Kinematic Fingerprints of a Nucleon-Triggered $V_{\mathcal{B}}$ in Rare $\eta^{(\prime)}\to\pi^0(\eta)\gamma\gamma$ Decays on Nucleon Targets
The nucleon-triggered vector boson $V_{\mathcal B}$ is a hypothetical particle motivated by a statistically significant discrepancy between the KLOE measurement, $\mathrm{BR}^{\eta\rightarrow\pi^{0}\gamma\gamma}_{\text{KLOE}} = \left(0.98\pm0.11_{\text{stat}}\pm0.14_{\text{syst}}\right)\times10^{-4}$, and the current world average, $\left(2.55\pm0.22\right)\times10^{-4}$, dominated by MAMI photoproduction data. This hypothetical new interaction is activated by external nucleons, causing deviations from the Standard Model predictions, while leaving leptonic measurements, such as KLOE and BESIII, unaffected. We embed this mechanism in representative MAMI- and JEF-like photoproduction settings, deriving distinctive and directly testable kinematic and cut-dependent signatures to quantify small differences between the two regimes. One such signature is a small predicted population of events beyond the nominal on-shell boundaries of the $\gamma\gamma$ and $\pi^0\gamma$ spectra, containing 4--23 events in a sample of 1200, depending on the selection window. These events correlate with the recoil proton kinematics, and the polar angle relative to the on-shell hypothesis can reach $\lesssim -5^\circ$ 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, $\langle M_{\pi^0\gamma\gamma}\rangle-m_\eta\simeq +2$--$7~\mathrm{MeV}$. Finally, we identify representative $V_{\mathcal B}$-induced topologies that could contribute to recently reported discrepancies in the angular distributions of $\gamma d\to\pi^0\eta d$ and $\gamma d\to\pi^0\pi^0 d$, leaving their quantitative investigation to future work.