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### Description

The sensitivity of the rare decays $\eta^{(\prime)}\to\pi^{0}\gamma\gamma$ and $\eta^{\prime}\to\eta\gamma\gamma$ to signatures of a leptophobic $B$ boson in the MeV-GeV mass range is analyzed in this work.

By adding an explicit $B$-boson resonance exchange, $\eta\to B\gamma\to\pi^{0}\gamma\gamma$, to the Standard Model contributions from vector and scalar meson exchanges,and employing experimental data for the associated branching ratios,it allows us to improve the current constraints on the $B$-boson mass $m_{B}$ and coupling to Standard Model particles $\alpha_{B}$.

From these constraints and the analysis of the available experimental $\gamma\gamma$ invariant mass distribution, we show that a $B$-boson signature in the resonant mass range $m_{\pi^{0}}\leq m_{B}\leq m_{\eta}$ is strongly suppressed and would be very difficult to experimentally identify, assuming that the leptophobic $B$ boson only decays to Standard Model particles.

In contrast, the limits outside this mass window are less stringent and the corresponding $t$- and $u$-channel signatures may still be observable in the data, as it occurs with the nonresonant Standard Model $\rho$, $\omega$ and $\phi$ meson exchanges.

In addition, we make use of experimental data from the $\eta^{\prime}\to\pi^{0}\gamma\gamma$ and $\eta^{\prime}\to\eta\gamma\gamma$ decays to explore larger $B$-boson masses.

Our results are relevant for the $B$-boson search programs at existing and forthcoming light-meson facilities, such as KLOE(-II) and Jefferson Lab Eta Factory experiments.