Semileptonic and Leptonic Decays at Belle II

This proceeding summarizes recent studies on semileptonic and leptonic $B$ decays using combined Belle and Belle II datasets to test lepton flavour universality and improve the precision of inclusive $|V_{ub}|$ and $|V_{cb}|$ determinations.

The Gist

Imagine the universe as a giant, high-speed dance floor where particles are constantly pairing up, spinning, and then breaking apart. The Belle II experiment is like a super-powered security camera system at this dance floor (located at the KEK laboratory in Japan), filming every move the particles make.

This paper is a report from Raynette van Tonder, a researcher who analyzed the footage to solve two major mysteries in the physics world:

1. Are all the "dancers" (leptons) treated equally?
2. How do we accurately count the "steps" (decay rates) to understand the rules of the universe?

Here is a breakdown of the findings using simple analogies.

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1. The "Fairness" Test: Lepton Flavor Universality

In the Standard Model (the rulebook of particle physics), there is a rule called Lepton Flavor Universality. It says that the heavy "dancers" (electrons, muons, and tau particles) should all behave exactly the same way when they interact with the force of the weak nuclear force, regardless of how heavy they are.

• The Analogy: Imagine a playground slide. The rule says a small child, a teenager, and a grown-up should all slide down at the exact same speed if the slide is frictionless.
• The Problem: Recent measurements suggest the "grown-up" (the heavy tau particle) might be sliding down faster or differently than the kids (electrons and muons). This has created a "tension" or a puzzle in physics.
• The Belle II Solution: The team looked at B-mesons (heavy particles) decaying into a D-meson and a lepton. They compared how often a B-meson turns into a tau vs. how often it turns into an electron or muon.
  • They used a technique called "Hadronic Tagging." Imagine trying to find a lost coin in a messy room. Instead of just looking for the coin, they first perfectly reconstruct the other half of the room (the "tag") to know exactly what was there. This allows them to calculate the missing piece with extreme precision.
  • The Result: Their new measurements are twice as precise as before. While the numbers are still a bit "wobbly" (due to limited data), they are currently consistent with the standard rulebook, though the tension with other experiments remains. They are essentially saying, "We haven't found a broken slide yet, but we are looking very closely."

2. Counting the Steps: The CKM Matrix (|Vub| and |Vcb|)

Physicists need to know the exact values of certain "coupling constants" (numbers that tell us how likely particles are to change into one another). These are called |Vub| and |Vcb|.

• The Analogy: Think of these numbers as the odds on a casino slot machine. If you know the exact odds, you can predict the game. But right now, there are two different groups of gamblers:

  • The "Exclusive" Gamblers: They count every single specific outcome (e.g., "I saw a cherry, a bar, and a 7").
  • The "Inclusive" Gamblers: They just count the total money won, regardless of the specific symbols.
  • The Puzzle: These two groups keep getting different answers for the odds. This is a major headache for physicists.

• The Belle II Solution:

  • Inclusive Approach: They looked at all the messy, chaotic ways a B-meson can decay into an electron and a neutrino. They used Neural Networks (AI) to filter out the "noise" (background events) and count the specific "signal" events.
  • The Result: They got a very precise number for the "odds" (|Vub|). Interestingly, their result sits right in the middle of the two conflicting groups, offering a new perspective that might help solve the puzzle.

3. The "Ghost" Hunt: B+ → µ+ν

This section is about a very rare, difficult-to-spot event where a B-meson decays into a muon and a neutrino.

• The Analogy: Imagine a magician pulling a rabbit out of a hat, but the rabbit is invisible (the neutrino) and the only thing you see is the hat shaking (the muon). It's incredibly hard to prove the rabbit was there because the background is full of other magicians doing similar tricks.
• The Challenge: The signal is just one muon. Everything else in the collision is "missing energy." It's like trying to find a single specific grain of sand on a beach during a storm.
• The Solution: They combined data from the old Belle experiment and the new Belle II experiment. They used a "back-to-back" trick: since the particles are created in pairs, if they find one B-meson, they know exactly where the other one should be.
• The Result: They found a "hint" of the rabbit (a 2.4 standard deviation signal), but it's not quite a "smoking gun" yet (which usually requires 5 standard deviations). However, they set the tightest limit yet on how often this happens. It's like saying, "We haven't seen the rabbit clearly, but we know for sure it's not appearing more than once in a million tries."

Summary: What Does This Mean?

This paper is a report card for the Belle and Belle II collaborations.

• They are getting sharper: Their tools (detectors and AI) are now so good that they can see details that were blurry before.
• They are competitive: Even with less data than some other experiments, their new methods make their results some of the most precise in the world.
• The Mystery Continues: While they haven't definitively broken the Standard Model yet, they have narrowed the search area. The tension between different ways of measuring particle "odds" is still there, but now we have a clearer map of where to look next.

As the Belle II detector continues to collect more data (like a camera recording more hours of the dance), the hope is that these "wobbly" measurements will eventually snap into focus, revealing if the universe is playing by the rules we think it is, or if there is a new, hidden dancer on the floor.

Technical Summary

1. Problem Statement and Motivation

The paper addresses two critical challenges in high-energy physics:

• Lepton Flavour Universality (LFU) Anomalies: There is a persistent tension (currently at $3.8\sigma$) between experimental measurements of the ratios $R(D^{(*)}) = \mathcal{B}(B \to D^{(*)}\tau\nu_\tau) / \mathcal{B}(B \to D^{(*)}\ell\nu_\ell)$ and Standard Model (SM) predictions. These ratios test whether the weak interaction couples equally to electrons, muons, and tau leptons.
• The $|V_{ub}|$ and $|V_{cb}|$ Puzzle: Determinations of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{ub}|$ differ significantly (by $\approx 3\sigma$) between inclusive methods (summing all final states) and exclusive methods (reconstructing specific decay modes). Resolving this discrepancy is vital for testing the consistency of the SM.

The paper presents recent results from the Belle and Belle II experiments using electron-positron collision data at the $\Upsilon(4S)$ resonance to address these issues through precise measurements of semileptonic and leptonic $B$-meson decays.

2. Methodology and Data Sets

The analyses utilize:

• Data Samples: The complete Belle dataset ($711 \text{ fb}^{-1}$) and Belle II Run 1 data ($365 \text{ fb}^{-1}$, collected 2019–2022), totaling $1076 \text{ fb}^{-1}$ for combined analyses.
• Tagging Techniques:
  • Hadronic Tagging (Full Event Interpretation - FEI): One $B$ meson is fully reconstructed in exclusive hadronic decay channels using a hierarchical multivariate algorithm. This allows for a "completeness constraint" (no extra tracks in the event), suppressing background and enabling the reconstruction of the signal $B$ meson's kinematics via missing mass squared ($M^2_{miss}$).
  • Inclusive Tagging: Used for $B^+ \to \mu^+\nu$, where the signal is identified by a high-momentum muon, and the rest of the event (ROE) is used to infer the tag-side $B$ kinematics.
• Background Suppression:
  • Neural Networks (MLP): Used to distinguish signal from dominant backgrounds like $B \to X_c \ell \nu$ and continuum processes ($e^+e^- \to q\bar{q}$).
  • Kinematic Variables: Lepton energy ($E_B^\ell$), hadronic invariant mass ($M_X$), and squared four-momentum transfer ($q^2$).
  • Event Shape Observables: Used in Boosted Decision Trees (BDT) to suppress continuum backgrounds.

3. Key Contributions and Results

A. Measurement of $R(D)$ and $R(D^*)$ (LFU Test)

• Analysis: A simultaneous measurement of $R(D)$ and $R(D^*)$ using hadronic tagging. The analysis reconstructs main $D^*$ decay modes (including $D^0\gamma$) and seven $D$ decay channels, covering 36% of $D^0$ and 29% of $D^+$ widths.
• Results:
  • $R(D^*) = 0.242 \pm 0.019 (\text{stat}) \pm 0.016 (\text{sys})$
  • $R(D) = 0.439 \pm 0.055 (\text{stat}) \pm 0.046 (\text{sys})$
  • Correlations: $-0.40$ (stat) and $-0.20$ (syst).
• Significance: The results are consistent with SM predictions within $0.5\sigma$ ($R(D^*)$) and $2.0\sigma$ ($R(D)$). The combined measurement agrees with the world average within $1.3\sigma$. The precision is improved by a factor of two compared to previous Belle II hadronic-tagged results, making this the most precise determination of $R(D^{(*)})$ using this method.

B. Inclusive $B \to X_u \ell \nu$ and $|V_{ub}|$ Determination

• Analysis: Measurement of partial branching fractions in three kinematic regions (covering 32% to 87% of phase space) to extract inclusive $|V_{ub}|$. The analysis uses a signal region with high classifier scores ($>0.87$) and a control region ($<0.60$) to correct for $B \to X_c \ell \nu$ modeling.
• Results:
  • The most inclusive measurement ($E_B^\ell > 1 \text{ GeV}$) yields:
    $$|V_{ub}| = (4.01 \pm 0.11 (\text{stat}) \pm 0.16 (\text{sys}) ^{+0.07}_{-0.08} (\text{theo})) \times 10^{-3}$$
  • This value was derived using the GGOU theoretical framework.
• Significance: This result is more precise than previous Belle and BaBar hadronic-tagged measurements. It agrees with the inclusive HFLAV average and $B \to \pi \ell \nu$ exclusive measurements but exceeds the HFLAV exclusive average, contributing data to the ongoing $|V_{ub}|$ tension debate.

C. Study of $B^+ \to \mu^+\nu$ (Purely Leptonic Decay)

• Analysis: The first $B^+ \to \mu^+\nu$ study at Belle II, combined with an updated Belle measurement. It utilizes an inclusive tagging approach and a binned maximum-likelihood fit to the muon momentum ($p_B^\mu$) in the $B$ rest frame.
• Results:
  • Measured Branching Fraction: $\mathcal{B}(B^+ \to \mu^+\nu) = (4.4 \pm 1.9 (\text{stat}) \pm 1.0 (\text{sys})) \times 10^{-7}$.
  • Significance: $2.4\sigma$ (consistent with the expected $2.3\sigma$).
  • Derived $|V_{ub}|$: $(3.92^{+0.77}_{-0.96} (\text{stat}) ^{+0.44}_{-0.49} (\text{sys}) \pm 0.03 (\text{theo})) \times 10^{-3}$.
  • Upper Limits: Due to low significance, 90% CL upper limits were set: $< 6.7 \times 10^{-7}$ (Frequentist) and $< 7.2 \times 10^{-7}$ (Bayesian).
• Significance: This is the most precise search for this decay to date. While the result is less precise than inclusive/exclusive semileptonic determinations, it provides a theoretically clean (negligible hadronic uncertainty) cross-check for $|V_{ub}|$.

4. Significance and Outlook

• Precision Improvement: The Belle II detector's performance, combined with advanced analysis techniques (FEI, ML-based background suppression, and improved modeling), has significantly reduced systematic uncertainties compared to previous generations of experiments.
• Resolving Tensions: The results provide stringent constraints on LFU and $|V_{ub}|$. While the current $R(D^{(*)})$ results are consistent with the SM, the precision allows for tighter future tests. The $|V_{ub}|$ results add a crucial data point to the inclusive vs. exclusive discrepancy, though the tension remains unresolved.
• Future Prospects: With Belle II continuing to collect data, these analyses are expected to improve in precision, potentially uncovering new physics or definitively resolving the long-standing tensions in CKM matrix element determinations.

In summary, this work represents a significant step forward in the precision measurement of $B$-meson decays, utilizing the full potential of the Belle and Belle II datasets to test the Standard Model's validity in the flavor sector.