Strong electromagnetic fields are expected to be generated in off-central relativistic heavy ion collisions, which can induce a splitting of the directed flow of charged particles and anti-particles ($\Delta v_1$). Such a splitting manifests even for neutral charmed mesons pairs ($D^0,\overline{D}^0$), hence being a direct probe of the formation of deconfined phase with charm quarks as degree of freedom.

In the limit of large $p_T$ and weak interaction with the QGP, a formula of $\Delta v_1(p_T,y_z)$ of charged particles and anti-particles as a function of $p_T$ and rapidity $y_z$ can be obtained, which is found to be related to the spectra of charged particles and the integrated effect of the Lorentz force. This formula is expected to be valid to heavy quarks and leptons at high $p_T$, where the modification to their equations of motion due to the interaction with both QGP and electromagnetic fields is small, and should have a general application. We also proposed a measurement of $\Delta v_1(p_T,y_z)$ of leptons from $Z^0$ decay and its correlation to that of $D$ mesons, which would be a strong probe determining whether the large splitting measured in experiments has the electromagnetic origin.