Pionic dispersion relations in the presence of a weak magnetic field

In this work, dispersion relations of ${\pi }^0$ and ${\pi }^{\pm }$ have been studied in vacuum in the limit of a weak external magnetic field using a phenomenological pion-nucleon $(\pi N)$ Lagrangian. For our purpose, we have calculated the results up to one loop order in self-energy diagrams with the pseudoscalar $(\mathrm{PS})$ and pseudovector $(\mathrm{PV})$ pion-nucleon interactions. By assuming a weak external magnetic field, it is seen that the effective mass of the pion gets explicit magnetic field dependence, and it is modified significantly for the case of PS coupling. However, for the PV coupling, only a modest increase in the effective mass is observed. These modified dispersion relations due to the presence of the external field can have substantial influence in the phenomenological aspect of the mesons, both in the context of neutron stars as well as in relativistic heavy ion collisions.

Figure 1

Figure (a) represents the one-loop self-energy diagram for ${\pi }^0$ and (b) represents the same for ${\pi }^{\pm }$.

Figure 2

Effective pion mass as a function of the magnetic field for PS coupling.

Figure 3

The left panel shows the dispersion relation for neutral pion with the $z$ component of momentum for pseudoscalar coupling. The right panel shows a similar plot for the case of charged pions.

Figure 4

Effective pion mass as a function of the magnetic field for PV coupling.

Figure 5

The left panel shows the dispersion relation for neutral pion with the $z$ component of momentum for pseudovector coupling. The right panel shows a similar plot for the case of charged pions.