Nucleon isovector couplings from $N_f=2$ lattice QCD

We compute the axial, scalar, tensor and pseudoscalar isovector couplings of the nucleon as well as the induced tensor and pseudoscalar charges in lattice simulations with $N_{\mathrm{f}}=2$ mass-degenerate nonperturbatively improved Wilson-Sheikholeslami-Wohlert fermions. The simulations are carried out down to a pion mass of 150 MeV and linear spatial lattice extents of up to 4.6 fm at three different lattice spacings ranging from approximately 0.08 fm to 0.06 fm. Possible excited state contamination is carefully investigated and finite volume effects are studied. The couplings, determined at these lattice spacings, are extrapolated to the physical pion mass. In this limit we find agreement with experimental results, where these exist, with the exception of the magnetic moment. A proper continuum limit could not be performed, due to our limited range of lattice constants, but no significant lattice spacing dependence is detected. Upper limits on discretization effects are estimated and these dominate the error budget.

Figure 1

Overview of the ensembles listed in Table 1. Colors encode the lattice spacings and symbols the lattice extents. The color and symbol labeling defined here will be used throughout in Secs. 3, 4, 5. The horizontal lines separate different volume ranges.

Figure 2

Effective nucleon masses Eq. (13) for five of our ensembles, computed from smeared-smeared two-point functions $C_{2\mathrm{pt}}(t_{\mathrm{f}})$.

Figure 3

The same as Fig. 2 for the pion effective mass.

Figure 4

The renormalized ratio equation (19) for the example of $g_A$ obtained on ensemble VIII ($m_{\pi }\approx 150\mathrm{MeV}$, $a\approx 0.071\mathrm{fm}$) for three different values of $t_{\mathrm{f}}$. The shaded region represents the result of a constant fit in the range $t/a\in [4,11]$ to the $t_{\mathrm{f}}=15a$ data.

Figure 5

The ratio of the three- over the two-point function for $g_A$ at $t_{\mathrm{f}}=17a$ on ensemble IX ($m_{\pi }\approx 490\mathrm{MeV}$, $a\approx 0.060\mathrm{fm}$) with different smearing methods.

Figure 6

The combination $C_{3\mathrm{pt}}(t,t_{\mathrm{f}})/(A_0{e}^{-m_N{t}_{\mathrm{f}}})$ with $t_{\mathrm{f}}/a\in \{9,12,15\}$ on ensemble VIII ($m_{\pi }\approx 150\mathrm{MeV}$, $a\approx 0.071\mathrm{fm}$), multiplied by the appropriate renormalization factors to give $g_S^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$. $A_0{e}^{-m_N{t}_{\mathrm{f}}}$ corresponds to the ground state contribution to $C_{2\mathrm{pt}}(t_{\mathrm{f}})$ obtained from a simultaneous fit according to Eqs. (14) and (15) to $C_{3\mathrm{pt}}$ and $C_{2\mathrm{pt}}$. The fit ranges were $t_{\mathrm{f}}/a\in [2,26]$ for $C_{2\mathrm{pt}}$ and $\delta t=2a$ for $C_{3\mathrm{pt}}$ where $B_1$ is set to zero. Also shown are the resulting fit curves for each $t_{\mathrm{f}}$. The shaded region indicates the fitted value of $g_S^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$ and the corresponding statistical uncertainty.

Figure 7

The same as Fig. 6 for $g_T$.

Figure 8

The same as Fig. 7 on ensemble IV ($m_{\pi }\approx 290\mathrm{MeV}$, $a\approx 0.071\mathrm{fm}$) and $t_{\mathrm{f}}/a\in \{7,9,11,13,15,17\}$.

Figure 9

The same as Fig. 5 for $g_T$.

Figure 10

Ratios of renormalized three- over two-point functions, giving $g_A$ in the limit $0\ll t\ll t_{\mathrm{f}}$ for four of our ensembles.

Figure 11

The same as Fig. 10 for $g_T^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$.

Figure 12

Results on $g_S^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$ obtained with the summation method equation (21) for different fit ranges $t_{\mathrm{f}}\in [t_{\mathrm{f},\min },t_{\mathrm{f},\max }]$ and $\delta t/a\in \{2,3\}$ on ensemble IV ($m_{\pi }\approx 290\mathrm{MeV}$, $a\approx 0.071\mathrm{fm}$). The error band corresponds to the result obtained with the fit method detailed in Sec. 2b, including our assignment of systematic errors. All data are normalized with respect to the $\overline{\mathrm{MS}}$ scheme. The error of the renormalization factor is smaller by more than one order of magnitude than any of the statistical errors displayed and can be neglected.

Figure 13

The same as Fig. 12 for $g_T^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$.

Figure 14

$g_V/Z_V\equiv g_V^{\text{lat}}(1+am{b}_V)$ as a function of $m_{\pi }^2$ for all ensembles. Symbols are as in Fig. 1. Shown as solid bands are the $1/Z_V$ values determined nonperturbatively [68] (updated in Table 3) for the three $\beta$ values.

Figure 15

$g_A$ as a function of $m_{\pi }^2$ for all ensembles. Symbols are as in Fig. 1: the square corresponds to $L{m}_{\pi }\approx 6.7$, circles to $L{m}_{\pi }>4.1$, stars to $L{m}_{\pi }\in [3.4,4.1]$ and the triangle to $L{m}_{\pi }\approx 2.8$. The line drawn to guide the eye represents the result of a linear fit to the four $m_{\pi }<430\mathrm{MeV}$ points with $L{m}_{\pi }>4.1$.

Figure 16

$g_A$ as a function of $m_{\pi }^2$: our results [RQCD, nonperturbatively improved (NPI) Wilson clover] in comparison to other results (fermion action used in brackets). $N_{\mathrm{f}}=2$: QCDSF [26] (NPI Wilson clover), Mainz [28] (NPI Wilson clover), ETMC [29] (twisted mass). $N_{\mathrm{f}}=2+1$: LHPC [23] (HEX-smeared Wilson clover), RBC/UKQCD [27] (domain wall). $N_{\mathrm{f}}=2+1+1$: ETMC [35] (twisted mass), PNDME [39] (Wilson clover on a HISQ staggered sea). Also indicated as a shaded area is the result from extrapolating our $g_A/F_{\pi }$ data to the physical point, see Sec. 4.

Figure 17

The combination $[m_{\pi }(L)-m_{\pi }]/m_{\pi }^3$ as a function of the linear lattice extent, in comparison with the leading order [76] [Eq. (28)] and NNLO [80] chiral perturbation theory expectations.

Figure 18

The ratio $g_A(L)/F_{\pi }(L)$ as a function of the linear lattice extent for three different pion masses. The error bands are the predictions of Eq. (32), multiplied by constants $g_A(\infty )/F_{\pi }$ to match the three data sets. The widths of the error bands are from varying the ratio $g_A^0/g_A(\infty )\in [0.9,1.1]$.

Figure 19

$g_A/F_{\pi }$ as a function of $m_{\pi }^2$ for all ensembles, together with a linear fit to the low mass points, omitting the smallest volume (ensemble VII). Symbols are as in Fig. 1.

Figure 20

$g_S^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$ as a function of $m_{\pi }^2$ for all ensembles. Symbols are as in Fig. 1. Also shown is a linear extrapolation in $m_{\pi }^2$ to the physical point.

Figure 21

$g_S^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$ as a function of $m_{\pi }^2$: our results (RQCD, NPI Wilson clover) in comparison to other results. $N_{\mathrm{f}}=2+1$: LHPC [32] (HEX-smeared Wilson clover). $N_{\mathrm{f}}=2+1+1$: PNDME [39] (Wilson clover on a HISQ staggered sea) and ETMC [40] (twisted mass). Also included is the linear extrapolation of our data points.

Figure 22

$g_T^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$ as a function of $m_{\pi }^2$ for all ensembles. Symbols are as in Fig. 1. Also shown is a linear extrapolation in $m_{\pi }^2$ to the physical point.

Figure 23

$g_T^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$ as a function of $m_{\pi }^2$: our results (RQCD, NPI Wilson clover) in comparison to other results. $N_{\mathrm{f}}=2$: ETMC [31] (twisted mass). $N_{\mathrm{f}}=2+1$: RBC/UKQCD [30] (domain wall), LHPC [32] (HEX smeared Wilson clover). $N_{\mathrm{f}}=2+1+1$: PNDME [39] (Wilson clover on a HISQ staggered sea), ETMC [31] (twisted mass). Also included is the linear extrapolation of our data.

Figure 24

${\tilde{g}}_T(Q^2)/g_V(Q^2)$ (left panel) and ${\tilde{g}}_T(Q^2)$ as functions of the virtuality $Q^2$ at $m_{\pi }\approx 290\mathrm{MeV}$ for three volumes (ensembles IV, V and VI).

Figure 25

The isovector induced tensor charge ${\tilde{g}}_T={\kappa }_{u-d}$ as a function of $m_{\pi }^2$. Symbols are as in Fig. 1. Also shown is a linear extrapolation in $m_{\pi }^2$ to the physical point.

Figure 26

The isovector anomalous magnetic moment ${\tilde{g}}_T$ as a function of $m_{\pi }^2$: our results (RQCD, NPI Wilson clover) in comparison to other results (fermion action used in brackets). $N_{\mathrm{f}}=2$: QCDSF [42] (NPI Wilson clover), Mainz [28, 46] (NPI Wilson clover), ETMC [43] (twisted mass). $N_{\mathrm{f}}=2+1$: LHPC [45] (HEX smeared Wilson clover), RBC/UKQCD [38] (domain wall). $N_{\mathrm{f}}=2+1+1$: ETMC [35] (twisted mass), PNDME [39] (Wilson clover on a HISQ staggered sea). Also included is the linear extrapolation of our data.

Figure 27

The ratio of form factors ${\tilde{g}}_P(Q^2)/g_A(Q^2)$, normalized with respect to the single pole dominance expectation, as a function of the virtuality $Q^2$. Data from all 11 ensembles are plotted on top of each other. Symbols are as in Fig. 1. Deviations from unity quantify violations of the pole dominance model.

Figure 28

Extrapolation of the induced pseudoscalar form factor to the muon capture point $Q^2=0.88m_{\mu }^2$ (vertical line) for three values of the pion mass (ensembles III, VI and VIII). The error bands correspond to fits according to Eq. (41).

Figure 29

Chiral extrapolation of the induced pseudoscalar coupling $g_P^{*}$. The error band corresponds to the parametrization equation (42). Symbols are as in Fig. 1.

Figure 30

The Goldberger-Treiman ratio $m_N{g}_A/F_{\pi }$ as a function of the squared pion mass. Symbols are as in Fig. 1. The line indicates a linear extrapolation of $L{m}_{\pi }>4.1$ data. The experimental values for $g_{\pi NN}$ (black triangles) are from Refs. [93, 94, 95].

Figure 31

The pseudoscalar charge $g_P^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$, defined in Eq. (46), as a function of the squared pion mass. Symbols are as in Fig. 1. The physical point (Phys.) is obtained dividing the experimental value of $m_N{g}_A$ by the $\overline{\mathrm{MS}}$-scheme quark mass of Ref. [78]. The $1/m_{\pi }^2$ curve is drawn to guide the eye.