Generalized Parton Distributions from Lattice QCD with Asymmetric Momentum Transfer: Unpolarized Quarks at Nonzero Skewness

Abstract

We extend the formalism of asymmetric frames of reference for generalized parton distributions (GPDs) to the case of nonzero skewness, i.e., including longitudinal momentum transfer. The framework, based on Lorentz-invariant amplitudes and previously developed and numerically implemented for unpolarized, helicity and transversity GPDs at zero skewness, gives efficient access to a broad range of kinematics, making full mapping of GPDs from the lattice realistic. The general-skewness formalism is tested using lattice data with both transverse and longitudinal or only longitudinal momentum transfer, the latter being a special case with a reduced number of independent amplitudes. We extract the amplitudes in coordinate space and express the GPDs $H$ and $E$ in terms of these amplitudes. This is followed by reconstruction of quasi-distributions and their matching to the light cone. We further identify and discuss the principal challenges for nonzero skewness GPDs.

Figures (26)

• Figure 1: Bare matrix elements $\Pi_0(\Gamma_0)$ at momentum combinations $\{P_{f3}, \Delta_1, \Delta_2, \Delta_3\}$ in units of $2\pi/L$. Upper and lower panels correspond to the real and imaginary parts, respectively.
• Figure 2: Bare matrix elements $\Pi_0(\Gamma_1)$ at momentum combinations $\{P_{f3}, \Delta_1, \Delta_2, \Delta_3\}$ in units of $2\pi/L$. Upper and lower panels correspond to the real and imaginary parts, respectively.
• Figure 3: Momentum-transfer dependence of amplitudes $A_1$ and $A_5$ at fixed skewness $\xi = -1/5$. $A_1^L$ from $\Delta_T=0$ case is included and noted in the legend (numerically, $A_1^L\approx A_1$, see the main text for details). The horizontal axis is $\lambda=zP_3$. Upper and lower panels correspond to the real and imaginary parts, respectively.
• Figure 4: Skewness dependence of amplitudes $A_1$ and $A_5$ at approximately fixed $t$ corresponding to different $\vec{\Delta}$ vectors for different $\xi$. Upper and lower panels correspond to the real and imaginary parts, respectively.
• Figure 5: Skewness dependence of renormalized amplitudes $A_1$ and $A_5$ at approximately fixed $t$ corresponding to different $\vec{\Delta}$ vectors for different $\xi$. Upper and lower panels correspond to the real and imaginary parts, respectively.