Electromagnetic and spin polarizabilities in lattice QCD

We discuss the extraction of the electromagnetic and spin polarizabilities of nucleons from lattice QCD. We show that the external field method can be used to measure all the electromagnetic and spin polarizabilities including those of charged particles. We then turn to the extrapolations required to connect such calculations to experiment in the context of finite volume chiral perturbation theory. We derive results relevant for lattice simulations of QCD, partially-quenched QCD and quenched QCD. Our results for the polarizabilities show a strong dependence on the lattice volume and quark masses, typically differing from the infinite volume limit by $\sim 10\%$ for current lattice volumes and quark masses.

Figure 1

Perturbative expansion of the hadron propagator in an external field.

Figure 2

Anomalous contributions to the polarizabilities. The crossed circle corresponds to the insertion of an operator from the Wess-Zumino-Witten Lagrangian, Eq. (32), and the crossed meson line corresponds to a hairpin interaction [76].

Figure 3

Born diagrams involving internal 44-plet states that give contribution to the polarizabilities.

Figure 4

Diagrams contributing to nucleon polarizabilities at order $Q^3$. The solid and dashed lines denote nucleons, and pions, respectively. Additionally, our results include a similar set of diagrams in which the internal 70-plet propagator is replaced by a 44-plet resonance.

Figure 5

Volume dependence of the proton polarizabilities. Here we show the ratio of the difference of the finite and infinite volume results to the infinite volume results for three values of the pion mass using the parameters described in the text. The curves terminate at $m_{\pi }L=3$.

Figure 6

Volume dependence of the neutron polarizabilities. The various curves are as in Fig. 5.

Figure 7

Volume dependence of the proton polarizabilities in quenched QCD at the lightest quark mass used in the lattice calculations of Refs. 7, 8. The central curves and shaded region correspond to the parameters quoted in the text.

Figure 8

As in Fig. 7, but for the neutron.

Figure 9

Volume dependence of the Thompson limit of photon neutron scattering. Notice the infinite volume limit is zero.