Consistent regularization and renormalization in models with inhomogeneous phases

Abstract

In many models in condensed matter and high-energy physics, one finds inhomogeneous phases at high density and low temperature. These phases are characterized by a spatially dependent condensate or order parameter. A proper calculation requires that one takes the vacuum fluctuations of the model into account. These fluctuations are ultraviolet divergent and must be regularized. We discuss different ways of consistently regularizing and renormalizing quantum fluctuations, focusing on momentum cutoff, symmetric energy cutoff, and dimensional regularization. We apply these techniques calculating the vacuum energy in the Nambu-Jona-Lasinio model in 1 þ 1 dimensions in the large-N c limit and in the 3 þ 1 dimensional quark-meson model in the mean-field approximation both for a one-dimensional chiral-density wave.

Chiral density wave versus pion condensation in the ( 1+1 )-dimensional NJL model

“…The coefficients are equal also if one uses momentum cutoff regularization. This is in contrast to three dimensions where only dimensional regularization [41] or Pauli-Villars regularization [17] yield β 2 ¼ β 3 due to the absence of surface terms. The tricritical point is given by the condition that the quadratic and quartic terms vanish, and the Lifschitz point is given by the condition that the quadratic and gradient terms vanish.…”

“…[41], we investigated systematically different regularization schemes in effective models with inhomogeneous phases. The vacuum energy of the NJL model in (1 þ 1) dimensions was calculated in the large-N c limit in the background of a chiral-density wave.…”

“…[41] to calculate the free energy to leading order in N c and map out the phase diagram in the μ − T plane both in and away from the chiral limit. We also consider the competition between a constant pion condensate and a chiral density wave at finite isospin.…”

Inhomogeneous chiral condensate in the quark-meson model

“…The residual q dependence in the limit Δ → 0 is then an artifact of the regulator which can be dealt with by introducing extra subtraction terms. Different regularization methods are discussed in some detail in [35,36].…”

First-order phase boundaries of the massive ( 1+1 )-dimensional Nambu–Jona-Lasinio model with isospin

“…But this is precisely the first line of the self-consistency relation (11) if we use the isospin up and down spinors as solutions of the Dirac-HF equation (12). The 2nd line of (11) with C = 0 is trivially fulfilled, so that we have indeed self-consistency for baryons in the massive isoNJL model.…”