A light DM model for large $B \to K + \mbox{invisible}$ and $K \to \pi + \mbox{invisible}$ decays and its implications for $B_s-\bar B_s$ mixing and neutron EDM

This paper proposes a light dark matter model within a two-Higgs-doublet framework to explain recent deviations in rare meson decays, demonstrating that while neutral Higgs contributions to the neutron EDM are suppressed by cancellations lifted by QCD evolution, charged scalar contributions and specific CP-violating phases can generate a neutron EDM consistent with current experimental bounds while also predicting non-negligible effects on $B_s-\bar B_s$ mixing.

The Gist

The Big Picture: A "Ghost" in the Machine

Imagine the Standard Model of physics as a very strict, well-organized library. Every book (particle) has a specific place, and the librarians (scientists) know exactly how many books should be on every shelf.

Recently, two very sensitive sensors (experiments called Belle II and NA62) started counting books in two specific sections: the "B-meson" section and the "Kaon" section. They noticed something weird. The shelves seemed to be missing more books than the library's rules predicted.

In the standard story, these missing books are neutrinos—tiny, invisible particles that zip right through the sensors without leaving a trace. But the number of missing books was slightly too high. This made the scientists wonder: What if the missing books aren't neutrinos at all? What if they are something else entirely?

This paper proposes a new theory: The missing books are actually Dark Matter. Specifically, a very light, invisible type of dark matter that acts like a "ghost" pair, sneaking out of the decay process just like neutrinos do.

The Solution: A New "Double-Deck" Library

To make this idea work, the authors built a new theoretical framework. Think of the Standard Model as a house with one floor (one Higgs boson). This new model adds a second floor (a second Higgs boson).

• The Old House: Had one Higgs boson that gave particles their mass.
• The New House: Has two Higgs bosons (let's call them H and A) and a charged cousin (H+).
• The Ghost: They also added a new, invisible particle called $\phi$ (phi). This is the Dark Matter candidate.

In this new house, the "ghost" ($\phi$) can be created in pairs whenever a heavy particle decays. Because the ghost is invisible, it looks exactly like a missing neutrino to our sensors. This perfectly explains why Belle II and NA62 are seeing more "missing energy" than expected.

The Plot Twist: The "Tug-of-War"

Now, here is where it gets tricky. Adding a second floor to the library changes how the books interact. The authors had to check if this new house would cause the whole building to collapse (i.e., violate other known laws of physics).

They focused on two specific "stress tests":

1. The Heavyweight Boxing Match ($B_s$ Mixing)

Imagine two heavy boxers, $B_s$ and its anti-particle, constantly swapping places. This is called "mixing."

• The Problem: In the new house, the new Higgs bosons (H and A) try to push these boxers around.
• The Tug-of-War: The H boson pushes one way, and the A boson pushes the other. If they are exactly the same weight, their pushes cancel out perfectly, and nothing happens.
• The Result: The authors found that if the H and A bosons are slightly different weights, they don't cancel out completely. They create a small, measurable shift in the boxing match. This shift is small enough to be hidden within current experimental errors, but it's big enough that future, more precise experiments might catch them. It's like hearing a faint creak in the floorboards that suggests a new room exists upstairs.

2. The Magnetic Compass (Neutron EDM)

Imagine a neutron as a tiny compass needle. In a perfect world, this needle points straight. But if there is "CP violation" (a subtle asymmetry in the laws of physics), the needle might tilt slightly, creating an Electric Dipole Moment (EDM).

• The Neutral Higgs Trap: When the neutral Higgs bosons (H and A) try to tilt the needle, they do a weird dance. One tries to tilt it left, the other right. At the high-energy level where they are created, they cancel each other out perfectly. The needle stays straight.
• The "Time Travel" Effect: However, as these forces travel down to the energy levels we can measure (like in a lab), the rules of the universe (QCD) change slightly. This is like a "renormalization group evolution." It's as if the cancellation gets "stuck" or "unlocked" as it travels through time. The perfect cancellation breaks, and the needle does tilt a little bit.
• The Charged Higgs Surprise: Then there is the charged Higgs boson (H+). It doesn't play by the cancellation rules. It tries to tilt the needle much harder—about 100 times harder than the neutral ones.
• The Grand Cancellation: Here is the clever part. The authors realized that if the "tilting" from the neutral Higgs and the "tilting" from the charged Higgs happen with opposite signs (one left, one right), they can cancel each other out again.
• The Conclusion: By carefully tuning the "phases" (the angles of the tilt), the model can produce a neutron EDM that is just barely small enough to hide from current experiments, but large enough to be detected by future, more sensitive ones.

The Takeaway

This paper is a detective story.

1. The Clue: Experiments see more invisible particles than expected.
2. The Suspect: A light Dark Matter particle.
3. The Alibi: A new "Two-Higgs" model that explains the clue.
4. The Test: The model was put under pressure by checking if it breaks the rules of $B_s$ mixing or the neutron's magnetic tilt.
5. The Verdict: The model survives! It creates small, interesting effects that are currently hidden but could be the smoking gun for new physics in the next few years.

It's like finding a secret door in a house you thought you knew perfectly. You can't see the room yet, but you can hear the floorboards creaking, and you know exactly where to look next.

Technical Summary

1. Problem Statement

Recent experimental results from Belle II ($B^+ \to K^+ \nu\bar{\nu}$) and NA62 ($K^+ \to \pi^+ \nu\bar{\nu}$) have reported central values for branching ratios that exceed Standard Model (SM) predictions.

• Belle II: Observed a $2.7\sigma$ excess in $B^+ \to K^+ \nu\bar{\nu}$, with a weighted average $2.1\sigma$ above the SM.
• NA62: Reported the first observation of $K^+ \to \pi^+ \nu\bar{\nu}$ with a central value exceeding the SM expectation by $\Delta B_K \approx 4.6 \times 10^{-11}$.

Since the final-state neutrinos are invisible, these signals are effectively $B^+ \to K^+ + \text{invisible}$ and $K^+ \to \pi^+ + \text{invisible}$. The authors investigate whether these anomalies can be explained by the production of light Dark Matter (DM) pairs ($\phi\phi$) instead of neutrinos, and they analyze the phenomenological consequences of such a model on other precision observables, specifically $B_s - \bar{B}_s$ mixing and the neutron Electric Dipole Moment (nEDM).

2. Methodology

A. Effective Field Theory (EFT) Framework

The authors first establish the low-energy requirements using a Low-Energy Effective Field Theory (LEFT). To accommodate the observed excesses, they introduce a light real scalar DM candidate $\phi$ ($m_\phi < 177$ MeV) coupled to quarks via dimension-6 operators:
$$\mathcal{L}_{qq\phi^2} \supset \frac{1}{2} \left[ C_{S,ij}^{d\phi} (\bar{d}_i d_j) + C_{P,ij}^{d\phi} (\bar{d}_i i\gamma_5 d_j) \right] \phi^2 + \dots$$
To explain the data, specific Wilson coefficients are required:

• $C_{S,bs}^{d\phi} \sim (2.5 - 6.0) \times 10^{-5} \, \text{TeV}^{-1}$ for $B^+ \to K^+ \phi\phi$.
• $C_{S,sd}^{d\phi} \sim (0.06 - 0.5) \times 10^{-7} \, \text{TeV}^{-1}$ for $K^+ \to \pi^+ \phi\phi$.
• Constraints from DM relic density and direct detection (Migdal effect) restrict the DM mass to $110-146$ MeV and fix specific combinations of couplings to up and down quarks.

B. UV-Complete Model: Type-III 2HDM

To realize these effective operators in a renormalizable framework, the authors propose a Type-III Two-Higgs-Doublet Model (2HDM) extended with a real scalar singlet $\phi$ (the DM candidate).

• Higgs Sector: Two doublets $H_1$ and $H_2$. $H_1$ acquires a VEV ($v \approx 246$ GeV) and behaves like the SM Higgs ($h$). $H_2$ has a zero VEV and contains heavy scalars: a neutral CP-even $H$, a neutral CP-odd $A$, and a charged scalar $H^\pm$.
• Yukawa Couplings: Both Higgs doublets couple to quarks, allowing for tree-level Flavor Changing Neutral Currents (FCNCs).
• DM Stability: A $Z_2$ symmetry ensures $\phi$ is stable.
• Mechanism: The exchange of $H$ and $A$ between quarks and $\phi$ generates the required effective operators for the rare decays. The model assumes specific non-zero Yukawa couplings ($Y_{2}^{bs}, Y_{2}^{sd}, Y_{2}^{dd}, Y_{2}^{uu}$) to match the benchmark Wilson coefficients.

3. Key Contributions and Analysis

The paper extends previous work by analyzing the correlated constraints imposed by this specific model on $B_s - \bar{B}_s$ mixing and the neutron EDM, which were not fully explored in previous effective operator studies.

A. $B_s - \bar{B}_s$ Mixing

• Mechanism: The exchange of neutral scalars $H$ and $A$ generates four-quark operators contributing to $B_s$ mixing.
• Cancellation: If $m_H = m_A$, the contributions from $H$ and $A$ cancel exactly. A mass splitting ($m_H \neq m_A$) is required to generate a non-zero New Physics (NP) contribution.
• Result: The NP contribution to the mass difference $\Delta m_s$ is proportional to $(1 - m_H^2/m_A^2)$.
  • The authors find that for allowed parameter spaces (where $m_H \approx 0.5 - 1$ TeV and $|Y_{2}^{dd}|$ is near perturbative limits), the NP contribution can reach $\sim -0.13$ to $-0.08 \, \text{ps}^{-1}$.
  • This is significant compared to the experimental uncertainty ($\Delta m_s^{\text{exp}} - \Delta m_s^{\text{SM}} \approx -0.465 \pm 0.63 \, \text{ps}^{-1}$), implying the model can be tested by future precision measurements.

B. Neutron Electric Dipole Moment (nEDM)

The study investigates CP-violating phases in the Yukawa couplings ($Y_{2}^{dd} \to |Y|e^{i\theta_{dd}}$, $Y_{2}^{uu} \to |Y|e^{i\theta_{uu}}$).

1. Neutral Higgs Contribution ($H, A$):

   • At the high scale, there is a significant cancellation between the quark Electric Dipole Moment (EDM) and Chromo-EDM (CEDM) contributions.
   • QCD Running: This cancellation is lifted by QCD Renormalization Group Evolution (RGE) from the heavy mass scale down to the hadronic scale ($\mu \sim 1$ GeV). The RGE factors for EDM ($\eta_{\text{EDM}}$) and CEDM ($\eta_{\text{CDM}}$) differ, resulting in a residual contribution:
     $$d_n^{\text{neutral}} \approx -0.17 \, d_n^{\text{EDM}}(\mu_h)$$
   • The contribution scales as $1/m_H^2$. For $m_H = 0.5$ TeV, the phases must be small ($\tan \theta \lesssim 10^{-1}$) to satisfy current bounds.

2. Charged Higgs Contribution ($H^\pm$):

   • The charged scalar contribution does not suffer from the H-A cancellation mechanism.
   • Numerically, the charged Higgs contribution is roughly two orders of magnitude larger than the neutral Higgs contribution for the same parameters.
   • To satisfy the nEDM bound ($|d_n| < 1.8 \times 10^{-26} e\cdot\text{cm}$), the combination of phases must be extremely small ($|\tan \theta_{uu} + \tan \theta_{dd}| \lesssim 10^{-3}$).

3. Total Cancellation:

   • Crucially, the authors demonstrate that the neutral and charged contributions can have opposite signs depending on the relative signs of the phases $\theta_{uu}$ and $\theta_{dd}$.
   • By tuning these phases, a cancellation between the large charged Higgs term and the neutral Higgs term can occur, allowing the model to generate a neutron EDM consistent with current bounds even with relatively large CP-violating phases.

4. Results

• Feasibility: The Type-III 2HDM with light scalar DM successfully reproduces the effective operators required to explain the Belle II and NA62 anomalies.
• $B_s$ Mixing: The model predicts a non-negligible contribution to $B_s - \bar{B}_s$ mixing ($\Delta m_s^{\text{NP}} \sim -0.1 \, \text{ps}^{-1}$), which is within current experimental limits but potentially detectable with improved precision.
• nEDM:
  • Neutral Higgs contributions are suppressed by RGE-lifted cancellations.
  • Charged Higgs contributions are dominant but can be cancelled by the neutral sector.
  • The model remains viable provided the CP-violating phases in the first-generation Yukawa couplings are tuned to allow for this cancellation.
• Constraints: The model satisfies constraints from $B_s \to \text{invisible}$, $B \to X_s \gamma$, and Kaon mixing ($K^0 - \bar{K}^0$).

5. Significance

• UV Completion: This work provides a concrete, renormalizable UV-complete framework (Type-III 2HDM + Scalar DM) for the effective operators proposed to explain recent flavor anomalies.
• Interplay of Observables: It highlights the critical interplay between rare meson decays, neutral meson mixing, and EDMs. Specifically, it shows that explaining the $B/K$ anomalies does not automatically rule out the model via $B_s$ mixing or nEDM, provided specific mass hierarchies ($m_H \neq m_A$) and phase cancellations are realized.
• RGE Effects: The paper emphasizes the importance of QCD RGE evolution in lifting the EDM/CEDM cancellation in neutral Higgs models, a general feature often overlooked in simplified analyses.
• Future Probes: The predicted contributions to $B_s$ mixing and the specific phase-dependent cancellations in nEDM offer distinct signatures for future experimental tests at Belle II, LHCb, and next-generation EDM experiments.