Deconfinement and Chiral Symmetry Restoration

We illustrate why color deconfines when chiral symmetry is restored in gauge theories with quarks in the fundamental representation, and while these transitions do not need to coincide when quarks are in the adjoint representation, entanglement between them is still present. Introduction. One of the long-standing puzzles in theoretical physics is the relation between confinement and chiral symmetry breaking. In pure Yang-Mills theory the ZN center of the gauge SU(N) group is a global symmetry. The Polyakov loop is a gauge invariant operator charged under ZN , whose expectation value vanishes at low temperatures, and is nonzero above a critical temperature Tχ, when the center symmetry is spontaneously broken. This feature, together with its relation to the infinitely heavy quark free energy makes the Polyakov loop suitable to be the order parameter for the deconfinement transition in pure gauge theory. We denote with χ the polyakov loop field which is usually denoted with l. When quarks with finite masses are added to the theory in the fundamental representation of the gauge group the ZN symmetry is not exact. QCD with massless quarks exhibits chiral symmetry. The order parameter, the chiral condensate, is zero above Tσ, where chiral symmetry is restored. Here σ is the interpolating field associated to the scalar component of the q̄q operator. For any finite quark mass chiral symmetry is explicitly broken. When quarks are in the adjoint representation the center group symmetry is intact. For realistic quark masses there are no exact symmetries, but one can still follow the behavior of the condensates. Analysis done on the lattice showed that with quarks in the fundamental representation deconfinement (a rise in the Polyakov loop), happens at the temperature where chiral symmetry is restored (chiral condensate decreases) Tχ = Tσ [1]. Lattice also revealed that when quarks are in the adjoint representation deconfinement and the chiral symmetry restoration do not happen at the same temperature, Tσ ≃ 8Tχ [2]. Despite the attempts to explain these behaviors [3], ∗ Speaker at the conference. Deconfinement and Chiral Symmetry Restoration 2 the underlying reasons are still unknown. Lattice simulations for two color QCD at nonzero baryon chemical potential observe deconfinement for 2 color QCD and 8 continuum flavors. Here also, the Polyakov loop rises when the chiral condensate vanishes, at the same value of the chemical potential [4]. Our goal is to provide a simple unified way to describe all of these features. We study the two color theory with Nf flavors in the chiral limit, since with only minor modifications of the effective Lagrangian we can discuss the theory with quarks in the fundamental and adjoint representation at nonzero temperature or quark chemical potential. The results presented here are based on our recent work [5] concerning the transfer of critical properties from true order parameters to non-critical fields, approach presented in [6, 7], envisioned first in [8]. For a complete review see [9]. The transfer of information is possible due to the presence of a trilinear interaction between the light order parameter and the heavy non-order parameter field, singlet under the symmetries of the order parameter field. Due to this interaction, the expectation value of the order parameter field in the symmetry broken phase induces a variation in the expectation value for the singlet field, and spatial correlators for the non-critical fields are infrared dominated. Fundamental Representation. In two color QCD with two massless quark flavors in the fundamental representation the global symmetry group is SU(2Nf ) which breaks to Sp(2Nf) . The chiral degrees of freedom are 2N 2 f −Nf − 1 Goldstone fields π , and a scalar field σ , which is the order parameter. For Nf = 2 the potential is [10]: Vch[σ, π ] = m 2 Tr [