While perturbative calculations are widely used in quantum field theory, their summation is always a formidable task, unreachable but in the simplest, not to say most simplistic, occurences. >From elementary particle to atomic and to solid state physics, resummation techniques have been developed to go beyond a fixed order perturbation estimate through the sampling and evaluation of an infinite class of diagrams shown to dominate in a limited kinematical region for some given observables. The result of such procedures is often an exponentiated factor as in the famous Sudakov case [1], exp(-Kg₂log₂z) where g is the coupling constant and z is the ratio of two different characteristic scales, which governs both QED and QCD calculations of exclusive form factors. As explained in detail in [2] we obtain for a specific case of exclusive scattering amplitude, namely the deeply virtual Compton scattering in the generalized cosh(kg log z) orken 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 previously emerged in field theoretic calculations. The process that we focus on is the most studied case of a class of reactions - exclusive hard hadronic processes - which are under intense experimental investigation. The result presented here provides an important stepping-stone for further developments enabling a consistent extraction of the quantities describing the 3-dimensional structure of the proton.

In the collinear factorization framework the scattering amplitude for exclusive processes such as deeply virtual Compton scattering (DVCS) has been shown [3] to factorize in specific kinematical regions, provided a large scale controls the separation of short distance dominated partonic subprocesses and long distance hadronic matrix elements, the generalized parton distributions (GPDs) [4]. The amplitude for the DVCS process

with a large virtuality q²=-Q², factorizes in terms of perturbatively calculable coefficient functions $C\left(x,\xi ,{\alpha }_s\right)$ and GPDs H(x,ξ,t), where the scaling variable in the generalized Bjorken limit is the skewness ξ defined as $\xi =\frac{Q^2}{\left(p+p^{\text{'}}\right).\left(q+q^{\text{'}}\right)}.$The calculation of first order perturbative corrections to the partonic amplitude has shown that terms of order $\frac{{\log }^2\left(x\pm \xi \right)}{x\pm \xi }$ play an important role in the region of small (x±ξ) i.e. in the vicinity of the boundary between the domains where the QCD evolution equations of GPDs take distinct forms (the so-called ERBL and DGLAP domains). We scrutinize these regions and demonstrate that they are dominated by soft fermion and gluon propagation. This explains why they can be exponentiated using quasi-eikonal techniques.

To set up our notations, let us remind the reader of the known results for the NLO corrections to the DVCS amplitude (1), specializing to the quark contribution to its symmetric part. After proper renormalization, it reads

where the quark coefficient function T^q reads [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 contributions are obtained from one another by a simple (x↔-x) interchange, we now 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, the extraction of the soft-collinear singularities which dominate the amplitude in the limit x → ± ξ is made easier if one uses the light-like gauge p₁. A = 0 with p₁ = q^' We argue (and verified) that in this gauge the amplitude is dominated by ladder-like diagrams. We expand any momentum in the Sudakov basis p₁p₂, as $k=\alpha p_1+\beta p_2+k_{}$ where p₂ is the light-cone direction of the two incoming and outgoing partons $\left(p_1^2=p_2^2=0,2p_1.p_2=s=Q^{2}2\xi \right)$. In this basis, $q_{\stackrel{\text{∗}}{y}}=p_1–2\xi p_2.$.

We now restrict our study to the limit x → + ξ . The dominant kinematics is given by a strong ordering both in longitudinal and transverse momenta, according to (see Fig. 1) :

This ordering is related to the fact that the dominant double logarithmic contribution for each loop arises from the region of phase space where both soft and collinear singularities manifest themselves. When x→ξ the left fermionic line is a hard line, from which the gluons are emitted in an eikonal way (which means that these gluons have their all four-components neglected in the vertex w.r.t. the momentum of the emitter), with an ordering in p₂ direction and a collinear ordering. For the right fermionic line, eikonal approximation is not valid, since the dominant momentum flow along p₂ is from gluon to fermion, nevertheless the collinear approximation can still be applied.

When computing the coefficient functions, one faces both UV and IR divergencies. On the one hand, the UV divergencies are taken care of through renormalization, which manifest themselves by a renormalization scale µ_R dependency. On the other hand, the IR divergencies remain, but factorization proofs at any order for DVCS justify the fact that they can be absorbed inside the generalized parton distributions and result in finite coefficient functions. In our study, we are only interested into finite parts. Thus, using dimensional regularization, in a factorization scheme like $\stackrel{}{\mathrm{MS}}$, any scaleless integral can be safely put to zero although it contains both UV and IR divergencies. Following this line of thought, we can thus safely deal with DVCS on a quark for our resummation purpose.

Finally, the issue related to the iε prescription in Eq. (5) is solved by computing the coefficient function in the unphysical region ξ > 1. After analytical continuation to the physical region 0≤ ξ ≤ 1, the final result is then obtained through the shift ξ→ξ-iε.

We define K_n as the contribution of a n-loop ladder to the coefficient function. Let us sketch the main steps of the derivation of K₁ and then generalize it for K_n.

2.2. The Ladder Diagram at Order ${\alpha }_s^n$

Let us now turn to the estimation of all ${log}^{2n}(x-\xi )$ 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_ί as :

Fig.(1). The ladder diagrams which contribute in the light-like gauge to the leading $a_s^n{1n}^{2n}\left(x\right)\left(x\right)$ terms in the perturbative expansion of the DVCS amplitude. The p2 and ᾕ momentum components are indicated. The dashed lines show the dominant momentum flows along the p2 direction.

16  ${\int }_0^{\xi –x}d{\beta }_1{\int }_0^{\xi –x–{\beta }_1}{d\beta }_2...{\int }_0^{\xi –x–{\beta }_1–..–{\beta }_{n–1}}d{\beta }_n$

The numerator for the n^th order box diagram is obtained as:

17  ${\left(\mathrm{Num}\right)}_2=4s{\left(x+\xi \right)}^n\frac{2k_1^2}{{\beta }_1}\left(1+\left(\frac{2\left(x–\xi \right)}{{\beta }_1}\right)\frac{2k_2^2}{{\beta }_2}\left(1+\left(\frac{2\left({\beta }_1+x–\xi \right)}{{\beta }_2}\right)...\frac{{2k}_n^2}{{\beta }_n}\right)\left(1+\left(\frac{2\left({\beta }_{n–1}+...+{\beta }_1+x–\xi \right)}{{\beta }_n}\right)\right)\right)$

and the denominators of propagators are, for ί=1...n,

18  $R_i^2\mathrm{}=–{\underset{}{k}}_i^2{\alpha }_i\mathrm{}({\beta }_1\dots {\beta }_{i}x–\xi \mathrm{})s\mathrm{,}{S}^2\mathrm{}=–{\underset{}{k}}_n^2\mathrm{}({\beta }_1\dots {\beta }_{n}x–\xi \mathrm{})s\mathrm{.}$

Using dimensional regularization and omitting scaleless integrals, the integral reads:

19  $\begin{array}{c}{\int }_0^{\xi –x}d{\beta }_1\dots {\int }_0^{\xi –x–\dots –{\beta }_{n–1}}d{\beta }_n{\int }_0^{}{d}^{d–2}{\underset{}{k}}_n\dots {\int }_0^{{\underset{}{k}}_2^2}{d}^{d–2}{\underset{}{k}}_1{\mathrm{}(–1)}^n\\ \frac{4s{\mathrm{}\left(2i\mathrm{}\right)}^n}{x–\xi }\frac{1}{{\beta }_{1}x–\xi }\dots \frac{1}{{\beta }_1\dots {\beta }_{n–1}x–\xi }\\ \frac{1}{{\underset{}{k}}_1^2}\dots \frac{1}{{\underset{}{k}}_n^2}\frac{1}{{\underset{}{k}}_n^2–\mathrm{}({\beta }_1\dots {\beta }_{n}x–\xi \mathrm{})s}\end{array}$

The integrals over ${\underset{\_}{k}}_1\cdots {\underset{\_}{k}}_n$ are performed similarly as in the one-loop case, resulting in:

20  $k_n=\frac{i}{4}{e}_q^2{\left(–i{C}_F{a}_s\frac{1}{{\left(2\right)}^2}\right)}^n\frac{4}{x–\xi +i}\frac{{\left(2i\right)}^n}{{\left(2n\right)}^{!}}{\log }^{2n}\left(\frac{\xi –x}{2\xi }–i\xi \right),$

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 the complete amplitude; we get with

In the absence of a next to leading logarithmic calculation, the minimal and most natural resummed formula which has the same O(α_s) expression as the full NLO result, reads, :

For several decades the effects of gluons on many high energy processes has been widely studied. Specifically the theory of "Color Glass Condensate" shows that at very high energies the behavior of the scattering amplitudes are dominated by gluons [6]. Recently it was also shown that even at moderate energies, there are significant O(α_s) corrections to scattering amplitudes due to gluonic contributions for spacelike and timelike virtual Compton scatterings [7]. With the above mentioned motivations performing a similar resummation procedure for gluon coefficient function of DVCS and TCS would result in a more trustful extraction gluon GPDs.

The authors confirm that this article content has no conflict of interest.