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

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 S1 ? 3 as a function of the length of the compact dimension, ?, and the fermion mass, m, for various N and Nf. We consider the effect of periodic boundary conditions [PBC(+)] on fermions as well as antiperiodic boundary conditions [ABC(-)]. With standard ABC(-) on fermions only the deconfined phase is found at one-loop for all representations considered. However, the use of PBC(+) produces a rich phase structure. These phases are distinguished by the eigenvalues of the Polyakov loop P. In the case of fundamental representation fermions [QCD(F,+)], a phase in which Re?Tr P is minimized (and negative) is favoured for all values of m?. For N odd charge conjugation () symmetry is spontaneously broken in this phase due to (1/N) effects. Minimization of the effective potential for QCD(AS/S,+) results in a phase where |Im?Tr P| is maximized, resulting in -breaking for all N and all values of m?, however, the partition function is the same up to (1/N) corrections as when ABC are applied. Therefore, regarding orientifold planar equivalence, we argue that in the one-loop approximation -breaking in QCD(AS/S,+) resulting from the application of PBC on fermions does not invalidate the large N equivalence with QCD(Adj,-). Similarly, with respect to orbifold planar equivalence, breaking of Z2 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 1$>Nf > 1 Majorana flavour confinement is obtained for sufficiently small m?, and deconfinement for sufficiently large m?. For N ? 3 these two phases are separated by one or more additional phases, some of which can be characterized as partially-confining phases.