Kinematic Fingerprints of a Nucleon-Triggered $V_{\mathcal{B}}$ in Rare $η^{(\prime)}\toπ^0(η)γγ$ Decays on Nucleon Targets

The nucleon-triggered vector-boson scenario $V_{\mathcal B}$ was motivated by a statistically significant discrepancy between the KLOE measurement, $\mathrm{BR}^{η\rightarrowπ^{0}γγ}_{\text{KLOE}} = \left(0.98\pm0.11_{\text{stat}}\pm0.14_{\text{syst}}\right)\times10^{-4}$, and the current world average, $\left(2.55\pm0.22\right)\times10^{-4}$, dominated by MAMI photoproduction data. In this model, the new interaction is activated by external nucleons, which causes deviations from the Standard Model predictions, while leaving leptonic measurements, such as KLOE and BESIII, unaffected. We quantify the residual kinematic and cut-dependent effects that make the MAMI- and JEF-like predictions close but not identical due to their different production-energy regimes. The cleanest discriminator of the model is a kinematically forbidden tail in the $γγ$ and $π^0γ$ spectra, and for $η\rightarrowπ^0γγ$, and we predict 4--23 events in a sample of 1200, depending on the selection window, in these regions. These events correlate with the kinematics of the recoil proton, and the polar angle relative to the on-shell hypothesis can reach $\approx-5^\circ$ or below at MAMI energies, but only $\approx -0.2^\circ$ at JEF. For identical selection windows, the JEF-like effective branching fraction exceeds the MAMI-like prediction by less than $1\%$, while widening the window increases both predictions by up to $\sim4\%$. Both energy regimes produce a similar upward shift of the mean reconstructed mass, $\langle M_{π^0γγ}\rangle-m_η\simeq 2$--$7~\mathrm{MeV}$. Finally, we identify representative $V_{\mathcal B}$-induced topologies that could contribute to recently reported discrepancies in the angular distributions of $γd\toπ^0ηd$ and $γd\toπ^0π^0 d$, leaving their quantitative investigation to future work.