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

It is well known that the traditional treatment of radiative corrections that utilizes the ``true'' neutrino momentum Pp_ν in the differential decay rate formula could lead to a ~α/π systematic error in certain observables due to the mistreatment of 4-body kinematics. I investigate the theory structure of one of the proposed solutions, the ``ν^'-formalism'', in the nonrecoil limit appropriate for neutron and nuclear β decays. I derive an elegant master formula for the 4-body phase space and use it to reanalyze the spectrum-dependent ``outer'' radiative corrections to the β decay of a polarized spin-half nucleus; a complete set of analytic expressions is provided for readers to straightforwardly obtain the final numerical results. I compare it to the ``recoil formalism'' where the energy of the recoil nucleus is fixed.