Disconnected quark loop contributions to nucleon observables in lattice QCD

TL;DR

The paper addresses the impact of disconnected quark loops on nucleon observables in lattice QCD and develops a GPU-accelerated workflow that combines the Truncated Solver Method (TSM) with the one-end trick in a $N_f=2+1+1$ twisted-mass framework on a $32^3\times64$ lattice. High-statistics results reveal that disconnected contributions are significant for several isoscalar quantities—most notably the sigma terms and the isoscalar axial charge—and are non-negligible for $G_A$ and $G_p$ at nonzero momentum, while some moments like $\langle x\rangle_{u+d}$ are consistent with zero within errors. The study demonstrates the feasibility and importance of including disconnected diagrams, validates results using both plateau and summation methods, and outlines plans to extend to physical pion mass ensembles. These findings highlight the necessity of accounting for disconnected contributions to achieve percent-level control in nucleon structure calculations and establish GPUs with variance-reduction techniques as a viable path forward for precision lattice QCD.

Abstract

We perform a high statistics calculation of disconnected fermion loops on Graphics Processing Units for a range of nucleon matrix elements extracted using lattice QCD. The isoscalar electromagnetic and axial vector form factors, the sigma-terms and the momentum fraction and helicity are among the quantities we evaluate. We compare the disconnected contributions to the connected ones and give the physical implications on nucleon observables that probe its structure.

Figures (14)

• Figure 1: The error on the isoscalar momentum fraction $\delta\langle x \rangle_{u+d}$ as a function of $N_{\rm HP}+N_{\rm LP}$ for 68000 measurements. The three leftmost points (red squares) correspond to $N_{\rm LP}=0$ and the three rightmost to $N_{\rm HP}=24$. The dotted line is the result of fitting to the Ansatz $1/\sqrt{a+\frac{b}{N_{\rm HP}+N_{\rm LP}}}$.
• Figure 2: The disconnected contribution to the ratio from which $\sigma_{\pi N}$ is extracted. On the upper panel we show the ratio as a function of the insertion time slice with respect to the mid-time separation ($t_{\rm ins}-t_s/2$) for source-sink time separations, $t_{\rm s}=$14$a$ (red filled circles), $t_{\rm s}=16a$ (blue filled squares), $t_{\rm s}=18a$ (green open squares) and $t_{\rm s}=20a$ (yellow filled triangles). In the central panel we show the summed ratio, for which the fitted slope yields the desired matrix element. On the lower panel we show the results obtained for the fitted slope of the summation method for various choices of the initial and final fit time slices. The star shows the choice for which the gray bands are plotted in the upper and central panels.
• Figure 3: The ratio from which the strange quark content of the nucleon, $\sigma_{s}$, is extracted. The notation is the same as that of Fig. \ref{['sigmaLight']}.
• Figure 4: The ratio from which the charm quark content of the nucleon, $\sigma_{c}$, is extracted. The notation is the same as that of Fig. \ref{['sigmaLight']}.
• Figure 5: The disconnected contribution to the renormalized ratio which yields the isoscalar axial charge of the nucleon, $g_A^{u+d}$. The upper panel shows the ratio as a function of the insertion time slice with respect to the mid-time separation ($t_{\rm ins}-t_s/2$) for source-sink separations $t_{\rm s}=8a$ (red filled circles), $t_{\rm s}=10a$ (blue filled squares), $t_{\rm s}=12a$ (green open squares) and $t_{\rm s}=14a$ (yellow filled triangles). The central panel shows the summed ratio and the lower panel the results obtained for the fitted slope of the summation method for various choices of the initial and final fit time slices as explained in the text. The star shows the choice of $t_i$, which yields the gray bands shown in the upper and central plots.