Spectral properties of the $\rho$ meson in a magnetic field

We calculate the rho meson mass in a weak magnetic field using effective $\rho \pi \pi$ interaction. It is seen that both ${\rho }^0$ and ${\rho }^{\pm }$ masses decrease with the magnetic field in vacuum. The $\rho$ meson dispersion relation has been calculated and shown to be different for ${\rho }^0$ and ${\rho }^{\pm }$. We also calculate the $\rho \pi \pi$ decay width and spectral functions of ${\rho }^0$ and ${\rho }^{\pm }$. The width is seen to decrease with $eB$ and the spectral functions become narrower.

Figure 1

Feynman diagrams for the one-loop self-energy of $\rho$.

Figure 2

Effective mass variations for $\rho \pi \pi$ coupling as a function of $eB$. Both of the modes show a decrease in effective mass of ${\rho }^0$ and ${\rho }^{\pm }$ with the increasing external field. The ratio plotted in the inset demonstrates the difference between the two modes.

Figure 3

Dispersion relations of ${\rho }^0$ and ${\rho }^{\pm }$ for two different values of $eB$ in the weak field regime. The left panel shows the variation with $k_z$ for a fixed value of $k_{\mathrm{per}}$. The right panel instead shows the dispersion as a function of $k_{\mathrm{per}}$ keeping the $k_z$ fixed.

Figure 4

The left panel shows the variation of decay width of $\rho$ mesons with external magnetic field. ${\mathrm{\Gamma }}_{\rho }$ for both ${\rho }^0$ and ${\rho }^{\pm }$ decreases with magnetic field but remains finite even at a maximum value of $eB=0.2{\mathrm{GeV}}^2$. Spectral functions are plotted in the right panel for $eB=\{0,0.1,0.2\}{\mathrm{GeV}}^2$. It becomes narrower and taller as it shifts towards the lower values of $k_0$.