Nucleon scalar and tensor charges using lattice QCD simulations at the physical value of the pion mass

TL;DR

This work reports a first-principles lattice QCD calculation of the nucleon scalar and tensor charges for light, strange, and charm quarks directly at the physical pion mass, including both connected and disconnected contributions. Using a $N_f=2$ twisted-mass Clover-improved setup with non-perturbative renormalization, the authors carefully control excited-state contamination via plateau, summation, and two-state analyses. They provide precise ${\overline{\rm MS}}$ values at 2 GeV for all charges, revealing nonzero strange and charm scalar contents and small tensor-disconnected contributions, with results broadly consistent with other lattice determinations. The study demonstrates robust techniques (one-end trick, TSM) for disconnected diagrams and sets the stage for multi-ensemble continuum and finite-volume explorations.

Abstract

We present results on the light, strange and charm nucleon scalar and tensor charges from lattice QCD, using simulations with $N_f=2$ flavors of twisted mass Clover-improved fermions with a physical value of the pion mass. Both connected and disconnected contributions are included, enabling us to extract the isoscalar, strange and charm charges for the first time directly at the physical point. Furthermore, the renormalization is computed non-perturbatively for both isovector and isoscalar quantities. We investigate excited state effects by analyzing several sink-source time separations and by employing a set of methods to probe ground state dominance. Our final results for the scalar charges are $g_S^u = 5.20(42)(15)(12)$, $g_S^d = 4.27(26)(15)(12)$, $g_S^s=0.33(7)(1)(4)$, $g_S^c=0.062(13)(3)(5)$ and for the tensor charges $g_T^u = 0.782(16)(2)(13)$, $g_T^d = -0.219(10)(2)(13)$, $g_T^s=-0.00319(69)(2)(22)$, $g_T^c=-0.00263(269)(2)(37)$ in the $\overline{\rm MS}$ scheme at 2~GeV. The first error is statistical, the second is the systematic error due to the renormalization and the third the systematic arising from possible contamination due to the excited states.

Figures (13)

• Figure 1: Diagrams of a connected (left) and disconnected (right) three-point function.
• Figure 2: Left: The correlation $r_c$ between the high precision and low precision propagators for several values of the twisted mass parameter. Right: Interpolation of our data for determining the optimal value of $n_{\rm iter}^{\rm LP}$ for the strange quark at $r_c\simeq 0.99$. A similar procedure was followed for the charm quark.
• Figure 3: The variance as a function of the sink-source time separation for the isovector scalar and the tensor charges.
• Figure 4: Top left: Ratio yielding $g_S^{u-d}$ as a function of $t_{\rm ins}-t_s/2$ for source-sink separations $t_s=0.94$ fm (green circle), $t_s=1.13$ fm (red square), $t_s=1.31$ fm (black diamond), $t_s=1.50$ fm (purple triangle) and $t_s=1.69$ fm (blue pentagon). The plateau value selected is shown by the short band with the color of the corresponding $t_s$ selected, with its starting and ending points indicating the fit range used. The other two bands spanning the whole range of the plot show the results we select for the summation method (light brown) and the two-state fit (gray). Top right: Summary of our results for $g_S^{u-d}$ from the plateau fits (left column) and the summation method and two-state fit (right column). With $t_s^{\rm low}$ we denote the smallest $t_s$ in the latter two fits. The open red and blue symbols denote the selected final results from the plateau and two-state fits. The red band is the statistical error of the plateau fit. Bottom left and right: Corresponding plots for the connected contributions to $g_S^{u+d}$.
• Figure 5: Disconnected contributions to $g_S^{u+d}$. The notation is as in Fig. \ref{['fig:gS_light_conn']}. The various values for $t_s$ shown for the plateau method are listed in the legend of the plots.