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

We study the two flavour non-local Nambu\textemdash Jona-Lasinio (NJL) model in the presence of a magnetic field and explore the chiral crossover in presence of a non-local form of the 't Hooft determinant term. Its coupling is governed by a dimensionless parameter $c$. This term is responsible for the explicit breaking of $U(1)_A$ symmetry. We have attempted a systematic analysis of the model parameters by fitting to self-consistent lattice QCD calculations. Three parameters of the model are fixed by $eB=0$ results from published lattice QCD on the chiral condensate, the pion decay constant ($F_\pi$), and the pion mass ($m_\pi$). The difference of the $u$ and $d$ quark condensates in the 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 evidence that $c$ depends on $eB$. The crossover temperature decreases with increasing $eB$ only for condensate values at the lower end of the allowed values (as already seen in~\cite{Pagura:2016pwr}) and $F_\pi$ at the upper end of the allowed values. We further check our model predictions by calculating the topological susceptibility with the fitted $c$ values and comparing it with lattice results. Since the topological susceptibility is related to the extent of the $U(1)_A$ symmetry breaking, we find that it is sensitive to the value of $c$.