Moments of Nucleon Generalized Parton Distributions in Lattice QCD

TL;DR

This paper tackles the challenge of obtaining moments of generalized parton distributions (GPDs) from lattice QCD by introducing a novel overdetermined set of lattice observables that maximizes statistical constraints on the generalized form factors (GFFs) $A_{ni}(t)$, $B_{ni}(t)$, and $C_n(t)$ at fixed $t$. The approach uses non-diagonal matrix elements of local twist-two operators, a 1-loop renormalization matching to the continuum, and a singular value decomposition to extract multiple GFFs from an overabundance of lattice measurements, thereby reducing uncertainties. In an exploratory calculation with unquenched Wilson fermions at heavy pion mass ($m_\2$ ~ 896 MeV) on a $16^3\times 32$ lattice, they determine $n=2$ GFFs up to $t\approx 3$ GeV$^2$ for the flavor non-singlet and observe a nearly vanishing singlet $B_{20}$, implying that the total quark angular momentum is dominated by $A_{20}$. They report that the connected contribution to the nucleon spin can be measured to ~1% accuracy, and the overall uncertainties on $A^{u\!\!+\!d}_{20}$, $A^{u-d}_{20}$, and $B^{u-d}_{20}$ are typically 5–10%, illustrating the method’s potential for informing the transverse structure of the nucleon. The work provides a foundation for extending to higher moments, spin-dependent GFFs, and chiral/continuum limits, with significant implications for connecting lattice results to experimental extractions of GPDs.

Abstract

Calculation of moments of generalized parton distributions in lattice QCD requires more powerful techniques than those previously used to calculate moments of structure functions. Hence, we present a novel approach that exploits the full information content from a given lattice configuration by measuring an overdetermined set of lattice observables to provide maximal statistical constraints on the generalized form factors at a given virtuality, t. In an exploratory investigation using unquenched QCD configurations at intermediate sea quark masses, we demonstrate that our new technique is superior to conventional methods and leads to reliable numerical signals for the n=2 flavor singlet generalized form factors up to 3 GeV^2. The contribution from connected diagrams in the flavor singlet sector to the total quark angular momentum is measured to an accuracy of the order of one percent.

Figures (4)

• Figure 1: Plateau plots of the ratios $R(\tau,P',P)$ for the $n = 2$ operators ${\cal O}^{\text{u-d}}_{\text{diag,1}}$, ${\cal O}^{\text{u-d}}_{\text{diag,2}}$, and ${\cal O}^{\text{u-d}}_{\lbrace 20 \rbrace}$.
• Figure 2: Generalized form factors obtained by simultaneous fits to $N$ external momentum combinations having virtuality $t_{\text{vlow}}$. As described in the text, the $N = 0$ points, denoted by triangles, use three operators at a single external momentum combination to determine the three form factors. The remaining points, denoted by squares, use six operators and $N$ external momentum combinations to determine the three form factors.
• Figure 3: Generalized form factors $A^{\text{u-d}}_{2 0}(t)$, $B^{\text{u-d}}_{2 0}(t)$, and $C^{\text{u-d}}_{2}(t)$ for all available virtualities obtained using the full set of operators and external momentum combinations. The dashed curves denote dipole fits to $A$ and $B$ to guide the eye and extrapolate to $t = 0$. The form factor $C$ is consistent with zero. Four data points, denoted by horizontal brackets, have been shifted by $\vert t\vert=0.2\,$GeV$^2$ to the right for clarity in plotting.
• Figure 4: Flavor singlet generalized form factors $A^{\text{u+d}}_{2 0}(t)$ and $B^{\text{u+d}}_{2 0}(t)$ with dipole fits denoted by dashed curves. Note that the singlet combination $B^{\text{u+d}}_{2 0}(t)$ is consistent with zero, so that the total quark angular momentum $J_q = \frac{1}{2} [A^{\text{u+d}}_{2 0}(0) + B^{\text{u+d}}_{2 0}(0) ]$ is dominated by $A$. Three data points, denoted by horizontal brackets, have been shifted by $\vert t\vert=0.2\,$GeV$^2$ to the right for clarity in plotting.