Neutron in a strong magnetic field: Finite volume effects

We investigate the neutron’s response to magnetic fields on a torus with the aid of chiral perturbation theory, and expose effects from nonvanishing holonomies. The determination of such effects necessitates nonperturbative treatment of the magnetic field; and, to this end, a strong-field power counting is employed. Using a novel coordinate-space method, we find the neutron propagates in a coordinate-dependent effective potential that we obtain by integrating out charged pions winding around the torus. Knowledge of these finite volume effects will aid in the extraction of neutron properties from lattice QCD computations in external magnetic fields. In particular, we obtain finite volume corrections to the neutron magnetic moment and magnetic polarizability. These quantities have not been computed correctly in the literature. In addition to effects from nonvanishing holonomies, finite volume corrections depend on the magnetic flux quantum through an Aharonov-Bohm effect. We make a number of observations that demonstrate the importance of nonperturbative effects from strong magnetic fields currently employed in lattice QCD calculations. These observations concern neutron physics in both finite and infinite volume.

Figure 1

Charged pion propagator in an external magnetic field. Depicted are the $\mathcal{O}(p^2)$ couplings to the magnetic field which must be summed to derive the propagator in strong-field power counting.

Figure 2

Pion loop contribution to the nucleon mass in zero magnetic field. The single line represents the free pion propagator, while double lines represent nucleon propagators.

Figure 3

Charged pion loop contributions to the neutron two-point function in an external magnetic field. Dashed lines represent charged pion propagators in an external magnetic field, while double lines represent nucleon propagators. The wiggly lines represent explicit couplings to the external field.

Figure 4

Spin-independent energy shift of the neutron as a function of magnetic field. The first fifteen nonvanishing perturbative approximations to the neutron energy in powers of the magnetic field are compared with the nonperturbative result given in Eq. (54). The dashed curve shows the nonperturbative magnetic field result, while the quadratic and quartic approximations to the energy are labeled.

Figure 5

Depiction of charged pion winding in the plane transverse to the magnetic field. The pion propagates a distance ${\nu }_{1}L$ in the ${\widehat{x}}_1$ direction, and ${\nu }_{2}L$ in the ${\widehat{x}}_2$ direction. This propagation is accompanied by two Wilson loops. By periodicity, the pion ends up at a point equivalent to where it started, and additionally must acquire an Aharonov-Bohm phase, $e^{iQAB}$.

Figure 6

Zeeman splitting of the neutron as a function of magnetic field. The first fifteen nonvanishing perturbative approximations to the Zeeman splitting in powers of the magnetic field are compared with the nonperturbative result which follows from Eq. (60). The dashed curve shows the nonperturbative magnetic field result, while the linear and cubic approximations to the splitting are labeled.

Figure 7

Finite volume effect on the ground-state neutron Zeeman splitting computed in weak and strong magnetic fields. Plotted versus $L$ are the finite volume corrections $\mathrm{\Delta }{\kappa }_n(L)$ and the strong-field generalization $\mathrm{\Delta }{\kappa }_n(B,L)$. These corrections are compared to the physical neutron magnetic moment, ${\kappa }_n=-1.91[{\mu }_N]$. Results are shown using three values of the pion mass, and include the effects of proton-pion and delta-pion intermediate states. The strong-field results are shown for a flux quantum of $N_{\mathrm{\Phi }}=3$.

Figure 8

Finite volume effect on the spin-independent ground-state neutron energy. Plotted versus the lattice size $L$ are the finite volume corrections $\mathrm{\Delta }{\beta }_M(L)$ and the strong-field generalization $\mathrm{\Delta }{\beta }_M(B,L)$. These finite-size effects are compared to ${\beta }_M\equiv \frac{e^2}{4\pi }\frac{g_A^2}{96\pi f^2{m}_{\pi }}=1.25\times {10}^{-4}[\mathrm{fm}{]}^3$, which we take as an estimate of the physical value. Results are shown using three values of the pion mass. Included are contributions from both proton-pion and delta-pion intermediate states. The strong-field results are evaluated for a flux quantum $N_{\mathrm{\Phi }}=3$.

Figure 9

Plot of the weak-field expansion parameter $|\xi |=|eB|/m_{\pi }^2$ as a function of the lattice size $L$. The magnetic field is quantized according to $eB{L}^2=2\pi N_{\mathrm{\Phi }}$, and we use the physical value of the pion mass. The expansion parameter is shown for the two lowest values of the flux quantum for QCD.