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Twisted-mass QCD, O(a) improvement and Wilson chiral perturbation theory

Sinya Aoki, Oliver Baer

TL;DR

The paper identifies a caveat in automatic $O(a)$ improvement for twisted-mass lattice QCD at maximal twist arising from the twist-angle definition, showing that the standard definition can require $m_q \gg a^2\Lambda_{\rm QCD}^3$ for automatic improvement to hold. It proposes an alternative twist-angle definition based on the average critical mass $\overline{M}_{cr}$ that guarantees automatic $O(a)$ improvement without restricting $m_q$, and confirms this inside Wilson Chiral Perturbation Theory by computing the pion masses with leading lattice-spacing terms. By comparing the two definitions, the work demonstrates that the new scheme yields $m_{\pi}$ free of $O(a)$ artifacts for all $m_q$, while the standard FR approach can retain $O(a)$ effects when the critical mass is not strictly odd in the Wilson parameter. The results have practical impact for simulations aiming at light quark masses and clarify the role of the critical mass and Aoki-phase structure in lattice QCD, suggesting that action choices and lattice artifacts can be managed without imposing stringent mass bounds.

Abstract

We point out a caveat in the proof for automatic O(a) improvement in twisted mass lattice QCD at maximal twist angle. With the definition for the twist angle previously given by Frezzotti and Rossi, automatic O(a) improvement can fail unless the quark mass satisfies m_q >> a^2 Lambda_QCD^3. We propose a different definition for the twist angle which does not require a restriction on the quark mass for automatic O(a) improvement. In order to illustrate explicitly automatic O(a) improvement we compute the pion mass in the corresponding chiral effective theory. We consider different definitions for maximal twist and show explicitly the absence or presence of the leading O(a) effect, depending on the size of the quark mass.

Twisted-mass QCD, O(a) improvement and Wilson chiral perturbation theory

TL;DR

The paper identifies a caveat in automatic improvement for twisted-mass lattice QCD at maximal twist arising from the twist-angle definition, showing that the standard definition can require for automatic improvement to hold. It proposes an alternative twist-angle definition based on the average critical mass that guarantees automatic improvement without restricting , and confirms this inside Wilson Chiral Perturbation Theory by computing the pion masses with leading lattice-spacing terms. By comparing the two definitions, the work demonstrates that the new scheme yields free of artifacts for all , while the standard FR approach can retain effects when the critical mass is not strictly odd in the Wilson parameter. The results have practical impact for simulations aiming at light quark masses and clarify the role of the critical mass and Aoki-phase structure in lattice QCD, suggesting that action choices and lattice artifacts can be managed without imposing stringent mass bounds.

Abstract

We point out a caveat in the proof for automatic O(a) improvement in twisted mass lattice QCD at maximal twist angle. With the definition for the twist angle previously given by Frezzotti and Rossi, automatic O(a) improvement can fail unless the quark mass satisfies m_q >> a^2 Lambda_QCD^3. We propose a different definition for the twist angle which does not require a restriction on the quark mass for automatic O(a) improvement. In order to illustrate explicitly automatic O(a) improvement we compute the pion mass in the corresponding chiral effective theory. We consider different definitions for maximal twist and show explicitly the absence or presence of the leading O(a) effect, depending on the size of the quark mass.

Paper Structure

This paper contains 18 sections, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Critical quark mass and twist angle
  3. Automatic O(a) improvement at maximal twist
  4. The critical mass
  5. Subtleties at $\omega=\pm\pi/2$
  6. Alternative definition for $\omega=\pm\pi/2$
  7. Wilson Chiral Perturbation Theory (WChPT)
  8. Chiral effective Lagrangian
  9. Gap equation
  10. Pion mass formulae
  11. VEV and pion masses
  12. Twist angle from the Ward-Takahashi identities
  13. Maximal twist and O($\rm\bf a$) improvement
  14. $\omega =\pi/2$
  15. Maximal twist of Frezzotti and Rossi
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Phase diagram for $N_f=2$ lattice QCD with Wilson fermions, where $M\equiv m_0 a + 4 r$. The parity and flavor symmetries are spontaneously broken in phase B.
  • Figure 2: Two different definitions for the twist angle. The angle $\omega$ has been defined in Ref.Frezzotti:2003ni, the angle $\omega^{\prime}$ corresponds to the definition in eq. (\ref{['eq:twQCD2']}).