Heavy quarkonium electric dipole transitions in non-relativistic quantum field theory

In this work we use the low-energy EFT called potential non-relativistic QCD (pNRQCD) to calculate the partial decay width of different $b\bar{b}$-states undergoing an electric dipole transition at next-to-next-to leading order (NNLO). Explicitly, the $\chi_{b J}$, for $J=0,1,2$, and the $h_b$ are investigated by computing the processes $\chi_{b J} \to \Upsilon + \gamma$ and $h_b \to \eta_b + \gamma$. Relativistic corrections of relative order $v^2$ to the leading electric dipole operator are included. The analysis separates those contributions that account for the electromagnetic interaction terms in the pNRQCD Lagrangian, which are $v^2$ suppressed, and those that account for quarkonium state corrections of relative order $v$ and $v^2$. Within the last ones, corrections come from higher order potentials ($\frac{1}{m}$ and $\frac{1}{m^2}$ terms), and from higher order Fock states which account for the coupling of the quark-antiquark state to other low-energy degrees of freedom and thus demand non-perturbative input. Finally, the experimentally known branching fractions are used to predict the total decay with of the respective initial states.