In this work, we evaluate the recoil corrections to the correlation coefficients in the neutron ${\beta }^{-}$ decay with polarized neutron and polarized electron, up to order $O\left(m_n^{-2}\right)$, where $m_n$ represents the mass of neutron. In our calculations, we express the mass of proton $m_p$ in terms of $m_n$ and other small quantities to ensure that only one larger parameter, $m_n$, remains in the differential cross section. Subsequently, we expand the differential cross section on $m_n^{-1}$, and finally integrate the solid angles of the antineutrino to obtain the analytic expressions for the angular correlation coefficients. Comparing the analytic results to those in a previous study [Ivanov, Hollwieser, Troitskaya, Wellenzohn, and Berdnikov, Phys. Rev. C 95, 055502 (2017); Phys. Rev. C 104, 069901(E) (2021)], we find that, at order $O\left(m_n^{-1}\right)$, three of the results are consistent, while two differ slightly. In addition, the contributions at order $O\left(m_n^{-2}\right)$ are presented when considering a general form of $Wnp$ interactions. Numerical results show that contributions from order $O\left(m_n^{-2}\right)$ and additional form factors of $Wnp$ interactions are required to achieve a precision of ${10}^{-5}$.

• Received 11 February 2023
• Accepted 16 January 2024

DOI:https://doi.org/10.1103/PhysRevC.109.025204