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We investigate the contributions of the hadronic structure of the neutron toradiative $O(\alpha E_e/m_N)$ corrections (or the inner $O(\alpha E_e/m_N)$ RC)to the neutron beta decay, where $\alpha$, $E_e$ and $m_N$ are thefine-structure constant, the electron energy and the nucleon mass,respectively. We perform the calculation within the effective quantum fieldtheory of strong low-energy pion-nucleon interactions described by the linear$\sigma$-model with chiral $SU(2) \times SU(2)$ symmetry and electroweakhadron-hadron, hadron-lepton and lepton-lepton interactions for theelectron-lepton family with $SU(2)_L \times U(1)_Y$ symmetry of the StandardElectroweak Theory (Ivanov et al., Phys. Rev. D99, 093006 (2019)). We show thatafter renormalization, carried out in accordance with Sirlin's prescription(Sirlin, Phys. Rev. 164, 1767 (1967)), the inner $O(\alpha E_e/m_N)$ RC are ofthe order of a few parts of $10^{-5} - 10^{-4}$. This agrees well with theresults obtained in (Ivanov et al., Phys. Rev. D99, 093006 (2019)).

physical review. d/physical review. d.

Radiative corrections of order $O(\alpha E_e/m_N)$ to Sirlin's radiative  corrections of order $O(\alpha/\pi)$, induced by the hadronic structure of  the neutron