-matrix, but they are q-dependent. (It should be noted further that a priori, there was no reason why these operators should belong to class I @ class @ class III which itself is a small subset of the whole set of local operators {F[A, c, 2; ~~1). It turns out that in discussing the technical aspect of the problem, one has to solve a functional differential equation. A special case of this equation had been solved in the discussion of renormalization of gauge invariant operators in Ref. [5, seie

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

237

I

Eq. (4.23) of [5]). The methods presented here considerably shorten the solution of this equation. This is briefly discussed at the end of the Appendix B. We believe that the methods of this paper can be carried over in a straightforward manner to the spontaneously broken gauge theoreis (where of course all the matrix elements may exist) and reach the same conclusion. The problem there, may be thought of as that of finding “local observables” of the theory. 2.

REVIEW

A. Preliminaq In this section we shall briefly explain our notations which are very close to those of Ref. [5] and copy down several known definitions and results that we will need. In case more details are needed, the reader is referred to Ref. [5] and references quoted therein. We shall use the summation integration-convention as in Ref. [ll], its suspension in a few places will be explicitly noted where it is not obvious. The fields Ai will generically stand for both gauge and “matter” fields unless otherwise noted. Here, i contains all attributes of the field, a group index, a space-time point, possibly a Lorentz index. The fermion fieldsI when explicitly needed will be denoted by $, . Greek letters will be used to indicate a combination of a group index and a spacetime point (only). Last several alphabets will be reserved for space-time points only. It will be conventional to denote by xi the space-time point in a generalized index i. We shall deal with unbroken gauge theories with fermions only even though we believe that the results of Sections 4 and 5 can be easily extended to the spontaneously broken gauge theories. It is well known that the Feynman rules for constructing the Green’s functions of a gauge theory can be deduced from the effective actionz5:

%dA, c, 4 = %,[A1- t{f,[4)” + W,,[Al

c, .

(2.1)

Here, -${fa[A]}2 are the gauge fixing terms; Co, c, are the Feynman-DeWittFaddeev-Popov ghost fields. LYo[A] is invariant under local gauge transformations of group G, which we shall take to be simple. Extension to any semisimple groups is obvious. Thus, ZO[.4] is invariant under the infinitesimal transformation Ai + Ai’ = Ai + (aim + g,ti”jAj) 8, G Ai + Di”[A] 0, .

(2.2)

f: are representations of the generators of G and 2; zz i’yy(X, ZZ 0 g, is the unrenormalized I8The

theory

will

- Xi)

i refers to a gauge field, i G (01,xi , p),

otherwise. coupling constant.

be assumed to be anomaly-free.

(2.3)

238

SATISH D. JOGLEKAR

We shall work in linear gauges defined by f&4]

= r],1’2 at/Ii

EJ;:“Ai

(2.4)

where r], is an arbitrary real positive parameter. M&4]

of Eq. (2.1) is defined by [12]

M&4] The BRS supertransformations

= J;I*D$[A].

[13, 141 consist of

&I$ = c,Di”[A] a, 6c, = --j&l] where 6h is an infinitesimal BRS transformations

(2.5)

6c, = - s .!TOfk9,W~ a

8X,

(2.7)

anticommuting

x-independent

6(- ~{f&4]>2 + CMC) = 0;

c-number. Under the

K=%efM c, 4) = 0.

We shall also find it useful to consider the transformations Si;, = 0. We define an operator go by

of Eq. (2.6) only with

6F[A, c, E] = 3-oF[A, c, C[ 6X where 8F refers to the change in F under transformations the operator identity cYo2= 0 which is equivalent to the group condition

(2.6)

(2.9)

of Eq. (2.6) and 9, satisfies (2.10)

on Di”[A], viz. (2.1 la)

i.e., t;Dj4 - &DjOi = f”8Y&Y. Equation (2.11) has a further consequence, derivable in a straightforward done in Appendix A, that13

(2.11b) manner as

(2.12)

Thus the noncommutative algebra of the intinitesimal gauge transformations (Eq. (2.11a)) is equivalent to a commutative algebra (Eq. (2.12)). I8 This observation is due to Professor B. W. Lee.

239

OPERATOR PRODUCTS IN GAUGE THEORY, I

It is convenient to define an operator B which projects the noncovariant part of a functional, as opposed to gO which projects the noninvariant part of a functional under Eq. (2.7) as done in Eq. (4.28) of Ref. [5]. It also satisfies (2.13)

92 = 0. In particular, expressed as

the fact that L&

= &!&/&Ii

is a gauge-covariant

functional

is

B. Generating Functionals of a Gauge Theory; WT Identities

In the following, until renormalization is discussed, we shall be dealing unrenormalized but dimensionally regularized quantities. We defineI

The generating functional

with

of the Green’s functions is given by

W[j, <, f, K, I] = I [d.4 de d?] exp 1’{S[A, c, F, K,

I]

+

jiAi

-t-

&,

-I-

Q,}

(2.16)

where t and { are anticommuting sources for the ghost fields. The generating functional for the connected Green’s functions is given by Z[j,

5,

f,

K,

I]

=

-iIn

W[,j, (, f, K, I].

(2.17)

We define expectation values of fields in the presence of sources (2.18) which themselves are functionals of all the sources. Then the generating fuctional for the proper vertices is given by

r[%Q,0, K,11= z[j, t, $3K,I] --j#i - &+i&, .--r;l,r$, . Sourcesj, 6, and 5 can be expressed as functionals of F, L?, 8, ji=-"r

Em= 6%

'

f,

-g, a

I4 We have changed the sign of I (and L) from that in Refs. [5. 141.

=

K,

I

(2.19)

by the relations (2.20)

gl I

240

SATISH

D.

IOGLEKAR

where the partial derivatives of r are always taken keeping the rest of the fields (y, Q, 0) and K, I fixed. With a similar convention, --sr 8Ki

62 -

K

w ;

6Z

(2.21)

sl,=sl,*

W satisfies the following constraint known as the quation of motion of the (anti)ghost field. It reads

i.e.,

W, further, satisfies the WT identity

[14]

1

which, with the help of Eq. (2.23) is equivalent

Wj, 4, %,K, 4 = 0

(2.24)

to

o

(2.25)

As a special case of Eq. (2.24) we have (see also Eq. (3.25)), 0 = 1 [dA dc dE](fs + ~iDi~c&)

(2.27a)

exp i(%~r f j&

= 1 [dA dc dz](fs + jiDiaA4G1) exp %=%re+ jA)

(2.27b)t

and also, 0 = j [dA de dE](fo” - &MC + jiDi”M,-,tf,)

exp i(-%r

-I- jA>,

(2.28)

ie., (2.29) where we have defined Z[j]

G -iln

W[j] 3 -iln

W[j, 5, [, K, II Ic=~=~=I=II

(2.30:

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

241

I

and for a functional F[A, c, E]15 1 :F[A, c, c]:‘j :: I_w[jl

I’ [dA dc d?] F[A, c. (;I exp ijY&

(2.31)

-+ ,jiAJ.

Equation (2.29) gives the variation of the unrenormalized Green‘s functions with respect to the unrenormalized gauge parameter. The theory is made finite by wavefunction and coupling constant renormalizations

&

._

g.ypz-”

,!a>.

Ki _ K,p,

2

=

when i refers to a gauge fieid,

Z(i)

I, = L,Z1’2,

70 = +-‘3

(2.321

&&AA,= f$p.

It is a well-known result [I l] that 2, 2, X can be chosen in successive loop approximations so that the resulting ris a finite functional in terms of renormalized quantities and satisfies renormalized analogs of Eqs. (2.23) and (2.25). Further W satisfies the generalized equation [5] of motion for Ai . For any local functional FJA, c, E]16*li 0 = j [dA dc dE] 13

F,[A, c, L;] +jiFi[A,

c, (:]I exp i{&f

+ ,jiAil.

(2.33)

C. The Green’s Furlctions of a Local Operator Consider a local operator density18 O,[A, c, C;] and the generating functional

= 1 [dA

L~C d?]

exp I’{S[A,

C,

C, K,

I]

4

,jiAi +

E&C,

t- C,,[, + N,O,[A,

C?

Cl:. (2.34)

IS The phrases “field operators” and “functionals of A, c, E” have been unfortunately used synonymously. Thus matrix elements of a functional F[A, c, E] should really be read as the matrix elements of an operator F[A, c, E]. I6 In deriving this, one sets to zero terms proportional to P(0) or higher derivatives of 6”(x) at x = 0. This is possible in dimensional regularization. Ii As a special case of Eq. (2.33) we have the equation of motion for A, : (k!Y&%l, + jJ,,[.t -= 0 which is a relation between (unrenormalized but regularized) Green’s functions to all orders, and should not be confused with the classical equation of motion which is only a special case of the above equation of motion for Ai . I8 When we shall want to consider a local functional depending on one specific point x only, we shall call it a local (operator) density and denote its dependence on the space-time point x by a subscript. Thus the Lagrange density is a local density for the action.

242

SATISH

Then the connected unrenormalized to be) generated bylg

D.

JOGLEKAR

Green’s functions of the operator 0, are (defined

1 SW 1 -__ w SNzIN=O= w[j, t, g, 4 11I [dA dc dE]O,[A, c, ?] (2.35)

where

Z[j, 5, E,4 1,Nl = --i In W, 5, CT, 4 1,Nl. The corresponding proper (SI’/SN,),=, where

vertices with one insertion

of 0,

(2.36)

are generated by

We further have (2.38)

Unrenormalized

proper vertices of 0 at a specific momentum

s

d*x e-iq’x(U’/SN,),,,

The renormalized Green’s INEo where

functions20

q correspond to

.

of 0,

are generated

by (l/ WcR))

X

(6 W(R)/GN,)

WtR’[J, 5, %, K, L, N] = s [dA dc dz’] exp i S[A, 1 + N,

(O&4

where J, 5;, c, K, L are renormalized Ji = Zt,‘,“j, ,

C,

C, K, I] + jiA, +

c, ?I + ; Z(+k’)[A,

$,c,

c, 2, K, d,l

.+ C,ta (2.39)

sources (see Eq. (2.32)) defined by 5, = 2”“5,

9

{a = ,l’2fa

counter-terms for 0, . Here Z(p) is expanded in and C, Z(p)O(p) are renormalization terms of a fictitious loop parameter a: Z(p) = X+.=1 a rz,(‘). Given these counterterms Is Note that this definition is an analog of covariant T* Green’s functions. It may have no relation to the S-matrix for the theory with a “Lagrangian” =.%‘e~r+ N,O. beyond the tree approximation. Further, this definition is relevant to the operator product expansion. e” The following is a brief summary and it could be misleading if extrapolated beyond the actual statements.

OPERATOR

PRODUCTS

up to (n - I)-loop approximation, are determined by constructing

IN

GAUGE

THEORY,

the counterterms

243

I

in the n-loop approximation

with the determined counterterms

and expressioning the r.h.s. as a functional of renormalized fields @, w, w and sources transformations, such as j - y * CD.) Then the divergent part of I’,’ (as determined according to certain conventions for determining the finite parts of renormalization constants) is local and has the structure K, L. (One has to go through

Then Eq. (2.41) determines are generated by {ly’), These counterterms

ZF)

and then the renormalized

proper

vertices

= {TN’>, - {r,‘}“,i”.

are then incorporated

in Eq. (2.39) by changing its exponent by,

We shall use the following terminology: (i) The Physical Green’s functions will refer to the Green’s functions (not necessarily renormalized) whose external lines are on-mass shell and attached to physical wavefunctions (polarization vectors, spinors, etc.). (ii) Physical matrix elements (or just matrix elements) of an operator are its truncated renormalized (external lines renormalized properly on-shell) physical Green’s functions. (We shall come to the problem of infrared divergences in the next subsection.) (iii) Gauge invariance will refer to the invariance property of a functional O[A] under the gauge transformations of Eq. (2.2). (iv) Gauge independence will refer to the independence of a c-number (such as the Green’s functions) under the variation of the gauge fixing term, which in this paper is restricted to varying q. (v) Operators with identical matrix elements are said to be “equivalent.”

244

SATISH D. JOGLEKAR

D. RenormaIization Conditions and the Infrared Divergences

Since the on-shell Green’s functions in gauge theories are generally infrared divergent, it is necessary to renormalize them off the mass shell. We shall perform subtraction at an off-mass-shell point which may be varied and brought arbitrarily close to the mass shell. (This is necessary because we will have to consider the on-shell limits of quantities involving these renormalized Green’s functions.) Thus, for a theory with fermions with mass m (which may be zero and gluons, we shall renormalize a proper vertex with 2m (n? = 0, l,...) external fermion lines with momenta (ql, q2,..., q2,,J and n (n = 0, l,...) gluon lines (omitting the m = 1, n = 0, i.e., the fermion propagator case) with momenta p1 , p2 ,..., pn at a renormalization point restricted by (i) pi2 = --p2; p2 > 0, (ii) qi2 = m2 - p2, (iii) every partial sum (C qd + Cpi) with 2r fermion momenta and s gluon momenta satisfying, for some fixed pcLo2, (C q + Cp)” < (2rm)2 for every ~2 < ~~2, and qi . qi all equal and fixed; (iv) pi . pj all equal, (v) qi . qi all equal. If we are considering a proper vertex with an insertion of an operator (of dimension D > 4) at an arbitrary momentum t we will have to impose instead of (iii) (vi)

(vii) (viii)

min{(C

q +

C P)“,

E

q +

E p +

t)2>

<

@rm)2

for

p2 <

po2

and

t2 < 0 (we allow t = 0), t * p all equal, t 9 q all equal.

(It is easy to modify the above conditions if there are fermion species with different masses.) A fermion propagator will be renormalized at q2 = m2 - p2. Of course, this means that if we are considering a proper vertex with one insertion of a local operator of dimension D (or an ordinary proper’vertex without an operator insertion, in which case D = 0) and we have renormalized it to (k - l)-loop approximation, then we express the kth loop contribution to it as a sum of terms Cr P& each term proportional to a polynomial Pi[{qj, (p}, t] of dimension (d - 2m - n) and multiplied by functions h of Lorentz invariants (which in four dimensions will be dimensionless) and subtract fi’s at the renormalization point;21 so that the pole part of fi is removed at the renormalization point and the value of the subtracted fi at the renormalization point is some T-independent finite number. (It may happen that for certain polynomials Pi the coefficients functions fi are finite as a result of certain symmetries and it may be necessary not to subtractfi at all to preserve the symmetry. In such cases, it is understood that they will not be subtracted.) In the limit p2 + 0, *I The finite parts of unsubtracted Feynman diagrams in the presence of divergences are always ambiguous. They may be made unique by making some conventions, though this is unnecessary.

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

I

245

our prescription would reduce to on-shell renormalization (except that the limit may not exist) with “standard” physical renormalization conditions. Thus the quantities that enter Eq. (1.1) or Eq. (1.2) are the renormalized truncated Green’s functions (with p2 = 0) in the above manner and evaluated off the mass-shell for pi2 = -p12, q12 = m2 - p12 and contracted with physical wavefunctions (defined by P . 4~) = 0, (4 - Cm” - p12)1/2)u(q) = 0, etc. and which are continuous in pr2 and exist at p12 = 0). We shall let pr2 = p2. [At any rate we must have p12 - pZ 0(p4) for the discussion to hold.] We have not yet specified how the finite parts of renormalization parts at the renormalization point are determined apart from saying that they are gauge independent. The loop (21) contributions to Green’s function of operators at zero momentum with two external lines will be set at zero at the renormalization point. For operators which carry a space-time index and quantum numbers of a field, the loop contribution to its Green’s function with one external line will be set to zero at the mass-shell for the field. We leave the rest of the finite parts arbitrary except to remark that they be chosen so that, in particular, the equation of motion and WT identities are preserved. E. Results on the Renormalization

of Gauge Invariant Operators

We shall briefly state and explain the results on the renormalization of gauge invariant operators as explained in Ref. [5]. Consider the following subset of operators of up to a given dimension22 and given Lorentz transformation property and perhaps other quantum numbers and symmetries. C/ass 1: Gauge invariant Class II:

operators not expressible as ZO,iFi[A].

Operators expressible as

if i refers to a gauge field and F = 0, otherwise, and FJ.A, c, Z’] is a local functional containing terms with equal number of ghosts and antighosts. g,, was defined in Eq. (2.9) and ,!?[A, c, (;I =TJA] + ?Mc; C,’ = +1’2)Ea . Class III:

Operators expressible as

where 1, is a local functional containing one more c than the number of Zs. Then 22Of a given dimension in the absenceof massiveparameters; up to a given dimension otherwise.

246

SATISH D. JOGLEKAR

(i) The operators of classes {I @ II @ III>, (II 0 III), (III) are separately closed under renormalization to all orders. This means that class I operators need only class I, class II, and class III renormalization counterterms, class II operators need only class II and class III renormalization counterterms, and class III operators are closed by themselves. (ii) The Green’s functions of class III operators with no external ghost lines vanish identically. (iii) The connected unrenormalized Green’s functions of a class II operator are generated by (0”)j

= j,(FJA,

c, E] - D&JkBZ;F,[A,

c, E]>j

(2.46)

to all orders. (iv) Any of these operators, 0, has the following WT for the Green’s functions

where W’ = 1 [dA dc d?] exp i{-%r + K~‘(D& + NO + j,Ai

+ hu

+ NQia)

+ la’(&go&,csc,

+ XJ

+ AX,)

(2.48)

where Qi = 9’FC(9 was defined above Eq. 2.13) for a class II operators andzero otherwise, X is nonzero only for a class III operator. Since all of these operators are closed under renormalization and since they all satisfy the above WT identity, it is easy to show that the generating functional of the renormalized Green’s functions of any of these operators also satisfy a WT identity of the same form

(v) The matrix elements of a class II operator-vanish if inserted at a nonzero momentum (except for sets of exceptional momenta). This is a consequence of form (2.46) for its renormalized Green’s functions. In such cases, the quantity multiplying j, in Eq. (2.46) does not have pole in the (momentum)2 entering via xi at the (mass)” of Ai ; except for terms which vanish when contracted with jjphyl because of a.,jTphy) = 0. At zero momentum their matrix elements may not vanish. * I (vi) We shall refer to Fi in the definition of O(n) as the “F-function” for On. 3. SOME USEFULRESULTS

The purpose of this section is to work out several isolated results needed in the discussion of the future sections. In Section 3A we shall introduce some convenient

OPERATOR PRODUCTS IN GAUGE THEORY, I

241

notations. In Section 3B we shall consider the gauge variation of a specific kind of Green’s functions. In Section 3C we shall derive a formula for the variation with 7 of the Green’s functions of an arbitrary local field operator. A. A Convenient Notation Here we shall establish some convenient notations and express certain constraints that the physical Green’s functions satisfy in these notations. We know that the S-matrix elements or the matrix elements of an operator are obtained from the appropriate renormalized on-shell Green’s functions by truncating the external lines and attaching the physical wavefunctions. This process is written in short hand as setting (the renormalized source) J( = J/PhY) :-I ~~;j~,io

(3.1)

in the Green’s functions, where vj” satisfies, ~$p~O = 0 which is the transversality (k . c(k) = 0) condition also

(3.2) for gauge field wavefunctions and

s?o,ij(go = 0) f&o + ~~~jcpj” = 0.

(3.3)

The arrow on 5!‘:1, indicates 23 that while considering a quantity such as Fj[J] -?$,f:,qjB the xi-integration be done before the xj-integration. Equation (3.3) with the help of Eq. (3.2), is just the free field equation for ~~0, and ensures that qI() contains only the on-shell Fourier components. 2h’$j simply truncates the external propagator. Now the renormalized fields @JJ] are related to Jj via the relation @[[.I] = SZ[J]/SJ, .

(3.4)

We define the symbol @!phy)I

=

@,[J]!

Jz-J(l’h>)

(3.5)

.

Then the WT identity of Eq. (2.27a), which as applied to the renormalized Green’s functions can be written as

.fy g (J=J,phy) = ~YDl”“Y)= f”[@‘““Y’] = 0. Now consider three approximation

(3.6)

[p, = @,.i = J]. There we have f”[,$““Y’]

23These notations are similar to those introduced 59.5/108/2-2

physical

= 0 by B. S. Dewitt.

(3.7) See Ref. [15].

248

SATISH

D.

JOGLEKAR

and further (3.8) and thus, using Eq. (3.7) jjPby)

=

-~o,i[Cp(phy)].

(3.9)

Now, for any functional &;i[y]; jd(phy’F.[p] = ~ioJ&ijF,[~] vanishes on account of Eq. (3.3), if F,[v] does not contain L pole at the (mass)2 of the field Ai in the (momentum)2 entering through xi ,24 Under these conditions on Fi 6p.

i[,$~“y’]

Fa[#““Y’]

= 0.

(3.10)

Thus, for example, (~~,iP)i)plsgtDh~B # 0, while (~o,itij~j)~=mcphu~ = 0, or s (S~o/8A,a(x)) F,*(x) eiwz d”x = 0 for q # 0 in general. Further, we note that Eqs. (3.7) and Eq. (3.8) are the only conditions that y = #*hy) satisfies. Further, they involve local functionals [&[y] and Zo,J~]] of pl. Therefore, given that a local functional satisfies,

it follows that it is expressible as,

where P

and IP2) can be chosen to be local.

B. Gauge Variation of the Green’s Functions In Section 5 we shall need the q-variation of the renormalized truncated connected Green’s functions whose n external lines are all on mass-shell but only (n - 1) lines are attached to physical wavefunctions. The renormalized Green’s functions are generated by

= s [dA dc dE] exp i(Tp,[A; gY] - +~Z-l(ai”A~)” + ~1’2Z-‘1”2)F~ ai”D,‘[A; g Y] Co + Z-‘1’2’JiAi)

(3.13)

where Y is defined by Y(g, 7, ,@) = xz-1z-‘1/2’ 24 In this case the direction of the arrow is irrelevant.

= go/g

(3.14)

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

and Y, Z, 2 are functions of g, 7, p2 (the renormalization connected Green’s functions are generated by

1

249

point). The corresponding

Z”)[J; g, ~1 = --i In W”)[J; g, 71 = Z[j; go I ToI.

(3.15)

In the following, we shall make it a convention that unless otherwise noted, a partial derivative of a function is to be computed with the rest of its arguments fixed. We may avoid writing these arguments whenever they are obvious. Thus, IV(“) ::.= W(‘)[J; g, q] while W = [j; go , qo]. Further renormalization points are also held fixed. Now,

(3.16)

while

We shall assume that g is defined by, say, minimal subtraction of one of the threepoint functions. [The dimensional parameter of the dimensions of mass that appears in dimensional regularization may be given some fixed value.] In this case it is known that25 (3. IX) ThusZ6

85See Ref. [4] in this connection. z6 We believe that the following relation and the discussion leading to it are implicitly in the discussion of Ref. [lo].

understood

250

SATISH D. JOGLEKAR

The first term in the curly bracket is given by Eq. (2.29) (3.20) It is a well-known result (Ref. [lo]) that if one were considering the &‘-matrix elements2’ (properly renormalized on mass-shell) then the contribution from the first term in the curly bracket on the right-hand side of Eq. (3.20) is canceled exactly by the second term which corresponds to taking into account the change in the wavefunction renormalization constant with a change of gauge. We are, however, going to consider the Green’s functions which are on-shell and have physical wavefunctions for all but oozeexternal lines. In a manner analogous to the discussion of Ref. [lo], we can easily write down the net contribution of the curly bracket in Eq. (3.19) (using Eq. (3.20)) which has poles in all the external momenta. It is shown diagrammatically in Fig. 1. It arises from ji(aiOLM;l&j_i(P~Y~ where ji refers to the source for the last line which is not constrained to be physical (ai@ji # 0) (Fig. la) and from a part of ji(t&AiMa;8tfp)j=jcphyr (Fig. Ib). The remaining contribution from the latter term is canceled on-shell (i.e., up to O(p2) off-shell) by the second term in the curly

j,Sp--

*

_

:i~~~~: ,._.,. ..:.::;:::::::::>:..

4

G 9

(4

(b)

-*-

Ghost

m

Propagator

FIG. 1. Contributions

propogator

to q-dependence

as a functional

of j

for Ai of the Green’s functions under consideration.

27Statements containing on-shell limits are to be understood in the sense of Section 2D.

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

251

I

bracket as in the case of the S-matrix element. The sum (1 a) + (lb) obviously has the structure Z:i*G,,[J(phy)] Xq[J(phg)] (i refers to the last external line with no physical wavefunction attached to it). Note that the other contributes to ji(Di~M,-,lfa>J,P~,Y~ of the form ,jl :D,‘~M;~,;,~:J(Phr,,~~[~(PhY)~ vanish identically on account of Eq. (3.6). C. Gauge Variation of the Green’s Functions of’ an Operator

The renormalized are generated by

connected Green’s functions

ZC’Kg*771 =:pq$TTj 3.

-i

of a local operator

O[A, c. ?]

[dA dc dL;]{O[A, c, C; gY] + C.T.1 exp ii...:

w&i; gn?rlol wi; go9701.

(3.21)

The exponent in Eq. (3.21) is identical to the corresponding one in Eq. (3.13) and C.T. refers to the renormalization counterterms for O[A, c, T] as introduced in Section 2C. We have assumed that 0 is independent of T,, . In a manner analogous to the previous subsection it is easy to see that

(3.22)

The first two terms on the right-hand side are just the connected parP of (a/+,) WN and the circular bracket in the third term is the connected part of (6 WN/Sji). Thus,

(3.23)

Now, WJj:

go, Q,] = s [dA dc d?](O[A, =

s

[dA dc d?](O’[A] -/- (C.T.)‘) exp i{P&

= 10’ [-i$-] 28Connected part of (AB)

c, T] f C.T.) exp i(~?‘~rr -+ j,~~:

+ (CT.1

is CAB) - (A)(B).

[-i&-l!

+ ,jiAd

W[j; g,, qJ

(3.24)

252

SATISH

D.

JOGLEKAR

where in obtaining (3.24) we have integrated the ghost field dependence of (0 + C.T.) by using formulas such as [dc dEj c& exp(iEMc) = -iMG’[A] s and written the resultant (nonlocal) functional of A as O’[A] + (C.T.)‘[A]. &

W,U;

go, rlol =

10’ [--i

+]

+ CT.1

[--i

+]I

6

WA

(3.25) Thus,

go 7 TCJ

(3.26)

+ $

]O [-i+]

+ (CT.)

[-+]I

W.

Now using Eq. (3.20) and Eq. (2.17) we obtain,

+ ; ((O’[A]

+ (C.T.)’ [A]) j&%;;~)jCon

+ (70 $10’

(3.27)

+ (C.T.Yl)j.

Thus, in the end

&,B-!-

=

MC

[0’ + (C.T.)‘])

f vD (& + j&O

i

(0’ + (C.T.)‘} M$+ + i ((0 + CT.} jiD~Mg~fy)co”n

+ C.T.) Adconn - Z~j1’2~

aZ:$2 0

E -iT

/ f + O(p2) go

J!-
(We have dropped primes in the third term in the curly bracket as they are of no consequence.) 4.

OPERATORS

WITH

~-INDEPENDENT

MATRIX

ELEMENTS

In this section, we shall take up the main problem of this paper, viz., to find the subset of all local functionszg {F[A, c, C; go, r),]} of a gauge theory whose matrix 2Q It is assumed that F[A, c, E, g, , v,,] has the following properties: (i) It is invariant under the global gauge transformations, (ii) has equal number of ghosts and antighosts; (iii) the total power of fields - power of g, = 2 for each term.

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

I

253

elements are gauge independent in the limiting sense explained in the Introduction and Section 2D. We shall go about it as follows. We shall first consider a local functional of gauge fields only, and show that for any local functional O[A, g, , VI,,] such that aO/$, has the structure

(where E, has the properties mentioned above Equation (3.10)) has v-independent matrix elements in the tree approximation. Then we shall consider such an O[A] and consider the variation (with 7) of its Green’s functions in the one-loop approximation. We shall show that a part of the change in the Green’s functions is removed by taking into account the variation of the wavefunction renormalization and by making suitable changes in the counterterms needed to renormalize 0. There is, however, a remainder. We shall then examine (i) under what conditions the remainder vanishes and if not (ii) under what conditions can the remainder be canceled by modifying 0 by adding to it local terms that do not alter its matrix elements in the tree approximation. The first possibility is realized for all the gauge-invariant operators while the study of the second possibility restricts greatly the gauge-variant local field operators for which the second possibility works. Once this restricted set is obtained in the one-loop approximation, the problem becomes a similar analysis of the following two questions to be studied in successive loop approximations. (i) What is the set of local operators whose matrix elements vanish up to n loop level (n = 1, 2,...): (ii) What subset of the set of operators whose matrix elements vanish up to n-loop approximation (n = 1, 2,...) which may have nonvanishing and T-independent matrix elements in the (n + 1)-loop order. One then has to take all the operators which may have gauge-independent matrix elements and show that they can indeed be renormalized to have gauge-independent matrix elements. This will follow from the analysis of the next section. Consider a local functional O[A] which does not depend on Q explicitly. In the tree approximation


qqj] 1-7(A):.

Now according to Eq. (3.28) in the tree approximation ( q‘qj;

i-S77-

(O[A]),;

= -

$


(4.1)

(Q = 7, j == J, QV=: @)

+ ~,ji(o,~Ml,%sO[*]~~:""". (4.2)

NOW

t?ivjz(phs’

=

0 and,

254

SATISH

D.

JOGLEKAR

vanishes in the tree approximation because the term multiplying ji does not contain a single particle pole with quantum numbers of Ai in the tree approximation. Also in the tree approximation,

sincefB[@hJ’] = 0, on account of Eq. (3.7). Thus (4.4)

in the tree approximation. This statement can be obvious by generalized local functionals which depend on v,, (explicitly) in such away, that, in the tree approximation

i.e., the local functional g

= &Y&t]

aO/aq, , according to Eq. (3.12) must have the form + ZO,,pk + local terms containing an equal number of ghosts (2 1) and antighosts

(4.5)

where flk has the property mentioned above, Eq. (3.10). We, therefore, have to go one-loop approximation to be able to find the desired constraints. What we have to do now is (i) consider a local functional O[A] with aO/aqO = 0 and see under what conditions its renormalized matrix elements will be v-independent; (ii) under what conditions can their q-independence be restored by modifying 0 by terms whose physical matrix elements in the tree approximations vanish, (iii) consider the q-dependence of the latter terms separately in the one-loop approximation. Thus, now consider a local functional O[A] of gauge fields only, satisfying aO/aq, = 0. Then the gauge variation of its renormalized Green’s functions is given noting by Eq. (3.28). W e write the Eq. (3.28) valid up to one-loop approximation {arlO/aqj, = 1, 2 = 1 + z, {C.T.), = 0, O’[A] = O[A],

- i {(Dk40,kM&,~)}l

aolaTO = 0

+ terms that vanish at J = Jcphy’. (4.6)

(These last terms vanish for the same reasons as the terms on the right-hand Eq. (4.2).)

side of

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

255

I

The first term on the righ-hand side of Eq. (4.6) is indeed the change in the renormalized connected Green’s functions of 0 on account of the change in the external line wavefunction renormalization constants. Now, consider the second term. If the operator insertion is made at nonzero momentum, then the on]y33 contribution from this term that survives for physical (truncated) Green’s functions have the form shown in Fig. 2a. They are equal to (up to factors)

(Here, 5?$&, =~ (Sz-(tp/&4, &I,) lAzO is the tree level inverse propagator.) Comparing the above two contributions to the similar contributions in the case of the gauge variation of ordinary Green’s functions, it is clear that they cancel exactly on mass shell, [That is, when Z is determined slightly off mass shell (‘at -p’) they add up to

r --

7

(b) 2. Contributions to the second term on the right-hand side of Eq. (4.6). [A square box contains connected Green’s functions while a circular box contains an TPI graph. A cross denotes operator insertion of 01. FIG.

terms of O(/U.~). Next, if the operator insertion is made at zero momentum, one has in addition, the the contribution of Fig. 2b. Here, the operator O[A] DLfiM;$, is inserted in a one-particle irreducible manner. This term has the form

The Fourier is a function

transform of the two-point (truncated) function of one-momentum p only and has the structure

in the curley bracket

(4.9) where XL’(p)

are linearly independent

polynomials

of dimension

(D - 2). (D being

256

SATISH

D.

JOGLEKAR

the dimension of OIA] which we can choose to be well defined without loss ofgenerality) and carries additional Lorentz indices 0 may have, andfi(pz) are dimensionless functions of p2 in four dimensions. 21 At the renormalization point, therefore, the term (4.8) has the structure

f ‘“h’~~(~) @,&I] (4.10) Y where X$(a) is up to factors, the Fourier transform of x:$(p). But this is also the form of the truncated physical Green’s functions of cIass II operators of Section 2E. Thus it is clear that this change in the matrix elements of 0 can be entirely canceled by (perhaps a part of) (~o(a/a~,)(C.T.)‘)j , in other words by introducing renormalization counter-terms of the form a~,,( g, 7) C Zu)Oj’. We thus, have to consider only the term -(i[2)(
WDlc”O,df&Ji

= V’t’L4 O&l ~B;11[4L=-iw

W
(4.11)

Using the WT identity in the form of Eq. (2.27b), the r.h.s. of Eq. (4.11) becomes

= {DkpMl O.&II &,f[AI~~=--iw~~).M-Q”MG>~W = ji
W

(4.12) and thus,

At J = J’phy), several of the contributions from the first term on the r.h.s. of (4.13) now vanish for the familiar reasons. The only nonvanishing contribution from this term will be renormalized away at J = J(phy) by a class II counterterm in our renormalization scheme. We therefore have to consider the second term on the r.h.s. of Eq. (4.13). Since this quantity already contains a loop integration, we have up to terms that are renormalized away,

-W~sO~dfa;lfy~ih = -Di”[vl ~,l,l[d +,j(o,B~d K,bl

O,,[yl)

+ terms which vanish at cp = gpcphy).

(4.14)

We separate the first term on the right-hand side into two parts as shown diagrammatically in Fig. 3. It is evident that on account of the two consecutive ghost propagators these diagrams may be divergent for n = 4 even when all the external lines are off mass-shell and for any momentum q at which 0 is inserted because of the integrand of the form $ (dnk/k4) f (k2....). Now our strategy will be to consider all possible local modifications of 0 that leave its tree matrix elements (generated by O[p, (phr)]) unchanged; and see if gauge variation

OPERATOR

FIG. 3. Contributions external sources.]

PRODUCTS

to the right-hand

IN

GAUGE

THEORY,

I

257

side of Eq. (4.14). [All quantities are in presence of

of any of these has a similar contribution in the one-loop approximation. show that that is not the case. Further, the contribution of the diagrams removed completely by adding renormalization counterterms as they are infinite series in y. There are three (apparently) independent, local modifications of 0 O[@hr)] unchanged. Add:

We shall cannot be (nonlocal) that leave

(i) A term of the kind T,Q,,JA] cg . (Terms containing four or more ghosts are irrelevant at one loop level.) (ii) A term of the kind &[A] &[A] ( in view of Eq. (3.7).) We can choose X,[A] to be independent of c and 3 in view of the availability of alternative (i) above. (iii) A term of the kind (&%rt/&4J I;;[A] where FJA] has the above-mentioned property Eq. (3.10). (This is equivalent to adding z&~FJA] if one takes into account (i) and (ii) above.) Now, alternatives (i) and (ii) are related by the WT identity of Eq. (2.27). Acting on both the sides of Eq. (2.27) by X,[ -i(S/Sj)], we obtain

Thus, using Eq. (3.25),

But, as explained before in dailing with the term ,ji(DiyM$fBO) in Eq. (4.6), the on-shell matrix elements of the last term only correspond to those of local counterterms of the form of class II operators. Therefore, in considering the renormalized

258

SATISH D. JOGLEKAR

matrix elements off+X, , a renormalization counterterm will wipe out the second term (on-mass shell) when renormalization is done according to the conditions of Section 2D. Thus the second alternative is a special case of the first. Further, the third alternative is inconsequential since according to Eq. (2.33) @-%~f/SAJ &[A],

= ji(Fi)j

(4.16)

and for an F with the property mentioned above Eq. (3.10) and the same discussion applies to this term as the term ji(Di,M&~), viz., this term is also renormalized away. Thus we need consider only the first alternative. Now (for a local Q&

=

iQ,oP> 71 MiitPl

since the term already contains a loop. Thus

meaning that the gauge variation has the same form as the original term. We shall soon see that such a ghost term modification cannot cancel the contribution of diagrams in Fig. 3 unless certain conditions are met. The integrand of this contribution has the form (d%/k4)f(k2,...). We shall show that f must have the structure30 f(k2, -..) - f(k2, -) near kZ = 0 (4.19) where f”(k2,...) does not have a (1 /k2) singularity. In the present case, f is a sum of tree diagrams; and as such it only has poles. Hence J will have no singularity in k*. While considering a similar argument in higher orders, f will in general have logarithmic singularity in k2. Here we note that in expressingf, we have chosen k to be one of the independent momenta and k2 to be one of the independent Lorentz invariants. In the present case, given that f is a function which may only have poles, it follows that (4.20) f(k*) = kv(k*,...) for all k*. (If j had logarithmic still infer that

singularities in k2, as will be the case in highet orders, we can f(k2) = k2j(k*,...)

for at least a range of k* in the neighborhood

of k2 = 0.)

30It may not be entirely obvious. But an argument that follows using the discontinuity is self-contained and yields, in particular, this conclusion.

of diagrams

OPERATOR

PRODUCTS

IN

GAUGE

THEORY.

We need therefore to have the Fourier transform a given number of external lines) contained in

259

I

of the Green’s

functions

(for

proportional to k" when the momenta entering via X, and x,, viz. k and k’ are equal and opposite. But the above quantity is symmetric in h and o, since the curly bracket is symmetric in y and 6 on account of Eq. (2.12). It is evident from the structure of the diagrams that (4.21) does not contain a pole in k2 or k’“. Hence WC can expand the Fourier transform of the above quantity as (4.22)

a + b(k + /if’“)

with a certain choice of independent variables which include k, k’, k”, k”. a does not contain terms proportional to k2. b does not contain poles in k’ or k’“. Also CI is symmetric in h- and k’. Now, since .f = a + b(k2 + k’2) !I;fk.,=Oz kZj=,

(4.23)

we have

We shall now prove the statements made above. To do this, we shall consider the discontinuities of graphs in Fig. 3. In discussing the discontinuities, it is convenient to consider an auxiliary set of diagrams in which a nonzero (q) momentum is injected via vertex denoted by y in Fig. 3. (Indeed, requiring that the diagrams in Fig. 3 vanish for arbitrary q, is equivalent to requiring that, the matrix elements of 0 be invariant under a space-time-dependent change of gauge parameter 6 17) J‘ (i .

Sq(x)( i: . A(x))‘!

d”.Y

ST)(s) cc P?QJ.

(4.24)

This is quite consistent with the demand of gauge independence. More importantly Eq. ( I. I) is valid for such a change in the gaugeterm. We now consider discontinuitjtes of the diagrams in Fig. 4 obtained by cutting them as shown. They are the only discontinuities whose integrand blows up as q --f 0 like l/(k q $ q2). If thesediagrams have to vanish, so must these discontinuitites, in particular.31 This willhappen again, only if [seeAppendix D] the rest of the integrand suppliesa factor of k2 or (k t q)“. Now the right-hand side of the cut has the form of a truncated Green’s function with gauge bosons (at least one) attached to ghost propagator viz., (a2),&;;‘[@](?“),,, . It is easy to show that this factor does not supply the necessary factor of k*. Thus, 31It is possibleto devisea consistenton-shell limiting process for considering such a discontinuity of this quantity.

260

SATISH

D. JOGLEKAR

(b) 4. Specific discontinuities of graphs in Fig. 3 under consideration. of the cut, there is at least one gauge boson line.] FIG.

[On the right-hand

side

the left-hand side, which is precisely the expression of (4.21) evaluated at k2 = kt2 = 0 (k, k’ here being the momenta of the lines cut in the diagram) must supply such factors. We thus learn then that a(k, k’) (which does not contain terms proportional to k2, k’2 must vanish identically. [For proof, See Appendix D.] Therefore, the Fourier trans form of the above quantity must have the structure32 b(k2 + k12)

(4.25)

where b does not contain a pole in k2 or k’2. Now consider the expression (4.21). It is (4.26) What we shall hsow in Appendix C using the form (4.26) for this term, is that the quantity in the curly bracket must have the form

Kci@l G,Pl + (o! ++ /?) + terms that vanish at @ =

@(PRY),

(4.27)

It is clear that with this form for the curly bracket, the quantity (4.26) will have a Fourier transform of the form (4.25). We have to show the converse. Here, T,, does not contain a pole in the (momentum)2 entering viz. x, . c+*This must be true for any choice of independent variables as long as kP and k’e are two of the Lorentz invariants chosen.

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

261

1

Now consider applying the method of Appendix B to express the local functional Dk70.,; uniquely as (4.28) DkrO,,, = M,,Z, -I- R, where Z, and R, are local. Then we have Dj” -&

?

(D&)

- gDiaakStfiZ,

- gD,‘h,.‘t$M;;‘R,

= M,,TTB + (LXt-) /3) + terms that vanish at @ = @fl’hy!

(4.29)

The only nonlocal term on the left-hand side is the last term and we may take the part of it that is symmetric in ((Yt) p) since the whole left-hand side is known to be so. We apply the decomposition of Appendix Bl to D,“&tkzi to get Dioia~Pt~i + (~ f-) 8, =

-f EaTM,, + gti”,~,,,t~i6,.” + (‘Y t) p).

(4.30)

Now, on the right-hand side of Eq. (4.29), the only nonlocal terms are of the form M,,T,,j + (CX t+ p). When we substitute Eq. (4.30) in the third term on the left-hand side, the term ,f50TMTn + (LXtt fl) generates a nonlocal term of the same form, whereas the term ( gt&@,t&8,B + (CXts fl)> does not. Therefore, this term must vanish at @ :=: @(phy) entirely. Thus,

[~~Rt%m@,, + (a t-) 811MT;I[@l

(4.31)

Ri[@l/Q,e(my~ =- 0.

By considering the above equation for successively larger lines, it is easy to show that the above equation implies that

number

R -i[@(NW] z 0 and since @ here refers to a field in the tree approximation the result of Eq. (3.12), that R, has the form

of

external

(.I)

(4.32) [CD

‘p] we obtain, using

(4.33) and thus, Eq. (4.23) becomes

Thus we have obtained an algebraic constraint expressed by the above equation, on O[A] in order that (perhaps with suitable ghost term modification discussed earlier) it may have gauge-independent renormalized matrix elements. The above is a necessan~ condition. We shall now solve it.

262

SATISH

D.

JOGLEKAR

First note that defining

simplifies the equation to

Di%

=fsV, + %,,cQkd.

(4.35)

Further, without loss of generality, we may assume that Vomdoes not contain L& with respect to x, (which means if the method of separation of Appendix BII is applied to Vaa[A] taking the momentum entering via x, and via fields in V as independent momenta, the unique separation would read Vg = Voa). Now we may assume without loss of generality that 6 does not cot ain f in the sense of Appendix BIII, when 0 is a local density. men 0 is not a density we shall say that it does not containfif there exists a local density o(x) not containingf such that 6 = J a(x) d4x.] W e can assume that 6 does not contain f because we have already considered the alternative of adding such a term to 0 and shown that it does not affect the necessity of Eq. (4.35). It is therefore possible to write Eq. (4.35) for a local density 0, as, Di’&$,i

= &V,& + 90,kQ& + a four-divergence.

(4.36)

We may assume that the four-divergence does not contain L?& with respect to 01. Now we shall further separate 0, according to the procedure in Appendix BII

where again, we may assume that 0, does not contain f. Then we have Diaoz,i = faV,& + Y&Qg

+ a four-divergence

(4.38)

where

c,(QiTc- Q&j = --gT,,

.

(4.39)

As explained in Appendix B, given that a, does not contain f and SOSl,with respect to x its gauge variation Diaor.i cannot contain 90,, with respect to both x and x, . Hence Qz = 0. Hence integrating

over x again,

(4.40)

OPERATOR

PRODUCTS

We now expand V$ in powers offwith

IN

GAUGE

THEORY,

I

263

respect to x, as done in Appendix BIII

vr3Ol= c e!...,Y.fy, n=o

*..f,, .

(4.42)

Now Eq. (4.41) reads (4.43)

go0 = &V, with V. = Vs”c,. Hence, using go2 = 0,

~ou3) v5 +hsob

= .

(4.44)

Now go(fs) V, = c,M,,VJC, contains a term c,,c,(M,~VB~(~)- M,,Vp’) which itself does not containfwith respect to X, as can be easily seen. This requires that

vsY(O) = &fAYVP’O’M-1 A 50.

(4.45)

But since ?$t’ is local, this requires that (4.46) with a local T(O) or v;(O) = i!&, .

(4.47)

Thus, in the earlier case [note: A4,, = M,, + O(f)]

Then Di* &

(” - ;f5T~‘fc)

= o(f”).

(4.49)

It is possible to continue this argument further, to show then that 8 - &faTkyo = O(f")

(4.50)

(0 - $f,T,'"!fm)j = 0 m

(4.5 I)

in which case

in the one-loop approximation because of Eq. (3.7), so that such terms in 0 are irrelevant at this level. In the latter case, we have Diao,; = f

(4.52)

264

SATISH

which can be trivailly

D.

JOGLEKAR

solved to yield 0 = $fAiAi

+ gauge-invariant

part

(4.53)

which is an operator of dimension two, and the above discussions tell us that it may have (or modified to have) gauge-independent matrix elements. We shall treat this case separately. Thus for operators of dimension greater than two we have the solution of Eq. (4.35) to be 0 = Ji&Tk

+ G.I. part + terms containing

(two or more)f,

(4.45)

each term is separately local. Further, as shown earlier, the terms containing fa are equivalent to local ghost terms for their renormalized matrix elements. There is a futher restriction on Tko which follows simply from the decomposition of Eq. (4.28). We note that

But R, does not contain Mea (in the sense of Appendix BI) by definition. Hence R, cannot contain a term of the type (a2),, . Now consider Qk@[A = 01. It is clear that .%&Qk”[A = O] w i 11 contain a term proportional to the square of the momentum entering via X, because F.T. (Zi$J - q”g,,” - q,, g, unless this term vanishes; i.e., Qk”[A = 0] =.%&[A = 0] = 0.

(4.56)

This requires that Qka[A = 0] = p(8) 3~ where [-(a) is a polynomial

(4.57)

in the derivative a. Now, since Q is related to T by Qk” = Di”T,*i Q/[A

- gt~~Ti )

(4.58)

= 0] = aiaT,,i iA+.

T,[A = 0] being zero as Ti must carry the quantum numbers of Aj and as the Lagrangian preserves the global gauge symmetry. This condition of Eq. (4.57) requires that in T,[A] if k refers to a gauge boson, the Lorentz indix contained in k must be carried by the gauge field (after contracting all dummy indices) in the term T* that is linear in A. Thus for example is allowed while, is not allowed, is not allowed.

OPERATOR

PRODUCTS

1N

GAUGE

THEORY,

I

165

Thus, we have restricted the operator O[A] initially under consideration to gauge invariant operators and operators of the form (with the above restrictions on T, b 0 = q,T,

-L- suitable ghost terms.

where the “suitable ghost terms” are to be chosen so that 0 has gauge-indepenent matrix elements in the one-loop approximation. It is not obvious that this can always be done: but as shown in the next section, this indeed is possible and we shall give one explicit expression for such a choice of the ghost term. Now, for the remainder of the problem is the following. (i) We must first tind all local functionals whose matrix elements to one loop vanish identically and examine the behavior of their matrix elements in the two-loop approximation; and show how this can be continued to all orders. (ii) Consider the set of operators so obtained which may have gauge-independent matrix elements and show that they indeed do have this property to all orders. Once the above-mentioned set of operators is obtained. the latter part will be obvious from the discussion of Section 5. First we shall find the set of local operators whose matrix elements vanish through one-loop approximation. We shall divide the set of local operators according to the number of ghosts they contain. The local operators containing four or higher ghosts evidently have this property. If the operator contains no ghosts, its matrix elements in the tree approximation must also vanish. This restricts them to the form given by (3.12). We already have considered them and shown that they are equivalent to local operator with ghosts once they are renormalized. Thus, we have to consider a local functional of the form TJ&[A] cg and investigate under what circumstances its physical matrix elements might vanish. Now ~:C$&[A] c,$~ = iQJ@] &I;,‘[@] and the problem of making this vanish is very similar to the discussion of the diagrams in Fig. 3 that we considered earlier. In a straightforward application of methods used there, we can conclude that in order that its matrix elements vanish QJA] must have the form or Indeed, these two choices are equivalent since

in either case. Further, it is easy to see that for any local term containing two ghosts Z;,Q,,[,4, ?,,I co (4.61) is independent of 7 at @ = @(phy) if and only if

266

SATISH

D.

JOGLEKAR

has the form of the two ghost operator just found earlier; viz.

This completes our discussion of one-loop approximation. thus far is that

What we have shown

(1) the local operators, whose matrix elements are nonvanishing and maybe gauge independent through one-loop approximation, must be (i) gauge invariant or (ii) must have the form (PO,Jk + ghost terms) where Tk has certain restrictions mentioned earlier or (iii) must have the form #2c,&&4] c, (modulo terms that have vanishing matrix elements through one-loop approximation). (2) The local operators whose matrix elements vanish through one-loop approximation are operators (&S/SC,) Q,,[A, rO] c, (i.e., of class III in the classification scheme of Section 2E) and operators containing four or more ghosts. There might be other operators with this property but they are equivalent to one of the operators mentioned above. Basically, we now have to extend the above results to all orders. We, thus, have to consider the local operators whose matrix elements vanish through one-loop approximation and examine them in the two-loop approximation. Among these are the class III operators, but we already know that their matrix elements vanish to all orders. Hence, we need not consider them any longer. This analysis is a straightforward generalization of the methods discussed so far. What one has to remember, however, is that one will in general, be dealing with nonlocal functionals of A (obtained by converting ghost terms using Eq. (3.25) and its generalizations). The conclusion is that the operators, whose matrix elements vanish up to n-loop approximation and who may have gauge-independent matrix elements in the (12+ I)-loop approximation, have the structure S$,T,[A,

c, 21 + terms containing

(2n + 2) ghosts

where Tk contains n ghosts and n antighosts, and is expressible as Tk = 6T/6(&?)), , We have thus found out a set of operators (S) (for which we have specified the form of the term containing the least number of ghosts) that may have gauge-independent and nonvanishing matrix elements. We shall now show, invoking the statements proven in Section 5, that this is possible for each such operator with a suitable choice of the higher ghost terms and that the totality of this set of operators is equivalent to a subset of class II operators. [That is, for every operator we have found that has q-independent matrix elements, there is a class II operator with the identical matrix elements to all orders.] Let us consider an operator in S with at least 2n ghosts. The term in it with 2n ghosts is identical to the corresponding tetm of a class II operator. Since such class II operators have q-indepdendent matrix elements as shown in Section 5 it is clear that

OPERATOR PRODUCTS IN GAUGE THEORY, I

267

there is a choice of higher ghost terms for this operator in S such that it has T-independent matrix elements. Suppose that there is another choice of the higher ghost terms such that the operator has q-independent matrix elements. Then clearly the difference between these choices is that they have n-independent matrix elements, and contains at least (2~ + 2) ghosts. Then either these vanish, in which case the two choices are equivalent, or the (2n + 2) ghost terms in the difference must have the form of the i2n + 2) ghost terms of a class II operator; from our earlier discussion in this section. This can be continued in n from IZ = 0 onwards. This process must end if we are considering a local operator of a definite dimension D (which we always can) because it cannot contain more than D ghosts. This proves therefore, our contention that any operator in S which has q-independent matrix elements can be written (is equivalent to) a sum of classII operators each containing at least 0. 2. 4.... ghosts in successionwhich itself is a class II operator. Indeed a closer examination in the next section will show that all the operators of the form (4.62) (i.e., T, containing ghosts) have vanishing matrix elements in our renormalization scheme. Thus the set of operators having q-independent and nonvanishing (at least one) matrix elementsis the one we have found by the examination only up to one-loop approximation itself. We also had to find the set of operators whose matrix elements vanish up to n-loop approximation for each n. This is so because in the consideration of gauge independence of operators in the (n + 1)-loop approximation, we had to know what sort of local operators one can add to them without altering their matrix elements up to the n-loop approximation. This analysis for n + co, leadsus also to a set of operators whose matrix elementsall vanish to ail orders. This set is listed in Section 6.

5. GAUGE

INDEPENDENCE

In this section we shall consider the set of local field operators constructed in the previous section and show that they indeed have gauge-independent physical matrix elements. Let us first consider the operators of class II given by the expressions (4.62) and (4.45). They are a subset of class II operators with a specific form for the function Fi in (2.44). As stated in Section 2E thse operators have their unrenormalized Green’s functions generated by <:O>j :- j,‘Fi - Diac,Tgil,~Fi,>,.

(5.1 l

Further, as stated there, these being special casesof class II operators need only classII and classIII renormalization counterterms: now the physical Green’s functions of the class III operators vanish identically according to Section 2E. Thus. renormalized Green’s functions of 0 corresponding to the unrenormalized ones in Eq. (5.1) will be generated by 0 -:- counterterms of class II needed to renormalize 0’ ,

(5.3)

268

SAT&H

D.

expressed as a functional of renormalized discussion on the counterterms for 0,

JOGLEKAR

quantities.

But according to the above

(0 + C.T.), = ji~ where Pi = Fi + C, Z$ “Fj!“)-functions” in the class II counterterms thus has the same algebraic structure as Eq. (5.1). Thus,

(5.3) for 0 and

(5.4) As noted in Section 2E, this structure implies that the truncated physical Green’s functions of this quantity vanishes at a nonzero momentum. Because of the above structure, the diagrams that contribute to the physical matrix elements of 0 are the ones in which the quantity in the angular brackets on the right-hand side has a single particle pole and hence the diagrams in the Green’s functions of 0 (at sero momentum) that survive on the mass shell have the now vamiliar structure of the two-point function of 0 to which an ordinary Green’s function is attached. Consider the twopoint function of the above quantity (i.e., (0)JR). This on-mass shell is fixed uniquely by the renormalization conditions irrespective of the gauge. Thus the gauge variation of the matrix elements of 0 comes entirely from the ordinary Green’s function which is attached to the two-point function. This ordinary Green’s function has all of its external lines on mass shell and has (n - 1) external lines contracted to physical wavefunctions. The gauge variation of such a quantity was computed in Section 3B and it was shown that the part that has one particle pole in all the external lines and had the structure 8i”G,s[J] X,[J]. The matrix elements of 0 are generated by

Thus the net gauge variation

of the matrix elements of 0 is generated by

J,F~,i$%,[Jl ~~[Jll,.,c,tw~

(5.6)

where

the whole (Incidently, note that if Fk,j is zero, i.e., if F contains at least twojelds, physical matrix element of the corresponding operator 0” vanishes idzntically as seen jkom Eq. (5.5) in this renormalization scheme.) We are only interested in F’s that are restricted by the conditions of Eqs. (4.57) and (4.58). (F was called T there.) Given this structure of F, the Lorentz index in aia is contracted to the one in Jk with the consequence that the whsle term vanishes because of e(p) . p = 0. Thus the matrix

elements of this subset of operators are q-independent. Note that there are other class II gauge-variant operators which do not possess the property of Eqs. (4.57) (4.58). In these cases, their matrix elements are in general q-dependent.

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

269

I

Next, consider the gauge-invariant operators (all, that are of class I and II). The renormalized matrix elements of a gauge-invariant operator O[A; g,,] are generated by the physical Green’s functions generated by

expressed as a functional of renormalized quantities where the class I counterterms are present only if 0 is a class I gauge-invariant operator. The result that the Zj’) are T,,-independent [6]

can be easily generalized to

:z 0

(5.9)

using Eq. (3.18). What we have to show now is that the qo-dependence of the class IL and class III counterterms is such that the epxression (5.7) is independent of 77at J = Phy), when expressed as a function of renormalized quantities, when the renormalization counterterms are chosen according to prescription in Section 2D. Using the closure property of {I @ II @ III} and of {II @ III} and of (III} operators, it is possible to rearrange the expression (5.7) as

where C:Oj,R == 0 + C.T. to renormalize O>, , and the expression (5.10) as the expression (5.7) is to be taken up to a specific loop approximation. Now in Eqs. (5.10) the last term vanishes (see Section 2E), while only those 0” in the next to the last term whose “F’ functions have a single particle pole in Ai have nonfanishing (O*l>,R, according to the discussion of the previous subsection. In particular, these “F-functions” do not contain ghosts. Hence (O[A; goI ;R = (O[A; g&R + c Z~“(g&,O,‘)R

where x’ is the last term means restricted sum.

-I- c’ 2-(g,,

qo)(O”)R

(5.11)

270

SATISH

D.

JOGLEKAR

Now let (0) R(n-l) denote the Green’s functions of an operator which is renormalized up to (n - l)-loop approximation. We shall consider the gauge variation in the n-loop approximation; viz.

and show that it has the structure t’ ob.2) (5.12) so that defming (a/a~)i~fAln

= bi

(5.13)

= 06-4

(5.14)

one would obtain the result @la~ww(“%z

where (0)R’“’ is the expression (5.11) with the renormalization constants 2, 2 determined up to n-loop approximation. To prove Eq. (5.12) we consider the WT identity satisfied by
WR(+J, 5,[,K’,L’,N]IK~=L~=o.

O(N2) = [Ji&i - tct-&

(2.49) an d evaluating the result at K’ = L’ =

Operating on it by (G/SN)&[ --i(8/8J)]@/a1$) N = 5 = [ = 0 we get,

+ C.T.) + j,&&&,lf

0 = j [dA dc dZ][I,Diac&‘(O

+ (0 + C.T.>(-fa2

(5.15)

+ ZMc)] exp i(=%r + jiAi}.

C.T. stands for counter-terms introduced &a has effectively the structure, &“[A]

+ Cs’&BQi”c,

up to (n - l)-loop

approximation.

Here, (5.16)

= c’ &$@$jm’[A]

m.

where we have omitted from the above sum other class II operators since they will not contribute at J = J(phy) anyway. Now, as in Eq. (3.28) +

(0 + C.T.)$+”

= 2 ““‘g + J,((O

‘) &


+ C.T.) A,)“‘:”

2+“)

az-up) arl,

1 + O(p2) 0”

(5.17)

OPERATOR

PRODUCTS

-t ix’1L

IN

GAUGE

THEORY,

aztid (olI>j

1

271

(5.18)

where in the last step we have used Eq. (5.15). The last term in Eq. (5.17) which is due to the change in the external line wavefunction renormalizations is, as usual, canceled by a part of the term ( jiDiwc&(O + C.T.))Conn) which has the structure of Fig. 2a. Now, consider the first term on the right-hand side of Eq. (5.18). At J = J(P”“) the only terms in & that contribute to the renormalized Green’s functions are the ones which do not contain fields, i.e., at J = J(P”Y) (5.19)

since Qia[.4 = 0] = a,a&l,i

. This quantity is precisely

C' z~l)(a/aT)(oj;;]~;R

(5.20)

according to the discussion on the T-variation of the matrix elements of the class II operators given earlier in this section. Further, noting that in Eq. (‘5.17) +

((0 + C.T.jR(n-1)j(7L+1j = O($)

(5.21)

according to the induction hypothesis. This provides us the identification of (aZ”/i.rl,,),r with a certain part of j,(D,%&(O + C.T.)) corm Using this identification, one sees the result Eq. (5.12) for {OR+l >12. Here xi d,i,{(Ot:,)jO is equal to the part of {( jiD&&(O + C.T.))rree}, that has the structure of Fig. 2b whose IPI diagram shown in Fig. 2b (between Ji and @[J]) contains all the n loops. Hence the result. Finally, we note that the renormalized matrix elements (renormalized according to the prescription of Section 2D of class II operators are generated by (accroding to (5.5)) (5.22) J,F;,,,~~[JllJ,J(Ph).) . Here, Fk,j (when it does not vanish) is simply a polynomial

in P. As a result the matrix

272

SATISH

D.

JOGLEKAR

elements of a class 11 operator have the general structure [see the restriction on Fk,j explained under Eq. (4.991 (5.23)

T T pa(Pi) sCPi ***Pn)

where P,(p,) is a polynomial in pi ; a denotes a species of external lines (quarks, gluons) and P, can be different for different species. [The above statement is to be interpreted as a limiting statement.] Thus, these operators are not independent physical entities in this sense. We note that in the discussion of this section we have made two tacit assumptions which we shall now prove. They both concern our renormalization conditions as applied to the renormalization of gauge-invariant operators. (I) In discussing the gauge invariance of operators inserted at zero momentum [At nonzero momentu, gauge invariance was far more obvious since the renormalization counterterms of class II then vanish at J = .Phy) and class III always do.] We have made an assumption that we could subtract proper vertices of a gaugeinvariant operator at zero momentum (q = 0), without encountering infrared divergences. This is indeed obvious that by staying off-mass shell for the external lines the proper vertices at q = 0 are not infrared divergent except for a set of exceptional momenta (for D > 4). Now, if one were to consider renormalization of an operator insertion for an arbitrary q, the above procedure of subtracting at q = 0 for the zero momentum insertion would not make sense if, when the proper vertex is expanded in powers of q up to (D - nB - &)th power, and the coefficient of any of these powers (not just zeroth) contained infrared divergences. [Here, as is conventional ng and nF denotes numbers of boson (ghosts included) and fermion external lines.] We shall next show that this is indeed not the case for a gauge-invariant operator renormalization and that in trying to determine the renormalization constants in front of the class II and class III operators in the counterterms by evaluating the relevant terms in the two-point function of a gauge-invariant operator at q = 0 one does not encounter infrared divergence.33 Only these counterterms enter the discussion of gauge independence. Consider a proper vertex, at the renormalization point stated in Section 2D, for an operator of dimension D with nB boson and nF fermion external lines. Let it be denoted by r(q; p1 . ..~~)*[s=n~++.]Weexpand D--ng-(3:2)rrp

m

Pl . ..pJ

z

1

qul

--e +

I=0

r) uv

+ r(r).

(5.24)

a=0

It is evident by power counting that for r > D - 4, ((a/aq,J **. (a/aq+) F),=, may be infrafed divergent at the renormalization point. Our object is to show that these quantities, i.e., ((alag,> ... (a/aq,j F),=, r > D - 4, do not need a class II or III counterterm at all thus avoiding the question infrated divergences in these completely. aa Except for a set of exceptional momenta for the external lines.

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

I

273

Since we know that the renormalization counterterms belong to class I @ class 11 @: class III, we have to show that there are no operators of dimension D that are total derivatives of order (D - 3) or greater and belong to one of the above classes, in which case fields in the operator carry only three or fewer dimensions. It is obvious that one cannot construct a class II or III operator with these properties because in order for a structure (&S/&4,) F, + ...) or (&/SC,) X, to be global invariant, at feast four dimensions must be carried by fields. Thus the class II counterterms which are the only ones that mattered in the discussion of gauge invariance can indeed be determined by subtraction at q = 0. Indeed it is only the renormalization counterterms that are gauge invariant and which can be written as i-,l .” aUDmn. (operator of dimension three) cause any infrared problem at q =:- 0. The renormalization constants only, of such counterterms, have to be determined for q # 0. However, they are gauge independent no matter at what q ( f 0) they are determined and do not enter the discussion of gauge independence of matrix elements. The second point that we have inadvertently assumed is somewhat subtle. On one hand we have said that to preserve the WT identity for gauge-invariant operators in the form of Eq. (2.47), it is essential to restrict the counterterms to the subset of operators class I 0 11 @ 111; and that this is possible. On the other hand, we have prescribed that the finite part of a global invariant operator at q = 0 be set equal to zero at the renormalization point for loop ( 3 I) contributions. A priori, it is not obvious that by using only the counterterms of the form of the two-point functions of class 1 !s 11 ix> 111 operators one can accomplish the above the subtraction on the two-point function. Basically, the problem is whether there are additional polynomial solutions for the WT identity for such a two-point function, not contained in the twopoint functions of the class I @ II (3 111 operators, which are simultaneous solutions of a set of WT identities for 2, 3,..., D-point functions and hence likely to be more restricted. We shall indeed show that there are no more particular solutions to the WT idendity for a two-point function for D > 4. We shall give the proof for the twopoint function with external gauge bosons. The WT identity has the form (See Eq. (3.9) of Ref. [5]r (5.25) We expand

=.= x s:j( p)pq p?) QdP)= )3 mp) X,&J”), rs’v’(p) 2 I

(5.26)

where Pi’ and(i) S are independent polynomials in p of degree D - 3 and D --~ 2, respectively. Then substituting these expansions, in the above WT identity, we learn that only those S$,)( p) enter expansion of r$?( p) for which

holds for some P:‘(p).

The particular

solution of the above equation

is unique.

274

SATISH D. JOGLEKAK

phat is, its two solutions differ by a solution p&,,(p) = 0.1 But as is easily seen

of the homogeneous

equation

(5.28) is a particular solution since Pci)(p) is a homogeneous polynomial of degree (D - 3). But this SE indeed is a two-point function of a class II operator. Thus all particular polynomial solutions of Eq. (5.27) are two-point functions of some class II operator.34 All polynomial solutions of the homogeneous equations p&(p) = 0 are twopoint functions of gauge-invariant operators as can be seen by explicit construction. [This latter statement is, strictly, not necessary for the discussion of gauge invariance however.] We have thus shown that all the polynomial forms that may enter a two-bluon proper vertex of a class I operator (in higher orders) are two-gluon proper vertices of class I + II operators. However, on account of the massless nature of the internal lines, it is not clear that such a two-gluon proper vertex at zero momentum will not contain poles in external momentum p, even though it is a proper diagram. A similar question will arise in the discussion of the renormalization of a class II operator. For example, consider the Green’s functions (with external gluons) of a class II operator. They are generated by jj(Fi[A,

C,

(5.29)

E] - Diac,F[A, C, C])j e

Consider an example

There is no reason why the proper vertex with one gluon in (Fi)j,

viz.

{S/&
should not contain in its Fourier transforma term of the structure pAp,,pyp0/p2, allowed by dimensions and Lorentz transformation property. However, we shall show that such pole structures are in fact absent in the total two-gluon vertex for the class II operators [e.g., even if (Fi) contained a term of the form mentioned above, the total expression (5.29) does not]. We shall also show that the two-gluon vertex of a gauge invariant operator (it must have the number of Lorentz indices - dimension 3 -1) contains such pole terms, they do not affect our conclusions regarding gauge independence. 34 Note that from the two-point function alone, one may not be able to determine uniquely the renormalization constant for a class II operator. This is so because it is possible that the two-point function of a class II operator coincides with that of a gauge invariant operator. Thus one will be able to determine a linear combination such as (Zr + Zn) from a two-point function. However, one can determine from the change in the two-point function with 7, the change in 2” itself because aZr/8v = O(@) according to Eq. (5.9). This is sufficient for our purpose.

OPERATOR

PRODUCTS

IN

GAUGE

THEORY.

I

215

Here we shall only deal with Fi [in (5.29)] containing no fermions. If Fi contains factors such as g,, , where p, v are free indices, they may be omitted from the discussion because they are overall factors for the class 11 operator. Hence we need consider F, consisting of 8, , A,“, cq , ZVonly. Clearly the (dimension of F,-number of Lorentz indices of Fi) = t is a nonnegative even number as c, F must occur in pairs. The onegluon vertex in (Fi> under consideration has (2 - t) more Lorentz indices than its dimensions. The possibility of poles inp arises only if 2 ~- t > 0, i.e., only if t = 0. In this case, in order that there is pole in the above two-point function, it is necessary that all the Lorentz indices in it are carried by p. As we shall now show, such terms in (Fi> do not contribute to the two-gluon function of the corresponding class II operator which is of form (5.29) (s

f-)

07).

Here Yi is the bracket in (5.29) and has the property %i4Yi =- 0. rif) are inverse propagators. We may use aiOYi = 0 to write this in terms of the inverse transverse propagator rifii , i.e., as

Evidently any terms in 8 Y,/& ljZO - &@(i.e., pI1 in momentum space) will not contribute to the above expression because ai’.Fo,si = 0. But as explained earlier, the aforementioned pole terms must contain ais. Hence such pole terms are absent in the above expression. Next consider the gauge invariant operators of gluons only, As before (the dimension, the number of Lorentz indices) = t’ is a nonnegative even integer. Possibility of pole in the two-gluon function arises only if t’ < 4, i.e., t’ = 0 or 2. If t’ = 0, the gauge-invariant operator is antisymmetric in two pairs of Lorentz indices as it must contain Fu,,Fno; as it takes at least one factor of g,, to construct a tensor out of p and g,, which is antisymmetric in two of the Lorentz indices. Hence in the t’ = 0 case (as well as t’ = 2 case) only single poles in p2 are possible. Next consider the WT identity P,c%P)

= Qo(P)~P28”cl

- P,PA.

In the t’ = 2 case, in order to have a pole in Q,,(p), all the Lorentz indices of p0 must be carried by p. Hence Q0 - pV and

One can convince oneself that in the t’ = 0 case also, Q,(p) Ipole- p, . Hence the above equation holds. Hence the renormalization constants for the class 11 operators whose two-gluon functions are not transverse can be determined at zero momentum in section (q = 0). These are the only operators whose matrix elements might be

276

SATISH D. JOGLEKAR

gauge dependent, and hence the only ones which enter the discussion of gauge independence. Finally, one notes that if the Fi function or the gauge-invariant operator contained fermions (which are assumed to be massive) a similar treatment can be given.

6. DISCUSSION AND CONCLUSIONS

A. We have shown the following. Consider a set of local functionals of gauge fields, mater fields (collectively denoted by A) and ghost fields (c, E) {F[A, c, E; g,, , Q]} such that (i) F is invariant under global gauge transformations, (ii) F has equal number of ghosts and antighosts in each term, (iii) the total power of all fields - power of g, = 2 for each term. (I) Then the subset of operators for which its Green’s functions renormalized and evaluated off-mass shell (as discussed in I) satisfy

is the set of linear combinations

of:

(i) class II operators all of whose tree matrix elements vanish. They are characterized by an “F-function” which has the property that J,@E;,ISA)IA=~=~=~

(ii) (iii)

@j[JIIJ=J(pw) = 0

Class III operators [their from is (&S/SC,) X&4, c, i?]] “Class IV” operators defined by

&X&4,c,(:I- E,poxa-fs E;]. We shall call this the set So . (II) The set of operators whose truncated functions satisfy

renormalized

physical Green’s

and for at least one matrix element lim (F[A, c, P; go~o])u~# 0 2-O+

is the set of linear combinations (i)

of the linearly independent

all class I operators,

operators

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

I

‘77

(ii) a subset of class II operators such that (a) no linear combination of then belongs to the set so discussed above and (b) in its F,-function, when k ~1 (01,CL,x,J refers to a gauge field, the Lorentz index p is carried by the gauge field kafter contracting all the dummy indices]. One can, of course, add to them an arbitrary linear combination S,, discussed above.

of operators in

B. We shall make some comments on our renormalization scheme for operators and say the minimal subtraction scheme, is which one precisely subtracts the pole part in the renormalization. This comparison is of importance since when the operator product expansion is proved for two gauge-invariant operators, it is likely to be in terms of the minimally subtracted operators rather than in terms of operators renormalized at an arbitrary off-mass-shell point as we did. ( 1) Both renormalization schemes restrict the renormalization counterterms for a gauge-invariant operator to the class I, II, III operators. They both preserve the WT identities. (2) Because of the (1), for every operator belonging to one of the three classes 0 renormalized according to our prescription (which is not unique), can be expressed in terms of a linear combination of operators of class I, II, III renoremalized according to a minimal subtraction scheme, i.e., (0):

= C ai(p2)(Oi>,R.

Here, 1 and 2 refer to the two renormalization schemes; and the sum on the r.h.s. runs over class I, II, III operators. (3) If we use minimal subtraction schemes for renormalization of gaugeinvariant operators at q = 0, their matrix elements do not have a simple gaugeindependence property such as Eq. (1.2), in general. Furthermore, all of the class I and II operators will have a nonvanishing matrix elements in general. It is easy to show that not all the class II operators are linearly independent near mass shell (in the limiting sense). It is also possible to show that though gauge-independent operators do not have a simple gauge-independence property at q = 0, linear combinations of class I, II, III can be formed that have this property. In adopting a different renormalization scheme, what we have done effectively is to choose precisely some of such linear combinations (which are, of course nonunique) as the renormalized gaugeinvariant operators. We have in addition isolated effectively those independent linear combinations of class II operators (in the minimal subtraction scheme) that have vanishing matrix elements. These linear combinations, in our renormalization scheme. become the class II operators in set S, . Tn short, in the renormalization schemes we have used here, we can talk directly in terms of operators that have a simple property (related to gauge dependence or otherwise) near mass shell. This, of course, is only an advantage in language.

278

SATISH

D. JOGLEKAR

(4) The ambiguities in our renormalization scheme correspond only to addition of arbitrary counterterms whose renormalized matrix elements vanish identically.

APPENDIX

A

Here we shall prove Eq. (2.12), given Eq. (2.1 la)

where we have used Eqs. (2.11a), (2.1 lb), and (2.5) in successive steps. Though, in the above proof we considered only linear gauges, ((8/6Ai) fp’ = 0) the statement is true for quadratic gauges also. The converse of the above statement (i.e., Eq. (2.12) => Eq. (2.11)) is also true. The proofs are fairly straightforward.

APPENDIX

B

Our purpose here is to give procedures which explain how to separate a local functional into certain pieces. (We shall soon be specific about them.) The procedures that are (1) well defined and (2) useful in the text. We shall be dealing with local functionals of Ai which have at least one space-time index. Let this functional be Fa[A]. F may carry additional indices. I.

We shall give a procedure for expression F, as

F, = M,,Z, i- R, in a unique manner. Let F, have a well-defined dimension d and let FE = i F,i,...iTAil c=?z

..- Air .

0-W

032)

Consider Fail..+, . We shall choose the momenta p, pl ,..., pnvl which are conjugate

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

I

279

to X,) xi ...xj to be the independent momenta (i.e., in particular, choose their dot productslto i: various independent Lorentz invariants) and express the Fourier transform as (suppressing indices) a’ + b’p2 where a’ and b’ are polynomials in the momenta and a does not contain a term proportional to p2. Now, let b be the projection of b’ obtained by a complete Bose symmetrization. Let b” = b’ - b. We then express this Fourier transform as

(a’ + b”p2) + bp2

where the two groups are separately Bose symmetric and b” is a sum of terms each of which is Bose antisymmetric at least with respect to one pair. The fact that a’ + b”p2 is Bose symmetric can be shown to imply that the above separation is unique, i.e., independent of which momenta are chosen as the independent ones provided p is one of them. To see this, suppose we had chosen (p, pz ,..., p,J as the independent momenta, we can separate a’ + b”p2 into two parts, one which is symmetric under p1 f-) pn and the other antisymmetric as a’ + b”p2 = (a + b,p2) + (a, + b,p”), respectively in an obvious notation. What we have to show is that when the above is expressed in terms of the new variables by substituting for pl, then thep2 terms in the result is antisymmetric in at least one pair. But under this substitution. we must have

(aI + bd) + (a2 + b2P3 - (aI + hp2) - (a2 + b2p2). Thus, the new p2 terms are (b, - b,). Given that (b, + b,) is antisymmetric in at least one pair, it is easy to show that (b, - b2) is also a sum of terms each of which is antisymmetric in at least one pair. Hence, etc. (Note that one needs to reexpress these coefficients b, , b2, in terms of (pl ... pn) to see the symmetry.) Our claim (not necessary for the uniqueness of our procedure but its usefulness) is, further, given a’ there is a unique b” with the above properties such that a(’ + b”p2) is Bose symmetric, in other words given the Fourier transform at p2 = 0, one can recover the complete first part (i.e., a’ f b”p2). For if there were two distinct b”, say b;lb$ with this property, then (b; - bIJ p2 and hence (b; - b@ must be totally Bose symmetric. Hence if we denote by S the Bose total-symmetrizer, b; - b; = S(b; - b;) 1 I-J since

Sb; = 0 = Sb;

showing that b” is unique, given it exists. Now in coordinate space, this decomposition corresponds to FL, ,..., i, = &I . .. .si, +
280

SATISH D. JOGLEKAR

the separation of Eq. (Bl) where the Fourier transforms of coefficient functions in R, are unique once they are given at pa = 0, p being the momentum conjugate to x, . If the function has more space-times indices, (say two) FaB one must also choose the momentum conjugate to x5 as one of the independent momenta. In this case the two possible separations Fas = MwZs f %B

are in general distinct (i.e., Rns # R& in general). II.

We shall explain how to express the functional Fa in a unique manner as Ex = .%,cQka+ &.

(B3)

We express Fa in (B2) and express the Fourier transforms now as a function of independent momenta (pl a.-p,). Let (pl .*. ,u,J be the set of Lorentz indices associated with Ai1 a** At”. We separate this coefficient into terms according to whether they are proportional to at least one pi2 or not. We further separate out from the terms that contain at least one factor of pi2, terms that are proportional to pi2 and pi&, for at least one i. Let these remaining terms be Cpi2T$’ where we have suppressed additional indices, We then write the Fourier transform of the coefficient as F (Pi2&o

- pi,,pio)

T,‘i’ + the remainder,

i.e., in coordinate space, as the Bose symetry of this separation is obvious,

where We then consider Fa - L&Q$ ,..., in-lAil a-. Aimel which now has at least (n + 1) gauge fields, and apply this process. As before, the process ends and yields the decomposition of Eq. (B3). III.

Finally, we shall give the procedure for expansion of a function as F, = F>) + c F$?. .,q,,&, ... &, VI=1

uniquely. This is indeed trivial, We can consider F, here to be monomial. Then one expresses the Fourier transform of the coefficient as in II and picks out the terms containing one or more factors of pill, . The coefficients of these terms yield Fourier transforms of the coefficient functions in F. Finally, we shall note the following result. Let O,[A] be a local density, not con-

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

I

281

taining 90,1, or f with respect to x. Then its gauge variation Di”O,,i cannot contain 9& or f with respect to X. The statement is obvious if one notes that (i) all the momenta caried by fields in 0, are independent momenta. Let them be p1 ... I),, with Lorentz indices of gauge fields pI .” pin cl,.... 111::. II) refer to gauge bosons). (ii) Then there are no terms proportional to pi1 ... ,D,!~or pIU, ... I’,,,,,,!, in the Fourier transforms of an n-point function. It is further easy to see that Di”Ozi does not contain YO,Ic with respect to I also. The above discussion in (II) was given for FJA] which is a function of gauge fields only. It is a straightforward matter to extend the discussion when F, contains fermions. (Then one looks for the pi terms instead of pi2 terms.) The above conclusions remain unchanged. Furthermore. these methods can be generalized to nonlocal functionals F, .

APPENDIX

C

Here we shall prove the result stated above in (4.27). Consider the Green’s function generated by the quantity (4.26) with the minimum number (n) of external lines. This has the structure in which these n external lines attached to the curly bracket in (4.26) and having no external lines are attached to M$’ and A4i; . We are given that this Green’s function has the form (the sign depends on the symmetry of group indices)

h,,(k,AZ’,...)k” :c (k H k’).

(c-1)

Let this be generated by (in coordinate space, with T” related to the Fourier transform of h,,‘) (i’l):, T,j’:‘[@] k (N tt p, when P)[@] (4.26) viz.

(CZ)

is expanded to OIJYi]. Let us denote by P3,J@,l the curly bracket in

Now construct (c-3)

Consider the quantity

282

SATISH

D.

JOGLEKAR

Now this together has the nonvanishing Green’s function of O[P+l] at J = J@‘y). Now the (n + 1) lines Green’s function of the first term on the right-hand side is given to be of the form b(k, k’,...) k2 f (k t) k’).

The second term (the curly bracket) has evidently this form. Thus from Paa we have constructed P$ such that the left-hand side of Eq. (C4) has zero physical Green’s functions up to O(J”), and Pare - P.$’ has the structure,

where, of course, it is understood that T(O)[@] does not contain a pole in the (momentum)2 entering via X, . This process can be continued indefinitely until we arrive at Eq. (4.27) viz. Pm0 = MtaTsE + (a H /3) + terms that vanish at @ = @phy).

D

APPENDIX

Here we shall give the proof of the statement that in order that the discontinuity of the graphs [discussed in the paragfraph above Eq. (4.25)] vanish, it is essential that the integrand and hence the Feynman diagram (see Fig. 4) on the left of the cut II at k2 = k12 = 0 vanish. Let this Feynman amplitude (truncated) be g(k, k’,...). It suffices to consider the case when there are two external gauge bosons attached to the ghost propagator on the right of the cut. Let this amplitude be h,,(k, k’; p1 , p2) with p1 + p2 = k + k’. Explicitly, h has the form (in Feynamn gauge) h = A k(k - PA + B kv(k (k - ~1)~ (k -

+c

k/\Kpl

+

~21, a,

+

~21, ~2)~ (~2

-

(PI +

and thus the integral under consideration

s

PAA guv - (~1 +

2~2L

gv/\l

P2Y

(Dl)

is

d.k d-k’ 6(k2) 6(k’2) 8(k,) e(k,‘) g(k, k’;...)

h,,(k, k’; p1 , p2).

034

We shall consider it for fixed 4-momenta diagrams so that

of the external lines on the left of the

k+k’=p,+p,=K=fixed.

(D3)

Further, g is known to be symmetric under k tt k’. We have to demonstrate that the integral (D2) vanishes implies that g = 0 at k2 = k’2 = 0.

OPERATOR PRODUCTS IN GAUGE THEORY, 1

283

To make the argument, we note that the ranges of integration variables in (D 1) are finite. Quantize the system in a box of length L. Then the integration in D will degenerate into discrete sum. Let N be the number of distinct choices {k, li’: k -1. h-’ : K, k 3 k’} [where the ordering k 3 k’ 3 k, >, k, for some Y and k, = k,’ for all

P < VI. Let {g, ; -C )I < Nj be the set of values of g for all the N choices. Then g = (91 ‘...> gN) defines a vector in an N-dimensional

vector space. So can we construct

h,,(p, , pZ) for a given pl, p2. Now, just as there were N choices in the set {k. k’: li + /i’ == K, k 2 k’) there will be N choices for {pl, pZ : p1 + pZ = K, p1 ;-Gpz;. [Note that the symmetry of g in k, k’ is eliminated from the problem by requiring k 3 k’, while p1 2: p2 takes care of symmetry of h in (p,~) H (P,v).] Thus we have constructed NN-vectors h = h,&&~ such that g-h=O.

04)

g=o

(D5)

Thus Eq. (D4) will imply

if we can show that the NN-vectors h are linearly independent. But from the continuum form for h it is obvious that there cannot be a linear dependence among h’s. First of all, in order to have an “irreducible” linear dependence

we must have all the h(p, , pZ) with nonvanishing coefficients in (D6) to have the poles in the same places for each component of h. [By irreducible we man that the left-hand side cannot be split into two parts nontrivially, such that each is zero.] Such a set is [note: k2 = 0, p12 = 0] (#:

k . ($’

- pf’) = 0 vi r,j>.

Jn order that Eq. (D6) be true for all components (i.e., for all k in the set) we must look for the subset of the above Q {p:‘: k(p;’

- p?‘) = 0 Vi #.j)

which is readily shown to be empty.

APPENDIX

E:

RENORMALIZATION

OF THE CLASS IV OPERATORS

We shall first derive the WT identity for the unrenormalized class IV operators. With the help of it we shall show that the set of operators of classes II, III, IV together close under renormalization. Further we shall show directly that their matrix elements vanish.

284

SATISH

D.

JOGLEKAR

Consider the WT identity

Operating on the above equation by

x,, r-i;, -i&i&] i& one obtains

03)

Now we can express (f&&) identity

= (fu>(&X,)

+ (j&XB)Conn

etc. and use the WT

to eliminate the disconnected terms. It is further convenient to express everything in terms of 2’ and r’ where W’ = eiz’ =

s

[dA dc dP] exp i{S[A, c, E, K,

I]

+

N,C,X,,

+

j,Ai + c’,& + &J,}. (E5)

Now

OPERATOR

PRODUCTS

IN

GAUGE

THEORY,

285

I

Similarly, &ToS,,,c,c,~~~,)Conn

and

Thus we have sr

sr s2r - j---652, SN Sl,

s2r

(0) = +iKm

We then use the WT identity for Z to simplify the last three terms on the r.h.s. and obtain

("

sr, s2r = i &pi ~--___ sNsKi szr

- if.,& SN

-

8Ki

sr s2r SQmSN Sl, sr s2r __ ___ 8K(

&pi

SN

sr +

- 81,

sr ___ ssz,

Now we consider the divergence in (0) in the one-loop approximation. {(o)}y

= i -$+ [

&

u-W

- gov,

s2r

8N - fo sQo sN - (E5) It is given by

4u-2~ (W

where

286

SATISH

D.

JOGLEKAR

is alocal polynomial of its arguments. Note that in defining the class IV operators we had not restricted it to contain Z and Ki in the combination @& + Ki and as such ~rivld’” will not generally contain K( and ~5’)~. The first square bracket is in fact the “correct” generalization of a class II operator when its F-function does not satisfy SF/ISK~ = i3ia(8F/iZ,). [This needs a proof which is left as an exercise.] The second term is a class III operator. Indeed the unrenormalized Green’s functions of any linear combination of class II and III operators have the form

(E7)

It is then clear from Eq. (E6) and the renormalized analog of Eq. (E7) that suitable class II and III counterterms can be chosen in each order so that the renormalized class IV operator has finite proper vertices. It should be remarked that the fact that a gauge invariant operator does not need class IV counterterms is only a peculiarity of the linear gauges. Though we have not persued fully the analysis of renormalization in quadratic gauges, it seems obvious, following Linn-Justin’s treatment [14] of quadratic gauges that (analogs of) class IV counterterms will be needed for the renormalization of a gauge invariant operator. Further, using expression [E3) and recalling our renormalization conditions, it is easy to show that the class IV operators have vanishing matrix elements in a manner analogous to the discussion of class II operators.

ACKNOWLEDGMENTS It is a pleasure to thank the Institute for Advanced Study, where and the theory groups of Fermilab and SLAC, for their hospitality.

most

of the work

was done,

REFERENCES 1. K. WILSON, Phys. Rev. 179 (1969), W. ZIMMERMAN, 1970 Braudeis Lectures, Vol. 1, Cambridge M.I.T. Press, 1970. 2. D. GROSS AND F. WJXZEK, Phys. Rev. D 8 (1925), 3633; D 9 (1974), 980; H. GEORGI AND H. POLITZER, Phys. Rev. D 9 (1973), 416. 3. M. K. GNLLARD AND B. W. LEE, Phy.s. Rev. Lett. 33 (1974), 108; ALTERELLI er al., University of Rome preprint. 4. H. KLUBERG-STERN AND J. B. ZUBER, Phys. Rev. D 12 (1975), 467. 5. S. JOOLEKAR AND B. W. LEE, Ann. Physics 97 (1976), 160. 6. H. KLUBERG-STERN AND J. B. ZUBER, Phys. Rev. D 12 (1975), 3159. 7. J. DD~ON AND J. C. TAYLOR, Oxford Preprint (74-74); W. DEANS AND J. DIXON, Oxford Preprint (75-20). 8. B. W. LEE AND J. ZINN-JUSTIN, Phys. Rev. D 5 (1972), 3121.

OPERATOR

9. 10. 11. 12. 13.

PRODUCTS

IN

GAUGE

THEORY,

I

287

S. JOGLEKAR, Operator products in gauge theories II, to appear. B. W. LEE AND J. ZINN-JUSTIN, Phys. Rev. D 7 (1973), 1049. B. W. LEE, Phys. Rev. D 9 (1974), 938. A list of references to the past works in the gauge theory may be found in Ref. [ll]. C. BECCHI, A. ROUET, AM) R. STORA, “Renormalization of Abelian Higgs Model,” CPT preprint, Marseille, 1974. 14. J. ZINN-JUSTIN, Lectures at “International Summer Institute for Theoretical Physics,” Bonn, 1974. 15. B. S. DEWITT, Phys. Rev. 162 (1967), 1195.