The phase between the three gluon and one photon amplitudes in quarkonium decays

The phase between three-gluon and one-photon amplitudes in ψ(2S) and ψ(3770) decays is analyzed. 1 Motivations It has been known that in J/ψ decays, the three gluon amplitude a3g and one-photon amplitude aγ are orthogonal for the decay modes 10 (90◦) [1], 1−0− (106 ± 10)◦ [2], 0−0− (89.6 ± 9.9)◦ [3], 1−1− (138± 37)◦ [4] and NN (89± 15)◦ [5]. J. M. Gérard and J. Weyers [6] augued that this large phase follows from the orthogonality of threegluon and one-photon virtual processes. The question arises: is this phase universal for quarkonium decays? How about ψ(2S), ψ(3770) and Υ(nS) decays? 2 Quarkonium produced in electron-positron colliding experiments Recently, more ψ(2S) data has been available. Most of the branching ratios are measured in ee colliding experiments. For these experiments, there are three diagrams [7, 8], as shown in Fig. 1, which contribute to the processes. Although such formulas were written in the early years after J/ψ was discovered, but e