Softening holographic nuclear matter

Abstract

Baryons in the holographic Witten-Sakai-Sugimoto model are described by instanton solutions on the flavor branes. A commonly used approximation for dense baryonic matter replaces the many-instanton solution by a simpler, spatially homogeneous, ansatz, which requires a discontinuity in the holographic direction of the non-abelian gauge field in order to account for topological baryon number. We point out that the simplest configuration with a single jump - often used in previous studies - results in matter at saturation density that is much stiffer than real-world nuclear matter. This is improved, although not completely remedied, by adding a second jump. We present a systematic discussion of all possible configurations up to four jumps, dynamically computing locations of and behavior at the discontinuities. We find solutions that continuously connect to those based on pointlike baryons, thus, for the first time, establishing a concrete link between the instantonic and homogeneous pictures. This is supported by translating the multi-jump profiles of the gauge field into gauge invariant multi-layer charge distributions. The most important of our novel configurations has a block-like structure in the bulk, becomes pointlike at low density and/or large coupling, and is energetically preferred over all previously studied configurations. Therefore, our work lays the ground for improved predictions from holography for dense nuclear matter in neutron stars.

Figures (11)

• Figure 1: Illustration of geometry and notation in both $u$ and $z$ coordinates. We allow for discontinuities of the non-abelian gauge field at the tip $z=0$ of the connected flavor branes and at dynamically determined locations $\pm z_k$. In the general derivation we include arbitrarily many discontinuities, keeping $d$ general, in the concrete numerical results we allow for up to four, i.e., $d\le 2$. For example, a 3-jump configuration has jumps at $z=0$ and $z=\pm z_1$, while a 4-jump configuration has jumps at $\pm z_1$ and $\pm z_2$ (and none at $z=0$).
• Figure 2: Sketch and terminology for the 4 different configurations at a discontinuity that correspond to stationary points of the free energy. The lines illustrate the non-abelian gauge field in the form of the function $h(z)$ with $z>0$ increasing in each panel from left to right.
• Figure 3: Onset chemical potential $\mu_0$, saturation density $n_0$, and incompressibility $K$ at saturation, as a function of the 't Hooft coupling $\lambda$ for the $D^0$ (left) and $D_L$ (right) phases. Each quantity is normalized by its physical value (the upper limit $K=300\, {\rm MeV}$ of the empirical range for the incompressibility), such that 1 (thin horizontal line) indicates the physical value. Each panel is created with the value of $M_{\rm KK}$ needed to reproduce the physical values for $n_0$ and $\mu_0$, i.e., $(\lambda,M_{\rm KK}) = (7.09,1000\, {\rm MeV})$ (left) and $(\lambda,M_{\rm KK}) = (17.8,520\, {\rm MeV})$ (right); the corresponding values of $\lambda$ are indicated by a thin vertical line. The dots on these lines mark the incompressibilities that are 8.3 times (left) and 5.6 times (right) larger than the (largest conceivable) physical value at saturation.
• Figure 4: Charge distribution $\rho(z)$ (solid) and gauge field $h(z)$ (dashed) for all phases with up to 2 discontinuities in $h(z)$, which is suitably rescaled for illustrative purposes. For all plots $\lambda=7.09$, while the densities are $\bar{n}_B \simeq 0.1,0.1,1.7,0.5,3.7$ for $D^0$, $D_L$, $D_R$, $D_S$, $D_A$, respectively. A yellow background indicates that the phase has a first-order baryon onset.
• Figure 5: Properties of phases with one (black) and two (blue) discontinuities, all at $\lambda = 7.09$: Location of the discontinuity $u_1$ on one half of the flavor branes (upper left panel), baryon density $\bar{n}_B$ (upper right panel), and free energy density $\Omega$ (lower panel), all as functions of the chemical potential $\bar{\mu}_B$. Dots indicate the points of the onset from the vacuum (and the point where $D_R$ connects to $D^0$). The $D_A$ configuration only exists at very large $\bar{\mu}_B$ (see upper left panel) and is outside the shown range of the two other panels. The vertical dashed lines in the upper panels indicate the first-order transitions between the vacuum and the $D^0$ phase (right panel) and between the $D^0$ and $D_L$ phases (both panels).