Double distributions and evolution equations

TL;DR

The paper develops a DD-based framework to describe nonforward parton information important for DVCS and hard exclusive processes, treating double distributions $F(x,y;t)$ as primary objects and linking them to nonforward distributions ${\cal F}_{\zeta}(X;t)$ via line integrals. It derives the evolution equations for DDs with kernels $R^{ab}$, showing their reduction to DGLAP and DA evolution and outlining how singlet mixing is handled; the kernels are constructed from universal light-ray evolution kernels. The author constructs simple, symmetry-respecting DD models $F^{(0,1,2)}$ that yield self-consistent nonforward distributions and demonstrates a small-$\zeta$ approximation ${\cal F}_{\zeta}(X;t=0) \approx f(X-\zeta/2)$, connecting DDs to Ji's off-forward distributions. The results provide a coherent framework to understand and evolve nonforward parton densities, with implications for DVCS amplitudes and hard exclusive processes, and lay groundwork for future numerical evolutions with realistic input distributions.

Abstract

Applications of perturbative QCD to deeply virtual Compton scattering and hard exclusive meson electroproduction processes require a generalization of usual parton distributions for the case when long-distance information is accumulated in nonforward matrix elements < p'| O(0,z) | p > of quark and gluon light-cone operators. In our previous papers we used two types of nonperturbative functions parametrizing such matrix elements: double distributions F(x,y;t) and nonforward distribution functions F_ζ(X;t). Here we discuss in more detail the double distributions (DD's) and evolution equations which they satisfy. We propose simple models for F(x,y;t=0) DD's with correct spectral and symmetry properties which also satisfy the reduction relations connecting them to the usual parton densities f(x). In this way, we obtain self-consistent models for the ζ-dependence of nonforward distributions. We show that, for small ζ, one can easily obtain nonforward distributions (in the X > ζregion) from the parton densities: F_ζ(X;t=0) \approx f(X-ζ/2).

Figures (8)

• Figure 1: Parton picture for double distributions.
• Figure 2: $a)$ Integration lines in the $(x,y)$-plane giving reduction of double distributions $F(x,y;t=0)$ to usual parton densities $f(x_1)$ and $f(x_2)$. $b)$ Symmetry line $y= (1-x)/2$ for double distributions.
• Figure 3: Parton interpretation of nonforward distributions. $a)$ Region $X>\zeta$. $b)$ Region $X<\zeta$.
• Figure 4: Relation between double distributions $F(x,y)$ and nonforward parton distributions ${\cal F}_{\zeta}(X)$. $a)$ Integration lines for three cases: $X_1 > \zeta$, $X = \zeta$ and $X_2 < \zeta$. $b,c)$ Comparison of integration lines for the nonforward parton distribution ${\cal F}_{\zeta}(X)$ and usual parton densities $f(X)$, $f(X')$ (shown in 2$b$) and $f(X)$, $f(X_2)$ with $X_2 = X'/\bar{\zeta}$ (shown in 2$c$).
• Figure 5: Nonforward parton distributions ${\cal F}_{\zeta}^{(1)} (X)$ for different values of the skewedness $\zeta=0.05$ (thin line), $\zeta=0.1$ (dashed line), $\zeta=0.2$ (dash-dotted line) and $\zeta=0.4$ (full line) in the "valence quark oriented" model specified by Eq.(6.11) for $a=0.5$.