Response of nucleons to external probes in hedgehog models. I. Electromagnetic polarizabilities

Electromagnetic polarizabilities of the nucleon are analyzed in a hedgehog model with quark and meson degrees of freedom. Semiclassical methods are used (linear response theory, quantization via cranking). It is found that in hedgehog models (Skyrmion, chiral quark models, Nambu-Jona-Lasinio model), the average electric polarizability of the nucleon, ${\alpha }_N$, is of the order $N_c$, and the splitting of the neutron and proton electric (proper) polarizabilities, $\delta \alpha ={\alpha }_n-{\alpha }_p$, is of the order $\frac{1}{N_c}$. We present a general argument why one expects $\delta \alpha >0\mathrm{}$ in models with a pionic cloud. Our model prediction for the sign and magnitude of $\delta \alpha$ is in agreement with recent measurements. The obtained value for ${\alpha }_N$, however, is roughly a factor of 3 too large. This is because of two problems with our particular model: a too strong pion tail and the degeneracy of $N$ and $\Delta$ states in the large-$N_c$ limit. This degeneracy also results in a very strong $N_c$ dependence of the paramagnetic part of the magnetic polarizability $\beta$, which is of the order $N_c^3$. We compare the large-$N_c$ results to the one-loop chiral perturbation theory predictions, and show the importance of $\Delta$ effects in pionic loops. We also investigate the role of nonminimal substitution terms in the effective Lagrangian on the polarizabilities of the nucleon.