Determine the physical continuum–resonance relative phase

Determine the physical relative phase between the nonresonant color-singlet NRQCD continuum amplitude and the tensor $X(6900)$ resonance amplitude in the $e^+e^-\to HH\gamma$ channels, with $H=J/\psi$ or $\eta_c$, in order to predict the interference-modified invariant-mass line shapes.

Background

The full amplitude is formed by coherently adding the continuum and resonance contributions with a phase ϕX\phi_X. The numerical analysis samples benchmark values of this phase rather than determining it dynamically. Because the interference term depends linearly on the resonance amplitude, the unresolved phase materially affects both the apparent resonance yield and the shape of the HHHH invariant-mass distribution, and its physical interpretation may differ between the two final states.

References

Since the relative phase is not determined within the present effective framework, we consider

— 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 “Coherent invariant-mass distribution”