24 December 1998

Ž .Physics Letters B 444 1998 411–420

Crossing symmetry and infrared safety of quark-antiquarkannihilation to all orders

M. Poljsak 1ˇJozef Stefan Institute, P.O.B. 3000, SI-1001 Ljubljana, SloÕeniaˇ

Received 28 July 1998Editor: L. Alvarez-Gaume

Abstract

Ž .The question of the degree of infrared IR safety of the Bloch-Nordsieck cross-section for quark-antiquark annihilationat high energy is reconsidered by using crossing symmetry. It is argued that this symmetry guarantees that the IR divergent

Ž .part of the relevant scattering amplitudes is invariant under the inversion of the anti quark velocity b in the centre-of-massframe, i.e., under b™by1. Employing this consequence of the crossing symmetry, a fairly simple explanation is given of

ŽŽ 2.2.why the power suppression factor O 1yb of IR divergences extends from two- to all-loops order in the strong-cou-Ž 2.pling expansion, once it is known that the logarithmic term ln 1yb is absent from the IR divergent part of the

cross-section. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 11.25.Db; 11.80.Fv; 12.38.Cy; 11.55.BqKeywords: QCD; Mass-singularities; Eikonal approximation; Crossing symmetry

1. Introduction

A necessary condition for factorization of theŽ .Drell-Yan DY cross-section into an on-shell cross-

section for quark-antiquark annihilation convolutedŽ .with functions describing the anti quark distribu-

tions inside incoming hadrons is that the partonicŽ .cross-section is rendered free of infrared IR diver-

Ž .gences by the Bloch-Nordsieck BN cancellation.This cancellation is known to be incomplete in QCD:to fourth order of perturbative expansion in thestrong coupling constant g, there is a subdominant IR

w xdivergence left over 1,2 . The divergence, however,m4Ž .is suppressed by a factor of O , where m is the2s

1 E-mail: mat.poljsak@ijs.si

1'mass of the quark and s its centre-of-mass en-2

Ž 4.ergy. So, to O g the necessary condition for fac-torization of the DY cross-section is fulfilled at

m2 m2Ž Ž .. Ž Žleading O ln and next-to-leading power O lns sm2 m2. Ž .., O of the inverse of the large energy scales s

's . The question I ask in this Letter is by whatm2 m2Ž .power of modulo possible factors of ln is thes s

IR divergent part of the BN cross section for quark-antiquark annihilation suppressed at higher orders ofperturbative expansion in g.

This question has already been addressed byŽ . w xFrenkel, Gatheral and Taylor FGT 3 . They were

m2Ž .able to establish suppression by a factor of O .s

w xThe question has also been dealt with by Bodwin 4Ž . w xas well as Collins, Soper and Sterman CSS 5 .

Working to all orders of perturbation theory andassuming that the scale of hard scattering is much

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01410-5

( )M. PoljsakrPhysics Letters B 444 1998 411–420ˇ412

larger than all the masses, they showed that theleading-power DY cross-section takes the factorisedform. Factorization of the DY cross-section at lead-ing power was proved to all orders also in a model

w xfor the incoming hadrons 6 .w xFGT 3 expressed their belief that stronger sup-

m4Ž .pression, O , of IR divergences in the BN cross-2s

section for quark-antiquark annihilation occurs to allw xorders. Basu, Ramalho and Sterman 7 suggested

that not only do IR divergent contributions to the DYcross-section, but also IR finite contributions from

m2Ž .spectators, factorise at next-to-leading power, O ,s

as they do at leading power. Neither statement wassubstantiated, however, in these references.

Reasoning based on coherent state representationw x8,9 of asymptotic states was also offered to suggestnext-to-leading power factorization, but it seems to

Ž .be inconclusive for two reasons: i it is doubtfulwhether the constructed initial states represent physi-

Ž w x. Ž .cal reality cf., e.g., Ref. 10 ; ii the scatteringamplitudes were expressed in terms of the quark

Žvelocity b as seen in the laboratory frame whichLm4 m2Ž .goes like 1yO as ™0, in contradistinction2s s

to the quark velocity in the centre-of-mass frame,m2 m2Ž . .b'b s1yO , ™0 without paying dueC M s s

attention to the possibility that the scattering ampli-Žtude may not be an analytic function of b I willL.discuss this problem in more detail in Section 2 .

An all-orders argument as to why the program offactorization can be extended to the level of next-to-

Žleading power but not to next-to-next-to-leading.one in many cases of inclusive cross-sections, in-

cluding the DY one, was given by Qiu and Stermanw x11 . Their argument implies that the answer to thequestion posed in the opening paragraph of this

Ž 4.Letter extends from O g to all orders. The authorsadmit, however, that they ’’make no pretense ofrigor’’, but believe that their ’’arguments are reason-ably convincing, and compare favorably to the all-orders understanding of leading-power factorizationin the 1980’s’’. So, there is still room for improve-

Žment upon their line of argument based mainly on.an analysis of the Landau equations , or for trying to

explore alternative avenues to the same conclusions.It is the purpose of this Letter to point out the

possibility of an alternative all-orders proof of thestatement that IR divergences in the BN cross-sec-tion for quark-antiquark annihilation at high energy

m4Ž .are suppressed by a factor of O . Barring niceties2s

of all-orders arguments, the argument presented be-low provides some grounds to believe that the expec-

w xtations of FGT 3 are firmly founded. It does notsubstitute the entire argument of Qiu and Stermanw x Ž11 or the suggestion of Basu, Ramalho and Ster-

w x.man 7 , but only so far as the power of suppressionof IR divergences in the partonic cross-section isconcerned. Also, I do not claim that my approach tothe question is more rigorous than the treatment byQiu and Sterman, but it does employ completelydifferent tools.

The question of the degree of IR safety of quark-antiquark annihilation, in addition to being interest-ing for an exact proof of the DY factorization theo-rem, is worth reexamining also because of recent

w xadvances in renormalon techniques 12–15 . Namely,it has been shown that single IR renormalon correc-tions contribute to the DY cross-section at

k'ŽŽ . . w xO L r s with ks2 13 , i.e., at the sameQC D

power where non-perturbative corrections, which aresignalled in perturbation theory via uncancelled IR

w xdivergences 16 , would intervene if the quark-masssuppression factor of these IR divergences were not

m4 m2Ž . Ž .of O but only of O . Setting aside the possi-2s s

bility that further renormalon corrections reduce thew xpower k from 2 to 1 14 , the IR divergence already

at the next-to-leading power would thus indicatenon-perturbative contributions from the non-vanish-ing quark masses that would need to be taken intoaccount in any systematic attempt at correlating ex-

w xperimental data phenomenologically 15 .The principal instrument which I shall use to

reconsider the question of the degree of IR safety ofthe BN cross-section for quark-antiquark annihilationat high energy is crossing symmetry. In the follow-ing section, I shall give the motive behind my deci-sion to think about this symmetry in answering thequestion. In Section 3, I shall demonstrate the powerof crossing symmetry by using it to reproduce, in arather economical way, the known result that the BNscenario fails to enact cancellation of IR divergencesin a complete sum of Feynman diagrams to two-looporder. In this derivation, and elsewhere in this Letter,I shall use the eikonal approximation to describeemission and absorption of soft gluons byŽ .anti quarks. In Section 4, I shall give reasons insupport of why crossing symmetry guarantees that

( )M. PoljsakrPhysics Letters B 444 1998 411–420ˇ 413

any quantity constructed out of eikonalized Feynmandiagrams with two external quark lines in such a waythat it is homogeneous in quark momenta of degree

Ž .zero is invariant under the inversion of the anti quarkvelocity b in the centre-of-mass frame, i.e., underb™by1. In Section 5, I shall explain why the

m4Ž .power of quark-mass suppression O extends from2s

two-loop to all-loops order in the strong couplingm2

expansion, once it is known that there are no ln s

terms present in the IR divergent part of the BNcross section for quark-antiquark annihilation. I willclose the Letter, in Section 6, with a few commentson the significance and limitations of the presentedarguments.

2. Dependence of the IR divergent part of the BNcross section for qq annihilation on the quarkvelocity

Ž 4.At two-loop order, O g , the IR divergent partof the BN cross-section for quark-antiquark annihila-tion into a lepton pair plus possibly at most two softgluons,

qqq™ llq llqsoft gluons, 2.1Ž .

w xtakes the form 2

1 1 1qbLIR 2s ss a C C y1 ln y1BN B S F A ž / ž /b 2b 1ybL L L

=1

. 2.2Ž .dy4

g 2Here a s , C and C are the Casimir operatorsS F A

4p

Ž .of the group SU 3 in fundamental and adjoint repre-sentation, respectively, d is the dimension of space-time continued analytically into the complex half-plane Re d)4 in order to regularise IR divergences.

Ž .The first factor on the r.h.s. of Eq. 2.2 is the BornŽ .cross-section for the process 2.1 ,

2 2 2 24pa sy4m 2m 2m ll2s sQ 1q 1q ,)B 2 ž / ž /3s s ssy4m ll

2.3Ž .

where Q is the ratio of the quark electric charge tothe charge of the electron, e, a is the electromag-

e2 X 2Ž . Ž .netic fine-structure constant as , ss pqp4p

is the square of the c.m. energy of the incomingquark having momentum p and antiquark havingmomentum pX, and m is the mass of the outgoingllŽ . IRanti lepton. The only other variable on which sBN

depends is the velocity of the antiquark, b , given inLŽthe frame of reference where the quark is at rest I

shall call it laboratory frame, which is the reason for.the subscript .L

In search of the degree of IR safety, we areinterested in the dependence of s IR on the quarkBN

mass. This dependence is hidden in s , alreadyBŽ .given explicitly in Eq. 2.3 , and in

24m1y( 4 64m ms

b s s1y qO ,L 2 2 3ž /m s s1y2

s

m2

™0. 2.4Ž .s

It is because of the factor 1rb y1 on the r.h.s. ofŽ .LŽ . Ž .Eq. 2.2 and the small mass or large energy limitŽ Ž .. IRof b Eq. 2.4 that s is suppressed by a factorL BN

m4Ž .of O as the quark mass tends to zero while s is2s

fixed.The same degree of IR safety would persist to

higher orders if s IR were given by a sufficientlyBNŽ .regular function of 1yb . To get an idea of whatL

should be understood by ’’sufficiently regular’’, ob-IR 2(serve that a term in s proportional to 1yb isBN L

m2 m2Ž .not acceptable, because it is of O as tends tos s

0. Since this term is not an analytic function ofŽ .1yb , one clue to the expected regularity is theL

analytic structure of s IR as a function of b .BN L

In order to learn something about the analyticproperties of s IR we may ask what should it lookBN

Ž .like as a function of the anti quark velocity in the24m(c.m. frame of reference, bs 1y . This veloc-s

Žity is related to b by the formula b s2br 1qL L2 .b . It is characteristic of this formula that its r.h.s.

is invariant under the inversion

1b™ . 2.5Ž .

b

( )M. PoljsakrPhysics Letters B 444 1998 411–420ˇ414

Ž . Ž . ŽFig. 1. Real a and virtual b gluon contributions to the maximally non-abelian part of the cross-section for annihilation of a quark having. Ž . Ž .momentum p and an antiquark momentum p’ into a virtual photon denoted by a circle . The Coulomb propagator of a gluon is

Ž . Ž .represented by a broken line, the transverse propagator by a curly, looping line. The right-hand parts of graphs a and b denote thecomplex conjugate of their pictures in a vertical mirror.

The connection between this transformation and ana-Ž .lyticity is that the inversion 2.5 is one of the

Ž .conformal more precisely, homographic transfor-mations of the complex b-plane. The question nowarises: Should s IR as a function of b be invariantBN

Ž .under the transformation 2.5 ?Ž .As judged from the behaviour under 2.5 of the

2 IR(undesired term 1yb in s , namelyL BN

1b™2 2

b1yb b y12 2( (1yb s ™ sy 1yb ,L L2 21qb b q1

2.6Ž .

Ž .as well as from the symmetry under 2.5 of the realŽ .part of the r.h.s. of Eq. 2.2 , the answer to the

question should probably be in the affirmative. In thefollowing, I am going to argue in favour of thissuggestion on the grounds of the crossing symmetryof scattering amplitudes. In order to do this, I shall

Ž .need some details from a derivation of Eq. 2.2 ,which I give in the following section.

3. Crossing symmetry and presence or absence ofIR divergences

IR Ž 4.There are several ways to calculate s at O gBNw x1,2,17 . For the purpose of this section, a most

w xconvenient approach to follow is that of Ref. 2 . It

uses the Coulomb gauge defined in the laboratoryŽ .system where the quark is at rest, ps m,0 , and

calculates Feynman diagrams directly. Compared toother possibilities, this choice simplifies the calcula-tion a lot. For example, in the non-relativistic limitwhen the velocity of the antiquark tends to zeroŽ .b ™0 there are only two diagrams depicted onL

Fig. 1 to deal with.Using standard Feynman rules together with the

eikonal approximation and averaging over spins ofthe quark and the antiquark, we can express thegraph on Fig. 1a algebraically in the form

AA s2ie2 g 4b 2 T T T T pPpX q2m2Ž .ei k L a b a b

=d3k d4kX

21yxŽ .H H3 4 2X< < < <2p k 2p kŽ . Ž .

=1 1

e eX X Xk yK q i yk q iX0 0p p0 0

=1

e< <y k qKy i Xp0

=1

, 3.1Ž .eX X < <k yK y k qKq i X0 p0

( )M. PoljsakrPhysics Letters B 444 1998 411–420ˇ 415

where T are the generators of the gauge groupaŽ .SU 3 and the following definitions have been used

< X < X X Xp k Pp kPpX < <b s , K s , Ks 'b x k ,X X XL L< <p p p0 0 0

y1-xscosu-1 . 3.2Ž . Ž .

The value of the diagram on Fig. 1b, using theeikonal approximation and averaging over spin statesof q and q, is

BB s2 e2 g 4 T T T T pPpX q2m2 p pXŽ .ei k a b a b 0 0

=

X4 4d k d k 1 2Xy pŽ .H H4 4 22X k q ie< <2p 2p kŽ . Ž .2Xp Pk 1 1Ž .

q X X X2 p Pk q ie ypPk q ie< <k

=1 1

. 3.3Ž .X X X Xp Pkq ie p Pk qp Pkq ie

Ž .If on the r.h.s. of Eq. 3.3 we perform integrationover k by the calculus of residues, completing the0

integration contour by a large semi-circle in thelower half of the complex k -plane, we get0

BB s2 i e2 g 4b 2 T T T T pPpX q2m2Ž .ei k L a b a b

=d3k d4kX

21yxŽ .H H3 4 2X< < < <2p k 2p kŽ . Ž .

=1 1

e eX X Xk yK q i yk q iX0 0p p0 0

=1

e< <y k yKq i Xp0

=1

. 3.4Ž .eX X < <k yK y k yKq i X0 p0

Ž . Ž .In the transition from 3.3 to 3.4 we have againŽ .used definitions 3.2 .

Changing the integration variable k into yk inŽ .3.4 and then noticing that the imaginary part in the

< < Xdenominators y k qK. ierp , where the upperŽ .0

sign refers to AA and the lower one to BB , doesei k ei k

not affect further integrations as far as IR divergentterms are concerned, we see from a comparison

Ž . Ž .between Eqs. 3.1 and 3.4 that

AA qAA ) sBB qBB) . 3.5Ž .ei k ei k ei k ei k

1IRTherefore, s , which is proportional toBN bL) ) ŽAA qAA qBB qBB the remaining factorŽ .ei k ei k ei k ei k

involves the Mandelstam variable s and leptonic.parameters only , is non-zero because either side of

Ž . Ž .Eq. 3.5 is non-zero in fact, IR divergent .Now, we shall verify that crossing of both the

Ž .quark and the antiquark line with ll ll lines whileleaving gluons intact has as a consequence that thereare no IR divergences in the BN cross-section for thereaction

llq ll™qqqqsoft gluons. 3.6Ž .Ž .The double crossing transformation is effected by

p™yp and pX™ypX . 3.7Ž .

Ž X .This transformation maps AA p, p into AAei k ei kŽ X . Ž . Ž .yp,yp . From Eq. 3.1 and definitions 3.2 wecan see that the sole effect of the crossing transfor-mation is to reverse the signs of imaginary parts of

Ž X.all denominators in AA p, p . Because of the fac-ei kŽtor i written as the second factor on the r.h.s. of Eq.

Ž ..3.1 , we can therefore write

AA yp ,ypX syAA ) p , pX 3.8Ž . Ž . Ž .ei k ei k

and consequently

AA yp ,ypX qAA ) yp ,ypXŽ . Ž .ei k ei k

X X)sy AA p , p qAA p , p . 3.9Ž . Ž . Ž .ei k ei k

On the other hand, the crossing transformationŽ . Ž Ž ..3.7 does not change the value of BB Eq. 3.4 .ei k

This can most easily be seen by starting from Eq.Ž . X X3.3 : The change of p and p into yp and yp ,respectively, together with the change of integrationvariables k™yk, kX

™ykX does not affect anyŽ .factor in Eq. 3.3 . Therefore,

BB yp ,ypX qBB) yp ,ypXŽ . Ž .ei k ei k

sBB p , pX qBB) p , pX . 3.10Ž . Ž . Ž .ei k ei k

( )M. PoljsakrPhysics Letters B 444 1998 411–420ˇ416

Ž . Ž . Ž .As a consequence of Eqs. 3.9 , 3.10 and 3.5 ,the IR divergent part of the BN cross-section forannihilation of a lepton pair into a quark, an anti-quark and possibly soft gluons, s IR , is vanishing:˜BN

1X XIRs A AA yp ,yp qBB yp ,ypŽ . Ž .˜BN eik ei k

bL

1X X

)qc.c. ss yAA p , p yAA p , pŽ . Ž .ei k ei kbL

X X)qBB p , p qBB p , pŽ . Ž .ei k ei k

1X X

)s yAA p , p yAA p , pŽ . Ž .ei k ei kbL

X X)qAA p , p qAA p , p s0. 3.11Ž . Ž . Ž .ei k ei k

4. Crossing symmetry implies symmetry underthe inversion of quark velocity

Ž .From the expressions for AA , Eq. 3.1 , BB ,ei k ei kŽ . Ž .Eq. 3.4 , and s , Eq. 2.3 , it is evident that theB

ratio s IRrs is a homogeneous function of theBN B

quark and antiquark momenta of degree zero if oneignores imaginary parts of denominators. Furthersteps in the argument for the symmetry of s IRrsBN B

under the inversion of quark velocity will relystrongly on this fact.

Ž . Ž .In deriving Eqs. 3.1 and 3.4 , I have employedthe Coulomb gauge. On the other hand, if one wishesto establish the analytic properties of individual dia-grams, one better uses a covariant gauge, becauseonly then it might be simple to discern these proper-ties. It is therefore important to notice that individualcontributions to s IRrs in a covariant gauge areBN B

also homogeneous of degree zero. For example, theexpressions for the contributions of Figs. 1a and 1b,with the broken line now representing the gluonpropagator in a covariant gauge, differ from the

Ž . Ž .expressions AA , Eq. 3.1 , BB , Eq. 3.4 , as far asei k ei k

their dependence on the momenta p and pX in thelaboratory system is concerned, only in the absence

2 Ž 2 .of the factors b and 1yx .L

So far momenta of the quark and the antiquarkwere given in the laboratory system. For the discus-sion in this section it will be more appropriate to usethe c.m. system. In this frame and in a covariant

gauge, there are two further differences in how theŽ .expressions corresponding to AA , Eq. 3.1 , andei k

Ž . X Ž .BB , Eq. 3.4 , depend on p and p : i The denomi-ei kX enator yk q i is replaced by the denominator0 p0

X X ek P p Ž . Ž .yk y q i ; ii The parameter b in Eqs. 3.20 Lp0p0

Ž .is replaced by b sb .C M

It is characteristic of the expressions so obtainedthat their dependence on p and pX is contained onlyin the parameter

< < < X <p pbs s 4.1Ž .X< < < <p p0 0

eand in the imaginary parts of denominators, i andp0e Ž .Xi . The values of the components of on-shellp0

X Ž .momenta p and p are such that bg 0,1 and theimaginary parts of denominators are positive.

The question I would like to answer now is howto extend s IRrs as a function of p and pX fromBN B

the domain described above to the domain obtainedŽ . X Xby the single crossing transformation p ™yp ,Ž X Xp™p. The transformation p ™p , p™yp would

.be equally good for our purpose.There are two ways to do this: One can

Ž .a replace the parameter b in the expression fors IRrs , by the formulaBN B

24mbs 1y , 4.2Ž .) 2XpqpŽ .

Ž .which is equivalent to Eq. 4.1 on the originaldomain of values of p and pX, and then applythe crossing transformation referring to the mo-mentum of the antiquark, i.e., pX

™ypX. As aŽ .result, the r.h.s. of Eq. 4.2 is transformed into

2X2(1y4m r pyp . This expression satisfies,Ž .when momenta p and pX are on-shell, the fol-lowing algebraic identity

11 y2 2 24m 4m2

1y s 1y2 2X Xpyp pqpŽ . Ž .4.3Ž .

( )M. PoljsakrPhysics Letters B 444 1998 411–420ˇ 417

1whose r.h.s. equals . Crossing transformationb

also changes the signs of the imaginary partsierpX of denominators.0

X e< < < <p pŽ .b express the integrands as functions of , , i p0X< < < <p p0 0eXand i in the same way as in the originalp0

domain, only now allowing the parameter pX to0

take negative values and lifting the constraints< < < X <p p-1, -1.

X< < < <p p0 0

When these two possible ways of analytic contin-uation of s IRrs are considered together with theBN B

general theorem of complex analysis that analyticcontinuation of a function is unique, if it exists, wecan draw the following conclusion: The ratios IRrs , considered now as a function of b takingBN B

Ž .values on the interval 0,` , is even under the trans-Ž .formation 2.5 , because the two proposed analytic

extensions must coincide.The conclusion just reached is a crucial point in

this Letter, so it deserves a few comments:1. The argument that led to the symmetry under the

inversion of quark velocity was based on theŽ .principle of analytic continuation in anti quark

momentum which is the basis also of crossingsymmetry. I have supposed that there are noobstacles to analytic continuation of the IR diver-gent parts of the BN cross-sections for the pro-cesses under discussion. In particular, I have as-sumed that the two paths, along which it wasproposed to make the extension of the originalfunction s IRrs , both lie within the same un-BN B

ramified Riemann surface over the complexŽ X. Žp, p space, containing the original domain acondition for uniqueness of an analytic extensionw x.18 . There is only circumstantial evidence tosuggest that the assumption is reasonable, as ex-emplified by the calculation reported in the previ-ous section, but I have no proof.

Ž 4.2. The argument presented to O g in favour ofŽthe statement that individual contributions of

pairs of Feynman diagrams or partial sums.thereof to the IR divergent part of the BN cross-

Ž . Ž . Ž .section for processes 2.1 , 4.4 and 3.6 , withthe Born cross-section factored out, are invariantunder the inversion of the quark velocity in thec.m. frame depended upon the fact that they arehomogeneous functions of quark momenta of or-

Ž .der zero for ds4 . It is important for an all-

orders argument to recall that the condition ofhomogeneity is fulfilled also to higher orders ofperturbative expansion in the strong coupling con-

Žstant because of eikonal scaling cf., e.g., Ref.w x.3 .

3. The conclusion literally is that the IR divergentpart of the BN cross-section for for the process

qq ll™qq llqsoft gluons, 4.4Ž .

divided by its Born value, s IRrs , obtained as˜BN B

an analytic continuation of s IRrs for the pro-BN BŽ .cess 2.1 , has to be symmetric under the transfor-

Ž . Ž .mation 2.5 . This is trivially correct for a com-plete sum of diagrams, because the BN cross-sec-

Ž .tion for the process 4.4 is known to be IR finitew x19 . In a covariant gauge, however, the conclu-sion presumably applies also to contributions of

Žindividual Feynman diagrams cf., in this respect,.remark 4. .

Since one can always retrieve the original func-tion from its analytic extension, one can apply thesame argument as before to establish that the ratio

IR Ž .s rs for the process 2.1 must also enjoy theBN BŽ .symmetry under 2.5 .

By an extension of the argument, involving thecrossing transformation in the other momentumvariable, p™yp, one can conclude that the ratioof the IR divergent part of the BN cross-section

Ž .for the process 3.6 and the corresponding Borncross-section also has to be even under the inver-

Ž .sion of the anti quark velocity in the c.m. frame.This result is obviously correct for the sum of alldiagrams contributing at a given order due to theabsence of IR divergences in the BN cross-section

w xfor leptoproduction of a quark-antiquark pair 20 .Ž 4.4. As a check on the proposed symmetry to O g ,

Ž .we can verify from Eq. 2.2 that the IR divergentpart of the BN cross-section for the the processŽ . Ž2.1 more precisely, the real part of

IR Ž .. Ž .s rs b is indeed invariant under 2.5 . Fur-BN B

thermore, we can avail ourselves of the calcula-tion performed to the precisely required detail in

w xRef. 21 to verify the symmetry also in the caseŽof scattering of quarks off virtual photons pro-

Ž ..cess 4.4 . Although the published results forw xdiagrams on Fig. 2 in Ref. 21 do not seem to beŽinvariant under b™1rb in the notation used in

( )M. PoljsakrPhysics Letters B 444 1998 411–420ˇ418

w xRef. 21 , the required symmetry transformation.reads r™1rr at first sight, when misprints are

corrected 2 the symmetry is actually restored.

5. How symmetry under the inversion of quarkvelocity constrains the degree of IR safety

Now, I would like to explore whether or not thesymmetry under the inversion of quark velocity inany way restricts the quark mass dependence of theIR divergent part of the BN cross section for quarkantiquark annihilation. Because in this cross-sectionthe quark mass always appears together with thesquare of the c.m. energy of the quark-antiquark

4m2

pair, s, in the combination zs , from here on-s

wards I shall trace the z-dependence of the cross-sec-tion instead of its m-dependence.

I shall assume that the IR divergent part of theBN cross-section for quark-antiquark annihilation in

2 As an example of typographical error, notice that the value ofthe integral

esinv d sinvp Ž . Ž .

4.5Ž .H1ycosv0

2w x Ž Ž . .is quoted in Ref. 21 just before Eq. 3.4 there to be q

e2ln2, e ™0, while in fact it equals minus this value. When thischange in sign is taken into account, some terms in the expressionfor the sum of diagrams on Figs. 2e and 2f also change sign sothat the correct expression reads

4 2 eg 4p DŽ . e

ds sy ds y C Cc B A F2 2 2e12pŽ .G 1q ež /2

=

2 22 1q r 1q r 1q ry y1q2ln2 ln y1 yž /½ e 2 r 1y r 4r

=1q r 1 1q r 1y r 2

2ln 1y r ln y ln q2ln2qŽ . ž / ž /1y r r 1y r 2 r

=1q r 1 ln 1y rxŽ .12ln y1y 1q r dxŽ .Hž /1y r 2 1q rx0

2ln 1q rx ln 1y xŽ . Ž .12q q 1q r dx 4.6Ž . Ž .H 2 2 51y rx 1y r x0

which can be verified, using the analytic properties of the diloga-1rithm function, to be invariant under r ™ .r

general takes the following form as far as z-depen-dence is concerned:

IR `sBN n 2z sRe z c qc ln zqc ln z ,Ž . Ž .Ý 0 n 1n 2 nsB ns0

c s0. 5.1Ž .00

This particular form is suggested by the knownz-dependence of low-order contributions of pairs of

w xFeynman diagrams 1–3,22 , as well as by the Frois-sart bound which restricts any cross-section to growwith increasing s not faster than ln2s. The valuec s0 summarizes the result that there is no IR00

divergence present in the case of massless quarksw x23 . I shall also assume that the coefficients c ,k n

ks0,1,2, are real, because the cross-section has tobe real.

Ž .From the dependence of the anti quark velocityin the c.m. system on the quark mass, b

2 '(s 1y4m rs s 1yz , we can infer that the in-Ž . Ž .version of the anti quark velocity, Eq. 2.5 , is

equivalent to the following transformation of theparameter z:

zz™y . 5.2Ž .

1yz

IR Ž .The requirement that s rs z should be invariantBN BŽ .under the transformation 5.2 can therefore be ex-

pressed by the equation:

s IR s IR zBN BNz s y . 5.3Ž . Ž .ž /s s 1yzB B

IR Ž .Since we are interested in the structure of s rs zBN BŽ .only for small values of z i.e., high energies , we

Ž .follow the z-dependence of the r.h.s. of Eq. 5.3Ž . Ž 2 . Žonly up to and including terms of O z modulo

2 .logarithmic factors of ln z and ln z , i.e.,

z™yzyz 2 q ...

12Re ln z ™Re ln z qzq z q ...Ž . Ž .

2

Re ln2 z ™Re ln2 z q 2 zqz 2 Re ln zŽ . Ž . Ž . Ž .

yp 2 qz 2 q ... 5.4Ž .

( )M. PoljsakrPhysics Letters B 444 1998 411–420ˇ 419

Ž .Eq. 5.3 introduces, to the accuracy specified byŽ .relations 5.4 , the following constraints on the coef-

Ž .ficients c in Eq. 5.1 :k n

0syp 2c20

c syc qc qp 2c01 01 10 21

12 2c sc yc q c qc qp c yp c02 02 01 10 20 21 222

c sc10 10

c syc q2c11 11 20

c syc qc qc y2c12 11 12 20 21

c sc20 20

c syc21 21

c sc yc . 5.5Ž .22 22 21

From the known absence of the ln z term inIRŽ . w xs z to all orders 3 , which in our notationBN

amounts to the condition c s0, and from Eqs.10Ž .5.5 it follows that the coefficients c , c , c , c01 11 20 21

and c have to vanish, while the coefficients c22 12

and c are unconstrained.02

This result makes it necessary that s IR is sup-BNm4Ž .pressed at high energies by a factor of O to all2s

orders. Notice that we deduced also vanishing of thecoefficient c to all orders; it is indeed absent from22

Ž Ž ..the cross-section to two-loop order see Eq. 2.2 .

6. Discussion

In this Letter it was pointed out that there isanother role for the crossing symmetry to play inQCD: For the IR divergent part of the physicalŽ .according to BN cross-section for quark-antiquark

Žannihilation into a lepton pair and processes relatedto this one by crossing either the quark line or the

.antiquark line or both lines with lepton lines thesymmetry implies that the cross-section divided bythe Born cross-section is invariant under the inver-sion of the quark velocity in the c.m. frame. As aconsequence of this, it was possible to deduce thatthe IR divergent part of the BN cross section forquark-antiquark annihilation is suppressed to all or-ders by the same power of the quark mass as attwo-loop order. This result verifies, by means that

w xare completely different from those used in Ref. 11 ,that a necessary condition for factorization of the

cross-section for Drell-Yan process at next-to lead-ing power is satisfied to all orders.

Crossing symmetry of the S-matrix is a generalresult of quantum theory, not merely a property ofFeynman diagrams. This means that the implicationof the crossing symmetry discussed here applies,unless there are some snags in the perturbative argu-ment, even beyond perturbation theory.

Because the argument in Section 5 relies on theŽvalidity of leading power suppression established in

w x w x.Ref. 3 and in Ref. 5 , the same limitations thatrelate to those proofs apply here too.

The argument for suppression of remaining IRdivergences in the BN cross-section for quark-anti-quark annihilation by the inverse fourth power of thelarge energy scale also relies, in an essential way, onthe use of the eikonal approximation. Although somepeople prefer to avoid this approximation, e.g., the

w xauthors of Ref. 5 , it is usually believed to be validfor detecting IR divergences.

The use of crossing symmetry required an analy-Ž .sis of functions of two complex vector variables

Ž .quark and antiquark momenta , which is notoriouslya difficult subject. In my analysis, I have assumedthat I had correctly traced the analytic structure of IRdivergences as functions of these variables from astudy of lowest-order examples. Although there maystill be technical difficulties in developing this sideof the argument in all its aspects, I feel, in view ofthe results obtained in this Letter, that the argumentis adequate to justify further study.

Acknowledgements

I acknowledge the financial support of the Min-istry for Science and Technology of the Republic ofSlovenia through Project No. J1-7462-106-98.

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