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

We present a lattice calculation of the charged/neutral pion mass difference $M_{π^{+}}-M_{π^{0}}$ at order $\mathcal{O}(α_{em})$ using the gauge configurations 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 $M_π \simeq 250-500~{\rm MeV}$. We employ the RM123 method and expand the path-integral around the isospin symmetric point at leading order in the electromagnetic coupling $α_{em}$. Making use of the recently proposed RTM scheme, we evaluate the full $\mathcal{O}(α_{em})$ contribution, with the inclusion of the disconnected diagram. At the physical point, after performing the continuum and infinite volume extrapolation, we obtain the value $M_{π^{+}}-M_{π^{0}}= 4.622~(95)~{\rm MeV}$ which is in good agreement with the experimental result $[ M_{π^{+}} - M_{π^{0}} ]^{exp.} = 4.5936(5)~{\rm MeV}$.