Production-Dependent Interpretation of the KLOE--MAMI $\eta \text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{\pi }^0\gamma \gamma$ Tension via the Nucleon-Triggered Leptophobic Vector $V_{\text{\hspace{0.17em}⁣}\mathcal{B}}$

The recent KLOE measurement ${\mathrm{B}\mathrm{R}}_{\text{KLOE}}^{\eta \to {\pi }^0\gamma \gamma }=(0.98\pm {0.11}_{\text{stat}}\pm {0.14}_{\text{syst}})\times {10}^{-4}$ is less than half the current world average, $(2.55\pm 0.22)\times {10}^{-4}$, dominated by MAMI photoproduction data. We show that this $\approx 5.5\sigma$ discrepancy can be resolved by the new leptophobic, nucleon-triggered vector particle $V_{\mathcal{B}}$ with $1.5\text{ GeV}\lesssim m_{V_{\text{\hspace{0.17em}⁣}\mathcal{B}}}\lesssim 5\text{ GeV}$, coupled via the effective operator $\left(\bar{N}N\right){\tilde{V}}_{\mathcal{B}}^{\mu \nu }{F}_{\mu \nu }P$. This interaction modifies the ${\eta }^{(\mathrm{\prime })}\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{\pi }^0\left(\eta \right)\gamma \gamma$ decay rates $\mathit{\text{only}}$ in the processes involving an external nucleon current, $\gamma p\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{\eta }^{(\mathrm{\prime })}p$ and ${\pi }^{-}p\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{\eta }^{(\mathrm{\prime })}n$, but leaves purely leptonic production channels, such as $e^{+}{e}^{-}\to \phi \to {\eta }^{(\mathrm{\prime })}\gamma$ at KLOE and $e^{+}{e}^{-}\to J\mathrm{/}\psi \to \gamma {\eta }^{(\mathrm{\prime })}$ at BESIII, Standard-Model-like. The same mechanism predicts a $\approx 10\mathrm{\%}$ nucleon-triggered enhancement of the ${\eta }^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{\pi }^0\gamma \gamma$ decay rate and a negligible shift for ${\eta }^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}\eta \gamma \gamma$, together with an ${\mathrm{A}}^2$-scaling boost if produced on heavy nuclei instead of protons. $V_{\text{\hspace{0.17em}⁣}\mathcal{B}}$ can be searched directly in $2\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}3$ photoproduction, for example, $\gamma p\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{V}_{\text{\hspace{0.17em}⁣}\mathcal{B}}{\pi }^{0}p$. An integrated experimental program that compares ${\eta }^{(\mathrm{\prime })}\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{\pi }^0(\eta )\gamma \gamma$ in the presence of external nucleon currents with purely leptonic production, and conducts direct photoproduction searches for a GeV-scale vector, can decisively confirm or exclude our nucleon-rescaled, leptophobic-vector interpretation of the KLOE–MAMI discrepancy.