Lattice calculation of the pion mass difference $M_{\ddot{I}{€}^{+}}-M_{\ddot{I}{€}^0}$ at order $\mathcal{O}(\hat{I}{\pm }_{em})$

Abstract

We present a lattice calculation of the charged/neutral pion mass difference $M_{\ddot{I}{€}^{+}}-M_{\ddot{I}{€}^0}$ at order $\mathcal{O}(\hat{I}{\pm }_{em})$ 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 ($a\simeq 0.062,0.082,0.089\mathrm{fm}$) and pion masses in the range $M_{\ddot{I}}€\simeq 250-500\mathrm{MeV}$. We employ the RM123 method and expand the path-integral around the isospin symmetric point at leading order in the electromagnetic coupling $\hat{I}{\pm }_{em}$. Making use of the recently proposed RTM scheme, we evaluate the full $\mathcal{O}(\hat{I}{\pm }_{em})$ contribution, with the inclusion of the disconnected diagram. At the physical point, after performing the continuum and infinite volume extrapolation, we obtain the value $M_{\ddot{I}{€}^{+}}-M_{\ddot{I}{€}^0}=4.622(95)\mathrm{MeV}$ which is in good agreement with the experimental result $[M_{\ddot{I}{€}^{+}}-M_{\ddot{I}{€}^0}{]}^{exp.}=4.5936(5)\mathrm{MeV}$.

13 pages, 5 figures, 1 table