Double logarithms resummation in exclusive processes : the surprising behavior of DVCS
aa r X i v : . [ h e p - ph ] S e p Double logarithms resummation in exclusiveprocesses : the surprising behavior of DVCS ∗ T. Altinoluk , B. Pire , L. Szymanowski and S. Wallon & UPMC Univ. Paris 06, facult´e de physique, 4 place Jussieu,75252 Paris Cedex 05, France
July 31, 2018
Abstract
Double logarithms resummation has been much studied in inclusiveas well as exclusive processes. The Sudakov mechanism has often be thecrucial tool to exponentiate potentially large contributions to amplitudesor cross-sections near phase-space boundaries. We report on a recent workwhere a very different pattern emerges : the DVCS quark coefficient func-tion C q ( x, ξ ) develops near the particular point x = ξ a non-alternate se-ries in α ns log n ( x − ξ ) which may be resummed in a cosh[ K √ α s log( x − ξ )]factor. This result is at odds with the known result for the correspondingcoefficient function for the pion transition form factor near the end point C q ( z ) although they are much related through a z → x/ξ correspondence.Preprint numbers: CPhT-PC-081-0813; LPT-ORSAY-13-62 While perturbative calculations are widely used in quantum field theory, theirsummation is always a formidable task, unreachable but in the simplest, notto say most simplistic, occurences. From elementary particle to atomic and tosolid state physics, resummation techniques have been developped to go beyonda fixed order perturbation estimate through the sampling and evaluation of an ∗ Presented at the Low x workshop, May 30 - June 4 2013, Rehovot and Eilat, Israel − Kg log z ), where g isthe coupling constant and z is the ratio of two different characteristic scales,which governs both QED and QCD calculations of exclusive form factors. Asexplained in detail in [2] we obtain for a specific case of exclusive scatteringamplitude, namely the deeply virtual Compton scattering in the generalizedBjorken regime, a very different resummed result of the form cosh( Kg log z )where z is a momentum fraction. To our knowledge, this form never previouslyemerged in field theoretic calculations. The process that we focus on is themost studied case of a class of reactions - exclusive hard hadronic processes -which are under intense experimental investigation. The result presented hereprovides an important stepping-stone for further developments enabling a con-sistent extraction of the quantities describing the 3-dimensional structure of theproton.In the collinear factorization framework the scattering amplitude for ex-clusive processes such as deeply virtual Compton scattering (DVCS) has beenshown [3] to factorize in specific kinematical regions, provided a large scale con-trols the separation of short distance dominated partonic subprocesses and longdistance hadronic matrix elements, the generalized parton distributions (GPDs)[4]. The amplitude for the DVCS process γ ( ∗ ) ( q ) N ( p ) → γ ( q ′ ) N ′ ( p ′ ) , (1)with a large virtuality q = − Q , factorizes in terms of perturbatively calculablecoefficient functions C ( x, ξ, α s ) and GPDs H ( x, ξ, t ), where the scaling variablein the generalized Bjorken limit is the skewness ξ defined as ξ = Q ( p + p ′ ) · ( q + q ′ ) .The calculation of first order perturbative corrections to the partonic amplitudehas shown that terms of order log ( x ± ξ ) x ± ξ play an important role in the region ofsmall ( x ± ξ ) i.e. in the vicinity of the boundary between the domains wherethe QCD evolution equations of GPDs take distinct forms (the so-called ERBLand DGLAP domains). We scrutinize these regions and demonstrate that theyare dominated by soft fermion and gluon propagation. This explains why theycan be exponentiated using quasi-eikonal techniques. To set up our notations, let us remind the reader of the known results forthe NLO corrections to the DVCS amplitude (1), specializing to the quarkcontribution to its symmetric part. After proper renormalization, it reads A µν = g µνT Z − dx " n F X q T q ( x ) H q ( x, ξ, t ) , (2)2here the quark coefficient function T q reads [5] T q = C q + C q + C qcoll log | Q | µ F , (3) C q = e q (cid:18) x − ξ + iε − ( x → − x ) (cid:19) , (4) C q = e q α S C F π ( x − ξ + iε ) (cid:26) log ( ξ − x ξ − iε ) − − ξ − xξ + x log (cid:18) ξ − x ξ − iǫ (cid:19)(cid:27) − ( x → − x ) . (5)The first (resp. second) terms in Eqs. (4) and (5) correspond to the s − channel(resp. u − channel) class of diagrams. One goes from the s − channel to the u − channel by the interchange of the photon attachments. Since these two con-tributions are obtained from one another by a simple ( x ↔ − x ) interchange, wenow restrict mostly to the discussion of the former class of diagrams.Let us first point out that in the same spirit as for evolution equations, theextraction of the soft-collinear singularities which dominate the amplitude inthe limit x → ± ξ is made easier if one uses the light-like gauge p · A = 0 with p = q ′ . We argue (and verified) that in this gauge the amplitude is dominatedby ladder-like diagrams. We expand any momentum in the Sudakov basis p , p , as k = α p + β p + k ⊥ , where p is the light-cone direction of the twoincoming and outgoing partons ( p = p = 0, 2 p · p = s = Q / ξ ). In thisbasis, q γ ∗ = p − ξ p .We now restrict our study to the limit x → + ξ . The dominant kinematicsis given by a strong ordering both in longitudinal and transverse momenta,according to (see Figure 1) : x ∼ ξ ≫ | β | ∼ | x − ξ | ≫ | x − ξ + β | ∼ | β | ≫ · · ·· · · ≫ | x − ξ + β + β + · · · + β n − | ∼ | β n | , (6) | k ⊥ | ≪ | k ⊥ | ≪ · · · ≪ | k ⊥ n | ≪ s ∼ Q , (7) | α | ≪ · · · ≪ | α n | ≪ . (8)This ordering is related to the fact that the dominant double logarithmic con-tribution for each loop arises from the region of phase space where both softand collinear singularities manifest themselves. When x → ξ the left fermionicline is a hard line, from which the gluons are emitted in an eikonal way (whichmeans that these gluons have their all four-components neglected in the vertexw.r.t. the momentum of the emitter), with an ordering in p direction and acollinear ordering. For the right fermionic line, eikonal approximation is notvalid, since the dominant momentum flow along p is from gluon to fermion,nevertheless the collinear approximation can still be applied.When computing the coefficient functions, one faces both UV and IR di-vergencies. On the one hand, the UV divergencies are taken care of throughrenormalization, which manifest themselves by a renormalization scale µ R de-pendency. On the other hand, the IR divergencies remain, but factorization3 Sfrag replacements x − ξ + β + · · · + β n ,k ⊥ + · · · + k ⊥ n p − ξ p p β n , k ⊥ n β n − , k ⊥ n − β , k ⊥ x + ξ + β + · · · + β n ,k ⊥ + · · · + k ⊥ n x + ξ + β + · · · + β n − ,k ⊥ + · · · + k ⊥ n − x + ξ + β , k ⊥ ( x + ξ ) p x − ξ + β + · · · + β n ,k ⊥ + · · · + k ⊥ n x − ξ + β + · · · + β n − ,k ⊥ + · · · + k ⊥ n − x − ξ + β , k ⊥ ( x − ξ ) p Figure 1:
The ladder diagrams which contribute in the light-like gauge to the leading α ns ln n ( ξ − x ) / ( x − ξ ) terms in the perturbative expansion of the DVCS amplitude. The p and ⊥ momentum components are indicated. The dashed lines show the dominantmomentum flows along the p direction. proofs at any order for DVCS justify the fact that they can be absorbed insidethe generalized parton distributions and result in finite coefficient functions. Inour study, we are only interested into finite parts. Thus, using dimensionalregularization, in a factorization scheme like M S , any scaleless integral can besafely put to zero although it contains both UV and IR divergencies. Followingthis line of thought, we can thus safely deal with DVCS on a quark for ourresummation purpose.Finally, the issue related to the iǫ prescription in Eq. (5) is solved by com-puting the coefficient function in the unphysical region ξ >
1. After analyticalcontinuation to the physical region 0 ≤ ξ ≤
1, the final result is then obtainedthrough the shift ξ → ξ − iǫ .We define K n as the contribution of a n -loop ladder to the coefficient func-tion. Let us sketch the main steps of the derivation of K and then generalizeit for K n . The ladder diagram at order α s . A careful analysis [2] shows that amongthe one loop diagrams and in the light-like axial gauge p .A = 0, the box diagramis dominant for x → ξ . Starting from the dominant part of the numerator ofthe Born term which is θ = − p , the numerator of the box diagram is tr (cid:8) p γ µ [ k + ( x − ξ ) p ] θ [ k + ( x + ξ ) p ] γ ν (cid:9) d µν . (9)4n the limit x → ξ , while the left fermionic line is hard with a large p momen-tum, the gluonic line is soft with respect to the left fermionic line. So we performsoft gluon approximation in the numerator by taking k + ( x + ξ ) p → ( x + ξ ) p .The dominant contribution comes from the residue of the gluonic propagator.Thus, the numerator of the on-shell gluon propagator, d µν , is expressed in termsof transverse polarizations, i.e. d µν ≈ − P λ ǫ µ ( λ ) ǫ ν ( λ ) . Writing the gluon polarization vectors in the light-like p · A = 0 gaugethrough their Sudakov decomposition ǫ µ ( λ ) = ǫ µ ⊥ ( λ ) − ǫ ⊥ ( λ ) · k ⊥ βs p µ , (10)allows us to define an effective vertex for the gluon and outgoing quark throughthe polarization sum X λ ǫ ⊥ ( λ ) · k ⊥ ǫ µ ( λ ) = − k µ ⊥ + 2 k ⊥ βs p µ . (11)The numerator, ( N um ) , is α − independent and reads − x + ξ ) β tr (cid:26) p (cid:18) k ⊥ − k ⊥ βs p (cid:19) [ k + ( x − ξ ) p ] p (cid:27) = − x + ξ ) s k ⊥ β (cid:20) x − ξ ) β (cid:21) . (12)We now calculate the integral over the gluon momentum k , using dimensionalregularization R d d k → s R dα dβ d d − k , ( k ⊥ = − k ). The Cauchy integrationof the gluonic pole which gives the dominant contribution reads − πi s Z ξ − x dβsβ Z ∞ d d − k ( N um ) L R S (cid:12)(cid:12)(cid:12) α = k βs (13)with the denominators L = − k + α ( β + x + ξ ) s , R = − k + α ( β + x − ξ ) s , S = − k + ( β + x − ξ ) s and k = − k + αβs . The relevant region of integrationcorresponds to small | β + x − ξ | . The β and k integrations results in our finalone-loop expression : K = i e q (cid:18) − i C F α s π ) (cid:19) x − ξ πi
2! log ( a ( x − ξ )) , (14)where we kept only the most singular terms in the x → ξ region and have nocontrol of the value of a within our approximation. To fix a , we match ourapproximated one-loop result with the full one-loop result (5). This amountsto cut the k integral at Q . The iǫ term is included according to the samematching. This leads to K = i e q (cid:18) − i C F α s π ) (cid:19) x − ξ + iǫ πi
2! log (cid:18) ξ − x ξ − iǫ (cid:19) , (15)which is the known result. This is a positive test of the validity of our approxi-mation procedure that we now generalize to the n -rung ladder.5 he ladder diagram at order α ns . Let us now turn to the estimation of alllog n ( x − ξ ) terms in the diagram shown on Fig. 1. Assuming the strong ordering(7, 8) in k ⊥ and α , the distribution of the poles generates nested integrals in β i as : Z ξ − x dβ Z ξ − x − β dβ · · · Z ξ − x − β −···− β n − dβ n . (16)The numerator for the n th order box diagram is obtained as:( N um ) n = − s ( x + ξ ) n k ⊥ β (cid:20) x − ξ ) β (cid:21) k ⊥ β (17) (cid:20)
1+ 2( β + x − ξ ) β (cid:21) · · · k ⊥ n β n (cid:20)
1+ 2( β n − + · · · + β + x − ξ ) β n (cid:21) , and the denominators of propagators are, for i = 1 · · · n , L i = α i ( x + ξ ) sR i = − k i + α i ( β + · · · + β i + x − ξ ) s ,S = − k n + ( β + · · · + β n + x − ξ ) s . (18)Using dimensional regularization and omitting scaleless integrals, the integralreads: Z ξ − x dβ · · · Z ξ − x −···− β n − dβ n Z ∞ d d − k n · · · Z k d d − k ( − n (19) × s (2 πi ) n x − ξ β + x − ξ · · · β + · · · + β n − + x − ξ × k · · · k n k n − ( β + · · · + β n + x − ξ ) s . The integrals over k · · · k n are performed similarly as in the one-loop case,resulting in: K n = i e q (cid:18) − i C F α s π ) (cid:19) n x − ξ + iǫ (2 πi ) n (2 n )! log n (cid:18) ξ − x ξ − iǫ (cid:19) , (20)where the matching condition introduced in one-loop case is extended to n − loops. Based on the results Eqs. (15, 20), one can build the resummed formula for thecomplete amplitude; we get with D = q α s C F π ∞ X n =0 K n = e q x − ξ + iǫ cosh (cid:20) D log (cid:18) ξ − x ξ − iǫ (cid:19)(cid:21) (21)= 12 e q x − ξ + iǫ "(cid:18) ξ − x ξ − iǫ (cid:19) D + (cid:18) ξ − x ξ − iǫ (cid:19) − D .
6n the absence of a next to leading logarithmic calculation, the minimal andmost natural resummed formula which has the same O ( α s ) expression as thefull NLO result, reads, :( C + C ) res = e q x − ξ + iǫ (cid:26) cosh (cid:20) D log (cid:18) ξ − x ξ − iǫ (cid:19)(cid:21) − D (cid:20) ξ − xx + ξ log (cid:18) ξ − x ξ − iǫ (cid:19)(cid:21)(cid:27) − ( x → − x ) . (22) For several decades the effects of gluons on many high energy processes has beenwidely studied. Specifically the theory of ”Color Glass Condensate” shows thatat very high energies the behavior of the scattering amplitudes are dominatedby gluons [6]. Recently it was also shown that even at moderate energies,there are significant O ( α s ) corrections to scattering amplitudes due to gluoniccontributions for spacelike and timelike virtual Compton scatterings [7]. Withthe above mentioned motivations performing a similar resummation procedurefor gluon coefficient function of DVCS and TCS would result in a more trustfulextraction gluon GPDs. We have demonstrated that resummation of soft-collinear gluon radiation effectscan be performed in hard exclusive reactions amplitudes. The resulting formulafor coefficient function stabilizes the perturbative expansion, which is crucialfor a trustful extraction of GPDs from experimental data. A related expressionshould emerge in various reactions, such as the crossed case of timelike Comptonscattering [8] and exclusive meson electroproduction.Giving these results, a question should be raised : what is the physics beyondthis result, or in other words, why is the Sudakov resummation [9] familiarto experts of hard exclusive processes not applicable here ? An even moreprecise question may be : how is our analysis compatible with the discussionof soft effects in the pion transition form factor, a quantity which has beenmuch discussed [10] recently thanks to the experimental results of BABAR andBELLE? Let us stress that the coefficient function of this quantity is identicalto the ERBL part of the coefficient function of the DVCS amplitude after arescaling z → x/ξ . Our result thus may be applied to the transition formfactor. In Ref. [11], it has been argued that the α s log (1 − z ) factor in the oneloop expression of the coefficient function had to be understood as the sum oftwo very distinct terms, one of them exponentiating in a Sudakov form factor.To advocate this fact, the authors allow themselves an excursion outside thecolinear factorization framework and use the familiar detour into the coordinatespace framework. Our procedure is different and we resum the complete oneloop result. In other words, one may ask to the authors of Ref. [11] : what7appens to the remnant term proportional to α s log (1 − z )? If indeed theusual resummation procedure of the transition pion form factor must be revisedfollowing our new results, one may ask whether the understanding of the mesonform factor [12] should also be reconsidered. Acknowledgments
We thank the organisors and the French CEA (IPhT and DSM) for support.This work is supported by the P2IO consortium, the Polish Grant NCN No.DEC-2011/01/B/ST2/03915, the French grant ANR PARTONS (ANR-12-MONU-0008-01), the Joint Research Activity ”Study of Strongly Interacting Matter”(HadronPhysics3, Grant Agreement n.283286) under the 7 th Framework Pro-gramm of the European Community, the European Research Council grantHotLHC ERC-2001- StG-279579, Ministerio de Ciencia e Innovac´ıon of Spaingrants FPA2009-06867-E, Consolider-Ingenio 2010 CPAN CSD2007-00042 andFEDER.
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