Phase diagrams of SU(N) gauge theories with fermions in various representations

TL;DR

This work performs a one-loop perturbative analysis of SU($N$) gauge theories with fermions in $F$, $Adj$, $S$, and $AS$ representations on $S^1\times\mathbb{R}^3$, mapping the phase diagram as a function of the compactification scale $\beta$ and fermion mass $m$ under periodic and antiperiodic boundary conditions. The authors derive the one-loop effective potential $V_{1\text{-loop}}$ in terms of Polyakov-loop traces ${\rm Tr}_R(P^n)$ and show how boundary conditions drive rich phase structures, including ${\cal C}$-breaking in AS/S with periodic bosons and various confinement regimes in adjoint theories, especially at large $N$. They further connect these perturbative results to orientifold and orbifold planar equivalences, showing that certain symmetry breakings cancel in the partition function and thus preserve equivalences in the one-loop limit. The findings illuminate how representation and boundary conditions control deconfinement, confinement, and chiral dynamics, with implications for large-$N$ gauge theory dualities and the analytic control of confinement mechanisms. Overall, the paper provides a detailed perturbative phase catalog for QCD-like theories and clarifies the conditions under which planar equivalences hold at small compactification scales.

Abstract

We minimize the one-loop effective potential for SU(N) gauge theories including fermions with finite mass in the fundamental (F), adjoint (Adj), symmetric (S), and antisymmetric (AS) representations. We calculate the phase diagram on S^1 x R^3 as a function of the length of the compact dimension, beta, and the fermion mass, m. We consider the effect of periodic boundary conditions [PBC(+)] on fermions as well as antiperiodic boundary conditions [ABC(-)]. The use of PBC(+) produces a rich phase structure. These phases are distinguished by the eigenvalues of the Polyakov loop P. Minimization of the effective potential for QCD(AS/S,+) results in a phase where | Im Tr P | is maximized, resulting in charge conjugation (C) symmetry breaking for all N and all values of (m beta), however, the partition function is the same up to O(1/N) corrections as when ABC are applied. Therefore, regarding orientifold planar equivalence, we argue that in the one-loop approximation C-breaking in QCD(AS/S,+) resulting from the application of PBC to fermions does not invalidate the large N equivalence with QCD(Adj,-). Similarly, with respect to orbifold planar equivalence, breaking of Z(2) interchange symmetry resulting from application of PBC to bifundamental (BF) representation fermions does not invalidate equivalence with QCD(Adj,-) in the one-loop perturbative limit because the partition functions of QCD(BF,-) and QCD(BF,+) are the same. Of particular interest as well is the case of adjoint fermions where for Nf > 1 Majorana flavour confinement is obtained for sufficiently small (m beta), and deconfinement for sufficiently large (m beta). For N >= 3 these two phases are separated by one or more additional phases, some of which can be characterized as partially-confining phases.

Figures (26)

• Figure 1: QCD: (Left) $V_{FUND(-)}$ for $N_f = 1$; (Right) $\langle {{\bar{\psi}} \psi} \rangle_{FUND(-)}$.
• Figure 2: $V_{FUND(+)}$ for $N_f = 1$. The dots correspond to minimization of the effective potential of QCD(F,+) with respect to the Polyakov loop angles $v_i$. The curves correspond to evaluation of the effective potential for $v_i = \pi \pm \frac{\pi}{N} [N,2]$, indicating that the anti-deconfined phase of Table \ref{['tabF']} is favoured for all $m \beta$. The curves corresponding to other phases in Table \ref{['tabF']} are not included for clarity, and because they would correspond to larger values of the effective potential.
• Figure 3: $N_f = 2$: (Left) $V_{AS(+)}$; (Right) $V_{SYMM(+)}$. The dots correspond to minimization of $V_{eff}$ in eq. ( \ref{['veffmu0']}) with respect to the $v_i$. The curves result from evaluation of eq. ( \ref{['veffmu0']}) for $v_i = \pm [N,2] \left( \frac{\pi}{2} \pm \frac{\pi}{2 N} \right) \pm \frac{1}{2} [N,4] [(N+1),2] \left( \frac{\pi}{2} \pm \frac{\pi}{N} \right) \pm \frac{1}{2} [(N+2),4][(N+1),2] \left( \frac{\pi}{2} \right)$, as in Table \ref{['tabAS']}. This indicates that in QCD(AS/S,+) for any $N$ the ${\cal C}$-breaking phase is favoured for all $m \beta$, except for $N = 2$ QCD(S), which is equivalent to QCD(Adj) [see also Figure \ref{['pt_adj_nc2_nf2']}].
• Figure 4: (Left) $V_{ADJ(+)}$ with $N_f = 1$ Majorana flavour. The dots correspond to minimization of $V_{eff}$ in eq. ( \ref{['veffmu0']}) with respect to the $v_i$. The curves correspond to the result of evaluating eq. ( \ref{['veffmu0']}) for $v_i = 0 \, \forall i$, the values corresponding to the deconfined phase. (Right) $\langle {\lambda \lambda} \rangle_{ADJ(+)}$ for $N_f = 1$ and various values of $N$.
• Figure 5: $N = 2$: $V_{ADJ(+)}$ for $N_f = 2$ Majorana or $V_{SYMM(+)}$ for $N_f = 1$ Dirac flavour