Skewness-dependent Moments of Pion GPD from Nonlocal Quark-Bilinear Correlators

TL;DR

This work computes the odd Mellin moments of the pion valence-quark GPD up to $\langle x^4\rangle$ for skewness in $\xi\in[-0.33,0]$ using lattice QCD and the OPE of nonlocal quark bilinears. It employs LaMET/SDF-inspired quasi-GPDs, a ratio renormalization scheme, and NNLL resummed perturbative matching to connect lattice matrix elements to light-cone GPD moments, enforcing polynomiality across multiple $\xi$ and $t$. Zero-skewness results reproduce known GFFs $A_{1,0}$, $A_{3,0}$, and $A_{5,0}$, while nonzero skewness yields $A_{3,2}$, $A_{5,2}$, and $A_{5,4}$, with reconstructed moments $H_3(\xi,t)$ and $H_5(\xi,t)$ showing suppression with increasing $-t$ and $\xi$. The study demonstrates the feasibility of extracting skewness-dependent pion GPD moments from lattice data, providing first-principles constraints for phenomenology and guiding future experimental programs, while acknowledging limitations from heavier pion masses and the absence of singlet/gluon contributions.

Abstract

We present lattice QCD calculations of the odd Mellin moments of pion valence-quark generalized parton distribution (GPD) up to fifth order, $\langle x^4\rangle$, and for the skewness range $[-0.33, 0]$ using operator product expansion of bilocal quark-bilinear operators. The calculations are performed on an ensemble with lattice spacing $a=0.04~\mathrm{fm}$ and valence pion mass $300$ $\mathrm{MeV}$, employing boosted pion states with momenta up to 2.428 GeV and momentum transfers reaching 2.748 GeV$^2$. We employ ratio-scheme renormalization and next-to-leading-logarithmic resummed perturbative matching. At zero skewness, our results are consistent with previous lattice studies. By combining matrix elements at multiple values of skewness and momentum transfer, skewness-dependent moments are obtained through simultaneous polynomiality-constrained fits.

Figures (15)

• Figure 1: The kinematic coverage presented in this study includes multiple combinations of the momentum transfer $-t$, the skewness $\xi$, and the final-state momentum in the $z$-direction $P^f_z$.
• Figure 2: Demonstration of fitting the ratio $R^{fi}(\mathbf{P}^f, \mathbf{P}^i; \tau, t_s)$ to obtain the bare ground-state matrix element.
• Figure 3: Comparison of the quasi-GPDs in coordinate space defined via the LI construction ($\widetilde{\mathcal{H}}_{\rm LI}$) and the $\gamma_0$ operator ($\widetilde{\mathcal{H}}$). Left: $P_z=1.937$ GeV, $-t=0.864$ GeV$^2$, right: $P_z=1.937$ GeV, $-t=2.748$ GeV$^2$.
• Figure 4: Renoramlized matrix elements $\widetilde{\mathcal{H}}^R$ as functions of $\lambda$ at four nonzero skewness values. In each panel, data points are distinguished by color, representing results at different momentum transfers $-t$.
• Figure 5: GFFs $A_{1,0}$ and $A_{3,0}$ at $\xi=0$, $P_z=1.937~\mathrm{GeV}$, and $-t=0.446~\mathrm{GeV}^2$, with different $\kappa$ values. Marker styles distinguish fixed-order perturbative matching (up to NNLO), LL resummation, and NLL resummation. Data at the same $z_{\text{max}}$ value are shifted by $\pm0.005$ fm for clarity. The shaded band highlights the plateau region where the fixed order, NLL-, and NNLL-resummed results agree and remain stable, indicating that $z_{\text{max}} = 0.24~\mathrm{fm}$ can be taken as a representative choice.