We calculate the radiative corrections of order $O(\alpha E_e/m_N)$ as next-to-leading order corrections in the large nucleon mass expansion to Sirlin’s radiative corrections of order $O(\alpha /\pi )$ to the neutron lifetime. The calculation is carried out within a quantum field theoretic model of strong low-energy pion-nucleon interactions described by the linear $\sigma$ model ($\mathrm{L}\sigma \mathrm{M}$) with chiral $SU(2)\times SU(2)$ symmetry and electroweak hadron-hadron, hadron-lepton and lepton-lepton interactions for the electron-lepton family with $SU(2{)}_L\times U(1{)}_Y$ symmetry of the standard electroweak model (SEM). Such a quantum field theoretic model is some kind a hadronized version of the Standard Model. From a gauge invariant set of the Feynman diagrams with one-photon exchanges we reproduce Sirlin’s radiative corrections of order $O(\alpha /\pi )$, calculated to leading order in the large nucleon mass expansion, and calculate next-to-leading corrections of order $O(\alpha E_e/m_N)$. This confirms Sirlin’s confidence level of the radiative corrections $O(\alpha E_e/m_N)$. The contributions of the $\mathrm{L}\sigma \mathrm{M}$ are taken in the limit of the infinite mass of the scalar isoscalar $\sigma$ meson. In such a limit the $\mathrm{L}\sigma \mathrm{M}$ reproduces the results of the current algebra [S. Weinberg, Phys. Rev. Lett. 18, 188 (1967)] in the form of effective chiral Lagrangians of pion-nucleon interactions with nonlinear realization of chiral $SU(2)\times SU(2)$ symmetry. In such a limit the $\mathrm{L}\sigma \mathrm{M}$ is also equivalent to Gasser-Leutwyler’s chiral quantum field theory or chiral perturbation theory with chiral $SU(2)\times SU(2)$ symmetry and the exponential parametrization of a pion-field [(G. Ecker, Prog. Part. Nucl. Phys. 35, 1 (1995)].

• Received 31 March 2019

DOI:https://doi.org/10.1103/PhysRevD.99.093006