Probing vector- vs scalar-mediator dark-matter scenarios in $B\to (K,K^*) M_X$ decays

This paper demonstrates that differential distributions in $B\to K M_X$ and $B\to K^* M_X$ decays can unambiguously distinguish between scalar and vector dark-matter mediator scenarios, while current data tightly constrain the vector mediator mass to below 3 GeV but leave the scalar mediator mass unconstrained.

The Gist

Imagine you are a detective trying to solve a mystery in the subatomic world. Recently, the Belle-II experiment (a giant particle detector in Japan) noticed something strange: when certain heavy particles called B-mesons decay, they seem to be missing more energy than physics textbooks say they should.

In the "Standard Model" (our current rulebook for how particles behave), these decays should produce a specific amount of invisible energy (neutrinos). But Belle-II saw 5.4 times more invisible energy than expected. It's like a magician pulling a rabbit out of a hat, but this time, the rabbit is invisible, and there are five of them instead of one.

The authors of this paper propose a new theory to explain this "missing energy": Dark Matter. They suggest that instead of just neutrinos, the B-mesons are decaying into pairs of invisible Dark Matter particles.

Here is the breakdown of their investigation, explained simply:

1. The Two Suspects: The "Scalar" vs. The "Vector"

To explain how these Dark Matter particles are created, the scientists imagine a "messenger" particle (a mediator) that carries the force between the known matter and the dark matter. They are testing two specific types of messengers:

• The Scalar Messenger (The "Ball"): Think of this as a simple, round ball. It has no direction or spin.
• The Vector Messenger (The "Arrow"): Think of this as an arrow. It has a direction and a specific spin.

The big question is: Which one is it? Is the messenger a ball or an arrow?

2. The Detective's Trick: Comparing Two Crime Scenes

The scientists realized they could tell the difference between the "Ball" and the "Arrow" by comparing two different types of decay events:

• Crime Scene A: The B-meson decays into a Kaon (a specific type of particle) and the invisible Dark Matter.
• Crime Scene B: The B-meson decays into a K-star (a slightly heavier, excited version of the Kaon) and the invisible Dark Matter.

The Analogy:
Imagine you are trying to figure out if a sound was made by a drum (Scalar) or a trumpet (Vector). You can't just listen to the sound; you have to see how the sound behaves in two different rooms.

• If the messenger is a Ball (Scalar), the "K-star" crime scene will look very different from the "Kaon" scene. The ratio of events will drop as the energy gets higher.
• If the messenger is an Arrow (Vector), the ratio will actually rise as the energy gets higher.

The paper shows that these two scenarios produce completely different shapes on a graph. By measuring the ratio of these two events, scientists can unambiguously identify whether the messenger is a ball or an arrow, regardless of other messy details.

3. The Weight Limit: How Heavy is the Messenger?

The paper also looked at the "weight" (mass) of these messengers.

• For the Scalar (Ball): The current data doesn't give a weight limit. The ball could be light or heavy; the data fits both.
• For the Vector (Arrow): The data puts a strict limit. If the arrow is too heavy (heavier than about 3 GeV, which is roughly 3 times the mass of a proton), the math breaks down and contradicts other known limits. Therefore, if the messenger is a vector, it must be light.

4. The Verdict: Both Suspects Could Be Innocent (or Guilty)

The authors ran their models against the actual data from Belle-II.

• Result: Both the "Ball" scenario and the "Arrow" scenario can perfectly explain the shape of the data Belle-II collected.
• Conclusion: We cannot rule out either one yet. Both scenarios allow us to calculate the properties of the Dark Matter (how heavy it is, how fast it moves) and fit the data beautifully.

Summary of Findings

1. The Shape Test: By comparing how often B-mesons turn into a Kaon vs. a K-star, we can tell if the invisible messenger is a Scalar (Ball) or a Vector (Arrow). The patterns are totally different.
2. The Weight Limit: If the messenger is a Vector (Arrow), it must be light (under 3 GeV). If it's a Scalar (Ball), there is no limit yet.
3. The Fit: Both theories work well with the current data, meaning we need more experiments to decide which "messenger" is actually doing the job.

In short: The paper provides a clear "litmus test" for future experiments to distinguish between two leading theories of Dark Matter, using the ratio of two specific particle decays as the deciding factor.

Technical Summary

Technical Summary: Probing Vector- vs. Scalar-Mediator Dark-Matter Scenarios in $B \to (K, K^*)MX$ Decays

Problem Statement
The paper addresses the significant excess in the branching ratio of $B^+ \to K^+ \nu \bar{\nu}$ decays recently observed by the Belle-II collaboration, which exceeds the Standard Model (SM) prediction by a factor of approximately $5.4 \pm 1.5$. While numerous new physics explanations have been proposed, this work specifically tests the hypothesis that the excess arises from the decay of $B$ mesons into a pair of invisible dark matter (DM) fermions ($\bar{\chi}\chi$) mediated by a new field $R$. The central problem is distinguishing between two specific top-philic mediator scenarios: a scalar mediator ($R = \phi$) and a vector mediator ($R = V$). The authors aim to determine if current and future data on $B \to KMX$ and $B \to K^*MX$ can unambiguously discriminate between these spin-0 and spin-1 mediator hypotheses and constrain the properties of the mediator and DM particles.

Methodology
The authors employ an effective field theory approach within the "top-philic" framework, where the mediator couples to the top quark and the DM fermions but has no direct tree-level coupling to other SM particles. The flavor-changing neutral current (FCNC) transition $b \to s \bar{\chi}\chi$ is induced via a top-W loop.

1. Theoretical Framework:

   • Scalar Scenario (S-scenario): The interaction is governed by a scalar field $\phi$. The effective Lagrangian for $b \to s \phi$ is derived, and the differential decay rates for $B \to K \bar{\chi}\chi$ and $B \to K^* \bar{\chi}\chi$ are calculated using QCD form factors ($f_0, A_0$). Finite-width effects of the mediator are included via a $q^2$-dependent width $\Gamma_\phi(q^2)$.
   • Vector Scenario (V-scenario): The interaction involves a vector field $V$. The effective Lagrangian for $b \to s V$ possesses a $V-A$ structure. Differential rates are calculated using form factors ($f_+, A_{1,2}, V$) and a $q^2$-dependent width $\Gamma_V(q^2)$.
   • Form Factors: Non-perturbative QCD inputs are taken from lattice QCD (for $B \to K$) and light-cone sum rules (for $B \to K^*$), parametrized according to established literature.

2. Discriminators:
   The authors define two key observables to distinguish the scenarios:

   • Differential Ratio: $\hat{R}^{(R)}_{K^*/K}(q^2) = \frac{d\Gamma(B \to K^* \bar{\chi}\chi)/dq^2}{d\Gamma(B \to K \bar{\chi}\chi)/dq^2}$. This ratio is shown to be independent of specific DM parameters (masses and couplings) and depends solely on the mediator spin and kinematic form factors.
   • Integrated Rate Ratio: $R^{(R)}_{K^*/K} = \frac{\Gamma(B \to K^* \bar{\chi}\chi)}{\Gamma(B \to K \bar{\chi}\chi)}$. This ratio is sensitive to the mediator mass ($M_R$) and width.

3. Data Analysis:
   The theoretical predictions are fitted to the differential distribution of the excess events in $B \to KMX$ measured by Belle-II. The analysis accounts for the experimental variable $q^2_{rec}$ used by Belle-II (due to the inability to fully reconstruct the $B$ meson direction) by averaging theoretical $q^2$ distributions. The fit extracts the DM parameters ($M_R, m_\chi, \Gamma_R^0$) and couplings.

Key Contributions and Results

1. Unambiguous Discrimination via Differential Distributions:
   The paper demonstrates that the differential ratio $\hat{R}^{(R)}_{K^*/K}(q^2)$ exhibits qualitatively different behaviors for scalar and vector mediators, independent of DM parameters.

   • In the Scalar scenario, the ratio decreases with $q^2$ and remains below 1 over a broad kinematic region.
   • In the Vector scenario, the ratio increases with $q^2$ and stays above 1 in almost the entire kinematically available region.
     This provides a robust signature to determine the mediator spin without needing precise knowledge of the DM mass or coupling strength.

2. Constraints on Mediator Mass:
   By combining the measured excess in $\Gamma(B \to KMX)$ with the existing upper limit on $\Gamma(B \to K^*MX)$, the authors derive constraints on the mediator mass:

   • Scalar Mediator: No constraints are imposed on the scalar mediator mass $M_\phi$ by the current integrated rate data; the scenario is consistent with the data for any $M_\phi$.
   • Vector Mediator: The data imposes a tight constraint on the vector mediator mass, requiring $M_V \lesssim 3$ GeV. Heavier vector mediators are excluded because they would predict a $B \to K^*MX$ rate exceeding the experimental upper limit.

3. Parameter Extraction:
   Both scenarios provide a good description of the Belle-II differential distributions in $B \to KMX$.

   • Scalar Fit: Yields $M_\phi = 2.4 \pm 0.4$ GeV, $\Gamma_\phi^0 = 2.9^{+1.1}_{-0.9}$ GeV, and $m_\chi = 0.42^{+0.2}_{-0.4}$ GeV.
   • Vector Fit: The best fit occurs at the boundary of the allowed region defined by the $K^*$ constraint. The authors fix $M_V = 3$ GeV, yielding $\Gamma_V^0 = 4.0^{+2.0}_{-1.5}$ GeV and $m_\chi = 0.6^{+0.10}_{-0.18}$ GeV.

4. Robustness of Observables:
   The study confirms that the theoretical ratios derived in Section III remain largely unaffected by the smearing procedures and detection efficiencies inherent in the experimental analysis. The "experimentally measurable" ratio $R_{K^*/K}(q^2_{rec})$ closely approximates the theoretical ratio $\hat{R}_{K^*/K}(q^2)$, validating the use of these ratios as direct probes of the mediator spin.

Significance
The paper claims that the proposed observables offer a definitive method to distinguish between scalar and vector mediator dark matter models explaining the Belle-II $B \to K \nu \bar{\nu}$ excess. Specifically, it establishes that:

• The shape of the differential distribution ratio $B \to K^*MX$ vs. $B \to KMX$ is a model-independent probe of the mediator spin.
• The integrated rate ratio places a strict upper bound on the mass of a vector mediator ($M_V \lesssim 3$ GeV), whereas the scalar scenario remains unconstrained by current data.
• Both scenarios can successfully describe the observed excess, but they predict distinct phenomenological signatures in the $B \to K^*MX$ channel, which serves as a critical test for future experimental verification.

The authors conclude that measuring these ratios in future Belle-II data will allow for the determination of the mediator spin and the ruling out of at least one of the dark matter scenarios. They also note that similar excesses are expected in $B \to (\pi, \rho)MX$ decays relative to SM predictions, providing additional signatures for top-philic DM scenarios.