Two Meson Systems with Ginsparg-Wilson Valence Quarks

TL;DR

This paper demonstrates that mixed-action chiral perturbation theory with Ginsparg-Wilson valence quarks produces NLO extrapolation formulae for mesonic observables that contain only physical low-energy constants when expressed in lattice-physical parameters. Through spurion analysis and explicit calculations of $\pi\pi$, $KK$, and $K\pi$ scattering lengths (and $f_K/f_\pi$) in MAχPT, they show that lattice-spacing and sea-quark mass artifacts can be absorbed into measured lattice quantities, reducing unphysical counterterms and enabling robust, single-spacings checks of MA theory. The results reveal that some observables (like $a_{KK}^{I=1}$) are independent of the unknown $C_ ext{Mix}$ in lattice-physical form, while others (notably $a_{K\pi}^{I=3/2}$) retain sensitivity to $C_ ext{Mix}$, necessitating its determination for precise extrapolations. Overall, the work provides a practical framework for chiral extrapolations in mixed-action simulations and argues for the broader utility of lattice-physical parameterizations in reducing systematic uncertainties across hadronic observables.

Abstract

Unphysical effects associated with finite lattice spacing and partial quenching may lead to the presence of unphysical terms in chiral extrapolation formulae. These unphysical terms must then be removed during data analysis before physical predictions can be made. In this work, we show that through next-to-leading order, there are no unphysical counterterms in the extrapolation formulae, expressed in lattice-physical parameters, for meson scattering lengths in theories with Ginsparg-Wilson valence quarks. Our work applies to most sea quark discretization, provided that chiral perturbation theory is a valid approximation. We demonstrate our results with explicit computations and show that, in favorable circumstances, the extrapolation formulae do not depend on the unknown constant C_Mix appearing at lowest order in the mixed action chiral Lagrangian. We show that the I=1 KK scattering length does not depend on C_Mix in contrast to the I=3/2 K-pi scattering length. In addition, we show that these observables combined with f_K / f_pi and the I=2 pi-pi scattering length share only two linearly independent sets of counterterms, providing a means to test the mixed action theory at one lattice spacing. We therefore make a prediction for the I=1 KK scattering length.

Figures (2)

• Figure 1: We plot the ratio, ${\Delta}(f_K / f_\pi)$ defined in Eq. (\ref{['eq:fKoverfpi']}) as a function of the unknown mixed meson mass splitting, $-(600 \textrm{ MeV})^2 \lesssim a^2 {\Delta}_\mathrm{Mix} \lesssim (800 \textrm{ MeV})^2$. The observed deviation from the continuum $\chi$PT formulae is on the order of 5%, which is important, but not significant enough to directly determine this unknown mass splitting from the MA lattice data of $f_K / f_\pi$Beane:2006kx alone.
• Figure 2: We plot the absolute values of the various NLO contributions to $m_\pi a_{\pi\pi}^{I=2}$ listed in Table \ref{['t:Fn']}. The NLO $\chi$PT contribution is given by $(a)$ (green), which demonstrates the large cancellation of the counterterm and the chiral log for light to medium pion masses. The long-dashed curve (red) is the 2-sea flavor hairpin effects, $(b)$, which are the same order of magnitude as $(a)$, for $m_\pi \lesssim 400\ \textrm{MeV}$. When the new 3-sea flavor hairpin effects, $(c)$ (blue), are added to the 2-sea flavor effects, one finds that the total 3-sea flavor hairpin effects, $(d)$ (black), are small compared to $(a)$ for $m_\pi \gtrsim 250$ MeV.