Phase structure and Hosotani mechanism in gauge theories with compact dimensions revisited

TL;DR

This work investigates the phase structure of $SU(3)$ gauge theory on spaces with one compact dimension, using perturbative one-loop potentials and PNJL-based effective potentials to probe spontaneous gauge-symmetry breaking via the Hosotani mechanism. Adjoint matter with periodic boundary conditions yields exotic phases, notably $SU(3)\to SU(2)\times U(1)$ (split) and $SU(3)\to U(1)\times U(1)$ (reconfined), with the split phase expanding when fundamental quarks are added; a pseudo-reconfined phase with negative Polyakov-loop expectation value can appear. Chiral properties, studied within the PNJL framework, show the chiral condensate gradually decreasing as the compact dimension shrinks, with no sharp chiral transition in the weak-coupling regime. The results, aligned qualitatively with lattice studies, provide a phase-structure guide for future simulations in both 4D and 5D compactified gauge theories and highlight observable signatures in Polyakov loops and KK spectra that can test Hosotani-type dynamics. Overall, the paper demonstrates how effective-field-theory tools can map rich gauge-breaking phases in compactified settings and connect them to lattice and phenomenological contexts.

Abstract

We investigate the phase structure of SU(3) gauge theory in four and five dimensions with one compact dimension by using perturbative one-loop and PNJL-model-based effective potentials, with emphasis on spontaneous gauge symmetry breaking. When adjoint matter with the periodic boundary condition is introduced, we have rich phase structure in the quark-mass and compact-size space with gauge-symmetry-broken phases, called the $SU(2)\times U(1)$ split and the $U(1)\times U(1)$ re-confined phases. Our result is qualitatively consistent with the recent lattice calculations. When fundamental quarks are introduced in addition to adjoint quarks, the split phase becomes more dominant and larger as a result of explicit center symmetry breaking. We also show that another $U(1)\times U(1)$ phase (pseudo-reconfined phase) with negative vacuum expectation value of Polyakov loop exists in this case. We study chiral properties in these theories and show that chiral condensate gradually decreases and chiral symmetry is slowly restored as the size of the compact dimension is decreased.

Figures (26)

• Figure 1: The one-loop effective potential of $SU(3)$ gauge theory on $R^{3}\times S^1$ with one adjoint fermion with PBC $[ {\cal V}_g + {\cal V}_a^{0}(N_{a}=1, m_a =0)] L^4$. (Right) The contour plot as a function of $q_1$ and $q_2$. Thicker region stands for deeper region of the potential. (Left) The effective potential as a function of $q_1$ with $q_2=0$. The global minima are located at $(q_{1},q_{2})=(\pm1/3,0)$.
• Figure 2: The one-loop effective potential of $SU(3)$ gauge theory on $R^{3}\times S^1$ with one PBC adjoint quark as a function of $q_1$ with $q_2=0$$[ {\cal V}_g + {\cal V}_a^{0}] L^4$, for $m L=1.2$ (reconfined), $1.6$ (reconfined$\leftrightarrow$split), $2.0$ (split$\leftrightarrow$deconfined) and $3.0$ (deconfined).
• Figure 3: Contour plot of the one-loop effective potential of $SU(3)$ gauge theory on $R^{3}\times S^1$ with one PBC adjoint quark $[ ( {\cal V}_g )_{pert} + {\cal V}_a^{0} ] L^4$, for $m L=1.6$ and $1.8$ ($SU(2)\times U(1)$ split phase) as a function of $q_1$ and $q_2$. Thicker region indicates deeper region of the potential.
• Figure 4: $L^{-1}$-$m$ phase diagram for $SU(3)$ gauge theory on $R^{3}\times S^1$ with one PBC adjoint quark based on one-loop effective potential. D stands for "deconfined ($SU(3)$)", S for "split ($SU(2)\times U(1)$)" and R for "re-confined ($U(1)\times U(1)$)" phases. Phase transitions are first-order.
• Figure 5: Schematic distribution plot of Polyakov loop $\Phi$ as a function of $\mathrm{Re}~\Phi$ and $\mathrm{Im}~\Phi$ for $SU(3)$ gauge theory on $R^{3}\times S^1$ with one PBC adjoint quark.