Leading quenching effects in the proton magnetic moment

We present the first investigation of the extrapolation of quenched nucleon magnetic moments in quenched chiral effective field theory. We utilize established techniques in finite-range regularization and incorporate finite-volume corrections to the relevant loop integrals. Finally, the contributions of dynamical sea quarks to the proton moment are estimated using a recently discovered phenomenological link between quenched and physical QCD.

Figure 1

Diagram providing the leading nonanalytic contribution of pions (dashed curve) to the nucleon (solid line) magnetic moment.

Figure 2

Double-hairpin ${\eta }^{\prime }$ vertex correction to nucleon magnetic moment.

Figure 3

Loop corrections, in units of ${\mu }_N$, evaluated with a dipole FRR with $\Lambda =0.8\mathrm{G}\mathrm{e}\mathrm{V}$. The dashed and solid curves are the pion-loop corrections of Fig. 1 in quenched and physical QCD, respectively. The dash-dotted curve is the ${\eta }^{\prime }$ vertex correction of quenched QCD, where the physical magnetic moment is used to define the renormalized coupling.

Figure 4

Fit of Eq. (15) to quenched lattice data. The solid squares (■) illustrate the FLIC fermion results [24, 25] and the open symbols describe nonperturbatively improved clover results [23], at $\beta =6.0$ (▽), $\beta =6.2$ (△) and $\beta =6.4$ (◇).

Figure 5

The dependence of the extrapolated magnetic moment determined at the physical pion mass on the dipole regulator parameter $\Lambda$. Variation in the quenched proton moment is similar to the statistical uncertainty of the FLIC simulation result at the lightest quark mass considered.

Figure 6

Finite-volume FRR-EFT fit to FLIC fermion results for fixed volume. Here the dipole-vertex regulator parameter is fixed to $\Lambda =0.8\mathrm{G}\mathrm{e}\mathrm{V}$.

Figure 7

The extrapolated magnetic moment on a finite volume, $V=(2.56\mathrm{f}\mathrm{m}{)}^3$, for varying regulator parameter $\Lambda$.

Figure 8

Correcting the finite-volume quenched approximation to the infinite-volume limit of full-QCD. The solid curve is the finite-volume quenched fit to the data as in Fig. 4. The dotted curve provides an estimate of the infinite-volume limit magnetic moment in the quenched approximation. The dashed curve shows estimates of the proton magnetic moment in full-QCD as described in the text.

Figure 9

Truncations of the Taylor expansion of the full-QCD expression of the magnetic moment, Eq. (10), to various orders in $m_{\pi }$. The leftmost curves display a succession of truncations from $m_{\pi }$ up to $m_{\pi }^{10}$. To reach convergence at the lightest FLIC data point considered here one requires terms up to $m_{\pi }^{26}$, while for QCDSF terms to $m_{\pi }^{50}$ are necessary. The dashed curve displays the full-QCD expression as displayed in Fig. 8.

Figure 10

Leading pion-loop contribution to nucleon magnetic moment from the $\Delta$ resonance.

Figure 11

The role of $\Delta$ contributions in the extrapolation of the proton magnetic moment. The solid curve displays the leading-order QQCD extrapolation of Eq. (16), without explicit decuplet contributions, as seen in Fig. 4. If Eq. (16) is extended to include the pion contributions with decuplet baryons in Eq. (20), one obtains the dotted curve.