Scrutinizing the pion condensed phase

Stefano Carignano,Luca Lepori,Massimo Mannarelli,Andrea Mammarella,Giulia Pagliaroli

doi:10.1140/epja/i2017-12221-x

Abstract

When the isospin chemical potential exceeds the pion mass, charged pions condense in the zero-momentum state forming a superfluid. Chiral perturbation theory provides a very powerful tool for studying this phase. However, the formalism that is usually employed in this context does not clarify various aspects of the condensation mechanism and makes the identification of the soft modes problematic. We re-examine the pion condensed phase using different approaches within the chiral perturbation theory framework. As a first step, we perform a low-density expansion of the chiral Lagrangian valid close to the onset of the Bose-Einstein condensation. We obtain an effective theory that can be mapped to a Gross-Pitaevskii Lagrangian in which, remarkably, all the coefficients depend on the isospin chemical potential. The low-density expansion becomes unreliable deep in the pion condensed phase. For this reason, we develop an alternative field expansion deriving a low-energy Lagrangian analog to that of quantum magnets. By integrating out the "radial" fluctuations we obtain a soft Lagrangian in terms of the Nambu-Goldstone bosons arising from the breaking of the pion number symmetry. Finally, we test the robustness of the second-order transition between the normal and the pion condensed phase when next-to-leading-order chiral corrections are included. We determine the range of parameters for turning the second-order phase transition into a first-order one, finding that the currently accepted values of these corrections are unlikely to change the order of the phase transition.

Highlights

Brief summary of the standard χPT results on the pion-condensed phaseThe O(p2) χPT Lorentz-invariant Lagrangian describing the interaction of pions with an external vector field, vμ, can be written as follows [2,28,31]
Introduction ulations atμI 2mπ have large errors and Nambu-Jona Lasinio (NJL) model results depend on the parameter sets employed.Systems with a nonvanishing isospin chemical potential, μI, are very good playgrounds for gaining insight on quantum chromodynamics (QCD) in the nonperturbative regime
The GP approximation breaks down deep in the pion condensed phase, because it corresponds to a low-density approximation and the number density of pions in the ground state grows with μI

Summary

Brief summary of the standard χPT results on the pion-condensed phase

The O(p2) χPT Lorentz-invariant Lagrangian describing the interaction of pions with an external vector field, vμ, can be written as follows [2,28,31]. Since the condensed bosons are electrically charged, the resulting phase is a superconductor [27] This symmetry breaking mechanism can be described by maximizing the O(p2) ground state Lagrangian. It can be shown that there exists a flat direction of the potential, which is a typical feature of the BEC phase because it is associated with the existence of NGBs. by a variational procedure it is possible to show that the unit vector n in Eq (4) has to be orthogonal to the direction taken by the vector field vμ in isospin space [27]. For γ > 1, it describes a mode propagating with the speed of sound (which can as well be obtained from the equation of state, Eq (12)) and should correspond to the phonon This interpolation is possible because the sound speed vanishes at the phase transition point.

Low energy description of the normal phase-BEC phase transition

Alternative description of the pion condensed phase

NLO corrections

Conclusions