Lattice QCD calculations have shown that the transition from hadrons to quarks and gluons is a rapid crossover at $T=155–160$ MeV at vanishing chemical potential. Many model calculations show that the transition is first order at a sufficiently high baryon chemical potential. It is then natural to expect the existence of a critical point where the crossover and first-order phase transition lines meet. We show how to embed a phase boundary that terminates at the critical point in a smooth background equation of state, using several different but closely related criteria, so as to yield the critical exponents and critical amplitude ratios expected of a transition in the three-dimensional Ising and liquid-gas universality classes. The crossover curves can be tuned to pass through experimental freeze-out data from heavy-ion collisions at the BNL Relativistic Heavy Ion Collider and the CERN Large Hadron Collider. The resulting equations of state can be used in hydrodynamic simulations of these collisions to probe the existence of a critical point and the corresponding first-order phase transition.

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• Equations of state of nuclear matter
• QCD phase transitions
• Relativistic heavy-ion collisions