Pion mass splitting and phase structure in Twisted Mass QCD

TL;DR

The paper develops a Wilson chiral perturbation theory description of two-flavor Twisted Mass QCD including $O(a^2)$ lattice artifacts to analyze vacuum orientation and the pion spectrum. By fixing the ratio $\\\eta=\\rho/\\chi$ and expanding in the quark mass, the authors map how the vacuum rotates with twist and lattice spacing, and how the pion masses split due to flavor breaking, with both phenomena governed by the same combination of LECs that also controls possible Aoki phases. The main contributions are explicit LO and NLO (in both $m$ and $a$) expressions for the vacuum orientation and for $m_{\\pi^{±}}^2$ and $m_{\\pi^0}^2$, including loop effects and renormalized LECs, and a detailed analysis of the critical region near $\\eta\\approx1$, $\\omega\\approx\\pi$. These results provide practical guidance for interpreting tmQCD lattice data and for extracting LEC combinations from pion spectroscopy, with direct relevance to phase structure and Aoki-phase searches in simulations.

Abstract

In the framework of Wilson Chiral Perturbation Theory, we study the effect induced by a twisted Wilson term, as it appears in Twisted Mass QCD (with 2 degenerate quarks). In particular we consider the vacuum orientation and the pion masses. The computations are done to NLO both in the mass and in the lattice spacing (i.e. to O(a^2)). There are no restrictions on the relative size of lattice artifacts with respect to the physical mass, thus allowing, in principle, to bridge between the physical regime and the unphysical one, where lattice artifacts tend to dominate. The inclusion of O(a^2) lattice artifacts can account for the splitting of degeneracy of the three pion masses. Moreover O(a^2) terms are necessary to model non trivial behaviors of the vacuum orientation such as possible Aoki phases. It turns out that these last two phenomena are determined by the same constant.

Figures (7)

• Figure 1: LO solution $\theta_{LO}$ as function of $\omega$.
• Figure 2: NLO solution for the vacuum orientation $\theta$ as a function of $\omega$ for various choices of the parameters (dashed-blue). NLO solution obtained by numerical minimization of the NLO potential (dotted-red). LO solution (full-black).
• Figure 3: NLO solution for the vacuum orientation $\theta$ as a function of $\eta$ for various choices of the parameters. Dashed-blue, dotted-red, full-black as before.
• Figure 4: NLO solution for the vacuum orientation $\theta$ as a function of $\omega$ for various choices of the parameters and $\eta\sim 1$. Dashed-blue, dotted-red, full-black as before. Instabilities are clearly displayed in the NLO analytical solution.
• Figure 5: Potential ${\cal V}_{\chi}(\theta)$ evolving from $\omega=\pi/2$ (thick-blue) to $\omega=\pi$ (thin-red). The other parameters are $\eta=9/10$, $\chi\eta^2 W'_{86}=-1/20$, $L_{86}=W_{86}=0$.