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We study the two flavour non-local Nambu\textemdash Jona-Lasinio (NJL) modelin the presence of a magnetic field and explore the chiral crossover inpresence of a non-local form of the 't Hooft determinant term. Its coupling isgoverned by a dimensionless parameter $c$. This term is responsible for theexplicit breaking of $U(1)_A$ symmetry. We have attempted a systematic analysisof the model parameters by fitting to self-consistent lattice QCD calculations.Three parameters of the model are fixed by $eB=0$ results from publishedlattice QCD on the chiral condensate, the pion decay constant ($F_\pi$), andthe pion mass ($m_\pi$). The difference of the $u$ and $d$ quark condensates inthe presence of a magnetic field ($eB$) is quite sensitive to $c$ and we fix$c$ using published lattice QCD results for this observable. We see no evidencethat $c$ depends on $eB$. The crossover temperature decreases with increasing$eB$ only for condensate values at the lower end of the allowed values (asalready seen in~\cite{Pagura:2016pwr}) and $F_\pi$ at the upper end of theallowed values. We further check our model predictions by calculating thetopological susceptibility with the fitted $c$ values and comparing it withlattice results. Since the topological susceptibility is related to the extentof the $U(1)_A$ symmetry breaking, we find that it is sensitive to the value of$c$.

Studying explicit $U(1)_A$ symmetry breaking in hot and magnetised two flavour non-local NJL model constrained using lattice results