Measurement of the ratio of the B$_\mathrm{c}^+$ $\to$ J/$\psi$$\tau^+\nu_\tau$ and B$_\mathrm{c}^+$ $\to$ J/$\psi$ $\mu^+\nu_\mu$ branching fractions using three-prong $\tau$ lepton decays

Using proton-proton collision data collected by the CMS experiment at 13 TeV, this study measures the ratio of $B_c^+ \to J/\psi \tau^+\nu_\tau$ to $B_c^+ \to J/\psi \mu^+\nu_\mu$ branching fractions via three-prong $\tau$ decays, finding a combined result consistent with the Standard Model prediction and providing no evidence for lepton flavor universality violation.

The Gist

The Big Picture: A Cosmic Tug-of-War

Imagine the universe has a set of strict rules called the Standard Model. Think of this rulebook like the rulebook for a very complex board game. One of the most important rules in this game is Lepton Flavor Universality.

In plain English, this rule says: "The heavy particles (like the Tau) and the light particles (like the Muon) should behave exactly the same way, just scaled down by their weight." It's like saying if you drop a bowling ball and a tennis ball in a vacuum, they should fall at the exact same speed. If the bowling ball suddenly fell slower or faster than the tennis ball, it would mean there is a hidden force or a new rule we don't know about yet.

This paper is a report card from the CMS experiment at CERN (the Large Hadron Collider), checking if the "bowling balls" (Taus) and "tennis balls" (Muons) are playing fair.

The Experiment: Catching a Rare Decay

The scientists are looking at a specific particle called the $B_c^+$ meson. You can think of this particle as a very unstable, short-lived "parent" that loves to break apart.

When it breaks apart, it usually produces:

1. A $J/\psi$ particle (which quickly turns into two muons, like a pair of twins).
2. A neutrino (a ghost particle that is invisible).
3. Either a Muon (the light particle) OR a Tau (the heavy particle).

The scientists want to measure the ratio: How often does the parent choose the heavy Tau compared to the light Muon?

• The Prediction: The Standard Model rulebook says the Tau should be chosen about 26% as often as the Muon (because it's heavier and harder to produce).
• The Mystery: Another team (LHCb) previously found a ratio of 71%, which is way higher than the rulebook predicts. This hinted that the rules might be broken.

The New Method: The "Three-Prong" Detective Work

In this new paper, the CMS team looked at the data from 2016 to 2018 (a massive amount of data, equivalent to 138 "inverse femtobarns"—a unit of collision volume).

They focused on a specific way the Tau particle decays.

• The Old Way: They looked for Taus that turned into a Muon (like a shape-shifter).
• The New Way: They looked for Taus that turned into three charged pions (three little particles flying out). This is called a "three-prong" decay.

The Analogy:
Imagine you are trying to find a specific type of rare bird in a forest.

• The Muons are easy to spot; they are like bright red parrots.
• The Taus are shy. Sometimes they turn into a red parrot (the old method), but sometimes they turn into a flock of three blue jays (the new "three-prong" method).
• The forest is full of other birds (background noise) that look like blue jays.

The CMS team built a sophisticated "bird-watching filter" (a computer algorithm called a Boosted Decision Tree or BDT). This filter looks at the flight patterns, speed, and direction of the birds to separate the real "three-blue-jay" Taus from the fake ones.

The Results: The Rules Hold Up

After sifting through millions of collisions and using their high-tech filter, the scientists counted the results:

1. The Raw Ratio: When looking only at the "three-prong" (three blue jay) method, the ratio came out to be 1.04. This looks weirdly high! It suggests the heavy Tau is being chosen just as often as the light Muon.
2. The Combined Result: However, science is about checking your work. The team combined this new "three-prong" result with their previous "red parrot" (leptonic) result.
3. The Final Verdict: When they put both methods together, the final ratio was 0.49.

What does this mean?

• The Standard Model predicted 0.258.
• The LHCb team previously found 0.71 (which was suspicious).
• The CMS team found 0.49.

While 0.49 is higher than the prediction, the "uncertainty" (the margin of error) is quite large. The result is consistent with the Standard Model. In other words, the data is not precise enough to say the rules are broken. It's like measuring a runner's time and getting "somewhere between 9 and 11 seconds." The rulebook says they should run 9.5 seconds. Your measurement includes 9.5, so you can't claim they broke the record yet.

The Conclusion

The paper concludes that no evidence of "Lepton Flavor Universality violation" was found.

To use a final metaphor: The universe is like a giant,精密 (precision) clock. Some people thought they heard a "tick" that was out of rhythm (the LHCb result). This new paper is another person listening to the clock with a different ear (the three-prong method). They heard a slightly different rhythm, but when they combined their ears, the clock still sounded like it was ticking according to the original design.

The Standard Model remains the best description we have, and the "heavy" and "light" particles are still playing by the same rules, within the limits of our current measurement tools.

Technical Summary

Technical Summary: Measurement of the $B_c^+ \to J/\psi \tau^+ \nu_\tau$ and $B_c^+ \to J/\psi \mu^+ \nu_\mu$ Branching Fraction Ratio

Problem and Motivation
The Standard Model (SM) describes physical processes up to the electroweak scale, with lepton flavor universality (LFU) emerging as a consequence of the model's structure rather than an underlying principle. While Yukawa couplings are the sole source of LFU violation in the SM, current measurements for the second and third lepton generations align with SM predictions. However, observations of potential LFU violation in $B$ meson decays would signal physics beyond the SM.

A specific point of tension exists in the ratio $R_{J/\psi} = \mathcal{B}(B_c^+ \to J/\psi \tau^+ \nu_\tau) / \mathcal{B}(B_c^+ \to J/\psi \mu^+ \nu_\mu)$. The LHCb Collaboration previously measured this ratio using the leptonic $\tau$ decay channel ($\tau^+ \to \mu^+ \nu_\mu \nu_\tau$), obtaining a value of $0.71 \pm 0.17 \text{ (stat)} \pm 0.18 \text{ (syst)}$, which exceeds the SM prediction of $0.258 \pm 0.004$ by approximately two standard deviations. The CMS Collaboration previously measured this ratio in the same final state using 2018 data, finding a result consistent with the SM but with large uncertainties. This paper presents a complementary measurement using the hadronic three-prong $\tau$ decay channel ($\tau^+ \to \pi^+ \pi^- \pi^+ \nu_\tau$) to test LFU in $B_c^+$ decays and combine results with the previous leptonic analysis.

Methodology
The analysis utilizes proton-proton collision data collected by the CMS detector at a center-of-mass energy of $\sqrt{s} = 13$ TeV during the 2016–2018 runs, corresponding to an integrated luminosity of $138 \text{ fb}^{-1}$.

• Event Selection: Signal events are reconstructed via the decay chain $B_c^+ \to J/\psi (\to \mu^+ \mu^-) \tau^+ \nu_\tau$, where the $\tau^+$ decays hadronically into three charged pions ($\tau_{had}^+$). The $J/\psi$ candidate is formed from two oppositely charged muons with specific transverse momentum ($p_T > 4$ GeV) and invariant mass constraints. The $\tau_{had}^+$ candidate is reconstructed as a triplet of charged hadrons (pions) with a net charge of $\pm 1$, forming a displaced vertex.
• Background Suppression: The dominant background consists of inclusive $B$ meson decays producing a $J/\psi$ and additional tracks. A Boosted Decision Tree (BDT) classifier, trained using the XGBoost algorithm, is employed to separate signal from background. The BDT utilizes 18 kinematic and topological variables. Two signal regions (SR) and sideband regions (SB) are defined based on the BDT output score ($sBDT$) to optimize signal significance while controlling background contamination.
• Signal Extraction: The signal yield is extracted using a maximum likelihood fit with the COMBINE tool. The fit utilizes the "unrolled" two-dimensional distribution of the invariant masses of the two possible $\rho(770)^0 \to \pi^+ \pi^-$ pairs ($m(\rho_1)$ vs $m(\rho_2)$) within the $\tau_{had}^+$ candidate. Genuine $\tau$ decays exhibit a distinct peak in these mass distributions compared to background events.
• Normalization and Systematics: The measurement relies on the $B_c^+ \to J/\psi \mu^+ \nu_\mu$ yield (from a separate leptonic $\tau$ analysis using 2018 data) as the denominator. Systematic uncertainties are treated as nuisance parameters in the fit. Key sources include the modeling of the $B_c^+$ form factors (using the HAMMER framework), the TAUOLA model for $\tau$ decays, pileup effects, and the normalization of the $B_c^+ \to J/\psi D_s^{(*)+}$ and other $B_c^+$ background processes. The background from non-$B_c^+$ $H_b \to J/\psi X$ processes is estimated directly from data in the sideband regions and extrapolated to the signal region using simulation-derived factors.

Key Contributions

• First Hadronic $\tau$ Measurement: This work provides the first measurement of $R_{J/\psi}$ using the hadronic three-prong $\tau$ decay mode in $B_c^+$ decays at the LHC.
• Combined Analysis: The result is combined with the previous CMS measurement based on the leptonic $\tau$ decay channel to provide a more precise determination of $R_{J/\psi}$.
• Comprehensive Systematic Treatment: The analysis includes a detailed evaluation of systematic uncertainties, particularly those related to the extrapolation of data-driven backgrounds and the modeling of $B_c^+$ decay dynamics.

Results

• Hadronic Channel: The measured ratio of branching fractions in the hadronic $\tau$ decay mode is $R_{had}^{J/\psi} = 1.04^{+0.50}_{-0.44}$.
• Combined Result: Combining the hadronic result with the previous leptonic $\tau$ analysis yields a final ratio of:
  $$R_{J/\psi} = 0.49 \pm 0.26$$
  (where the uncertainty includes statistical, systematic, and theoretical components).
• Consistency: The combined result is consistent with the Standard Model prediction of $0.258 \pm 0.004$. The paper notes that the result is also compatible with the previous CMS leptonic measurement ($0.17 \pm 0.33$) and the LHCb result, though the LHCb central value remains higher.

Significance
The paper concludes that no evidence of lepton flavor universality violation is found in this measurement. The result, $R_{J/\psi} = 0.49 \pm 0.26$, is consistent with the SM prediction within the current experimental uncertainties. This measurement represents a significant step in testing LFU in $B_c^+$ decays by utilizing a distinct decay topology (hadronic $\tau$) to cross-check previous findings and reduce model dependencies associated with specific decay channels. The authors emphasize that while the result is consistent with the SM, the uncertainties remain large enough that further data is required to definitively confirm or refute the tension observed in previous LHCb measurements.