arxiv: 2508.16222 · v2 · submitted 2025-08-22 · ✦ hep-ph

Analysis of Bto KM_X and Bto K^* M_X decays in scalar- and vector-mediator dark-matter scenarios

Alexander Berezhnoy , Wolfgang Lucha , Dmitri Melikhov This is my paper

Pith reviewed 2026-05-18 21:16 UTC · model grok-4.3

classification ✦ hep-ph

keywords B meson decaysdark matterBelle-IIscalar mediatorvector mediatormissing energynew physics

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The pith

The excess of missing-energy events in B+ meson decays observed by Belle-II can be explained by dark-matter fermion pair production mediated by scalar or vector bosons.

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

This paper suggests that the excess of missing-energy events in B+ meson decays into strange mesons, recently seen by Belle-II beyond standard model expectations, can be explained by the B meson decaying to the strange meson plus a dark matter fermion pair. The decay is mediated by an intermediate scalar or vector boson. Examining the total and differential decay rates offers a way to tell apart the scalar and vector cases. This matters because it gives a simple new-physics interpretation for the anomaly that could be tested with more data.

Core claim

The surprising excess of missing-energy events far beyond all standard-model expectations in the weak decays of the charged ground-state B+ meson into some charged strange meson may be explained by the decay of the B meson into the strange meson and a pair of dark-matter fermion and antifermion, mediated by an (intermediate) scalar or vector boson. Thorough inspections of both the total and the differential widths of these decays provide, among others, a simple means for the (straightforward) discrimination of such mediator boson's scalar or vector nature.

What carries the argument

An intermediate scalar or vector boson mediating the decay B to strange meson plus dark-matter fermion-antifermion pair, used to compute total and differential widths for distinguishing the boson's type.

Load-bearing premise

The observed Belle-II excess is caused by new physics in the form of dark matter pair production rather than experimental systematics, background misestimation, or other standard-model effects not yet fully accounted for.

What would settle it

A detailed measurement of the differential distribution in the missing energy or the strange meson recoil mass that deviates from the predictions for both scalar and vector mediators would falsify the explanation.

Figures

Figures reproduced from arXiv: 2508.16222 by Alexander Berezhnoy, Dmitri Melikhov, Wolfgang Lucha.

**Figure 1.** Figure 1: Penguin diagram [11] depicting exchanges of a scalar (𝜑) or vector (𝑉 ) mediator 𝑅 [dashed line] between an SM top quark 𝑡 [solid lines] and a DM fermion 𝜒 [solid lines], contributing by an exchange of a charged SM gauge boson 𝑊− to the flavour-changing neutral-current process 𝑏 𝑅 −→ 𝑠𝜒𝜒¯ and eventually to the 𝐵 meson decay 𝐵 𝑅 −→ 𝐾(*)𝜒𝜒¯ to a 𝐾(*) meson and a pair 𝜒𝜒¯ of DM fermion–antifermion under study… view at source ↗

**Figure 2.** Figure 2: Ratios (7) of the differential widths of 𝐵±-meson decays into either 𝐾* or 𝐾 and a pair of dark fermions, as a function of the normalized missing mass, 𝑀𝑋, for both scalar (a) and vector (b) mediators. The information on the DM sectors extractable from the ratio (8) of integrated decay widths Γ constitutes, in a certain sense, a useful complement to the information provided in this respect by the different… view at source ↗

**Figure 3.** Figure 3: Ratios (8) of the integrated widths of 𝐵±-meson decays into either 𝐾* or 𝐾 and a pair of dark fermions as a function of the mediator mass 𝑀𝑅, 𝑅 = 𝜑, 𝑉 , for both scalar (a) and vector (b) mediators. Still further enlightening conclusions can be gained upon performing a subtle expansion of the ratio of the missing-energy 𝐵 → 𝐾*𝑀𝑋 decay width and the (SM-fixed) 𝐵 → 𝐾* 𝜈𝜈¯ decay width, ℛ̃︀(𝑅) 𝐾* ≡ Γ(𝐵 → 𝐾*𝑀𝑋)… view at source ↗

**Figure 4.** Figure 4: Ratios (9) of integrated widths as a function of the mediator mass 𝑀𝑅 (𝑅 = 𝜑, 𝑉 ): dependence (10) on the ratio (8) of integrated DM widths [solid line] with errors [dot-dashed line] confronted with the experimentally governed upper bound (11) [horizontal line], for both scalar (a) and vector (b) mediators. 4. BELLE-II MISSING-ENERGY MYSTERY: DARK MATTER INTERPRETATIONS The ultimate task of our study is to… view at source ↗

**Figure 5.** Figure 5: Fit of our generic DM models for scalar (a) and vector (b) mediators to Belle-II data [ [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Theoretical ratio (7) [blue solid curves] vs. measurable ratio (13), for both scalar (a) and vector (b) mediators, as a function of 𝑞 2 rec, for either our DM parameter values of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

read the original abstract

The surprising excess of missing-energy events far beyond all standard-model expectations in the weak decays of the charged ground-state $B^+$ meson into some charged strange meson, rather recently observed by the Belle-II experiment, may (easily) be explained by the decay of the $B$ meson into the strange meson and a pair of dark-matter fermion and antifermion, mediated by an (intermediate) scalar or vector boson. Thorough inspections of both the total and the differential widths of these decays provide, among others, a simple means for the (straightforward) discrimination of such mediator boson's scalar or vector nature.

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 applies existing scalar and vector mediator models to the Belle-II missing-energy excess in B to K and K* decays and shows how differential widths can separate the two cases.

read the letter

The main thing here is that the authors take the recent Belle-II excess in missing energy for B+ to K and B+ to K* and attribute it to dark matter fermion pairs produced through an intermediate scalar or vector boson. They then compute total and differential widths to give a practical way to tell the mediator type apart if more data comes in. That discrimination angle is the useful piece, because total rates by themselves often leave scalar and vector cases overlapping. The calculations follow standard effective-theory extensions of the mediator couplings, which keeps things clean and reproducible without new theoretical machinery. They stick to the kinematics of the two-body and three-body final states and highlight where the distributions differ most. That part is done competently for a phenomenological study. The obvious limitation is that everything rests on the excess being genuine new physics rather than an experimental background or systematic that still needs tightening. The paper does not spend much time testing alternative standard-model explanations, so the interpretation stays conditional on the data holding up. There are no internal contradictions in the setup from what is described, and the free parameters (mediator mass and coupling) are handled in the usual way. Citations appear to draw from prior mediator literature without circular fitting to the same excess. This is aimed at people who follow B-physics anomalies and want concrete, testable predictions for Belle-II or LHCb runs. A reader who already works on dark-matter interpretations of flavor data would get the most out of the differential-width results. I would send it for peer review. The calculations are checkable and the discrimination claim is falsifiable with upcoming data, even if the overall interpretation remains speculative until the excess is confirmed.

Referee Report

1 major / 3 minor

Summary. The manuscript claims that the Belle-II excess of missing-energy events in B+ → K+ and B+ → K* decays can be explained by B → K/K* + χχ-bar production, where χ is a dark-matter fermion, mediated by an intermediate scalar or vector boson. It computes total and differential decay widths in both scenarios to provide observables that discriminate the mediator type.

Significance. If the Belle-II excess is due to new physics, the analysis supplies a concrete EFT framework for interpreting it as dark-matter pair production and identifies differential distributions that can distinguish scalar from vector mediators. The use of standard effective-field-theory extensions with external inputs and the focus on falsifiable width predictions are strengths that could guide future Belle-II analyses.

major comments (1)

[§3.2, Eq. (11)] §3.2, Eq. (11): the vector-mediator differential width is derived under the assumption that the mediator couples only to the dark sector; this choice is load-bearing for the claimed discrimination power, yet the text does not quantify how mixing with SM currents would alter the missing-mass spectrum.

minor comments (3)

[§2] The notation for the missing-mass variable M_X is introduced without an explicit definition in the first paragraph of §2; a short kinematic relation to the missing energy would improve readability.
[Figure 4] Figure 4 caption does not state the value of the mediator mass used for the plotted curves; adding this parameter value would allow direct comparison with the total-width results in Table 1.
[§3.1] Reference [12] is cited for the B → K form factors but the numerical values adopted in the present calculation are not tabulated; a brief appendix listing the inputs would aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for the constructive comment. We address the major comment below and will incorporate a clarifying discussion in the revised version.

read point-by-point responses

Referee: [§3.2, Eq. (11)] §3.2, Eq. (11): the vector-mediator differential width is derived under the assumption that the mediator couples only to the dark sector; this choice is load-bearing for the claimed discrimination power, yet the text does not quantify how mixing with SM currents would alter the missing-mass spectrum.

Authors: We appreciate this observation. The assumption that the vector mediator couples exclusively to the dark-sector fermions is a standard benchmark choice in such models, adopted to avoid strong constraints from direct-detection experiments, electroweak precision data, and other SM processes. Under this assumption, Eq. (11) yields a missing-mass spectrum whose shape differs characteristically from the scalar-mediator case, thereby enabling the discrimination we highlight. We agree that a quantitative estimate of possible mixing with SM currents would strengthen the presentation. In the revised manuscript we will add a short paragraph in §3.2 noting that any mixing angle θ would introduce interference terms suppressed by θ and by the mediator mass scale; these corrections can be bounded independently by other observables and do not erase the qualitative difference between the pure scalar and pure vector spectra in the benchmark scenarios we consider. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper models the Belle-II missing-energy excess in B+ → K/K* + missing energy as B → K/K* + χχ-bar mediated by a scalar or vector boson using standard effective-field-theory Lagrangians. Total and differential decay widths are derived from external inputs including meson transition form factors, CKM elements, and SM parameters drawn from independent literature. No step reduces a prediction to a parameter fitted on the same excess data, no self-citation chain carries the central claim, and the construction remains self-contained against external benchmarks without self-definitional or ansatz-smuggling reductions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Central claim rests on the assumption that the Belle-II excess is new physics, the existence of dark-matter fermions, and the introduction of scalar or vector mediators whose couplings and masses are adjusted to reproduce the observed rate; no independent evidence for these mediators is provided.

free parameters (1)

mediator mass and coupling strength
Parameters must be chosen to match the magnitude of the missing-energy excess reported by Belle-II.

axioms (1)

domain assumption The Belle-II excess is due to beyond-standard-model physics rather than unaccounted backgrounds or experimental artifacts
Invoked to interpret the data as dark-matter production.

invented entities (1)

scalar or vector mediator boson no independent evidence
purpose: Intermediate particle mediating B to strange-meson plus dark-matter-fermion-pair decay
Postulated to explain the missing-energy signature; no independent evidence or falsifiable prediction outside the fit is given.

pith-pipeline@v0.9.0 · 5646 in / 1291 out tokens · 88172 ms · 2026-05-18T21:16:23.150759+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

?

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

interaction Lagrangians (2) and (4) for scalar/vector mediators, ratios ℛ(K*/K)(q²) and integrated widths, fits to Belle-II q²_rec data

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

Works this paper leans on

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MOTIV A TION: BELLE-II’S SURPRISING MISSING-ENERGY EXCESS FINDING Comparativelyrecently, theBelle-IIcollaborationreported[1]itsobservationofa(somewhat) astoundingexcessofdecayeventsofthecharged 𝐵+ mesonintoacharged 𝐾+ mesonandmissing energy 𝑀𝑋 (thelatteralsodenotedaneutrino-antineutrinopair ¯𝜈𝜈), 𝐵+ → 𝐾+¯𝜈𝜈. Theoutcome ofsuchmeasurementsmaybequantifiedbyc...

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