The QCD phase diagram

A phase diagram aims to describe the behaviour of matter at different conditions depending for example from variables such as pressure and temperature.

In the QCD phase diagram the interactions between particles are ruled by the quantum chromodynamics. The QCD phase transition scheme, in fact, is yet under studies and one of the targets of the research is therefore to quantitatively map it out.

It can be useful to consider a comparison between the phase diagram for water and QCD matter gas

The well-known phase diagram of water (on the left) and (on the right) a QCD phase diagram.

In classical thermodynamics, phase diagrams present three broad regions, one for each phase of the matter (gas, liquid, solid), typically separated by phase transition lines.

In correspondence of the triple point, all the three phases can exist.

The critical point, defined by the critical temperature, T_C, and critical energy density, \epsilon_C, indicates the condition at which the vapour pressure curve terminates and liquid and gas phases can coexist.

Of course, in classical thermodynamics and especially in the case of water, all features are experimentally well established with great accuracy.

The QCD phase diagram is usually built as a function of the baryo-chemical potential \mu_B vs the temperature T according to phenomenological models and mainly to lattice QCD (L-QCD) calculations.
This is because, in thermodynamics, the chemical potential \mu quantifies the variation of the internal energy of the system U after the introduction of an additional particle, \mu=\frac{\partial U}{\partial N}.

The baryo-chemical potential \mu_B is the chemical potential for a single baryon and it expresses the energy needed to increase the baryon quantum number, that is a globally conserved property of dense hadronic matter.
The addition of a baryon-antibaryon pair, then, would not affect the global energy of the system that can be expressed by the first law of thermodynamics as dE=-PdV+TdS+\mu_BdB, since the baryon number B remains unchanged.

A variation of \mu_B, instead, would change the global energy of the system, its pressure or its temperature.
The baryo-chemical potential is also connected with the baryon density number n_B and it can represent the pressure P.
As a consequence, it is closely linked to the hadronic density \rho_B.
Furthermore, \mu_B is a measurable quantity and, unlike the density, it remains continuous during the phase transition.

Various effective theories and phenomenological models are the basis of the schematic phase diagram of QCD that is shown in the right side of Figure above.

The transition to the QGP can be induced either by increasing the temperature or the density.
At low temperatures and high values of the chemical potential, nuclear matter consists of an interacting and degenerate highly compressed Fermi gas of quarks.
The interaction among the quarks can be attractive in specific combinations of colours states, leading to the formation of quark-quark pairs which determine a colour superconducting phase.

While in the case of early Universe the transition from QGP to hadrons is supposed at high temperature and vanishing chemical potential, it is thought that in the neutron stars, due to gravitational collapse, the QGP state should be formed for high values of the baryo-chemical potential and temperature close to zero.
The \Lhc achieves experimental conditions close to those supposed for the primordial stages of the Universe, with high T and low \mu_B, thus allowing to investigate in a region of highest interest in the QGP phase transition scheme.
Thus, such conditions are also the frame for most L-QCD calculations and results.

The search for experimental evidences of the creation of a deconfined phase is linked to the nature of the transition between confined and deconfined phases and then
to the prediction for observables in a QGP creation experiment.
The nature of such transition depends also on the number of quark flavours involved and on their masses.

A phase transition is classified according to the free energy F as a function of the temperature.
If a discontinuity occurs, then a phase transition is expected.
The free energy F is defined as F=U-TS, where U is the free internal energy and S the entropy of the thermodynamic system.

A n^{th} order phase transition implies that \frac{\partial^nF}{\partial T^n} is discontinuous while \frac{\partial^{n^\prime}F}{\partial T^{n^\prime}}, with n^\prime<n, is continuous.
A first order transition implies a discontinuity in the free energy derivative \frac{\partial F}{\partial T}=S that means that a latent heat is associated to the transition.
In second order transitions, there are discontinuities only for derivatives of orders higher than the first.
In a crossover region, the transition takes place without discontinuities for the free energy and its derivatives and an abrupt change in the phase transition from a phase to another is not observed.

In a QCD phase transition scheme, the first order phase transition line divides the hadronic matter area from the quark-gluon plasma state.
At the crossover point, the confined phase of hadronic gas and the deconfined state of QCD can cohexist.

Analogies with classical thermodynamics and other arguments based on a variety of models could suggest that one should expect that the phase transition line ends
at a critical point as a first-order phase transition is a function of temperature at finite \mu_B.

Analogies with classical thermodynamics and other arguments based on a variety of models could suggest that as a first-order phase transition is a function of temperature at finite \mu_B, one should expect that the phase transition line to end at a critical point.

Furthermore, L-QCD calculations at non-zero baryo-chemical potential suggest the existence of a tri-critical point (\mu_{B,c};T_c) at which the first order transition becomes a crossover for \mu_B<\mu_{B,c} and T<T_c.
However, the existence of a critical point is not yet established experimentally.

The Figure below shows the result of L-QCD calculations for hadronic matter phase and shows the dependence of phase transitions on the number of flavours for \mu_B=0 and m_u=m_d.

Dependence of the phase transition orders on quark masses and on number of flavours considered in the QCD lagrangian. The results obtained in the calculation are expressed in the lagrangian in terms of a observable, i.e. the ratio of the lightest pseudo-scalar and vector meson masses (m_{PS} / m_{V}). Temperatures for deconfinement and for chiral symmetry restoration in the case of two distinct phase transitions are also reported. [Picture from F. Karsch, see references below]

In the case of three flavours, the order of the transition varies depending on mass values that u, d or s quarks assume.
The tri-critical value for the mass of the s quark is indicated by m_s^{tric}.
Beyond such value, the transition becomes of second order, while m_{u,d} is set to zero.

Quantum chromodynamics is highly dependent on values of parameters entering the lagrangian.
In particular, number of quark flavours and quark masses can affect some symmetries (e.g. the restoring of the chiral symmetry) and then the order of the phase transitions.
The greatest effects are expected by variations of the lighter quark masses which dominate the dynamic of the system.
In the low \mu_B values region, transitions strongly depend on the number of quark flavours and on the masses of light quarks.
It is still unclear whether the transition shows discontinuities for realistic values of the light quark masses, or whether it is a crossover.
Recent calculations (see the work from Z. Fodor and S.D. Katz and the one from S. Ejiri) indicate that the transition is a crossover for values of \mu_B \leq 400\ MeV.

Further indications instead come from lattice calculations results.


References and further readings:
J. Letessier and J. Rafelski, Hadrons and Quark–Gluon Plasma. Cambridge monographs on Particle Physics, nuclear Physics and Cosmology, 2002.
G. M. Garcı̀a, Advances in Quark Gluon Plasma, [arXiv:nucl-ex/1304.1452v1], 2013.
W. Greiner, S. Scram, E. Stein, Quantum Chromodynamics – Second Edition. Springer, 2002.
R. S. Bhalerao, Relativistic Heavy–Ion Collisions, [arXiv:nucl-th/1404.3294v1], 2014.
Z. Fodor, S. D. Katz, Critical point of QCD at finite T and \mu, lattice results for physical quark masses, JHEP 0404, 2004.
F. Karsch, Lattice Results on QCD Thermodymamics, [arXiv:hep-ph/0103314v1].
S. Ejiri et al, Study of QCD thermodynamics at finite density by Taylor expansion, Prog. Theor. Phys. Suppl., 153, 2004

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