Determination of electric dipole transitions in heavy quarkonia using potential non-relativistic QCD

The electric dipole transitions ${\chi }_{bJ}(1P)\to \gamma \mathrm{{\rm Y}}(1S)$ with $J=0,1,2$ and $h_b(1P)\to \gamma {\eta }_b(1S)$ are computed using the weak-coupling version of a low-energy effective field theory named potential non-relativistic QCD (pNRQCD). In order to improve convergence and thus give firm predictions for the studied reactions, the full static potential is incorporated into the leading order Hamiltonian; moreover, we must handle properly renormalon effects and re-summation of large logarithms. The precision we reach is $k_{\gamma }^3\mathrm{/}(mv{)}^2\times \mathcal{O}(v^2)$, where $k_{\gamma }$ is the photon energy, $m$ is the mass of the heavy quark and $v$ its velocity. Our analysis separates those relativistic contributions that account for the electromagnetic interaction terms in the pNRQCD Lagrangian which are $v^2$ suppressed and those that account for wave function corrections of relative order $v^2$. Among the last ones, corrections from $1\mathrm{/}m$ and $1\mathrm{/}{m}^2$ potentials are computed, but not those coming from higher Fock states since they demand non-perturbative input and are ${\mathrm{\Lambda }}_{\text{QCD}}^2\mathrm{/}(mv{)}^2$ or ${\mathrm{\Lambda }}_{\text{QCD}}^3\mathrm{/}(m^3{v}^4)$ suppressed, at least, in the strict weak coupling regime. These proceedings are based on the forthcoming publication \cite{steinbeisser}