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

TL;DR

The paper develops Wilson chiral perturbation theory (WChPT) that incorporates $a^2$ lattice artifacts for $N_f=2$ degenerate quarks, deriving the one-loop quark-mass dependence of $m_ pi^2$, $m_{ m AWI}$, and $f_ pi$ with additive mass renormalization and novel logarithmic corrections. It applies the resulting formulas to CP-PACS lattice data, performing a simultaneous fit of $m_ pi^2$ and $m_{ m AWI}$ across four lattice spacings, and finds that WChPT describes the data better than standard ChPT, with stable determinations of some parameters and larger uncertainties for others. The authors also derive a resummed version of the formulas to tame the most singular logs, demonstrating improved behavior near the chiral limit. Overall, the work shows that including $O(a^2)$ effects in the chiral effective theory yields a more accurate description of lattice QCD results with Wilson-type fermions and provides a framework for controlled chiral extrapolations in this setting.

Abstract

We have derived the quark mass dependence of $m_π^2$, $m_{\rm AWI}$ and $f_π$, 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 $m_π^2$ 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.

Figures (4)

• Figure 1: The WChPT fits for $m_\pi^2$ and $m_{\rm AWI}$ at each $\beta$. Results are shown for $m_\pi^2/m_{\rm AWI}$ as a function of $m_{\rm AWI}$. For comparison the standard ChPT fits ($w_1=w_0=0$) are also included.
• Figure 2: The fit parameters as a function of $a$ or $g^2$.
• Figure 3: The WChPT fits for $m_\pi^2/m_{\rm AWI}$ as a function of $m_R$ and $a$. Results are shown for $m_\pi^2/m_{\rm AWI}$ as a function of $m_{R}$.
• Figure 4: Left: The relative size of the next-to-leading contribution to the leading one in the WChPT as a function of the quark mass $m_R$ at $\beta$=1.8,1.95,2.1 and 2.2, together with the one in the continuum limit (ChPT). Right: Same quantities in the resummed WChPT.