The signature of charge dependent directed flow observables by electromagnetic fields in heavy ion collisions

We discuss the generation of the directed flow v1(pT, yz) induced by the electromagnetic field as a function of pT and yz. Despite the complex dynamics of charged particles due to strong interactions generating several anisotropies in the azimuthal angle, it is possible at pT> m to directly correlate the splitting in v1 of heavy quarks with different charges to some main features of the magnetic field, and in particular its values at formation and freeze-out time. We further found that the slope of the splitting dΔv1/dyz|yz=0 of positively and negatively charged particles at high pT can be formulated as dΔv1/dyz|yz=0=-α∂lnf∂pT+2α-βpT, where f is the pT spectra of the charged particles and the constants α and β (order of MeV) are constrained by the y component of magnetic fields and the sign of α is simply determined by the difference Δ [tBy(t)] in the center of colliding systems at the formation time of particles and at the time when particles leave the effective range of electromagnetic fields or freeze out. The formula is derived from general considerations and is confirmed by several related numerical simulations; it supplies a useful guide to quantify the effect of different magnetic field configurations and provides an evidence of why the measurement of Δ v1 of charm, bottom and leptons from Z decay and their correlations are a powerful probe of the initial e.m. fields in ultra-relativistic collisions.