New physics searches via angular distributions of $\bar{B} \to D^* (\to D \pi) \tau (\to \ell \nu_\tau \bar{\nu}_\ell) \bar{\nu}_\tau$ decays

This paper proposes a method to probe new physics in $\bar{B} \to D^* \tau \bar{\nu}_\tau$ decays by analyzing the angular distribution of the full cascade $\bar{B} \to D^* (\to D \pi) \tau (\to \ell \nu_\tau \bar{\nu}_\ell) \bar{\nu}_\tau$, demonstrating through simulated data that this approach can constrain right-handed and tensor current couplings with approximately 5–6% precision while incorporating recent lattice QCD form factor correlations.

The Gist

The Big Picture: Solving the "B-Anomaly" Mystery

Imagine the Standard Model of particle physics as a massive, incredibly detailed instruction manual for how the universe works. For decades, this manual has been perfect. But recently, scientists have noticed a few pages that don't quite add up. These are called "B-anomalies."

Specifically, when heavy particles called B-mesons decay (break apart), they seem to be producing heavy particles called Tau leptons more often than the manual predicts. It's like a vending machine that is supposed to give you a soda 10% of the time, but lately, it's giving you a soda 15% of the time. Something new might be inside the machine, or the machine's instructions are wrong.

This paper is about building a better tool to figure out what is causing that extra soda.

The Problem: The "Ghost" Particle

To understand the decay, scientists usually look at the angles at which the pieces fly apart. Think of it like a pool game: if you hit the cue ball, you can predict where the other balls will go based on the angle of the hit.

However, in this specific decay ($\bar{B} \to D^* \tau \bar{\nu}_\tau$), there is a major problem. One of the pieces flying out is a Tau particle, and the Tau immediately breaks apart into even smaller pieces, including neutrinos.

• Neutrinos are like ghosts. They have no electric charge and almost no mass. They pass through detectors without leaving a trace.
• Because the Tau decays into these invisible ghosts, scientists cannot see the Tau's original direction or speed.
• The Analogy: Imagine trying to figure out how a magician threw a ball, but the ball exploded into smoke and invisible dust the moment it left his hand. You can't see the ball's path, so you can't calculate the angle of the throw.

The Solution: The "Shadow" Strategy

The authors of this paper came up with a clever workaround. Instead of trying to see the invisible Tau, they look at the visible shadow it leaves behind.

1. The Tau decays into a lighter lepton (an electron or a muon) and the invisible ghosts.
2. While we can't see the Tau, we can see the electron or muon.
3. The authors realized that if you look at the angle of this electron/muon relative to the other visible pieces (the $D^*$ meson), you can mathematically reconstruct the "missing" information.

The Analogy: It's like trying to figure out the wind speed on a day you can't see the wind. You can't see the air, but you can see how the leaves on a tree are shaking. By measuring the leaves' movement carefully, you can calculate the wind's speed and direction, even though the wind itself is invisible.

The Method: A New "Camera Angle"

In the past, scientists tried to measure angles in the "Tau's rest frame" (imagining the Tau is standing still). But since the Tau is a ghost, we don't know where it is standing still.

This paper proposes a new camera angle: The W-Boson Rest Frame.

• Think of the W-boson as the "parent" of the Tau.
• The authors calculated a complex mathematical map (an "angular distribution") that translates the visible electron's movement into the language of the W-boson.
• This map allows them to ignore the invisible ghosts and focus entirely on the visible particles, creating a clear picture of what's happening.

The Simulation: Testing the Tool

Since this specific measurement hasn't been done perfectly in real life yet, the authors built a virtual laboratory.

1. The Toy Factory: They used a computer to generate 2,000 fake decay events based on the best current theories.
2. The Stress Test: They asked, "If new physics exists, can our new tool find it?"
3. The Results:
   • They tested three types of "New Physics" (Right-handed currents, Pseudoscalar currents, and Tensor currents). Think of these as three different types of "saboteurs" trying to mess with the vending machine.
   • The Good News: Their new method is very sensitive.
     • For the "Right-handed" saboteur, they can detect a deviation as small as 5%.
     • For the "Tensor" saboteur, they can detect a deviation of about 6%.
   • The Secret Weapon: They found that combining their new angle measurements with existing data about the strength of the weak force ($V_{cb}$) makes the tool even sharper, cutting the error rate in half for some cases.

Why This Matters

Currently, we have a hunch that "New Physics" is hiding in these decays, but we don't have enough evidence to prove it.

This paper provides a blueprint for the future. It tells experimentalists at labs like Belle II (a particle collider in Japan) exactly how to set up their detectors and analyze the data.

• The Metaphor: Imagine you are a detective trying to solve a crime. You know the criminal is there, but you can't see them. This paper is like a new type of fingerprint powder that reveals the invisible prints on the glass. It doesn't catch the criminal yet, but it gives the police the exact technique they need to catch them the next time they visit.

Summary

• The Problem: We can't see the "Tau" particle because it hides in invisible neutrinos.
• The Fix: We measure the visible "children" (electrons/muons) and use math to reconstruct the invisible parent's behavior.
• The Result: This new method is highly sensitive and could be the key to proving that the Standard Model is incomplete, potentially opening the door to a whole new understanding of the universe.

Technical Summary

1. Problem Statement

The paper addresses the challenge of probing New Physics (NP) in the semileptonic decay $\bar{B} \to D^* \tau \bar{\nu}_\tau$, a process central to the $R(D^{(*)})$ anomaly (the observed deviation in lepton flavor universality ratios $R(D)$ and $R(D^*)$ from Standard Model predictions).

• The Core Difficulty: In standard angular analyses of $B$ decays, the angular distribution is defined in the rest frame of the decaying particle. However, the $\tau$ lepton decays into a charged lepton ($\ell = e, \mu$) and at least two neutrinos ($\tau \to \ell \nu_\tau \bar{\nu}_\ell$). Because neutrinos are undetected, the $\tau$ momentum and its rest frame cannot be fully reconstructed. Consequently, the helicity angles of the $\tau$ decay products cannot be measured directly, hindering the extraction of NP couplings via angular distributions.
• The Gap: Previous attempts to solve this focused on two-body $\tau$ decays (e.g., $\tau \to \pi \nu$). This paper extends the analysis to the more common three-body leptonic decay $\tau \to \ell \nu_\tau \bar{\nu}_\ell$, which offers higher statistics but introduces complex kinematic constraints due to the missing neutrinos.

2. Methodology

The authors develop a theoretical framework and a statistical analysis strategy to extract NP parameters from the observable final states ($D^* \to D\pi$ and $\ell$).

A. Theoretical Framework

• Effective Hamiltonian: The analysis uses a low-energy effective Hamiltonian containing dimension-six four-fermion operators for the $b \to c \ell \nu$ transition. This includes Standard Model (SM) left-handed currents and potential NP contributions: Right-handed ($C_{VR}$), Scalar ($C_S$), Pseudoscalar ($C_P$), and Tensor ($C_T$) currents.
• Reference Frame Strategy: To bypass the unknown $\tau$ rest frame, the authors calculate the angular distribution in the $W$-boson rest frame (the virtual $W$ mediating the $b \to c$ transition).
  • The hadronic part ($D^* \to D\pi$) is analyzed in the $D^*$ rest frame.
  • The leptonic part ($\tau \to \ell \nu \nu$) is analyzed in the $W$ rest frame.
• Amplitude Calculation: The amplitude is separated into hadronic and leptonic helicity amplitudes using the completeness relation of $W$-boson polarization vectors. This allows the calculation to be performed in Lorentz-invariant quantities.
• Phase Space Integration: A critical technical contribution is the derivation of the kinematic limits for the charged lepton energy ($E_\ell$) in the $W$ rest frame. Due to the unobserved neutrinos, the integration range for $E_\ell$ and the angle between the $\tau$ and $\ell$ ($\theta_{\tau\ell}$) is non-trivial. The authors identify two distinct integration regions (Range 1 and Range 2) based on the condition $(p_\tau - p_\ell)^2 > 0$.
• Observable Angular Distribution: The final differential decay rate is expressed in terms of 11 angular coefficients ($J_i$) dependent on the angles $\theta_D$, $\theta_\ell$, and $\chi_\ell$. These coefficients are functions of the NP Wilson coefficients and form factors. The distribution is split into contributions from the two kinematic ranges.

B. Statistical Analysis (Sensitivity Study)

• Simulated Data: Since no experimental data exists for this specific angular distribution, the authors generate simulated data using "truth" values derived from existing fits to $\bar{B} \to D^* \ell \nu_\ell$ data (Belle collaboration) and lattice QCD inputs. They assume a future dataset of 2000 events (representative of combined Belle and Belle II data).
• Unbinned Maximum Likelihood Fit: Instead of binned histograms (which lose information), they employ an unbinned maximum likelihood method.
• Global Fit Strategy: The fit minimizes a total $\chi^2$ function comprising:
  1. Angular Coefficients: Constraints from the simulated angular distribution.
  2. Lattice QCD Data: Constraints on the $B \to D^*$ form factors (using BGL parametrization) from Fermilab-MILC and JLQCD collaborations.
  3. Branching Ratio: Constraints from the world average of $\mathcal{B}(\bar{B} \to D^* \tau \bar{\nu}_\tau)$.
  4. $|V_{cb}|$ Constraint: An external constraint on the CKM matrix element $|V_{cb}|$ from CKMfitter.

3. Key Contributions

1. Analytical Derivation: The paper provides the first complete analytical expression for the angular distribution of $\bar{B} \to D^*(\to D\pi)\tau(\to \ell\nu\bar{\nu})\bar{\nu}$, explicitly handling the three-body $\tau$ decay kinematics and the resulting two-range integration.
2. Kinematic Limits: It rigorously derives the limits for the charged lepton energy in the $W$ rest frame, a necessary step for constructing the probability density function (PDF).
3. Feasibility Demonstration: It demonstrates that despite the missing neutrinos, the angular distribution retains sufficient sensitivity to distinguish between different NP scenarios (Right-handed, Pseudoscalar, Tensor) through interference terms with the SM amplitude.
4. Correlation Analysis: It quantifies the correlations between NP Wilson coefficients, form factors, and $|V_{cb}|$, showing how external constraints (like $|V_{cb}|$) significantly improve sensitivity.

4. Results

The sensitivity study (assuming 2000 events) yields the following statistical uncertainties for the NP Wilson coefficients (assuming they are real):

• Right-Handed Current ($C_{VR}$):
  • Without $|V_{cb}|$ constraint: $\sim 10\%$ uncertainty.
  • With $|V_{cb}|$ constraint: Uncertainty reduces to $\sim 4.5\%$. This is a significant improvement, making this a powerful probe for right-handed NP.
• Tensor Current ($C_T$):
  • Without $|V_{cb}|$ constraint: $\sim 8\%$ uncertainty.
  • With $|V_{cb}|$ constraint: Uncertainty reduces to $\sim 6\%$.
• Pseudoscalar Current ($C_P$):
  • Sensitivity remains around $\sim 20\%$. The $|V_{cb}|$ constraint has little effect due to weak correlations between $C_P$ and $|V_{cb}|$.
• Imaginary Coefficients: Sensitivity to imaginary parts of $C_P$ and $C_T$ is poor ($\sim 200\%$) because they depend on fewer observables (specifically $J_7$).
• $|V_{cb}|$ Tension: The study notes that without the external $|V_{cb}|$ constraint, the fitted value of $|V_{cb}|$ tends to be higher than the global average. This is attributed to the discrepancy between the theoretical SM prediction for the branching ratio (which is lower) and the experimental world average; the fit artificially inflates $|V_{cb}|$ to match the observed rate.

5. Significance

• Belle II Physics Case: This work provides a concrete physics case for the Belle II experiment. It shows that even with the "invisible" neutrinos in $\tau$ decays, angular analyses are viable and can provide competitive constraints on NP.
• Complementarity: The sensitivity achieved (4.5% for $C_{VR}$ and 6% for $C_T$) is comparable to global fits that use a wide variety of observables ($R(D^{(*)})$, polarization fractions, etc.). This suggests that angular distributions alone can be a robust, independent test of the SM.
• Theoretical Robustness: By utilizing the latest lattice QCD form factors and the BGL parametrization, the analysis minimizes theoretical uncertainties, ensuring that any observed deviation is likely due to NP rather than hadronic modeling errors.
• Future Outlook: The paper concludes that with the anticipated increase in data from Belle II, this method will provide one of the most stringent tests of the Standard Model in the charged current sector, potentially resolving the $R(D^{(*)})$ anomaly or confirming the presence of New Physics.