Dense and Hot QCD at Strong Coupling

We present a novel framework for the equation of state of dense and hot quantum chromodynamics (QCD), which focuses on the region of the phase diagram relevant for neutron star mergers and core-collapse supernovae. The model combines predictions from the gauge/gravity duality with input from lattice field theory, QCD perturbation theory, chiral effective theory, and statistical modeling. It is therefore, by construction, in good agreement with theoretical constraints both at low and high densities and temperatures. The main ingredients of our setup are the nonperturbative V-QCD model based on the gauge/gravity duality, a van der Waals model for nucleon liquid, and the DD2 version of the Hempel-Schaffner-Bielich statistical model of nuclear matter. By consistently combining these models, we also obtain a description for the nuclear to quark matter phase transition and its critical end point. The parameter dependence of the model is represented by three (soft, intermediate, and stiff) variants of the equation of state, all of which agree with observational constraints from neutron stars and their mergers. We discuss resulting constraints for the equation of state, predictions for neutron stars, and the location of the critical point.

Popular Summary

Quantum chromodynamics (QCD) is the theory of the strong nuclear force and an important building block of the standard model of particle physics. However, solving QCD at intermediate temperatures and densities—a few times the typical density found in atomic nuclei—is a long-standing open problem; even the precise phase structure of QCD in this region is not known. This region, which is relevant to neutron star mergers and core-collapse supernovae, is thought to include a critical end point of the nuclear-to-quark matter transition. The location of this point will be narrowed down by upcoming collider experiments. But improved theoretical predictions in this region are urgently needed. To that end, we develop a novel framework for the equation of state of dense and hot QCD.

We bridge the gap in theoretical predictions at intermediate densities by using the gauge/gravity duality, which maps the strongly coupled dynamics of QCD to classical analysis in higher-dimensional gravity. The model combines predictions from the gauge/gravity duality with input from lattice field theory, QCD perturbation theory, and traditional nuclear theory methods. It is therefore, by construction, in good agreement with theoretical constraints at low and high densities and temperatures and allows one to make predictions for the critical point in the QCD phase diagram.

Our predictions for the critical point can be tested in collider experiments. Our equation-of-state models can be used in neutron star merger simulations, which then can be checked against existing and future gravitational wave detections.

Figure 1

A schematic diagram that shows the construction for the temperature dependence in the model. See text for details.

Figure 2

Contours of the speed of sound squared in $\beta$ equilibrium for soft (left), intermediate (middle), and stiff (right) EOS. Solid black lines separate the mixed phase from NM and QM phases; yellow stars mark the locations of critical points whose numerical values are listed in Table 1.

Figure 3

Latent heat as function of $T$ is shown for the three variants of EOS. The location of the critical end point is determined via the condition $\mathrm{\Delta }\epsilon (T_c,n_{bc})=0$.

Figure 4

Pressure as function of the baryon number density in $\beta$ equilibrium for different values of the temperatures. The lower (upper) bounds of the colored bands represent the soft (stiff) model, while central curves correspond to the intermediate version.

Figure 5

Pressure for the intermediate model for different values of the temperatures. Solid curves are $\beta$ equilibrium, while upper and lower bounds of the hatched colored bands represent maximum and minimum values of the pressure at given temperature and density.

Figure 6

Thermal index in $\beta$ equilibrium. Notation for the bands as in Fig. 4.

Figure 7

Mass-radius relation of nonrotating stars.

Figure 8

Comparison of cold V-QCD EOS and EOS of hadron gas with excluded volume correction. Left: normalized pressure ($p/{\mu }_b^4$) in terms of baryon chemical potential ${\mu }_b$. Right: pressure in terms of baryon number density $n_b$ in units of $n_s$. In both panels, stiff, intermediate, and soft variants are denoted by blue, green, and red curves, respectively, and hadron gas with excluded volume of $v_0=0.56{\mathrm{fm}}^{-3}$, $v_0=1{\mathrm{fm}}^{-3}$, and $v_0=1.5{\mathrm{fm}}^{-3}$ are exhibited by solid, dotted, and dashed black curves. In addition, ideal hadron gas result is shown with the gray solid curve in the panel on the right.

Figure 9

Comparison of the three variants of V-QCD finite EOS and the chosen set of the other existing finite-temperature nuclear theory EOS: energy density dependence of $\beta$-equilibrium pressure at low temperature. Orange and blue bands show the uncertainties in pressure in nuclear theory [154] and perturbative QCD (pQCD) [155], respectively. Blue band indicates all EOS satisfying astrophysical constraints [156].

Figure 10

Comparison of the three variants of V-QCD finite EOS and the chosen set of the other existing finite-temperature nuclear theory EOS: baryon number density dependence of $\beta$-equilibrium pressure at low temperature (top left) and intermediate temperature (top right), temperature dependence of $\beta$-equilibrium pressure at $n_b=3n_s$ (bottom left), charge fraction dependence of pressure at $n_b=3n_s$ and $T=50\mathrm{MeV}$ (bottom right).