Generalized Parton Distributions from Lattice QCD with Asymmetric Momentum Transfer: Tensor Case

TL;DR

This work develops a Lorentz-invariant amplitude framework to extract the four leading-twist quark transversity GPDs $H_T$, $E_T$, $\widetilde{H}_T$, and $\widetilde{E}_T$ from lattice QCD calculations performed in an asymmetric momentum-transfer frame at zero skewness $\xi=0$. By decomposing the nonlocal tensor operator into twelve Lorentz-invariant amplitudes $A_{Ti}$ and relating them to light-cone GPDs, the authors enable frame-independent access to $x$-dependent transversity GPDs through a two-step process: (i) compute matrix elements in both symmetric and asymmetric frames, and (ii) reconstruct the $x$-dependence using Backus-Gilbert and apply one-loop matching to connect lattice results to the $\overline{\mathrm{MS}}$-scheme light-cone GPDs at 2 GeV. The results show robust signals for $H_T$, $E_T$, and the combination $E_T+2\widetilde{H}_T$, while $\widetilde{E}_T$ vanishes at $\xi=0$ as expected, and demonstrate frame-independence of the Lorentz-invariant amplitudes. This approach provides a practical path to probing the transverse spin structure of the proton from first principles and lays groundwork for extending to nonzero skewness and systematic studies of lattice artifacts.

Abstract

The calculation of generalized parton distributions (GPDs) in lattice QCD was traditionally done by calculating matrix elements in the symmetric frame. Recent advancements have significantly reduced computational costs by calculating these matrix elements in the asymmetric frame, allowing us to choose the momentum transfer to be in either the initial or final states only. The theoretical methodology requires a new parametrization of the matrix element to obtain Lorentz-invariant amplitudes, which are then related to the GPDs. The formulation and implementation of this approach have already been established for the unpolarized and helicity GPDs. Building upon this idea, we extend this formulation to the four leading-twist quark transversity GPDs ($H_T$, $E_T$, $\widetilde{H}_T$, $\widetilde{E}_T$). We also present numerical results for zero skewness using an $N_f=2+1+1$ ensemble of twisted mass fermions with a clover improvement. The light quark masses employed in these calculations correspond to a pion mass of about 260 MeV. Furthermore, we include a comparison between the symmetric and asymmetric frame calculations to demonstrate frame independence of the Lorentz-invariant amplitudes. Analysis of the matrix elements in the asymmetric frame is performed at several values of the momentum transfer squared, $-t$, ranging from 0.17 GeV$^2$ to 2.29 GeV$^2$.

Figures (22)

• Figure 1: Bare matrix elements $\Pi_{3j}(\Gamma_0)$ in the symmetric frame (top) and in the asymmetric frame (bottom), for $|P_3|=1.25$ GeV and $-t=0.69$ GeV$^2$ ($-t=0.65$ GeV$^2$) for the symmetric (asymmetric) frame. The left (right) panel corresponds to the real (imaginary) part. The notation in the legend is $j,~\{P_3,\vec{\Delta}\}$ in units of $2\pi/L$.
• Figure 2: Bare matrix elements $\Pi_{3j}(\Gamma_k; \Delta_j=0)$ in the symmetric frame (top) and in the asymmetric frame (bottom), for $|P_3|=1.25$ GeV and $-t=0.69$ GeV$^2$ ($-t=0.65$ GeV$^2$) for the symmetric (asymmetric) frame. The left (right) panel corresponds to the real (imaginary) part. The notation in the legend is $(j,k),~\{P_3,\vec{\Delta}\}$ in units of $2\pi/L$.
• Figure 3: Bare matrix elements $\Pi_{3j}(\Gamma_k; \Delta_j\neq 0)$ in the symmetric frame (top) and in the asymmetric frame (bottom), for $|P_3|=1.25$ GeV and $-t=0.69$ GeV$^2$ ($-t=0.65$ GeV$^2$) for the symmetric (asymmetric) frame. The left (right) panel corresponds to the real (imaginary) part. The notation in the legend is $(j,k),~\{P_3,\vec{\Delta}\}$ in units of $2\pi/L$.
• Figure 4: Comparison of amplitudes $A_{T2}$ (blue), $zA_{T3}$ (red), and $A_{T10}$ (black) with the real (imaginary) part on the left (right). $A^s_{Ti}$ (filled symbols) correspond to $-t^a=0.69~\mathrm{GeV}^2$, while $A^a_{Ti}$ (open symbols) to $-t^s=0.65~\mathrm{GeV}^2$.
• Figure 5: Frame comparison of amplitudes $A_{T4}$, $z A_{T5}$, $z^2 A_{T7}$, $z A_{T12}$ for the real (imaginary) parts on the left (right). $A^s_{Ti}$ (filled symbols) correspond to $-t^a=0.69~\mathrm{GeV}^2$, while $A^a_{Ti}$ (open symbols) to $-t^s=0.65~\mathrm{GeV}^2$.