Abstract
We investigate QCD with adjoint Dirac fermions on ℝ3 × 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 → 0 and m → ∞) reproduce known results in the literature. In the center- symmetric phase at small L, we show that Ünsal’s bion-induced confinement mechanism is at work but is substantially weakened at ϕ = 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 = 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 ℝ3 × S 1 with twisted boundary conditions, which is important for semiclassical analysis of monopoles.
Highlights
We investigate QCD with adjoint Dirac fermions on R3 × S1 with generic boundary conditions for fermions along S1
As for the center symmetry breaking at φ = 0, at present we do not know which phase diagram of figure is the right one in QCD(adj), so we have indicated our ignorance by the annotation “gap open?” in figure
QCD(adj) on R3×S1 with the thermal boundary condition has been simulated on the lattice to test relationship between chiral symmetry breaking and confinement
Summary
We discuss center and gauge symmetry breaking at small S1 using perturbation theory and semiclassical methods. For QCD(adj) with NfW ≥ 2 Weyl fermions of one chirality, the gauge invariance of the partition function requires that φ be an integer multiple of π/(N NfW ). This will happen automatically if we combine all the Weyl fermions into NfD(= NfW /2) Dirac fermions, and impose a twisted boundary conditions (1.1) on the latter For this reason we will only consider Dirac fermions in the rest of this paper.. Because the determination of the conformal window itself is not the purpose of this work, in sections 2.2 and 3 we will use effective models for specific values of N and NfD where chiral symmetry breaking and confinement are assumed to occur on R4.
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