The cancellation of nonlocal divergences in light-cone theories

Nuclear Physics B328 (1989) 527-544 North-Holland, Amsterdam

THE CANCELLATION OF NONLOCAL DIVERGENCES IN LIGHT-CONE THEORIES O. PIGUET*

D@artement de Ph.vsique Th~orique, Uniuersit~ de Gen~ue, Bd. d'Yt,ov 32, CH-1211 Gen~ve, Switzerland

G. POLLAK** and M. SCHWEDA**

lnstitut fgt'r Theoretische Physik, Technische UniL,ersitiit Wien, Karlsplatz 13, A -1040 Wien, Austria Received 31 March i989 Revised 19 June 1989)

The nonlocal divergences arising in gauge theories quantized in the light-cone gauge are regularized with the help of a "cut-off" playing the role of a gauge parameter. It follows from the nonphysical character of this parameter that the nonlocal divergences drop out in the physical quantities. The result is valid to all orders of perturbation theory and does not require the existence of any invariant regularization scheme. Our considerations are based on an extended BRS-symmetry.

1. Introduction It is known [1-3] that 4-dimensional gauge theories quantized in the light-cone (LC) gauge develop divergences which are nonlocal (nonpolynomial in the external momenta): the usual power-counting arguments [4] are not allowed. These divergences arise in the perturbative calculation of one-particle-irreducible (1PI)-vertex functions but do not affect the Green functions. The arguments found in the literature [3] leading to the latter conclusion rely on the existence of a gauge-invariant regularization scheme, namely dimensional regularization. A more general proof is much desirable, in view of the fact that most theories of physical interest (with chiral fermions, etc.) to the best of our knowledge do not admit any regularization compatible with all their symmetries. * Partially supported by the Swiss National Science Foundation. ** Partially supported by "Fonds zur F6rderung der Wissenschaftlichen Forschung" under contract no. P6827. 0550-3213/89/$03.50'~: Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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O. Piguet et al. / Nonlocal dit, ergences

We shall show in this paper that it is possible to regularize the nonlocal singularities with the help of a gauge condition which interpolates between the LC gauge

n~A~=

0,

n 2 = 0,

(1.1)

and the covariant gauge

O"A, =

0.

(1.2)

The interpolation parameter ~ is shown to be a gauge parameter: the physical quantities do not depend on it. We prove this by establishing the invariance of the theory under "extended" BRS-transformations [5, 6] under which the gauge parameters, including ~, transform into Grassmann parameters. Only local counterterms are needed for any finite value of ~, because the usual power-counting arguments now apply. Thus the gauge parameter ~ plays the role of a regulator for the nonlocal divergent counterterms. The limit ~ ~ oo corresponding to the pure LC gauge is of course singular. However, the nonphysical character of ~ means that this LC singularity cancels in the physical quantities. Our formalism* is similar to the one in ref. [7], where a parametrized gauge is considered in the framework of ordinary axial gauges (i.e. with n z =~ 0). Our proof is quite general. It has the advantage of avoiding any explicit diagram computation. Let us note that such computations, already complicated in the formalism of ref. [7], would become horrendous in the present case, resulting in a very difficult characterization of the ~--* oo singularities and of the final cancellation in physical quantities. In order to exhibit the complexity of the ~-structure, however, we give the results of some simple one-loop integrals, together with estimations for ~ -o o0 in appendix A. The paper is organized as follows. In sect. 2 we present a pure Yang Mills (YM) theory (the introduction of matter is straightforward) in the announced gauge which interpolates between the covariant and the LC gauge. As a special example we compute an integral e.g. appearing in the one-loop gluon self-energy- in two ways: first with the propagators taken directly in the LC gauge (~ = oo) and with the Leibbrandt Mandelstam (LM) prescription [1, 8] and secondly with the propagators in the interpolating gauge (~ finite). In the first instance we observe nonlocal divergences, whereas the second computation shows how the ~-gauge improves the situation: the corresponding Feynman integral becomes finite, but it diverges like log~ in the limit ~-o oo. However, the simplest matrix element of physical interest, the time-ordered vacuum expectation value of the gauge-invariant object F2(x)F2(y), to one-loop order, does not depend on ~: the limit ~ ~ oo exists. This is a hint that also in more complicated physical matrix elements the cancellation of the singular ~-structure takes place, as it will be shown in sect. 4. Extended * We thank H. Skarke for suggesting to interpolate between the LC gauge and a covariant gauge.

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BRS-invariance is introduced in sect, 3 at the classical level in order to control the dependence on ~ as well as on the other gauge parameters n ". The renormalization is discussed in sect. 4. It is based on the B P H Z L regulator-free scheme [9,10] for massless theories, which is shown to apply in our ~-gauge. In particular the usual power-counting and renormalized action principle hold. This allows the implementation of extended BRS-invariance to all orders, entailing the gauge independence of the theory. Some one-loop basic integrals and their asymptotic behaviour for ~ ~ are given in appendix A. In appendix B we derive all needed propagators,

2. T h e i n t e r p o l a t i n g g a u g e

We start from a pure massless YM theory. At the tree level one has

I~II,

IP = /~inv -1-

(2.1)

t*

--- -- ~~ Tr ~d4x F ~ F~, J

F~u=Tr/d4x(BNUA - c

(2.2)

NUD, c+),

(2.3)

where we use the matrix notations =+..

,

D,=Ou+ig[,A,],

,

,

,

F,~=O~A~-O,Au+g[A~,Av].

(2.4)

The total lagrangian is invariant under the nilpotent BRS-transformations [10,11]

sA, = D~c+ ,

sc = B,

sc+= ic+c+ ,

s 2 = 0.

(2.5)

B is a multiplier field for imposing the gauge condition, c+ are the F a d d e e v - P o p o v ( ~ / / ) ghost fields. The gauge-defining Nu in (2.3) would be the derivative operator 0" in the case of a covariant (Landau) gauge, or a constant vector n ~ in the case of a non-covariant (axial or light-cone) gauge. In order to interpolate between the L a n d a u gauge and the LC gauge, we take for N" the following expression:

N"=

(2.6)

The 4-vector n~ = (n °, n), with n 2 = 0, is the LC gauge direction, whereas the dual vector n.* = (n °, n) is needed in order to define the LM prescription. Note that

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O. Piguet et a L / Nonlocal divergences

the dual vector appears already in the classical action. The nonlocality of this action will be eliminated in sect. 3 with the help of auxiliary fields. The limit ~ ~ 0 describes the covariant gauge (1.1), and the limit ~ ~ ~ the LC gauge (1.2): one can see this explicitly on the gluon propagator

A~ a b

__

--'~

- i 8 ~b [g~ k 2 + it

(k~,n~ + k.n~,)~kn* k 2 - ~knkn* + it(1 + ~)

]

k2k~k~ ( k 2 + ~knkn* + it(1 +

~))2 • (2.7)

We can observe that the LM /e-prescription [1] emerges quite naturally in the LC limit from the usual prescription "k 2 + it". In calculating the n,n~-part of the gluon self-energy one gets the following expression: 1

n~n~jd2~k2f

1

+ it ( p + k) 2 + it

× A g ( k , ~)Ag(p + k, ~ ) [ k 2 ( p

+ k) 2 - k2p 2 - p 2 ( k ~_p)2 +p4], (2.8)

where Ag(k,~) = lim

~+o k2 + ~(kn*)(kn) + ie(l +,~)

(2.9)

"

In order to get some preliminary insight in the general situation, let us consider the second term in eq. (2.8), i.e. the integral

J(P, 4)

=

fd2 k Z(p, k, a) = fd2%

+ k,

(p+k)2+it

"

(2.10)

We compute (2.10) first with ~ set to ~ in the integrand: this corresponds to the usual LC situation. The result is

fd2~kI( p, k, ~ )

= -i~2F(2 - ~)

2 pn* + finite terms, nn* pn

where we have made explicit its nonlocal infinite part.

(2.11)

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O. Piguet et al. / Nonlocal divergences

On the other hand, the integral (2.10) for any finite value of the parameter ; is calculated with the techniques presented in appendix A. The result

J(P';)=iw°~(ip4+P3)2;2I'(3-°°)yoldY3Lff

×

1

(1 -- ]/3 -{- ;]/2)( 1 q- ;)]/3 V3d]/2 (1 + ; ( ] / 2 + ] / 3 ) ) 3

(1 -- ]/3 q- ;V2)( 1 q- ;)V3 1 +;(]/2+]/3) (p24+p2) + (1 - ]/3)]/3p 2

]¢o--3 ,

(2.12)

is seen to be finite: this sustains our purpose to use ; as a regulator for the nonlocal divergences. Of course the behaviour at ; ---, oc is singular:

J ( p , ;) ~ -ivr 2

2

pn* --log;.

nn* pn

(2.13)

It corresponds to the nonlocal divergence (2.11). (Other integrals present in one-loop calculations are finite in ; or diverge like ;.) We may thus hope to replace the nonlocal divergences of the LC gauge through the singular behaviour at infinite ; of a local theory. But, ; being a gauge parameter, we expect that physical quantities - S-matrix elements or Green functions of gauge invariant operators - do not depend on it: in other words we expect that in the LC gauge their possible nonlocal infinities cancel. The simplest example is provided by the two-point function (0[T(F F""](x](F

F " " ] ( y ) l 0)

(2.14)

in the one-loop approximation. Its ;-independence follows simply from that of the field-strength propagator (01TF~(x)roa(y)[O )

(2.15)

in the tree approximation. In addition, in the presence of fermions one can show at the one-loop level that the mass counterterm to the fermion self-energy

3m = gt( p ) N(1)( p )u( P )[o,]shell

(2.16)

is gauge independent as well. It will be the aim of the rest of the paper to demonstrate this gauge independence for the ;-regularized theory to all orders.

3. Redefinition of the model; local gauge fixing and extended BRS-symmetry In order to impose the gauge condition N"A, = 0 in a BRS-invariant manner we introduce the multiplier field B and the associated antighost field c leading to the

O. Piguet et al. / Nonlocal divergences

532

classical action

+n" A , - 7

2(n,a)2e •

(3.1)

The corresponding BRS-transformations are given below in eq. (3.4). In eq. (3.1) we have introduced a further gauge-breaking term parametrized with c~. It will be needed in the subtraction procedure discussed in sect. 4. In the next step we introduce two auxiliary fields K and D in order to rewrite the bosonic part of (3.1) in a local form:

F~b°"=Tr fd4x[K(~(n*O)D-OA)+BD+ B(nA)+~e~K2] II

(3.2)

We finally get a local version of the whole q~F/-part:

F4,l~=Trfd4xs[K (~(n*O)D-(OA))+c_D+c (nA)+½aK K],

(3.3)

after introducing the ghost fields K and D+ and requiring invariance of eq. (3.3) under the nilpotent BRS-transformations

sA.=D~c+,

sc =B,

sc+=ic+c+,

sK = K,

sD =

s 2 = 0.

D+ ,

(3.4)

Here the suffixes _+ indicate the q~//-charge. Fields without suffixes are q)H-neutral. The dimensions and the q~H-charges are collected in table 1. In order to describe the composite BRS-variations of the fields A, and c+ one introduces sources p" and o as usual [10]:

(3.5) The total action Fcl = /-'inv q- Fq~/7 q- F s

(3.6)

TABLE l field

A~

B

c

dimension q~H-charge

1 0

3 0

3 - 1

c+

D

D+

0 1

t 0

1 1

K

K 2 1

2 0

O. Piguet et al. / Nonlocal divergences

533

is again BRS-invariant if so s= so = 0. The symmetry content of the model is characterized by the following nonlinear Slavnov identity for the classical action:

Now we are going to define the extended BRS-symmetry [5, 6, 12]. At the tree level, we have 0iFcl

= Tr

fd"x s ( A i ) .

(3.8)

The index i describes the various differentiations with respect to ~, n,, n s* and a and the A i are the insertions

A~=K

(n*O)D,

A :c

A s,

A j=K

~OSD,

A ~ = ½ K K.

(3.9)

Eq. (3.8) suggests to define an extended BRS-symmetry under which the gauge parameters ~, n s, n~ and a transform into anticommuting Grassmann parameters:

s~ = ~+ ,

sn s = n~+ ,

sn*" = n*",

sa = a+ ,

s 2 = 0,

(3.10)

with the restrictions coming from the light-cone [12,13] n s = (no, n ) - (]n], n) n s* = (n 0, - n). This extended invariance leads to a modified OH-action I~n=Trfd4x[K(~(n*0)D-0A)-K

+BD-c -K

~(n*O)D++K

9 sDsc +

D ++ B(nA)-c_#'D~,c+ + laK2

~+(n*O)D-K

~(n*+O)D-c ( n + A ) + ~ a + K K ] .

(3.]1)

Our model is now characterized by the following identities: (i) Extended Slavnov identity: With Fcl = Finv + Fq, n + Fs one has

5¢ (Fcl)

Tr

jrd4x [SFc' I 8rd 8Fd 8re' 8re' 8Fd [3---pu 8As + 8o 8c+ + B ~ c + D ~ + K 3 K

8rd --

arc, , arc, aG] +~+~7+n+o~ 3n~-+a+-~-a ]=0.

(3.12)

O. Piguet et al. / Nonlocal divergences

534

(ii) Gauge conditions: The dependence on the various Lagrange multiplier fields is prescribed by the linear gauge conditions

8r l (3.13)

8B - nrAr + D,

8C1

1

(3.14)

8K = ~(n*O ) D - OA + o~K + 5e~+K ,

rc,

-

8D+

(3.15)

~(n*O)K_+ c

(iii) Ghost equations: The Slavnov identity (3.12) together with the gauge conditions implies the "ghost equations" G1~1 = - n ~ A r - D + ,

(3.16)

G2/'~1 = - ~ ( n * O ) D + - ~ + ( n * O ) D - ~ ( n * + O ) D - } e ~ K ,

(3.17)

- B- ~(n*O)K- ~+(n*O)K_- ~(n*O)K,

(3.18)

8Do

with 8 G 1 =

n r

-

80 r

8 +

- -

8c

'

8 8 G2 - - 0 r + -80 r 8K

(3.19)

Solving these gauge conditions and ghost equations one arrives at F~l= T ' d ( A , c + , ~ , ~ , n , n * , a , ~ + , n + , n * , a + )

+ Tr

fd%[K(~(n*O)D- OA)1 2 +seeK -K

K ((n* O)D++ BD - c_D++ B ( n A )

~+(n*O)D-K ~(n*O)D-c

(n+A)+~c~+K K ] ,

(3.2o) where P is independent of the multiplier and ghost fields, c+ excepted, and depends on 0 and c only through the combination ~r = p r _ nrc + OrK .

(3.21)

(One sees here again, through the presence of both ghosts K and c_, how the ~-gauge interpolates between the covariant and the LC gauge, to which correspond the multiplier fields K and B, respectively.)

535

O. Piguet et al. / Nonlocal divergences

The Slavnov identity (3.12) then reduces to Tr f d 4 x [

8pd 8P~, + 8P~, 61"~, -- +

[

8A.

8c+

8T'~, + n +.- -8Pd +a

W

8.'

6Fd ] = 0 .

+ W. J

(3.22)

We know [5] that the fulfillment of an extended Slavnov identity like (3.22) ensures the gauge-parameter independence of the physical quantities. This concludes the discussion in the classical approximation.

4. Renormalization

The renormalization program has to organize the radiative corrections in a consistent way, such that unitarity of the S-matrix is guaranteed, that physics is gauge independent, etc. In our case this amounts to show that it is always possible, by adding a finite number of local counterterms, to construct from the classical action /'~l obeying the extended Slavnov identity (3.12), a vertex functional F obeying the same identity at all orders of perturbation theory. The general strategy is based on the renormalized action principle [14, 15]: it identifies the changes of the Green functions under an infinitesimal variation of fields or parameters, with a local insertion. The dimension of this insertion has an upper bound resulting from the power-counting theorem [16]. To our knowledge, the latter theorem is proved up to now only for theories which are explicitly Lorentz-invariant. We therefore begin by generalizing the proof as given in ref. [16] to the case of the ~-gauge. We want to adapt the BPHZL subtraction scheme of refs. [9, 16] to our needs. In order to avoid spurious infrared singularities emerging in the course of the ultraviolet subtractions, we first introduce, as usual, auxiliary mass terms in the classical action

rm..=M,'Trfd4x[½A.A"+Kc+],

(4.1)

where M, = M(s - 1), and the parameter s participates to the subtractions we shall discuss later on (s is set to 1 at the end of the calculations). This leads to massive propagators, the list of which is given in appendix B. We have there given the value 1 to the gauge parameter a in order to get propagators with denominators which are quadratic in the momentum or squares of quadratic expressions. We can then define the e-prescription for these denominators in a way analogous to that of refs. [9, 16]: /)cov = k 2 - M~2 + i e ( M f + k2), D .......... = k Z + ~ ( k n * ) ( k n ) - M ? + i e ( M ~ + k Z + ~ ( k n ) 2 ) ,

(4.2)

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O. Piguet et a L /

Nonlocal divergences

where, in Minkowski space, k 2 = k~ -

k z ,

(kn*)(kn)

= (kono)

2 -

(kn) 2.

(4.3)

Finally the momentum space integrand associated to a connected Feynman graph is subtracted according to the rules of Zimmermann's forest formula [16,17]. The subtraction operators we use correspond to the "naive" degrees of divergence, i.e. to the degrees defined by giving to the fields their canonical dimensions: this means that the subtractions performed here are in one-to-one correspondence with those which would have been performed in a covariant case. This procedure leads to integrands which are formally convergent, i.e. the degree of divergence of the integral and of each subintegral is negative. Nevertheless, one knows that in the pure LC gauge (4 = oo) integrability is not guaranteed: (nonlocal) divergences may still occur. This is linked to the fact that, in this gauge, the asymptotic behaviour of the noncovariant propagators in momentum space is anisotropic: e.g. k n behaves like k to the power zero in directions orthogonal to n~. Such a pathological behaviour is not present for finite, positive 4. One sees indeed in eqs. (4.2) that the noncovariant denominators are obtained from the covariant ones through the following replacement of the Minkowski metric: k 2 ._, ~ l ~ k ~ k

~'

00=l+Og, The space metric

-~ij

,7,j = -(aij+O,,,j).

(4.4)

as well as the corresponding "euclidean metric" gop, ~ ~0~ '

g i ) = -- ~ i j

(4.5)

are positive definite (for nonnegative 4). It is therefore not difficult to convince oneself that the usual arguments [16] based on the existence of euclidean lower and upper bounds for every minkowskian integral are still valid. We thus conclude that, also in our f-gauge, the BPHZL subtraction procedure leads to absolutely convergent integrals for e > 0 which tend towards tempered (but not Lorentz-covariant) distributions in the limit e-o 0. At the same time we can infer that the renormalized action principle holds with "canonical" power counting. We can now turn to the renormalizability's proof of the gauge theory in the interpolating f-gauge, together with the proof of the f-independence for the physical quantities as expressed by the extended Slavnov identity (3.12). Our first task is to control the breaking of Lorentz invariance due to the presence of the vectors n" and n *~, and in particular to show that the physical quantities are not affected by this breaking. This problem was solved in ref. [18] for the case of the axial gauge: we shall therefore not repeat the argumentation, which applies here

O. Piguet et al. / Nonlocal die~ergences

537

without any substantial modification. The result is a Ward identity which describes in a precise way the breaking of Lorentz invariance. The practical consequence for us is simply that all expressions we have to deal with in the following - local field polynomials, e.g. counterterms - are formally Lorentz invariant, if one considers n" and n *" as well as their BRS-transforms n~+ and n *~' as Lorentz vectors. Before dealing now with the Slavnov identity, we begin by assuming the validity of the gauge conditions (3.13 3.15) and of the ghost equations (3.16-3.18): their p r o o f is trivial due to their linearity. It is therefore sufficient to prove the - nonlinear - Slavnov identity in the form (3.12) with F defined from F, as Fd was defined in (3.20) from F~. But the latter problem, as it is well known [5, 10,11], amounts to solve the consistency condition bA = 0,

(4.6)

where J ( A , c+, ~/, a, pi, p + ) is a local functional of dimension 4 and ghost number 1, and b is the nilpotent differential operator - the linearized version of the Slavnov operator (3.12):

+

J

[ 8A~ 8VlI'

+

8"q~' 8A.

+

86'+ 80

80 8c+

+p+--

0

Opi •

(4.7)

Here pi stands for all gauge parameters ~, c~, n i (n o and n *~ are not independent), and p'+ for their BRS-variations. The nilpotency b2 = 0

(4.8)

follows from the fact that the classical action itself obeys the Slavnov identity. In the absence of the chiral gauge anomaly, the general solution of the cohomology problem (4.6) is known [5, 11] to be trivial, i.e. there exists a local functional /~ of dimension 4 and ghost number 0 such that A = bz].

(4.9)

This means that the Slavnov identity can be maintained at each order of perturbation theory. In order to complete our discussion, we still have to look for all possible invariant counterterms L: bL = 0

(4.10)

that one can freely add to the action at each order without spoiling the Slavnov identity. The general solution of (4.10) for L as a local field polynomial of

538

O. Piguet et al. / Nonlocal divergences

dimension 4 and ghost number 0 reads

L = XgLg+ b L ( A , c+,~,o, p, p+), = Tr f d 4 x F . . F

.

(4.11)

(4.12)

We have moreover [5]

op,x

(p) -- 0

(4.13)

for the coefficient of the cohomologically nontrivial term Lg in eq. (4,11): it corresponds to the coupling constant renormalization. In the gauge dependent part bL, the local functional L consists of all field monomials of dimension 4 and q)H-charge - 1 , depending on n, and n~* in a formally Lorentz-invariant way. Their coefficients correspond to the unphysical parameters occurring in the (generalized) field amplitude renormalizations [6, 7,12]. All the coefficients appearing in (4.11) must be fixed by a suitable set of normalization conditions [5]. Let us conclude this section by stressing again that our proof of the extended Slavnov identity (3.12) obeyed by the renormalized vertex functional P - which means the gauge independence of the renormalized physical quantities and thus their existence in the light-cone limit f--* o e - was purely algebraic. It is in particular independent of the explicit details of the subtraction procedure used, once the usual power-counting rules are shown to hold for the f-regularized theory. The resort to a particular subtraction scheme, namely the BPHZL one, was only motivated by the fact that general proofs of the power-counting theorem and of the renormalized action principle were first given in this scheme. Other schemes [15] could as well have been considered.

5. Conclusions

The basic result of this paper is a constructive proof of the renormalizability of YM theory in a regularized version of the LC gauge. Our considerations are based on the BPHZL subtraction scheme. The gauge parameter f interpolating between the covariant gauge and the LC gauge may be regarded as a regulator for the nonlocal divergences in 1PI-vertex functions. With the help of extended BRS-symmetry, we have shown that f-dependent structures leading to nonlocal divergences for f --' ve are completely decoupled from the physical quantities to all orders of perturbation theory. We want to thank H. Skarke for suggesting to use an interpolating gauge, for computational help in early stages of the work and for stimulating discussions. We also thank C. Becchi, P. Breitenlohner and G. Eeibbrandt for very useful discussions.

539

O. Piguet et al. / Nonlocal divergences

Appendix A Here we want to calculate the minkowskian integral i(1)

1 ~( pn* + kn*) ~kn* [dd2'°k ( p + k) 2 (p + k) 2 + ~(pn* + kn*)(pn + kn) k 2 + ~kn*kn

(A.1)

with the techniques presented in ref. [2] and study its behaviour for large values of ~. In order to get euclidean integrals, we use po=ip4,

n~,=(l,0,0,1),

k o = i k 4,

n~=(1,0,0,-a).

(A.2)

Exponential parametrization yields I °) = - i f d k 4 dk 3 d2°'-2k ~ ~2(ip4 + ik 4 + P3 + k3)(ik4 + k3)

3<£°Cdalda2da3exp{-eq[(p4 + ka)2+ (P3 + k3) 2+ (P± + k ± ) 2] -a2[(l+~)((p4+k4)?+(p3+k3)2)+(p±

+ k ± ) 2]

(A.3)

-a3[(1 + ~)(k42 + k 2) + k2]}. Introducing some abbreviations 1~1 ~ O/l '

B=

/91 4- /~2

,

O¢1 -1- Ol2

A

/~1 q- /~2 + B3

,

(A.4)

O/1 q- 0/2 q- 0/3

and shifting the integration variables

ka=Bp,+k

,

k;=Bp,+k,,

(A.5)

we get, using standard gaussian integrals G ~_ O~2 q_ G3)I I (l) = iqr'~(ip4 + p3)2~2fo°Od % d a 2 d a

3

w (1 - B ) e

B1 + & + B3

× e x p [ - B ( 1 - B)(p42 +p2) _ A(1 - A ) p 2 ] .

(A.6)

540

O. Piguet et al. / Nonlocal divergences

After the usual change of variables a 1 = t(1 - "}/2-- ~/3),

0~2 = tY2,

a3 = t'f3

(A.7)

this reads

fo fo

l(1)=ivr'°(ip4+p3)2~ 2 ~ d t

ldT~

fol_~,3dT212tl_l (1-

(1Y3 + ~q-( ~T2)( T 2 + Y13 )-b) ~)T3 3

t

(1 - v, + ~,~)(1 + f ) ~ ×exp -t

(A.S)

1 q-~(y2q-Y3)

and dt-integration yields

1 I-(1 -- ")/3q- t~2)( 1 -{-~)~3 /(l)=dTr~(ip4-bp3)2~2F(3--o))f 0 d'/3fo '3672 ~ 1 + - ~ T 2 + ~ ) 7

x

[ (1 -'~3q-~'/2)(1 q-~)~/3 I + ~(~,,+ ~,,)

~/ 2 ]co-3

(p~+d) + (I-~_,),p~

(A.9)

Thus the integral is more convergent than naive power counting would demand. This is due to the fact that two quadratic terms have cancelled each other in the c o m p u t a t i o n of (A.6). To study the behaviour of I (1) for large values of ~, we p e r f o r m a change of the integration variable ")'2-+ ~:

x =

1

1 + ~(~ + y~) '

dx = -

~dT2

(1 + ~(~ + ~,))~ "

(A.10)

F u r t h e r m o r e we are allowed to put ~o = 2 from now on:

i(1) = i ~

_

P3)

fl~

fl/(l

+ ~y3) "

,.

( d + P 3 ) ( l + })(I - x(1 + }) Y3)

pg +p3 JoeY'Jl/(,+~) aXg(p2 +p2)(l +~)(1- x(l +()TB)+(1-.I,3)p~ _i~(ip4+p,)2rl~

rl/(1+,,,>

[

~(1-~,)p~

1 (A.11)

O. Piguet et al. / Nonlocal divergences

541

where the first term evaluates to

p~+p23

dY3 l - ~ y 3

=iqr2 p~+p~ ( l o g ~ + O ( 1 ) ) .

1+~

(a.12)

To show that the rest of (A.11) remains finite for ~ ---, oc, we give an upper bound for its absolute value. Integrating over x one has an integrand of the form A / ( B + p2 C). The inequalities

1

x < - 1 + ~`/3

~

1 - `/3

0< - -

~< 1 - x ( 1 + ~ ) ` / 3 ,

(A.13)

1 + ~`/3

show that A, B and C are positive quantities. Therefore it is allowed to put p2 = 0 in the denominator to obtain an upper bound. Now changing x ---, u: (1 q- ~)2`/3

1 + ~ b/ --

du -

(1 -- X ( 1 q- ~ ) ` / 3 ) '

1 - `/3

1 - `/3

dx,

(A.14)

we can estimate

fo~d`/3~(l+~)/H+~v~) /1+~ du

1 - `/3

`/3

~

1

el

rl+~

du -(1 q..~)2 U ~ ]J0 d`/3] J(1+$)/(1+ ~Y3)

1 - `/3

`/3

~

1

(1 + ~)2 f~(A.15)

because of u >/(1 + ~)/(1 + ~`/3)>/1. Integration of the r.h,s, of (A.15) over u furnishes 1 - "/3

2fo dz~

`/3

~

(

1

(1 + ~)3/2 1

vla~3

1

-`/3)

2 (1 + ~)3/2 f0 d`/3 1 -F ~`/3

(A.16)

The last estimate follows from

1

1

Thus the rest of (A.11) is finite for ~ --, ~ .

1

~3

(A.17)

542

O. Piguet et al. / Nonlocal divergences

We have seen that for large values of

1 (1) -

- (logf)r(3

-

,~)

2 pn* nn* pn

(A.18)

Instead of a pole term (,0 - 2) 1 we observe a logarithmic divergence in 4. Other integrals, however, show a different behaviour. E.g. for large values of ( we have

1(2)=

1 ~kn* fJ dz~k (p + k) z k 2 + ~kn*kn

2pn*

F(2 - o a ) - - ,

n/q*

(A.19)

whereas

1 (3) =

+ kn*) ~kn* fd2,Ok( p + k) 2 + ~(pn* ~(pn* + kn*)(pn + kn) k 2 + ~kn*kn

- U'(2

-

,~)~(pn*/nn*)

2.

(A.20)

•(3) is divergent (while it is zero if the limit ~ ~ ~ is taken before integration), but the divergence is local. For I (2), the limit ~ ~ o0 commutes with the integration. The only rule obeyed by all Feynman integrals in this interpolating gauge is that pole terms ( ~ o - 2) i must not have nonlocal coefficients, This just reflects that power counting is valid for finite values of ~.

Appendix B Here we derive the free propagators in the presence of the auxiliary mass term (4.1) that is needed to avoid infrared singularities in the procedure of BPHZL-subtractions. We set to zero the coupling constant g, the sources P% o for the composite operators and the Grassmann-valued parameters ~+, n4, n * , a + . The bilinear part of the action

rbilin=Trfd4x[-¼(O,A~-O~A,) 2+!M2A21'',',~'A~" 1 2 + K( ~( n*O )D - OA) + B( D + nA) + ~aK

- K ~(n*O)D++K (02+M~2)c+-c D + - c (nO)c+]

(B.1)

543

o. Piguet et al. / Nonlocal di~rergences

then leads to the system of equations

o 2A. - a . ( O A ) + M?A. + a . K + . . 8 = - y . , - (OA) + a K + ~(n*O)D = - J x , -~(n*O)K+

B= -JD,

(B.2)

( nA ) + D = - J s .

Expressing A~, in terms of j~,, Jx, JD and J8 and using 62Z,. (TA,(x)A~(y))

.aA,(x)

= - 8+(x) 8j,(y) -

l 8j~(y~ '

(B.3)

we get after some algebra

,[

= 0=+M? g..-(O...+ 0o..)(O~+M?)~(.*0) N~

+OuO(02+M2)(°~-I) N,

-

nut'"

M~2(n*a)2] N~

8(x-y),

(B.4)

where N~= ( c9~2)2+ 2 0 7 M 2 + ( ~ -

1 ) 0 2 M 2 + e~M~,

o~=o=+~(.a)(.*o).

(B.5) (B.6)

The auxiliary mass term (4.1) has generated a very complicated denominator N~. If we set c~ = 1, however, the propagator becomes much simpler:

(TA (x)A,,(y))

i [

~(,,*a)

M~2(n,0)2] , . , . (07 + M~) 2 8(x-y~. (B.7)

thus allowing for the further reasoning in sect. 4. For the ghost sector, the system of

544

O. Piguet et al. / Nonlocal dit~ergences

equations reads --~(F/*0)D++(

02-{- M a 2 ) c + = --JK

-

,

D+-(nO)c+= -j,. ,

-~(n*O)K

-c

=

--jD,

,

(O2+M?)K +(nO)c =-j,.+.

(B.S)

There are two propagators that can form ghost loops,

= - i o f f + M 2 8 ( x - y ) ,

(B.9)

i = 3~ + M 2 6 ( x - y) •

(B.10)

the others are irrelevant because of their trivial couplings.

References [1] G. Leibbrandt, Phys. Rev. D29 (1984) 1699 [2] G. Leibbrandt, Rev. Mod. Phys. 59 (1987) 1067 [3] A. Bassetto, M. Dalbosco and R. Soldati, Phys. Rev. D36 (1987) 3138; A. Bassetto, G. Nardelli and R. Soldati, Mod. Phys. Lett. A3 (i988) 1663 [4] S. Weinberg, Phys. Rev. 118 (1960) 838 [5] O. Piguet and K. Sibold, Nucl. Phys. B253 (1985) 517; O. Piguet, "Comments on the gauge (in)dependence of gauge theories", UGVA-DPT 1988/12-600 [6] P. Gaigg, O. Piguet, A. Rebhan and M. Schweda, Phys. Lett. B175 (1986) 53 [7] S.E. Nyeo, Phys. Rev. D36 (1987) 2512 [8] S. Mandelstam, Nucl. Phys. B213 (1983) 149 [9] J.H. Lowenstein and W. Zimmermann, Nucl. Phys. B86 (1975) 77; J.H. Lowenstein, Commun. Math. Phys. 47 (1976) 53 [10] O. Piguet and A. Rouet, Phys. Rep. C76 (1981) 1 [11] C. Becchi, A. Rouet and R. Stora, Ann. Phys. (NY) 98 (1976) 287 [12] O. Piguet, M. Schweda and H. Skarke, Phys. Lett. B 210 (1988) 159; A. Andrasi and J.C. Taylor, DAMTP 86-II (unpublished) [13] H. Skarke and P. Gaigg, Phys. Rev. D38 (1988) 3205 [14] Y,M.P. Lain, Phys. Rev. D6 (1972) 2145, 2161; T.E. Clark and J.H. Lowenstein, Nucl. Phys. Bl13 (1976) 109 [15] J.P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977) 55 [16] W. Zimmermann, Commun. Math. Phys. i1 (1968) i; J.H. Lowenstein and W. Zimmermann, Commun. Math. Phys. 44 (1975) 73; J.H. Lowenstein and E. Speer, Commun. Math. Phys. 47 (1976) 43 [17] W. Zimmermann, Lectures on elementary particle physics and QFT, 1970 Brandeis Summer Inst. for Theor. Phys., Vol. 1, p. 397 (MIT Press, Cambridge, MA) [18] H. Balasin, O. Piguet, M. Schweda and M. Stierle, Phys. Lett. B215 (1988) 328