Higher-order multipole amplitudes in charmonium radiative transitions

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Abstract

Using $24\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\equiv}\ensuremath{\psi}(2S)$ decays in CLEO-c, we have searched for higher multipole admixtures in electric-dipole-dominated radiative transitions in charmonia. We find good agreement between our data and theoretical predictions for magnetic quadrupole ($M2$) amplitudes in the transitions ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\chi}}_{c1,c2}$ and ${\ensuremath{\chi}}_{c1,c2}\ensuremath{\rightarrow}\ensuremath{\gamma}J/\ensuremath{\psi}$, in striking contrast to some previous measurements. Let ${b}_{2}^{J}$ and ${a}_{2}^{J}$ denote the normalized $M2$ amplitudes in the respective aforementioned decays, where the superscript $J$ refers to the angular momentum of the ${\ensuremath{\chi}}_{cJ}$. By performing unbinned maximum likelihood fits to full five-parameter angular distributions, we found the following values of $M2$ admixtures for ${J}_{\ensuremath{\chi}}=1$: ${a}_{2}^{J=1}=(\ensuremath{-}6.26\ifmmode\pm\else\textpm\fi{}0.63\ifmmode\pm\else\textpm\fi{}0.24)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ and ${b}_{2}^{J=1}=(2.76\ifmmode\pm\else\textpm\fi{}0.73\ifmmode\pm\else\textpm\fi{}0.23)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, which agree well with theoretical expectations for a vanishing anomalous magnetic moment of the charm quark. For ${J}_{\ensuremath{\chi}}=2$, if we fix the electric octupole ($E3$) amplitudes to zero as theory predicts for transitions between charmonium $S$ states and $P$ states, we find ${a}_{2}^{J=2}=(\ensuremath{-}9.3\ifmmode\pm\else\textpm\fi{}1.6\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ and ${b}_{2}^{J=2}=(1.0\ifmmode\pm\else\textpm\fi{}1.3\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$. If we allow for $E3$ amplitudes we find, with a four-parameter fit, ${a}_{2}^{J=2}=(\ensuremath{-}7.9\ifmmode\pm\else\textpm\fi{}1.9\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, ${b}_{2}^{J=2}=(0.2\ifmmode\pm\else\textpm\fi{}1.4\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, ${a}_{3}^{J=2}=(1.7\ifmmode\pm\else\textpm\fi{}1.4\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, and ${b}_{3}^{J=2}=(\ensuremath{-}0.8\ifmmode\pm\else\textpm\fi{}1.2\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$. We determine the ratios ${a}_{2}^{J=1}/{a}_{2}^{J=2}={0.67}_{\ensuremath{-}0.13}^{+0.19}$ and ${a}_{2}^{J=1}/{b}_{2}^{J=1}=\ensuremath{-}{2.27}_{\ensuremath{-}0.99}^{+0.57}$, where the theoretical predictions are independent of the charmed quark magnetic moment and are ${a}_{2}^{J=1}/{a}_{2}^{J=2}=0.676\ifmmode\pm\else\textpm\fi{}0.071$ and ${a}_{2}^{J=1}/{b}_{2}^{J=1}=\ensuremath{-}2.27\ifmmode\pm\else\textpm\fi{}0.16$.

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