Abstract
Solutions and their evolutions of the quark gap equation are studied within the Nambu–Jona–Lasinio model, which is a basic issue for studying the QCD phase structure and locating the possible critical end point. It is shown that in the chiral limit case of the vacuum, chiral symmetry will hold if the coupling strength G is small, then the system only has the Wigner solution at M,=,0. If increasing G, two symmetric minima will appear as the positive and “negative” Nambu solutions, however, the solution M,=,0 now corresponds to a maximum instead of a minimum of the thermodynamical potential, so is not a physically stable state anymore (we call it “pseudo-Wigner solution”). Besides, it is shown that as the current quark mass m increases, the pseudo-Wigner solution will become negative, and disappear together with the negative Nambu solution if m is large enough. Similar things happen if we increase the temperature or quark chemical potential mu . Some interesting phenomenon is, from some mu a second local minimum will show up. As mu increases gradually, it will be stabler than the Nambu solution, survives even the Nambu solution disappears, and approaches m, which are just the features of the Wigner solution we expect.
Highlights
The fundamental theory of strongly interacting quarks and gluons, Quantum Chromodynamics (QCD), is an important part of the Standard Model of particle physics
Drastic changes in the early Universe, such as first-order phase transition, can produce a stochastic gravitational wave background [1], researches of QCD phase transitions are helpful for the investigation of the structure of symmetries in the early Universe; on the other hand, neutron star mass limit at 2M might support the existence of a critical end point (CEP) [2]
This is just the Wigner solution we need, which describes a chiral symmetry partially restored state compared to the Nambu solution, while the pseudoWigner solution and negative Nambu solution can only be regarded as mathematical solutions
Summary
The QCD phase diagram in the temperature (T ) and quark chemical potential (μ) plane is one of the related open problems that has attracted lots of interests, both theoretically and experimentally. The search for the position or even the existence of the CEP is one of the hot topics [4,5,6,7,8,9,10], so does the study about equation of state that provides information on pressure, entropy, energies and other thermodynamic variables of interest [11,12,13], but thanks to the non-perturbative property of QCD at this region, the results are usually quite model dependant Essential understanding of these kinds of questions needs us to know different phases and how they change with some parameters (such as T and μ) first, which is usually explored via solutions as well as their evolutions of the quark gap equation.
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