Dark Matter emission at Belle II and NA62 in Minimal Flavor Violation framework

This paper demonstrates that while the Minimal Flavor Violation framework with a single dark matter multiplet can naturally explain either the $K^+ \to \pi^+ \nu \bar{\nu}$ or $B^+ \to K^+ \nu \bar{\nu}$ excess observed at Belle II and NA62, it cannot simultaneously account for both anomalies, thereby highlighting the specific constraints and testability of flavored dark matter models.

The Gist

The Big Picture: Hunting for Invisible Ghosts

Imagine the universe is a giant, busy party. We know most of the guests (the particles we can see, like electrons and quarks), but we suspect there are invisible "ghosts" floating around too. These ghosts are Dark Matter. We can't see them, but we know they must be there because they have mass and gravity.

Physicists at two major experiments, NA62 (in Europe) and Belle II (in Japan), have been watching specific "parties" where heavy particles decay (break apart). They noticed something strange: sometimes, these particles seem to lose more energy than they should. It's like watching a magician pull a rabbit out of a hat, but the rabbit is invisible, and the hat is lighter than it was before.

This paper asks: Could these missing energy clues be the ghosts (Dark Matter) we've been looking for?

The Rulebook: "Minimal Flavor Violation" (MFV)

To solve this mystery, the authors use a specific rulebook called Minimal Flavor Violation (MFV).

Think of the Standard Model (our current understanding of physics) as a strict dance club with a very specific dress code. Everyone has to dance in a specific pattern based on their "flavor" (like up, down, strange, charm, etc.).

• The MFV Rule: If new guests (Dark Matter) show up, they must follow the exact same dress code and dance steps as the regular guests. They can't just do whatever they want; they have to respect the existing hierarchy.
• The Twist: In this specific dance club, the rules are so strict that they accidentally make the lightest new guest stable. They can't leave the party or disappear. This makes them perfect candidates for Dark Matter.

The Mystery: Two Different Excesses

The experiments found two "excesses" (more events than expected):

1. The Kaon Mystery (NA62): A particle called a Kaon decaying into a Pion and missing energy. The number of events is slightly higher than the Standard Model predicts.
2. The B-Meson Mystery (Belle II): A heavier particle called a B-meson decaying into a Kaon and missing energy. This one is even more exciting; the number of events is significantly higher than expected (about 2.7 times the "standard" noise).

The authors wanted to see if their "MFV Dance Club" (with Dark Matter guests) could explain both mysteries at the same time using just one type of Dark Matter guest.

The Investigation: Scalar vs. Fermion

The authors tested two types of potential Dark Matter guests:

1. Scalar DM: Think of these as "ghosts" that are like invisible balls (spin-0).
2. Fermion DM: Think of these as "ghosts" that are like invisible spinning tops (spin-1/2).

They ran the numbers to see if a single type of ghost, with a single mass, could explain why both the Kaon and the B-meson were losing extra energy.

The Results: A "Goldilocks" Problem

The findings were a bit disappointing for a simple solution, but very interesting for theory:

• The Low-Mass Problem: If the Dark Matter is very light (like a feather), it fits the Kaon data perfectly. However, it creates too much "noise" for the B-meson data. It's like a key that fits the front door lock but breaks the back door lock.
• The High-Mass Problem: If the Dark Matter is heavier (like a brick), it fits the B-meson data perfectly. But now, it's too heavy to explain the Kaon data. It's like a key that fits the back door but is too big for the front door.
• The Conclusion: You cannot use one single type of Dark Matter with one single mass to explain both mysteries simultaneously within this strict rulebook.

The Solution? A "Two-Guest" Party

To explain both mysteries at once, the authors suggest you would need to invite two different types of Dark Matter guests, or have one type that comes in two different "sizes" (masses) for different flavors.

• Imagine needing a small ghost for the Kaon party and a large ghost for the B-meson party.
• While this is possible, it makes the theory less "minimal" (less simple/economical). The authors decided to stop there, noting that while a simple one-guest solution doesn't work, the idea of "flavored" Dark Matter is still a very strong and testable theory.

Summary

This paper is a detective story. The detectives (physicists) tried to solve two missing-energy crimes using a single suspect (one type of Dark Matter) following a strict rulebook (MFV).

• Verdict: The single suspect is innocent of doing both crimes at once.
• New Lead: To solve both, you likely need a team of suspects with different sizes.
• Takeaway: Even though the simple version didn't work, the framework remains a powerful tool for understanding how Dark Matter might interact with the particles we can see. The experiments at Belle II and NA62 are successfully narrowing down the search.

Technical Summary

Technical Summary: Dark Matter emission at Belle II and NA62 in Minimal Flavor Violation framework

Problem Statement
Recent experimental data from the NA62 and Belle II collaborations have revealed intriguing excesses in rare flavor-changing neutral current (FCNC) decays. Specifically, NA62 reports a central value for the $K^+ \to \pi^+ \nu\bar{\nu}$ branching ratio that is approximately 50% higher than the Standard Model (SM) prediction, while Belle II observes a $\sim 2.7\sigma$ excess in the $B^+ \to K^+ \nu\bar{\nu}$ channel. These anomalies suggest the potential emission of invisible particles, possibly dark matter (DM), rather than standard neutrinos. The central problem addressed in this work is whether these simultaneous excesses can be explained within the Minimal Flavor Violation (MFV) framework, where new physics respects the SM flavor structure, and where a new QCD-singlet field $\chi$ transforming under the global quark flavor symmetry $SU(3)^3$ serves as a stable DM candidate.

Methodology
The authors investigate the emission of flavored dark matter in meson decays ($d_i \to d_j + \not{E}$) within the MFV paradigm. The analysis proceeds through the following steps:

1. Theoretical Framework: The study considers two types of DM candidates: a scalar multiplet $S$ and a fermion multiplet $\psi$. The interactions are governed by effective operators of dimension-6 (for scalars) and dimension-6/7 (for fermions), constructed to be invariant under the $U(3)^5$ flavor symmetry, with breaking controlled solely by SM Yukawa spurions. The authors focus on the lowest-dimensional representations, specifically $(3,1,1)$ and $(1,1,3)$, which allow for flavor-violating interactions in the down-quark sector.
2. Mass Degeneracy: In the leading order of the MFV expansion, the components of the DM multiplet are degenerate in mass. Mass splittings arise only at higher orders via Yukawa insertions. The analysis assumes this degeneracy, meaning a single DM mass parameter $m_\chi$ characterizes the multiplet.
3. Phenomenological Calculation: The authors calculate the decay rates for $K \to \pi \chi\bar{\chi}$ and $B \to K^{(*)} \chi\bar{\chi}$ processes. They utilize form factors derived from lattice QCD and experimental data (Refs. [87, 91–94]) to model the hadronic matrix elements. The differential decay rates are computed for both scalar and fermion DM scenarios.
4. Statistical Analysis: A comprehensive likelihood function is constructed, incorporating all publicly available experimental data:
   • NA62: $K^+ \to \pi^+ \nu\bar{\nu}$ (2016–2022 dataset) and searches for $K^+ \to \pi^+ X$.
   • KOTO: $K_L \to \pi^0 \nu\bar{\nu}$ limits.
   • Belle II: $B^+ \to K^+ \nu\bar{\nu}$ (using both Hadronic Tagging Analysis and Inclusive Tagging Analysis).
   • Belle and BaBar: Upper limits on $B \to \pi/\rho \nu\bar{\nu}$ and $B \to K^{(*)} \nu\bar{\nu}$.
     The likelihood includes nuisance parameters for background normalizations and SM branching ratio uncertainties. The improvement of the BSM hypothesis over the SM is quantified using the test statistic $\Delta \equiv -2 \ln (\hat{\mathcal{L}}_{BSM} / \hat{\mathcal{L}}_{SM})$.

Key Contributions and Results
The paper presents a systematic scan of the parameter space (DM mass $m_\chi$ and coupling coefficients) for both scalar and fermion DM candidates.

• Scalar DM ($S \sim (3,1,1)$):

  • For low DM masses ($m_S \lesssim 125$ MeV), the model can accommodate the NA62 excess, improving the fit by up to $\sim 3\sigma$. However, the coupling strength required to explain the Belle II excess in this mass range is excluded by NA62 constraints.
  • For higher masses ($m_S \gtrsim 150$ MeV), the model can explain the Belle II excess (improving the fit by $\sim 2\text{--}3\sigma$ at $m_S \sim 0.5$ GeV) but fails to account for the NA62 excess, as the NA62 signal is left unexplained.
  • The $S \sim (1,1,3)$ representation yields qualitatively identical results.

• Fermion DM ($\psi \sim (3,1,1)$):

  • The behavior mirrors the scalar case. Low-mass fermions ($m_\psi \lesssim 125$ MeV) can fit the NA62 data but are constrained from explaining the Belle II excess.
  • Higher-mass fermions ($m_\psi \gtrsim 150$ MeV) can fit the Belle II excess but leave the NA62 anomaly unexplained.
  • The $\psi \sim (1,1,3)$ case yields similar qualitative conclusions.

• Unified Explanation: The central finding is that a unified explanation of both the $K^+ \to \pi^+ \nu\bar{\nu}$ and $B^+ \to K^+ \nu\bar{\nu}$ excesses cannot be achieved within a minimal setup containing only a single DM multiplet with nearly degenerate masses. The constraints from Kaon decays and the requirements for B-meson decays are mutually exclusive in this minimal framework.

Significance and Claims
The authors conclude that while the MFV framework provides a theoretically robust mechanism for stabilizing flavored dark matter and can individually address the anomalies in either the Kaon or B-meson sectors, it fails to provide a simultaneous solution in its minimal form.

• Theoretical Implication: To reconcile both excesses, the model would require non-minimal extensions, such as introducing an additional DM multiplet with a different mass for each flavor or generating significant mass splittings within a single multiplet. The authors note that while such extensions are consistent with MFV, they move away from strict minimality and are not pursued in this work.
• Experimental Relevance: The study underscores the intricate interplay between MFV-based model building and precision flavor experiments. It demonstrates that flavored dark matter remains a testable paradigm, capable of addressing collider-scale anomalies, but highlights the tension between different flavor sectors when constrained by minimal assumptions.
• Methodological Value: The paper provides a rigorous likelihood analysis incorporating the latest datasets from NA62, KOTO, Belle II, Belle, and BaBar, offering a comprehensive phenomenological assessment of the MFV DM scenario against current experimental bounds.