Delayed versus accelerated quarkonium formation in a magnetic field

Formation time of heavy quarkonia in a homogeneous magnetic field is analyzed by using a phenomenological ansatz of the vector current correlator. Because the existence of a magnetic field mixes vector quarkonia ($J/\psi , {\psi }^{\prime }$) and their pseudoscalar partners (${\eta }_c, {\eta }_c^{\prime }$), the properties of the quarkonia can be modified through such a spin mixing. This means that the formation time of quarkonia is also changed by the magnetic field. We show the formation time of vector quarkonia is delayed by an idealized constant magnetic field, where the formation time of the excited state becomes longer than that of the ground state. As a more realistic situation in heavy-ion collisions, effects by a time-dependent magnetic field are also discussed, where delayed formation of $J/\psi$ and ${\psi }^{\prime }$ and very early formation of ${\eta }_c$ and ${\eta }_c^{\prime }$ are found.

Figure 1

Mixing angles of charmonia in a magnetic field, which are defined by Eqs. (13) and (14).

Figure 2

Examples of fraction $F_i$ and distribution $P_i$ for charmonia in a constant magnetic field. Upper: $J/\psi$ in vacuum and $J/\psi -{\eta }_c$ at $eB=0.2{\mathrm{GeV}}^2$. Lower: ${\psi }^{\prime }$ in vacuum and ${\psi }^{\prime }-{\eta }_c^{\prime }$ at $eB=0.2{\mathrm{GeV}}^2$.

Figure 3

Quarkonium formation time ${\langle {\tau }_{\mathrm{form}}\rangle }_i$ in a constant magnetic field.

Figure 4

Normalized fraction $F_i^{\mathrm{nor}}$ and distribution $P_i^{\mathrm{nor}}$, defined by Eqs. (17, 18, 19, 20), for charmonia in a time-dependent magnetic field at RHIC. The black dotted line is the time dependence of magnetic field, given by Eq. (16). Upper: $J/\psi$ and ${\psi }^{\prime }$. The gray dashed (solid) line stands for $F_i$ ($P_i$) in vacuum. Lower: ${\eta }_c$ and ${\eta }_c^{\prime }$.