Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Twisted mass chiral perturbation theory at next-to-leading order

Stephen R. Sharpe, Jackson M. S. Wu

TL;DR

This work develops a next-to-leading order twisted mass chiral perturbation theory (tm\chiPT) framework for two-flavor tmLQCD with a counting $m_q \sim a\Lambda_{\rm QCD}^2$, showing that automatic $O(a)$ improvement at maximal twist persists when a non-perturbative twist angle is used. It provides explicit NLO expressions for pion masses, decay constants, and parity-conserving and parity-violating matrix elements, demonstrating how discretization effects enter via a controlled set of low-energy constants (LECs) and how they can be disentangled from physical observables. The paper extends the GSM analysis to the Aoki regime ($m_q \sim a^2\Lambda_{\rm QCD}^3$), where $O(a)$ improvement holds under certain twist definitions, and discusses phase structure, mass splittings, and the role of parity-violating quantities as diagnostics for discretization errors. Overall, it offers a comprehensive EFT toolkit to interpret tmLQCD data, determine discretization LECs, and guide reliable continuum extrapolations.

Abstract

We study the properties of pions in twisted mass lattice QCD (with two degenerate flavors) using chiral perturbation theory (ChPT). We work to next-to-leading order (NLO) in a power counting scheme in which m_q ~ a Λ_QCD^2, with m_q the physical quark mass and a the lattice spacing. We argue that automatic O(a) improvement of physical quantities at maximal twist, which has been demonstrated in general if m_q >> a Λ_QCD^2, holds even if m_q ~ a Λ_QCD^2, as long as one uses an appropriate non-perturbative definition of the twist angle. We demonstrate this with explicit calculations, for arbitrary twist angle, of all pionic quantities that involve no more than a single pion in the initial and final states: masses, decay constants, form factors and condensates, as well as the differences between alternate definitions of twist angle. We also calculate the axial and pseudoscalar form factors of the pion, quantities which violate flavor and parity, and which vanish in the continuum limit. These are of interest because they are not automatically O(a) improved at maximal twist. They allow a determination of the unknown low energy constants introduced by discretization errors, and provide tests of the accuracy of ChPT at NLO. We extend our results into the regime where m_q ~ a^2 Λ_QCD^3, and argue in favor of a recent proposal that automatic O(a) improvement at maximal twist remains valid in this regime.

Twisted mass chiral perturbation theory at next-to-leading order

TL;DR

This work develops a next-to-leading order twisted mass chiral perturbation theory (tm\chiPT) framework for two-flavor tmLQCD with a counting , showing that automatic improvement at maximal twist persists when a non-perturbative twist angle is used. It provides explicit NLO expressions for pion masses, decay constants, and parity-conserving and parity-violating matrix elements, demonstrating how discretization effects enter via a controlled set of low-energy constants (LECs) and how they can be disentangled from physical observables. The paper extends the GSM analysis to the Aoki regime (), where improvement holds under certain twist definitions, and discusses phase structure, mass splittings, and the role of parity-violating quantities as diagnostics for discretization errors. Overall, it offers a comprehensive EFT toolkit to interpret tmLQCD data, determine discretization LECs, and guide reliable continuum extrapolations.

Abstract

We study the properties of pions in twisted mass lattice QCD (with two degenerate flavors) using chiral perturbation theory (ChPT). We work to next-to-leading order (NLO) in a power counting scheme in which m_q ~ a Λ_QCD^2, with m_q the physical quark mass and a the lattice spacing. We argue that automatic O(a) improvement of physical quantities at maximal twist, which has been demonstrated in general if m_q >> a Λ_QCD^2, holds even if m_q ~ a Λ_QCD^2, as long as one uses an appropriate non-perturbative definition of the twist angle. We demonstrate this with explicit calculations, for arbitrary twist angle, of all pionic quantities that involve no more than a single pion in the initial and final states: masses, decay constants, form factors and condensates, as well as the differences between alternate definitions of twist angle. We also calculate the axial and pseudoscalar form factors of the pion, quantities which violate flavor and parity, and which vanish in the continuum limit. These are of interest because they are not automatically O(a) improved at maximal twist. They allow a determination of the unknown low energy constants introduced by discretization errors, and provide tests of the accuracy of ChPT at NLO. We extend our results into the regime where m_q ~ a^2 Λ_QCD^3, and argue in favor of a recent proposal that automatic O(a) improvement at maximal twist remains valid in this regime.

Paper Structure

This paper contains 14 sections, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The effective chiral Lagrangian
  3. Chiral perturbation theory for generic small masses
  4. The vacuum and Feynman rules
  5. Currents and densities in the twisted basis
  6. Defining the twist angle
  7. Constructing the physical currents and densities
  8. Pion masses
  9. Decay constants and PCAC quark mass
  10. Parity conserving matrix elements
  11. Parity violating matrix elements
  12. Summary of results in GSM regime
  13. The Aoki regime
  14. Conclusion

Figures (1)

  • Figure 1: Generic types of Feynman diagrams that contribute to the two pion matrix elements of the physical axial current. The arrows label only the momentum flow. The direct term comes from the two pion vertices in the axial current. The pole term arises from contracting the single pion term in the axial current with the three pion vertices in $\mathcal{L}_\chi$.