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

Abstract: The nucleon-triggered vector-boson scenario $V_{\mathcal B}$ was motivated by a statistically significant discrepancy between the KLOE measurement, $$\mathrm{BR}^{{η\rightarrowπ^{{0}γγ}_{\text{KLOE}}}} = \left(0.98\pm0.11_{\text{stat}}\pm0.14_{\text{syst}}\right)\times10<sup>{-4}$$, and the current world average, $\left(2.55\pm0.22\right)\times10<sup>{-4}$, 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 $π<sup>0γ$ spectra, and for $η\rightarrowπ<sup>0γγ$, 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 $\approx-5<sup>\circ$ or below at MAMI energies, but only $\approx -0.2<sup>\circ$ 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 $γd\toπ<sup>0ηd$ and $γd\toπ^0π⁰ d$, leaving their quantitative investigation to future work.

• The paper develops off-shell predictions for a nucleon-triggered vector boson in eta to pi-zero gamma-gamma decay, including a fitted effective coupling of (8.54 plus or minus 1.08) times 10 to the minus 5 GeV to the minus 8.
• The model predicts a positive reconstructed-mass shift of about 1.5–7.4 MeV and a branching fraction near 2.63–2.75 times 10 to the minus 4, largely independent of MAMI-like or JEF-like beam energy.
• The paper identifies forbidden kinematic regions as a decisive test: approximately 4–23 events per 1,200 may cross these Rubicons, with recoil-proton angle correlations reaching about 5 degrees at MAMI energies.

Motivation: a production-dependent discrepancy

The paper addresses the statistically significant tension surrounding the rare decay $\eta\to\pi^0\gamma\gamma$. The KLOE-2 measurement, $$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$, is consistent with the earlier KLOE result but differs from the current world average of $(2.55\pm0.22)\times10^{-4}$ — dominated by MAMI photoproduction data — by approximately $5.5\sigma$, i.e. a factor of about 2.6. The nucleon-target measurements (MAMI photoproduction and Crystal Ball@AGS charge exchange) are mutually consistent at roughly $(2.2$–$2.7)\times10^{-4}$, while leptonic-production environments (KLOE via $e^+e^-\to\phi\to\eta\gamma$, BESIII via $J/\psi$ decay) agree with Standard Model predictions based on Vector Meson Dominance (VMD) plus the Linear Sigma Model (L$\sigma$M). The paper builds on the previously proposed nucleon-triggered vector-boson scenario $V_{\mathcal B}$ (2608.13537), which postulates a leptophobic vector boson with mass $$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$0–$$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$1 whose effective interaction,

$$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$2

is activated only by an external nucleon. This operator is generated at tree level by a mediator realization involving a nucleophilic scalar $$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$3, a dark pion $$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$4, and a mixed dark anomaly term of hidden-valley type; the vacuum condition $$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$5 forbids the nucleon-independent mixing that would affect KLOE or BESIII.

The present work goes beyond the on-shell approximation used previously ($$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$6) and quantifies the off-shell kinematic fingerprints of the mechanism, their dependence on production energy (MAMI-like $$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$7 versus JEF-like $$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$8), and on invariant-mass selection cuts.

Pole–continuum decomposition and off-shell kinematics

The total amplitude is decomposed into a Standard Model pole term, its interference with $$\mathrm{BR}=(0.98\pm0.11_{\rm stat}\pm0.14_{\rm syst})\times10^{-4}$$9, and the pure continuum $(2.55\pm0.22)\times10^{-4}$0. Using the Sokhotski–Plemelj identity in the narrow-width limit, both the resonant SM term and the interference are shown to be localized at the $(2.55\pm0.22)\times10^{-4}$1 pole; the principal-value part of the interference is odd in $(2.55\pm0.22)\times10^{-4}$2 and cancels under symmetric mass windows, an approximation the authors adopt explicitly. Consequently, the only unsuppressed contribution extending beyond the on-shell support is the continuum term $(2.55\pm0.22)\times10^{-4}$3, which permits four-momentum exchange between the reconstructed three-body subsystem $(2.55\pm0.22)\times10^{-4}$4 and the recoil proton, so that $(2.55\pm0.22)\times10^{-4}$5 varies continuously within the selection window $(2.55\pm0.22)\times10^{-4}$6.

Using recursive phase-space factorization, the effective continuum width is computed as an average of the running width $(2.55\pm0.22)\times10^{-4}$7 over the window weighted by the production two-body phase space. Since $(2.55\pm0.22)\times10^{-4}$8 grows with $(2.55\pm0.22)\times10^{-4}$9 faster than the production phase space decreases, the mean reconstructed mass is predicted to shift upward, $5.5\sigma$0.

Kinematic Rubicon as a discriminator

The central observable is what the paper terms the "kinematic Rubicon": regions of the $5.5\sigma$1 and $5.5\sigma$2 invariant-mass spectra that lie beyond the on-shell endpoints and therefore have exactly zero support for a genuine on-shell pole. Analytically, the thresholds for simultaneous crossing of two Rubicons require $5.5\sigma$3 values of 146–310 MeV depending on the channel, so for the adopted windows $5.5\sigma$4 the three forbidden regions are mutually exclusive and their event fractions add without double counting. A further analytic result shows that crossing the $5.5\sigma$5 Rubicon forces the second photon to be soft, with maximum energy proportional to the off-shell displacement $5.5\sigma$6, producing a phase-space suppression of order $5.5\sigma$7; no such suppression applies to the $5.5\sigma$8 Rubicon. Numerically this asymmetry is confirmed: the $5.5\sigma$9 forbidden fraction exceeds the $(2.2$0 one by roughly an order of magnitude.

Numerical predictions

The single model parameter is refitted to the MAMI diphoton spectrum, yielding $(2.2$1, about 10% below the value reported previously owing to a corrected numerical bug — a correction the authors state does not alter qualitative conclusions. In the on-shell approximation, the interference-dominated pole component accounts for $(2.2$2 of the Dalitz-integrated rate for $(2.2$3, $(2.2$4 for $(2.2$5, and $(2.2$6 for $(2.2$7; thus the branching-fraction enhancement arises mainly through interference, while the subdominant continuum generates all off-shell signatures.

Key quantitative results for $(2.2$8:

Quantity	JEF-like (11 GeV)	MAMI-like (1.4 GeV)
Branching ratio range ($(2.2$9–100 MeV)	$2.7)\times10^{-4}$0–$2.7)\times10^{-4}$1	$2.7)\times10^{-4}$2–$2.7)\times10^{-4}$3
Mean mass shift $2.7)\times10^{-4}$4	$2.7)\times10^{-4}$5 to $2.7)\times10^{-4}$6 MeV	$2.7)\times10^{-4}$7 to $2.7)\times10^{-4}$8 MeV
Combined Rubicon fraction $2.7)\times10^{-4}$9	$e^+e^-\to\phi\to\eta\gamma$0–$e^+e^-\to\phi\to\eta\gamma$1	$e^+e^-\to\phi\to\eta\gamma$2–$e^+e^-\to\phi\to\eta\gamma$3
Rubicon events per 1200	$e^+e^-\to\phi\to\eta\gamma$4–$e^+e^-\to\phi\to\eta\gamma$5	$e^+e^-\to\phi\to\eta\gamma$6–$e^+e^-\to\phi\to\eta\gamma$7

For identical selection windows, JEF- and MAMI-like effective branching fractions differ by less than 1%, while widening the window raises both by up to ~4%. For $e^+e^-\to\phi\to\eta\gamma$8, the effects are suppressed by roughly two orders of magnitude — fewer than one Rubicon-crossing event even in a 3500-event sample at $e^+e^-\to\phi\to\eta\gamma$9 — because the $J/\psi$0 width increment (~eV scale) is small against the keV-scale resonant $J/\psi$1-exchange SM amplitude. The $J/\psi$2 channel is negligible. These results imply that $J/\psi$3 is the decisive discovery channel, combining an order-one rate enhancement, a measurable several-MeV upward mass shift, and roughly ten forbidden-region events in a realistic sample.

Recoil-proton correlations and search strategy

Because the continuum topology transfers four-momentum to the recoil proton, Rubicon-crossing events correlate with recoil-proton kinematics through the relation between $J/\psi$4, $J/\psi$5, and $J/\psi$6. The angular displacement relative to the on-shell hypothesis scales strongly with beam energy regime: it can reach approximately $J/\psi$7 or larger at MAMI-like energies near the upper end of the accessible $J/\psi$8 interval, but only about $J/\psi$9 to $\sigma$0 at JEF-like energies. Since detector-induced photon-energy migration is generically uncorrelated with recoil-proton angles, this correlation offers a discriminator against experimental artifacts. The paper recommends reanalyzing existing MAMI data with $\sigma$1 left unconstrained — noting that the original MAMI extraction effectively imposed an on-shell-$\sigma$2 signal template — and performing parallel on-shell and unconstrained extractions at JEF, together with a scan over $\sigma$3, since imposing $\sigma$4 would suppress precisely the cut dependence that discriminates pole from continuum.

Connection to coherent deuteron photoproduction

The same effective operator permits $\sigma$5-mediated two-nucleon-exchange amplitudes in coherent $\sigma$6 and $\sigma$7, channels exhibiting unexpectedly flat deuteron angular distributions that conventional impulse and rescattering calculations fail to reproduce. The paper presents representative topologies in which two insertions of the $\sigma$8 interaction connect the two nucleon lines, redistributing momentum transfer, but explicitly defers the quantitative calculation and interference assessment to future work; these channels are offered as qualitative, not established, corroboration.

Limitations and open questions

Several caveats bear directly on the results. The neglect of the principal-value interference term rests on the assumption that $\sigma$9 varies slowly across the resonance and on symmetric mass windows; asymmetric cuts would reintroduce it. The prediction that JEF and MAMI branching fractions agree to better than 1% assumes identical selection windows, whereas real experiments use different criteria — the model predicts approximate equality of extracted branching fractions, so a KLOE-like JEF result would rule out or severely constrain the scenario, while a MAMI-compatible result would confirm only the qualitative pattern, not the model itself. The claim that detector migrations are uncorrelated with recoil-proton angles is asserted rather than demonstrated for specific detector systems. The deuteron-channel connection remains schematic, and the predicted enhancement of the effective branching fraction with nuclear mass number $V_{\mathcal B}$0 (relevant to prospective GlueX heavy-target tests) is stated without a quantitative calculation. Finally, the fitted coupling carries a ~13% uncertainty propagated from the MAMI spectrum, and the entire framework presumes the KLOE–MAMI discrepancy is physical rather than systematic.

Conclusion

This work converts the nucleon-triggered $V_{\mathcal B}$1 hypothesis from an on-shell rate argument into a set of falsifiable, cut-dependent kinematic predictions: a positive mean reconstructed-mass shift of 2–7 MeV, a 3–4% growth of the effective yield with window half-width, and — most distinctively — 4–23 events per 1200 in spectrally forbidden regions correlated with recoil-proton angles up to $V_{\mathcal B}$2 at MAMI kinematics. Because a genuine on-shell pole has identically zero support there, observation or exclusion of these events at the predicted rate constitutes the cleanest available test of the model, with the ongoing JEF program and a BESIII measurement of $V_{\mathcal B}$3 providing complementary, largely orthogonal discrimination.