Towards a Stability Analysis of Inhomogeneous Phases in QCD

The possible occurrence of crystalline or inhomogeneous phases in the QCD phase diagram at large chemical potential has been under investigation for over thirty years. Such phases are present in models of QCD such as the Gross-Neveu model in 1+1 dimensions, Nambu-Jona-Lasinio (NJL) and quark meson models. Yet, no unambiguous confirmation exists from actual QCD. In this work, we propose a new approach for a stability analysis that is based on the two-particle irreducible effective action and compatible with full QCD calculations within the framework of functional methods. As a first test, we reproduce a known NJL model result within this framework. We then discuss the additional difficulties which arise in QCD due to the non-locality of the quark self-energy and suggest a method to overcome them. As a proof of principle and as an illustration of the analysis, we consider the Wigner-Weyl solution of the quark Dyson-Schwinger equation (DSE) within a simple truncation of QCD in the chiral limit and analyse its stability against homogeneous chiral-symmetry breaking fluctuations. For temperatures above and below the tricritical point we find that the boundary of the instability region coincides well with the second-order phase boundary or the left spinodal, respectively, obtained from the direct solutions of the DSEs. Finally, we outline how this method can be generalized to study inhomogeneous fluctuations.