Measurements of electroweak penguins and $B$ decays to final states with missing energy at Belle and Belle II

This paper presents results from the Belle and Belle II experiments on rare electroweak penguin $B$ decays involving missing energy, specifically analyzing $b\to s \ell^+\ell^-$, $b\to s\tau^+\tau^-$, and $b\to s\nu\bar \nu$ transitions using a 1.3 ab$^{-1}$ dataset collected at the $\Upsilon(4S)$ resonance.

The Gist

Imagine the universe as a giant, high-stakes billiard table. In this game, particles are the balls, and they bounce off each other, breaking apart, and recombining in predictable ways. Physicists at the Belle II experiment in Japan are like super-observant referees watching these collisions, hoping to spot a ball that behaves in a way the rulebook (the Standard Model of physics) says it shouldn't.

Here is a breakdown of what this paper is about, using simple analogies:

The Setting: A Factory for "B" Particles

The researchers are working at the SuperKEKB collider, which is essentially a particle factory. It smashes electrons and positrons together to create a specific type of particle called the $\Upsilon(4S)$. Think of this particle as a unstable egg that almost always cracks open to reveal two "B-mesons" (a particle and its anti-particle) flying out in opposite directions.

The team has collected a massive amount of data (over a billion of these pairs). This is their "gold mine" for looking for rare events.

The Goal: Finding the "Invisible"

Most of the time, when these B-mesons decay (break apart), they leave behind a clear trail of debris (electrons, muons, pions) that the detectors can see. But the scientists are looking for something much sneakier: Missing Energy.

Imagine you see a car crash, but when you look at the wreckage, a whole chunk of the car is gone. You know it must be there because of the laws of physics, but you can't see it. In particle physics, this "missing chunk" is often a neutrino or a tau particle that slips through the detectors like a ghost.

The paper presents results on three specific types of "ghostly" searches:

1. The "Lepton" Mystery ($b \to s \ell^+ \ell^-$)

• The Analogy: Imagine a B-meson is a magician who usually pulls a rabbit out of a hat. Sometimes, it pulls out a pair of rabbits (an electron and a positron, or a muon and an anti-muon).
• The Problem: Other experiments (like LHCb) have seen these rabbits behaving strangely, suggesting the magician might be using a trick not in the official rulebook.
• The Belle II Result: Belle II looked at the sum of all these tricks. They found that the rabbits are behaving exactly as the rulebook predicts. There is no evidence of "New Physics" here yet; the magic is standard.

2. The "Heavy Ghost" Hunt ($b \to s \tau^+ \tau^-$)

• The Analogy: The Tau ($\tau$) is like a heavy, clumsy cousin of the electron. It's much harder to spot because it decays quickly and often leaves behind invisible neutrinos.
• The Theory: Some theories suggest that if "New Physics" exists, it might prefer to interact with these heavy Taus more than with light electrons.
• The Belle II Result: They looked for B-mesons decaying into a Kaon and a pair of Taus. They found zero evidence of this happening. However, they set a new, stricter "speed limit" (an Upper Limit). They proved that if this rare event does happen, it's at least four times rarer than anyone thought before. It's like saying, "We didn't find the ghost, but we proved it's not hiding in this room."

3. The "Pure Ghost" Search ($b \to s \nu \bar{\nu}$)

• The Analogy: This is the ultimate ghost hunt. The B-meson decays into a Kaon and two neutrinos. Neutrinos are the ultimate "invisible" particles; they pass through the entire Earth without stopping. The only way to know they were there is to notice that energy is missing from the final count.
• The Context: Recently, there was a hint that this specific decay might be happening more often than expected.
• The Belle II Result: They looked at the data and found no significant signal. They set the world's best limit on how often this can happen. It's like saying, "We checked the whole house for the ghost, and while we didn't find it, we now know exactly how quiet the house must be for it to remain hidden."

4. The "Re-Interpretation" Tool

• The Analogy: Imagine the scientists found a suspicious footprint (the $B^+ \to K^+ \nu \bar{\nu}$ excess mentioned in the paper). Instead of just saying "It's a bear" or "It's a wolf," they built a universal translator.
• The Innovation: They created a new mathematical tool that allows other scientists to take this single footprint and test it against any theory they want. Is it a bear? A wolf? A dragon? This tool lets the whole physics community quickly check their own theories against the data without having to re-do the entire experiment.

The Bottom Line

This paper is a report card from the "ghost hunters" of the particle world.

1. Did they find New Physics? Not yet. The data mostly matches the current rulebook (Standard Model).
2. Did they learn anything? Yes! They have ruled out many "what-if" scenarios by proving that certain rare events are even rarer than we thought.
3. What's next? They have provided better tools and stricter limits, which helps theorists narrow down where the "New Physics" might actually be hiding.

In short, they didn't find the treasure, but they drew a much more accurate map of where the treasure isn't, which is just as valuable for future explorers.

Technical Summary

1. Problem and Motivation

The paper addresses the search for Flavor-Changing Neutral Currents (FCNC) processes, specifically rare $B$ meson decays involving electroweak penguins. In the Standard Model (SM), these processes are forbidden at the tree level and occur only via loop diagrams, making them highly suppressed and sensitive probes for New Physics (NP).

The primary motivations are:

• Resolving Tensions: Recent LHCb measurements of exclusive $B \to K^{(*)}\ell^+\ell^-$ decays show $2-4\sigma$ tensions with SM predictions in the $q^2$ region of $1 < q^2 < 6 \text{ GeV}^2$. However, inclusive measurements have smaller theoretical uncertainties and have historically been consistent with the SM. Clarifying this discrepancy is crucial.
• Missing Energy Signatures: Decays involving neutrinos (missing energy) are difficult to reconstruct but are powerful NP probes. Recent evidence of an excess in $B^+ \to K^+ \nu\bar{\nu}$ decays (2.7$\sigma$ tension) necessitates further investigation into inclusive $B \to X_s \nu\bar{\nu}$ and related channels.
• Lepton Flavor Universality (LFU): Testing if NP couples differently to the third generation ($\tau$ leptons) compared to electrons and muons.

2. Methodology and Experimental Setup

The analysis utilizes data from the Belle and Belle II experiments at the SuperKEKB collider (KEK, Japan).

• Dataset: A combined integrated luminosity of approximately 1.1 ab$^{-1}$ (711 fb$^{-1}$ from Belle + 365 fb$^{-1}$ from Belle II Run 1). Note: Belle II Run 2 data (2024) was not used.
• Production Environment: $e^+e^-$ collisions at the $\Upsilon(4S)$ resonance ($\sqrt{s} = 10.58$ GeV), which decays almost exclusively into $B\bar{B}$ pairs.
• Reconstruction Strategy (B-tagging):
  • Exclusive Tagging: One $B$ meson ($B_{tag}$) is fully reconstructed in hadronic or semileptonic decay channels.
  • Signal Inference: Using the known initial state ($\Upsilon(4S)$) and the reconstructed $B_{tag}$, the kinematics, flavor, and charge of the signal $B$ meson ($B_{sig}$) are inferred. This allows for the recovery of missing information (neutrinos) in the signal side.
• Background Suppression:
  • Multivariate Classifiers: Feature Tokenizer Transformers and Boosted Decision Trees (BDT) are used to suppress backgrounds from semileptonic $B/D$ decays, misidentified hadrons, and charmonium resonances.
  • Kinematic Cuts: Variables such as invariant mass ($m_{K\ell}$), missing mass, and extra calorimetric energy ($E_{extra}$) are optimized.

3. Key Contributions and Results

A. Inclusive $B \to X_s \ell^+\ell^-$ ($\ell = e, \mu$)

• Method: The signal is reconstructed as a combination of two opposite-charged leptons and an $X_s$ system (sum of 20 exclusive modes covering ~71% of the inclusive branching fraction).
• Improvements: Signal modeling was significantly improved over previous Belle/BaBar measurements by updating exclusive branching fractions, improving resonance descriptions above 1.1 GeV, and calibrating fragmentation using $B \to X_s \gamma$ and $B \to X_s J/\psi$ data. This reduced systematic uncertainties by a factor of 2–3.
• Results:
  • The differential branching fraction $dB/dq^2$ shows good agreement with SM predictions, particularly in the $1 < q^2 < 6 \text{ GeV}^2$ region where exclusive modes show tensions.
  • Branching Fractions:
    • $\mathcal{B}(B \to X_s \ell^+\ell^-)_{1-6} = (1.36 \pm 0.23^{+0.13}_{-0.10}) \times 10^{-6}$.
    • Total Inclusive $\mathcal{B}(B \to X_s \ell^+\ell^-) = (3.91 \pm 0.49^{+0.34}_{-0.26}) \times 10^{-6}$.
  • LFU Ratio: First measurement of $R_{X_s} = \mathcal{B}(B \to X_s \mu^+\mu^-)/\mathcal{B}(B \to X_s e^+e^-) = 0.74 \pm 0.19 \pm 0.04$, consistent with the SM.

B. $b \to s \tau^+\tau^-$ Transitions

• Context: $\tau$ leptons are unique probes for NP models that enhance couplings to the third generation. SM branching fractions are $\sim 10^{-7}$, currently unobservable, but NP could enhance them by orders of magnitude.
• $B \to K^+ \tau^+\tau^-$ Search:
  • Reconstructed using leptonic $\tau$ decays only.
  • Result: No significant signal. Set a World Best Upper Limit (UL): $\mathcal{B} < 0.56 \times 10^{-3}$ (90% CL), improving the previous limit by a factor of four.
• $B \to K_S^0 \tau^+\tau^-$ Search:
  • First search for this channel. Reconstructed targeting one-prong $\tau$ decays ($e, \mu, \pi, \rho$). Events categorized by final state ($\ell\ell, \ell h, h\ell, \rho\ell$, etc.).
  • Result: No significant signal. Set a World First UL: $\mathcal{B} < 0.84 \times 10^{-3}$ (90% CL).

C. Inclusive $B \to X_s \nu\bar{\nu}$ Search

• Context: The SM predicts $\mathcal{B} \approx 3.48 \times 10^{-5}$. Previous best limit was $6.4 \times 10^{-4}$ (ALEPH). This channel is sensitive to different NP parameters than exclusive $B \to K^{(*)}\nu\bar{\nu}$.
• Method: Sum of 30 exclusive modes. Signal extracted via a binned maximum likelihood fit to the BDT score and reconstructed invariant mass $M(X_s)$.
• Results: No significant signal observed. New World Best Upper Limits at 90% CL:
  • $\mathcal{B} < 2.2 \times 10^{-5}$ for $0 < M(X_s) < 0.6$ GeV.
  • $\mathcal{B} < 9.5 \times 10^{-4}$ for $0.6 < M(X_s) < 1.0$ GeV.
  • $\mathcal{B} < 3.12 \times 10^{-4}$ for $M(X_s) > 1$ GeV.
  • Combined UL: $\mathcal{B}(B \to X_s \nu\bar{\nu}) < 3.2 \times 10^{-4}$.

D. Reinterpretation of $B^+ \to K^+ \nu\bar{\nu}$

• Goal: To provide a model-agnostic framework for interpreting the recent 2.7$\sigma$ excess in $B^+ \to K^+ \nu\bar{\nu}$.
• Method: Developed a novel reweighting technique to construct a model-agnostic likelihood. The method reweights the SM signal density based on $q^2$ (squared di-neutrino mass) to test various NP models.
• Results:
  • Interpreted within Weak Effective Theory (WET) including dimension-6 operators (Scalar, Vector, Tensor).
  • The data prefers a non-zero tensor contribution and a larger vector contribution compared to the SM.
  • The WET signal has a significance of 3.3$\sigma$ against the background-only hypothesis.
  • The analysis releases the joint number density distribution and likelihood tools to facilitate future reinterpretations.

4. Significance

• Theoretical Clarification: The inclusive $B \to X_s \ell^+\ell^-$ results, showing consistency with the SM, help contextualize the tensions seen in exclusive LHCb measurements, suggesting the anomalies may be localized to specific exclusive modes or kinematic regions rather than a global inclusive failure of the SM.
• Experimental Milestones: The paper establishes world-leading limits for $B \to K \tau^+\tau^-$ and $B \to X_s \nu\bar{\nu}$, and performs the first search for $B \to K_S^0 \tau^+\tau^-$.
• Methodological Innovation: The development of the model-agnostic reweighting method for $B^+ \to K^+ \nu\bar{\nu}$ provides the community with a robust tool to test a wide range of New Physics scenarios without re-running complex analyses.
• Future Physics: These results constrain the parameter space for NP models attempting to explain LFU violations and the $B \to K \nu\bar{\nu}$ excess, guiding future theoretical developments and experimental searches at Belle II and the LHC.

Note: The paper explicitly states that the results presented are preliminary and that some references are pending publication.