Determine the timelike X(6900) electromagnetic transition form factor

Determine the timelike $X(6900)\gamma^*\gamma$ transition form factor at the kinematic point $k^2=s>0$ relevant to radiative double-charmonium production, thereby replacing the phenomenological off-shell parametrization used for the resonance amplitude.

Background

The radiative process involves one time-like virtual photon with squared momentum k2=sk^2=s, whereas the available diphoton decay width constrains only the on-shell X(6900)γγX(6900)\gamma\gamma coupling. Consequently, the production calculation introduces a phenomenological factor Fexp⁡(s)=exp⁡(−s/ΛX2)F_{\exp}(s)=\exp(-s/\Lambda_X^2) with the benchmark choice ΛX=MX\Lambda_X=M_X. A determination of the timelike transition form factor would remove this principal modeling assumption and improve predictions for the resonance and interference contributions.

References

The diphoton width fixes only the on-shell normalization of the electromagnetic vertex. In the present process, the initial photon is time-like, with $k2=s>0$, whereas the final-state photon is on shell. Since the timelike $X\gamma*\gamma$ transition form factor is not presently known, we adopt the following phenomenological parametrization:

— Resonance-continuum interference with a possible tensor $X(6900)$ in radiative double-charmonium production at $\sqrt{s}=10.58~{\rm GeV}$  (2608.23408 - Jia et al., 24 Aug 2026) in Section II, subsection “Effective couplings and off-shell form factor”