Radiative $E1$ transitions between $^3P_1$ and $^3S_1$ quarkonium states

In this work we study the E1 decay processes, $^3P_1$ $\rightarrow$ $^3S_1\gamma$, and $^3S_1$ $\rightarrow$ $^3P_1\gamma$ in the framework of Bethe-Salpeter equation and calculate their decay widths. We have used algebraic forms of Salpeter wave functions obtained through analytic solutions of mass spectral equations for ground and excited states of $^3S_1$, and $^3P_1$ equal mass quarkonia in approximate harmonic oscillator basis to do analytic calculations of their decay widths. These decay widths have been compared with data and other models.