Spontaneous Flavor and Parity Breaking with Wilson Fermions

TL;DR

The paper tackles whether Wilson fermions at nonzero lattice spacing exhibit spontaneous flavor and parity breaking (the Aoki phase) and how this connects to continuum chiral symmetry breaking. It formalizes this by augmenting the continuum chiral Lagrangian with lattice-spacing terms and deriving a two-parameter effective potential whose sign determines the phase: either an Aoki phase with two massless pions and a width $\Delta m_0 \sim a^3$, or a non-breaking phase with all pions heavy at $\mathcal{O}(a)$. The authors show that, for unimproved Wilson fermions, numerical evidence supports the Aoki phase, and they provide testable predictions for pion masses and the flavor-parity condensate within this phase; the framework also applies to non-perturbatively improved Wilson fermions. The work clarifies how lattice flavor-parity breaking maps onto continuum chiral dynamics and informs lattice QCD simulations, including domain-wall fermion considerations, about the interplay between discretization effects and chiral symmetry restoration.

Abstract

We discuss the phase diagram of Wilson fermions in the $m_0$--$g^2$ plane for two-flavor QCD. We argue that, as originally suggested by Aoki, there is a phase in which flavor and parity are spontaneously broken. Recent numerical results on the spectrum of the overlap Hamiltonian have been interpreted as evidence against Aoki's conjecture. We show that they are in fact consistent with the presence of a flavor-parity broken ``Aoki phase''. We also show how, as the continuum limit is approached, one can study the lattice theory using the continuum chiral Lagrangian supplemented by additional terms proportional to powers of the lattice spacing. We find that there are two possible phase structures at non-zero lattice spacing: (1) there is an Aoki phase of width $Δm_0 \sim a^3$ with two massless Goldstone pions; (2) there is no symmetry breaking, and all three pions have an equal non-vanishing mass of order $a$. Present numerical evidence suggests that the former option is realized for Wilson fermions. Our analysis then predicts the form of the pion masses and the flavor-parity breaking condensate within the Aoki phase. Our analysis also applies for non-perturbatively improved Wilson fermions.

Figures (5)

• Figure 1: The phase diagram proposed by Aoki: $g$ is the gauge coupling and $m_0$ the dimensionless bare quark-mass. The continuum-like phases are labeled A, and the flavor and parity broken phase B. The phase diagram is symmetric under $m_0\leftrightarrow -(m_0+8)$. The continuum limit of particular interest is that at $m_0=0$, $g=0$.
• Figure 2: The potential ${\cal V}_\chi$ vs. $A$ for $c_2>0$ and $|A_m|>1$. The vacuum is at $A_0=1$.
• Figure 3: The potential ${\cal V}_\chi$ vs. $A$ for $c_2>0$ and $|A_m|<1$. The vacuum is at $A_0=A_m$.
• Figure 4: Pion masses as a function of $\epsilon$ for $c_2>0$. The curves are labeled by the flavor of the corresponding pion.
• Figure 5: Pion masses as a function of $\epsilon$ for $c_2<0.$