A PNJL Model for Adjoint Fermions with Periodic Boundary Conditions

TL;DR

This work extends Polyakov-Nambu-Jona Lasinio (PNJL) modeling to SU(N) gauge theories with adjoint fermions under periodic boundary conditions on $R^{3}\times S^{1}$, combining the perturbative one-loop potential for the Polyakov loop with NJL-type chiral dynamics. The resulting effective potential $V_{\text{eff}}(m,P)$ yields a rich phase structure, including confined, deconfined, skewed, and reconfined phases, and demonstrates that confinement at small $L$ can arise when the constituent mass is light. For SU(3) with two adjoint flavors, the authors map phase diagrams in terms of the four-fermion coupling $\kappa$ and bare mass $m_{0}$, showing that large-$L$ and small-$L$ confinement can be smoothly connected within a PNJL framework, and that lattice results by Cossu and D'Elia are broadly reproduced. The study highlights that extending the model with irrelevant four-fermion operators may be necessary to realize a single, connected confined phase across all $L$, with potential implications for volume independence in large-$N$ limits.

Abstract

Recent work on QCD-like theories has shown that the addition of adjoint fermions obeying periodic boundary conditions to gauge theories on $R^{3}\times S^{1}$ can lead to a restoration of center symmetry and confinement for sufficiently small circumference $L$ of $S^{1}$. At small $L$, perturbation theory may be used reliably to compute the effective potential for the Polyakov loop $P$ in the compact direction. Periodic adjoint fermions act in opposition to the gauge fields, which by themselves would lead to a deconfined phase at small $L$. In order for the fermionic effects to dominate gauge field effects in the effective potential, the fermion mass must be sufficiently small. This indicates that chiral symmetry breaking effects are potentially important. We develop a Polyakov-Nambu-Jona Lasinio (PNJL) model which combines the known perturbative behavior of adjoint QCD models at small $L$ with chiral symmetry breaking effects to produce an effective potential for the Polyakov loop $P$ and the chiral order parameter $\barψψ$. A rich phase structure emerges from the effective potential. Our results are consistent with the recent lattice simulations of Cossu and D'Elia, which found no evidence for a direct connection between the small-$L$ and large-$L$ confining regions. Nevertheless, the two confined regions are connected indirectly if an extended field theory model with an irrelevant four-fermion interaction is considered. Thus the small-$L$ and large-$L$ regions are part of a single confined phase.

Figures (7)

• Figure 1: The normalized order parameter $\langle Tr_F P \rangle/3$ for pure $SU(3)$ gauge theory as a function of temperature. The deconfinement transition is first-order, and occurs at $T_d = 270\,MeV$.
• Figure 2: The constituent mass $m$ and $\langle Tr_F P \rangle$ for two-flavor QCD with fundamental representation fermions with antiperiodic boundary conditions as a function of temperature. The order parameters are normalized by dividing by their values at $T=0$ and $T=\infty$, respectively.
• Figure 3: The constituent mass $m$ and $\langle Tr_F P \rangle$ for two-flavor QCD with adjoint representation fermions with antiperiodic boundary conditions as a function of temperature for one choice of $\kappa$ and $\Lambda$. The order parameters are normalized by dividing by their values at $T=0$ and $T=\infty$, respectively.
• Figure 4: The constituent mass $m$ and $\langle Tr_F P \rangle$ for two-flavor QCD with adjoint representation fermions with periodic boundary conditions as a function of $L^{-1}$ for one choice of $\kappa$ and $\Lambda$, with $m_0 = 0$. The order parameters are normalized by dividing by their values at $L=\infty$ and $L=0$, respectively. C, D, S and R refer to the confined phase, deconfined phase, skewed phase and reconfined phase respectively.
• Figure 5: The phase diagram for two-flavor QCD with adjoint representation fermions with periodic boundary conditions as a function of $L^{-1}$ and Lagrangian mass $m_0$ for one choice of $\kappa$ and $\Lambda$. C, D, S and R refer to the confined phase, deconfined phase, skewed phase and reconfined phase respectively. The Lagrangian mass is measured in MeV.