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Twisted mass lattice QCD

A. Shindler

TL;DR

The paper analyzes Wilson twisted mass lattice QCD (Wtm QCD), focusing on how the twisted mass term enables automatic $O(a)$ improvement and simplifies renormalization, while introducing parity and flavour breaking at finite lattice spacing. It establishes the theoretical equivalence between tmQCD and standard QCD in the continuum (and under chiral-symmetry-preserving regulators), and develops the Symanzik framework to quantify discretization errors, including $O(a)$ and $O(a^2)$ effects. It extends the formalism to non-degenerate quarks, discusses transfer-matrix positivity, and presents practical strategies for achieving maximal twist through the critical mass, with substantial numerical evidence supporting improved scaling behavior, especially for parity-even observables. The work also surveys numerical tests, restoration of symmetries at full twist, and methods to ease renormalization patterns for phenomenologically relevant operators, highlighting the practical impact for lattice QCD simulations of light and heavy quarks.

Abstract

I review the theoretical foundations, properties as well as the simulation results obtained so far of a variant of the Wilson lattice QCD formulation: Wilson twisted mass lattice QCD. Emphasis is put on the discretization errors and on the effects of these discretization errors on the phase structure for Wilson-like fermions in the chiral limit. The possibility to use in lattice simulations different lattice actions for sea and valence quarks to ease the renormalization patterns of phenomenologically relevant local operators, is also discussed.

Twisted mass lattice QCD

TL;DR

The paper analyzes Wilson twisted mass lattice QCD (Wtm QCD), focusing on how the twisted mass term enables automatic improvement and simplifies renormalization, while introducing parity and flavour breaking at finite lattice spacing. It establishes the theoretical equivalence between tmQCD and standard QCD in the continuum (and under chiral-symmetry-preserving regulators), and develops the Symanzik framework to quantify discretization errors, including and effects. It extends the formalism to non-degenerate quarks, discusses transfer-matrix positivity, and presents practical strategies for achieving maximal twist through the critical mass, with substantial numerical evidence supporting improved scaling behavior, especially for parity-even observables. The work also surveys numerical tests, restoration of symmetries at full twist, and methods to ease renormalization patterns for phenomenologically relevant operators, highlighting the practical impact for lattice QCD simulations of light and heavy quarks.

Abstract

I review the theoretical foundations, properties as well as the simulation results obtained so far of a variant of the Wilson lattice QCD formulation: Wilson twisted mass lattice QCD. Emphasis is put on the discretization errors and on the effects of these discretization errors on the phase structure for Wilson-like fermions in the chiral limit. The possibility to use in lattice simulations different lattice actions for sea and valence quarks to ease the renormalization patterns of phenomenologically relevant local operators, is also discussed.

Paper Structure

This paper contains 24 sections, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic properties
  3. Twisted mass QCD in the continuum
  4. Beyond the classical theory
  5. Wilson twisted mass QCD
  6. Symmetries
  7. Chiral symmetry at $\omega={\pi \over 2}$
  8. Tree-level
  9. Transfer matrix
  10. Renormalization
  11. Correlation functions
  12. Exceptional configurations
  13. Non-degenerate quarks
  14. Continuum actions
  15. Lattice actions
  16. ...and 9 more sections

Figures (2)

  • Figure 1: $f_{\rm PS}$ vs. $a^2$ at at a fixed value of the pseudoscalar mass $M_{\rm PS} \! \simeq \! 1.2 M_{\rm K^\pm}$ for non-perturbatively (NP) improved Wilson fermions ($\square$) Garden:1999fg and non-perturbatively (NP) improved Wilson twisted mass ($\blacksquare$) DellaMorte:2001tu. The continuum extrapolation by ref. Garden:1999fg is also shown.
  • Figure :