Measurement of the ratio of branching fractions $\mathcal{B}(B_c^+ \to J/\psi \tau^+ \nu_{\tau})/\mathcal{B}(B_c^+ \to J/\psi \mu^+ \nu_{\mu})$

Using 5.4 fb$^{-1}$ of LHCb proton-proton collision data at 13 TeV, the paper reports a measurement of the branching fraction ratio $\mathcal{R}(J/\psi) = 0.51 \pm 0.12\text{(stat)} \pm 0.08\text{(syst)}$, which is consistent with Standard Model predictions within 1.8 standard deviations.

The Gist

Imagine the universe as a giant, high-speed racetrack where tiny particles zoom around at nearly the speed of light. At CERN, the European Organization for Nuclear Research, scientists use a massive machine called the Large Hadron Collider (LHC) to smash protons together, creating a chaotic but fascinating shower of new particles.

This paper is a report from the LHCb collaboration, a team of scientists who act like ultra-precise detectives at this racetrack. Their job is to catch specific, short-lived particles called $B_c^+$ mesons and watch how they fall apart.

The Mystery: Do All Particles Play by the Same Rules?

In the Standard Model of physics (our current best rulebook for how the universe works), there is a rule called Lepton Flavor Universality. Think of this rule like a strict bouncer at a club who says, "It doesn't matter who you are—whether you're a muon (a heavy electron cousin) or a tau (an even heavier cousin)—you get the same VIP treatment."

According to this rule, when a $B_c^+$ meson decays, it should be equally likely to produce a muon or a tau, once you adjust for their different weights. However, in recent years, other experiments have seen a "glitch" in the system: it seems like the heavier tau particles are showing up more often than the rulebook predicts. This has scientists wondering if the rulebook is missing a page or if there's a new, undiscovered force at play.

The Experiment: A High-Stakes Coin Flip

To test this, the LHCb team looked at a specific type of decay. Imagine the $B_c^+$ meson is a parent particle that splits into two main parts:

1. A $J/\psi$ particle (which is like a stable, recognizable "child" that the scientists can easily spot).
2. A lepton (either a muon or a tau) and a neutrino (a ghostly particle that is almost impossible to catch).

The scientists wanted to count how many times the parent chose the tau route versus the muon route. They calculated a ratio, which they call $R(J/\psi)$.

• If the rulebook is perfect, this ratio should be around 0.26.
• If the "glitch" is real and taus are favored, the ratio would be higher.

The Detective Work: Sorting the Noise

The challenge is that the racetrack is incredibly noisy. For every real decay the scientists want to see, there are millions of other particle collisions that look similar but aren't what they're looking for. It's like trying to find a specific red marble in a bucket of sand while the bucket is being shaken violently.

To solve this, the team used data from 2016–2018 (a huge amount of data, equivalent to 5.4 "inverse femtobarns"—a unit of collision volume). They built a sophisticated filter system:

• The "Unpaired" Muon: They looked for a specific signature: a $J/\psi$ (which breaks into two muons) plus one extra muon. This extra muon is the clue.
• The Ghostly Clue: Since the tau particle decays into a muon and two invisible neutrinos, the scientists couldn't see the tau directly. Instead, they looked at the "missing energy" and the way the particles moved to guess if a tau was there.
• The Bouncer's List: They used computer algorithms (like a smart bouncer) to reject fake signals, such as random muons that just happened to be near each other, or particles that were misidentified.

The Results: A Step Closer, But Not a Breakthrough

After sorting through millions of collisions, the team found their answer:

• The Measured Ratio: They found $R(J/\psi) = 0.51$.
• The Uncertainty: Because the data is complex, there is a margin of error. The true value is likely between 0.31 and 0.71 (roughly speaking).
• The Comparison: The Standard Model predicts a value of about 0.26.

The result of 0.51 is higher than the prediction, which is exciting. However, because of the "margin of error" (the statistical uncertainty), the result is only 1.8 standard deviations away from the prediction.

Here is the simple analogy for what that means:
If the Standard Model prediction is a target, the scientists' result is a dart throw that landed somewhat close to the bullseye but not quite on it. In the world of particle physics, to claim a "discovery" (a new law of physics), you need to be 5 standard deviations away from the target. This result is a "hint" or a "nudge," but it's not a slam-dunk proof yet. It's consistent with the old rules, but it leaves the door open for the possibility that the rules might need a slight tweak.

Why This Matters

This measurement is an improvement over previous attempts. The scientists reduced the "noise" (systematic errors) significantly, making their measurement much sharper than before. They also used better theoretical calculations (from a field called Lattice QCD) to know exactly what the "target" should look like.

In summary:
The LHCb team took a closer look at how heavy particles decay. They found a slight tendency for heavier particles (taus) to appear more often than the standard rulebook predicts, but the evidence isn't strong enough yet to say the rulebook is wrong. It's a fascinating clue that keeps the mystery of "Lepton Flavor Universality" alive, urging scientists to keep collecting data and refining their tools.

Technical Summary

Technical Summary: Measurement of the ratio of branching fractions $R(J/\psi)$

Problem and Context
Semileptonic decays of $b$-hadrons serve as precision tests for Lepton Flavor Universality (LFU), a fundamental Standard Model (SM) principle asserting equal couplings for all charged leptons to gauge bosons. Previous measurements of the ratios $R(D)$ and $R(D^*)$ have shown a persistent excess of semitauonic decays, currently standing at approximately 3.8 standard deviations above SM predictions. This paper addresses the extension of these studies to the $B_c^+$ meson, specifically measuring the ratio of branching fractions:
$$R(J/\psi) \equiv \frac{\mathcal{B}(B_c^+ \to J/\psi \tau^+ \nu_\tau)}{\mathcal{B}(B_c^+ \to J/\psi \mu^+ \nu_\mu)}$$
While a previous LHCb measurement using Run 1 data (2011–2012) yielded $0.71 \pm 0.17 \text{ (stat)} \pm 0.18 \text{ (syst)}$, and CMS reported a combined value of $0.49 \pm 0.26$, significant theoretical progress has been made. Recent Lattice QCD (LQCD) calculations of the $B_c^+ \to J/\psi \ell^+ \nu_\ell$ form factors predict an SM value of $R(J/\psi) = 0.2597 \pm 0.0027$ assuming LFU. This paper presents a new measurement using Run 2 data to test these improved theoretical inputs against experimental results.

Methodology
The analysis utilizes proton-proton collision data recorded by the LHCb detector at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of $5.4 \text{ fb}^{-1}$ collected between 2016 and 2018. The sample size for $b$-hadrons is approximately four times larger than the previous LHCb measurement due to the increased production cross-section at higher energies.

• Decay Reconstruction: Both the signal ($B_c^+ \to J/\psi \tau^+ \nu_\tau$) and normalization ($B_c^+ \to J/\psi \mu^+ \nu_\mu$) channels are reconstructed via the $J/\psi \to \mu^+ \mu^-$ decay and the purely leptonic $\tau^+ \to \mu^+ \nu_\mu \nu_\tau$ decay. This results in a visible signature of three muons ($\mu^+ \mu^- \mu^+$) for both channels. The muon not originating from the $J/\psi$ is termed the "unpaired muon."
• Selection and Background Suppression: Events are selected using hardware and software triggers requiring high-quality muon tracks. Offline selections include strict particle identification to suppress hadrons misidentified as muons (the leading background), isolation criteria to reduce partially reconstructed decays ($B_c^+ \to J/\psi H_c X$), and kinematic constraints on the $J/\psi \mu^+$ invariant mass ($3.2 < m < 6.4 \text{ GeV}/c^2$).
• Fit Strategy: The composition of the selected sample is determined using a multidimensional binned maximum-likelihood fit. The fit utilizes three variables:
  1. The squared missing mass ($m^2_{\text{miss}}$).
  2. The $B_c^+$ candidate decay time ($\tau$).
  3. A categorical variable $Z$, which maps intervals of the unpaired muon energy in the $B_c^+$ rest frame ($E^*_\mu$) and the squared four-momentum transfer ($q^2$) to integers 0–7.
• Background Modeling:
  • Feed-down: Contributions from $B_c^+ \to \psi(2S)\ell^+\nu_\ell$ and $B_c^+ \to \chi_{c(1,2)}\ell^+\nu_\ell$ are modeled using simulation, with yields constrained by theoretical predictions.
  • Partially Reconstructed: $B_c^+ \to J/\psi H_c X$ backgrounds are modeled using a "cocktail" of two-body, quasi-two-body, and three-body decays, normalized using the control sample $B_c^+ \to J/\psi D_s^+$.
  • Combinatorial: Random combinations of $J/\psi$ candidates with unrelated muons are modeled using data-driven templates and sideband fits.
  • Misidentification (MisID): The dominant background arises from $J/\psi$ combined with a hadron misidentified as a muon. This is modeled using purified hadron samples weighted by misidentification probabilities derived from control data.
• Systematic Uncertainties: Sources include simulation statistical limitations, form factor modeling (weighted to match LQCD synthetic data), misID composition estimation, and efficiency ratios. The analysis employs alternative weighting procedures and fit configurations to evaluate these uncertainties.

Key Contributions

• Data Volume: The analysis benefits from a fourfold increase in the $b$-hadron sample size compared to the previous LHCb measurement, leveraging the higher $b$-quark production cross-section at 13 TeV.
• Improved Background Suppression: New selection criteria, particularly regarding isolation and particle identification, significantly reduced background processes, leading to a marked reduction in systematic uncertainty.
• Theoretical Integration: The measurement incorporates the latest LQCD form factor calculations as Gaussian priors in the fit, allowing for a more precise extraction of the signal yield and a direct comparison with the refined SM prediction.
• Robustness Checks: The study validates background modeling through independent cross-checks (e.g., extrapolating from inclusive $B_c^+ \to J/\psi H_c X$ candidates) and evaluates the impact of different misID template strategies.

Results
The measured ratio of branching fractions is:
$$R(J/\psi) = 0.51 \pm 0.12 \text{ (stat)} \pm 0.08 \text{ (syst)}$$
The total uncertainty is $0.14$. The result is consistent with the previous LHCb Run 1 measurement. The significance of the $B_c^+ \to J/\psi \tau^+ \nu_\tau$ signal is estimated to be 3.6 standard deviations, accounting for relevant systematic uncertainties.

Significance
The measured value of $R(J/\psi) = 0.51 \pm 0.14$ is within 1.8 standard deviations of the Standard Model prediction of $0.2597 \pm 0.0027$ (assuming LFU). While the central value remains higher than the SM prediction, the result is statistically consistent with the SM within the current experimental precision. The paper highlights that the measurement represents a significant reduction in systematic uncertainty compared to previous efforts, driven by improved selection criteria and better theoretical inputs. The result does not confirm the LFU violation seen in $R(D^{(*)})$ measurements but remains compatible with the SM prediction within $1.8\sigma$.