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The effective photon-quark-antiquark ($\gamma q \overline{q}$) vertexfunction is evaluated at finite temperature in the presence of an arbitraryexternal magnetic field using the two-flavor gauged Nambu--Jona-Lasinio (NJL)model in the mean field approximation. The lowest order diagram contributing tothe magnetic form factor and the anomalous magnetic moment (AMM) of the quarksis calculated at finite temperature and external magnetic field using theimaginary time formalism of finite temperature field theory and the Schwingerproper time formalism. The Schwinger propagator including all the Landau levelswith non-zero AMM of the dressed quarks is considered while calculating theloop diagram. Using sharp as well as smooth three momentum cutoff, weregularize the UV divergences arising from the vertex function and theparameters of our model are chosen to reproduce the well known phenomenologicalquantities at zero temperature and zero magnetic field, such as pion-decayconstant ($f_\pi$), vacuum quark condensate, vacuum pion mass ($m_\pi$) as wellas the magnetic moments of proton and neutron. We then study the temperatureand magnetic field dependence of the AMM and constituent mass of the quark. Wefound that, the AMM as well as the constituent quark mass are large at thechiral symmetry broken phase in the low temperature region. Around thepseudo-chiral phase transition they decrease rapidly and at high temperaturesboth of them approach vanishingly small values in the symmetry restored phase.

Thermomagnetic modification of the anomalous magnetic moment of quarks using the NJL model