Adjoint quarks and fermionic boundary conditions

We study quenched SU(2) lattice gauge theory with adjoint fermions in a wide range of temperatures. We focus on spectral quantities of the Dirac operator and use the temporal fermionic boundary conditions as a tool to probe the system. We determine the deconfinement temperature through the Polyakov loop, and the chiral symmetry restoration temperature for adjoint fermions through the gap in the Dirac spectrum. This chiral transition temperature is about four times larger than the deconfinement temperature. In between the two transitions we find that the system is characterized by a non-vanishing chiral condensate which differs for periodic and anti-periodic fermion boundary conditions. Only for the latter (physical) boundary conditions, the condensate vanishes at the chiral transition. The behavior between the two transitions suggests that deconfinement manifests itself as the onset of a dependence of spectral quantities of the Dirac operator on boundary conditions. This picture is supported further by our results for the dual chiral condensate. 1. Introductory remarks Confinement and chiral symmetry breaking are two outstanding properties of Quantum Chromodynamics (QCD), shaping all of nuclear physics. The emergence of both these phenomena depends on non-perturbative mechanisms, but it is an open question if and how the respective mechanisms are related, or what they are in detail. Each phenomenon is connected with a particular symmetry becoming broken or restored in certain limits of the theory. From the moment on, at which the existence of phase transitions in QCD was realized, the question whether deconfinement and chiral symmetry restoration are related to different transition temperatures, Tdec and Tch, was posed. Because of the dual role of quarks, with quarks being confined on one hand, and their role in the (chiral) hadron dynamics on the other hand, this question was asked in the first instance about fermions in the fundamental representation. For this case a consensus [1] based on lattice gauge theory calculations has formed that, at least as long as no finite baryonic chemical potential μ is involved, the temperature driven phase transition happens at roughly the same temperature, Tdec ≃ T (f) ch , where we use the superscript (f) to indicate the fundamental representation. 1 Fermions, and thus implicitly also chiral symmetry, play a role also outside QCD, e.g., for model building beyond the standard model [2]. In particular in many of those theories [3], like supersymmetry and technicolor, fermions in other representations appear, in particular adjoint ones. There is no a priori reason to expect the same transition temperature for such fermion representation. Such gauge theories with adjoint fermions have been investigated since the early days of lattice simulations [4, 5, 6, 7, 8, 9]. In this case, it is well established that, at least for the gauge groups investigated so far, the deconfinement temperature Tdec and the chiral restoration temperature T (a) ch of adjoint quarks do not coincide, the latter being generally significantly larger than the former. Aside from the practical considerations of theories beyond the standard model, this requires that any mechanism proposed as an explanation for the equality Tdec ≃ T (f) ch must at the same time provide an explanation for the inequality Tdec 6= T (a) ch . Such attempts have been made already. E. g., to capture the characteristic temperatures T (r) ch of chiral symmetry restoration for some representation r, a hypothetical Casimir scaling law has been proposed [10, 11], For most quark masses, including likely the physical ones, there is a crossover instead of a genuine phase transition [1] in full QCD. Thus, there is no qualitative distinction of the lowand high-temperature phase, despite their historic names. In quenched QCD (”gluodynamics”), however, a second or first order phase transition exists.