Measurement of inclusive $B \to X_u \ell \nu$ partial branching fractions and $|V_{ub}|$ at Belle II

Using 365 fb$^{-1}$ of data from the Belle II experiment, this study measures the partial branching fraction of inclusive charmless semileptonic $B$ meson decays with lepton energies above 1 GeV to determine the CKM matrix element $|V_{ub}|$ as $(4.01 \pm 0.19 ^{+0.07} _{-0.08}) \times 10^{-3}$, a result consistent with the world average from previous inclusive measurements.

The Gist

Imagine the universe as a giant, high-speed racetrack where tiny particles called B-mesons are the race cars. These cars are unstable and break apart (decay) almost instantly. Physicists at the Belle II experiment in Japan are like race officials trying to understand exactly how these cars fall apart to learn the fundamental rules of the universe.

This paper is about a specific, very tricky type of crash they are studying: when a B-meson breaks apart into a lepton (a light particle like an electron or muon), a neutrino (a ghost particle that is invisible), and a cloud of other particles (called $X_u$).

Here is the story of their investigation, explained simply:

1. The Mystery: The "Ghost" in the Machine

The main goal is to measure a number called $|V_{ub}|$. Think of this number as a "strength dial" on a control panel. It tells us how strongly two specific types of quarks (the building blocks of matter) like to talk to each other.

• The Problem: There are two different ways to measure this dial. One way says the dial is set to 3.4. The other way says it's set to 4.0.
• The Conflict: In physics, if two ways of measuring the same thing disagree by that much, it usually means we are missing something important. It could be a mistake in our math, or it could be a hint of "new physics" (something we don't know yet).
• The Mission: The Belle II team wants to take a fresh, super-precise look at the "inclusive" way of measuring (counting all the crashes, not just specific ones) to see who is right.

2. The Challenge: Finding a Needle in a Haystack

Imagine you are at a massive party (the particle collider).

• The Signal: You are looking for a specific guest wearing a red hat (the B-meson decaying into a lepton and a neutrino).
• The Background: The problem is that for every one "red hat" guest, there are 50 other guests wearing blue hats (a different, more common type of decay called $B \to X_c \ell \nu$).
• The Ghost: The "neutrino" is like a ghost. It leaves the party without being seen. You can't see it, but you know it's there because the other particles don't add up to the total energy of the room.

The team has to find the red hats in a crowd of 50 blue hats, while also accounting for a ghost that vanished.

3. The Strategy: The "Tag Team" and the "Net"

To solve this, the Belle II team uses a clever two-part strategy:

A. The Tag Team (The "Tag B")
When two B-mesons are created, they are born as a pair. The team catches one of them (the "Tag B") and reconstructs its entire history. Because they know exactly what the Tag B was doing, they can use the laws of physics (conservation of energy and momentum) to figure out exactly what the other B-meson (the "Signal B") was doing, even though it's breaking apart into invisible ghosts. It's like knowing exactly what one twin ate for lunch so you can deduce what the other twin ate, even if you didn't see them.

B. The High-Tech Net (Machine Learning)
Once they have the data, they have to filter out the 50 blue hats. They use Artificial Intelligence (AI)—specifically neural networks—to act as a super-smart bouncer.

• The AI looks at the shape of the crash, the energy of the particles, and the angles.
• It learns that "red hat" crashes look slightly different from "blue hat" crashes.
• The AI throws out 98% of the blue hats while keeping 25% of the red hats. This is a huge improvement over older methods!

4. The Results: A Clearer Picture

After analyzing 365 "inverse femtobarns" of data (which is a fancy way of saying "a massive amount of collision data"), they found:

1. The Measurement: They successfully measured how often these specific crashes happen.
2. The Dial Setting: Using their new, precise measurement and some advanced math theories, they calculated the strength dial ($|V_{ub}|$) to be 4.01.
3. The Verdict: This result agrees with the "inclusive" measurements from the past (the 4.0 group) and disagrees with the "exclusive" measurements (the 3.4 group).

5. Why This Matters

Think of the Standard Model of physics as a giant, complex puzzle. The fact that the "inclusive" and "exclusive" measurements of this dial don't match is a crack in the puzzle.

• If the crack is just a measurement error, we need better tools.
• If the crack is real, it might mean there are new particles or forces we haven't discovered yet that are messing with the math.

By using a larger dataset, better AI, and a more precise method, the Belle II team has confirmed that the "4.0" value is likely correct. This puts more pressure on physicists to figure out why the other method (the 3.4 value) is giving a different answer.

In a Nutshell

The Belle II team acted like master detectives. They used a "tag team" strategy to track invisible ghosts, used AI to filter out a crowd of 50 imposters for every real suspect, and confirmed that the "strength dial" of the universe is set to 4.0. This result keeps the mystery alive, urging scientists to keep digging to find the new physics hiding in the discrepancy.

Technical Summary

1. Problem Statement

The precise determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{ub}|$ is critical for testing the unitarity of the CKM matrix within the Standard Model. Currently, a significant discrepancy exists (approximately $3\sigma$) between values of $|V_{ub}|$ extracted from exclusive decays (e.g., $B \to \pi \ell \nu$) and inclusive decays ($B \to X_u \ell \nu$):

• Exclusive Average: $(3.43 \pm 0.12) \times 10^{-3}$
• Inclusive Average: $(4.06 \pm 0.16) \times 10^{-3}$

Inclusive measurements are theoretically challenging because the final state hadronic system $X_u$ is not fully reconstructed, and the signal must be separated from a background that is $\sim 50$ times larger, originating from CKM-favored semileptonic decays to charm ($B \to X_c \ell \nu$). Furthermore, kinematic selections required to suppress this background break the "inclusivity" assumption of the Heavy Quark Expansion (HQE), necessitating complex theoretical frameworks involving non-perturbative shape functions.

2. Methodology

Dataset and Detector:

• Data: 365 fb$^{-1}$ of $e^+e^- \to \Upsilon(4S) \to B\bar{B}$ data collected by the Belle II experiment at SuperKEKB (on-resonance), plus 43 fb$^{-1}$ of off-resonance data for continuum background studies.
• Tagging: The analysis employs hadronic tagging. The "tag-side" $B$ meson ($B_{\text{tag}}$) is fully reconstructed in hadronic decay channels using the Full Event Interpretation (FEI) algorithm. This allows for the kinematic constraint of the "signal-side" $B$ meson despite the presence of an undetected neutrino.

Event Reconstruction and Selection:

• Signal Definition: The signal consists of a lepton ($\ell = e, \mu$) and a hadronic system $X_u$ (Rest-of-Event, ROE). The neutrino momentum is inferred from missing momentum.
• Background Suppression:
  • Continuum ($e^+e^- \to q\bar{q}$): Suppressed using a Multi-Layer Perceptron (MLP) trained on event shape variables (thrust, Fox-Wolfram moments).
  • $B \to X_c \ell \nu$: Suppressed via a dedicated MLP trained on kinematic variables (missing mass squared $M^2_{\text{miss}}$, total event charge, $D^*$ reconstruction from slow pions, etc.) and a veto on kaons in the ROE.
  • Fake/Secondary Leptons: Suppressed via track quality cuts, particle identification (PID), and $J/\psi$ conversion vetoes.
• Kinematic Regions: Three distinct phase-space regions are defined to measure partial branching fractions, varying in acceptance (87%, 57%, and 31% of the full phase space) based on cuts on lepton energy ($E_\ell^B$), hadronic mass ($M_X$), and momentum transfer ($q^2$).

Signal Extraction and Modeling:

• Template Fit: A binned maximum likelihood fit is performed using the PYHF framework.
• Simultaneous Fit: The signal region is fitted simultaneously with a control region ($CR_{0,\text{low}}$) enriched in $B \to X_c \ell \nu$ events. This allows for the simultaneous extraction of the signal yield and the correction of the $B \to X_c \ell \nu$ background shape, which is known to be poorly modeled in simulation.
• Systematics: Extensive systematic uncertainties are treated as nuisance parameters, including modeling of $B \to X_u \ell \nu$ (DFN, BLNP frameworks), $B \to X_c \ell \nu$ form factors, tracking efficiency, PID, and continuum calibration.

3. Key Contributions

1. High-Precision Inclusive Measurement: This is the first inclusive $|V_{ub}|$ measurement using the full Belle II dataset to date, achieving competitive precision compared to previous Belle and BaBar results despite using roughly half the integrated luminosity. This is attributed to improved FEI reconstruction efficiency and the use of separate neural networks for continuum and charm suppression.
2. Advanced Background Handling: The paper introduces a robust method for correcting $B \to X_c \ell \nu$ modeling mismatches by fitting the signal and a data-enriched control region simultaneously. This addresses a major source of systematic uncertainty in inclusive analyses.
3. Multi-Region Analysis: The measurement is performed in three different kinematic regions, allowing for a cross-check of theoretical predictions (BLNP, DGE, GGOU) across different phase-space boundaries.
4. Bias Correction: The authors implemented a rigorous validation procedure to correct for biases arising from differences in $B \to X_c \ell \nu$ composition between control and signal regions, quantifying the associated uncertainties.

4. Results

Partial Branching Fractions:
The partial branching fraction for the broadest region ($E_\ell^B > 1.0$ GeV) is measured as:
$$\Delta \mathcal{B}(B \to X_u \ell \nu) = (1.54 \pm 0.08 \text{ (stat.)} \pm 0.12 \text{ (syst.)}) \times 10^{-3}$$
This result has a relative uncertainty of 9.6%, making it competitive with the most precise inclusive measurements from BaBar (8.1%) and significantly more precise than previous hadronic tagging measurements from Belle (11.4%).

Determination of $|V_{ub}|$:
Using the measured branching fraction and the Gambino, Giordano, Ossola, Uraltsev (GGOU) theoretical framework (chosen for consistency with HFLAV averages), the value of $|V_{ub}|$ is determined as:
$$|V_{ub}| = (4.01 \pm 0.19 \text{ (total exp.)} ^{+0.07}_{-0.08} \text{ (theo.)}) \times 10^{-3}$$

• The total experimental uncertainty is 0.19.
• The theoretical uncertainty is 0.07–0.08.
• The result is consistent with the current HFLAV inclusive average ($4.06 \pm 0.16$) but remains higher than the exclusive average ($3.43 \pm 0.12$).

Theoretical Framework Comparison:
The analysis also extracted $|V_{ub}|$ using the BLNP and DGE frameworks. While the central values varied slightly depending on the theoretical model and phase-space region, all results remained consistent within uncertainties and aligned with the inclusive HFLAV average.

5. Significance

• Resolving the $|V_{ub}|$ Puzzle: The result reinforces the tension between inclusive and exclusive determinations of $|V_{ub}|$. By providing a high-precision inclusive measurement with a different experimental technique (hadronic tagging vs. inclusive tagging) and improved background modeling, it helps rule out experimental artifacts as the sole cause of the discrepancy.
• Validation of Theoretical Frameworks: The consistency of $|V_{ub}|$ across three different kinematic regions and three different theoretical models (BLNP, DGE, GGOU) supports the validity of the Heavy Quark Expansion and shape function approaches in describing inclusive decays, even in restricted phase spaces.
• Belle II Capability: This measurement demonstrates the maturity of the Belle II detector and analysis chain, showing that it can achieve world-leading precision in complex inclusive analyses, paving the way for future high-precision tests of the Standard Model.

In conclusion, this paper provides a state-of-the-art inclusive measurement of $|V_{ub}|$ that confirms the existing tension with exclusive measurements, highlighting the need for further theoretical and experimental investigation into the sources of this discrepancy.