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We present a lattice calculation of the charged/neutral pion mass difference$M_{\pi^{+}}-M_{\pi^{0}}$ at order $\mathcal{O}(\alpha_{em})$ using the gaugeconfigurations produced by the Extended Twisted Mass Collaboration with$N_{f}=2+1+1$ dynamical quark flavours at three values of the lattice spacing($a \simeq 0.062, 0.082, 0.089~{\rm fm}$) and pion masses in the range $M_{\pi}\simeq 250-500~{\rm MeV}$. We employ the RM123 method and expand thepath-integral around the isospin symmetric point at leading order in theelectromagnetic coupling $\alpha_{em}$. Making use of the recently proposed RTMscheme, we evaluate the full $\mathcal{O}(\alpha_{em})$ contribution, with theinclusion of the disconnected diagram. At the physical point, after performingthe continuum and infinite volume extrapolation, we obtain the value$M_{\pi^{+}}-M_{\pi^{0}}= 4.622~(95)~{\rm MeV}$ which is in good agreement withthe experimental result $[ M_{\pi^{+}} - M_{\pi^{0}} ]^{exp.} = 4.5936(5)~{\rmMeV}$.

Lattice calculation of the pion mass difference $M_{\pi^{+}}-M_{\pi^{0}}$ at order $\mathcal{O}(\alpha_{em})$