Colour-Octet Effects in Radiative $\Upsilon$ Decays

Phys.Rev.D59:074006,1999 We investigate the effects of colour-octet contributions to the radiative $\Upsilon$ decay within the Bodwin, Braaten and Lepage NRQCD factorization framework. Photons coming both from the coupling to hard processes (`direct') and by collinear emission from light quarks (`fragmentation') are consistently included at next-to-leading order (NLO) in $\alpha_s$. An estimate for the non-perturbative matrix elements which enter in the final result is then obtained. By comparing the NRQCD prediction at NLO for total decay rates with the experimental data, it is found that the non-perturbative parameters must be smaller than expected from the na\"\i ve scaling rules of NRQCD. Nevertheless, colour-octet contributions to the shape of the photon spectrum turn out to be significant.