to the hyperfine splitting in the hydrogen atom \[[See e.g. ref. [@LPR72] and references therein.]{}\]

### The Adler–Weisberger Sum Rule

The sum rule in question [@Adl65; @Wei66] relates the axial coupling constant $g_{\mbox{\scriptsize\rm A}}$ of $\beta$–decay to integrals of pion–nucleon total cross–sections: $$1-g_{\mbox{\scriptsize\rm
A}}^2=\frac{2f_{\pi}^2}{\pi}\int_{0}^{\infty}\frac{d\nu}{\nu}
\left[\sigma_{\mbox{\scriptsize\rm tot}}^{\pi^{-}{\mbox{\scriptsize\rm
p}}}(\nu)-
\sigma_{\mbox{\scriptsize\rm tot}}^{\pi^{+}{\mbox{\scriptsize\rm p}}}(\nu)
\right],$$ where $f_{\pi}\simeq 92\,{\mbox{\rm MeV}}$ denotes the pion decay coupling constant. The reference amplitude in this case is the forward pion–nucleon scattering amplitude illustrated in Fig. \[fig:Fig1b\]

which has two invariant amplitudes: $$T_{ij}(\nu)=\delta_{ij}\,T^{(+)}(\nu)+\frac{1}{2}\left[\tau_{i},
\tau_{j}\right]
\,\frac{M_{\mbox{\scriptsize\rm p}}\nu}{f_{\pi}^2}T^{(-)}(\nu),$$ where $i,j$ are isospin indices. The basic dispersion relation is the one obeyed by the isospin–odd amplitude $T^{(-)}(\nu)$, and as in the case of the Compton scattering spin–flip amplitude, it is expected that $T^{(-)}(\nu)$ obeys an unsubtracted dispersion relation. The optical theorem relates ${\mbox{\rm Im}}T^{(-)}(\nu)$ to the difference of $\pi^-$–p and $\pi^+$–p total cross–sections. On the other hand, the calculation of the low–energy behaviour of the amplitude ${\mbox{\rm Re}}T^{(-)}(\nu)$ can be done using baryon chiral perturbation theory [@ManlesH; @PiclesH]. The relevant Feynman diagrams are those of the Born approximation shown in Fig. \[fig:Fig1c\]

with the result $$\lim_{\nu{\rightarrow}0}{\mbox{\rm Re}}T^{(-)}(\nu)=1-g_{\mbox{\scriptsize\rm A}}^2.$$ The seagull graph contributes the factor 1, each one of the other graphs a factor 