⚛️ phenomenology

Probing dark-matter scenarios of Belle II excess in $B^+\to K^+M_X$ with $K^+\to \pi^+ M_X$ decays

This paper proposes that the Belle II excess in $B^+\to K^+M_X$ decays, potentially caused by top-philic dark matter, can be further tested and constrained by $K^+\to \pi^+ M_X$ decays, which offer unique sensitivity to the dark matter fermion mass $m_\chi$ that the $B$ -meson decays lack.

Original authors: Alexander Berezhnoy, Wolfgang Lucha, Dmitri Melikhov

Published 2026-10-20

📖 5 min read🧠 Deep dive

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 vast, invisible architecture of the universe, a significant portion of matter remains hidden from our eyes. This missing substance, known as dark matter, does not emit light, nor does it interact with the forces that hold atoms together in the way ordinary matter does. Scientists have long suspected that this elusive material is not just a passive background player but might occasionally reveal itself through subtle interactions with the visible world. To find it, researchers turn to the most unstable and energetic particles nature provides: heavy particles called B mesons. These particles, created in high-energy collisions, decay almost instantly into lighter particles. By studying how they break apart, physicists look for tiny discrepancies between what standard physics predicts and what actually happens. If a B meson decays into a visible particle and nothing else, leaving a gap in the energy balance, it could signal the escape of invisible dark matter particles. This is the hunting ground where recent experiments have found something unexpected, sparking a new chapter in the search for the universe's hidden mass.

A team of researchers has taken a fresh look at a surprising signal detected by the Belle II experiment in Japan. In 2023, this experiment observed that B mesons were decaying into a kaon (a lighter cousin of the B meson) and missing energy far more often than the standard model of particle physics allows. The standard model predicts that these decays should happen very rarely, involving only neutrinos, which are nearly massless, ghost-like particles. The Belle II data, however, showed a rate more than five times higher than expected. While this could be a statistical fluke, it is also a tantalizing hint of new physics. The researchers behind this new study proposed a specific explanation: that the missing energy is not just neutrinos, but pairs of dark matter particles. They focused on a scenario where these dark matter particles are "top-philic," meaning they interact most strongly with the top quark, the heaviest known elementary particle. In this model, the decay happens through a complex loop involving the top quark and a hypothetical carrier particle, or mediator, that bridges the gap between the visible world and the dark sector.

The team's work involved a clever cross-check using a different type of particle decay to test this top-philic idea. If the dark matter explanation for the B meson excess is correct, the same underlying mechanism should also affect the decay of a charged kaon into a pion and missing energy. While the B meson decay is sensitive to the properties of the heavy mediator particle, the kaon decay is uniquely sensitive to the mass of the dark matter particles themselves. The researchers calculated exactly how the decay rates and energy distributions would look for two different versions of this dark matter model: one where the mediator is a scalar particle and another where it is a vector particle. They then compared their theoretical predictions with the latest experimental data from the NA62 experiment, which measures kaon decays with extreme precision.

The results of this comparison offer a clear path forward for distinguishing between different possibilities. For the model involving a scalar mediator, the data from the kaon decays aligns well with the predictions derived from the B meson excess, provided the dark matter particles are very light, with a mass close to zero. In this scenario, the extra events would appear as a smooth, gentle increase across the entire range of possible energies, consistent with what the NA62 experiment has currently observed. However, the picture changes dramatically for the model involving a vector mediator. Here, the mass of the dark matter particle plays a critical role. If the dark matter particles are very light, the model predicts a significant excess of events that the NA62 data does not see, effectively ruling out that specific combination of parameters. Conversely, if the dark matter particles have a mass slightly above a certain threshold, the model predicts a sharp spike in events at higher energies.

This distinction provides a powerful tool for future discovery. The researchers found that the B meson data alone was not precise enough to pin down the exact mass of the dark matter particles, leaving a wide range of possibilities open. The kaon decay data, however, acts as a filter. If future experiments with more data confirm a smooth, low-level excess, it would strongly support the scalar mediator model with very light dark matter. If, instead, the data reveals a distinct peak of events at higher energies, it would point toward the vector mediator model with dark matter particles having a mass between 0.11 and 0.15 GeV. Until more data arrives, the top-philic dark matter hypothesis remains a viable and testable explanation for the Belle II anomaly, with the kaon decay serving as the crucial next step to either confirm the nature of these invisible particles or guide theorists toward a different solution. The search continues, not with a single experiment, but with the careful, cross-referenced scrutiny of how different particles behave when they vanish into the dark.

Technical Summary: Probing Dark-Matter Scenarios of Belle II Excess in $B^+ \to K^+ M_X$ with $K^+ \to \pi^+ M_X$ Decays

Problem Statement
The Belle II collaboration reported a significant excess in the missing-energy decay $B^+ \to K^+ M_X$ (where $M_X$ denotes invisible particles), with a measured branching ratio of $(2.3 \pm 0.7) \times 10^{-5}$ , approximately 5.4 times the Standard Model (SM) prediction for $B^+ \to K^+ \bar{\nu}\nu$ . One proposed explanation for this anomaly is the production of dark matter (DM) fermions ( $\chi\bar{\chi}$ ) via a mediator field $R$ that couples preferentially to the top quark ("top-philic" scenarios). While previous analyses successfully extracted mediator parameters ( $M_R$ , couplings) from the $B$ -decay data, the sensitivity to the DM fermion mass $m_\chi$ within the $B \to K M_X$ channel was found to be low. This paper investigates whether the related flavor-changing neutral current (FCNC) decay $K^+ \to \pi^+ M_X$ can provide complementary constraints, specifically offering higher sensitivity to $m_\chi$ .

Methodology
The authors analyze the $K^+ \to \pi^+ M_X$ decay within two specific top-philic DM scenarios previously established to explain the Belle II excess:

S-Scenario: A scalar mediator $R_s$ couples to the top quark and DM fermions via a Lagrangian involving scalar interactions.
V-Scenario: A vector mediator $R_v$ couples to the top quark and DM fermions via a Lagrangian involving vector/axial-vector interactions.

In both scenarios, the $K^+ \to \pi^+ \bar{\chi}\chi$ transition proceeds via a top-W boson loop, analogous to the $B^+ \to K^+ \bar{\chi}\chi$ process. The methodology involves:

Parameter Extraction: Utilizing parameters ( $M_R$ , $m_\chi$ , couplings $g_{Rtt}$ , $g_{R\chi\chi}$ ) derived from fitting the Belle II $B^+ \to K^+ M_X$ data.
Theoretical Calculation: Deriving the differential decay rates $d\Gamma/dq^2$ for $K^+ \to \pi^+ \bar{\chi}\chi$ in both scenarios. The calculations incorporate the relevant CKM factors ( $V_{td}V_{ts}^*$ vs. $V_{tb}V_{ts}^*$ ), quark masses, and form factors ( $f_+^{K\to\pi}$ and $f_0^{K\to\pi}$ ).
Relation of Parameters: Establishing the relationship between the $B$ -decay governing parameters ( $g_{bsR}$ ) and the $K$ -decay governing parameters ( $g_{sdR}$ ) via CKM ratios and quark mass scaling.
Comparison with Data: Comparing the theoretical predictions (SM + DM contributions) against recent NA62 experimental measurements of the $K^+ \to \pi^+ M_X$ differential distribution.

Key Contributions and Results

Sensitivity to $m_\chi$ : The paper demonstrates that while $B \to K M_X$ decays are relatively insensitive to the precise value of $m_\chi$ (allowing a broad range for the S-scenario and a moderate range for the V-scenario), the $K^+ \to \pi^+ M_X$ decay exhibits strong kinematic sensitivity to $m_\chi$ for light fermions ( $m_\chi < (M_K - M_\pi)/2$ ).
S-Scenario Findings:
- The DM contribution is maximized when $m_\chi = 0$ .
- The predicted branching ratio for $K^+ \to \pi^+ \bar{\chi}\chi$ is $2.83 \times 10^{-11}$ , resulting in a $\sim 30-40\%$ excess over the SM prediction, distributed homogeneously across the allowed $q^2$ range.
- This smooth distribution is consistent with current NA62 measurements, which show no significant deviation from the SM but allow for such an excess within uncertainties.
V-Scenario Findings:
- The scenario exhibits a pronounced dependence on $m_\chi$ . The region $m_\chi < 0.11$ GeV is effectively excluded by NA62 data.
- For $m_\chi > 0.11 - 0.12$ GeV, the differential distribution shows a strong enhancement (peak) in the high- $q^2$ region ( $q^2 > 0.07$ GeV $^2$ ).
- Current NA62 data does not definitively confirm this enhancement, but the authors note that if higher-statistics data reveals a peak in this region, it would strongly support the V-scenario with $m_\chi \approx 0.11 - 0.15$ GeV.
Parameter Constraints: Using the parameters extracted from Belle II, the authors provide parameter-free predictions for the $K$ -decay sector. The analysis yields specific constraints on the mediator mass and fermion mass, suggesting that $K^+ \to \pi^+ M_X$ can either extract a precise value for $m_\chi$ or set a lower limit on it, provided theoretical and experimental uncertainties are reduced.

Significance and Claims
The paper claims that the $K^+ \to \pi^+ M_X$ decay serves as a critical, independent probe for top-philic dark matter scenarios motivated by the Belle II anomaly. Its primary significance lies in its ability to break the degeneracy in the DM fermion mass $m_\chi$ that persists in $B$ -decay analyses.

The authors modestly conclude that while current NA62 data is compatible with SM predictions, the potential for an observable excess exists. If future high-statistics data confirms an excess:

A smoothly distributed excess would favor the S-scenario with a very light DM fermion ( $m_\chi \approx 0$ ).
A peaked excess at high $q^2$ would favor the V-scenario with a specific DM fermion mass range ( $m_\chi \approx 0.11 - 0.15$ GeV).

Thus, the $K^+ \to \pi^+ M_X$ channel is presented not as a definitive discovery tool at this stage, but as a vital discriminator for the specific nature of the new physics if the Belle II excess is confirmed to be of dark matter origin.

Technical Summary: Probing Dark-Matter Scenarios of Belle II Excess in B+→K+MXB^+ \to K^+ M_XB+→K+MX​ with K+→π+MXK^+ \to \pi^+ M_XK+→π+MX​ Decays

More like this

Technical Summary: Probing Dark-Matter Scenarios of Belle II Excess in $B^+ \to K^+ M_X$ with $K^+ \to \pi^+ M_X$ Decays