Study of $B^+ \to μ^+ ν_μ$ decays at Belle and Belle II

This paper presents a combined measurement from the Belle and Belle II experiments of the rare leptonic decay $B^+ \to \mu^+ \nu_\mu$, yielding a branching fraction of $(4.4 \pm 1.9 \pm 1.0) \times 10^{-7}$ and providing the most precise search for this process to date.

The Gist

The Cosmic Needle in a Haystack: A Tale of Two Detectors

Imagine you are trying to listen for a single, specific whisper in the middle of a roaring heavy metal concert. That is essentially what physicists at the Belle and Belle II experiments are doing. They are looking for a very rare "whisper"—a specific type of particle decay—to see if the universe follows the rules we think it does, or if there’s a secret "ghost" hiding in the music.

Here is a breakdown of what this massive scientific paper is actually saying.

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1. The Goal: Checking the "Recipe" of the Universe

Everything in the universe is built according to a recipe called the Standard Model. One of the ingredients in this recipe is a particle called the $B^+$ meson. Occasionally, this particle "decays" (breaks apart) into a muon and a neutrino.

The Standard Model predicts exactly how often this should happen. If we observe it happening more or less often than predicted, it’s like finding an extra ingredient in a cake that shouldn't be there. That extra ingredient would be "New Physics"—proof of things like dark matter or new forces we haven't discovered yet.

2. The Challenge: The "Invisible" Problem

The problem is that one of the products of this decay, the neutrino, is a "ghost particle." It flies through detectors without leaving a trace.

Imagine you are watching a high-speed car crash through a window. You see a car (the muon) fly off in one direction, but you can't see the other car that hit it because it’s invisible. To figure out what happened, you have to use math and physics to "reconstruct" the invisible car based on how the visible car moved. This paper describes a highly advanced way of doing that math using data from two massive particle detectors in Japan.

3. The Method: The "Tagging" Strategy

Because the signal is so rare and the "noise" (other particle collisions) is so loud, the scientists use a trick called Inclusive Tagging.

Think of it like this: You are looking for one specific person (the signal) in a massive, crowded stadium. Instead of searching every face, you look at the person sitting next to them. If you can identify the "friend" (the other $B$ meson produced in the collision) with 100% certainty, you can narrow your search down to just the people sitting in that specific section. This makes finding the "whisper" much easier.

4. The Results: What did they find?

• The Measurement: They measured the decay rate and found it to be $(4.4 \pm 1.9 \pm 1.0) \times 10^{-7}$. In plain English: out of every 10 million $B^+$ mesons, only about 4 or 5 do this specific decay. It is incredibly rare.
• The Verdict: The result is very close to what the Standard Model predicted. It didn't "break" the recipe, but it was precise enough to set new boundaries.
• Hunting for Ghosts: They also searched for Sterile Neutrinos—hypothetical "super-ghosts" that might explain why the universe exists. They didn't find them this time, but they proved that if these ghosts exist, they must be even more elusive than we previously thought.

5. Why does this matter?

Even though they didn't find "New Physics" this time, this paper is a massive achievement in precision.

In science, knowing exactly where the boundaries are is just as important as finding something new. By tightening the limits on what is not possible, they are narrowing the search area for the next generation of physicists. They have essentially cleaned the lens of the cosmic microscope, making it sharper for the next time we look for the secrets of the universe.

Technical Summary

Technical Summary: Study of $B^+ \to \mu^+ \nu_\mu$ Decays at Belle and Belle II

1. Problem and Motivation

The leptonic decay $B^+ \to \mu^+ \nu_\mu$ is a rare process in the Standard Model (SM), proceeding via the annihilation of $\bar{b}$ and $u$ quarks into a virtual $W^+$ boson. This decay is highly suppressed due to both the smallness of the CKM matrix element $|V_{ub}|$ and helicity suppression.

Because of this suppression, the decay is extremely sensitive to New Physics (NP). Potential NP contributions could include:

• Charged Higgs Bosons: Predicted in many Supersymmetric extensions (Two-Higgs-Doublet Models), which could lift helicity suppression.
• Leptoquarks: Which could provide alternative mediation pathways.
• Sterile Neutrinos: Which would alter the kinematics of the final state.

The experimental challenge lies in the signature: a single high-momentum muon accompanied by missing energy (the neutrino), making it difficult to distinguish from large backgrounds like semileptonic $b \to u$ and $b \to c$ transitions and continuum $e^+e^- \to q\bar{q}$ processes.

2. Methodology

This study performs a combined analysis of data from the Belle (KEKB) and Belle II (SuperKEKB) experiments, totaling an integrated luminosity of $1076\text{ fb}^{-1}$.

• Inclusive Tagging Approach: To maximize efficiency, the researchers used an inclusive tagging method. They reconstructed the "tag-side" $B$ meson using the remaining particles in the event (Rest of the Event, ROE). By exploiting the known beam energy and the back-to-back kinematics of $\Upsilon(4S) \to B\bar{B}$ decays, the four-momentum of the "signal-side" $B$ meson is determined, allowing the muon momentum ($p_B^\mu$) to be boosted into the $B$ rest frame.
• Event Selection & Background Suppression:
  • A Boosted Decision Tree (BDT) was trained using event-shape variables (e.g., Fox-Wolfram moments, thrust axis) to suppress continuum backgrounds.
  • The analysis was divided into eight mutually exclusive categories (four from Belle, four from Belle II) based on the BDT output to separate signal-enriched regions from background-dominated regions.
• Statistical Framework: A binned maximum-likelihood fit was performed on the $p_B^\mu$ spectrum. The fit included templates for signal, semileptonic $b \to u$ and $b \to c$ transitions, rare decays, and continuum backgrounds. Systematic uncertainties were incorporated as nuisance parameters (NPs) affecting both the normalization and the shape of the templates.
• Validation: The modeling of the ROE and selection efficiencies was validated using the control channel $B^+ \to \bar{D}^0 \pi^+$.

3. Key Contributions

• First Belle II Result: This paper presents the first measurement of $B^+ \to \mu^+ \nu_\mu$ using Belle II data.
• Updated Belle Measurement: It provides an updated Belle result that supersedes previous publications by improving background modeling.
• Combined Precision: By combining the two datasets, the study achieves the most precise search for this decay to date.
• Multi-faceted Search: Beyond the branching fraction, the analysis includes a model-dependent search for Weak Annihilation (WA) processes and a search for stable sterile neutrinos ($m_N \in [0, 1.5]\text{ GeV}$).

4. Results

• Branching Fraction: The measured branching fraction is:
  $$\mathcal{B}(B^+ \to \mu^+ \nu_\mu) = (4.4 \pm 1.9_{\text{stat}} \pm 1.0_{\text{syst}}) \times 10^{-7}$$
  The observed significance is $2.4\sigma$ above the background-only hypothesis, consistent with the SM expectation.
• Upper Limits: Due to the low significance, the authors set the following 90% limits:
  • Frequentist: $\mathcal{B}(B^+ \to \mu^+ \nu_\mu) < 6.7 \times 10^{-7}$
  • Bayesian: $\mathcal{B}(B^+ \to \mu^+ \nu_\mu) < 7.2 \times 10^{-7}$
• CKM Element: The value of $|V_{ub}|$ was extracted as $|V_{ub}| = (3.92^{+0.77}_{-0.96} \text{ (stat)} ^{+0.44}_{-0.49} \text{ (sys)} \pm 0.03 \text{ (theo)}) \times 10^{-3}$, which is consistent with current world averages.
• New Physics Constraints:
  • The results exclude significant regions of the parameter space for Type II and Type III Two-Higgs-Doublet Models (2HDM).
  • The search for sterile neutrinos yielded no significant signal, providing improved limits on the squared mixing parameter $|U_{\mu N}|^2$.
• Semileptonic Decays: The partial branching fraction for $B \to X_u \ell \nu_\ell$ ($p_B^\mu > 2.2\text{ GeV}$) was measured at $(2.72 \pm 0.05 \pm 0.29) \times 10^{-4}$.

5. Significance

This work represents a milestone in flavor physics, pushing the sensitivity of leptonic $B$ decays to unprecedented levels. By providing the most stringent limits on $B^+ \to \mu^+ \nu_\mu$ and sterile neutrinos, the paper significantly constrains the landscape of physics beyond the Standard Model. Furthermore, the methodology established for combining Belle and Belle II data sets provides a roadmap for future high-precision measurements at the SuperKEKB collider.