Embedding a critical point in a hadron to quark-gluon crossover equation of state

Lattice QCD simulations have shown unequivocally that the transition from hadrons to quarks and gluons is a crossover when the baryon chemical potential is zero or small. Many model calculations predict the existence of a critical point at a value of the chemical potential where current lattice simulations are unreliable. We show how to embed a critical point in a smooth background equation of state so as to yield the critical exponents and critical amplitude ratios expected of a transition in the same universality class as the liquid-gas phase transition and the three-dimensional Ising model. There are only two independent critical exponents; the relations $\alpha +2\beta +\gamma =2$ and $\beta (\delta -1)=\gamma$ arise automatically, as does a relation between the two critical amplitudes. The resulting equation of state has parameters that may be inferred by hydrodynamic modeling of heavy-ion collisions in the Beam Energy Scan II at the BNL Relativistic Heavy Ion Collider or in experiments at other accelerators.

Figure 1

The choice of critical curve as described in the text. The critical temperature is taken to be 100 MeV and the critical chemical potential to be 750 MeV.

Figure 2

The dimensionless function $R$ with parameters given in the text. The critical point is indicated by the solid dot.

Figure 3

Isotherms of pressure versus density. The numbers label the temperature with $T_c=100$ MeV.

Figure 4

Temperature versus density separating the two phases.