Skewed and double distributions in pion and nucleon

TL;DR

The paper analyzes non-forward matrix elements of twist-2 QCD light-ray operators for the pion and nucleon, comparing skewed and double distribution representations and showing that a complete twist-2 decomposition yields a two-component skewed distribution with distinct behavior in different X-regimes.Using an effective chiral theory based on the instanton vacuum, it computes pion skewed and double distributions at a low normalization point, illustrating the necessity of including longitudinal (r·z) contributions and introducing a D(y) term that leads to the region |X|<ξ/2 being governed by a distribution amplitude–like component.The authors extend the discussion to crossing relations with the two-pion distribution amplitude, derive explicit connections between moments of skewed distributions and 2πDA coefficients, and show how measurements of γ*γ → ππ or γ*N → ππ could provide direct information on both skewed and usual quark distributions in the pion.Across a resonance-exchange perspective, the work explains how even-spin t-channel exchanges generate the D-term-like structure, linking non-forward QCD matrix elements to hadronic resonances and soft-pion theorems, and offering a framework for constructing realistic, symmetry-respecting parametrizations of generalized parton distributions.

Abstract

We study the non-forward matrix elements of twist-2 QCD light-ray operators and their representations in terms of skewed and double distributions, considering the pion as well as the nucleon. We point out the importance of explicitly including all twist-2 structures in the double distribution representation, which naturally leads to a ``two-component'' structure of the skewed distribution, with different contributions in the regions |X| > xi/2 and |X| < ξ/2. We compute the skewed and double quark distributions in the pion at a low normalization point in the effective chiral theory based on the instanton vacuum. Also, we derive the crossing relations expressing the skewed quark distribution in the pion through the distribution amplitude for two--pion production. Measurement of the latter in two-pion production in gamma^* gamma and gamma^* N reactions could provide direct information about the skewed as well as the usual quark/antiquark-distribution in the pion.

Figures (5)

• Figure 1: The range of the variables $x$ and $y$ in the double distribution, Eq.(\ref{['double_naive']}). The reduction to the skewed distribution, $H(X, \xi)$, is achieved by integrating the double distribution over the line $x + y \xi /2 = X$, cf. Eq.(\ref{['reduction_naive']}), shown here for the case that $X > \xi /2$ ( thick line).
• Figure 2: Schematic representation of resonance exchange contributions to the non-forward matrix element of the twist--2 operator in the pion, Eq.(\ref{['me_def']}). The upper blob denotes the distribution amplitude of the spin--$J$ resonance, Eq.(\ref{['R_da']}).
• Figure 3: Diagrams in the effective low--energy theory contributing to the skewed quark distribution at a low normalization point. The dashed line denotes the pion field, the solid line the quark propagator with the dynamical quark mass, $[i\partial\space/\space - M F^2(\partial^2 )]^{-1}$, and the filled circles the quark--pion vertices contained in the effective action, Eqs.(\ref{['action']}) and (\ref{['U_gamma5']}), which include a form factor $F(\partial^2 )$ for each quark line. Diagram (a) contributes only to the isoscalar distribution, and vanishes in the forward limit ($r \rightarrow 0$).
• Figure 4: The contributions from diagrams (a) and (b) ( cf. Fig.\ref{['fig_diagrams']}) to the isoscalar skewed quark distribution in the pion, $H^{I=0}(X, \xi )$, at a low normalization point, as functions of $X$, for a value of $\xi = 1$. (Here $t = 0$). Dashed lines: Results obtained neglecting the momentum dependence of the dynamical quark mass, cf. Eqs.(\ref{['H_1']}) and (\ref{['H_2']}). Solid lines: The corresponding contributions obtained when including the form factors, $F(\partial^2 )$. Note that the contribution from diagram (a) is non-zero only in the region $-\xi /2 < X < \xi /2$. The momentum dependence of the dynamical quark mass forces this contribution to vanish at the end points, $X = \pm\xi / 2$.
• Figure 5: The total isoscalar skewed quark distribution in the pion, $H^{I=0} (X, \xi )$, for $\xi = 1$, being the sum of the two contributions (a) and (b) shown in Fig.\ref{['fig_H12']}. Dashed line: Result obtained neglecting the momentum dependence of the dynamical quark mass, cf. Eq.(\ref{['H_tot']}). Solid lines: Distribution obtained when including the form factors, $F(\partial^2 )$. Due to the vanishing of contribution (a) at $X = \pm \xi/2$ the total distribution is continuous at these points.