Flavour changing charged current decays at LHCb

This paper presents three recent LHCb results on flavor-changing charged current decays: the first measurement of the branching fraction ratio $\mathcal{R}(D^{**})$ using $B^{-} \to D^{**0} \tau^{-} \bar{\nu}_{\tau}$, the determination of the branching fraction for $\Lambda \to p \mu^{-} \bar{\nu}_{\mu}$, and the extraction of form-factor parameters from $B^0 \to D^{*-} \mu^{+} \nu_{\mu}$ decays.

The Gist

Imagine the universe is built on a set of strict rules, like a grand cosmic recipe book called the Standard Model. One of the most important rules in this book is Lepton Flavour Universality. Think of this rule like a strict bouncer at a club who treats every guest exactly the same, regardless of their name. In physics, the "guests" are particles called leptons (specifically electrons, muons, and tau particles). The rule says: "If you are a muon or a tau, you interact with the force-carrying particles (the 'gauge bosons') exactly the same way an electron does, except for the fact that you might be heavier."

If the bouncer starts treating a heavy guest differently than a light one, that's a huge clue that there's a secret, hidden rulebook (New Physics) we haven't discovered yet.

The LHCb experiment at CERN is like a high-speed camera crew trying to catch these particles in the act of breaking the rules. They focus on heavy particles containing a "bottom" quark (b-hadrons) as they decay, or break apart. Here is a breakdown of the three main stories this paper tells, using simple analogies:

1. The "Heavy Hitters" Check: $R(D^{**})$

The Scenario:
Usually, when scientists measure how often a bottom particle turns into a tau particle versus a muon (to check if the bouncer is fair), they look at specific, well-known outcomes. However, sometimes the bottom particle decays into a "messy" intermediate state involving excited versions of other particles (called $D^{**}$ resonances). These are like the "background noise" or the "crowd" that usually gets in the way of the main measurement.

The Discovery:
Instead of ignoring this noise, the LHCb team decided to measure it directly for the first time. They looked at a specific decay where a bottom particle turns into an excited particle ($D^{**}$) and a tau.

• The Analogy: Imagine trying to count how many people enter a VIP room, but there's a side hallway where people are also getting dressed up. Usually, you ignore the side hallway. Here, the team went into the side hallway, counted the people, and found 123 specific events.
• The Result: They found that this "side hallway" decay happens about 13% as often as the muon version of the same decay. This matches the Standard Model's prediction perfectly. It's like confirming that even in the messy, crowded side hallway, the bouncer is still treating everyone fairly.

2. The "Lambda" Test: $\Lambda \to p \mu^- \bar{\nu}_\mu$

The Scenario:
The team also looked at a different type of particle called a "Lambda" baryon (a heavy cousin of the proton). They wanted to see how often this particle decays into a proton and a muon compared to how often it decays into a proton and an electron.

• The Analogy: Think of the Lambda particle as a factory machine that can produce two types of products: "Muons" or "Electrons." The Standard Model predicts the machine should produce Muons about 15% as often as Electrons.
• The Discovery: Using data from 2016–2018, the team counted the products coming off the assembly line. They found the machine produces Muons at a rate of roughly 17.5% compared to Electrons.
• The Result: This is a very precise measurement (twice as accurate as the previous best record). The result is compatible with the Standard Model, meaning the factory machine is working exactly as the recipe book says it should. It also helps scientists check the "unitarity" of the CKM matrix (a mathematical check to ensure the math of particle mixing adds up to 100%).

3. The "Shape Shifter" Analysis: $B^0 \to D^{*-} \mu^+ \nu_\mu$

The Scenario:
In this third story, the team didn't just count how often a decay happens; they looked at how it happens. When a $B^0$ particle decays into a $D^*$ particle and a muon, the particles fly off at specific angles.

• The Analogy: Imagine throwing a spinning top. You can describe the throw by how fast it spins, which way it leans, and the angle of the throw. In physics, these are called "angles" and "form factors" (which describe the shape and internal structure of the particles).
• The Discovery: The team used a massive amount of data (3.0 fb$^{-1}$) to map out these angles in five different dimensions at once. They tested three different mathematical "blueprints" (called BGL, CLN, and BLPR) to see which one best describes the shape of the decay.
• The Result: All three blueprints agreed with each other and with the most advanced computer simulations (Lattice QCD). The team extracted the "form factors" with improved precision. This is like creating a 3D model of the decay that is sharper and clearer than any model made before.

The Big Picture

The paper concludes that the LHCb experiment is playing a crucial role in the global effort to understand particle physics. By measuring these rare decays and checking the angles and rates, they are confirming that the Standard Model is holding up strong.

• They found the first evidence of a specific "side-hallway" decay ($D^{**}$).
• They set a new world record for measuring a specific Lambda decay.
• They created the most detailed map yet of how a $B^0$ particle spins and flies apart.

So far, the "bouncer" is still treating everyone fairly, and the "factory machines" are running exactly as the recipe book predicts. No new physics has been found in these specific measurements, but the precision of these measurements is essential for spotting the tiny cracks in the rulebook that might appear in future experiments.

Technical Summary

1. Problem Statement and Motivation

The Standard Model (SM) predicts Lepton Flavour Universality (LFU), asserting that electroweak gauge bosons couple universally to all charged leptons ($e, \mu, \tau$), with differences arising solely from lepton masses. Any deviation from this principle would signal New Physics (NP).

Recent measurements by Belle, BaBar, and LHCb have revealed tensions with SM predictions in:

• LFU Ratios: $R(D)$ and $R(D^*)$ show a $\sim 3.8\sigma$ discrepancy between SM predictions and experimental averages.
• CKM Matrix Elements: Discrepancies of $2\text{--}3\sigma$ exist between values of $|V_{ub}|$ and $|V_{cb}|$ extracted from exclusive versus inclusive $b$-hadron decays.

Semileptonic $b$-hadron decays are ideal for testing these anomalies due to their large branching fractions and theoretically clean hadronic matrix elements (dominated by tree-level diagrams). This paper presents three specific analyses addressing backgrounds, hyperon decays, and form-factor extractions to refine these tests.

---

2. Methodology and Key Contributions

The paper details three distinct analyses performed using LHCb data collected between 2011 and 2018.

A. Measurement of the Branching Fraction Ratio $R(D^{**})$

• Context: In measurements of $R(D^*)$ using hadronic $\tau$ decays, decays into higher excited charmed meson resonances ($D^{**}$) constitute a significant background ($\sim 3.5\%$).
• Objective: Perform the first measurement of the ratio $R(D^{**}_{1,2})$ to constrain this background and test LFU in the $D^{**}$ sector.
• Methodology:
  • Dataset: $9 \text{ fb}^{-1}$ (2011–2018).
  • Signal Channel: $B^- \to D^{**0} \tau^- \bar{\nu}_\tau$, where $D^{**0}$ refers to the sum of narrow states $D_1(2420)^0$ and $D_2(2460)^0$ (collectively $D^{**0}_{1,2}$), and $\tau$ decays hadronically.
  • Normalization: $B^- \to D^{**0}_{1,2} D_s^-$ (branching ratio taken from a previous LHCb amplitude analysis).
  • Analysis Technique: A three-dimensional binned template fit using:
    1. Mass difference $\Delta m = m(D^{**0}) - m(D^{*+})$.
    2. Invariant mass of the leptonic system $q^2$.
    3. Output of a Boosted Decision Tree (BDT) trained to reject $D_s$ backgrounds.
• Key Contribution: First evidence for the $B^- \to D^{**0}_{1,2} \tau^- \bar{\nu}_\tau$ decay mode.

B. Measurement of the Branching Fraction $\mathcal{B}(\Lambda \to p \mu^- \bar{\nu}_\mu)$

• Context: Semileptonic hyperon decays ($s \to u$ transition) are sensitive to scalar and tensor NP contributions. They also allow for the determination of $|V_{us}|$ to test CKM unitarity.
• Objective: Measure the branching fraction of $\Lambda \to p \mu^- \bar{\nu}_\mu$ to improve precision over previous results and calculate the LFU ratio $R_{\mu/e}$.
• Methodology:
  • Dataset: $5.4 \text{ fb}^{-1}$ (2016–2018).
  • Normalization: $\Lambda \to p \pi^-$ (known branching fraction $\approx 0.641$).
  • Analysis Technique:
    • Signal extraction via a binned maximum-likelihood fit in bins of corrected mass $m_{\text{corr}}(p\pi)$ and invariant mass $m(p\pi)$.
    • The fit utilizes a 2D plane binning scheme to separate signal from background.
• Key Contribution: A world-leading precision measurement of the $\Lambda \to p \mu^- \bar{\nu}_\mu$ branching fraction, improving precision by a factor of two compared to BESIII.

C. Extraction of Form-Factor Parameters in $B^0 \to D^{*-} \mu^+ \nu_\mu$

• Context: The differential decay rate of $B^0 \to D^{*-} \mu^+ \nu_\mu$ depends on decay angles ($\theta_D, \theta_\ell, \chi$) and $q^2$, making it sensitive to NP.
• Objective: Extract hadronic form-factor parameters using a multi-dimensional angular analysis.
• Methodology:
  • Dataset: $3.0 \text{ fb}^{-1}$ (2011–2012).
  • Analysis Technique: A five-dimensional binned fit to:
    1. Three decay angles ($\theta_D, \theta_\ell, \chi$).
    2. Squared momentum transfer ($q^2$).
    3. Squared missing invariant mass ($m^2_{\text{miss}}$).
  • Models: The analysis assumes three form-factor parameterizations: BGL (Boyd-Grinstein-Lebed), CLN (Caprini-Lellouch-Neubert), and BLPR.
  • Optimization: The Bayesian Information Criterion (BIC) is used to optimize the truncation order of the BGL coefficient series.
• Key Contribution: The first LHCb angular analysis of this mode, providing form-factor parameters with improved precision and verifying unitarity constraints.

---

3. Results

1. $R(D^{**}_{1,2})$ Measurement:

   • Signal Yield: $123 \pm 23$ events.
   • Significance: $3.5\sigma$ (First evidence for this decay).
   • Visible Branching Fraction: $\mathcal{B}(B^- \to D^{**0}_{1,2} \tau^- \bar{\nu}_\tau) \times \mathcal{B}(D^{**0}_{1,2} \to D^{*+}\pi^-) = (5.1 \pm 1.7) \times 10^{-4}$.
   • LFU Ratio: $R(D^{**}_{1,2}) = 0.13 \pm 0.04$.
   • Conclusion: Consistent with the SM prediction of $0.09 \pm 0.02$.

2. $\Lambda \to p \mu^- \bar{\nu}_\mu$ Branching Fraction:

   • Measured Value: $\mathcal{B}(\Lambda \to p \mu^- \bar{\nu}_\mu) = (1.46 \pm 0.10) \times 10^{-4}$ (Total uncertainty 6.9%).
   • LFU Ratio ($R_{\mu/e}$): Calculated using the electron mode value from literature: $R_{\mu/e} = 0.175 \pm 0.012$.
   • Conclusion: Compatible with the SM prediction of $0.153 \pm 0.008$.

3. Form-Factor Parameters ($B^0 \to D^{*-} \mu^+ \nu_\mu$):

   • The measured form-factor values are consistent across the BGL, CLN, and BLPR models.
   • Results agree with recent Lattice QCD calculations.
   • The BGL parameters satisfy unitarity constraints, and the analysis achieves improved precision over previous measurements.

---

4. Significance

• Background Control: The first measurement of $R(D^{**})$ provides crucial constraints on backgrounds for $R(D^*)$ measurements, helping to clarify the origin of the existing LFU anomalies.
• Precision Hyperon Physics: The result for $\Lambda \to p \mu^- \bar{\nu}_\mu$ represents the most precise determination to date, offering a new, independent probe for $|V_{us}|$ and NP in the $s \to u$ sector.
• Theoretical Validation: The angular analysis of $B^0 \to D^{*-} \mu^+ \nu_\mu$ validates the consistency of different form-factor parameterizations and lattice QCD inputs, reinforcing the theoretical framework used to interpret $b \to c$ transitions.
• Overall Impact: These results confirm LHCb's central role in the flavor physics program, providing complementary data to $B$-factories (Belle/BaBar) and tightening constraints on potential New Physics scenarios.