Measurement of the direct photon momentum spectrum in Υ(1S), Υ(2S), and Υ(3S) decays

Using data taken with the CLEO III detector at the Cornell Electron Storage Ring, we have investigated the direct photon spectrum in the decays $\ensuremath{\Upsilon}(1\mathrm{S})\ensuremath{\rightarrow}\ensuremath{\gamma}gg$, $\ensuremath{\Upsilon}(2\mathrm{S})\ensuremath{\rightarrow}\ensuremath{\gamma}gg$, $\ensuremath{\Upsilon}(3\mathrm{S})\ensuremath{\rightarrow}\ensuremath{\gamma}gg$. The latter two of these are first measurements. Our analysis procedures differ from previous ones in the following ways: (a) background estimates (primarily from ${\ensuremath{\pi}}^{0}$ decays) are based on isospin symmetry rather than a determination of the ${\ensuremath{\pi}}^{0}$ spectrum, which permits measurement of the $\ensuremath{\Upsilon}(2\mathrm{S})$ and $\ensuremath{\Upsilon}(3\mathrm{S})$ direct photon spectra without explicit corrections for ${\ensuremath{\pi}}^{0}$ backgrounds from, e.g., ${\ensuremath{\chi}}_{bJ}$ states, (b) we estimate the branching fractions with a parametrized functional form (exponential) used for the background, and c) we use the high-statistics sample of $\ensuremath{\Upsilon}(2\mathrm{S})\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\Upsilon}(1\mathrm{S})$ to obtain a tagged sample of $\ensuremath{\Upsilon}(1\mathrm{S})\ensuremath{\rightarrow}\ensuremath{\gamma}+X$ events, for which there are no QED backgrounds. We determine values for the ratio of the inclusive direct photon decay rate to that of the dominant three-gluon decay $\ensuremath{\Upsilon}\ensuremath{\rightarrow}ggg\text{ }\text{ }({R}_{\ensuremath{\gamma}}=B(gg\ensuremath{\gamma})/B(ggg))$ to be ${R}_{\ensuremath{\gamma}}(1\mathrm{S})=(2.70\ifmmode\pm\else\textpm\fi{}0.01\ifmmode\pm\else\textpm\fi{}0.13\ifmmode\pm\else\textpm\fi{}0.24)%$, ${R}_{\ensuremath{\gamma}}(2\mathrm{S})=(3.18\ifmmode\pm\else\textpm\fi{}0.04\ifmmode\pm\else\textpm\fi{}0.22\ifmmode\pm\else\textpm\fi{}0.41)%$, and ${R}_{\ensuremath{\gamma}}(3\mathrm{S})=(2.72\ifmmode\pm\else\textpm\fi{}0.06\ifmmode\pm\else\textpm\fi{}0.32\ifmmode\pm\else\textpm\fi{}0.37)%$, where the errors shown are statistical, systematic, and theoretical model dependent, respectively. Given a value of ${Q}^{2}$, one can estimate a value for the strong coupling constant ${\ensuremath{\alpha}}_{s}({Q}^{2})$ from ${R}_{\ensuremath{\gamma}}$.