Finite-size effects on multibody neutrino exchange

We use an effective Lagrangian to study the multibody massless neutrino exchange inside a finite neutron star. Following Schwinger, we show how the total interaction energy density is computed by comparing the values of the zero-point energy of the neutrino sea with and without the star. Here we extend a previous one-dimensional toy computation by using a three-dimensional spherical model of the star. We find that there is a nonvanishing of the zero-point energy density difference between the inside and the outside due to the refraction index at the stellar boundary and the resulting nonpenetrating waves. This effect is shown analytically and numerically to be the dominant one and to lead to an infrared-safe total energy density, thus confirming that there is no need for the neutrino to be massive. Altogether, the energy due to neutrino exchange is of the order of ${10}^{-8}-{10}^{-13}\hspace{0.5em}\mathrm{GeV}$ per neutron, i.e., negligible with respect to the neutron mass density.