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

We evaluate by means of lattice QCD calculations the low-energy constant ℓ7 which parametrizes strong isospin effects at next-to-leading order (NLO) in SU(2) chiral perturbation theory. Among all low-energy constants at NLO, ℓ7 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 Δm=(md−mu)/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 ℓ7 from either the pion mass splitting Mπ+−Mπ0 induced by strong isospin breaking at order O((Δm)2) (mass method), or from the coupling of the neutral pion π0 to the isoscalar operator (¯uγ5u+¯dγ5d)/√2 at order O(Δm) (matrix element method). In this pilot study we limit the analysis to a single ensemble generated by the Extended Twisted Mass Collaboration (ETMC) with Nf=2+1+1 dynamical quark flavors, which corresponds to a lattice spacing a≃0.095  fm and to a pion mass Mπ≃260  MeV. We find that the matrix element method outperforms the mass method in terms of resulting statistical accuracy. Our determination, ℓ7=2.5(1.4)×10−3, is in agreement and improves previous calculations.