Resonance-continuum interference with a possible tensor $X(6900)$ in radiative double-charmonium production at $\sqrt{s}=10.58~{\rm GeV}$
We study radiative double-charmonium production, $e^+e^-\to HH\gamma$ with $H=J/\psi$ and $\eta_c$, at $\sqrt{s}=10.58~{\rm GeV}$, including a possible tensor contribution from the fully charmed state $X(6900)$. The nonresonant amplitudes are calculated within the color-singlet non-relativistic QCD framework, while the $J^{PC}=2^{++}$ resonance is combined coherently with the continuum. We find that continuum--resonance interference significantly modifies the invariant-mass distributions near $M_{HH}\simeq M_X$. The $J/\psi J/\psi\gamma$ channel exhibits a pronounced phase-dependent resonant structure, with a near-resonance cross section of $0.038$--$0.111~{\rm fb}$. In contrast, the $\eta_c\eta_c\gamma$ channel is continuum dominated and shows only a localized interference effect with an extremely suppressed production rate, rendering its experimental observation extremely challenging. These channel-dependent features arise from the different effective decay structures and demonstrate the importance of an amplitude-level treatment of the $X(6900)$ contribution.