### The Electromagnetic Pion Mass Difference

The most remarkable application of the Weinberg sum rules is the calculation of the $\pi^{+}-\pi^{0}$ electromagnetic mass difference [@Lowetal67]. In the chiral limit and to lowest order in the electromagnetic interactions, there appears a new term with no derivatives in the low–energy effective chiral Lagrangian [@EGPR89] $$\label{eq:effem}
{{\cal L}}_{\mbox{\scriptsize\rm eff}}=\cdots +e^2 C\,{\mbox{\rm tr}}\left(Q_{R}UQ_{L}U^{\dagger}\right),$$ where $U$ is the matrix field which collects the octet of pseudoscalar Goldstone fields \[[See Pich’s lectures [@PiclesH] in this school.]{}\] and $Q_{R}=Q_{L}=\mbox{\rm diag}[2/3, -1/3, -1/3]$ the right– and left–charges associated with the electromagnetic couplings of the light quarks. Expanding $U$ in powers of the pseudoscalar fields there appear quadratic terms like $$\label{eq:effemex}
{{\cal L}}_{\mbox{\scriptsize\rm eff}}=\cdots -2e^2 C\frac{1}{f_{\pi}^2} (\pi^+
\pi^- + K^+ K^-) + \cdots ,$$ showing that, in the presence of electromagnetic interactions, the charged pion and kaon fields become massive: $$(m_{K^+}^{2}-m_{K^0}^{2})\vert_{\mbox{\scriptsize\rm
EM}}=(m_{\pi^+}^{2}-m_{\pi^0}^{2})\vert_{\mbox{\scriptsize\rm
EM}}=
\frac{2e^2 C}{f_{\pi}^2}.$$ In fact, the main contribution to the physical $\pi^{+}$–$\pi^{0}$ mass difference is of electromagnetic origin because quark masses do not contribute significantly to the $\pi^{+}$–$\pi^{0}$ mass splitting.

It can be formally shown \[[See e.g. refs. [@BdeR91; @KU97] and references therein.]{}\] that the coupling constant $C$ of the effective term in eq. (\[eq:effem\]) which results 