Universality of the chiral soliton lattice and its interaction with quark matter

Abstract

In this paper, we show that the chiral soliton lattice (ChSL) is, in a precise sense, a universal feature of the low-energy limit of QCD minimally coupled to Maxwell theory. Here, we disclose that not only can the ChSL be obtained from the gauged Skyrme model in $3+1$ dimensions, including the back-reaction of the Maxwell $U(1)$ gauge field, we also demonstrate that the ChSL remains unchanged when higher-order terms arising from QCD, specifically the sub-leading corrections in the 't Hooft large $N_c$ expansion, are included. By considering a suitable ansatz adapted to describe topological solitons at finite baryon density in a constant magnetic field, the generalized Skyrme model coupled to the Maxwell theory is reduced to the effective Lagrangian of the ChSL phase, which describes a lattice of domain walls made of hadrons. One of the key points in this construction is the fact that even when the usual topological charge density vanishes, the presence of the Callan-Witten term in the topological charge density allows for a non-vanishing baryon number. In the present approach, the magnetic field can be external, as is usually assumed for the ChSL, or it can be self-consistently generated by the hadronic layers themselves. Finally, we show how our formulation allows us to study the coupling of the ChSL with quark matter. In particular, we derive the exact analytical spectrum of the Dirac equation in the high-density limit, providing a microscopic characterization of the fermionic excitations within the inhomogeneous hadronic background provided by the ChSL. The comparison of the present spectrum of the Dirac operator within the ChSL with the spectrum of the usual Dirac operator in a constant magnetic field discloses the fundamental role of both the quark-Skyrmion coupling and the hadronic profile in opening a gap and generating a shift in the spectrum itself.