Relativistic and binding energy corrections to direct photon production inϒdecay

A systematic gauge-invariant method, which starts directly from QCD, is used to calculate the rate for an $\Upsilon$ meson to decay inclusively into a prompt photon. An expansion is made in the quark relative velocity $v$, which is a small natural parameter for heavy quark systems. Inclusion of these $O\left(v^2\right)$ corrections tends to increase the photon rate in the middle $z$ range and to lower it for larger $z$, a feature supported by the data.