Lattice determination of the pion mass difference $M_{\pi^{+}} - M_{\pi^{0}}$ at order $\mathcal{O}(\alpha_{em})$ and $\mathcal{O}( (m_{d}-m_{u})^{2})$ including disconnected diagrams

TL;DR

This paper uses lattice QCD with twisted mass simulations to precisely compute the pion mass difference and determine the low-energy constant , including disconnected diagrams, improving accuracy over previous estimates.

Contribution

It introduces a rotated twisted-mass scheme to evaluate disconnected diagrams in lattice QCD, enabling more precise calculations of pion mass differences and low-energy constants.

Findings

Pion mass difference  consistent with experimental value.

Determined  as 2.5(1.4)  10^{-3}, aligning with phenomenology.

Improved evaluation of disconnected diagrams using RTM scheme.

Abstract

We present our preliminary results concerning the charged/neutral pion mass difference $M_{\pi^{+}} - M_{\pi^{0}}$ at order $\mathcal{O}(\alpha_{em})$ in the QED interactions, and for $M_{\pi^{+}} - M_{\pi^{0}}$ at order $\mathcal{O}\left( (m_{d}-m_{u})^{2}\right)$ in the strong isospin-breaking term. The latter contribution provides a determination of the $\rm{SU}(2)$ chiral perturbation theory low-energy constant $\ell_{7}$, whose present estimate is affected by a rather large uncertainty. The disconnected contributions appearing in the diagrammatic expansion of $M_{\pi^{+}} - M_{\pi^{0}}$, being very noisy, are notoriously difficult to evaluate and have been neglected in our previous calculations. By making use of twisted mass Lattice QCD simulations and adopting the RM123 method, we will show that taking profit from our recently proposed rotated twisted-mass (RTM) scheme, tailored to improve the signal on these kinds of observables, it is possible to evaluate the disconnected diagrams with good precision. For the QED induced pion mass difference, we obtain, after performing the extrapolation towards the continuum and thermodynamic limit and at the physical point, the preliminary value $M_{\pi^{+}}-M_{\pi^{0}} = 4.622~(95)~{\rm MeV}$, that is in good agreement with the experimental result. For the determination of the low-energy constant $\ell_{7}$, our result $\ell_{7} = 2.5~(1.4)\times 10^{-3}$, which is limited so far to a single lattice spacing, is in agreement and improves phenomenological estimates.