The behaviors of the solutions of the quark gap equation in the NJL model with the self-consistent mean field approximation

In this paper, we use the self-consistent mean field approximation to study the Quantum Chromodynamics (QCD) phase transition. In the self-consistent mean field approximation of the Nambu–Jona-Lasinio (NJL) model, a parameter [Formula: see text] is introduced, which reflects the weight of “direct” channel and the “exchange” channel and needs to be determined by experiments (as mentioned in a recent work [T. Zhao, W. Zheng, F. Wang, C.-M. Li, Y. Yan, Y.-F. Huang and H.-S. Zong, Phys. Rev. D 100, 043018 (2019)], the results with [Formula: see text] are in good agreement with astronomical observation data on the latest binary neutron star merging. This indicates that the contribution of “exchange” channel should be considered, and [Formula: see text] is a possible choice). By comparing the results with different parameter [Formula: see text]’s ([Formula: see text], [Formula: see text] and [Formula: see text]), we study the influence of “exchange” channel on the behavior of the solutions of the quark gap equation and the critical point of chiral phase transition. Our results show that the second-order chiral phase turns to the crossover from the chiral limit to the non-chiral limit around [Formula: see text] in the case of [Formula: see text]. The difference of the quark mass with different [Formula: see text]’s mainly occurs in the intermediate temperatures for the different fixed chemical potentials. At zero temperature and the chemical potential [Formula: see text] there will be two solutions (including a meta-stable solution) of gap equation with [Formula: see text], and as [Formula: see text] increases it will be only one solution left (the meta-stable solution will disappear until [Formula: see text]). Besides, the discrepancy of the critical temperature (above which the pseudo-Wigner solution and negative Nambu solution will disappear) in the three cases of [Formula: see text] will become large when the chemical potential increases.

An operator-theoretical study on the BCS-Bogoliubov model of superconductivity near absolute zero temperature

In the preceding papers the present author gave another proof of the existence and uniqueness of the solution to the BCS-Bogoliubov gap equation for superconductivity from the viewpoint of operator theory, and showed that the solution is partially differentiable with respect to the temperature twice. Thanks to these results, we can indeed partially differentiate the solution and the thermodynamic potential with respect to the temperature twice so as to obtain the entropy and the specific heat at constant volume of a superconductor. In this paper we show the behavior near absolute zero temperature of the thus-obtained entropy, the specific heat, the solution and the critical magnetic field from the viewpoint of operator theory since we did not study it in the preceding papers. Here, the potential in the BCS-Bogoliubov gap equation is an arbitrary, positive continuous function and need not be a constant.

The BCS energy gap at low density

We show that the energy gap for the BCS gap equation is $$\begin{aligned} \varXi = \mu \left( 8 {\mathrm{e}}^{-2} + o(1)\right) \exp \left( \frac{\pi }{2\sqrt{\mu } a}\right) \end{aligned}$$ Ξ = μ 8 e - 2 + o ( 1 ) exp π 2 μ a in the low density limit $$\mu \rightarrow 0$$ μ → 0 . Together with the similar result for the critical temperature by Hainzl and Seiringer (Lett Math Phys 84: 99–107, 2008), this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential V. The results hold for a class of potentials with negative scattering length a and no bound states.

An Operator-theoretical Study on the BCS-Bogoliubov Model of Superconductivity Near Absolute Zero Temperature

In the BCS-Bogoliubov model of superconductivity, no one gave a proof of the statement that the solution to the BCS-Bogoliubov gap equation is differentiable with respect to the temperature. But, without such a proof, one differentiates the solution and the thermodynamic potential with respect to the temperature twice, and one obtains the entropy and the specific heat at constant volume of a superconductor. In the preceding papers, the present author showed that the solution is indeed differentiable with respect to the temperature twice. Thanks to these results, we in this paper differentiate the thermodynamic potential with respect to the temperature twice so as to obtain the entropy and the specific heat at constant volume from the viewpoint of operator theory. Here, the potential in the BCS-Bogoliubov gap equation is a function and need not be a constant. We then show the behavior near absolute zero temperature of the entropy, the specific heat, the solution and the critical magnetic field. Mathematics Subject Classification 2020. 45G10, 47H10, 47N50, 82D55.

Dynamical quark mass generation in QCD3 within the Hamiltonian approach in Coulomb gauge

We investigate the equal-time (static) quark propagator in Coulomb gauge within the Hamiltonian approach to QCD in d = 2 spatial dimensions. Although the underlying Clifford algebra is very different from its counterpart in d = 3, the gap equation for the dynamical mass function has the same form. The additional vector kernel which was introduced in d = 3 to cancel the linear divergence of the gap equation and to preserve multiplicative renormalizability of the quark propagator makes the gap equation free of divergences also in d = 2.

Pairon Green’s function for high-Tc superconductors

Considering the many-body quantum dynamics, the pairon Green’s function has been developed via a Hamiltonian that encompasses the contribution of pairons, pairon-phonon interactions, anharmonicities, and defects. To obtain the renormalized pairon energy dispersion, the most relevant Born–Mayer–Huggins potential has been taken into account. The Fermi surface for the representative [Formula: see text] high-[Formula: see text] superconductor has been obtained via renormalized pairon energy relation. This revealed the [Formula: see text]-shape superconducting gap with a nodal point along [Formula: see text] direction. Further, the superconducting gap equation has been derived using the pairon density of states. The developed gap equation is the function of temperature, Fermi energy, and renormalized pairon energy. The temperature variation of the gap equation is found to be in good agreement with the BCS gap equation. Also, this reveals the reduced gap ratio ([Formula: see text] for [Formula: see text]) in the limit (5–8) of the reduced gap ratio designated for high-[Formula: see text] superconductors.

Supersymmetric NJL-Type Model for a Real Superfield Composite

The Nambu–Jona-Lasinio (NJL) model is a classic theory for the strong dynamics of composite fields and symmetry breaking. Supersymmetric versions of the NJL-type models are certainly of interest too. Particularly, the case with a composite (Higgs) chiral superfield formed by two (quark) chiral superfields has received much attention. Here, we propose a prototype model with a four-chiral-superfield interaction, giving a real superfield composite. It has a spin-one composite vector field with properties being somewhat similar to a massive gauge boson of spontaneously broken gauge symmetry. As such, it is like the first supersymmetric analog to non-supersymmetric models with spin-one composites. The key formulation developed here is the picture of quantum effective action as a superfield functional with parameters like constant superfields, having explicit supersymmetric and Grassmann number dependent supersymmetry breaking parts. Following the standard non-perturbative analysis for NJL-type models, the gap equation analysis shows plausible signature of dynamical supersymmetry breaking which is worth more serious analysis. With an extra superfield model Lagrangian included, comparison between the models and their non-supersymmetric counterparts is discussed, illustrating the notion of supersymmetrization is nontrivial in the setting.

Solving the Gap Equation of the NJL Model through Iterations: Unexpected Chaos

We explore the behavior of the iterative procedure to obtain the solution to the gap equation of the Nambu-Jona-Lasinio (NLJ) model for arbitrarily large values of the coupling constant and in the presence of a magnetic field and a thermal bath. We find that the iterative procedure shows a different behavior depending on the regularization scheme used. It is stable and very accurate when a hard cut-off is employed. Nevertheless, for the Paul-Villars and proper time regularization schemes, there exists a value of the coupling constant (different in each case) from where the procedure becomes chaotic and does not converge any longer.