First direct lattice calculation of the chiral perturbation theory low-energy constant ${\ell }_7$

Abstract

We evaluate by means of lattice QCD calculations the low-energy constant ${\ell }_7$ which parametrizes strong isospin effects at NLO in $\mathrm{SU}(2)$ chiral perturbation theory. Among all low-energy constants at NLO, ${\ell }_7$ 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 $\hat{I}”m=(m_d-m_u)/2$. 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 ${\ell }_7$ from either the pion mass splitting $M_{\ddot{I}{€}^{+}}-M_{\ddot{I}{€}^0}$ induced by strong isospin breaking at order $\mathcal{O}\left((\hat{I}”m{)}^2\right)$ (mass method), or from the coupling of the neutral pion $\ddot{I}{€}^0$ to the isoscalar operator $\left(\bar{u}{\hat{I}}_5^{3}u+\bar{d}{\hat{I}}_5^{3}d\right)/\sqrt{2}$ at order $\mathcal{O}(\hat{I}”m)$ (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 flavours, which corresponds to a lattice spacing $a\simeq 0.095\mathrm{fm}$ and to a pion mass $M_{\ddot{I}}€\simeq 260\mathrm{MeV}$. We find that the matrix element method outperforms the mass method in terms of resulting statistical accuracy. Our determination, ${\ell }_7=2.5(1.4)\times 10^{-3}$, is in agreement and improves previous calculations.

17 pages, 3 figures