Direct flow of heavy mesons as unique probe of the initial Electro-Magnetic fields in Ultra-Relativistic Heavy Ion collisions

In Ultra-relativistic Heavy-Ion Collisions (HICs) very strong initial electro-magnetic (E.M.) fields are created: the order of magnitude of the magnetic field is about $10^{19} \, Gauss$, the most intense field in the Universe, even larger than that of a magnetar. These fields rapidly decrease in time, inducing a drift of particles in the reaction plane. The resulting flow is odd under charge exchange and this allows to distinguish it from the large vorticity of the bulk matter due to the initial angular momentum conservation. Conjointly charm quarks, thanks to their large mass $M_{c}>>\Lambda_{QCD}$, are produced in hard partonic processes at formation time $\tau_f \approx 1\,/\,( 2M_{HQ} )$ which is comparable with the time scale when the E.M. field attains its maximum value. Moreover, with a mass of $M_c \approx 1.3 \,$ GeV there should be no mixing with the chiral magnetic dynamics and the condition $M_c \gg T$ allows charm quarks to have sufficiently large thermalization time, so that they can probe the entire phase-space evolution of the QGP retaining the initial kick given by the E.M. field. We show that such E.M. field entails a transverse motion of charm quarks resulting in a splitting of directed flow $v_1$ of $ D$ and $\bar{D}$ mesons of few percent, i.e. much larger compared to the measured pion one.