Pseudo-neutrino versus recoil formalism for 4-body phase space and applications to nuclear decay

Abstract

It is well-known that the traditional treatment of radiative corrections that utilizes the "true" neutrino momentum $\vec{p}_\nu$ in the differential decay rate formula could lead to a $\sim \alpha/\pi$ systematic error in certain observables due to the mistreatment of 4-body kinematics. We investigate the theory structure of one of the proposed solutions, the "$\nu'$-formalism", in the non-recoil limit appropriate for neutron and nuclear beta decays. We derive an elegant master formula for the 4-body phase space and use it to re-analyze the spectrum-dependent "outer" radiative corrections to the beta decay of a polarized spin-half nucleus; a complete set of analytic expressions is provided for readers to straightforwardly obtain the final numerical results. We compare it to the "recoil formalism" where the energy of the recoil nucleus is fixed.