The phase diagram of twisted mass lattice QCD

TL;DR

This work analyzes the phase structure of two-flavor twisted mass lattice QCD by deriving an effective chiral theory that includes discretization effects up to next-to-leading order. The authors establish a Symanzik-based continuum Lagrangian and construct the corresponding SU(2) chiral Lagrangian, identifying coefficients that encode the interplay of the untwisted mass, twisted mass, and lattice spacing. They demonstrate two distinct phase diagrams controlled by the sign of $c_2$: for $c_2>0$ an Aoki phase arises in the untwisted limit but is suppressed by nonzero twisting, while for $c_2<0$ a first-order transition extends into the twisted-mass plane and ends at two symmetric second-order points where $m_{\pi_3}^2=0$, with a precise 90-degree relation linking the two scenarios. The results provide explicit condensate and pion-mass behaviors across the phase diagrams, offering practical guidance for lattice simulations and action choices.

Abstract

We use the effective chiral Lagrangian to analyze the phase diagram of two-flavor twisted mass lattice QCD as a function of the normal and twisted masses, generalizing previous work for the untwisted theory. We first determine the chiral Lagrangian including discretization effects up to next-to-leading order (NLO) in a combined expansion in which m_π^2/(4πf_π)^2 ~ a Λ(a being the lattice spacing, and Λ= Λ_{QCD}). We then focus on the region where m_π^2/(4πf_π)^2 ~ (a Λ)^2, in which case competition between leading and NLO terms can lead to phase transitions. As for untwisted Wilson fermions, we find two possible phase diagrams, depending on the sign of a coefficient in the chiral Lagrangian. For one sign, there is an Aoki phase for pure Wilson fermions, with flavor and parity broken, but this is washed out into a crossover if the twisted mass is non-vanishing. For the other sign, there is a first order transition for pure Wilson fermions, and we find that this transition extends into the twisted mass plane, ending with two symmetrical second order points at which the mass of the neutral pion vanishes. We provide graphs of the condensate and pion masses for both scenarios, and note a simple mathematical relation between them. These results may be of importance to numerical simulations.

Figures (5)

• Figure 1: Phase diagram of tmLQCD. $\alpha$ and $\beta$ are proportional to the untwisted and twisted mass, respectively, in units proportional to the lattice spacing squared [see eqs. (\ref{['eq:alphadef']},\ref{['eq:betadef']})]. The sign of the coefficient $c_2$ determines whether flavor symmetry is spontaneously broken in the standard Wilson theory. The solid lines are first order transitions across which the condensate is discontinuous, with second-order endpoints. Figures \ref{['fig:c2gt0Am']}-\ref{['fig:c2lt0']} show the dependence of the condensate and pion masses along the horizontal dashed lines. See text for further discussion.
• Figure 2: The global minimum of the potential, $A_m$, as a function of $\alpha$, for $c_2 > 0$ and $\beta = 0,\,1,\,2,\,3$.
• Figure 3: Mass of the pions as a function of $\alpha$, for $c_2 > 0$ and $\beta = 0,\,1,\,2,\,3$.
• Figure 4: Global minimum $A_m$ and pion masses as a function of $\alpha$, for $c_2 < 0$ and $\beta = 0,\,1,\,2,\,3$. The dashed lines are for $\pi_{1,2}$ and the solid lines are for $\pi_3$.
• Figure 5: Mass of the pions as a function of $\beta$, for $c_2 < 0$ and $\alpha = 0,\,1,\,2,\,3$.