${\hat{I}}^{3}W$-exchange correction beyond the forward-angle limit in neutron ${\hat{I}}^2$ decay

Abstract

In this work, we discuss the ${\hat{I}}^{3}W$-exchange contributions in neutron ${\hat{I}}^2$ decay with an elastic intermediate state, beyond the forward-angle limit (FAL). By decomposing the one-$W$-exchange and ${\hat{I}}^{3}W$-exchange amplitudes in terms of $16$ independent Pauli-spinor structures, we calculate the ${\hat{I}}^{3}W$-exchange corrections to the relevant coefficients. Our numerical results show that the relative corrections to the Fermi Born term $C_{\text{Born}}^{\text{F}}$ and the Gamow-Teller Born term $C_{\text{Born}}^{\text{GT}}$ are enhanced by about 8\% and 18\%, respectively. In particular, we find a non-zero contribution to $C_{\text{Born}}^{\text{GT}}$ from the axial-vector current, which is identically zero in the FAL. The corresponding effect on the extracted $V_{ud}$ from the neutron lifetime is also analyzed, and we find the correction to be at the $10^{-4}$ level.