Analysis of $B\to KM_X$ and $B\to K^* M_X$ decays in scalar- and vector-mediator dark-matter scenarios

This paper proposes that the recently observed excess of missing-energy events in $B^+$ decays to charged strange mesons can be explained by the production of dark-matter fermion pairs mediated by scalar or vector bosons, and demonstrates that analyzing the total and differential decay widths offers a straightforward method to distinguish between these two mediator scenarios.

The Gist

Imagine the universe as a giant, busy dance floor. Most of the dancers we can see and hear are the "Standard Model" particles—the familiar protons, electrons, and neutrinos. But for a long time, physicists have suspected there are invisible dancers in the crowd, moving to a different rhythm. We call this invisible crowd Dark Matter.

Recently, a very sensitive camera at a particle accelerator called Belle-II took a picture of a specific dance move: a heavy particle called a B-meson breaking apart. The camera saw something strange. The B-meson was turning into a lighter, strange particle (a K-meson) and... disappearing energy. It was as if the B-meson danced, handed a gift to the K-meson, and then the rest of the energy vanished into thin air.

The amount of "missing energy" was much higher than the Standard Model predicted. It was like the dance floor was losing energy at a rate that shouldn't be possible. This paper asks: Could this missing energy be the invisible Dark Matter dancers?

Here is the breakdown of the paper's investigation, using simple analogies:

1. The Mystery: A Missing Gift

In the standard dance, when a B-meson decays, it usually leaves behind a pair of invisible neutrinos (ghosts that are hard to catch). The Belle-II experiment found that the "missing energy" was about 5.4 times larger than expected.

The authors propose that instead of just neutrinos, the B-meson is actually decaying into a K-meson and a pair of Dark Matter particles (let's call them "Dark Ghosts"). But how do these Dark Ghosts get involved? They can't just appear out of nowhere; they need a messenger to carry the energy from the visible world to the dark world.

2. The Two Messengers: The Ball vs. The Bat

The paper investigates two types of messengers that could carry this energy:

• The Scalar Messenger (The Ball): Imagine a messenger that is a simple, round ball. It has no direction or spin.
• The Vector Messenger (The Bat): Imagine a messenger that is a long bat. It has a specific direction and spin.

The scientists wanted to know: Is the missing energy being carried by a Ball or a Bat?

3. The Detective Work: Comparing Two Dance Moves

To figure out which messenger is being used, the authors looked at two slightly different versions of the dance:

• Dance A: The B-meson turns into a K-meson (a simple, round particle).
• Dance B: The B-meson turns into a K-meson* (a slightly more complex, spinning particle).

They calculated what would happen if the messenger was a Ball versus if it was a Bat. They found a clear difference in the "footprints" left behind:

• If the messenger is a Ball (Scalar): The ratio of Dance B to Dance A starts high and slowly drops down as the missing energy increases. It's a smooth, gentle slide.
• If the messenger is a Bat (Vector): The ratio starts low, shoots up to a peak (like a hill), and then drops sharply.

The Analogy: Imagine you are trying to guess if a package was delivered by a slow, steady truck (Ball) or a fast, bouncy motorcycle (Bat). By looking at how the package bounces on the road (the ratio of the two dances), you can tell exactly which vehicle delivered it. The paper claims this "bouncing pattern" is a perfect way to tell the two messengers apart, regardless of how heavy the Dark Matter particles are.

4. The Results: What the Data Says

The authors took the actual data from the Belle-II experiment and tried to fit their "Ball" and "Bat" models to it.

• The Fit: Both models (Ball and Bat) could actually explain the data. The "Ball" model worked with a messenger mass of about 2.4 GeV, and the "Bat" model worked with a mass of about 3 GeV.
• The Constraint: However, when they checked the rules of the dance floor (specifically, limits on how much energy can be lost in the K*-meson dance), they found a problem for the "Bat." The "Bat" messenger can only work if it is very light (under 3 GeV). If it were heavier, it would break the rules of the Standard Model. The "Ball" messenger has more freedom.

5. The Conclusion

The paper concludes that the surprising "missing energy" seen by Belle-II can be explained by Dark Matter.

• The Main Tool: The most powerful way to solve the mystery of which messenger is involved is to compare the two different dance moves (K vs. K*). The pattern of the energy loss acts like a fingerprint.
• The Outcome: If future experiments measure this specific ratio, they will know instantly if the universe is using a "Ball" (Scalar) or a "Bat" (Vector) to hide Dark Matter.

In short, the paper provides a simple test (comparing two specific particle decay rates) to distinguish between two very different theories of how Dark Matter interacts with our visible world, using the strange "missing energy" events recently spotted by the Belle-II experiment as the starting point.

Technical Summary

Technical Summary: Analysis of $B \to KMX$ and $B \to K^*MX$ Decays in Scalar- and Vector-Mediator Dark-Matter Scenarios

Problem Statement
The paper addresses a recent experimental anomaly reported by the Belle-II collaboration: an excess of missing-energy events in the decay of charged $B^+$ mesons into a charged strange meson ($K^+$) and invisible particles ($MX$), denoted as $B^+ \to K^+ MX$. The measured branching ratio, $\mathcal{B}(B^+ \to K^+ \bar{\nu}\nu) = (2.3 \pm 0.7) \cdot 10^{-5}$, is approximately $5.4$ times larger than the Standard Model (SM) prediction. While this excess could potentially arise from SM neutrino-antineutrino pairs, the authors investigate the hypothesis that it results from the production of a pair of dark matter (DM) fermions ($\bar{\chi}\chi$) mediated by a new boson. The central challenge is to determine whether the mediator boson ($R$) is a scalar ($\phi$) or a vector ($V$) particle and to establish if such a scenario can consistently explain the observed data while respecting other experimental constraints.

Methodology
The authors analyze $B$-meson decays into a strange meson ($K$ or $K^*$) and a DM pair ($\bar{\chi}\chi$) via a "top-philic" mediator. The interaction is modeled through a penguin-type process where the mediator couples to SM top quarks and DM fermions. The study considers two specific interaction Lagrangians:

1. Scalar Mediator ($\phi$): Coupling to the top quark and DM fermions via a Yukawa-like interaction.
2. Vector Mediator ($V$): Coupling to vector and axial-vector currents of the top quark and DM fermions.

The authors calculate both differential decay widths ($d\Gamma/dq^2$) and integrated decay widths ($\Gamma$) for the processes $B \to K \bar{\chi}\chi$ and $B \to K^* \bar{\chi}\chi$. They utilize form factors parameterized by lattice QCD and light-cone sum rules to handle nonperturbative QCD effects.

Key to their methodology is the construction of two specific ratios to discriminate between mediator types:

1. Differential Ratio ($\hat{\mathcal{R}}_{K^*/K}^{(R)}(q^2)$): The ratio of differential widths for $B \to K^* \bar{\chi}\chi$ to $B \to K \bar{\chi}\chi$.
2. Integrated Ratio ($\mathcal{R}_{K^*/K}^{(R)}$): The ratio of total decay widths for the same processes.

The authors also incorporate experimental reconstruction variables ($q^2_{rec}$) used by Belle-II to ensure theoretical predictions are directly comparable to experimental data. They perform $\chi^2$ fits to the Belle-II $q^2_{rec}$ distribution data to extract numerical values for DM parameters (masses and couplings).

Key Contributions and Results

• Discrimination via Differential Ratios: The authors demonstrate that the differential decay-width ratio $\hat{\mathcal{R}}_{K^*/K}^{(R)}(q^2)$ is independent of the specific numerical values of DM parameters (couplings and masses) because these factors cancel out. However, the shape of this ratio is highly sensitive to the spin of the mediator:

  • For scalar mediators, the ratio decreases monotonically from $\approx 1$ to $0$ as the missing mass increases.
  • For vector mediators, the ratio initially rises to a maximum (at $M_X/(M_B - M_{K^*}) \approx 0.93$) before dropping sharply to $0$.
    This provides a robust, model-independent tool for distinguishing the mediator's nature.

• Discrimination via Integrated Ratios: The integrated width ratio $\mathcal{R}_{K^*/K}^{(R)}$ exhibits a distinct dependence on the mediator mass ($M_R$):

  • For scalar mediators, the ratio remains below $1$ for all mediator masses.
  • For vector mediators, the ratio is strictly greater than $1$ for all mediator masses.

• Constraints on Vector Mediators: By combining the Belle-II measurement of $B \to K \bar{\chi}\chi$ with the upper bound on the SM-like decay $B \to K^* \bar{\nu}\nu$ (from Belle), the authors derive a constraint on the ratio of total widths. This compatibility check imposes a severe restriction on vector mediators, limiting their mass to a low range: $M_V \lesssim 3$ GeV. Scalar mediators face no such stringent mass constraint from this specific comparison.

• Fit to Experimental Data: The authors successfully fit the Belle-II $q^2_{rec}$ distribution using both scalar and vector mediator scenarios.

  • For the scalar scenario, the best-fit parameters are $M_\phi \approx 2.4$ GeV, $\Gamma_\phi \approx 2.9$ GeV, and $m_\chi \approx 0.42$ GeV.
  • For the vector scenario, adhering to the mass constraint ($M_V = 3$ GeV), the fit yields $m_\chi \approx 0.6$ GeV.
  • Table 1 in the paper lists the specific coupling strengths and masses derived from these minimum-$\chi^2$ fits.

• Experimental Robustness: The study confirms that the theoretical differential ratio $\hat{\mathcal{R}}_{K^*/K}^{(R)}(q^2)$ remains stable even when recalculated using the experimental reconstruction variable $q^2_{rec}$ and accounting for detection efficiencies. The predicted ratios align closely with the measurable experimental ratios, validating the utility of this observable for future discrimination.

Significance and Claims
The paper claims that the simultaneous analysis of $B \to KMX$ and $B \to K^*MX$ decays offers a straightforward and effective means to discriminate between scalar and vector dark matter mediators. The primary significance lies in the identification of observables (specifically the ratios of differential and integrated widths) that are insensitive to the unknown numerical values of DM couplings and masses but are uniquely determined by the spin of the mediator.

The authors assert that their analysis provides a "simple means" for the "straightforward discrimination" of the mediator's nature. Furthermore, they demonstrate that the observed Belle-II excess can be reproduced by both scalar and vector mediator models, provided the vector mediator's mass is constrained to be low ($\lesssim 3$ GeV). The work underscores the importance of measuring the $B \to K^* MX$ channel alongside $B \to K MX$ to resolve the nature of the missing-energy excess and to narrow down the viable candidates for dark matter models.