Kaon condensation in baryonic Fermi liquid at high density

We formulate kaon condensation in dense baryonic matter with antikaons fluctuating from the Fermi-liquid fixed point. This entails that in the Wilsonian renormalization group (RG) approach, the decimation is effectuated in the baryonic sector to the Fermi surface while in the meson sector to the origin. In writing the kaon-baryon (KN) coupling, we take a generalized hidden local symmetry Lagrangian for the meson sector endowed with a “mended symmetry” that has the unbroken symmetry limit at high density in which the Goldstone $\pi$, scalar $s$, and vectors $\rho$ (and $\omega )$ and $a_1$ become massless. The vector mesons $\rho$ (and $\omega$) and $a_1$ can be identified as emergent (hidden) local gauge fields and the scalar $s$ as the dilaton field of the spontaneously broken scale invariance at chiral restoration. In matter-free space, when the vector mesons and the scalar meson—whose masses are much greater than that of the pion—are integrated out, then the resulting KN coupling Lagrangian consists of the leading chiral order [$O\left(p^1\right)$] Weinberg-Tomozawa term and the next chiral order [$O\left(p^2\right)$] ${\Sigma }_{\text{KN}}$ term. In addressing kaon condensation in dense nuclear matter in chiral perturbation theory, one makes an expansion in the “small” Fermi momentum $k_F$. We argue that in the Wilsonian RG formalism with the Fermi-liquid fixed point, the expansion is on the contrary in $1/k_F$ with the “large” Fermi momentum $k_F$. The kaon-quasinucleon interaction resulting from integrating out the massive mesons consists of a “relevant” term from the scalar exchange (analog to the ${\Sigma }_{\text{KN}}$ term) and an “irrelevant” term from the vector-meson exchange (analog to the Weinberg-Tomozawa term). It is found that the critical density predicted by the latter approach, controlled by the relevant term with the irrelevant term suppressed, is three times less than that predicted by chiral perturbation theory. This would make kaon condensation take place at a much lower density than previously estimated in chiral perturbation theory.

Figure 1

Only the orthogonal component of the momentum to the Fermi surface scales as $l\to sl$, where $\delta {\vec{k}}_{\parallel }$ is the parallel component of the momentum to the Fermi surface.

Figure 2

One-loop graphs contributing to $h$. The process corresponds to $\text{f(ermion)}(\epsilon ,\vec{k})+\text{m(eson)}(\omega ,\vec{q})\to \text{f(ermion)}({\epsilon }^{\prime \prime },{\vec{k}}^{\prime \prime })+\text{m(eson)}({\omega }^{\prime \prime },{\vec{q}}^{\prime \prime })$.

Figure 3

One-loop graph contributing to $m_{\varphi }$.

Figure 4

RG flows of the parameters, $m_{\varphi }^2\left(t\right)$ and $h\left(t\right)$, are shown in the parameter space. The lines are drawn with the fixed value of $A$ and the different values of $c_{a_h}$. The black solid line stands for the “critical line (or surface)” that delineates the parameter spaces.

Figure 5

The kinematic configuration in the momentum space.

Figure 6

One-$\varphi$-exchange graphs for (a) the four-Fermi interaction and (b) the KN-KN interaction.