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Chirally improving Wilson fermions II. Four-quark operators

R. Frezzotti, G. C. Rossi

TL;DR

This work develops a lattice QCD framework using maximally twisted (twisted-mass) Wilson fermions to compute CP-conserving ΔS=1,2 matrix elements with no wrong-chirality or parity mixing and with automatic $O(a)$ improvement. By separating sea and valence quarks and employing carefully chosen valence actions (including OS-type replicas and ghost fields), the authors guarantee a positive determinant, controlled renormalization, and the absence of problematic operator mixing for key four-quark operators such as ${\cal Q}_{VV+AA}^{\Delta S=2}$ and ${\cal Q}^{\pm}_{VA+AV}$, enabling multiplicative renormalization and clean continuum limits. The paper provides concrete lattice discretizations and symmetry-based proofs showing no mixing with lower- or equal-dimension operators (aside from harmless logarithmic pieces) and demonstrates how to extract $B_K$ and $K\to\pi$ or $K\to\pi\pi$ amplitudes from correlators with automatic $O(a)$ improvement. These advances offer a practical route to first-principles, nonperturbative determinations of kaon weak matrix elements with controlled discretization effects, potentially matching or surpassing GW-based approaches in computational efficiency. The results also outline future directions, including hybrid GW/tm schemes and Schrödinger functional implementations, to further reduce lattice artifacts and enable precise phenomenology of the unitarity triangle.

Abstract

In this paper we discuss how the peculiar properties of twisted lattice QCD at maximal twist can be employed to set up a consistent computational scheme in which, despite the explicit breaking of chiral symmetry induced by the presence of the Wilson and mass terms in the action, it is possible to completely bypass the problem of wrong chirality and parity mixings in the computation of the CP-conserving matrix elements of the $ΔS=1,2$ effective weak Hamiltonian and at the same time have a positive determinant for non-degenerate quarks as well as full O($a$) improvement in on-shell quantities with no need of improving the lattice action and the operators.

Chirally improving Wilson fermions II. Four-quark operators

TL;DR

This work develops a lattice QCD framework using maximally twisted (twisted-mass) Wilson fermions to compute CP-conserving ΔS=1,2 matrix elements with no wrong-chirality or parity mixing and with automatic improvement. By separating sea and valence quarks and employing carefully chosen valence actions (including OS-type replicas and ghost fields), the authors guarantee a positive determinant, controlled renormalization, and the absence of problematic operator mixing for key four-quark operators such as and , enabling multiplicative renormalization and clean continuum limits. The paper provides concrete lattice discretizations and symmetry-based proofs showing no mixing with lower- or equal-dimension operators (aside from harmless logarithmic pieces) and demonstrates how to extract and or amplitudes from correlators with automatic improvement. These advances offer a practical route to first-principles, nonperturbative determinations of kaon weak matrix elements with controlled discretization effects, potentially matching or surpassing GW-based approaches in computational efficiency. The results also outline future directions, including hybrid GW/tm schemes and Schrödinger functional implementations, to further reduce lattice artifacts and enable precise phenomenology of the unitarity triangle.

Abstract

In this paper we discuss how the peculiar properties of twisted lattice QCD at maximal twist can be employed to set up a consistent computational scheme in which, despite the explicit breaking of chiral symmetry induced by the presence of the Wilson and mass terms in the action, it is possible to completely bypass the problem of wrong chirality and parity mixings in the computation of the CP-conserving matrix elements of the effective weak Hamiltonian and at the same time have a positive determinant for non-degenerate quarks as well as full O() improvement in on-shell quantities with no need of improving the lattice action and the operators.

Paper Structure

This paper contains 35 sections, 86 equations.

Table of Contents

  1. Introduction
  2. Sea and valence quarks
  3. Symmetries and renormalization
  4. Sea quark determinant
  5. Renormalization constants and O($a$) improvement
  6. O($a$) improvement via Wilson averaging
  7. O($a$) improved results from a single simulation
  8. Some conclusive observations
  9. The $B_K$ parameter
  10. A convenient lattice discretization of $C_{K'{\cal Q}K}$
  11. Proof of the absence of mixing in ${\cal Q}_{VV+AA}^{\Delta S=2}$
  12. The $\Delta I=1/2$ rule from $K\to\pi\pi$ amplitudes
  13. Kaon decay amplitudes
  14. Correlation functions and valence quarks
  15. A convenient lattice discretization of the $C_{\pm,K\pi\pi}^{(4s10v)}$ correlators
  16. ...and 20 more sections