⚛️ phenomenology

$B_{(s)}\to V M_X$ decays as probes of dark-matter scenarios of Belle II enhancement in $B\to KM_X$ decays

This paper proposes that $B_{(s)}\to V M_X$ decays serve as a definitive probe to distinguish between scalar and vector dark-matter mediator scenarios, which were previously fitted to explain the Belle II $B\to K M_X$ excess, by predicting distinct enhancement factors of approximately 1.6 and 3.5 respectively compared to Standard Model expectations.

Original authors: Alexander Berezhnoy, Wolfgang Lucha, Dmitri Melikhov

Published 2026-09-14

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CC BY 4.0

Original authors: Alexander Berezhnoy, Wolfgang Lucha, Dmitri Melikhov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, particles called B mesons are unstable travelers that decay, or break apart, into lighter particles almost instantly. Physicists have long studied these decays to understand the fundamental rules of nature. According to the Standard Model, the best theory we have for how particles interact, a specific type of B meson decay should produce a pair of invisible neutrinos. These neutrinos are so elusive that they escape detection, leaving behind a "missing energy" signature in experiments. For decades, the predicted rate of these events matched the theoretical calculations, and experiments confirmed that the universe behaved exactly as expected. However, the story changed recently when a new experiment called Belle II, located in Japan, began collecting data. The researchers observed a surprising number of these missing-energy events that was far higher than the Standard Model predicted. This excess suggests that something else might be happening: perhaps the B mesons are not just decaying into neutrinos, but are instead transforming into pairs of dark matter particles, the mysterious substance that makes up most of the universe's mass but remains invisible to our telescopes.

A team of physicists has now taken a closer look at this anomaly to see if the dark matter explanation holds up under scrutiny. They focused on two specific possibilities for how this dark matter might be produced. In one scenario, the B meson decays through a scalar mediator, a type of force-carrying particle that acts like a bridge between the visible world and the dark sector. In the other scenario, the bridge is a vector mediator, which behaves differently in its interactions. The researchers used the data from the Belle II experiment to fine-tune the properties of these dark matter particles, such as their mass and how strongly they interact with ordinary matter. They found that both scenarios could successfully explain the excess events seen in the decay of B mesons into a K meson and invisible particles. The key to their work was not just explaining the past data, but using those findings to make precise predictions for other types of decays that have not yet been fully explored.

The team turned their attention to two related decay processes: one where the B meson turns into a K-star meson and invisible particles, and another where a heavier B-s meson turns into a phi meson and invisible particles. By applying the rules derived from the initial dark matter explanation, they calculated what should happen in these new channels. The results revealed a clear difference between the two scenarios. If the dark matter is produced via the scalar mediator, the rate of these new decays should be about 1.6 times higher than what the Standard Model predicts. If the vector mediator is the correct explanation, the increase would be much more dramatic, with the decay rate jumping to roughly 3.5 times the expected value. This distinction is crucial because it offers a way to tell the two theories apart. The researchers noted that while the exact numbers carry some uncertainty due to the complex nature of the particles involved, the gap between the two predictions is large enough that future experiments should be able to distinguish between them.

To make these predictions useful for experimentalists, the team calculated exactly how many extra events the Belle II detector should see if these dark matter scenarios are true. They accounted for the specific way the detector works, including how it records the energy of the particles and how it filters out background noise. Their analysis shows that if the dark matter explanation is correct, the detector should record a significant number of additional events in the K-star and phi meson channels. The difference between the two dark matter models is so pronounced that it acts like a fingerprint; observing the rate of these decays will tell scientists which type of mediator, if any, is responsible for the anomaly. The work provides a clear roadmap for the next phase of research, turning a puzzling excess of data into a testable hypothesis that could finally reveal the nature of dark matter.

Technical Summary: $B(s) \to V MX$ Decays as Probes of Dark-Matter Scenarios of Belle II Enhancement in $B \to K MX$ Decays

Problem Statement
The paper addresses the recent observation by the Belle II collaboration of an excess in the missing-energy decay $B^+ \to K^+ MX$ , where the branching ratio significantly exceeds Standard Model (SM) predictions for $B^+ \to K^+ \bar{\nu}\nu$ . While numerous new-physics scenarios have been proposed to explain this anomaly, the authors focus on the hypothesis that the excess arises from the decay of $B$ mesons into pairs of invisible dark-matter (DM) fermions ( $\bar{\chi}\chi$ ). Specifically, the study investigates "top-philic" scenarios where the DM particles couple to the top quark via a mediator field $R$ , which can be either a scalar ( $R_S$ ) or a vector ( $R_V$ ). The central problem is to determine whether this specific DM mechanism, constrained by $B \to K MX$ data, can predict observable enhancements in related decay modes, specifically $B \to K^* MX$ and $B_s \to \phi MX$ , and to assess the viability of these modes as discriminators between scalar and vector mediator scenarios.

Methodology
The authors employ a theoretical framework based on effective Hamiltonians describing the flavor-changing neutral current (FCNC) transition $b \to s \bar{\chi}\chi$ .

Model Definition: The analysis considers two distinct DM scenarios:
- S-scenario: A scalar mediator $R_S$ .
- V-scenario: A vector mediator $R_V$ .
  In both cases, the mediator couples to the SM via a top-quark loop (penguin diagram) and to the DM sector via a coupling $g_{R\chi\chi}$ .
Parameter Extraction: The DM parameters (mediator mass $M_R$ , width $\Gamma_R$ , DM mass $m_\chi$ , and couplings $g_{R\chi\chi}$ and $g_{bsR}$ ) were previously extracted by fitting the differential distributions of the Belle II excess events in $B \to K MX$ [35, 40]. These fixed parameters are used here without further adjustment.
Prediction of New Modes: Using the extracted parameters and form-factor parameterizations for the hadronic matrix elements ( $B \to K^*$ and $B_s \to \phi$ ), the authors calculate the differential and integrated branching fractions for $B \to K^* MX$ and $B_s \to \phi MX$ .
Experimental Simulation: To facilitate comparison with experimental data, the theoretical differential distributions are transformed from the invariant mass squared $q^2$ to the reconstructed variable $q^2_{rec}$ , accounting for the partial reconstruction techniques and efficiency functions ( $\epsilon(q^2)$ ) used by the Belle II experiment.

Key Contributions

Parameter-Free Predictions: The paper provides essentially parameter-free predictions for the branching ratios of $B \to K^* MX$ and $B_s \to \phi MX$ based solely on the DM parameters derived from the $B \to K MX$ excess.
Discrimination Power: The study demonstrates that the ratio of enhancement factors between the SM and the DM scenarios differs significantly between the scalar and vector cases, offering a clear method to distinguish between the two mediator types.
Differential Distributions: The authors present detailed differential distributions ( $d\Gamma/dq^2$ and $d\Gamma/dq^2_{rec}$ ) for the new decay modes, allowing for a direct comparison with the kinematic shapes expected in the SM versus the DM scenarios.

Results
The analysis yields the following integrated branching ratios (using central values for parameters):

Standard Model Baselines:
- $Br(B \to K^* \bar{\nu}\nu)_{SM} = 0.84 \times 10^{-5}$
- $Br(B_s \to \phi \bar{\nu}\nu)_{SM} = 0.89 \times 10^{-5}$
Scalar Mediator (S-scenario):
- $Br(B \to K^* \bar{\chi}\chi)_S = 1.34 \times 10^{-5}$ (Enhancement factor $\sim 1.6$ )
- $Br(B_s \to \phi \bar{\chi}\chi)_S = 1.49 \times 10^{-5}$ (Enhancement factor $\sim 1.7$ )
Vector Mediator (V-scenario):
- $Br(B \to K^* \bar{\chi}\chi)_V = 2.76 \times 10^{-5}$ (Enhancement factor $\sim 3.3$ )
- $Br(B_s \to \phi \bar{\chi}\chi)_V = 3.23 \times 10^{-5}$ (Enhancement factor $\sim 3.6$ )

The differential distributions for the V-scenario exhibit a much stronger enhancement relative to the SM compared to the S-scenario. The authors also provide expected event yields for Belle II, assuming the same detection efficiency as the $B \to K MX$ channel, showing that the excess should be visible in the $B \to K^* MX$ data.

Significance and Claims
The paper claims that the decays $B \to K^* MX$ and $B_s \to \phi MX$ serve as "clear probes" of the underlying dark-matter scenario. The primary significance lies in the ability to discriminate between scalar and vector mediators: while both scenarios predict an enhancement over the SM, the magnitude of the enhancement is distinct (factor of $\sim 1.6$ for scalar vs. $\sim 3.5$ for vector).

The authors are modest regarding theoretical uncertainties, acknowledging that form-factor uncertainties (currently estimated at $\ge 10\%$ ) lead to branching ratio uncertainties of approximately $20\%$ . However, they argue that even with this uncertainty, the large difference in enhancement factors between the S- and V-scenarios allows for a robust discrimination once experimental data for these modes becomes available. The paper does not propose new experimental setups but rather provides the theoretical benchmarks necessary for the Belle II collaboration to interpret future measurements of these specific decay channels.

Technical Summary: B(s)→VMXB(s) \to V MXB(s)→VMX Decays as Probes of Dark-Matter Scenarios of Belle II Enhancement in B→KMXB \to K MXB→KMX Decays

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Technical Summary: $B(s) \to V MX$ Decays as Probes of Dark-Matter Scenarios of Belle II Enhancement in $B \to K MX$ Decays