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 pi+ - M-pi 0 at order O(alpha(em)) using the gauge configurations produced by the Extended Twisted Mass Collaboration with N-f = 2 + 1 + 1 dynamical quark flavors at three values of the lattice spacing (a similar or equal to 0.062, 0.082, 0.089 fm) and pion masses in the range M-pi similar or equal to 250-500 MeV. We employ the RM123 method and expand the path integral around the isospin symmetric point at leading order in the electromagnetic coupling alpha(em). Making use of the recently proposed rotated twisted-mass scheme, we evaluate the full O(alpha(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 pi+ - M-pi 0 = 4.622(95) MeV which is in good agreement with the experimental result [M pi+ - M-pi 0 ](exp) = 4.5936(5) MeV.