Lattice determination of the QCD low-energy constant $\ell_{\scriptscriptstyle{7}}$

Abstract

We provide a non-perturbative determination of the scheme- and scale-independent low-energy constant $\ell_{\scriptscriptstyle{7}}$, appearing in the QCD effective chiral Lagrangian at next-to-leading order, by means of lattice QCD simulations with $N_{\scriptscriptstyle{\rm f}}=2+1$ quark flavors. We adopt staggered fermions and extract $\ell_{\scriptscriptstyle{7}}$ 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 $\ell_{\scriptscriptstyle{7}} \,\times \, 10^3 = 1.98(48)_{\scriptscriptstyle{\rm stat}}(26)_{\scriptscriptstyle{\rm syst}} = 1.98(54)_{\scriptscriptstyle{\rm tot}}$ agrees with and substantially improves on previous determinations.

Figures (8)

• Figure 1: Left panel: lattice estimator $\ell_{{7}}^{{{(\rm eff)}}}(t)$ and individual contributions of the connected/disconnected diagrams to this function for the ensemble with $R\simeq0.1421$ and $a=0.095$ fm. Right panel: extraction of $\ell_{{7}}$ from the large-time-separation plateau exhibited by $\ell_{{7}}^{{{(\rm eff)}}}(t)$.
• Figure 2: Study of finite-size effects on $\ell_{{7}}$ for our ensemble with $a=0.0964$$\mathrm{fm}$ and $M_\pi=263(4)$$\mathrm{MeV}$. They are invisible within our statistical errors for $M_\pi L \ge 4$. Same conclusions are reached after subtracting finite-size effects estimated from NLO $\chi\mathrm{PT}$: $\ell_{{7}}(L)\to\ell_{{7}}(L)/[1+R_{\ell_{{7}}}(L)]$, see Sec. \ref{['sec:FSE']}.
• Figure 3: Continuum limit extrapolations of $\ell_{{7}}$ for the 3 values of the parameter $R=m_{{\ell}}/m_{{\rm s}}^{{{(\mathrm{phys})}}}$ explored. Left panels report best fits as a function of $a^2$, while right panels report best fits as a function of $w_0^2\Delta$. Continuous lines (with their related continuous shaded areas) correspond to linear fits using all available data points, dashed lines represent linear fits restricted to the 3 finest lattice spacings, and dotted lines (with their related dashed shaded areas) represent quadratic fits. The star points represent our final results for the continuum limits, are obtained from the linear extrapolations in $a^2$ (no star point is represented for extrapolations in $w_0^2\Delta$, as these fits are only used to assess systematic errors). Since the total error bar, given by the sum in quadrature of the statistical and systematic uncertainties, is barely distinguishable from the statistical one for the two largest values of $R$, it has been slightly shifted horizontally for clarity.
• Figure 4: Chiral limit extrapolation of the corrected values of $\ell_{{7}}(R)$ according to a linear function of the ratio $R=m_{{\ell}}/m_{{\rm s}}^{{{(\mathrm{phys})}}}$. The full star point at $R=0$ stands for the extrapolated value in the chiral limit, according to the best fit of the corrected data (dashed line). The two error bars plotted for the chiral extrapolation represent, respectively, the statistical and the sum of the statistical and systematic uncertainties. For the sake of comparison we also show the non-corrected data and their naive linear extrapolation, depicted as a dashed shaded band.
• Figure 5: Continuum limits of the pion masses extracted from interpolating operators with different taste structure for all the gauge ensembles employed in this study.