The density corrections, in terms of the isospin chemical potential µ I , to the mass of the pions are studied in the framework of the SU (2) low energy effective chiral lagrangian. The pion decay constant f π (T, µ I ) is also analized. As a function of temperature for µ I = 0, the mass remains quite stable, starting to grow for very high values of T , confirming previous results. However, there are interesting corrections to the mass when both effects (temperature and chemical potential) are simultaneously present. At zero temperature the π ± should condensate when µ I = ∓m π . This is not longer valid anymore at finite T . The mass of the π 0 acquires also a non trivial dependence on µ I due to the finite temperature.

We apply a recently constructed model of analytic QCD in the Operator Product Expansion (OPE) analysis of the τ lepton decay data in the V +A channel. The model has the running coupling A1(Q 2 ) with no unphysical singularities, i.e., it is analytic. It differs from the corresponding perturbative QCD coupling a(Q 2 ) at high squared momenta |Q 2 | by terms ∝ (1/Q 2 ) 5 , hence it does not contradict the ITEP OPE philosophy and can be consistently applied with OPE up to terms of dimension D = 8. In evaluations for the Adler function we use a Padé-related renormalizationscale-independent resummation, applicable in any analytic QCD model. Applying the Borel sum rules in the Q 2 plane along rays of the complex Borel scale and comparing with ALEPH data of 1998, we obtain the gluon condensate value (αs/π)G 2 = 0.0055 ± 0.0047 GeV

We study the thermo-magnetic properties of the strong coupling constant G and quark mass M entering the Nambu-Jona-Lasinio model. For this purpose, we compute the quark condensate and compare it to lattice QCD (LQCD) results to extract the behavior of G and M as functions of the magnetic field strength and temperature. We find that at zero temperature, where the LQCD condensate is found to monotonically increase with the field strength, M also increases whereas G remains approximately constant. However, for temperatures above the chiral/deconfinement phase transitions, where the LQCD condensate is found to monotonically decrease with increasing field, M and G also decrease monotonically. For finite temperatures, below the transition temperature, we find that both G and M initially grow and then decrease with increasing field strength. To study possible consequences of the extracted temperature and magnetic field dependence of G and M , we compute the pressure and compare to LQCD results, finding an excellent qualitative agreement. In particular, we show that the transverse pressure, as a function of the field strength, is always negative for temperatures below the transition temperature whereas it starts off being positive and then becomes negative for temperatures above the transition temperature, also in agreement with LQCD results. We also show that for the longitudinal pressure to agree with LQCD calculations, the system should be described as a diamagnet. We argue that the turnover of M and G as functions of temperature and field strength is a key element that drives the behavior of the quark condensate going across the transition temperature and provides clues for a better understanding of the inverse magnetic catalysis phenomenon.