arxiv: 2512.13776 · v2 · submitted 2025-12-15 · ✦ hep-ph · hep-ex· hep-lat· nucl-th

Recognition: 2 theorem links

· Lean Theorem

Improved Standard-Model predictions for η^{(prime)}to ell^+ ell^-

Noah Messerli , Martin Hoferichter , Bai-Long Hoid , Simon Holz , Bastian Kubis

Authors on Pith no claims yet

Pith reviewed 2026-05-16 21:48 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-latnucl-th

keywords eta decayeta prime decaydilepton branching fractiontransition form factorstandard model predictionhadronic light-by-lightnew physics bounds

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The pith

Using dispersive transition form factors, the paper computes updated Standard Model branching fractions for the rare eta and eta-prime decays to dileptons.

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

The paper establishes refined predictions for the branching fractions of η and η' decaying into electron or muon pairs. These decays are highly suppressed in the Standard Model by chirality structure and the need for a loop connecting the lepton pair to the two-photon matrix element. The authors apply transition form factors previously derived in dispersive analyses for hadronic light-by-light scattering, now including robust imaginary parts from subleading channels for the first time. This yields concrete numbers that can be compared directly to experiment to search for deviations from the Standard Model. The results include a mild tension in the η to muon-pair channel.

Core claim

The central claim is that the branching fractions are Br[η→e⁺e⁻]=5.37(4)(2)[4]×10^{-9}, Br[η→μ⁺μ⁻]=4.54(4)(2)[4]×10^{-6}, Br[η'→e⁺e⁻]=1.80(2)(3)[3]×10^{-10}, and Br[η'→μ⁺μ⁻]=1.22(2)(2)[3]×10^{-7}, obtained by evaluating the η(′) transition form factors in dilepton kinematics with an improved treatment of asymptotic contributions and subleading imaginary parts, and that the η→μ⁺μ⁻ result shows a 1.6σ tension with experiment while bounds on physics beyond the Standard Model can be derived from any discrepancies.

What carries the argument

The η(′) transition form factor, a single scalar function encoding the two-photon matrix element, whose dispersive representation supplies both real and imaginary parts across the relevant kinematics.

If this is right

The branching ratios serve as benchmarks for experimental tests of the Standard Model in rare pseudoscalar decays.
Any significant deviation from these values would allow derivation of bounds on physics beyond the Standard Model.
The inclusion of subleading imaginary parts reduces theoretical uncertainty in the normalized branching fractions.
The mild tension observed for η→μ⁺μ⁻ motivates refined experimental measurements and further theoretical cross-checks.
The same form-factor framework can be used to improve predictions for related processes such as the two-photon widths.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The predictions could be incorporated into global fits that combine multiple observables to constrain specific beyond-Standard-Model scenarios such as leptoquarks.
Higher-precision data from upcoming experiments could either confirm the tension or bring the measurement into agreement with the calculation.
The dispersive method may be extended to other rare decays of light pseudoscalars where similar two-photon loops appear.

Load-bearing premise

The transition form factors and their subleading imaginary parts from prior hadronic light-by-light analyses remain accurate when applied to the dilepton decay kinematics.

What would settle it

An experimental measurement of Br[η→μ⁺μ⁻] that differs from 4.54×10^{-6} by more than roughly three standard deviations would falsify the central prediction.

read the original abstract

The rare decays $\eta^{(\prime)}\to\ell^+\ell^-$, $\ell\in\{e,\mu\}$, are highly suppressed in the Standard Model, both by their chirality structure and the required loop attaching the lepton line to the $\eta^{(\prime)}\to\gamma^*\gamma^*$ matrix element. The latter is described by a single scalar function, the transition form factor, which has recently been studied in great detail for $\eta^{(\prime)}$ in the context of the pseudoscalar-pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon. Based on these results, we evaluate the corresponding prediction for the $\eta^{(\prime)}$ dilepton decays, supplemented by an improved evaluation of the asymptotic contributions including pseudoscalar mass effects. In particular, the dispersive representation for the $\eta^{(\prime)}$ transition form factors allows us, for the first time, to perform a robust evaluation of the imaginary parts due to subleading channels besides the dominant two-photon cut. Our final results are $\text{Br}[\eta\to e^+e^-]=5.37(4)(2)[4]\times 10^{-9}$, $\text{Br}[\eta\to \mu^+\mu^-]=4.54(4)(2)[4]\times 10^{-6}$, $\text{Br}[\eta'\to e^+e^-]=1.80(2)(3)[3]\times 10^{-10}$, and $\text{Br}[\eta'\to \mu^+\mu^-]=1.22(2)(2)[3]\times 10^{-7}$, where the errors refer to the uncertainty in the normalized branching fraction, the one propagated from $\text{Br}[\eta^{(\prime)}\to\gamma\gamma]$, and the total uncertainty, respectively. The branching fraction for $\eta\to\mu^+\mu^-$ exhibits a mild $1.6\sigma$ tension with experiment, and we explore the bounds that can be derived on physics beyond the Standard Model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives updated SM predictions for eta and eta-prime dilepton decays by applying recent dispersive form factors with better subleading cuts, plus a mild tension in one channel.

read the letter

The punchline is that this paper gives updated Standard Model predictions for the eta and eta-prime decays to lepton pairs, using refined dispersive transition form factors, and notes a mild tension in the eta to muon pair mode. They apply the form factors developed for pseudoscalar contributions to hadronic light-by-light scattering in the muon g-2 calculation. The improvements include a better asymptotic contribution with mass effects and, for the first time, a solid evaluation of imaginary parts from subleading channels in the relevant loop integral. This leads to the specific numbers: 5.37(4)(2)[4]×10^{-9} for eta to e+e-, 4.54(4)(2)[4]×10^{-6} for eta to mu+mu-, and the eta-prime equivalents. The tension at 1.6 sigma in the muon channel is used to set bounds on new physics. The paper handles the uncertainties transparently, separating the contribution from the normalized branching fraction and from the two-photon decay rate. The approach is not circular because the form factors come from independent prior work. A soft spot is the assumption that the form factors and their subleading parts transfer accurately to the dilepton kinematics. The paper addresses this by improving the treatment of imaginary parts, but any residual uncertainty from the original analysis would propagate. It's not a big problem given the overlap in kinematics. This work is for people doing precision calculations in flavor physics or looking to constrain BSM models with rare pseudoscalar decays. It would be useful in a reading group focused on hadronic form factors or rare decays. I recommend it for peer review because the calculation is grounded and the results are relevant for ongoing experimental programs.

Referee Report

0 major / 1 minor

Summary. The manuscript computes improved Standard-Model branching-ratio predictions for the rare decays η(′) → ℓ⁺ℓ⁻ (ℓ = e, μ) by employing dispersive representations of the η(′) transition form factors previously determined in the context of hadronic light-by-light scattering for (g−2)μ. An improved treatment of the asymptotic contributions, including pseudoscalar mass effects, and a first robust evaluation of imaginary parts from subleading channels are presented, yielding Br[η→e⁺e⁻] = 5.37(4)(2)[4] × 10^{-9}, Br[η→μ⁺μ⁻] = 4.54(4)(2)[4] × 10^{-6}, Br[η'→e⁺e⁻] = 1.80(2)(3)[3] × 10^{-10}, and Br[η'→μ⁺μ⁻] = 1.22(2)(2)[3] × 10^{-7}, with a noted 1.6σ tension in the η → μ⁺μ⁻ channel.

Significance. These predictions furnish high-precision SM benchmarks that can be used to test the Standard Model and to derive bounds on new physics. The transparent propagation of uncertainties from the form-factor normalization and the two-photon branching ratios, together with the use of dispersive methods that incorporate subleading imaginary parts, enhances the robustness of the results. The mild tension observed in one channel invites further experimental scrutiny.

minor comments (1)

[Abstract] Abstract: the total-uncertainty bracket notation [4] is introduced without an immediate inline definition; a parenthetical clarification of the three error components would improve readability for non-specialist readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the clear summary of our results, and the recommendation to accept the manuscript. We are pleased that the referee recognizes the robustness of the dispersive framework, the improved treatment of asymptotic contributions, and the transparent uncertainty propagation.

Circularity Check

0 steps flagged

No significant circularity; form factors from independent prior work on distinct observable

full rationale

The derivation chain takes η(′) transition form factors from prior dispersive analyses performed for hadronic light-by-light scattering in (g−2)μ, a separate observable. These inputs are not refitted or redefined here; the paper instead applies them to dilepton kinematics with an improved treatment of asymptotic contributions and subleading imaginary parts. No equation reduces by construction to the output branching ratios, no self-definitional loop exists, and no load-bearing step collapses to a self-citation whose validity is established only inside this manuscript. The central predictions therefore remain externally anchored and falsifiable against the distinct (g−2)μ data set.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central predictions rest on the validity of the dispersive representation of the transition form factor taken from prior g-2 work, standard QED loop structure, and normalization to measured two-photon branching fractions.

free parameters (2)

normalized branching fraction uncertainty
Uncertainty component tied to the two-photon decay rate used for normalization
asymptotic contribution parameters
Parameters controlling the high-energy behavior including pseudoscalar mass effects

axioms (2)

domain assumption Dispersive representation of the η(′)→γ*γ* transition form factor
Assumes the form factor satisfies dispersion relations with the same analytic properties as used in the prior hadronic light-by-light analysis
standard math Standard Model QED and QCD framework for the rare decay amplitude
Relies on the usual chiral and electromagnetic structure of the effective interaction

pith-pipeline@v0.9.0 · 5690 in / 1673 out tokens · 46195 ms · 2026-05-16T21:48:43.049811+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The rare decays η(′)→ℓ⁺ℓ⁻ ... described by a single scalar function, the transition form factor, which has recently been studied ... in the context of ... hadronic light-by-light scattering
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

dispersive representation for the η(′) transition form factors ... imaginary parts due to subleading channels

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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