Chiral perturbation theory with Wilson-type fermions including $a^2$ effects: $N_f=2$ degenerate case

TL;DR

This paper extends chiral perturbation theory to include $a^2$ effects from Wilson-type fermions for $N_f=2$ degenerate quarks, deriving formulas for meson masses and decay constants that fit lattice QCD data well.

Contribution

It introduces a comprehensive chiral perturbation theory framework incorporating $a^2$ effects specific to Wilson fermions, including additive mass renormalization and singular terms.

Findings

Derived formulas fit CP-PACS lattice data for $m_^2$ and $m_{ m AWI}$.

Identified $a^2$-induced singular $ ext{log } m_q$ terms.

Resummed singular terms to improve next-to-leading order accuracy.

Abstract

We have derived the quark mass dependence of $m_{\pi}^2$, $m_{\rm AWI}$ and $f_{\pi}$, using the chiral perturbation theory which includes the $a^2$ effect associated with the explicit chiral symmetry breaking of the Wilson-type fermions, in the case of the $N_f=2$ degenerate quarks. Distinct features of the results are (1) the additive renormalization for the mass parameter $m_q$ in the Lagrangian, (2) $O(a)$ corrections to the chiral log ($m_q\log m_q$) term, (3) the existence of more singular term, $\log m_q$, generated by $a^2$ contributions, and (4) the existence of both $m_q\log m_q$ and $\log m_q$ terms in the quark mass from the axial Ward-Takahashi identity, $m_{\rm AWI}$. By fitting the mass dependence of $m_\pi^2$ and $m_{\rm AWI}$, obtained by the CP-PACS collaboration for $N_f=2$ full QCD simulations, we have found that the data are consistently described by the derived formulae. Resumming the most singular terms $\log m_q$, we have also derived the modified formulae, which show a better control over the next-to-leading order correction.