Vacuum structure of Yang-Mills theory as a function of $θ$

TL;DR

The paper addresses how SU(N) Yang–Mills theory organizes its vacuum structure as a function of the topological angle θ. By employing center-stabilized YM on $\mathbb{R}^3\times S^1$, it achieves a controllable semiclassical framework where the vacuum landscape comprises N branches, yet only ≈N/2 are locally stable for any given θ. The authors identify θ-vacua via magnetic line operators carrying GNO charge, reveal spinodal points where metastable vacua lose stability as θ varies, and show that these spinodal features are robust in the calculable regime with hints of persistence in the full $\mathbb{R}^4$ theory. The results connect the multi-branched θ-dependence to magnetic holonomies and nonperturbative dynamics dominated by monopole-instantons and magnetic bions, and they illuminate how condensates reflect the underlying vacuum structure. Overall, the work provides a concrete semiclassical account of the θ-dependence of YM, clarifying the stability and interpretation of θ-vacua and their physical consequences.

Abstract

It is believed that in $SU(N)$ Yang-Mills theory observables are $N$-branched functions of the topological $θ$ angle. This is supposed to be due to the existence of a set of locally-stable candidate vacua, which compete for global stability as a function of $θ$. We study the number of $θ$ vacua, their interpretation, and their stability properties using systematic semiclassical analysis in the context of adiabatic circle compactification on $\mathbb{R}^3 \times S^1$. We find that while observables are indeed N-branched functions of $θ$, there are only $\approx N/2$ locally-stable candidate vacua for any given $θ$. We point out that the different $θ$ vacua are distinguished by the expectation values of certain magnetic line operators that carry non-zero GNO charge but zero 't Hooft charge. Finally, we show that in the regime of validity of our analysis YM theory has spinodal points as a function of $θ$, and gather evidence for the conjecture that these spinodal points are present even in the $\mathbb{R}^4$ limit.

Figures (6)

• Figure 1: [Color Online.] An sketch of the phase structure of $SU(N)$ YM theory for $N=4$ as a function of $\theta$. The vertical axis depicts the vacuum energy density. The gray dashed curves indicate the energy density associated with the four distinct $\theta$-extrema. The colored bold curves indicate the vacuum energy density associated with the thermodynamically stable vacuum branch for any given $\theta$. The black dots at the cusps label the locations of quantum phase transitions. These phase transitions are associated with changes in the expectation values of GNO 't Hooft magnetic holonomies, as discussed in the main text.
• Figure 2: [Color Online.] A sketch of the $\theta$ angle dependence of the lowest-dimension non-trivial parity even and odd condensate $\langle \frac{1}{N}\,{\rm tr}\, F_{\mu \nu} F_{\mu \nu} \rangle(\theta)$ and $\langle \frac{1}{N}\,{\rm tr}\, F_{\mu \nu} \widetilde{F}_{\mu \nu} \rangle(\theta)$ for $N=4$ in $SU(N)$ YM theory in the semiclassical confining domain. The continuous bold red curve shows the values of the parity-even condensate, while the saw-tooth bold blue curves denote the values of the parity-odd condensate. The grey dashed curves mark the values of the condensates on the $N$ different $\theta$ extrema.
• Figure 3: [Color Online.] Plot of the mass of the lightest excitation of YM theory with $N=4$ at small $NL\Lambda$ as a function of $\theta$, normalized to its value at $\theta = 0$, with magnetic bion effects taken into account. The thick black curve at the top marks the behavior of the gap along the thermodynamically stable branch. Each of the four colored curves show the dependence of the mass on $\theta$ in a given $k$-extremum, with blue, red, green, orange corresponding to $k=0,3,2,1$ respectively. The black dots mark spinodal points, and the fact that there are two distinct spinodal points near $\theta = 2\pi k, k \in \mathbb{Z}$ is due to magnetic bions.
• Figure 4: [Color Online.] Spinodal curves (blue) and phase transition lines (vertical red line) for center-stabilized $SU(2)$ YM theory in the $L\Lambda$-$\theta$ plane. The vertical red line at $\theta = \pi$ indicates a first-order phase transition between two $\theta$-vacua. The two blue curves mark the regions of spinodal instability of the $\theta$-vacua, where a metastable branch reaches the limit of local stability. The behavior of the spinodal curves for $L\Lambda \lesssim 1$ follows from \ref{['eq:spinodal_theta_0']} and \ref{['eq:spinodal_theta_2pi']}, and hence is indicated by a solid curve, while the behavior for $L\Lambda \gtrsim 1$ is a conjecture and is indicated by the dashed portion of the dashed curve.
• Figure 5: [Color Online.] A sketch of the $\theta$ angle dependence of the dual photon mass $m_{\sigma}$ in $SU(2)$ adjoint QCD with $n_{\rm adj}=2$ flavors of adjoint Majorana fermions in the $k=0$$\theta$-vacuum, as a function of the fermion mass $m_{\rm adj}$ in units of the mass gap at vanishing adjoint quark mass, $m_0 \sim m_W e^{-S_0}$. For small enough fermion mass, the $\theta$-dependence is very mild, in the sense that the $k=0$$\theta$-vacuum is stable for all $\theta$. The dual photon mass is normalized to its value in the chiral limit. But once $m_{\rm adj}$ exceeds $m_0$, the $\theta$-dependence becomes more dramatic: there is a region where the $k=0$ vacuum becomes locally unstable.