The Roberge-Weiss transition for QCD in a magnetic background

We investigate how a magnetic background field influences the location and the nature of the Roberge-Weiss (RW) finite temperature transition for $N_f = 2+1$ QCD with physical quark masses. To that purpose, we perform numerical simulations of the finite temperature theory, discretized through stout staggered quarks and the tree-level improved Symanzik pure gauge action, considering two different values of the Euclidean temporal extent in lattice units, $N_t = 6, 8$. The RW transition temperature $T_{RW}$ decreases with $eB$, in particular it follows closely the behavior of the pseudo-critical QCD crossover temperature $T_{pc}$, so that $T_{RW} (eB) - T_{pc}(eB)$ is practically constant, within errors, for magnetic fields up to $eB \sim 1$ GeV$^2$; consistent results are found from the drop of the chiral condensate, which signals chiral symmetry restoration, leading also to the phenomenon of inverse magnetic catalysis above the transition. Moreover, we find that the magnetic field turns the RW transition from second order to first order, with a tri-critical magnetic field in-between 1 and 2.4 GeV$^2$, i.e. for magnetic fields substantially lower than those for which the standard QCD transition turns to first order.