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"Hard-scattering" approach to very hindered magnetic-dipole transitions in quarkonium

For a class of hindered magnetic dipole ($M1$) transition processes, such as $Υ(3S)\to η_b+γ$ (the discovery channel of the $η_b$ meson), the emitted photon is rather energetic so that the traditional approaches based on multipole expansion may be invalidated. We propose that a "hard-scattering" picture, somewhat analogous to the pion electromagnetic form factor at large momentum transfer, may be more plausible to describe such types of transition processes. We work out a simple factorization formula at lowest order in the strong coupling constant, which involves convolution of the Schrödinger wave functions of quarkonia with a perturbatively calculable part induced by exchange of one semihard gluon between quark and antiquark. This formula, without any freely adjustable parameters, is found to agree with the measured rate of $Υ(3S)\to η_b+γ$ rather well, and can also reasonably account for other recently measured hindered $M1$ transition rates. The branching fractions of $Υ(4S)\to η_b^{(\prime)}+γ$ are also predicted.