We calculate quark number density and susceptibility under one-loop correction in the mean-field potential. The calculation shows continuous increase in the number density and susceptibility up to the temperature $T = 0.4 \rm{GeV}$. Then the values of number density and susceptibility approach the very weakly result with higher values of temperature. The result indicates that the calculated values fit well with increase in temperature to match the lattice QCD simulations of the same quantities.

The effect of two-loop correction in quark–gluon plasma (QGP) droplet formation is studied by introducing the two-loop correction factor in the mean-field potential. The correction factor leads to the stability in the droplet formations of QGP at different parametrisation factors of the QGP fluid. This also shows that the gluon parameter factor shifts to a larger value from its earlier value of one-loop correction in attaining the stability of the droplets. The results show a decrease in the observable QGP droplet sizes which are found to be 1.5–2.0 fm radii with the two-loop correction. It indicates that the dynamics of the QGP droplet and the stability of the droplet withthe two-loop correction factor can be controlled by the fluid parameter in the model.

We present analytical results of the equation of state (EOS) described by a model of thermal quark mass and magnetic potential term introduced in Polyakov–Nambu–Jona–Lasinio (PNJL) model for two-flavour quarks. Under the influence of thermal quark mass and magnetic field term in the potential, the calculated results of EOS using the model are enhanced in a good pattern up to the temperature $T=2.2T_c$ MeV and can follow result similar to the unmagnetised field when the temperature is increased beyond $T=2.2T_c$ MeV. The result shows that the thermodynamic behaviour agrees well with the standard properties of quantum chromodynamics (QCD) thermodynamics and enhance the result up to 2.2$T_c$ from the earlier predicted results and show the same behaviour beyond 2.2$T_c$ MeV.

We extend the work of two-loop correction in the mean-field potential to study the thermodynamical properties of quark–gluon plasma (QGP). In the study,we look forward to determine bulk thermodynamic properties like pressure, entropy density, specific heat, quark number density, quark number fluctuations and speed of sound. The determinations are shown in figures and the figures are found almost similarly enhanced to our earlier results of one-loop correction. It means that the model with the inclusion of two-loop corrections can provide the entire thermodynamic information about the phase structure of QCD and the parameter used in the loop correction plays a pivotal role in forming stable droplets having less effect to calculate all the thermodynamical parameters. Still,the result can be considered in presenting QGP phase structure as it shows its representation within the range of standard thermodynamic properties and enhance the energy density and pressure from the lattice result.

We extend our earlier work of boson star (BS) rotation after incorporating the cosmological constant and coupling term of the curvature of the Universe with the field of boson star. Due to the cosmological constant and coupling term in the Lagrangian density of the boson star, the oscillation amplitudes of the boson star are modified for the entire study and the amplitudes are found to increase from the result obtained earlier. The introduction of cosmological constant and coupling of curvature prove the hypothetical assumption of the existence of the bosonstar.