Probing the electromagnetic fields in ultra-relativistic collisions with leptons from $Z^0$ decay and charmed mesons

Ultra-relativistic heavy-ion collisions are expected to generate a huge electromagnetic field, $eB \approx 10^{18}\, Gauss$, that induces a splitting of the directed flow, $v_1=\left\langle p_x/p_T \right\rangle $, of charged particles and anti-particles. Such a splitting for charmed meson manifests even for neutral particle/anti-particles pairs ($D^0, \overline{D}^0$), hence being also a unique probe of the formation of the quark plasma phase. For the first time here we show that specific time evolution of the electromagnetic field may generate a $v_1(D^0) - v_1( \overline{D}^0)$ as huge as the one recently observed at LHC against early expectations. Within this new research topic we point out a novel measurement: the directed flow $v_1$ of leptons from $Z^0$ decay and its correlation to the $D$ mesons one. Such a correlation between the $\Delta v_1$ of $l^{+} - l^{-}$ and $D^0-\overline{D}^0$ would provide a strong probe of the electromagnetic origin of the splitting and hence of the formation of a quark plasma phase with charm quarks as degrees of freedom. The case of the $v_1({l^{\pm}})$ presents features due to the peculiar form of the $p_T$ spectrum never appreciated before in the study of heavy-ion collisions. We specifically predict a sign change of the $\Delta v_1(p_T)$ of leptons at $p_T=45$ GeV$/c$, that can be traced back to a universal relation of $\Delta v_1$ with the slope of the $p_T$ particle distribution and the integrated effect of the Lorentz force.