We investigate the contributions of the hadronic structure of the neutron to radiative $O(\alpha E_e/m_N)$ corrections [or the inner $O(\alpha E_e/m_N)$ RC] to the neutron beta decay, where $\alpha$, $E_e$, and $m_N$ are the fine-structure constant, the electron energy, and the nucleon mass, respectively. We perform the calculation within the effective quantum field theory of strong low-energy pion-nucleon interactions described by the linear $\sigma$ model 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 theory [Ivanov et al., Phys. Rev. D 99, 093006 (2019)]. We show that after 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 of the order of a few parts of ${10}^{-5}–{10}^{-4}$. This agrees well with the results obtained in [Ivanov et al., Phys. Rev. D 99, 093006 (2019)].

中文翻译：

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由中子的强子结构引起的O阶(αEe/mN)对Sirlin的O阶(α/π)辐射修正的辐射修正

我们研究了中子的强子结构对辐射的贡献 $哦(\alpha {乙}_{\mathrm{电子}}/{米}_N)$ 更正 [或内部 $哦(\alpha {乙}_{\mathrm{电子}}/{米}_N)$ RC] 到中子β衰变，其中 $\alpha$, ${乙}_{\mathrm{电子}}$， 和 ${米}_N$分别是精细结构常数、电子能量和核子质量。我们在由线性描述的强低能 pion-核子相互作用的有效量子场理论中执行计算$\sigma$ 手性模型 $秒你(2)\times 秒你(2)$ 电子-轻子家族的对称性和电弱强子-强子、强子-轻子和轻子-轻子相互作用 $秒你(2{)}_{升}\times 你(1{)}_{是}$标准电弱理论的对称性 [Ivanov等人。，物理。修订版 D 99 , 093006 (2019)]。我们证明重整化后，按照Sirlin的处方进行[Sirlin, Phys. Rev. 164 , 1767 (1967)]，内部$哦(\alpha {乙}_{\mathrm{电子}}/{米}_N)$ RC 是几个部分的顺序 ${10}^{-5}\mathrm{——}{10}^{-4}$. 这与[Ivanov et al。，物理。修订版 D 99 , 093006 (2019)]。