Vector condensation in QCD

The properties of the QCD vacuum at quark chemical potentials on the order of the pion mass and at zero temperature are by now quite well understood. The insight into this non-perturbative regime of QCD is obtained using chiral perturbation theory at nonzero quark chemical potentials [1–8]. The results apply not only to ordinary QCD but also to two color QCD and to QCD with quarks in the adjoint representation of the color group. Some of these realizations of QCD does not have a sign problem at finite chemical potential and hence can be studied by standard lattice methods as well. This has lead to a fruitful interplay between lattice calculations and chiral perturbation theory (see [6] for an extensive list of references). The central observation of all these studies is the condensation of specific goldstone channels. The critical chemical potentials at which the condensation occur can be understood in terms of the Bolzman weight in the partition function since the effective theory at lowest order describes a non-interacting Bose gas. For example, non-zero densities in QCD with Nc = Nf = 2 set in for isospin chemical potential μI > mπ or baryon chemical potential μB > mπ [3]. This is so since in this theory there is no goldstone mode with non-zero third component of isospin and non-zero baryon charge. The effects at next to leading order in the chiral expansion have been studied in two colour QCD at finite baryon chemical potential [7]. The renormalized theory at next to leading order confirms that the critical baryon chemical potential is μB = mπ. In addition the critical exponents at the phase transition was calculated. Surprisingly, they remain at the mean field values. The inclusion of a set of massive spin-one vectors, Aν , in lowest order chiral perturbation theory for two-color QCD was studied in [9]. In this case the effective Lagrangian must respect global SU(2Nf) transformations and is given by