  We compute the electric dipole transitions $\chi_{bJ}(1P)\to
\gamma\Upsilon(1S)$, with $J=0,1,2$, and $h_{b}(1P)\to \gamma\eta_{b}(1S)$ in a
model-independent way. We use potential non-relativistic QCD (pNRQCD) at weak
coupling with either the Coulomb potential or the complete static potential
incorporated in the leading order Hamiltonian. In the last case, the
perturbative series shows very mild scale dependence and a good convergence
pattern, allowing predictions for all the transition widths. Assuming
$\Lambda_{\text{QCD}} \ll mv^2$, the precision that we reach is
$k_{\gamma}^{3}/(mv)^{2} \times \mathcal{O}(v^{2})$, where $k_{\gamma}$ is the
photon energy, $m$ is the mass of the heavy quark and $v$ its relative
velocity. Our results are: $\Gamma(\chi_{b0}(1P)\to \gamma\Upsilon(1S)) =
28^{+2}_{-2}~\text{keV}$, $\Gamma(\chi_{b1}(1P)\to \gamma\Upsilon(1S)) =
37^{+2}_{-2}~\text{keV}$, $\Gamma(\chi_{b2}(1P)\to \gamma\Upsilon(1S)) =
45^{+3}_{-3}~\text{keV}$ and $\Gamma(h_b(1P)\to \gamma\eta_b(1S)) =
63^{+6}_{-6}~\text{keV}$.
