A note on the extended Mandelstam-Leibbrandt prescription for gauge theories in temporal gauge

Physics Letters B 320 (1994) 74-82 North-Holland PHYSICS LETTERS B

A note on the extended Mandelstam-Leibbrandt prescription for gauge theories in temporal gauge

M. Polj~ak Jo~ef Stefan Instttute, University of Ljubljana, 61111 L]ubljana, Slovema

Received 16 September 1993 Editor: R. Gatto

The extension to the temporal gauge of the Mandelstam-Lelbbrandt prescription for treatment of spurious poles of the gauge- field propagator in the hght-eone gauge is apphed at lowest order of radiative corrections to the calculation of the vacuum expectation value of the Wilson line integral along a contour with a cusp. The results show that in tins definition of the temporal gauge a cusp renormalizatlon constant arises non-locally: to calculate it one has to take into account the interactions of all parts of the contour, in contradistinction to the Feynman gauge, where one needs to consider the Wilson line integral along just the immediate neighbourhood of the cusp.

There have b e e n qui te a few inves t iga t ions [ 1 -9 ] on the poss ibi l i ty to develop gauge field theories i n the h o m o g e n e o u s t empora l a n d space-like axial gauge f rom the ideas of M a n d e l s t a m [ 10 ] a n d L e i b b r a n d t [ 11 ] on the l ight-cone gauge. The bas ic idea is to regularize the non-phys ica l poles 1 / (k. n ) a n d 1 / (k. n ) 2 o f the gauge- field propaga tor

i (g k~n~+k~n~ (n2+ak2)k~,k~ D ~ ( k ) = ~ ~, k.n + (k .n )2 y , (1 )

where n is the gauge-fixing vector a n d ot the gauge parameter , in such a way that the Wick ro ta t ion in the complex k°-p lane is no t obst ructed. In L e i b b r a n d t ' s vers ion, the idea is real ized b y the p resc r ip t ion

1 ( k . n * ) a = l i m - - , a = l , 2 (2)

(k.n) a ,so k.n k .n*+i~ /

while i n the so-called M a n d e l s t a m ve r s ion ~1 by

1 = l i m / 1 ) a (k.n) a ~o\k.n+i~lsignk.n* , a = l , 2 , (3 )

where n* is an auxi l ia ry four -vec tor which in the l ight-cone case is par i ty conjugate to the gauge-fixing vector n=(n°,n) , i .e .n* =(n °, -n) .

There are two schools of thought how best to proceed outs ide the l ight-cone gauge. One o f t hem [2,4 ] teaches tha t there is some f ixed re la t ion be tween n a n d n* also w h e n n2v~O, the precise fo rm o f which depends on whe ther n is t ime- l ike or space-like. The o ther school [ 3,5,6 ] advocates tha t n and n* outs ide the l ight-cone need no t be re la ted to each o ther b e y o n d the cons t r a in t (n . n* ) 2 _ n 2n'2 > 0. Here I shall be conce rned in some deta i l

~1 Actually, the original Mandelstam prescription [ 10 ] apphed only for a = 1 and in place of sign k n * contained k. n * in the particular Lorentz frame where n = ( 1, 0, 0, 1 ). Why the RHS of eq. ( 3 ), which is eqmvalent to the RHS of eq. (2) in the sense of the theory of distributions, is more suitable is explained e.g. m ref. [ 8 ] or ref. [ 3 ].

74 0370-2693/94/$ 07.00 © 1994 Elsevxer Science B.V. All rights reserved.

SSDI 0370-2693 (93 )E1390-J

Volume 320, number 1,2 PHYSICS LETTERS B 6 January 1994

only with the latter choice for the temporal gauge for which I will use n = ( 1, 0, ~ 0, 0) and n*= (0, 0, 0, 1 ), and or--0.

The extensions of the Mandelstam-Leibbrandt (ML) prescription outside the light-cone gauge seem to pro- vide a promising definition of the temporal and space-like axial gauge. They were partially justified within the formalism of canonical quantization [ 7 ]. It turned out that the structure of counterterms in the non-light-like case is more complicated than in the light-cone case [4-6 ], yet the S-matrix for quark-quark scattering to one- loop order came out independent of both n and n* [ 6 ]. It was also confirmed [ 8 ] that the vacuum expectation value of the Wilson line integral,

W[F]= (0ltr ~J-exp(-ig ~ A.dx) I 0), F

(4)

along a rectangle with two sides parallel to the time axis reproduces the Feynman gauge result, at least in the limit when the length of the time-like sides of the rectangle tends to infinity, as required by gauge invariance of Wilson loops.

The most characteristic consequence of the use of the extended ML prescription appears to be the non-local nature of counterterms needed for carrying out the renormalization of Green's functions. Although Burnel and Caprasse [ 9 ] questioned any inherent need for non-local counterterms in the axial gauge - in the particular case of gluon self-energy they traced the origin of non-local counterterms in previous calculations with the extended ML prescription down to a decomposition formula for products of spurious poles which they claim not to be valid - it has been pointed out [ 12 ], however, that the prescription which they used to establish the correctness of their claim is not identical with the extended ML prescription. Bagan and Santiafiez [ 12 ] have verified in the light-cone gauge that the UV parts of the relevant integrals do not depend on whether the decomposition formula is used or not. Since the temporal gauge is less singular than the light-cone gauge, the latter authors assumed that they can rely on the splitting formula in their calculation of the vacuum polarization tensor in the temporal- planar gauge. And as a result they obtained a non-local ultraviolet divergence. Therefore, non-locality seems genuinely to be tied up with the extended ML prescription.

In addition to Green's functions, also the vacuum expectation values of the Wilson line integral along closed paths in space-time, eq. (4), have proved to be very useful objects of gauge-field theories. The range of their applications includes a derivation of the potential between two static colour charges [ 13 ], an elucidation of the infrared behaviour of the structure functions [ 14 ] as well as proofs of factorization of hard processes in perturbative QCD [ 15 ]. It is known from the renormalization of Wilson loops in covariant gauges that, apart from the conventional charge, wave-function and mass renormalization (the latter is not needed if dimensional re- gularization is used), some supplementary subtractions are needed, in general, if paths are nondifferentiable (contain cusps) or selfintersecting [ 16,17 ].

The purpose of this letter is to examine whether there is some element of non-locality also in these extra divergences when one considers them in the extended ML definition of the temporal gauge. In particular, I will investigate the cusp renormalization constant for the vacuum expectation value of the Wilson line integral along the following contour F (fig. 1 a): two straight-line segments meet end to end at a point Po to form an acute angle 70 which is subtended by a differentiable curve in such a manner that the resulting contour is smooth except at Po- One of the straight-line segments is parallel with the time-axis and the other is also time-like. I opt to consider such a loop, because its cusp renormalization constant is complex, so that both the real and the imaginary part may separately serve as an indicator of non-locality of cusp renormalization in the temporal gauge. In order to avoid any misunderstanding, I should say that in the explicit calculation which I will carry out the extended ML prescription will be realized by the use of eq. ( 3 ) with n = ( 1, 0, 0, 0 ), n* = (0, 0, 0, 1 ), and without any ghosts.

In order to clarify whether there is any parallel between the cusp renormalization of Wilson loops and the renormalization of Green's functions, a few words are in order about the 7 dependence of the cusp renormalization constants Z. A cusp renormalization constant is defined such that WR[F] = Z ( 7 ) I~[F] is a finite func-

75


x~

F2

ro

Po ~3 V Po ~

(b)

Fig. 1. Closed contours along which the Wilson line integral is considered: (a) A loop F wth one cusp is composed out of two straight-line segments Fo and F1 that meet at a cusp of angle 7o and have their other ends connected smoothly via a chfferentia- ble curve Fz; (b) An (N+ 1 )-sided polygonal approximaUon to the loop F.

tion when IP[F] is obtained from the unrenormalized W[F], eq. (4), by a conventional charge and wave function renormalization. The following argument, applicable in covariant gauges, is sometimes offered [ 17 ] to show that the dependence of Z on F simply via the angles y is a natural generalization of the renormalization of Green's functions G(pI , P2, ..., P,,) by factors which are polynomials in the momentap. The Green's function G depends on a finite number of continuous variables and is renormalized for all (Pl, P2, ..., Pn) by a finite number of constant counterterms. The vacuum expectation value of the Wilson operator depends on a continuously infinite number of continuous variables (i.e., a loop) and is renormalized for all loops F by a finite number of counterterms depending on continuous variables (the angles y). In other words, the renormalized theory of Green's functions is made unique by the specification of a finite number of normalization conditions GR (P11, ~ ..... P-~) =Kn, on a finite number of Green's functions, whereas the renormalized theory of loop functions is made unique by the specification of a continuously infinite number of normalization conditions, WR [Fr ] = K , one for each value of angles ?. In each case, the dependence on each continuous variable is renormalized by a finite number of parameters; it is just that for loops there are continuously infinitely many continuous variables. And just as observables in the Green's functions theory must be independent of the choice of the normalization momenta if,, the observables in the loop function theory must be independent of the choice of the normalization loops ~ . This parallel between the renormalization of Wilson loops and Green's functions sug- gests that, since Green's functions contain non-local divergences in the extended ML definition of the temporal gauge, there may be a trace of non-locality also in the contour divergences of the Wilson loops when considered in the same definition of the temporal gauge. It is the main aim of this letter to show that this is the case indeed.

Before I consider the renormalization of Wilson loops in the temporal gauge with the extended ML prescription, I will briefly review the renormalization in the Feynman gauge. A fact, most relevant for what will be discussed later, is that one does not need to consider the whole loop if one is interested only in calculating a cusp renormalization constant, but it suffices to calculate radiative corrections to the Wilson line operator along just some portion of the loop (an open contour) in the immediate neighbourhood of the cusp containing the cusp as an inner point. Although a cusp renormalization constant is gauge independent and the angle at the cusp is the only geometric characteristic of the loop on which the cusp renormalization constant depends, this fact is not

76


trivial, because the Wilson line integral along an open contour is gauge variant. Because I shall often refer to this fact in later parts of this letter, I will give it the name of locality of renormalization of Wilson loops in the Feynman gauge.

In order to show that this fact is true let us approximate the loop in fig. la with an (N+ 1 )-sided polygon C by choosing Npoints on the curved part F2 of the loop (with the points of contact ofF2 and the two straight line segments Fo and F1 included) (fig. lb) . The resulting contour has N + 1 cusps. Among them there is the cusp of the original path as well as artificial cusps introduced by the approximation. Now, consider the Wilson line integral along the polygon C. To lowest order of radiative corrections

g 2

C C

(5)

I f we use the momentum representation for the gauge-field propagator,

f d2°~k D~,~(x-y) = j ~ exp [ik. (x-y) ]Ou~(k),

it is convenient to introduce the functionals [ 18 ]

f ~ ( k ) = dxUexp(ik.x)= dxUexp(ik.x)-- = ~ f~,(k), l~O twO

C C~

(6)

(7)

where C, is a side of the polygon (C0=Fo and C 1 = F 1 ), SO that

( 2 n ) ~=o~=o "

In the Feynman gauge, the divergent part (div) of W (2) [C] is given by

div =½g2div [ r)vey-m (9)

The double sum over the sides of the polygon in eq. (8) has reduced to a single sum in eq. (9), because the interaction of the sides with no point of contact is not divergent in the Feynman gauge. In eq. (9) we understand that CN+ 1 = Co.

On the other hand, the divergent part of W (2) [C] is given also by minus the sum of the divergent part of the renormalization constants associated with the cusps at the vertices of the polygon. Namely, when there are severaldifferent cusps 70, 71, -.., 7N, the multiplicative renormalization constant factorizes [ 17]: Z(7o, 71, ..., 7N) = Z (7o) Z (71).--Z (TN), which in the one-loop order reads

N N Z(2)(70,~l, '",TN) = l'-I [l+Z(2)(Ti)+O(g4)]=1+ ~, Z(2)(Tz)+0(g4)"

t=0 l=0

In accordance with the definition of locality of a cusp renormalization constant in the Feynman gauge, the renormalization constant of the cusp at the vertex Pj of the polygon should also be equal to (among other pos- sible expressions) the negative of the pole part of the vacuum expectation value of the Wilson line integral along the contour Cju Cj+ 1. If we take into account that Cj and Cj+I are external lines in this open Wilson line, while they were internal lines in the Wilson loop in eq. (8), so that their self-energies are halved w.r.t, the self-energies contributing to the Wilson loop, we get

77

V o l u m e 320, n u m b e r 1,2 P H Y S I C S L E T T E R S B 6 J a n u a r y 1994

N div W~.m..[C] = - E Z°)(TA

.1=0

N ¢ d2~°k l~2di v ~ / DFeynman/ l~e/z ¢ ' v # v ,u v 1 # v = -~f c~+, f c,+, ) (10) ~ ~ 1 ~ _ ~ 2 ~ u~ tvc~Jc~+fc~fc~+x+fc,+~fc~+ j = 0 "~ ~x.j~)

Since eq. (10) is algebraically equivalent to eq. (9), we have confirmed that the use of locality of the cusp renormalization constant in the Feynman gauge is justified. The artificial cusps have angles 7, whose supplementary angles ~=rc -? ,~ 1/N are vanishing for N--,c¢. But the limit is not trivial since the approximation has introduced additional cusp divergences. However, the cusp divergence is proportional to ~ cot ~,- 1 [ 16 ], which for a small angle ~, tends to ~2, so that the total cusp divergence corresponding to all the artificial cusps, is equal to 52 ~= 1Z (2) ( 7t ) ~ ~2N-~ 1/N and therefore vanishing in the limit N--, c~.

Consequently, in the limit N--. oe there is only one term left on the RHS of eq. (10):

2) , div W ~ y m ~ [ F ] = ½ g 2 d i v d ~ k DVe~ m"~ (~-zf~of f .o+f~off-~+f~ff .o+!zf~f f -1) (11)

which means that in the Feynman gauge the contour F2 does not contribute to the cusp renormalization constant Z (2) (7o). Its value, from eq. ( 11 ) or ref. [ 161, is

g2 Z ( 2 ) ( 7 o ) = ~ [ (7~'--70) cot 70+11 2-o9'1 (12)

in Euclidean space, or

gZ F1 {. X + f l o - 2 i r c ) - 2 ] 1 (13) z'2'(a°) = - 1 - ~ [ ~ ~,m 1 --2-~o 2-o9

in Minkowski space, where flo = tanh 0o and 00 is the Minkowski space angle of the cusp at the origin (7o = - i0o). Now, let us turn our attention to the axial gauge. The first point to appreciate is that those terms of the

propagator ( 1 ) with the tensor structures kun~ + k,n u and kuk, do not contribute to W (2) [ C ] in eq. (8) as long as the integrals involving these two tensor structures are well-defined. This is a consequence of the fact that for any continuous and closed curve C, differentiable or not,

kuf~( k ) =k u ~ dx u exp (ik.x) = - i ~ dxUOu exp (ik.x) ~ O . (14)

C C

A consequence of this is that, to lowest order of radiative corrections, the Wilson loop takes the same value in the temporal gauge as in the Feynman gauge, in accordance with gauge invariance of Wilson loops, provided the prescription for the spurious singularities of the gauge-field propagator makes the integrals in eq. (8) existent. In order to learn whether or not the cusp renormalization constant of this Wilson loop is local also in the temporal gauge with the extended ML prescription, I now calculate the RHS of eq. ( 11 ), replacing the propagator there by the temporal gauge propagator ( 1 ) in which the non-physical poles are treated by eq. (3).

The starting point of my calculation are the functionals f ~o (k) and f ~1 (k), defined in eq. (7). In the coordi- nate system shown in fig. 1 a, the only non-zero component of the functionalf ~0 is

0

f dxO exp(ikoxO) = 1 -exp( ikoLo) (15) /Oo= J iko LO

where Lo is the length of the side Fo. The orientation of the loop F is chosen to be counter-clockwise. The functionalsf~l may be obtained by transforming from the Lorentz frame (x °, x 3) to the frame ( ~ , ~ ) with the x-ZLaxis along the line F1 (remember that it is time-like),

78


=Xo coth 0o - x 3 sinh 0o ,, ~ = -Xo sinh 0o +x3 cosh 0o,

where 00 is the angle between the side r I and the time-axis so that

(16)

cosh 80 cosh 8o {exp [i(ko cosh 80 - k 3 sinh 8o)Ll1 -- 1) f o _ i ~ [exp(i~L1 ) - 1 ] = i(ko cosh 80 -k3 sinh 80)

s in o0 -

f~"- i-koo [exp(ikoL1)-ll=flof°r,, f~x=f2l=O. (17)

Here L~ is the length of the side 1"1. The first three terms on the RHS ofeq. ( 11 ) " W-,~m~ temp wath Du~ replaced by Du. give no contribution, as expected

from the temporal gauge with the gauge-fixing vector parallel to Fo. Therefore, if the cusp divergence of the Wilson loop arises locally, as is the case in the Feynman gauge, it should be given by

d2a'k r~temp 4"~ ,e--7 lg2di v f d2C°k i lg2div (~n)E~,-, u, j r , J r , = (-~<o k2~_ie ( 1 - f l g ) f ° , ( k ) f ° , ( k )

_ ~g2div f J d2°'k i 1 ( ko-flok3)f°r, ( k )f°l ( k ) ( 2 - ~ ' k2~-ie [ko~

+ ~g2div f d2~°k i 1 (22-~°~ k2q-ie [ko z] (ko-flok3)2f°,(k)f°, (k)" (18)

The first expression on the RHS of eq. (18), the self-energy of F1 in the Feynman gauge, takes the value (g2/87rz) [ 1 / ( 2 - o 9 ) ] . The second expression on the RHS of eq. (18 ), calculated with the extended ML prescription, reads

__ig2 f J dZ~°k 1 1-coscoLl(ko-flok3) (19) (2~) 2°~ k2+ie (/Co +i~/sign k3) (ko -flok3) '

where Co = cosh 0o = ( 1 -f12) --1/2 Integration over/Co results in the following expression: 1 oo

(4n)o,/,(o)_ 1 ) 2 d x ( 1 - x 2 ) °~-z dK 1¢2 + r/~-'-" ~ --1 0

× (1 - e x p ( - ~coLt ) cos coL~florx+i exp( -tlcoL1 ) sin coLlfloXX 0 ( - x ) floxx+iq

_ 1 - exp ( - tlcoL~ ) cos coL~floroc-i exp ( - rlCoLl ) sin coLlflo~.x O(x)) flo x x - it/

1 oo

+ f dx(l_x2)o~_2~ ~2co-- 3 dx ~ [ 1 - cos coL1 x( 1 + flox) + i sin coLi x( 1 + flox) ] --1 0

1 ~

+2itt f d x (1 - X 2 ) w-2 f K2°)- 4 l+floX signx dxK-T~+~ 2 [1-coscoLl~(l+flox)+isincoLltC(l+floX)] --1 0

+&-~-Po]. (20) The terms in expression (20) which have no factors of 8-functions result from the residues at the physical poles

79


of the propagator ko = + [k[ -T- ie. Their Laurent expansion contains a pole in 2 - co whose coefficient is equal to (g2/8n2) (1/flo) In [ (1 + flo)/( 1 - rio)] in the limit t/~0. Even if one keeps I/> 0 until the integral over x is evaluated, one gets the same residue at the pole o9=2.

The rest of expression (20) tends in the limit 7/--, 0 to 1 c~

- (4r~)°'F(og-1) floX d/¢/¢ 2°J-s (coscoLlfloxx-l+isincoLlfloXlC) . (21) 0 0

The imaginary part of (21) is equal to - ( ig2 /16r r 2) [ 1 / ( 2 - o 2 ) ] , i.e. one half of the value obtained in the Feynman gauge, eq. (13). Surprisingly, the real part of (21 ) has a double pole at co-2! If we keep ~/> 0 to the end of integrations, we get for the first double integral in expression (21 ) (terms containing the 0-functions) the following value:

1F2(2--O9, ~-o9, 2-0) ; (ltlc0LlflO) 2) g2(c°Ll)4-E~° exp(-tlc°L1)fl~-2C°l'(2-og) ~ _ ~ 1

( 4 ~ ) ~o

_ i F ( 3 _ o 9 ~ z c tF2(2-o9 ,2-o9 , ~; (½tlCoLlflO)2))+O(1), o9~2 , (22)

where IF2 is a hypergeometric function. Again the Laurent expansion of (22) in terms of o2- 2 starts with (g2/ 16 rc2flo ) (09- 2 ) -2. From the derivation of (22) it is seen that this double pole is engendered by the single non- physical pole of the gauge-field propagator.

The question now arises whether the remaining third expression on the RHS of eq. (18) cancels this double pole or makes evident that the evaluation of the cusp renormalization constant in the extended ML definition of the temporal gauge requires consideration of parts of the loop far away from the cusp. Performing the integral over/Co in the last expression on the RHS of eq. (18), which in the extended ML definition of the temporal gauge reads

f d2°~k 1 1-coscoLl(ko-flok3) ½ig2d(~)2C°k25rie (ko +it/sign k3) 2 '

we get 1 oo

2 (4r0O~F(o9_ 1 ) d x ( 1 - x 2 ) °~-2 dx x2~o-3 --1 0

(1-coscoLlX(l+flox)+isincoLllc(l+floX) 1-coscoLllC(1-flox)+isincoL~x(1-floX)) X (x- i t / s ign x) 2 + (x+ i~? sign x) 2

i i R72t°-2 g2t/ dx (1 - -x2) °J-2 d/l~ (/~72 ~_ ~/2) 2 + (41r)~°F(og- 1 )

--I 0

× { [ 1 - exp(--tlCoL~) cos coL~floXlc+i exp( -qcoL~ ) sin coLzfloXtC] O(--x)

+ [ 1 -- exp ( -- tlcoL 1 ) cos CoLlfloXlC- i exp ( - ~CoLl ) sin coLxfloXlC] O(x) }

icoLlg2 exp(-tlCoLl ) /172 ~o-2 j dx(1 X2) m-2

- - (4n)C°F(m - 1 ) - dr 1¢2.~. ~2 --1 0

× [ (cos coLlfloXX- i sin coLtfloXX) O( - x) + (cos coL~floXX+ i sin coL~floXtC) O(x) ] . (23)

Here the expression inside the large parentheses comes from the physical poles ko = -+ I kl • iE while the other

80


terms come from the double pole at ko = - i~/sign k3. The leading te rm of the Laurent series in o9 - 2 o f expression (22) is o f the form (g2/16zc2) [ a ( r / ) / ( 2 - o 9 ) ] where the Taylor deve lopment o f a ( t / ) is 1 +flff~ + O ( q ) . The first term, 1, originates f rom the large parentheses in (22) , while the second term, tiff 1, f rom the last integral over x and x.

The complete result for the would-be local cusp divergence in the extended ML def ini t ion of the tempora l gauge, i.e. expression (18) , reads

": l-I 1 ( 1 ) ] 1-Un2 L17 ° --(2_o9)2 + -~o[--l+7~E+21n2+lnn+ln(L,so)2--ini+~olnl+fl°~ - -2 ~ 1 + O ( 1 ) ,

o9->2, (24)

where 7E is the Euler constant and so = sinh 0o = 17o ( 1 - fl~) - 1/z. A compar i son o f the result (24) with expression (13) conveys the following message: in the tempora l gauge

with the extended ML prescr ip t ion for the non-physical poles of the gauge-field propagator , the contour renormal iza t ion constant o f a loop with a cusp does not arise f rom radia t ive correct ions o f only the immedia t e neighbourhood o f the cusp ( local ly) , as is the case in the F e y n m a n gauge. Since eq. (14) guarantees that eventual ly the F e y n m a n gauge and the tempora l gauge accounts o f the cusp renormal iza t ion constant agree in lowest order o f rad ia t ive corrections, the discrepancy between eqs. (13 ) and (24) means that, in the tempora l gauge with the extended M L prescript ion, the radia t ive correct ions involving the curved par t Fz o f the loop, which is remote f rom the cusp, are not negligible, in cont radis t inc t ion to the s i tuat ion in the Fe ynma n gauge. In this sense, non- local i ty o f renormal iza t ion o f Green ' s funct ions in the tempora l gauge w i th / t i e ex tended M L prescr ipt ion ex- tends to renormal iza t ion of cusp divergences.

I am grateful to J.C. Taylor for ini t ia t ing my interest in this subject. This work was f inancial ly suppor ted by the Minis t ry for Science and Technology o f the Republ ic o f Slovenia Gran t No. C 1-0507-106/92.

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A note on the extended Mandelstam-Leibbrandt prescription for gauge theories in temporal gauge