from the integration of a virtual photon of euclidean momentum squared $Q^2$ in the presence of the strong interactions in the chiral limit, is given by an integral of the correlation function $\Pi_{LR}(Q^2)$ defined in eqs. (\[eq:lrtpf\]) and (\[eq:lritpf\]): $$\label{eq:piem} 
C=
\frac{-1}{8\pi^2}\,\frac{3}{4}\,
\int_0^\infty dQ^2\,Q^2\Pi_{LR}(Q^2).$$ The constraints in eqs. (\[eq:ope2\]) and (\[eq:ope4\]) guarantee the convergence of the integral in the ultraviolet. Inserting the resonance dominance ansatz in eqs. (\[eq:Vresdomqcd\]) and (\[eq:Aresdomqcd\]) for the vector and axial–vector spectral functions in the dispersion integral representation in eq. (\[eq:udrwsr\]) results in $$\label{eq:lrtpfrd}
\Pi_{LR}(Q^2)=-\frac{f_{\pi}^2}{Q^2}-\frac{f_{A}^2 M_{A}^2} {M_{A}^2
+Q^2}Ê+ \frac{f_{V}^2 M_{V}^2} {M_{V}^2 +Q^2}.$$ With this expression inserted in the integrand of the r.h.s. in eq. (\[eq:piem\]) and using the 1st and 2nd Weinberg sum rules in eqs. (\[eq:tsc\]) one easily obtains $$(m_{\pi^+}^{2}-m_{\pi^0}^{2})\vert_{\mbox{\scriptsize\rm
EM}}=
\frac{\alpha}{\pi}\frac{3}{4}\frac{M_{A}^2 M_{V}^2}{M_{A}^2 -M_{V}^2}
\log\frac{M_{A}^2}{M_{V}^2}.$$ The authors of ref. [@Lowetal67] also use $M_{A}=\sqrt{2}M_{V}$ which results in $$(m_{\pi^+}^{2}-m_{\pi^0}^{2})\vert_{\mbox{\scriptsize\rm
EM}}=
\frac{\alpha}{\pi}\frac{3}{2}\,M_{\rho}^2\,\log 2.$$ Numerically, this corresponds to a mass difference $\Delta m_{\pi}\vert_{\mbox{\scriptsize\rm EM}}\simeq 5.2\,{\mbox{\rm MeV}}$ to be compared with the experimental mass difference $m_{\pi^+}-m_{\pi^0}=4.6\,{\mbox{\rm MeV}}$.

Equation (\[eq:piem\]) can nowadays be viewed as a rigorous QCD sum rule in the chiral limit. The sum rule relates the low–energy constant $(m_{\pi^+}^{2}-m_{\pi^0}^{2})\vert_{\mbox{\scriptsize\rm EM}}$ to an integral of the real part of 