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We present the first and complete dispersion relation analysis of the innerradiative corrections to the axial coupling constant $g_A$ in the neutron$\beta$-decay. Using experimental inputs from the elastic form factors and thespin-dependent structure function $g_1$, we determine the contribution from the$\gamma W$-box diagram to a precision better than $10^{-4}$. Our calculationindicates that the inner radiative corrections to the Fermi and theGamow-Teller matrix element in the neutron $\beta$-decay are almost identical,i.e. the ratio $\lambda=g_A/g_V$ is almost unrenormalized. With this result, wepredict the bare axial coupling constant to be{$\mathring{g}_A=-1.2754(13)_\mathrm{exp}(2)_\mathrm{RC}$} based on the PDGaverage $\lambda=-1.2756(13)$

Dispersion relation analysis of the radiative corrections to $g_A$ in the neutron $\beta$-decay