First direct lattice calculation of the chiral perturbation theory low-energy constant $ell_7$

We evaluate by means of lattice QCD calculations the low-energy constant $ell_{7}$ which parametrizes strong isospin effects at NLO in $rm{SU}(2)$ 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 $Delta m= (m_{d}-m_{u})/2$. 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 $M_{pi^{+}}-M_{pi^{0}}$ induced by strong isospin breaking at order $mathcal{O}left((Delta m)^{2}right)$ (mass method), or from the coupling of the neutral pion $pi^{0}$ to the isoscalar operator $left(bar{u}gamma_{5}u + bar{d}gamma_{5} dright)/sqrt{2}$ at order $mathcal{O}(Delta m)$ (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 flavours, which corresponds to a lattice spacing $asimeq 0.095~{rm fm}$ and to a pion mass $M_{pi}simeq 260~{rm MeV}$. 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.