Adjoint QCD on $\mathbb{R}^3\times S^1$ with twisted fermionic boundary conditions

TL;DR

This work analyzes QCD with adjoint fermions on $\text{R}^3\times S^1$ under twisted boundary conditions, mapping how center and chiral symmetries realize themselves as the circle size $L$, twist angle $\phi$, and fermion mass $m$ vary. By combining perturbative one-loop potentials, semiclassical monopole/bion arguments, and a PNJL-type chiral model with a phenomenological gluonic potential, the authors uncover a rich phase diagram featuring confined, split, and deconfined phases, and reveal a strong φ-dependence of both center and chiral dynamics. They show that the Ünsal bion confinement mechanism is substantially weakened for $\phi\neq0$ due to a linear monopole interaction, leading to exponential suppression of mass gaps and string tensions; the Polyakov loop and chiral condensate become strongly intertwined in the twisted setting, and a deconfined window can appear between weak- and strong-coupling confining regimes depending on $m_{\rm dyn}$ relative to the YM scale. The PNJL analysis also indicates possible loss of adiabatic continuity between small- and large-$L$ confining phases for certain parameter choices, with implications for resurgence and IR renormalons, and motivates future lattice tests to map the φ–$L$–$m$ phase structure in QCD(adj).

Abstract

We investigate QCD with adjoint Dirac fermions on $\mathbb{R}^3\times S^1$ with generic boundary conditions for fermions along $S^1$. By means of perturbation theory, semiclassical methods and a chiral effective model, we elucidate a rich phase structure in the space spanned by the $S^1$ compactification scale $L$, twisted fermionic boundary condition $φ$ and the fermion mass $m$. We found various phases with or without chiral and center symmetry breaking, separated by first- and second-order phase transitions, which in specific limits ($φ=0$, $φ=π$, $L\to 0$ and $m\to \infty$) reproduce known results in the literature. In the center-symmetric phase at small $L$, we show that Unsal's bion-induced confinement mechanism is at work but is substantially weakened at $φ\ne 0$ by a linear potential between monopoles. Through an analytic and numerical study of the PNJL model, we show that the order parameters for center and chiral symmetries (i.e., Polyakov loop and chiral condensate) are strongly intertwined at $φ\ne 0$. Due to this correlation, a deconfined phase can intervene between a weak-coupling center-symmetric phase at small $L$ and a strong-coupling one at large $L$. Whether this happens or not depends on the ratio of the dynamical fermion mass to the energy scale of the Yang-Mills theory. Implication of this possibility for resurgence in gauge theories is briefly discussed. In an appendix, we study the index of the adjoint Dirac operator on $\mathbb{R}^3\times S^1$ with twisted boundary conditions, which is important for semiclassical analysis of monopoles.

Figures (13)

• Figure 1: ${\cal V}(N_{c}=2, N_f^D=1)$ in \ref{['VNc2Nf1']} at $m=0$ for three values of $\phi$. A first-order phase transition is seen to occur at $\phi\simeq 0.326\pi$.
• Figure 2: The $mL$--$\phi$ phase diagram (Left:$N_{c}=2$, $N_f^D=1$. Right:$N_{c}=3$, $N_f^D=1$). The symbols C, D and S refer to the confining phase, the deconfined phase and the split phase, respectively. In both figures, the transitions are first order.
• Figure 3: Contour plots of ${\cal V}(N_{c}=3, N_f^D=1)$ in \ref{['VNc3Nf1']} at $m=0$ and $\phi \in \{0,~0.248\pi,~0.280\pi,$$0.326\pi,~0.400\pi,~\pi\}$. Phase transitions occur at $\phi\simeq 0.248\pi$ and $\phi\simeq 0.326\pi$, from the confining to the split phase and then to the deconfined phase, respectively.
• Figure 4: The non-perturbative potential ${\cal V}_{\textrm{g}}^{\rm np}$ for $N_{c}=2$ as a function of $q_{1}$ for $1/LM = 0.35$, $0.39$, and $0.45$. A second-order phase transition occurs at $1/LM\simeq 0.390$.
• Figure 5: Phase diagrams for $N_c=2$ and $N_f^D=1$ with varying $m/M$. The blue line with triangles ($\triangle$) denotes a second-order phase transition, and the red line with asterisks ($\ast$) a first-order phase transition. Symbols C and D are defined as before. Spontaneous chiral symmetry breaking is not considered at this stage.