Renormalization-group Flow Analysis of Meson Condensations in Dense Matter

We present a renormalization-group (RG) flow argument for s-wave kaon condensation in dense nuclear-star matter predicted in chiral perturbation theory. It is shown that it is the relevant mass term together with any attractive interaction for the kaon in medium that triggers the instability. We show that a saddle point of multi-dimensional RG flow can imply a phase transition. Pion condensation is also analyzed along the same line of reasoning. The idea of renormalization group (RG) has been used extensively both in condensed matter physics and particle physics, especially for the critical phenomena and various situations involving scaling behavior. Recently, Shankar[1] and Polchinski[2] showed that one can use the RG idea even for the phenomena including a scale, such as the mass gap, as in BCS superconductivity and charge density wave etc. The key point is that these phenomena could be understood as an instability from the Fermi liquid identified as the fixed point of the RG flow. They showed for the BCS case as an example that when two incoming momenta sum to zero, the corresponding four-Fermi interaction is marginally relevant #1 in the RG sense, so that the interaction causes an instability that pushes away the system from the Fermi liquid. The new insight gained in this approach is that one can identify in a clear and simple way the dynamics and kinematics that lead to the phase transitions. In this paper, we extend this approach to the condensation of negatively charged kaons (K) in dense nuclear medium as in neutron stars by considering the role of the quadratic term in the effective potential entering in kaon-nucleon interactions. (The K+ meson does not condense and the neutral kaons K0 and K0 are not relevant in neutron stars.) Recently Lee et al [3, 4] have shown by chiral perturbation theory (χPT ) treated to in-medium two-loop order (corresponding to next-to-next-to leading order) that kaons can condense in dense nuclear-star matter at a matter density ρ <∼ 4 ρ0 where ρ0 is the normal nuclear matter density. However to the order considered, many terms are involved and it is not transparent which mechanism is in action for triggering the condensation process. In this note, we present a renormalization group flow analysis to show what drives the process of kaon condensation and in particular to indicate the basic mechanism involved. The conclusion is that kaons must condense in s-wave, although the analysis cannot give the critical density. For the purpose of elucidating the basic concept, we find it sufficient to study a toy model which we believe captures the essence of the physics involved in kaon condensation. The corresponding action, S, can be decomposed into three parts: SK for the free kaon, SN for the nucleon and SKN for kaon-nucleon interactions. We assume that nucleons in nuclear matter are in Fermi-liquid state with the Fermi energy μF and the Fermi momentum kF . This state might arise from chiral Lagrangians as some sort of “Q-balls” or nontopological solitons. (For a discussion on this, see ref.[5].) For our purpose, it is crucial that the nuclear matter arises as a Fermi liquid[6]. Defining ψ as the nucleon field fluctuating around the Fermi surface such that the momentum integral has a cut-off ΛN , kF − ΛN < |~k| < kF + ΛN , (1) Throughout this article, we put in italic the terms “relevant,” “marginal,” and “irrelevant” when used in the RG sense.