High Density Quark Matter under Stress

TL;DR

This work analyzes how SU(3) flavor breaking, via a nonzero electron chemical potential $μ_e$ and finite strange quark mass $m_s$, affects high-density CFL quark matter. It builds a chiral EFT for CFL Goldstone modes, performs perturbative matching to QCD to fix $f_π$ and $O(M^4)$ operators, and uses linear response to show that $μ_e$ and $m_s^2/(2p_F)$ act as flavor gauge fields that induce kaon or pion condensation at scales well below the gap $Δ$. The main results are precise onset conditions $μ_e \sim \sqrt{m m_s}\,Δ/p_F$ and $m_s \sim m^{1/3}\,Δ^{2/3}$, with $f_π^2 = m_D^2$ and the $O(M^4)$ term fixed by matching; these findings illuminate the phase structure of dense quark matter and the role of Goldstone modes in neutron-star cores. Overall, the paper provides a coherent EFT framework linking microscopic QCD to the macroscopic behavior of stressed CFL matter and its potential astrophysical manifestations.

Abstract

We study the effect of SU(3) flavor breaking on high density quark matter. We discuss, in particular, the effect a non-zero electron chemical potential and a finite strange quark mass. We argue that these perturbations trigger pion or kaon condensation. The critical chemical potential behaves as $μ_e\sim\sqrt{m m_s} Δ/p_F$ and the critical strange quark mass as $m_s \sim m^{1/3} Δ^{2/3}$, where $m$ is the light quark mass, $Δ$ is the gap, and $p_F$ is the Fermi momentum. We note that parametrically, both the critical $μ_e$ and $m_s^2/(2p_F)$ are much smaller than the gap.

Figures (4)

• Figure 1: Schematic picture of weak decays in normal (a) and superfluid (b) quark matter with three quark flavors. We assume that initially the density of all quark flavors is the same, so that $\epsilon_{F,s} \simeq \epsilon_{F,ud} + m_s^2/(2p_F)$. Solid and open circles show particles $(p)$ and holes $(h)$. In (a) a strange particle decays into an up quark, an electron and a neutrino, leaving behind a strange hole. In the left panel of (b) a strange particle decays into an up quark, a down particle-hole pair, an electron and a neutrino. The remaining $(pp)(hh)$ configuration has the quantum numbers of a $K^+$. In the right panel we show the decay of a strange quark into a $(pp)(hh)$ configuration with the quantum numbers of a $K^0$.
• Figure 2: Masses of $K^\pm$ and $K^0,\bar{K}^0$ excitations in the color-flavor locked phase. We show the excitation energies as a function of $m_s$ for $p_F=500$ MeV. The gap $\Delta=67$ MeV and the pion decay constant $f_\pi=104$ MeV were determined to leading order in perturbation theory. The solid and dashed curve show the masses of the $(K^+,K^0)$ and $(K^-,\bar{K}^0)$ states. The dotted curve shows the kaon masses calculated from the leading order $O(m_q)$ term. The short dashed curve shows the pion masses.
• Figure 3: Diagrams contributing to the two-point functions of two $L$ currents (Fig. a), one $L$ and one $R$ current (Fig. b), and one $L$ and one color current (Fig. c). The squares denote the anomalous fermion self energy while the triangle denotes a resummed gluon propagator.
• Figure 4: Fig. a) shows the diagram in the microscopic theory which is matched against the $MM^\dagger \Sigma M^\dagger M\Sigma$ term in the chiral theory. Fig. b) shows the diagrams which are matched against the $(MM^\dagger)^2$ term.