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The $η_cγγ^*$ transition form factor in the decay $η_c\to γ\ell^+\ell^-$ and in the crossed channels $γe^-\to η_c e^- $ and $e^+e^-\toη_cγ

The Dalitz distribution and the effective mass spectrum of the lepton pair produced in the decay $η_c\to γ\ell^+ \ell^-$, as well as the differential cross sections for the crossed channels $γe^-\to η_c e^- $ and $e^+ e^-\to η_c γ$, are calculated in terms of the sole electromagnetic transition form factor for the vertex $η_c\toγγ^*$, where $γ^*$ is a virtual photon with a space-like (for $γe^- \to η_c e^- $) or a time-like (for $e^+ e^-\to η_c γ$ and $η_c\toγ\ell^+\ell^-$) four-momentum. Explicit parametrisations of the transition form factor are found in two different approaches, namely in Vector Dominance Model and in a QCD-inspired consideration on the basis of the process $c \bar{c}\to γ\ell^+ \ell^-$, with zero relative three-momentum of the annihilating quarks.