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

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 nonperturbatively 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 $g_S^u=5.20(42)(15)(12)$, $g_S^d=4.27(26)(15)(12)$, $g_S^s=0.33(7)(1)(4)$, and $g_S^c=0.062(13)(3)(5)$ and for the tensor charges $g_T^u=0.794(16)(2)(13)$, $g_T^d=-0.210(10)(2)(13)$, $g_T^s=0.00032(24)(0)$, and $g_T^c=0.00062(85)(0)$ in the $\overline{\mathrm{MS}}$ 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 estimating the contamination due to the excited states, when our data are precise enough to probe the first excited state.

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 $n_{\mathrm{iter}}^{\mathrm{LP}}$ 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 $t_{\mathrm{ins}}-t_s/2$ for source-sink separations $t_s=0.94\mathrm{fm}$ (green circles), $t_s=1.13\mathrm{fm}$ (red squares), $t_s=1.31\mathrm{fm}$ (black diamonds), $t_s=1.50\mathrm{fm}$ (purple triangles) and $t_s=1.69\mathrm{fm}$ (blue pentagons). 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 $t_s^{\mathrm{low}}$ 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 bottom 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. 4. The various values for $t_s$ shown for the plateau method are listed in the legend of the plots.

Figure 6

The strange (top) and charm (bottom) scalar charges. The notation is as in Fig. 4. The various values of $t_s$ shown for the plateau method for each observable are listed in the corresponding legend of the plots.

Figure 7

The isovector (top) and connected isoscalar (bottom) nucleon tensor charge, following the notation of Fig. 4. The various values of $t_S$ shown for the plateau method are listed in the legend of the plots.

Figure 8

Disconnected contributions to $g_T^{u+d}$. The notation is as in Fig. 4. The various values of $t_s$ shown for the plateau method are listed in the legend of the plots. The error bands of the summation method and two-state fit are not included in the plot on the left panel as their very large uncertainties preclude a sensible comparison.

Figure 9

The purely disconnected strange (top) and charm (bottom) tensor charges. The notation is as in Fig. 4. The various sink times shown for the plateau method for each observable are listed in the corresponding legend of the plots. As in Fig. 8, we do not include the error bands of the summation method and the two-state fit in the plots in the left panel.

Figure 10

Comparison of our results (red circles) for $g_S^{u-d}$ (left) and $g_T^{u-d}$ (right) with a number of other recent lattice QCD results (blue squares) and with phenomenology (green triangles). With filled squares we denote extrapolated values at the physical pion mass, whereas with the open squares we show the lattice results from the various collaborations at their lowest pion mass, for the cases with $m_{\pi }^{\mathrm{low}}\le 150\mathrm{MeV}$. The solid error bars denote statistical errors whereas the dashed error bars show the statistical and systematic uncertainties added in quadrature. The red vertical band showing our value and its errors is to help guide the eye.

Figure 11

Comparison of our results using the physical ensemble (red circles) for the connected $g_S^{u+d}$ (left) and $g_T^{u+d}$ (right) with lattice results from the ETMC using TMFs on $N_f=2+1+1$ gauge configurations [18] (blue square); the PNDME using $N_f=2+1+1$ staggered fermions (green triangles) from Ref. [69] for $g_S^{u+d}$ and Ref. [7] for $g_T^{u+d}$; and from the LHPC [59], using $N_f=2+1$ clover fermions, domain-wall fermions and a mixed action approach (magenta, brown and light blue diamonds, respectively). The solid error bars in our results denote statistical errors whereas the dashed error bars show the statistical and systematic uncertainties added in quadrature.

Figure 12

Comparison of our results for the disconnected contribution to $g_S^{u+d}$ (left) and $g_T^{u+d}$ (right) with lattice results from the ETMC, employing TMFs on $N_f=2+1+1$ gauge configurations [18] (blue square) and the PNDME Collaboration, using clover valence fermions on a $N_f=2+1+1$ HISQ quark sea (green triangles) from Ref. [70] for $g_S^{u+d}$ and Ref. [19] for $g_T^{u+d}$. The solid error bars in our results denote statistical errors whereas the dashed error bars show the statistical and systematic uncertainties added in quadrature.

Figure 13

Comparison of our results using the physical ensemble (red circles) for the strange charges $g_S^s$ (left) and $g_T^s$ (right) with the results from the PNDME Collaboration, using $N_f=2+1+1$ staggered fermions (green triangles) from Ref. [70] for $g_S^s$ and Ref. [19] for $g_T^s$.