Phases of Gauge Theories

TL;DR

The work addresses how gauge theories exhibit diverse phase structures, including confinement, deconfinement, and chiral transitions, across finite-temperature and compactified geometries. It develops and employs a synthesis of semiclassical methods on $R^{3}\times S^{1}$, center-symmetry analysis, and large-$N$ techniques to map confinement mechanisms via monopoles, calorons, and Polyakov-loop dynamics, while connecting to conformal windows and dualities. Key contributions include demonstrating confinement at small $L$ through double-trace deformations or periodic adjoint fermions, elucidating spatial-string tensions, and articulating the role of topological objects in determining phase structure and mass gaps. The findings have broad significance for understanding nonperturbative gauge dynamics, guiding beyond-the-Standard-Model model-building (e.g., walking/CFT scenarios), and informing lattice approaches with volume-reduction strategies and semiclassical control.

Abstract

One of the most fundamental questions we can ask about a given gauge theory is its phase diagram. In the standard model, we observe three fundamentally different types of behavior: QCD is in a confined phase at zero temperature, while the electroweak sector of the standard model combines Coulomb and Higgs phases. Our current understanding of the phase structure of gauge theories owes much to the modern theory of phase transitions and critical phenomena, but has developed into a subject of extensive study. After reviewing some fundamental concepts of phase transitions and finite-temperature gauge theories, we discuss some recent work that broadly extends our knowledge of the mechanisms that determine the phase structure of gauge theories. A new class of models with a rich phase structure has been discovered, generalizing our understanding of the confinement-deconfinement transition in finite-temperature gauge theories. Models in this class have space-time topologies with one or more compact directions. On R^3 x S^1, the addition of double-trace deformations or periodic adjoint fermions to a gauge theory can yield a confined phase in the region where the S^1 circumference L is small, so that the coupling constant is small, and semiclassical methods are applicable. In this region, Euclidean monopole solutions, which are constituents of finite-temperature instantons, play a crucial role in the calculation of a non-perturbative string tension. We review the techniques use to analyze this new class of models and the results obtained so far, as well as their application to finite-temperature phase structure, conformal phases of gauge theories and the large-N limit.

Figures (14)

• Figure 1: The Landau-Ginsburg free energy density $f(\phi)$ for the case $r<0$ where the symmetry $\phi\rightarrow-\phi$ is spontaneously broken.
• Figure 2: The Polyakov loop is associated with the worldline of a heavy particle.
• Figure 3: The Polyakov loop two-point function determines the electric string tension.
• Figure 4: The Polyakov loop $Tr_{F}P$ as a function of $T$ in $SU(3)$ using a phenomenological model Meisinger:2001cqNishimura:2009me.
• Figure 5: Schematic version of the Columbia plot showing phase structure as a function of the $m_{ud}$ and $m_{s}$ quark masses