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.Comment: 9 pages, 11 figures. Improved analysis separating the vacuum,
magnetic and thermo-magnetic contributions and re-scaling of the model's
critical temperature. Added analysis on the behavior of the longitudinal and
transverse pressures and its relation to the diamagnetic properties both
below and above the critical temperature. Version to appear in Phys. Rev.
