Time-reversal breaking in QCD4, walls, and dualities in 2 + 1 dimensions

Abstract

We study SU(N ) Quantum Chromodynamics (QCD) in 3+1 dimensions with N f degenerate fundamental quarks with mass m and a θ-parameter. For generic m and θ the theory has a single gapped vacuum. However, as θ is varied through θ = π for large m there is a first order transition. For N f = 1 the first order transition line ends at a point with a massless η particle (for all N ) and for N f > 1 the first order transition ends at m = 0, where, depending on the value of N f , the IR theory has free NambuGoldstone bosons, an interacting conformal field theory, or a free gauge theory. Even when the 4d bulk is smooth, domain walls and interfaces can have interesting phase transitions separating different 3d phases. These turn out to be the phases of the recently studied 3d Chern-Simons matter theories, thus relating the dynamics of QCD 4 and QCD 3 , and, in particular, making contact with the recently discussed dualities in 2+1 dimensions. For example, when the massless 4d theory has an SU(N f ) sigma model, the domain wall theory at low (nonzero) mass supports a 3d massless CP N f −1 nonlinear σ-model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.

A symmetry breaking scenario for QCD3

“…It is important to mention that we need to add to the sigma model Lagrangian a certain Wess-Zumino term, which we will discuss in detail soon. In the special case k = N f /2 − 1 this result is consistent with the study of domain walls in four dimensions [53]. This proposal for N * (N, k) > N f > 2k implies the dualities (1.6), (1.7) for lower values of N f and shifted values of k, as we explain in the bulk of the paper.…”

“…As we lower the quarks masses, the theory on the domain wall undergos a phase transition. Our scenarios below about the dynamics of the 2+1-dimensional theory are consistent by the transitions on these domain walls [53].…”

“…This will be developed in great detail in [53]. Briefly, pure Yang-Mills theory in 3+1 dimensions at θ = π has a first order transition associated with the spontaneous breaking of time-reversal.…”

Spontaneous CP breaking in QCD and the axion potential: an effective Lagrangian approach

“…These results can be seen as a rather straightforward generalization of those of [28][29][30] to the case of a generic value of χ YM and of [32,33] to the case of a generic quark mass matrix (the equal mass case is indeed quite special since it is always in the CP broken phase except in the case of a single light flavor). In [33] the issue of CP breaking in QCD was also addressed for finite N , and the theories residing on the resulting domain walls were studied.…”

“…In [33] the issue of CP breaking in QCD was also addressed for finite N , and the theories residing on the resulting domain walls were studied.…”