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Employing a field dependent three-momentum cut-off regularization technique,we study the phase structure and mesonic masses using the $2$-flavourNambu-Jona Lasinio model at finite temperature and density in presence ofarbitrary external magnetic field. This approach is then applied to incorporatethe effects of the anomalous magnetic moment(AMM) of quarks on constituentquark mass and thermodynamic observables as a function of temperature/baryonicdensity. The critical temperature for transition from chiral symmetry broken tothe restored phase is observed to decrease with the external magnetic field,which can be classified as inverse magnetic catalysis, while an oppositebehaviour is realized in the case of a vanishing magnetic moment, implyingmagnetic catalysis. These essential features are also reflected in the phasediagram. Furthermore, the properties of the low lying scalar and neutralpseudoscalar mesons are also studied in presence of a hot and dense magnetizedmedium including AMM of the quarks using random phase approximation. Fornon-zero values of magnetic field, we notice a sudden jump in the mass of theGoldstone mode at and above the Mott transition temperature which is found todecrease substantially with the increase in magnetic field when the AMM of thequarks are taken into consideration.

Effect of anomalous magnetic moment of quarks on the phase structure and mesonic properties in the NJL model