Ginsparg-Wilson Pions Scattering in a Sea of Staggered Quarks

TL;DR

This work derives mixed-action chiral perturbation theory expressions for $I=2$ $\pi\pi$ scattering with Ginsparg-Wilson valence quarks on staggered sea quarks, enabling correct chiral and continuum extrapolations of lattice data. A key finding is that, for this channel, the power-law finite-volume dependence matches Lüscher’s form even in a nonunitary theory, allowing extraction of infinite-volume scattering parameters from mixed-action simulations. The authors show that when the amplitude is written in lattice-physical parameters, most PQ and lattice-spacing artifacts cancel, leaving only hairpin-related corrections through $\tilde{\Delta}_{PQ}$ and a single low-energy constant $l_{\pi\pi}$. In the $2$ and $2+1$ flavor cases, explicit expressions for the $I=2$ pion scattering length are provided, highlighting cancellations of many lattice artifacts and giving practical formulas for extrapolating mixed-action lattice data to the physical point. These results offer a concrete framework for using mixed-action lattice QCD to determine physical $\pi\pi$ scattering parameters with controlled systematic errors.

Abstract

We calculate isospin 2 pion-pion scattering in chiral perturbation theory for a partially quenched, mixed action theory with Ginsparg-Wilson valence quarks and staggered sea quarks. We point out that for some scattering channels, the power-law volume dependence of two pion states in nonunitary theories such as partially quenched or mixed action QCD is identical to that of QCD. Thus one can extract infinite volume scattering parameters from mixed action simulations. We then determine the scattering length for both 2 and 2+1 sea quarks in the isospin limit. The scattering length, when expressed in terms of the pion mass and the decay constant measured on the lattice, has no contributions from mixed valence-sea mesons, thus it does not depend upon the parameter, C_Mix, that appears in the chiral Lagrangian of the mixed theory. In addition, the contributions which nominally arise from operators appearing in the mixed action O(a^2 m_q) Lagrangian exactly cancel when the scattering length is written in this form. This is in contrast to the scattering length expressed in terms of the bare parameters of the chiral Lagrangian, which explicitly exhibits all the sicknesses and lattice spacing dependence allowed by a partially quenched mixed action theory. These results hold for both 2 and 2+1 flavors of sea quarks.

Figures (4)

• Figure 1: One-loop diagrams contributing to the $\pi\pi$ scattering amplitude. Diagrams (a)--(c) are the $s$-, $t$-, and $u$-channel diagrams, respectively, while diagram (e) represents wavefunction renormalization.
• Figure 2: Example quark flow for a one-loop $t$-channel graph. This diagram illustrates the presence of meson loops composed of purely valence-valence mesons which are not canceled by valence-ghost loops. Different colors (shades of grey) represent different quark flavors.
• Figure 3: Example hairpin diagrams contributing to pion scattering. The propagator with a cross through it indicates the quark-disconnected piece of the $\bar{\eta}$ propagator, Eq. (\ref{['eq:su42hairprop']}).
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