Evolution and models for skewed parton distributions

TL;DR

This work develops a double-distribution framework to model skewed parton distributions (SPDs) by factorizing DDs into forward densities and a universal longitudinal-profile function, enabling SPDs to be constructed from forward densities via simple averaging for small skewedness. It demonstrates that the Gegenbauer moments of SPDs become effectively independent of skewedness (the Shuvaev connection) and that the α-profile of DDs is determined by the forward density’s small-$x$ behavior, with an evolution-stable relation $\\tilde{f}_m(\\alpha;\\mu)=\\rho_{m+1}(\\alpha) \\tilde{f}_m(\\mu)$. The authors provide a practical leading-log evolution algorithm and perform numerical SPDs evolution, revealing patterns where evolution narrows or widens the DD profile to align with forward densities and showing enhanced SPDs at the border for higher $Q^2$, especially in the singlet sector. Overall, the results confirm the self-consistency of the DD/SPD picture, highlight the primacy of forward densities in constraining SPDs at small skewedness, and suggest direct DD evolution as a fruitful future step.

Abstract

We discuss the structure of the ``forward visible'' (FW) parts of double and skewed distributions related to usual distributions through reduction relations. We use factorized models for double distributions (DDs) f(x, alpha) in which one factor coincides with the usual (forward) parton distribution and another specifies the profile characterizing the spread of the longitudinal momentum transfer. The model DDs are used to construct skewed parton distributions (SPDs). For small skewedness, the FW parts of SPDs H(x, xi) can be obtained by averaging forward parton densities f(x- xi alpha) with the weight rho (alpha) coinciding with the profile function of the double distribution f(x, alpha) at small x. We show that if the x^n moments f_n (alpha) of DDs have the asymptotic (1-alpha^2)^{n+1} profile, then the alpha-profile of f (x,alpha) for small x is completely determined by small-x behavior of the usual parton distribution. We demonstrate that, for small xi, the model with asymptotic profiles for f_n (alpha) is equivalent to that proposed recently by Shuvaev et al., in which the Gegenbauer moments of SPDs do not depend on xi. We perform a numerical investigation of the evolution patterns of SPDs and gave interpretation of the results of these studies within the formalism of double distributions.

Figures (12)

• Figure 1: $a)$ Support region and symmetry line $y = \bar{x}/2$ for $y$-DDs $\tilde{F}(x,y;t)$. $b)$ Support region for $\alpha$-DDs $\tilde{f} (x, \alpha)$.
• Figure 2: $a)$ Parton picture in terms of $y$-DDs; $b)$ meson-like contribution; $c)$ Polyakov-Weiss contribution; $d)$ parton picture in terms of $\alpha$-DDs.
• Figure 3: Integration lines for integrals relating SPDs and DDs.
• Figure 4: Valence quark distributions: untilded NFPDs $F^{q}_\zeta(x)$ (left) and OFPDs $H ^1_V(\tilde{x},\xi)$ (right) with $a= 0.5$ for several values of $\zeta$, namely, 0.1, 0.2, 0.4, 0.6, 0.8 and corresponding values of $\xi=\zeta / (2-\zeta)$. Lower curves correspond to larger values of $\zeta$.
• Figure 5: Singlet quark distributions: tilded NFPDs $\tilde{F}^{S}_\zeta(x)$ (left) and OFPDs $H ^1_S(\tilde{x},\xi)$ (right) for several $\zeta$ values 0.2, 0.4, 0.6 and corresponding values of $\xi=\zeta / (2-\zeta)$. Lower curves correspond to larger values of $\zeta$. Forward distribution is modeled by $(1-x)^3/x$.