Nucleon electromagnetic form factors from lattice QCD using a nearly physical pion mass

We present lattice QCD calculations of nucleon electromagnetic form factors using pion masses $m_{\pi }=149$, 202, and 254 MeV and an action with clover-improved Wilson quarks coupled to smeared gauge fields, as used by the Budapest-Marseille-Wuppertal Collaboration. Particular attention is given to the removal of the effects of excited-state contamination by calculations at three source-sink separations and the use of the summation and generalized pencil-of-function methods. The combination of a calculation at the nearly physical mass $m_{\pi }=149\mathrm{MeV}$ in a large spatial volume ($m_{\pi }{L}_s=4.2$) and the removal of excited-state effects yields agreement with experiment for the electric and magnetic form factors $G_E(Q^2)$ and $G_M(Q^2)$ up to $Q^2=0.5{\mathrm{GeV}}^2$.

Figure 1

Summary of pion masses and volumes in existing lattice calculations of nucleon structure. Open symbols are used to indicate cases where results were described by the authors as “preliminary.”

Figure 2

Two-point correlators $C_{2\text{pt}}(t,\overrightarrow{p})$ and $C_{2\text{pt}}^{\text{GPoF}}(t,\overrightarrow{p})$ for the 149 MeV ensemble, divided by their ground-state contributions, as determined from a two-state fit to the former with $t/a\in [3,14]$ and a one-state fit to the latter with $t/a\in [2,12]$. The left plot has $\overrightarrow{p}=\overrightarrow{0}$, and the right plot is averaged over $a\overrightarrow{p}$ equivalent to $\frac{2\pi }{48}(2,1,1)$, which is the largest used for computing matrix elements on this ensemble. The GPoF correlators are shifted to show that at large times their dominant contribution is from the time-displaced nucleon interpolating operator.

Figure 3

Vector current renormalization factor $Z_V$ versus bare quark mass, for the coarse ensembles. The band is from a fit assuming a linear relationship.

Figure 4

Comparison of different methods to extract the ground-state isovector form factors $F_1^v(Q^2)$ and $F_2^v(Q^2)$. The upper plots show the $m_{\pi }=149\mathrm{MeV}$ ensemble and the lower plots show the $m_{\pi }=254\mathrm{MeV}$, $32^3\times 48$ ensemble.

Figure 5

Dipole fits to $F_1^v(Q^2)$ with varying $Q_{\max }^2$. The upper-right plot shows the dependence on $Q_{\max }^2$ of the isovector Dirac radius derived from the fits.

Figure 6

Dipole fits to $F_2^v(Q^2)$ with varying $Q_{\max }^2$. The last two plots show the dependence on $Q_{\max }^2$ of the fit parameters.

Figure 7

Isovector electric and magnetic form factors. Each plot contains the curve with error band from the fit to experiment in Ref. [69] and the summation data from the $m_{\pi }=149\mathrm{MeV}$ ensemble.

Figure 8

Isovector Dirac radius $(r_1^2{)}^v$, determined on each lattice ensemble using different analysis methods for computing form factors.

Figure 9

Chiral extrapolation of the isovector Dirac radius $(r_1^2{)}^v$, determined on each lattice ensemble using the summation method. Two experimental points are shown, where $(r_E^2{)}^p$ is taken from either the CODATA 2010 result [3] used by the Particle Data Group (PDG) [76] or the measurement from spectroscopy of muonic hydrogen [77]. Both points use the PDG value for $(r_E^2{)}^n$. The magenta band results from fitting with the formula from Appendix pp1, whereas the orange band results from including an additional term proportional to $m_{\pi }^2$.

Figure 10

Chiral extrapolation of the isovector anomalous magnetic moment ${\kappa }_{\text{norm}}^v$, determined on each lattice ensemble using the summation method. The experimental point is from the PDG [76].

Figure 11

Chiral extrapolation of the product of the isovector anomalous magnetic moment and Pauli radius, ${\kappa }_{\text{norm}}^v(r_2^2{)}^v$, determined on each lattice ensemble using the summation method. We show two experimental values, where the radii are taken either from the 2012 PDG [76] or from the dispersion analysis in Ref. [10] (the difference mostly comes from different values for the proton magnetic radius).

Figure 12

Dipole fits to $F_1^s(Q^2)$ with varying $Q_{\max }^2$. The upper-right plot shows the dependence on $Q_{\max }^2$ of the isoscalar Dirac radius derived from the fits.

Figure 13

Line fits to $F_2^s(Q^2)$ with varying $Q_{\max }^2$. The last two plots show the dependence on $Q_{\max }^2$ of the fit parameters.

Figure 14

Isoscalar electric and magnetic form factors. Each plot contains the curve with error band from the fit to experiment in Ref. [69] and the summation data from the $m_{\pi }=149\mathrm{MeV}$ ensemble.

Figure 15

Isoscalar Dirac radius $(r_1^2{)}^s$, determined on each lattice ensemble using the summation method. Two experimental points are shown, where $(r_E^2{)}^p$ is taken from either the CODATA 2010 result [3] used by the PDG [76] or the measurement from spectroscopy of muonic hydrogen [77]. Both points use the PDG value for $(r_E^2{)}^n$.

Figure 16

Isoscalar anomalous magnetic moment ${\kappa }_{\text{norm}}^s$, determined on each lattice ensemble using the summation method.

Figure 17

Product of the isoscalar anomalous magnetic moment and Pauli radius, ${\kappa }_{\text{norm}}^s(r_2^2{)}^s$, determined on each lattice ensemble using the summation method. We show two experimental values, where the radii are taken either from the 2012 PDG [76] or from the dispersion analysis in Ref. [10] (the difference mostly comes from different values for the proton magnetic radius).

Figure 18

Proton electric and magnetic form factors, and their ratio. Each plot contains the curve with error band from the fit to experiment in Ref. [69] and the summation data from the $m_{\pi }=149\mathrm{MeV}$ ensemble. For the third plot, the lattice values for ${\mu }^p{G}_E^p/G_M^p$ are scaled using the proton magnetic moment from experiment, and not from the fits to lattice data.

Figure 19

Comparison of different methods to extract the ground-state isovector form factors $F_1^v(Q^2)$ and $F_2^v(Q^2)$. The upper plots show the $m_{\pi }=202\mathrm{MeV}$ ensemble and the lower plots show the $m_{\pi }=254\mathrm{MeV}$, $24^3\times 48$ ensemble.

Figure 20

Comparison of different methods to extract the connected-quark-contraction contribution to the ground-state isoscalar form factors $F_1^s(Q^2)$ and $F_2^s(Q^2)$. From top to bottom, data from the $m_{\pi }=149\mathrm{MeV}$, 202 MeV, 254 MeV ($32^3\times 48$), and 254 MeV ($24^3\times 48$) lattice ensembles are shown.

Figure 21

Isovector anomalous magnetic moment ${\kappa }_{\text{norm}}^v$, determined on each lattice ensemble using different analysis methods for computing form factors.

Figure 22

Isovector Pauli radius $(r_2^2{)}^v$, determined on each lattice ensemble using different analysis methods for computing form factors.

Figure 23

Isoscalar Dirac radius $(r_1^2{)}^s$, determined on each lattice ensemble using different analysis methods for computing form factors.

Figure 24

Isoscalar anomalous magnetic moment ${\kappa }_{\text{norm}}^s$, determined on each lattice ensemble using different analysis methods for computing form factors.

Figure 25

Product of the isoscalar anomalous magnetic moment and Pauli radius, ${\kappa }_{\text{norm}}^s(r_2^2{)}^s$, determined on each lattice ensemble using different analysis methods for computing form factors.