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Consider QCD with a single generation of massless quarks (u, d). This is probably the simplest variant of QCD which bears some relation to the real world. The theory has the following exact global symmetries:

  • U(1) acting on u and d (baryon number)
  • Chiral U(1) acting in opposite matter on left-hand and right-handed quarks. It is destroyed by an anomaly so we will not consider it further
  • SU(2) isospin rotation mixing u and d
  • Chiral SU(2). It is spontaneously broken
  • Charge conjugation C
  • Parity (spatial reflection) P
  • Time reversal T

A thermal equilibrium of the model is characterized by 3 parameters:

  • Temperature
  • Chemical potential associated to baryon number. Alternatively, baryon number density. We can use C to fix its sign
  • Chemical potential associated to isospin. Alternatively, isospin density. It is a vector but using isospin rotation symmetry we can align it along a prescribed axis, so we are left with a positive scalar parameter

Hence, the theory has a 3-dimensional phase diagram

How does the 3D phase diagram look like? Which phases do we have? Which phase transitions? What is the type of each phase transition?

Above a certain temperature T, chiral symmetry is restored. Btw, is this the same phase transition which breaks confinement? Above this T, we can introduce a 4th parameter, namely the chemical potential associated to chiral isospin. I suppose there a certain inequality governing the maximal value of this parameter as a function of the temperature?

How does the 4D phase diagram look like?

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The situation is well represented in the following very pictorial picture

enter image description here

but this is a very active field of study. It is interesting to note that a real proof of existence for the critical endpoint (CEP, indicated as a critical point in the figure), both from a theoretical and numerical point of view, does not exist yet. The reason, at least for the lattice computations, arises from the infamous sign problem. When you discretize the action of the QCD with a chemical potential, this contribution becomes imaginary. So, several ways out have been devised but, as far as I can tell, none is taken for accepted widely. Finally, CEP is not really a critical point but a cross over point. This behaves as a true critical point for a phase transition if you take zero mass for quarks and zero chemical potential.

Note added: Just today appeared a paper by Owe Philipsen exactly on this matter (see here). The title is "Status of the QCD phase diagram from lattice calculations".

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