First direct lattice calculation of the chiral perturbation theory low-energy constant ℓ7

Abstract

We evaluate by means of lattice QCD calculations the low-energy constant ${\ensuremath{\ell}}_{7}$ which parametrizes strong isospin effects at next-to-leading order (NLO) in SU(2) chiral perturbation theory. Among all low-energy constants at NLO, ${\ensuremath{\ell}}_{7}$ is the one known less precisely, and its uncertainty is currently larger than 50%. Our strategy is based on the RM123 approach in which the lattice path-integral is expanded in powers of the isospin breaking parameter $\mathrm{\ensuremath{\Delta}}m=({m}_{d}\ensuremath{-}{m}_{u})/2$. In order to evaluate the relevant lattice correlators we make use of the recently proposed rotated twisted-mass (RTM) scheme. Within the RM123 approach, it is possible to cleanly extract the value of ${\ensuremath{\ell}}_{7}$ from either the pion mass splitting ${M}_{{\ensuremath{\pi}}^{+}}\ensuremath{-}{M}_{{\ensuremath{\pi}}^{0}}$ induced by strong isospin breaking at order $\mathcal{O}((\mathrm{\ensuremath{\Delta}}m{)}^{2})$ (mass method), or from the coupling of the neutral pion ${\ensuremath{\pi}}^{0}$ to the isoscalar operator $(\overline{u}{\ensuremath{\gamma}}_{5}u+\overline{d}{\ensuremath{\gamma}}_{5}d)/\sqrt{2}$ at order $\mathcal{O}(\mathrm{\ensuremath{\Delta}}m)$ (matrix element method). In this pilot study we limit the analysis to a single ensemble generated by the Extended Twisted Mass Collaboration (ETMC) with ${N}_{f}=2+1+1$ dynamical quark flavors, which corresponds to a lattice spacing $a\ensuremath{\simeq}0.095\text{ }\text{ }\mathrm{fm}$ and to a pion mass ${M}_{\ensuremath{\pi}}\ensuremath{\simeq}260\text{ }\text{ }\mathrm{MeV}$. We find that the matrix element method outperforms the mass method in terms of resulting statistical accuracy. Our determination, ${\ensuremath{\ell}}_{7}=2.5(1.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, is in agreement and improves previous calculations.