Nucleon axial and pseudoscalar form factors using twisted-mass fermion ensembles at the physical point

We compute the nucleon axial and pseudoscalar form factors using three $N_f=2+1+1$ twisted-mass fermion ensembles with all quark masses tuned to approximately their physical values. The values of the lattice spacings of these three physical point ensembles are 0.080, 0.068, and 0.057 fm and spatial sizes 5.1, 5.44, and 5.47 fm, respectively, yielding $m_{\pi }L>3.6$. Convergence to the ground-state matrix elements is assessed using multistate fits. We study the momentum dependence of the three form factors and check the partially conserved axial-vector current (PCAC) hypothesis and the pion pole dominance (PPD). We show that in the continuum limit, the PCAC and PPD relations are satisfied. We also show that the Goldberger-Treimann relation is approximately fulfilled and determine the Goldberger-Treiman discrepancy. Our final results are $g_A=1.245(28)(14)$ for the nucleon axial charge, $⟨r_A^2⟩=0.339(48)(06){\text{fm}}^2$ for the axial radius, $g_{\pi NN}\equiv \underset{Q^2\to -m_{\pi }^2}{lim}{G}_{\pi NN}(Q^2)=13.25(67)(69)$ for the pion-nucleon coupling constant, and $G_P(0.88m_{\mu }^2)\equiv g_P^{*}=8.99(39)(49)$ for the induced pseudoscalar form factor at the muon capture point.

Figure 1

Nucleon effective mass using the three physical point ensembles. The dashed line is the value of the nucleon mass $m_N=0.938\mathrm{GeV}$.

Figure 2

Nucleon effective mass versus the time separation (left column) and results for the nucleon mass obtained via fits that have a model probability (fit prob.) larger than 1% (right column). The horizontal bands spanning both the left and right panels are the results of the model average among all fits using two states (red points and red band) and three states (blue points and blue band). The most probable fit is depicted with open symbols. In the left panel, we show for all ensembles the result of the most probable fit using two states (red curve) and three states (blue curve) over the range used in the fit. Panels from top to bottom are for the cB211.72.64, cC211.60.80, and cD211.54.96 ensembles, respectively.

Figure 3

The ratio $R^{\prime }$ of Eq. (42) that yields $g_A$ versus $t_{\mathrm{ins}}-t_s/2$ (left column) and versus $t_s$ for $t_{\mathrm{ins}}=t_s/2$ (middle column). In the header of the figure, we give the symbols used to denote the various values of $t_s/a$. In the right column, we show the value of the nucleon isovector axial charge obtained via fits, as in Eq. (41), versus the fit probability, using the notation of Fig. 2. In the left and middle panels, the curves correspond to the fit results, which have the largest probability among all two-state (blue) or three-state (red) fits.

Figure 4

The same as in Fig. 3, but for $G_A(Q^2)$ obtained via ${\overline{\mathrm{\Pi }}}_{AP}(Q^2,0)$ for the lowest nonzero value of $Q^2$.

Figure 5

${\overline{\mathrm{\Pi }}}_0(Q^2)$ for the lowest nonzero value of $Q^2$ using the notation of Fig. 3.

Figure 6

${\overline{\mathrm{\Pi }}}_{AP}(Q^2)$ for the lowest nonzero value of $Q^2$ using the notation of Fig. 3.

Figure 7

$G_5(Q^2)$ for the lowest nonzero value of $Q^2$ using the notation of Fig. 3.

Figure 8

Cumulative distributions of the results from three-state fits weighted by the fit probability. The black curve represents the cumulative distribution; the colored curves are the Gaussian distributions associated with each fit; and the dashed red curve depicts the Gaussian distribution associated with the outcome of the model average given in Eq. (55). For each panel, all distributions are normalized with the maximum value of the cumulative distribution, and we have doubled the height of the colored curves for visualization purposes.

Figure 9

Continuum limit of the nucleon isovector axial charge $g_A$ (top), radius $r_A$ (middle), and nucleon mass $m_N$ (bottom) using a linear extrapolation in $a^2$. The dashed line in the top panel is the experimental value $g_A=1.27641(56)$ [67] and in the bottom panel is the nucleon mass $m_N=938\mathrm{MeV}$.

Figure 10

Nucleon $E_N(\overrightarrow{p})=\sqrt{m_N^2+{\overrightarrow{p}}^2}$ (top) and first excited state $E_{1,\overrightarrow{0}}$ (middle) at sink and $E_{1,\overrightarrow{p}}$ (bottom) at source as a function of ${\overrightarrow{p}}^2$. These are determined by performing a two-state fit to the two- and three-point functions. To guide the eye, we depict the expected dispersion relations: the energy of a boosted nucleon $E_N$ (top); a noninteracting $\pi N$ state with either total momentum zero, $N(-\overrightarrow{p})+\pi (\overrightarrow{p})$ (middle), or with momentum $\overrightarrow{p}$, $N(\overrightarrow{0})+\pi (\overrightarrow{p})$ (bottom); and the energy or the first excited state of the nucleon, the Roper with $m_R=1.44\mathrm{GeV}$, either at rest [$R(0)$, middle] or moving with momentum $\overrightarrow{p}$ [$R(\overrightarrow{p})$, bottom].

Figure 11

Results for the three form factors obtained using a two- (red squares) or a three-state (blue crosses) fit analysis. From left to right, we show results for the cB211.72.64, cC211.60.80, and cD211.54.96 ensembles. From top to bottom, we show results for the axial, induced pseudoscalar, and pseudoscalar form factors. In the lower panel of each plot, we include the difference $\mathrm{\Delta }$ defined in Eq. (58) between the results extracted using two- and three-state fits. The difference $\mathrm{\Delta }$ is normalized such that errors are unity, and the dashed line represents a three standard deviation difference. Three-state fits are stable for $Q^2<0.465{\mathrm{GeV}}^2$ for all three ensembles. For larger values, the fits become unstable. We display these points by using lighter blue color, and we thus do not use them in the extraction of form factors.

Figure 12

The axial form factor obtained on each of the three ensembles using two-state fits (blue circles, orange downwardpointing triangles, and green triangles). The continuum limit form factor (red curve and band) and value for the axial charge (red cross) are obtained via dipole fits within the one-step approach. Also shown are the form factor curves obtained at the three values of the lattice spacing (blue, orange, and green curves, respectively).

Figure 13

The axial charge (top) and radius (bottom) obtained via a dipole fit to each ensemble (blue circles, orange downwardpointing triangles, and green triangles). The filled red asterisks and corresponding bands show the continuum limit using the two-step approach, i.e. via fits linear in $a^2$ to the axial charge and radius obtained at each value of $a^2$. At $a^2=0$, we compare with results obtained from a one-step approach (open stars), as described in the text.

Figure 14

Results for the axial charge and radius extracted using the dipole ansatz, versus $Q_{\max }^2$, i.e. the maximum value of $Q^2$ included in the fit. We show separately dipole fits to results obtained from two-state (red squares) and three-state (blue diamonds) fits to the three- and two-point correlation functions. For the two-state fit case, we include a variation in which the values of the form factors at $Q^2=0$ are not included in the fit (open squares).

Figure 15

The same as in Fig. 13, but using the first-order $z$-expansion to fit the $Q^2$-dependence.

Figure 16

The axial charge (top) and radius (bottom) obtained via first-order $z$-expansion fits to each ensemble. The notation is the same as in Fig. 13.

Figure 17

Results for the axial charge and radius extracted using the first-order $z$-expansion, versus $Q_{\max }^2$. The notation is the same as in Fig. 14.

Figure 18

Results for the axial charge and radius as a function of $Q_{\max }^2$ obtained from using the $z^3$-expansion to parametrize the $Q^2$-dependence. For each $Q_{\max }^2$, we depict five points having prior width $w=1$, 2, 3, 4, 5. The points are shifted to the right as $w$ increases with an increasing symbol size.

Figure 19

The axial charge (top) and radius (bottom) obtained via third-order $z$-expansion fits and model average for each ensemble. The notation is the same as that in Fig. 13.

Figure 20

Results for the isovector axial charge and radius extracted using four different approaches as described in the text. Numerical values are given in Table 9.

Figure 21

Continuum limit of $G_A(Q^2)$ using the $z^3$-expansion and data from the two-state fit analysis of the correlators up to $Q^2=1{\mathrm{GeV}}^2$ for the three ensembles with the symbols as indicated in the header of the figure.

Figure 22

Continuum limit of $G_A(Q^2)$ using the $z^3$-expansion and data from the three-state fit analysis of the correlators up to $Q^2=0.47{\mathrm{GeV}}^2$ for the three ensembles with the symbols as indicated in the header of the figure.

Figure 23

Results on $G_A(Q^2)$ at the continuum limit when fitting data extracted from the two-state (red band) and three-state (blue band) fit analysis of the correlators. The darker blue curve indicates up to which $Q^2$ we had data for the three-state fit analysis. The yellow band is when we added systematic errors to the parameters that define the red curve as discussed in the text. The parameters of the fit are given in Eq. (78).

Figure 24

$G_A(Q^2)/G_P(Q^2)$ (top) and $r_{\mathrm{PPD},2}$ (bottom) for the three ensembles. The blue, orange, and green curves show the results of combined linear fit in $Q^2$ for the cB211.72.64, cC211.60.80, and cD211.54.96, respectively, using the form of Eq. (80) to take into account cutoff effects. The red band is the continuum extrapolation after performing the model average as described in the text. The dashed line shows the expected value of the ratios if PPD close to the pion pole is satisfied, namely a slope of $1/4m_N^2$ from Eq. (79) for the first ratio and unity for the second.

Figure 25

The ratio $r_{\mathrm{PPD},1}$ for the cB211.72.64 (blue circles), cC211.60.80 (orange downward-pointing triangles), and cD211.54.96 (green upward-pointing triangles) ensembles when using the unitary pion mass $m_{\pi }^{\mathrm{TM}}$ (open symbols) and the OS pion mass $m_{\pi }^{\mathrm{OS}}$ (filled symbols) to remove the pion pole. The pion mass values are given in Table 10. The dashed line shows the expected value $r_{\mathrm{PPD},1}=1$ based on PPD.

Figure 26

Induced pseudoscalar coupling, $g_P^{*}$, and pion-nucleon coupling, $g_{\pi NN}$, from a $z^3$-expansion fit as a function of the largest $Q^2$ used in the fit, $Q_{\max }^2$, and the width of the priors. For each $Q_{\max }^2$, we depict five points having prior width of $w=1$, 2, 3, 4, 5. The points are shifted to the right as $w$ increases with an increasing symbol size.

Figure 27

Results on $G_P(Q^2)$ for each ensemble (blue band for the cB211.72.64, orange band for the cC211.60.80, and green band for the cD211.54.96 ensemble) and at the continuum limit (red band) using the $z^3$-expansion to fit the $Q^2$-dependence of the data determined from the two-state fit analysis up to $Q^2=1{\mathrm{GeV}}^2$. The inner panel shows a zoom-in of the region marked by the red square.

Figure 28

Results on $G_P(Q^2)$ using the $z^3$-expansion to fit the $Q^2$-dependence of the data determined from the three-state fit analysis of the correlators up to $Q^2=0.47{\mathrm{GeV}}^2$. The notation is the same as that for Fig. 27.

Figure 29

Results on ${\tilde{G}}_5(Q^2)$ as defined in Eq. (86) for each ensemble (blue band for the cB211.72.64, orange band for the cC211.60.80, and green band for the cD211.54.96 ensemble) and at the continuum limit (red band) using the $z^3$-expansion to fit the $Q^2$-dependence of the data determined from the two-state fit analysis up to $Q^2=1{\mathrm{GeV}}^2$. The inner panel shows a zoom-in of the region marked by the red square.

Figure 30

Results on ${\tilde{G}}_5(Q^2)$ as defined in Eq. (86) using the $z^3$-expansion to fit the $Q^2$-dependence of the data determined from the three-state fit analysis of the correlators up to $Q^2=0.47{\mathrm{GeV}}^2$. The notation is the same as that of Fig. 29.

Figure 31

Results per gauge ensemble for $G_P(Q^2)$ (left) and ${\tilde{G}}_5(Q^2)$ (right) when using the data for $G_{\mathrm{w}\mathrm{pole}}(Q^2)$ (open symbols) compared to those when using $G_{\text{improved}}(Q^2)$ (filled symbols) of Eq. (92) by correcting for the pole OS pion mass. The continuum limit form factors (red band) are those determined in Figs. 27 and 29 using the data for $G_{\mathrm{w}\mathrm{pole}}$.

Figure 32

Results on $(Q^2+m_{\pi }^2)G_P(Q^2)$ at the continuum limit when fitting data extracted from the two-state (red band) and three-state (blue band) fit analysis of the correlators. The darker blue curve indicates up to which $Q^2$ we had data for the three-state fit analysis. The yellow band is when we added systematic errors to the parameters that define the red curve as discussed in the text. The parameters of the fit are given in Eq. (94).

Figure 33

Results on $(Q^2+m_{\pi }^2){\tilde{G}}_5(Q^2)$ at the continuum limit when fitting data extracted from the two-state (red band) and three-state (blue band) fit analysis of the correlators. The darker blue curve indicates up to which $Q^2$ we had data for the three-state fit analysis. The yellow band is when we added systematic errors to the parameters that define the red curve as discussed in the text. The parameters of the fit are given in Eq. (96).

Figure 34

Top: $r_{\mathrm{PCAC}}$ as defined in Eq. (24). The blue, orange, and green curves are the result of the combined fits to $G_A(Q^2)$, $(m_{\pi }^2+Q^2)G_P(Q^2)$, and $(m_{\pi }^2+Q^2){\tilde{G}}_5(Q^2)$ for the cB211.72.64, cC211.60.80, and cD211.54.96 ensembles, respectively. The fits are done using the $z^3$-expansion to fit the $Q^2$-dependence of the data determined from the two-state fit analysis up to $Q^2=1{\mathrm{GeV}}^2$. The red curve and band show the results at the continuum limit. The yellow band shows the errors on the curve when systematic effects are taken into account, i.e. using the final parametrization of the form factors given in Eqs. (78), (94), and (96) for $G_A(Q^2)$, $(m_{\pi }^2+Q^2)G_P(Q^2)$, and $(m_{\pi }^2+Q^2){\tilde{G}}_5(Q^2)$, respectively. Bottom: same as top panel, but using three-state fit data and the respective $z^3$-expansion to fit the $Q^2$-dependence up to $Q^2\sim 0.5{\mathrm{GeV}}^2$.

Figure 35

$r_{\mathrm{PPD},1}$ as defined in Eq. (25). The notation is the same as that in the top panel of Fig. 34.

Figure 36

From left to right, we show our lattice QCD results from this work on $g_A$, $⟨r_A^2⟩$, $g_P^{*}$, and $g_{\pi NN}$ (red stars). The open circles show results extracted using only the cB211.72.64 [19] and the PCAC and PPD relations to extract $g_P^{*}$ and $g_{\pi NN}$ from $G_A(Q^2)$.

Figure 37

From left to right, we show recent lattice QCD results on $g_A$, $⟨r_A^2⟩$, $g_P^{*}$, and $g_{\pi NN}$. Our results are shown with the red star and red error band. The blue triangles show the recent results by PNDME [23], the green triangles show the results by RQCD [18, 78], the yellow squares show the results by NME [48], the gray diamonds show the results by the Mainz group [22], and the magenta square shows the results by CalLat [14]. The cyan circle shows the FLAG21 average of lattice results published at the time of the report [79].

Figure 38

Top: $G_A(Q^2)$ determined within this work using the parameters $a_{2\text{−}\mathrm{state}}$ (red solid line and band) and when including the systematic uncertainty as the difference in the central values of $a_{2\text{−}\mathrm{state}}$ and $a_{3\text{−}\mathrm{state}}$ (yellow band). We compare to the fit to the deuterium bubble-chamber data [25] shown by the green dashed line with error band and with the fit to the recent $\mathrm{MINER}\nu \mathrm{A}$ antineutrino-hydrogen data [1] shown by the blue dot-dashed line with error band. Bottom: $G_A(Q^2)$ determined within this work compared to two recent lattice QCD calculations: (i) by the Mainz group [22] shown with the gray dashed line with its error band and (ii) by PNDME [23] shown with the blue dashed line with its error band.

Figure 39

Results on $G_P(Q^2)$ determined directly from our lattice data using the parameters $a_{2\text{−}\mathrm{state}}$ (red solid line and band) and when including the systematic uncertainty as the difference in the central values of $a_{2\text{−}\mathrm{state}}$ and $a_{3\text{−}\mathrm{state}}$ (yellow band) are compared to those obtained by using PPD given in Eq. (20) and to our data on $G_A(Q^2)$ (dark blue and light blue when including the systematic error).