International Journal of Modern Physics A

We review some recent results related to the notion of volume independence in SU(N) Yang–Mills theories. The topic is discussed in the context of gauge theories living on a d-dimensional torus with twisted boundary conditions. After a brief introduction reviewing the formalism for introducing gauge fields on a torus, we discuss how volume independence arises in perturbation theory. We show how, for appropriately chosen twist tensors, perturbative results to all orders in the 't Hooft coupling depend on a specific combination of the rank of the gauge group (N) and the periods of the torus (l), given by lN^2/d, for d even. We discuss the well-known relation to noncommutative field theories and address certain threats to volume independence associated to the occurrence of tachyonic instabilities at one-loop order. We end by presenting some numerical results in 2+1 dimensions that extend these ideas to the nonperturbative domain.

We construct a lattice gauge theory using reduced staggered fermions and gauge fields which provides a nonperturbative realization of a complete technicolor model; one which treats both strong and weakly coupled gauge sectors on an equal footing. We show that the model is capable of developing a Higgs phase at nonzero lattice spacing via the formation of fermion condensates. We show further that while the broken symmetry associated with this phase has a vector character in the lattice theory it is realized as an axial symmetry in the continuum limit in agreement with the Vafa–Witten theorem. We discuss our result in the context of universality.

There has been substantial progress in understanding confinement in a class of four-dimensional SU(N) gauge theories using semiclassical methods. These models have one or more compact directions, and much of the analysis is based on the physics of finite temperature gauge theories. The topology R³ × S¹ has been most often studied using a small compactification circumference L such that the running coupling g²(L) is small. The gauge action is modified by a double-trace Polyakov loop deformation term, or by the addition of periodic adjoint fermions. The additional terms act to preserve Z(N) symmetry and thus confinement. An area law for Wilson loops is induced by a monopole condensate. In the continuum, the string tension can be computed analytically from topological effects. Lattice models display similar behavior, but the theoretical analysis of topological effects is based on Abelian lattice duality rather than on semiclassical arguments. In both cases, the key step is reducing the low-energy symmetry group from SU(N) to the maximal Abelian subgroup U(1)^N-1 while maintaining Z(N) symmetry.

SU(N_c) lattice gauge theories with N_f flavors of massless staggered fermions are studied in the presence of quark chemical potential μ. A recent exact result that in the strong coupling limit (vanishing inverse gauge coupling β) and for sufficiently large μ the theory is in a chiral symmetric phase is here extended into the finite gauge coupling region. These and similar results are obtained by means of a cluster expansion which combines a fermion spacelike hopping expansion and a strong coupling plaquette expansion. The cluster expansion is shown to converge for sufficiently large μ and small β at any temperature T. All expectations of chirally noninvariant local fermion operators vanish identically, or, equivalently, their correlations cluster exponentially within the expansion. This implies the absence of spontaneous breaking of (any part of) the global chiral symmetries in the theory. The resulting phase at low T may be described as a "quarkyonic" matter phase. There is thus absence of any superfluidity or color superconductivity in this strong coupling regime. Implications for the phase diagram of SU(N_c) theories in the region of large chemical potential are discussed.

We review the present status of the Anderson transition in the spectrum of the Dirac operator of QCD-like theories on the lattice. Localized modes at the low end of the spectrum have been found in SU(2) Yang–Mills theory with overlap and staggered valence fermions as well as in N_f = 2+1 QCD with staggered quarks. We draw an analogy between the transition from localized to delocalized modes in the Dirac spectrum and the Anderson transition in electronic systems. The QCD transition turns out to be in the same universality class as the transition in the corresponding Anderson model. We also speculate on the possible physical relevance of this transition to QCD at high temperature and the possible finite temperature phase transition in QCD-like models with different fermion contents.

The existence of strongly-interacting infrared (IR) fixed points in gauge theories can be studied using field theories defined on a lattice. Besides their academic interest, it has been argued that these theories could be used as a starting point to construct viable models of electroweak symmetry breaking (EWSB). A quantitative description of the large-distance dynamics is mandatory in order to develop predictive models for phenomenology. In these notes, we discuss the theoretical foundations of IR conformality, and present possible ways to study the problem using the lattice formulation of gauge theories. An overview of current results and methods is discussed.

We discuss our results on QCD with a number of fundamental fermions ranging from zero to sixteen. These theories exhibit a wide array of fascinating phenomena which have been under close scrutiny, especially in recent years, first and foremost is the approach to conformality. To keep this review focused we have chosen scale generation, or lack thereof, as a guiding theme, however the discussion will be set in the general framework of the analysis of the phases and phase transitions of strong interactions at zero and nonzero temperature.