Detailed study of the dual symmetry of dense quark matter within the effective action formalism

Abstract

The symmetry properties of the phase diagram of dense quark matter composed of $u$ and $d$ quarks with three colors have been investigated in the framework of massless (3+1)-dimensional Nambu--Jona-Lasinio (NJL) and QCD models. It turns out that in the presence of baryon $μ_B$, isospin $μ_I$, chiral $μ_5$ and chiral isospin $μ_{I5}$ chemical potentials, the Lagrangians of these models are invariant under the so-called dual transformation ${\cal D}$. In this paper, it has been shown that the path integration measure of the corresponding partition functions is also invariant under ${\cal D}$. Consequently, the entire NJL model (or QCD) thermodynamic potentials are dually symmetric. In particular, it means that in the total $(μ_B,μ_I,μ_5,μ_{I5})$-phase portraits of these models the chiral symmetry breaking (CSB) and charged pion condensation (PC) phases are arranged dually conjugate (or symmetrically) to each other. Dual symmetry between CSB and charged PC phases of quark matter is a fundamental property of massless dense QCD; it should be observed in the framework of any approximation to the QCD phase diagram, and have important implications for studies of the phase structure.