Searching for neutral state in the rare decay $J/\psi \rightarrow e^+ e^- \phi$

Ahmed Rashed

REVIEW 3 major objections 5 minor 31 references

Searching for neutral state in the rare decay $J/\psi \rightarrow e^+ e^- \phi$

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The rare decay $J/psi \to e^+e^-\phi$ cannot expose a dark photon or dark $Z$: allowed mixing keeps the new-physics signal below the experimental sensitivity.

desk verdict The negative dark-sector result is probably right, but the calculation as written doesn't show it: the new-state propagator is dropped without justification, and Eq. (3) is garbled. read the letter →

arxiv 2412.09926 v1 pith:EQFKDQGL submitted 2024-12-13 hep-ph

Ahmed Rashed This is my paper · ORCID

classification hep-ph

keywords charmoniumdecaydarkphotonZvectormesondominancerarebranchingratioStandardModelnewphysics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether the rare decay $J/\psi \to e^+e^-\phi$ could expose a neutral state from a dark sector: a dark photon or a dark $Z$. It computes the Standard Model branching ratio to be about $2.28 \times 10^{-8}$, dominated by electromagnetic transitions through vector meson dominance, with hadronic loop contributions from $f_0(980)$, $\eta$, and $\eta'$ smaller by at least three orders of magnitude. It then adds dark photon and dark $Z$ amplitudes at the largest mixing values current constraints allow, $\varepsilon = 2 \times 10^{-2}$ and $\varepsilon_Z = 10^{-5}$, and finds the total branching ratio stays far below the experimental upper limit of $1.2 \times 10^{-7}$. The paper concludes that, with present data, this channel cannot distinguish the Standard Model from these two dark-sector models, so observing a signal from them here is out of reach.

What carries the argument

The machinery is the $J/\psi \to e^+e^-\phi$ transition amplitude built from vector meson dominance: the $J/\psi$ and $\phi$ each couple to a virtual photon through effective couplings $g_{\psi\gamma}$ and $g_{\phi\gamma}$, and the photon converts into the $e^+e^-$ pair. The dark photon and dark $Z$ amplitudes take the same Lorentz structure as the electromagnetic amplitude, with each fermion-photon vertex replaced by an $\varepsilon$- or $\varepsilon_Z$-scaled vertex and with the dark mediator propagator treated as effectively massless. Because of that similarity and because the dark-state width is far smaller than its mass, the new-physics rate is independent of the mediator mass $M_x$ over the range $2m_\mu \lesssim M_x \lesssim M_B - M_K$ considered here. The result is a tiny additive shift to the SM branching ratio that current constraints prevent from reaching the experimental limit.

What would settle it

Look for a narrow enhancement in the $e^+e^-$ invariant-mass distribution of $J/\psi \to e^+e^-\phi$ at a high-statistics experiment, or a branching ratio approaching the $1.2 \times 10^{-7}$ upper limit; either would contradict the paper's claim that no dark photon or dark $Z$ signal is observable. Equivalently, recompute the amplitude with a finite mediator mass $M_x$ and check whether any allowed mass makes the total branching ratio reach the experimental limit.

Watch

Extended reading notes

Core claim

The central result is a null result with a specified mechanism. The SM amplitude for $J/\psi \to e^+e^-\phi$ is determined by vector-meson-dominance couplings $g_{\psi\gamma} = 0.150~\mathrm{GeV}^2$ and $g_{\phi\gamma} = 0.013~\mathrm{GeV}^2$, yielding a branching ratio of about $2.28 \times 10^{-8}$, while hadronic loops are suppressed by more than three orders of magnitude. The dark photon and dark $Z$ amplitudes are copies of this electromagnetic amplitude with each photon-fermion vertex rescaled by $\varepsilon$ or by the dark-$Z$ couplings, and at the benchmark values $\varepsilon = 2 \times 10^{-2}$ and $\varepsilon_Z = 10^{-5}$ the total branching ratio remains essentially unchanged. Because existing constraints on the mixing parameters are so tight, the paper concludes that a large enhancement up to the experimental limit of $1.2 \times 10^{-7}$ is not possible, making this decay an unpromising discovery channel for the dark photon and dark $Z$.

Load-bearing premise

The calculation assumes the new particle acts as a massless mediator, so the dark photon and dark $Z$ rates do not depend on the mediator mass; if the mass is comparable to the momenta in the decay, the rates and the sensitivity conclusion could change.

Editorial extensions

If this is right

The Standard Model prediction for $J/\psi \to e^+e^-\phi$ is stable at about $2.28 \times 10^{-8}$, since hadronic loop contributions are suppressed by more than three orders of magnitude.
The experimental limit of $1.2 \times 10^{-7}$ is only about five times the SM rate, yet dark photon and dark $Z$ contributions at currently allowed mixing stay far below that five-fold headroom.
Observing a signal from either dark state in this channel would require mixing parameters that are already excluded, so a positive detection is not expected.
If an excess is observed, it would have to come from a different new-physics mechanism rather than the dark photon or dark $Z$ as modeled here.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: computing the dark photon or dark $Z$ amplitude with a finite mediator mass, especially near the $J/\psi$ mass, could change both the rate and the invariant-mass shape even if the integrated effect stays small.
Beyond the paper: the same vertex-rescaling treatment applies to lepton-flavored analogs such as $J/\psi \to \mu^+\mu^-\phi$, so the null conclusion likely carries over to those channels while the constraints remain unchanged.
Beyond the paper: because the dark $Z$ couplings are chiral, angular or Dalitz-plot observables might discriminate it from the Standard Model even at the same total rate; the paper integrates over the full rate and does not test this.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: the SM part of this paper is fine—it reproduces Ref. [30]'s calculation of J/ψ→e+e−φ and confirms hadronic loops are negligible. The only new claim is that dark photon and dark Z contributions can't lift the branching ratio to the BESIII limit. That's probably true, but the calculation as presented doesn't support it.

The dark photon amplitude is just the EM amplitude with each vertex multiplied by ε and the same 1/q² photon propagators. That's a massless-propagator approximation, and the paper doesn't justify it. The statement in Sec. 4 that the NP contribution is independent of M_X because the vertex structure is similar and the width is small is simply not an argument. In the plotted M_X range, the invariant mass of the e+e− pair runs from threshold up to about 2.08 GeV, and m_φ is about 1.02 GeV, so for M_X in this range the propagator 1/(q²−M_X²) can pass through a pole. The true rate won't be flat in M_X, and the sensitivity conclusion could change by orders of magnitude near those points. That's load-bearing, because the entire NP section exists to make that negative statement.

There's also a practical issue: Eq. (3), the dark Z amplitude, is garbled—undefined symbols (Pψ, ¯g, etc.) and inconsistent Lorentz structure. And the benchmarks ε=2×10⁻² and εZ=10⁻⁵ are asserted rather than derived; at least ε=2×10⁻² looks like it's already excluded by dedicated dark photon searches, though if so that only strengthens the paper's point.

What's genuinely useful: the SM prediction and the demonstration that this channel is a clean probe. That part is worth citing. But the NP analysis is not in a publishable state. I'd desk reject with an invitation to resubmit after deriving the full amplitude with massive mediators and comparing to actual limits. There's a real, modest result in here—it just isn't in this version.

Referee Report

3 major / 5 minor

Summary. The paper reproduces the SM prediction for J/psi -> e+e- phi (branching ratio 2.28e-8, dominated by EM transitions, hadronic loops negligible) and then estimates the additional contribution from a dark photon or dark Z, taking benchmark mixings epsilon=2e-2 and epsilon_Z=1e-5 from the author's earlier constraint analysis. Because the resulting total branching ratio remains below the BESIII upper limit 1.2e-7, the paper concludes that this decay channel cannot discriminate between the SM and these two dark-sector models. The NP estimate is built by rescaling the SM amplitude with factors epsilon or epsilon_Z while retaining the SM propagators, and the central quantitative claim is that the NP contribution is mass-independent and negligible.

Significance. The SM part of the paper is not new but is a clear reproduction of Ref. [30], and the demonstration that hadronic loops are suppressed is useful context. If a fully derived dark-photon/dark-Z amplitude had been computed correctly, the paper would provide a valuable null-result stating that J/psi -> e+e- phi cannot probe these mediators under current constraints. However, as it stands the NP calculation is a scaling exercise rather than a derivation, so the main claim is not established; the paper would need a substantial rewrite of Section 4 and a justification of the used couplings before the result could be relied upon.

major comments (3)

[Sec. 4, Eq. (2), M_A, Figs. 2-3] The dark photon amplitude M_A in Eq. (2) is obtained from M_EM by multiplying each fermion-photon vertex by epsilon and keeping the denominators of the SM amplitude unchanged. This assumes 1/(q^2-M_X^2) is effectively 1/q^2 for every virtual line, but no such approximation is derived; the statement in Sec. 4 that the width of the new state is much smaller than its mass would not justify replacing the propagator by a massless one. In the plotted mass range, one virtual leg has q^2=m_12^2, which can equal M_X^2 for M_X below about 2.08 GeV, and another leg has q^2=m_phi^2, so M_X near m_phi is also near a pole. At these points the true dark-photon propagator is not suppressed and the integrated branching ratio can differ from the flat curves of Figs. 2 and 3 by orders of magnitude. Because the flatness of the curves is exactly the content of the 'no enhancement' conclusion, this is a load-bearing error.
[Sec. 4, Eq. (3)] Equation (3) is not a usable expression for the dark Z contribution. It contains undefined objects (P_psi, P_phi, gbar_nu nu', C_W, and the relation of g_L and g_R to the couplings of Sec. 2), and the propagator-like factors i(P_psi_nu P_psi_nu'/M_x^2 - gbar_nu nu')/(P_psi^2+i Gamma_x M_x^2 - M_x^2) appear to place the external J/psi momentum on the internal line. The expression therefore cannot be derived from the Lagrangian of Sec. 2, and the dark Z result cannot be checked or reproduced. This is not a presentation issue; it is the whole dark Z calculation.
[Sec. 4 and Sec. 1] The benchmark values epsilon=2e-2 and epsilon_Z=1e-5 are asserted rather than derived in this paper. The text refers to constraints 'discussed in Ref.' without completing the reference and then points to [31], but it does not show how the bounds in Ref. [31] translate into these specific values. Since the plotted NP rates are just the SM rates rescaled by these parameters, the conclusion that the NP contribution is far below the experimental limit is essentially a restatement of the input couplings. The paper should either derive the values or clearly frame the plots as a scan over representative points.

minor comments (5)

[Abstract and Sec. 3] The abstract and Sec. 3 state a 'branching ratio' of 2.28e-8 keV; a branching ratio is dimensionless. Table 1 lists the width as 2.12e-6 keV and the branching ratio as 2.28e-8, so the unit should be removed.
[Sec. 4] The notation for the dark Z mixing parameter is inconsistent: epsilon_Z is used in Sec. 2, while Sec. 4 and the figure captions use epsilon' with epsilon'=1e-5; please unify the notation.
[Sec. 4] The text says the M_X range is 'MB-MK and 2m_mu', but Figs. 2-3 extend from 0.2 to 4.5 GeV; the exact scanned range and the rationale for the gaps around the phi and J/psi masses need to be stated.
[Sec. 1] The sentence 'All these constraints were discussed in Ref.' is incomplete and should give the reference number and a short summary of the constraints.
[Sec. 3] The hadronic loop contributions are presented only through effective couplings and a table; since one of the paper's claims is that hadronic effects are negligible, a reference to the explicit loop amplitudes (or an appendix) is needed to make the table reproducible.

Circularity Check

2 steps flagged · score 6.0 of 10

The NP null result is the input mixing parameters squared, and the claimed M_X-independence is put in by the form of the amplitude.

fitted input called prediction [Section 4, dark-photon amplitude M_A and Fig. 2 caption]
"In Fig. 2 and Fig. 3 we show the contribution of the dark photon and dark ZD to the decay channel J/ψ → e+e−φ with ε = 2 × 10−2 and ε′ = 0 ... The contribution is almost negligible because of the stringent constraints on the model parameters."

The dark-photon amplitude M_A in Section 4 is exactly the EM amplitude M_EM of Section 3 with each -ieγ vertex changed to -ieεγ, keeping the same photon and vector-meson propagators. Hence M_A = ε^2 M_EM by construction, and the plotted branching ratio is (1+ε^2)^2 BR_SM. Since ε is imported from the constraints of Ref. [31] rather than determined by any J/ψ observable, the 'negligible NP contribution' is the input mixing parameter squared; the prediction reduces to the input by construction.
self definitional [Section 4, text after Figs. 2 and 3]
"The NP contribution is independent of the new state mass as the Lorentz structure of the vertex is similar to the EM contribution and the decay width of the NP states is way smaller than the mass."

This is presented as a derived property, but the written M_A amplitude contains no ZD propagator 1/(q^2 - M_X^2) and no M_X-dependent coupling; it is the EM amplitude with ε-rescaled vertices. The mass independence therefore holds by definition of the written amplitude, not by the stated reasoning about width. The flat curves in Figs. 2-3 are consequences of the omitted M_X-dependent propagator, so the claimed mass independence is an ansatz restated as a result.

full rationale

The SM EM decay calculation in Section 3 is self-contained and not circular: the branching ratio 2.28e-8 and the suppression of hadronic loops follow from the quoted couplings and Table 1. The circularity is confined to the NP part of Section 4. The dark-photon amplitude is written as the EM amplitude with each -ieγ replaced by -ieεγ and with no ZD propagator, so the NP contribution is proportional to the input ε^2 and the conclusion that the NP signal is negligible restates the imported constraint ε = 2e-2. Similarly, the sentence claiming independence of the new-state mass is not derived: the displayed amplitude has no standard 1/(q^2 - M_X^2) factor, so the flat M_X curves are an artifact of the omitted propagator. This makes the NP portion partially circular, though the SM background and the experimental-limit comparison keep the paper from being wholly circular. A separate physics concern, not itself circularity, is that a properly included massive propagator could resonate and alter the sensitivity conclusion.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper's central claim depends on the validity of VMD couplings from Ref. [30], the constraint scan from Ref. [31] (self-cited), and the assumption that the NP amplitude factorizes as ε^2 times the EM amplitude with massless propagators. The dark photon and dark Z are pre-existing constructs, not invented here.

free parameters (2)

ε (dark photon kinetic mixing) = 2 × 10^-2 (benchmark, not fitted)
Used in Fig. 2 as a representative value; claimed consistent with constraints from Ref. [31] but no derivation of why this is the maximum allowed value at each mass is shown.
εZ (dark Z mass mixing) = 10^-5 (benchmark, not fitted)
Used in Fig. 3 as a representative value; not derived from the constraint scan in this paper.

assumptions (5)

domain assumption VMD couplings gψγ = 0.150 GeV^2 and gφγ = 0.013 GeV^2 from Ref. [30] describe J/ψ-photon and φ-photon transitions.
Central to the SM amplitude; taken from earlier work, not derived here.
domain assumption Hadronic loop couplings (gψf0, gφf0, gψη, gψη') from Ref. [30] are correct.
Used to claim hadronic contributions are negligible (Table 1).
ad hoc to paper The dark photon and dark Z amplitudes are obtained by replacing the photon-fermion vertex with ε-scaled vertices while keeping massless photon propagators.
Section 4, MA and Eq. (3); the mediator mass is dropped, which makes the contribution independent of Mx. The paper justifies this by claiming the vertex Lorentz structure is similar and widths are small, but no derivation or approximation criterion is given.
domain assumption The benchmark values ε=2×10^-2 and εZ=10^-5 represent the maximum allowed values from the constraints in Ref. [31].
Section 4; the paper does not show how these values follow from the constraints.
ad hoc to paper The decay width of ZD is much smaller than its mass so it can be neglected in the propagator.
Section 4 statement; no quantitative check at the masses considered (0.2-4.5 GeV).

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Cite this review

Pith. "Pith review of Searching for neutral state in the rare decay $J/\psi \rightarrow e^+ e^- \phi$." pith.science (2026). https://pith.science/paper/EQFKDQGL

@misc{pith2026241209926,
  author       = {Pith},
  title        = {Pith review of: Searching for neutral state in the rare decay $J/\psi \rightarrow e^+ e^- \phi$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQFKDQGL}},
  note         = {Machine review of arXiv:2412.09926}
}

abstract

We investigate the decay process $J/\psi \rightarrow e^+ e^-\phi$, where the relatively clean electromagnetic (EM) transitions dominate at leading order at the tree level, while hadronic contributions arise only through hadronic loop transitions. The branching ratio of $J/\psi \rightarrow e^+ e^-\phi$ was estimated to be approximately $2.28 \times 10^{-8}$ keV where the hadronic effects are negligible compared to the EM contributions. The upper limit on the branching fraction was set to be $B(J/\psi \rightarrow e^+ e^-\phi) < 1.2 \times 10^{-7}$ by BESIII collaboration. In this paper we investigate the existence of a neutral state such as dark photon and dark $Z$. Our analysis shows that the constraints on the dark photon and dark $Z$ are stringent in the way that large enhancement to the SM branching ratio up to the present experimental limit is not possible. Therefore, observing a signal for the dark photon and dark $Z$ in this decay channel is changeable.

Figures

Figures reproduced from arXiv: 2412.09926 by the authors.

**Figure 1.** The EM transitions of J/ψ → e +e −φ. The decay J/ψ → γφ is forbidden by C-parity conservation unless new physics beyond the Standard Model (BSM) exists to break this rule. However, when the photon is virtual and converts into an e +e − pair, the decay J/ψ → e +e −φ becomes allowed and measurable. Electromagnetic (EM) Contribution: The leading contribution is electromagnetic (EM), as shown in [PITH_FULL_IMAGE:figur… view at source ↗

**Figure 2.** The contribution of dark photon to the BR of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** The contribution of dark ZD to the BR of J/ψ → e +e −φ with ε ′ = 10−5 and with ε = 0. The black line represents the experimental limit, the green line represents the SM contribution, and the red line represents the total SM+NP branching ratio. 5 Conclusion In conclusion, we have analyzed the decay process J/ψ → e +e −φ, where electromagnetic (EM) transitions dominate at leading order with negligible contributions f… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [30]

Study of the rare decay $J/\psi\rightarrow e^+e^- \phi$

X.-D. Guo, S.-R. Xue, H.-W. Ke, X.-Q. Li and Q. Zhao, Study of the rare decay J/ψ →e+e−φ, Chin. Phys. C 40 (2016) 073104 [1512.04149]

work page Pith review arXiv 2016

[2] [31]

Datta, A

A. Datta, A. Hammad, D. Marfatia, L. Mukherjee and A. Ras hed, Dark photon and dark Z mediated B meson decays, JHEP 03 (2023) 108 [2210.15662]. 11 J/ Ψ , p f0/η/η ′ φ, p 3 γ γ e+, p 1 e−, p 2 J/ Ψ , p f0/η/η ′ φ, p 3 γ γ e+, p 1 e−, p 2

arXiv 2023

[3] [1]

BESIII collaboration, Determination of the number of J/ψ events with J/ψ → inclusive decays, Chin. Phys. C 36 (2012) 915 [ 1207.2865]

work page Pith review arXiv 2012

[4] [2]

Belle collaboration, Measurement of inclusive D(s), D0 and J/psi rates and determ ination of the B*(s) anti-B*(s) production Phys. Rev. Lett. 98 (2007) 052001 [ hep-ex/0608015]

work page Pith review arXiv 2007

[5] [3]

Belle collaboration, Measurement of e+ e- — > pi+ pi- J/psi cross-section via initial state radiation at B elle, Phys. Rev. Lett. 99 (2007) 182004 [ 0707.2541]

arXiv 2007

[6] [4]

Guberina, J.H

B. Guberina, J.H. Kuhn, R.D. Peccei and R. Ruckl, Rare Decays of the Z0, Nucl. Phys. B 174 (1980) 317

work page 1980

[7] [5]

Holdom, Two U(1)’s and Epsilon Charge Shifts, Phys

B. Holdom, Two U(1)’s and Epsilon Charge Shifts, Phys. Lett. B 166 (1986) 196

1986

[8] [6]

Gopalakrishna, S

S. Gopalakrishna, S. Jung and J.D. Wells, Higgs boson decays to four fermions through an abelian hidde n sector, Phys. Rev. D 78 (2008) 055002 [ 0801.3456]

arXiv 2008