Magnetic form factors of the octet baryons from lattice QCD and chiral extrapolation

We present a $2+1$-flavor lattice QCD calculation of the electromagnetic Dirac and Pauli form factors of the octet baryons. The magnetic Sachs form factor is extrapolated at six fixed values of $Q^2$ to the physical pseudoscalar masses and infinite volume using a formulation based on heavy-baryon chiral perturbation theory with finite-range regularization. We properly account for omitted disconnected quark contractions using a partially quenched effective field theory formalism. The results compare well with the experimental form factors of the nucleon and the magnetic moments of the octet baryons.

Figure 1

Locations of lattice simulation results in the $m_l\text{−}{m}_s$ plane. The red star denotes the physical point and the dashes indicate the flavor-symmetric line where $m_l=m_s$. Our primary simulation trajectory, illustrated by the dotted line, corresponds to the line of constant singlet quark mass $(2m_K^2+m_{\pi }^2)$ at ${\kappa }_0=0.120900$ (simulations 1–3 in Table 1). The solid line indicates the physical value of the singlet mass.

Figure 2

Quark contributions to the Dirac form factor $F_1$ of the hyperons at the lightest simulation pion mass $(m_{\pi },m_K)=(310,520)\mathrm{MeV}$. (a) Doubly represented quark contributions. (b) Singly represented quark contributions. The charges of the relevant quarks have been set to unity. The lines show dipolelike fits [Eq. (10)].

Figure 3

Quark contributions to the Pauli form factor $F_2$ of the hyperons at the lightest simulation pion mass $(m_{\pi },m_K)=(310,520)\mathrm{MeV}$. (a) Doubly represented quark contributions. (b) Singly represented quark contributions. The charges of the relevant quarks have been set to unity. The lines show dipole fits [Eq. (11)].

Figure 4

Dirac and Pauli radii for the nucleon from recent $2+1$ and $2+1+1$-flavor lattice simulations [26, 27, 31, 32, 33], compared with the results of this work. (a) Isovector Dirac radii. (b) Isovector Pauli radii.

Figure 5

Loop diagrams which contribute to $G_M$ at leading order. Single, double, dashed and wavy lines represent octet baryons, decuplet baryons, mesons and photons, respectively.

Figure 6

$Q^2$ distribution for the lattice simulation results. Colors indicate the $Q^2$ bin groupings; each bin corresponds to a single value of the three-momentum transfer in lattice units.

Figure 7

Generalized dipole fits [Eq. (37)] upon which the binning corrections are based. The three fits shown correspond to the three different pseudoscalar mass points along the primary simulation trajectory (simulations 1–3 in Table 1). Quarks have unit charge.

Figure 8

Illustration of the quality fit for the data set at $Q^2\approx 0.26{\mathrm{GeV}}^2$, the lowest $Q^2$ bin. Each point denotes one of the lattice simulation results, e.g., $G_M^{p,u}$, $G_M^{p,d}$ …, at one of the sets of pseudoscalar masses.

Figure 9

Up quark (connected) contribution to the proton magnetic form factor for quarks with unit charge. Each set of results (top to bottom) represents the fit at a different (increasing) $Q^2$ value. The lines show these fits evaluated along the trajectory which holds the singlet pseudoscalar mass $(m_K^2+m_{\pi }^2/2)$ fixed to its physical value.

Figure 10

Isovector nucleon magnetic form factor compared to the Kelly parametrization of experimental results [55]. The small (solid blue) points show the results including all lattice simulations, while the large error bars (pale green) show the results including only lattice simulations along the primary simulation trajectory (see Table 1).

Figure 11

(a) Isovector sigma baryon magnetic form factor and (b) cascade baryon magnetic form factor with dipolelike fits [Eq. (37)]. The red stars indicate the experimental isovector magnetic moments.

Figure 12

Extrapolated (connected part of the) (a) proton and (b) neutron magnetic form factors, compared with Kelly parametrization [55] of experimental measurements.

Figure 13

Dipolelike [from Eq. (37), dashed red band] and general [from Eq. (39), solid blue band] fits to the proton magnetic form factor. The quality of fit is comparable for both fits.

Figure 14

Connected part of the (a) doubly and (b) singly represented quark contributions to the baryon magnetic form factors for $Q^2\approx 0.26{\mathrm{GeV}}^2$. The charges of the relevant quarks have been set to unity.

Figure 15

$Q^2$ dependence of unrenormalized fit parameters, defined in Eqs. (31) and (32).

Figure 16

Connected part of the octet baryon magnetic form factors. The red stars indicate the experimental magnetic moments. The lines show dipolelike fits [Eq. (37), dashed green; Eq. (39), solid blue].