Joyce C. Myers

New phases of SU(3) and SU(4) at finite temperature

The addition of an adjoint Polyakov loop term to the action of a pure gauge theory at finite temperature leads to new phases of SU(N) gauge theories. For SU(3), a new phase is found which breaks Z(3) symmetry in a novel way; for SU(4), the new phase exhibits spontaneous symmetry breaking of Z(4) to Z(2), representing a partially-confined phase in which quarks are confined, but diquarks are not. The overall phase structure and thermodynamics is consistent with a theoretical model of the effective potential for the Polyakov loop based on perturbation theory.

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

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

Finite volume phases of large-Ngauge theories with massive adjoint fermions

>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.

QCD with chemical potential in a small hyperspherical box

The phase structure of QCD-like gauge theories with fermions in various representations is an interesting but generally analytically intractable problem. One way to ensure weak coupling is to define the theory in a small finite volume, in this case S3 × S1. Genuine phase transitions can then occur in the large N theory. Here, we use this technique to investigate SU(N) gauge theory with a number Nf of massive adjoint-valued Majorana fermions having non-thermal boundary conditions around S1. For Nf = 1 we find a line of transitions that separate the weak-coupling analogues of the confined and de-confined phases for which the density of eigenvalues of the Wilson line transform from the uniform distribution to a gapped distribution. However, the situation for Nf > 1 is much richer and a series of weak-coupling analogues of partially-confined phases appear which leave unbroken a p subgroup of the centre symmetry. In these p phases the eigenvalue density has p gaps and they are separated from the confining phase and from one-another by first order phase transitions. We show that for small enough mR (the mass of the fermions times the radius of the S3) only the confined phase exists. The large N phase diagram is consistent with the finite N result and with other approaches based on 3 × S1 calculations and lattice simulations.

New phases ofSU(3)andSU(4)at finite temperature

To leading order in perturbation theory, we solve QCD, defined on a small three sphere in the large N and Nf limit, at finite chemical potential and map out the phase diagram in the (μ, T) plane. The action of QCD is complex in the presence of a non-zero quark chemical potential which results in the sign problem for lattice simulations. In the large N theory, which at low temperatures becomes a conventional unitary matrix model with a complex action, we find that the dominant contribution to the functional integral comes from complexified gauge field configurations. For this reason the eigenvalues of the Polyakov line lie off the unit circle on a contour in the complex plane. We find at low temperatures that as μ passes one of the quark energy levels there is a third-order Gross-Witten transition from a confined to a deconfined phase and back again giving rise to a rich phase structure. We compare a range of physical observables in the large N theory to those calculated numerically in the theory with N = 3. In the latter case there are no genuine phase transitions in a finite volume but nevertheless the observables are remarkably similar to the large N theory.

Numerical study of the two color attoworld

The addition of an adjoint Polyakov loop term to the action of a pure gauge theory at finite temperature leads to new phases of SU(N) gauge theories. For SU(3), a new phase is found which breaks Z(3) symmetry in a novel way; for SU(4), the new phase exhibits spontaneous symmetry breaking of Z(4) to Z(2), representing a partially-confined phase in which quarks are confined, but diquarks are not. The overall phase structure and thermodynamics is consistent with a theoretical model of the effective potential for the Polyakov loop based on perturbation theory.

Exotic phases of finite temperature SU(N) gauge theories

We consider QCD at very low temperatures and non-zero quark chemical potential from lattice Monte Carlo simulations of the two-color theory in a very small spatial volume (the attoscale). In this regime the quark number rises in discrete levels in qualitative agreement with what is found analytically at one loop on S3 × S1 with

Weak coupling large-N transitions at finite baryon density

{R_{{S^3}}} \ll \Lambda_{QCD}^{ - 1}

Deconfinement transitions of large N QCD with chemical potential at weak and strong coupling

[1]. The detailed level degeneracy, however, cannot be accounted for using weak coupling arguments. At each rise in the quark number there is a corresponding spike in the Polyakov line, also in agreement with the perturbative results. In addition the quark number susceptibility shows a similar behaviour to the Polyakov line and appears to be a good indicator of a confinement-deconfinement type of transition.

QCD sign problem as a total derivative

We calculate the phase diagrams at high temperature of SU(N) gauge theories with massive fermions by minimizing the one-loop effective potential. Considering fermions in the adjoint (Adj) representation at various N we observe a variety of phases when Nf⩾2 Majorana flavours and periodic boundary conditions are applied to fermions. Also the confined phase is perturbatively accessible. For N=3, we add Fundamental (F) representation fermions with antiperiodic boundary conditions to adjoint QCD to show how the Z(3)-symmetry breaks in the confined phase.