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We present a calculation of the hindered M$1$ $\Upsilon(2S) \to \eta_b(1S)\gamma$ decay rate using lattice non-relativistic QCD. The calculation includesspin-dependent relativistic corrections to the NRQCD action through$\mathcal{O}(v^6)$ in the quark's relative velocity, relativistic correctionsto the leading order current which mediates the transition through the quark'smagnetic moment, radiative corrections to the leading spin-magnetic couplingand for the first time a full error budget. We also use gluon field ensemblesat multiple lattice spacing values, all of which include $u$, $d$, $s$ and $c$quark vacuum polarisation. Our result for the branching fraction is$\mathcal{B}(\Upsilon(2S)\to\eta_b(1S)\gamma) = 5.4(1.8)\times 10^{-4} $, whichagrees with the current experimental value.

Hindered M1 Radiative Decay of $\Upsilon(2S)$ from Lattice NRQCD