Twisted-mass lattice QCD with mass non-degenerate quarks

TL;DR

The paper extends twisted-mass lattice QCD to mass-non-degenerate quark pairs at maximal twist ($ω=\pi/2$), showing the fermion determinant remains real and positive for $m_q\neq 0$ and that $O(a)$ discretization effects are automatically eliminated or obtainable without Wilson averaging. It develops non-singlet Ward-Takahashi identities in this setting, clarifies the renormalization structure with currents and densities ($Z_V$, $Z_A$, $Z_P$, $Z_S^0$) and introduces mass combinations $m_q^{(\pm)}=\hat m_q \pm \hat \epsilon_q$ for the non-degenerate case. The work demonstrates that $O(a)$ improvement persists under mass splitting ($\epsilon_q\neq 0$) and that the determinant positivity condition becomes $m_q^2>\epsilon_q^2$, with a detailed positivity proof and a practical bound on $Z_P/Z_S$. It discusses the implications for Monte Carlo simulations and outlines potential applications to CP-conserving weak Hamiltonians ($\Delta S=2$, $\Delta S=1$) with reduced wrong-chirality mixings, using regularizations with differently twisted Wilson terms. Overall, the framework provides a robust, $O(a)$-improved, positive-determinant lattice formulation for non-degenerate quark sectors with practical routes for operator matrix-element calculations.

Abstract

The maximally twisted lattice QCD action of an $SU_f(2)$ doublet of mass degenerate Wilson quarks gives rise to a real positive fermion determinant and it is invariant under the product of standard parity times the change of sign of the coefficient of the Wilson term. The existence of this spurionic symmetry implies that O($a$) improvement is either automatic or achieved through simple linear combinations of quantities taken with opposite external three-momenta. We show that in the case of maximal twist all these nice results can be extended to the more interesting case of a mass non-degenerate quark pair.