Kaon Condensation in “Nuclear Star” Matter

The critical density for kaon condensation in “nuclear star” matter is computed up to two-loop order in medium (corresponding to next-to-next-to-leading order in chiral perturbation theory in free space) with a heavy-baryon effective chiral Lagrangian whose parameters are determined from KN scattering and kaonic atom data. To the order considered, the kaon self-energy has highly non-linear density dependence in dense matter. We find that the four-Fermi interaction terms in the chiral Lagrangian play an important role in triggering condensation, predicting for “natural” values of the four-Fermi interactions a rather low critical density, ρc < 4ρ0. In a recent paper, Brown and Bethe [1] suggested that if kaon condensates develop at relatively low matter density in the collapse of large stars, then low-mass black holes are more likely to form than neutron stars of the mass greater than 1.5 times the solar mass M⊙. Ever since the seminal paper of Kaplan and Nelson[2], there have been numerous investigations on kaon condensation in dense neutron star matter as well as in nuclear matter based both on effective chiral Lagrangians[3, 4, 5, 6, 7] and on phenomenological off-shell meson-nucleon interactions[8, 9]. The results have been quite confusing: While the chiral Lagrangian calculations generally predict a relatively low critical density, ρc ∼ (2− 4)ρ0, the phenomenological approaches have indicated that a kaon condensation at such a low density may be incompatible with kaon-nucleon data and in some versions seem to exclude any condensation at al. It is now understood [10, 11] that the main difference in the two approaches lies in terms higher than linear in density in the energy density of the matter. In this paper, we report the first higher-order chiral perturbation calculation of the critical density with a chiral Lagrangian that when calculated to one loop order (that is to O(Q2) relative to the leading order), correctly describes s-wave kaon-nucleon amplitude near threshold and that includes four-Fermi interactions constrained by kaonic atom data. Our prediction for critical density is ρc ≈ (3− 4)ρ0. To implement spontaneously broken chiral symmetry in the computation, we take the JenkinsManohar heavy-baryon chiral Lagrangian [12] as extended in [13] to O(Q3) to describe s-wave kaon nucleon scattering to one loop order in chiral perturbation theory (ChPT). In addition to the usual octet and decuplet baryons and the octet pseudo-Goldstone fields, the Λ(1405) was found to figure importantly in the kaon-nucleon process. This field which provides repulsion at threshold in Kp scattering was introduced in [13] as an elementary field. By fitting the coefficients of the resulting chiral Lagrangian by empirical kaon nucleon s-wave scattering data at low energy, it was shown there that higher order chiral corrections can systematically be calculated while preserving the “naturalness” condition for on-shell scattering amplitudes. By a straightforward off-shell extension, we have predicted an off-shell kaon-nucleon amplitude that could be applied to kaonic atom [14] as well as kaon condensation phenomena. The predicted off-shell amplitude was found to be in fair agreement with the phenomenological fit obtained by Steiner [15]. A simple way of understanding the result so obtained is to use the chiral counting appropriate for the meson-baryon system. In heavy-baryon formalism (HBF), we can order the relevant observables as a power series in Q, say,