QCD at θ\sim π

TL;DR

This work analyzes QCD at $\theta \sim \pi$ for $N_f=2$ and $N_f=3$ by incorporating $O(m^2)$ corrections into the chiral Lagrangian. It demonstrates that for $N_f=3$ there are two degenerate vacua at $\theta=\pi$ separated by a domain wall with a calculable surface tension $\sigma$, establishing a first-order CP-violating transition, and it estimates the metastable-vacuum decay rate via $\Gamma \propto \exp\left\{ - \frac{27}{2}\pi^2 \frac{\sigma^4}{(\Delta\mathcal{E})^3} \right\}$ with $\Delta\mathcal{E}\sim m\Sigma\sqrt{3}\,|\phi|$. In the two-flavor case, a positive low-energy constant $l_7$ lifts degeneracy, yielding a wall with tension $\sigma = \frac{m\Sigma}{F_\pi}\sqrt{32\,l_7}$ and a domain-wall profile $\alpha(x)$, along with a metastable-decay rate $\Gamma \propto \exp\left\{ -12^3 \pi^2 l_7^2 \frac{m\Sigma}{F_\pi^4 |\phi|^3} \right\}$; the coexistence region shrinks as $m \to 0$. The Schwinger model (2D QED) with two light fermions provides a solvable analogue, where the light sector reduces to a sine-Gordon model and a domain-wall (soliton) structure at $\theta=\pi$ emerges, illustrating the same qualitative domain-wall physics and supporting positivity arguments for the low-energy constants.

Abstract

Taking into account the quadratic in mass terms in the effective chiral lagrangian, we show that, at θ\sim π, the theory with 2 light quarks of equal mass involves two degenerate vacuum states separated by a barrier. For three flavors, the energy barrier between two vacua appears already in the leading order in mass. This corresponds to the first order phase transition at θ= π. The surface energy density of the domain wall separating two different vacua is calculated. In the immediate vicinity of the phase transition point, two minima of the potential still exist, but one of them becomes metastable. The probability of the false vacuum decay is estimated.

Figures (4)

• Figure 1: Stationary point of $E(\alpha, \beta)$ for different $\theta$. Solid lines are the minima, dotted lines are the maxima, and dashed line are the saddle points
• Figure 2: Connected and disconnected contributions to the quark current correlators.
• Figure 3: Stationary point of $E(\alpha)$ for $N_f = 2$ in the region of small $|\phi| = |\theta - \pi|$. Solid lines are minima and dotted lines are maxima.
• Figure 4: Some graphs contributing in the effective lagrangian in the second order. Crosses stand for the insertion of light vertex $\cos (\sqrt{2\pi} \phi_-)$ and solid lines describe heavy Schwinger bosons. The graphs with even number of the lines contribute in $\kappa_0$ and the graphs with odd number in $\kappa_1$.