Opposite-parity contaminations in lattice nucleon form factors

The recently introduced parity expanded variational analysis (PEVA) technique allows for the isolation of baryon eigenstates at finite momentum free from opposite-parity contamination. In this paper, we establish the formalism for computing form factors of spin-$1/2$ states using PEVA. Selecting the vector current, we compare the electromagnetic form factors of the ground state nucleon extracted via this technique to a conventional parity-projection approach. Our results show a statistically significant discrepancy between the PEVA and conventional analyses. This indicates that existing calculations of matrix elements of ground state baryons at finite momentum can be affected by systematic errors of $\sim 20\%$ at physical quark masses. The formalism introduced here provides an effective approach to removing these systematic errors.

Figure 1

The contribution of the doubly represented quark flavor to the electric form factor of the ground-state nucleon at $m_{\pi }=156\mathrm{MeV}$, for the lowest-momentum kinematics, providing $Q^2=0.1655(48){\mathrm{GeV}}^2$. Our fits to the plateaus are illustrated by shaded bands. We plot the conventional analysis with open markers and dashed fit lines and the new PEVA analysis with filled markers and solid fit lines. The source is at time slice 16, and the current is inserted at time slice 21. Both the conventional and PEVA fits are from time slice 24–27.

Figure 2

The contribution of the singly represented quark flavor to the electric form factor of the ground-state nucleon. The conventions used in this plot are the same as used in Fig. 1. The kinematics are also the same, with $m_{\pi }=156\mathrm{MeV}$, and $Q^2=0.1655(48){\mathrm{GeV}}^2$. Both the conventional and PEVA fits are from time slice 24–27.

Figure 3

The contribution of the doubly represented quark flavor to the electric form factor of the ground-state nucleon at $m_{\pi }=570\mathrm{MeV}$, for the lowest-momentum kinematics, providing $Q^2=0.1444(44){\mathrm{GeV}}^2$. The conventions used in this plot are the same as used in Fig. 1. The PEVA fits start at time slice 26, whereas the conventional fits start at time slice 28. Note that this plot has been scaled up significantly to make the difference between the two fits visible.

Figure 4

Contributions of $u_p$ to the ground state $G_E(Q^2)$ at $m_{\pi }=156\mathrm{MeV}$ for $\mathit{p}=|{\mathit{q}}_{\min }|(-1,0,0)$ and ${\mathit{p}}^{\prime }=|{\mathit{q}}_{\min }|(1,0,0)$, providing $Q^2=0.690(20){\mathrm{GeV}}^2$. The conventions used in this plot are the same as in Fig. 1. The PEVA fit starts from time slice 23, but the conventional analysis starts from time slice 24.

Figure 5

Contributions of $d_p$ to the ground state $G_E(Q^2)$. Pion mass and kinematics are as in Fig. 4 above. The conventions used in this plot are the same as in Fig. 1. The PEVA fit starts from time slice 23, but the conventional analysis starts from time slice 24.

Figure 6

Contributions from individual quark flavors to the electric form factor of the ground-state nucleon at $m_{\pi }=156\mathrm{MeV}$. The shaded regions are dipole fits to the form factor, with lines indicating the central values. The $y$-axis intercept is fixed to one, as we are using an improved conserved vector current and the quarks are taken with unit charge. The errors on these fits are small enough that the shaded bands are barely distinguishable from the central lines. The fits correspond to a charge radius of 0.684(19) fm for the doubly represented quark ($u_p$) and 0.659(21) fm for the singly represented quark ($d_p$).

Figure 7

$G_E(Q^2)$ for the ground-state proton and neutron at $m_{\pi }=156\mathrm{MeV}$. The shaded region corresponds to a dipole fit to the proton form factor, with a charge radius of 0.691(19) fm.

Figure 8

Quark-mass dependence of charge radii from dipole fits to the isovector combination $G_{Ep}(Q^2)-G_{En}(Q^2)$. We see a clear trend to larger radii as the pion mass approaches the physical point, represented by the dashed vertical line.

Figure 9

The contributions to the magnetic form factor from single quarks of unit charge for the ground-state nucleon at $m_{\pi }=156\mathrm{MeV}$ for the lowest-momentum kinematics, providing $Q^2=0.1655(48){\mathrm{GeV}}^2$. We plot the conventional analysis with open markers and the new PEVA analysis with filled markers. Our fits to the plateaus are illustrated by shaded bands, with the central value indicated by dashed lines for the conventional analysis, and solid lines for the PEVA analysis. The plateau regions for both analyses are consistent, starting from time slice 23 for all four fits, but the value of the conventional plateau for the singly represented quark ($d_p$) has a magnitude approximately 35% lower than the PEVA plateau.

Figure 10

The contributions to the magnetic form factor from single quarks of unit charge for the ground-state nucleon at $m_{\pi }=570\mathrm{MeV}$ for the lowest-momentum kinematics, providing $Q^2=0.1444(44){\mathrm{GeV}}^2$. The plateau regions for both analyses are consistent, starting from time slice 23 for all four fits, but the value of the conventional plateau for the singly represented quark ($d_p$) has a magnitude slightly lower than the PEVA plateau.

Figure 11

Ratios of conventional plateaus to PEVA plateaus for ground state $G_M(Q^2)$ at $m_{\pi }=156\mathrm{MeV}$. For clarity, ratios with large statistical errors have been excluded from the plot. If the plateaus were consistent, the points should be distributed about 1.0. For the doubly represented quark flavor ($u_p$), some kinematics match this expectation, but others sit more than one standard deviation low. The singly represented quark flavor ($d_p$) appears to show an even larger effect, with the ratios shifting even further away from unity albeit with larger statistical errors.

Figure 12

Quark-flavor contributions to ground state $G_M(Q^2)$ at $m_{\pi }=156\mathrm{MeV}$. The shaded regions are dipole fits to the form factor, corresponding to magnetic radii of 0.514(30) fm for the doubly represented quark flavor ($u_p$) and 0.85(11) fm for the singly represented quark flavor ($d_p$).

Figure 13

$G_M(Q^2)$ for the ground-state proton and neutron at $m_{\pi }=156\mathrm{MeV}$. The shaded region corresponds to a dipole fit to the form factor, with a magnetic radius of 0.551(29) fm for the proton and 0.618(31) fm for the neutron. We also include the isovector combination ($G_{Mp}(Q^2)-G_{Mn}(Q^2)$), which is insensitive to disconnected loop corrections.

Figure 14

Quark-mass dependence of charge radii obtained from dipole fits to the isovector magnetic moment ($G_{Mp}(Q^2)-G_{Mn}(Q^2)$). As in Fig. 8, the dashed line corresponds to the physical pion mass.

Figure 15

${\mu }_{\mathrm{Eff}}(Q^2)$ for individual quark flavors in the ground state nucleon at $m_{\pi }=156\mathrm{MeV}$. The narrow shaded bands are constant fits to the effective magnetic moment. They correspond to magnetic moment contributions of 1.734(56) ${\mu }_N$ for the doubly represented quark and $-0.616(44)$ ${\mu }_N$ for the singly represented quark.

Figure 16

Pion-mass dependence of contributions to the ground-state magnetic moment from the doubly represented quark sector ($u_p$) and the singly represented quark sector ($d_p$). The vertical dashed line shows the physical pion mass.

Figure 17

Pion-mass dependence of the extracted magnetic moment for the ground-state proton and neutron. To cancel out any disconnected loop contributions, we plot the isovector combination ${\mu }_p-{\mu }_n$. As the physical point is approached, the trend in this combination approaches but does not quite reach the physical value of $4.70{\mu }_N$ [19], represented by a black star.

Figure 18

Pion-mass dependence of the extracted magnetic moment for the isovector combination with finite-volume corrections from Ref. [26]. At the physical point, we present chiral extrapolations of the PEVA and conventional results with filled and open stars respectively. The PEVA result agrees well with the physical value of $4.70{\mu }_N$ [19], represented by a black star.