In this work, we discuss the $\gamma W$-exchange contributions in neutron $\beta$ decay with an elastic intermediate state, beyond the forward-angle limit (FAL). By decomposing the one-$W$-exchange and $\gamma W$-exchange amplitudes in terms of 16 independent Pauli-spinor structures, we calculate the $\gamma W$-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 nonzero 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.

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• Beta decay
• Cabibbo–Kobayashi–Maskawa matrix