Lattice determination of the QCD low-energy constant ${\ell }_7$

Abstract

We provide a non-perturbative determination of the scheme- and scale-independent low-energy constant ${\ell }_7$, appearing in the QCD effective chiral Lagrangian at next-to-leading order, by means of lattice QCD simulations with $N_{\mathrm{f}}=2+1$ quark flavors. We adopt staggered fermions and extract ${\ell }_7$ from the pion mass splitting by suitably generalizing the method introduced in [Phys. Rev. D 104 (2021) 074513] for the Wilson discretization. Adopting 12 gauge ensembles with 3 different values of the pion mass, and 4 different values of the lattice spacing, we are able to achieve controlled extrapolations towards the continuum, infinite volume, and chiral limits. Our final result ${\ell }_7\times 10^3=2.79(58{)}_{\mathrm{stat}}(19{)}_{\mathrm{syst}}=2.79(61{)}_{\mathrm{tot}}$ agrees with and substantially improves on previous determinations.

14 pages, 8 figures. v2: 16 pages, 11 figures. New appendix reporting \ell_3 and \ell_4 determinations. Corrected a mistake in F_π, resulting in a 1.5-sigma shift of \ell_7. Conclusions unchanged. v3: references added