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

TL;DR

This paper presents a lattice QCD calculation of the pion mass difference at order _{em}, achieving results consistent with experimental data by including electromagnetic effects and disconnected diagrams.

Contribution

The study introduces a novel lattice calculation of the pion mass difference at _{em} using the RTM scheme and includes disconnected diagrams for improved accuracy.

Findings

Calculated pion mass difference: 4.622(95) MeV

Results agree with experimental value 4.5936(5) MeV

Used multiple lattice spacings and pion masses for extrapolation

Abstract

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