Axial, scalar, and tensor charges of the nucleon from $2+1+1$-flavor Lattice QCD

We present results for the isovector axial, scalar, and tensor charges $g_A^{u-d}$, $g_S^{u-d}$, and $g_T^{u-d}$ of the nucleon needed to probe the Standard Model and novel physics. The axial charge is a fundamental parameter describing the weak interactions of nucleons. The scalar and tensor charges probe novel interactions at the TeV scale in neutron and nuclear $\beta$-decays, and the flavor-diagonal tensor charges $g_T^u$, $g_T^d$, and $g_T^s$ are needed to quantify 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 highly improved staggered quarks action with $2+1+1$ dynamical flavors. These ensembles span three lattice spacings $a\approx 0.06,0.09$, and 0.12 fm and light-quark masses corresponding to the pion masses $M_{\pi }\approx 135,225$, and 315 MeV. High-statistics estimates on five ensembles using the all-mode-averaging method allow us to quantify all systematic uncertainties and perform a simultaneous extrapolation in the lattice spacing, lattice volume, and light-quark masses for the connected contributions. Our final estimates, in the $\overline{\mathrm{MS}}$ scheme at 2 GeV, of the isovector charges are $g_A^{u-d}=1.195(33)(20)$, $g_S^{u-d}=0.97(12)(6)$, and $g_T^{u-d}=0.987(51)(20)$. The first error includes statistical and all systematic uncertainties except that due to the extrapolation Ansatz, which is given by the second error estimate. Combining our estimate for $g_S^{u-d}$ with the difference of light quarks masses $(m_d-m_u{)}^{\mathrm{QCD}}=2.67(35)\mathrm{MeV}$ given by the Flavor Lattice Average Group, we obtain $(M_N-M_P{)}^{\mathrm{QCD}}=2.59(49)\mathrm{MeV}$. Estimates of the connected part of the flavor-diagonal tensor charges of the proton are $g_T^u=0.792(42)$ and $g_T^d=-0.194(14)$. Combining our new estimates with precision low-energy experiments, we present updated constraints on novel scalar and tensor interactions, ${\epsilon }_{S,T}$, at the TeV scale.

Figure 1

Comparison of the unrenormalized $g_A^{u-d}$ data obtained using all HP measurements (left) with the AMA method (right). The parameters for these five ensembles are given in Table 1. The increase in statistics with the AMA method significantly improves the resolution of the data at the various source-sink separations, $t_{\mathrm{sep}}$. In each case, the solid line within the grey error band is the $t_{\mathrm{sep}}\to \infty$ result given by the 2-state fit using Eqs. (9) and (10), and the colored lines are its values for different $t_{\mathrm{sep}}$ plotted in the same color as the data. On the $a12m220L$ ensemble, the HP data was generated only with $t_{\mathrm{sep}}=10$.

Figure 2

Comparison of the unrenormalized $g_S^{u-d}$ data obtained using all HP measurements (left) with the AMA method (right). The rest is the same as in Fig. 1.

Figure 3

Comparison of the unrenormalized $g_T^{u-d}$ data obtained using all HP measurements (left) with the AMA method (right). The rest is the same as in Fig. 1.

Figure 4

Comparison of the unrenormalized $g_V^{u-d}$ data obtained using all HP measurements (left) with the AMA method (right). The rest is the same as in Fig. 1.

Figure 5

Illustration of the decrease in the errors in the 2- and 3-point functions calculated on the $a06m220$ lattices with the number $N$ of LP measurements made per configuration. (Top) The 2-point nucleon correlation function at time separations 12, 16, 20. (Bottom) The data for the four charges at the midpoint $\tau =10$ of the $t_{\mathrm{sep}}=20$ calculation. The errors are normalized using the $N=64$ estimates and decrease by a factor of about 3.7 between $N=4$ and 64 measurements compared to the expected factor of 4 for uncorrelated data.

Figure 6

Effective-mass plot for the nucleon on the $a09m130$ ensemble. The solid line and the error band are the estimate of $M_0$ from the fits to the AMA (red squares) data. The dashed blue line and the error band give the HP (blue circles) estimate. The fit ranges used correspond to Case 3 defined in Table 5. Results for all the nine ensembles are given in Table 4.

Figure 7

Effective-mass plots for the nucleon on the $a12m220S$, $a12m220$, $a12m220L$, and $a12m220L$ with AMA ensembles. The two bands with errors are the results for $M_0$ from the 2-state fit for the $a12m220$ (blue dashed line) and $a12m220L$ with AMA (red solid line) ensembles. These two estimates differ by a combined $1\sigma$. The estimates of the masses and the amplitudes for all four calculations are summarized in Table 4. The fit ranges used correspond to Case 3 defined in Table 5.

Figure 8

The 2-state fit to the unrenormalized axial charge $g_A^{u-d}$ data for the seven ensembles at different values of the lattice spacing and pion mass. The grey error band and the solid line within it is the $t_{\mathrm{sep}}\to \infty$ estimate obtained using the 2-state fit. The result of the fit for each individual $t_{\mathrm{sep}}$ is shown by a solid line with the same color as the data points. Note that the data with $t_{\mathrm{sep}}=16$ in the two $a06$ ensembles are not used in the fit.

Figure 9

The 2-state fit results for the unrenormalized scalar charge $g_S^{u-d}$ data. The rest is the same as in Fig. 8.

Figure 10

The 2-state fit results for the unrenormalized tensor charge $g_T^{u-d}$ data. The rest is the same as in Fig. 8.

Figure 11

Data for the four renormalization constants $Z_A$, $Z_S$, $Z_T$, and $Z_V$ in the $\overline{\mathrm{MS}}$ scheme at $\mu =2\mathrm{GeV}$ as a function of the lattice momenta $|q|$ used in the RI-sMOM scheme. The data are organized as follows: (left column) $a=0.12\mathrm{fm}$, (middle column) $a=0.09\mathrm{fm}$, and (right column) $a=0.06\mathrm{fm}$ ensembles. In each panel, we show the data at the two values of $M_{\pi }$ analyzed and use blue squares to label the $M_{\pi }\approx 310\mathrm{MeV}$ and red diamonds for the $M_{\pi }\approx 220\mathrm{MeV}$ ensembles. The dashed line is the linear part of the fit $c/q^2+Z+dq$ used in method A to extract the $Z$’s. Data in the region of $q^2$ used to extract the $Z$’s using method B are shown using green ($M_{\pi }\approx 310\mathrm{MeV}$) and orange ($M_{\pi }\approx 220\mathrm{MeV}$) symbols.

Figure 12

Data for the ratios of renormalization constants $Z_A/Z_V$, $Z_S/Z_V$, $Z_T/Z_V$ in the $\overline{\mathrm{MS}}$ scheme at $\mu =2\mathrm{GeV}$ as a function of the lattice momenta $|q|$ used in the RI-sMOM scheme. The rest is the same as in Fig. 11.

Figure 13

Data for the renormalization constants in the $\overline{\mathrm{MS}}$ scheme at $\mu =2\mathrm{GeV}$ as a function of the lattice momenta $|q|$ used in the regularization independent symmetric momentum (RI-sMOM) scheme. (Top) The data for $Z_A$, $Z_S$, and $Z_T$ for all combinations of $(p_1{)}_{\mu }$ and $(p_2{)}_{\mu }$ calculated are plotted using green circles. The subset of points that minimize $\sum _{\mu }(p_1{)}_{\mu }^4/((p_1{)}^2{)}^2+\sum _{\mu }(p_2{)}_{\mu }^4/((p_2{)}^2{)}^2$ for each $q^2$ are shown using red diamonds. The value and error using method B discussed in the text is given by the average and variance of the points shown in black over the range $9.2–11.2{\mathrm{GeV}}^2$. (Bottom) The data for the ratios $Z_A/Z_V$, $Z_S/Z_V$, and $Z_T/Z_V$ using the same symbols.

Figure 14

The 9-point fit using Eqs. (13) and (14) to the data for the renormalized isovector charges, $g_A^{u-d}$, $g_S^{u-d}$, and $g_T^{u-d}$, in the $\overline{\mathrm{MS}}$ scheme at 2 GeV. The result of the simultaneous extrapolation to the physical point defined by $a\to 0$, $M_{\pi }\to M_{{\pi }^0}^{\mathrm{phys}}=135\mathrm{MeV}$, and $L\to \infty$ are marked by a red star. The error bands in each panel show the simultaneous fit as a function of a given variable holding the other two at their physical value. The data are shown projected on to each of the three planes. The overlay in the middle figures with the dashed line within the grey band is the fit to the data versus $M_{\pi }^2$ neglecting dependence on the other two variables. The symbols used for data points from the various ensembles are defined in Table 1.

Figure 15

The 9-point fit using Eqs. (13) and (14) to the data for the renormalized isovector charges obtained using $Z_A{g}_A^{\mathrm{bare},u-d}$, $Z_S{g}_S^{\mathrm{bare},u-d}$, and $Z_T{g}_T^{\mathrm{bare},u-d}$. The rest is the same as in Fig. 14.

Figure 16

The 9-point extrapolation of the isovector charges including the chiral logarithm term, defined in Eqs. (11) and (12), in the fit Ansatz. The rest is the same as in Fig. 14.

Figure 17

The 8-point fit to the isovector charges neglecting the $a12m220S$ point. The rest is the same as in Fig. 14.

Figure 18

The 9-point simultaneous fits to the connected contributions to $g_T^u$ and $g_T^d$ versus $a$, $M_{\pi }^2$, and $M_{\pi }L$ using Eq. (13). The dependence on the three variables $M_{\pi }^2$, $a$, or $M_{\pi }L$ is small. The data symbols are defined in Table 1.

Figure 19

The 9-point simultaneous fits to the connected contributions to $g_A^u$ and $g_A^d$ versus $a$, $M_{\pi }^2$, and $M_{\pi }L$ using Eq. (13). The data for $g_A^u$ show a notable dependence on the lattice spacing $a$. The rest of the panels show little dependence on either $M_{\pi }^2$ or $M_{\pi }L$. The data symbols are defined in Table 1.

Figure 20

The 9-point simultaneous fits to the connected contributions to $g_S^u$ and $g_S^d$ versus $a$, $M_{\pi }^2$, and $M_{\pi }L$ using Eq. (14) plus a $c_3{M}_{\pi }^2$ term. The data show significant dependence on $M_{\pi }^2$ and are much larger in magnitude compared to $g_{A,T}^u$ and $g_{A,T}^d$. The data symbols are defined in Table 1.

Figure 21

Comparison of the nucleon effective mass obtained on the $a06m310$ and $a06m220$ ensembles with different smearing sizes. The lattice parameters are summarized in Table 18.

Figure 22

Comparison of the excited-state contamination in the extraction of unrenormalized isovector charges $g_{A,S,T,V}^{u-d}$ from the $a06m310$ ensemble. The plots on the left are with smearing parameters $\{\sigma =6.5,N_{\mathrm{GS}}=70\}$ and on the right with $\{\sigma =12,N_{\mathrm{GS}}=250\}$. Both calculations were done using the AMA method with parameters summarized in Table 18.

Figure 23

Comparison of the excited-state contamination in the extraction of unrenormalized isovector charges $g_{A,S,T,V}^{u-d}$ from the $a06m220$ ensemble. The plots on the left are with smearing parameters $\{\sigma =5.5,N_{\mathrm{GS}}=70\}$ and on the right with $\{\sigma =11,N_{\mathrm{GS}}=230\}$. Both calculations were done using the AMA method with parameters summarized in Table 18.

Figure 24

(Top) A summary plot showing the current estimates of $g_A^{u-d}$ from lattice-QCD calculations with $2+1+1$, $2+1$, and 2 flavors. The data are taken from the following sources: PNDME’16 (this work); LHPC’12 [44]; LHPC’10 [30]; RBC/UKQCD’08 [45]; RQCD’14 [46]; QCDSF/UKQCD’13 [47]; ETMC’15 [48]; CLS’12 [49]; RBC’08 [50]. (Bottom) The experimental results have been taken from the following sources: Mund’13 [2]; Mendenhall’12 [1]; Liu’10 [51]; Abele’02 [52]; Mostovoi’01 [53]; Liaud’97 [54]; Yerozolimsky’97 [55]; Bopp’86 [56]. The blue band highlights the 2014 PDG average value 1.2723(23) [5]. Note the change in scale between the upper (lattice QCD) and the lower (experimental) panels. The lattice-QCD estimates in red indicate that estimates of excited-state contamination, or discretization errors, or chiral extrapolation were not presented. When available, systematic errors have been added to statistical ones as outer error bars marked with dashed lines.

Figure 25

Fits to $F_{\pi }$ (left panel) and $g_A/F_{\pi }$ (right panel) data given in Table 24 assuming a linear dependence on $M_{\pi }^2$ for both. The lattice scale of these HISQ ensembles used to convert $M_{\pi }$ and $F_{\pi }$ to physical units was determined using $r_1$ [22]. Since the extrapolated $F_{\pi }$, red star, matches the experimental value at the physical point, the value of the renormalization factor independent ratio $g_A/F_{\pi }$ remains about 7% below the experimental result shown as a black star.

Figure 26

A summary plot showing estimates for $g_S^{u-d}$ from lattice QCD and phenomenology. The data are taken from the following sources: PNDME’16 (this work); LHPC’12 [57]; PNDME’11 [7]; RQCD’14 [46]. The estimates based on the conserved vector current and phenomenology are taken from Gonzalez-Alonso [58] and Adler [59]. The rest is the same as in Fig. 24.

Figure 27

A summary plot showing estimates for $g_S^{u-d}$ from lattice QCD and phenomenology. The data are taken from the following sources: PNDME’16 (this work); PNDME’15 [11]; LHPC’12 [57]; RBC/UKQCD’10 [62]; ETMC’15 [48]; RQCD’14 [46]; and RBC’08 [50]. The phenomenological estimates are taken from the following sources: Kang’15 [63]; Goldstein’14 [64]; Pitschmann’14 [65]; Anselmino [66]; Bacchetta’13 [67]; and Fuyuto [68]. The rest is the same as in Fig. 24.

Figure 28

Left panel: current 90% C.L. constraints on $({\epsilon }_S)$ and $({\epsilon }_T)$ from beta decays ($\pi \to e\nu \gamma$ and $0^{+}\to 0^{+}$) and the LHC ($pp\to e\nu +X$) at $\sqrt{s}=8\mathrm{TeV}$. Right panel: prospective 90% C.L. constraints on $({\epsilon }_S)$ and $({\epsilon }_T)$ from beta decays and the LHC ($pp\to e\nu +X$) at $\sqrt{s}=14\mathrm{TeV}$. The low-energy constraints correspond to 0.1% measurements of $B$, $b$ in neutron decay and $b$ in ${}^6\mathrm{He}$ decay. In both panels we present the low-energy constraints under two different scenarios for the scalar and tensor charges $g_{S,T}$: quark model [75] (large dashed contour) and lattice QCD results given in Table 22 (smaller solid contour).