$b\to s\ell^+\ell^- (\ell = e, \mu, \tau)$ and $b\to s\nu\bar\nu$ at Belle and Belle II

Using a combined dataset of $1.2~\mathrm{ab}^{-1}$ from Belle and Belle II, this paper explores rare $B$ meson decays involving leptons and neutrinos, including the first inclusive measurement of $B\to X_s\nu\bar\nu$ and searches for $\tau^+\tau^-$ final states.

The Gist

The Cosmic Detective Agency: Hunting for "Ghost Particles"

Imagine you are a detective trying to solve a mystery in a high-stakes casino. The casino is the universe, and the "players" are subatomic particles like B-mesons.

Most of the time, these particles follow a very strict rulebook called the Standard Model. It’s like the house rules of the casino: if you bet $X$, you should get $Y$ back. But physicists suspect there might be "cheaters" in the room—new, undiscovered forces or particles (called New Physics) that are subtly changing the game.

This paper, written by scientists at the Belle and Belle II experiments, describes how they are using massive particle colliders to catch these cheaters in the act.

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1. The Problem: The "Invisible" Players

The specific mystery they are investigating involves a process called $b \to s$ transitions. Think of this as a specific type of card trick where a "b-quark" transforms into an "s-quark."

The tricky part? Sometimes, during this trick, particles called neutrinos or taus appear. These are the "ghosts" of the particle world. They are incredibly hard to see because they don't interact with much; they fly right through detectors like a ghost walking through a wall.

The Analogy: Imagine you are watching a magician perform a trick. You see him throw a ball into the air, but instead of it coming down, it just... vanishes. You can't see where it went, but you know it's gone because the magician's hand is now empty. In physics, we call that "missing energy." By measuring exactly how much "weight" or "energy" is missing from the room, we can figure out if a ghost particle was there.

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2. The Strategy: The "Tagging" Method

To catch a ghost, you need to be incredibly organized. The scientists use two main detective techniques called Tagging:

• The "Full Background Check" (Hadronic Tagging): This is like identifying every single person in a crowded room. If you know exactly who everyone else is, and you see one person suddenly vanish, you can be 100% sure they were the one who left. It’s very accurate, but it’s exhausting and takes a long time to check everyone.
• The "Quick Scan" (Inclusive Tagging): This is like looking for a general pattern of movement. It’s much faster, but it’s a bit messy and harder to be certain about.

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3. The Findings: What did they find?

The researchers looked at several different "tricks" (decay channels):

• The Tau Search ($B \to K \tau \tau$): They looked for decays involving "tau" particles (the heavy, moody cousins of electrons). They didn't find any "cheaters" (New Physics) yet, but they set very strict limits. It’s like saying, "We didn't see anyone stealing from the vault, but we've now installed cameras so good that if anyone even touches the handle, we'll catch them."
• The Neutrino Search ($B \to K \nu \nu$): This is the "holy grail." They re-examined previous data and found that the results slightly favor a version of reality where there is a bit more "vector" and "tensor" activity than the standard rulebook predicts. This is like noticing the dealer is slightly more likely to deal an Ace than the rules say they should be. It’s not a "smoking gun" yet, but it’s a very interesting clue.
• The Inclusive Search ($B \to X_s \nu \nu$): They performed the first-ever search for a broad category of these "ghostly" decays. Again, no "cheaters" were caught, but they set the world's strictest boundaries on how much these ghosts can exist.

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The Bottom Line

The scientists haven't found a "New Physics" monster under the bed just yet. However, they have built a much better "motion sensor" for the universe.

By setting these incredibly precise limits, they are narrowing down the hiding places for New Physics. They are telling the world: "If there is a new force or a new particle out there, it isn't acting this loudly. It must be much more subtle than we thought."

They are cleaning up the crime scene, making it harder and harder for the "cheaters" of the universe to hide.

Technical Summary

This technical summary outlines the research presented by Meihong Liu on behalf of the Belle and Belle II collaborations regarding rare $b \to s$ transitions.

1. Problem Statement

The research focuses on Flavour-Changing Neutral Currents (FCNC), specifically the quark transition $b \to s$. In the Standard Model (SM), these processes are forbidden at the tree level and can only occur through loop diagrams (e.g., electroweak penguins). Because these transitions are highly suppressed in the SM, they are extremely sensitive probes for New Physics (NP).

The paper specifically targets two types of rare decays that are experimentally challenging due to "missing energy" from neutrinos ($\nu$):

• Leptonic decays involving taus: $b \to s\tau^+\tau^-$ (e.g., $B^0 \to K^{*0}\tau^+\tau^-$, $B^0 \to K^0_S\tau^+\tau^-$, and $B^+ \to K^+\tau^+\tau^-$). These are sensitive to third-generation lepton couplings.
• Decays involving neutrinos: $b \to s\nu\bar{\nu}$ (e.g., $B^+ \to K^+\nu\bar{\nu}$ and the inclusive $B \to X_s\nu\bar{\nu}$).

2. Methodology

The study utilizes data from the Belle and Belle II experiments, totaling $1.2\text{ ab}^{-1}$ of $e^+e^- \to B\bar{B}$ collisions at the $\Upsilon(4S)$ resonance. The "clean" environment of $e^+e^-$ collisions allows for the reconstruction of the "companion" $B$ meson (the $B_{\text{tag}}$), which enables the inference of missing energy from the signal $B$ meson ($B_{\text{sig}}$).

Two primary tagging methods are employed:

• Inclusive Tagging: Uses machine learning to exploit signal features; it offers high efficiency but suffers from high background and simulation dependence.
• Exclusive Tagging: Uses the Full Event Interpretation (FEI) algorithm to reconstruct the $B_{\text{tag}}$ via:
  • Hadronic Tagging: Fully reconstructs specific hadronic decays. It provides high purity and tight kinematic constraints but has low efficiency ($<1\%$).
  • Semileptonic Tagging: Reconstructs $B \to D^{(*)}\ell\nu$ decays. It has higher efficiency due to larger branching fractions but lower purity due to the undetected neutrino in the tag side.

For the $b \to s\nu\bar{\nu}$ reinterpretation, the authors use Weak Effective Theory (WET) to parameterize NP through Wilson coefficients (vector, scalar, and tensor).

3. Key Contributions and Results

A. $b \to s\tau^+\tau^-$ Transitions (Hadronic Tagging)

• $B^0 \to K^{*0}\tau^+\tau^-$: Using Belle II data, the study improved the Belle constraint by a factor of two, setting an upper limit of $\mathcal{B}(B^0 \to K^{*0}\tau^+\tau^-) < 1.8 \times 10^{-3}$ at 90% CL.
• $B^0 \to K^0_S\tau^+\tau^-$: This represents the first search for this specific decay mode. By combining Belle and Belle II data, an upper limit of $8.4 \times 10^{-4}$ at 90% CL was set.
• $B^+ \to K^+\tau^+\tau^-$: Using a cut-based strategy on leptonic $\tau$ decays, the study set an upper limit of $5.6 \times 10^{-4}$ at 90% CL, providing the most stringent constraint to date.

B. $b \to s\nu\bar{\nu}$ Transitions

• $B^+ \to K^+\nu\bar{\nu}$ Reinterpretation: The authors reinterpreted previous Belle II evidence using a model-agnostic approach. They found that the data is better described by an enhanced vector contribution and a non-zero tensor contribution to the Wilson coefficients ($C_V$ and $C_T$), rather than the SM prediction.
• Inclusive $B \to X_s\nu\bar{\nu}$: This is the first study of this inclusive channel using the "sum of exclusive" method. No significant signal was observed, and an inclusive upper limit of $3.2 \times 10^{-4}$ at 90% CL was established.

4. Significance

The findings are significant for several reasons:

1. New Physics Constraints: The results provide the most stringent limits to date on several rare $b \to s$ decay channels, narrowing the parameter space for theoretical models like Leptoquarks or Charged Higgs bosons.
2. Methodological Advancement: The successful application of FEI-based hadronic tagging and the "sum of exclusive" method for inclusive decays demonstrates the growing capability of Belle II to handle complex, missing-energy final states.
3. Open Science: By releasing the likelihood for the $B^+ \to K^+\nu\bar{\nu}$ reinterpretation, the collaboration enables the broader physics community to perform rigorous, model-dependent tests of new physics scenarios.