We calculate the three-loop thermodynamic potential of QCD at finite temperature and chemical potential(s) using the hard-thermal-loop perturbation theory (HTLpt) reorganization of finite temperature and density QCD. The resulting analytic thermodynamic potential allows us to compute the pressure, energy density, and entropy density of the quark-gluon plasma. Using these we calculate the trace anomaly, speed of sound, and second-, fourth-, and sixth-order quark number susceptibilities. For all observables considered we find good agreement between our three-loop HTLpt calculations and available lattice data for temperatures above approximately 300 MeV.

We have evaluated the electromagnetic spectral function and its spectral properties by computing the one-loop photon polarization tensor involving quarks in the loop, particularly in a strong field approximation compared to the thermal scale. When the magnetic scale is higher than the thermal scale the lowest Landau level (LLL) becomes effectively (1+1) dimensional strongly correlated system that provides a kinematical threshold based on the quark mass scale. Beyond this threshold the photon strikes the LLL and the spectral strength starts with a high value due to the dimensional reduction and then falls off with increase of the photon energy due to LLL dynamics in a strong field approximation. We have obtained analytically the dilepton production rates from LLL considering the lepton pair remains unaffected by the magnetic field when produced at the edge of a hot magnetized medium or affected by the magnetic field if produced inside a hot magnetized medium. For the later case the production rate is of O[|eB| 2 ] along with an additional kinematical threshold due to lepton mass than the former one. We have also investigated the electromagnetic screening by computing the Debye screening mass and it depends distinctively on three different scales (mass of the quasiquark, temperature and the magnetic field strength) of a hot magnetized system. The mass dependence of the Debye screening supports the occurrence of a magnetic catalysis effect in the strong field approximation.

Considering the general structure of the two point functions of quarks and gluons, we compute the free energy and pressure of a strongly magnetized hot and dense QCD matter created in heavy-ion collisions. In the presence of a strong magnetic field we found that the deconfined QCD matter exhibits a paramagnetic nature. One gets different pressures in directions parallel and perpendicular to the magnetic field due to the magnetization acquired by the system. We obtain both longitudinal and transverse pressures, and magnetization of hot deconfined QCD matter in the presence of the magnetic field. We have used hard thermal loop approximation for the heat bath. We obtained completely analytic expressions for pressure and magnetization under certain approximations. Various divergences appearing in free energy are regulated using appropriate counterterms. The obtained anisotropic pressure may be useful for a magnetohydrodynamics description of a hot and dense deconfined QCD matter produced in heavy-ion collisions.

Based on transversality condition of gauge boson self-energy we have systematically constructed the general structure of the gauge boson two-point functions using four linearly independent basis tensors in presence of a nontrivial background, i.e., hot magnetized material medium. The hard thermal loop approximation has been used for the heat bath to compute various form factors associated with the gauge boson's two point functions both in strong and weak field approximation. We have also analyzed the dispersion of a gauge boson (e.g., gluon) using the effective propagator both in strong and weak magnetic field approximation. The formalism is also applicable to QED. The presence of only thermal background leads to a longitudinal (plasmon) mode and a two-fold degenerate transverse mode. In presence of a hot magnetized background medium the degeneracy of the two transverse modes is lifted and one gets three quasiparticle modes. In weak field approximation one gets two transverse modes and one plasmon mode. On the other hand, in strong field approximation also one gets the three modes in Lowest Landau Level. The general structure of two-point function may be useful for computing the thermo-magnetic correction of various quantities associated with a gauge boson.

We consider our recently obtained general structure of two point (self-energy and propagator) functions of quarks and gluons in a nontrivial background like a heat bath and an external magnetic field. Based on this we have computed free energy and pressure of quarks and gluons for a magnetized hot and dense deconfined QCD matter in weak field approximation. For heat bath we have used hard thermal loop perturbation theory (HTLpt) in presence of finite chemical potential. For weak field approximations, the results are completely analytic and gauge independent but depends on the renormalization scale in addition to the temperature, chemical potential and the external magnetic field. We also discuss the modification of QCD Debye mass of such matter for an arbitrary magnetic field. An analytic expression for Debye mass is also obtained for both strong and weak field approximation. It is found to exhibit some interesting features depending upon the three different scales, i.e, the thermal quark mass, temperature and the strength of the magnetic field. The various divergences appearing in the quark and gluon free energies are regulated through appropriate counter terms. In weak field approximation, the low temperature behaviour of the pressure is found to strongly depend on the magnetic field than that at high temperature. We also discuss the specific problem with one-loop HTLpt associated with the over-counting of certain orders in coupling.