Radiative corrections to neutron and nuclear $\beta$-decays: a serious kinematics problem in the literature

TL;DR

This paper identifies a critical kinematics error in existing radiative correction calculations for neutron and nuclear beta decays, showing that proper 4-body kinematics significantly alters the correction magnitude.

Contribution

It highlights the mistake of using 3-body decay kinematics in bremsstrahlung corrections and demonstrates the importance of correct 4-body kinematics for accurate results.

Findings

Incorrect neutrino-type corrections underestimate radiative effects.

Proper recoil-type calculations yield larger correction values.

The kinematic error impacts precision measurements in beta decay experiments.

Abstract

We report a serious kinematics problem in the bremsstrahlung photon part of the order-$\alpha$ outer (model independent) radiative correction calculations for those neutron (and nuclear beta) decay observables (like electron-neutrino correlation parameter measurement) where the proton (recoil particle) is detected. The so-called neutrino-type radiative correction calculations, which fix the neutrino direction in the bremsstrahlung photon integrals, use 3-body decay kinematics to connect the unobserved neutrino direction with the observed electron and proton (recoil particle) momenta. But the presence of the bremsstrahlung photon changes the kinematics from 3-body to 4-body one, and the accurate information about the recoil particle momentum is lost due to the integration with respect to the photon momentum. Therefore the application of the abovementioned 3-body decay kinematics connection for the radiative correction calculations, rather prevalent in the literature, is not acceptable. We show that the correct, so-called recoil-type radiative correction calculations, which fix the proton (recoil particle) momentum instead of the neutrino direction and use rather involved analytical, semianalytical or Monte Carlo bremsstrahlung integration methods, result usually in much larger corrections than the incorrect neutrino-type analytical methods.