Lattice QCD at the physical point meetsSU(2)chiral perturbation theory

Abstract

We perform a detailed, fully correlated study of the chiral behavior of the pion mass and decay constant, based on $2+1$ flavor lattice QCD simulations. These calculations are implemented using tree-level, $O(a)$-improved Wilson fermions, at four values of the lattice spacing down to 0.054 fm and all the way down to below the physical value of the pion mass. They allow a sharp comparison with the predictions of $SU(2)$ chiral perturbation theory ($\ensuremath{\chi}\mathrm{PT}$) and a determination of some of its low energy constants. In particular, we systematically explore the range of applicability of next-to-leading order (NLO) $SU(2)$ $\ensuremath{\chi}\mathrm{PT}$ in two different expansions: the first in quark mass ($x$ expansion), and the second in pion mass ($\ensuremath{\xi}$ expansion). We find that these expansions begin showing signs of failure for ${M}_{\ensuremath{\pi}}\ensuremath{\gtrsim}300\text{ }\text{ }\mathrm{MeV}$, for the typical percent-level precision of our ${N}_{f}=2+1$ lattice results. We further determine the LO low energy constants (LECs), $F=88.0\ifmmode\pm\else\textpm\fi{}1.3\ifmmode\pm\else\textpm\fi{}0.2$ and ${B}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=2.61(6)(1)\text{ }\text{ }\mathrm{GeV}$, and the related quark condensate, ${\mathrm{\ensuremath{\Sigma}}}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=(272\ifmmode\pm\else\textpm\fi{}4\ifmmode\pm\else\textpm\fi{}1\text{ }\text{ }\mathrm{MeV}{)}^{3}$, as well as the NLO ones, ${\overline{\ensuremath{\ell}}}_{3}=2.6(5)(3)$ and ${\overline{\ensuremath{\ell}}}_{4}=3.7(4)(2)$, with fully controlled uncertainties. We also explore the next-to-next-to-leading order (NNLO) expansions and the values of NNLO LECs. In addition, we show that the lattice results favor the presence of chiral logarithms. We further demonstrate how the absence of lattice results with pion masses below 200 MeV can lead to misleading results and conclusions. Our calculations allow a fully controlled, ab initio determination of the pion decay constant with a total 1% error, which is in excellent agreement with experiment.