Looking forward to $B^+\to τ^+ ν_τ$ and $B_c^+\to τ^+ ν_τ$

This paper presents a RapidSim feasibility study demonstrating that the LHCb experiment can observe the decays $B^+\to \tau^+ \nu_\tau$ and $B_c^+\to \tau^+ \nu_\tau$ during Run 3 by utilizing direct pixel hits from its VELO detector to overcome missing momentum and vertex limitations, thereby enabling early experimental constraints on these key channels without waiting for next-generation accelerators.

The Gist

Imagine the Large Hadron Collider (LHC) as a massive, high-speed particle racetrack where protons zoom around and crash into each other. When they crash, they create a chaotic explosion of new particles, some of which are rare and fleeting, like the B-mesons and tau leptons mentioned in this paper.

The scientists in this study are playing a game of "Where's Waldo?" They are trying to find two very specific, rare events:

1. A B-plus meson turning into a tau and a neutrino.
2. A B-c-plus meson turning into a tau and a neutrino.

The Problem: The Invisible Ghosts

The difficulty is that these particles decay (fall apart) almost instantly, and they produce neutrinos in the process. Neutrinos are like ghosts; they pass right through detectors without leaving a trace. Because these "ghosts" carry away energy and momentum, it's very hard to prove the original particle existed just by looking at the debris left behind. It's like trying to figure out what a car looked like by only seeing the skid marks, while the car itself drove off into a fog.

The Solution: The Ultra-Close-Up Camera

The researchers propose a clever trick using a special camera called the VELO (Vertex Locator). Think of the VELO as a high-speed security camera placed incredibly close to the racetrack—only 5.1 millimeters away from the proton beams.

Usually, when a particle is created in a crash, it travels a tiny distance before it decays. In the past, scientists assumed this distance was too short to catch on camera. But because the VELO is so close, there's a good chance the particle will actually hit the camera sensor before it decays.

• The Analogy: Imagine a sprinter starting a race. Usually, you only see them at the starting line and then at the finish line. But if you have a camera placed just a few inches past the starting line, you can snap a photo of the sprinter while they are running. That single photo tells you exactly which direction they were heading and how fast they started.

By catching this "hit" on the sensor, the scientists can reconstruct the path of the particle much more accurately, even with the missing "ghost" neutrinos. This extra clue helps them separate the real signal from the background noise (other particles that look similar but aren't what they are looking for).

The Simulation: A Digital Rehearsal

Before running the experiment on real data, the team used a software tool called RapidSim. Think of this as a flight simulator for particle physics. They ran thousands of virtual crashes to see if their "camera trick" would actually work.

They simulated:

• The rare "signal" events they want to find.
• The common "background" events that look like the signal but are actually just noise (like other particles decaying into three pions).

They applied strict rules to their simulation, such as requiring a "hit" on the camera sensor between the crash point and the decay point. This acted as a filter, removing most of the fake signals.

The Results: We Don't Have to Wait

The simulation showed that with the data LHCb is currently collecting (during "Run 3" of the LHC), they have enough statistical power to find these particles.

• For the B-c-plus meson: This is a "holy grail" discovery that many scientists thought would require waiting for a brand-new, massive collider in the 2030s. This paper claims that with the current data, they can see it sooner, likely by mid-2026.
• For the B-plus meson: The data is already good enough to measure this decay very precisely.

Why Does This Matter?

Finding these particles is like checking the rules of a game. The Standard Model is the current "rulebook" of physics. These specific decays are sensitive to any "cheating" or new physics (called Beyond the Standard Model) that might be happening.

The paper concludes that by using this "close-up camera" technique, the LHCb experiment can provide the first real experimental constraints on these decays right now. This helps scientists understand why certain particles behave the way they do and whether there are new, undiscovered forces at play, without having to wait for the next generation of particle accelerators.

Technical Summary

Technical Summary: Feasibility of Observing $B^+ \to \tau^+\nu_\tau$ and $B_c^+ \to \tau^+\nu_\tau$ at LHCb Run 3

Problem Statement
The purely leptonic decays $B^+ \to \tau^+\nu_\tau$ and $B_c^+ \to \tau^+\nu_\tau$ are critical probes for physics Beyond the Standard Model (BSM). While the branching fraction for $B^+ \to \tau^+\nu_\tau$ has been measured with some precision, it remains compatible with Standard Model predictions without a definitive discovery. Conversely, the $B_c^+ \to \tau^+\nu_\tau$ decay has not yet been experimentally measured. Observing these decays at a hadron collider like the LHC is notoriously difficult due to the presence of multiple neutrinos in the final state, which leads to significant missing momentum and vertex information. This kinematic ambiguity makes distinguishing signal events from background in the busy environment of proton-proton collisions a major challenge. Furthermore, the $B_c^+ \to \tau^+\nu_\tau$ channel is considered a key objective for future facilities (such as the FCC and CEPC), with first constraints often projected for the 2030s or beyond.

Methodology
This paper presents a feasibility study utilizing the RapidSim simulation software to demonstrate that the LHCb experiment can observe these decays using data collected during Run 3 of the LHC. The core of the proposed strategy relies on exploiting the unique geometry of the upgraded LHCb Vertex Locator (VELO).

• VELO-Hit Identification: The VELO's silicon sensors are positioned as close as 5.1 mm from the LHC proton beams. Given the lifetimes of $B$ and $B_c$ mesons and the $\tau$ lepton (order of 1 ps), there is a non-negligible probability that these charged particles traverse a VELO sensor before decaying. The study focuses on identifying "VELO-hits"—direct pixel hits in the VELO associated with the charged $B^+$, $B_c^+$, or $\tau^+$ particles traveling between the Primary Vertex (PV) and the $\tau$ decay vertex (TV).
• Signal Topology: The analysis targets the $\tau^+ \to \pi^+\pi^-\pi^+\bar{\nu}_\tau$ decay mode. The presence of at least one VELO-hit between the PV and TV provides a measurable signal that constrains the minimum flight distance and offers a more accurate estimate of the initial flight direction of the $B$ meson than the line connecting the PV and the reconstructed $\tau$ vertex alone.
• Background Modeling: The study simulates three primary background categories that mimic the three-pion signature:
  1. $D \to \tau\nu$ (specifically $D^+ \to \tau^+\nu_\tau$ and $D_s^+ \to \tau^+\nu_\tau$).
  2. $B \to D3\pi$ (involving $B_h \to D_h \pi^+\pi^+\pi^-$ and $B_h \to D_h^* \pi^+\pi^+\pi^-$).
  3. $B \to DY$ (involving various $B$ decays to $D$ and $D_s$ mesons, where intermediate particles escape reconstruction).
     A conservative isolation factor ($\epsilon_{iso} = 10\%$) is applied to backgrounds with additional undetected particles.
• Selection and Fitting: Events are filtered based on geometric acceptance and kinematic cuts (e.g., invariant mass of pion pairs and the three-pion system, transverse momentum). The sensitivity is evaluated using an extended two-dimensional binned maximum-likelihood fit on 2,000 pseudo-experiments. The fit observables are:
  1. Corrected Mass ($m_{corr}$): Defined using the three-pion momentum transverse to the $B$-meson flight direction (approximated by the line to the first VELO-hit).
  2. Boosted Decision Tree (BDT) Score: Trained to discriminate signal from background using kinematic and topological variables.

Key Contributions

• Novel Use of VELO Geometry: The paper demonstrates that utilizing the proximity of the VELO sensors to the beam pipe to identify intermediate charged particle tracks (VELO-hits) significantly reduces limitations caused by missing momentum and vertex resolution.
• Run 3 Feasibility: The study provides a quantitative assessment showing that the necessary statistical power to observe these decays is achievable with the integrated luminosity expected from LHC Run 3, without waiting for next-generation colliders.
• Systematic Uncertainty Analysis: The study explicitly models the impact of systematic uncertainties (ranging from 0% to 100% of statistical uncertainty) on the measurement precision, offering a realistic view of the experimental challenges.

Results
Based on the simulation and sensitivity study:

• $B_c^+ \to \tau^+\nu_\tau$:
  • With only statistical uncertainties, an observation (defined as $3\sigma$ significance) is achievable with an integrated luminosity slightly above 10 fb$^{-1}$.
  • When including a systematic uncertainty equal to the statistical uncertainty, a dataset of slightly more than 20 fb$^{-1}$ is required to claim a $5\sigma$ observation.
  • Given the luminosity collected by LHCb since 2024, the paper projects that such a dataset will be available by mid-2026.
• $B^+ \to \tau^+\nu_\tau$:
  • The study indicates that a relative precision smaller than 5% is achievable across all considered luminosity scenarios.
  • This suggests that a precise measurement of this decay is already feasible with data collected to date.

Significance
The paper concludes that the LHCb experiment is positioned to provide the first experimental constraints on the $B_c^+ \to \tau^+\nu_\tau$ decay and a precise measurement of $B^+ \to \tau^+\nu_\tau$ within the current decade. These results offer complementary information to ongoing tests of lepton flavor universality in $R(D^{(*)})$ and the broader search for physics beyond the Standard Model. The study asserts that the community does not need to wait for the 2030s or future facilities to address these key objectives in flavor physics.