Radiative $E1$ transitions between ${}^3{P}_1$ and ${}^3{S}_1$ quarkonium states

In this work we study the E1 decay processes, ${}^3{P}_1$ $\to$ ${}^3{S}_1\gamma$, and ${}^3{S}_1$ $\to$ ${}^3{P}_1\gamma$ 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 ${}^3{S}_1$, and ${}^3{P}_1$ 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.