Iso-vector and Iso-scalar Tensor Charges of the Nucleon from Lattice QCD

Abstract

We present results for the iso-vector and flavor diagonal tensor charges $g^{u-d}_T$, $g^{u}_T$, $g^{d}_T$, and $g^{s}_T$ needed to probe novel tensor interactions at the TeV scale in neutron and nuclear $β$-decays and the contribution of the quark electric dipole moment (EDM) to the neutron EDM. The lattice QCD calculations were done using nine ensembles of gauge configurations generated by the MILC collaboration using the HISQ action with 2+1+1 dynamical flavors. These ensembles span three lattice spacings $a \approx 0.06, 0.09$ and $0.12 $ fm and three quark masses corresponding to the pion masses $M_π\approx 130, 220$ and $310 $ MeV. Using estimates from these ensembles, we quantify all systematic uncertainties and perform a simultaneous extrapolation in the lattice spacing, volume and light quark masses for the connected contributions. The final estimates of the connected nucleon (proton) tensor charge for the iso-vector combination is $g_T^{u-d} = 1.020(76) $ in the $\bar{\text{MS}}$ scheme at $2$ GeV. The additional disconnected quark loop contributions needed for the flavor-diagonal matrix elements are calculated using a stochastic estimator employing the truncated solver method with the all-mode-averaging technique. We find that the size of the disconnected contribution is smaller than the statistical error in the connected contribution. This allows us to bound the disconnected contribution and include it as an additional uncertainty in the flavor-diagonal charges. After a continuum extrapolation, we find $g_T^{u} = 0.774(66) $, $g_T^{d} = -0.233(28) $ and $g_T^{u+d} = 0.541(67) $. The strangeness tensor charge, that can make a significant contribution to the neutron EDM due to the large ratio $m_s/m_{u,d}$, is $g_T^{s}=0.008(9)$ in the continuum limit.

Figures (12)

• Figure 1: The connected (left) and disconnected (right) three-point diagrams needed to calculate the matrix elements of bilinear quark operators in the nucleon state.
• Figure 2: The data for $g_T^{u-d}$ and the results of the simultaneous fit using multiple $t_{\rm sep}$ using the ansatz given in Eq. \ref{['eq:2pt_3pt']} to isolate the excited state contribution. The seven figures are arranged as follows: the $M_\pi \approx 310$ MeV ensembles (top), $M_\pi \approx 220$ MeV ensembles (middle) and the $M_\pi \approx 130$ MeV ensemble (bottom). The solid black line and the grey band are the ground state ($t_\text{sep} \to \infty$) estimate and error. The fits evaluated for different $t_\text{sep}$ are also shown by solid lines.
• Figure 3: Data for $Z_T$ (upper) and $Z_T/Z_V$ (lower) after translation to the $\overline{\text{MS}}$ scheme at $2$ GeV as a function of the lattice momentum $q$ . The lattice calculation was done on five ensembles in the RI-sMOM scheme. The $a=0.12$ fm ($a=0.09$ fm) fit are to the combination of a12m310 and a12m220 ( a12m310 and a12m220) data as there is no detectable dependence on the quark mass. The $a=0.06$ fm fit is to the a06m310 ensemble data. The data were fit using the ansatz $c/q^2 + Z + \alpha q$ and the dot-dashed, dotted and dashed lines show the extrapolation $Z + \alpha q$ for the $a=0.06$, $0.09$ and $0.12$ fm data.
• Figure 4: Comparision of the simultaneous fits versus $a$, $M_\pi^2$ and $M_\pi L$ to the isovector charge, $g_T^{u-d}$, data using Eq. \ref{['eq:chiralfit']} (top) with the simpler version without the chiral logarithms given in Eq. \ref{['eq:extrap']} (bottom). The data symbols are defined in Table \ref{['tab:ens']}. The fit is given by the red line and the physical value after extrapolation to the continuum limit ($a\rightarrow 0$), physical pion mass ($M_\pi \rightarrow M_{\pi^0}^{{\rm phys}}$) and infinite volume ($L \rightarrow \infty$) is marked by a red star. The error band is shown as a function of each variable holding the other two at their physical value. The data are shown projected on to each of the three planes.
• Figure 5: Simultaneous extrapolation to the physical point ($a\rightarrow 0$, $M_\pi \rightarrow M_{\pi^0}^{{\rm phys}}$, and $L \rightarrow \infty$) using Eq. \ref{['eq:extrap']}, of the connected contributions to the flavor diagonal nucleon (proton) tensor charges, $g_T^u$ (upper) and $g_T^d$ (lower), renormalized in the $\overline{{\rm MS}}$ scheme at $2\mathop{\rm GeV}\nolimits$. The physical values given by the fit are marked by a red star. The rest is the same as in Fig. \ref{['fig:chiral_gT_compare']}.