Prospects for discovering strongly decaying doubly heavy $T_{bc}$ tetraquark states at LHCb

This paper evaluates the discovery potential of the $J^P=0^+$ $T_{bc}$ tetraquark decaying into $B^- D^+$ at LHCb, finding that a $5\sigma$ observation is feasible during Run 4 for optimistic production cross sections but would require the full Run 5 dataset for more realistic estimates, while remaining unobservable under conservative scenarios.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle smasher. Every second, it smashes protons together, creating a chaotic explosion of subatomic debris. Among this debris, physicists are looking for a very specific, rare "gem": a new type of particle called the $T_{bc}$ tetraquark.

This paper is essentially a treasure map for the LHCb experiment, calculating exactly how much "digging" (data collection) they need to do to find this gem, and how likely they are to succeed.

Here is the breakdown of the paper's findings in simple terms:

1. The Target: A Rare Four-Quark Gem

Most particles are like simple Lego structures made of two or three pieces (quarks). The $T_{bc}$ is a rare "tetraquark," a structure made of four pieces: a heavy bottom quark, a heavy charm quark, and two lighter ones.

• The Analogy: Imagine looking for a specific, four-piece Lego castle in a pile of billions of random bricks.
• The Challenge: This castle is unstable. If it's heavy enough, it falls apart almost instantly into two other particles (a $B$ meson and a $D$ meson). The scientists are looking for the "shadow" of this castle in the debris.

2. The Noise: The "Background" Problem

The biggest problem isn't just finding the castle; it's that the pile of debris is full of fake castles.

• The Analogy: Imagine trying to hear a single person whispering in a stadium full of people shouting. The "shouting" is the background noise created when the collider accidentally produces a $B$ meson and a $D$ meson separately, which just happen to fly close to each other.
• The Paper's Work: The authors built a very detailed computer model to predict exactly how much "shouting" (background noise) there will be. They used two methods:
  1. Single Scattering (SPS): Like two people accidentally bumping into each other and dropping their items.
  2. Double Scattering (DPS): Like two separate pairs of people in the same stadium dropping items at the same time by pure coincidence. This is the main source of noise.

3. The Three Scenarios: How Rich is the Treasure?

Since no one knows exactly how often the $T_{bc}$ gem is created, the authors tested three different "treasure maps":

• Scenario A: The Optimist's Map (103 nb)
  • The Guess: The gem is very common.
  • The Result: If this is true, the LHCb experiment will find it very soon, likely by the end of their current data collection phase (Run 4). They would need about 50 units of data (femtobarns) to be 100% sure.
• Scenario B: The Realist's Map (18 nb)
  • The Guess: The gem is moderately common (based on scaling from similar discoveries).
  • The Result: This is the most likely scenario. Finding it will be harder. They will likely see "strong hints" (3-sigma evidence) with the full dataset, but to be 100% certain (5-sigma discovery), they will need to wait for the full Run 5 dataset (300 units of data).
• Scenario C: The Pessimist's Map (0.3 nb)
  • The Guess: The gem is extremely rare.
  • The Result: Even with the maximum amount of data the LHCb can collect (300 units), the signal would be too weak to see. It would be like trying to find a single grain of sand in a desert using a metal detector.

4. The "Signal-to-Noise" Ratio

The paper calculates that the "noise" (background) depends on a factor called $\sigma_{eff}$.

• The Analogy: Think of this as the "crowdedness" of the stadium. If the stadium is less crowded (a higher $\sigma_{eff}$), the accidental coincidences are fewer, and the whisper is easier to hear. If the stadium is packed (low $\sigma_{eff}$), the whisper is drowned out.
• The authors tested different levels of crowdedness and found that even in the best-case "less crowded" scenarios, the amount of data required is significant.

5. The Verdict

The paper concludes that:

1. Discovery is possible: If the $T_{bc}$ particle exists with a "moderate" production rate, the LHCb experiment has a very good chance of finding it by the time they finish collecting data in Run 5.
2. It depends on luck: If the particle is extremely rare (the pessimist's map), current technology and data limits might not be enough to see it.
3. A Guide for the Future: Even if they don't find it, this study tells the scientists exactly how to set their detectors and how much data they need to collect to either find the gem or prove it doesn't exist at certain production rates.

In summary: The authors have drawn a detailed map showing that if the $T_{bc}$ particle is "common enough," the LHCb team should be able to spot it in the next few years of data collection. If it's "too rare," they might need to build even bigger machines or wait for even more data.

Technical Summary

Technical Summary: Prospects for Discovering Strongly Decaying Doubly Heavy $T_{bc}$ Tetraquark States at LHCb

Problem Statement
The discovery of exotic hadrons, such as the doubly charmed tetraquark $T_{cc}^+$, has established that hadrons containing two heavy quarks can be produced promptly at hadron colliders. However, the $T_{bc}$ tetraquark (composed of $bcud$), which contains two different heavy flavors (bottom and charm), remains unobserved. A critical gap exists in the quantitative, data-driven assessment of the actual discovery potential for $T_{bc}$ at the LHCb experiment. Specifically, it is unclear under what specific conditions—regarding integrated luminosity, production cross section, and branching fraction—a $5\sigma$ discovery can be achieved amidst the formidable background of associated $B$ and $D$ meson production. The primary challenge lies in the quantitative understanding of prompt production mechanisms and the associated irreducible backgrounds, particularly distinguishing the signal from Single Parton Scattering (SPS) and Double Parton Scattering (DPS) processes.

Methodology
The study employs a phenomenological approach to evaluate the discovery potential of the $T_{bc}$ state with quantum numbers $J^P = 0^+$ decaying via the channel $T_{bc} \to B^- D^+$. The analysis is conducted for proton-proton collisions at $\sqrt{s} = 13$ TeV.

• Background Modeling: The background is constructed based on the associated production of $B$ and $D$ mesons.
  • SPS: Simulated using MadGraph5_aMC@NLO (MG5_aMC) and Pythia8.3. Two theoretical schemes are considered to account for uncertainties: a nominal Next-to-Leading Order (NLO) calculation in the 3-flavor number scheme (3FNS) and a conservative Leading Order (LO) calculation in the 4-flavor number scheme (4FNS) with charm-gluon initial states.
  • DPS: Simulated by independently generating $b\bar{b}$ and $c\bar{c}$ samples and randomly pairing events. To reduce model dependence, the DPS sample is reweighted event-by-event to match the rapidity difference ($\Delta y$) distributions observed in LHCb Run 2 data. The effective DPS cross section ($\sigma_{\text{eff}}$) is varied across representative values of 5, 15, and 30 mb.
• Signal Generation: The $T_{bc}$ signal is modeled as a Breit-Wigner resonance convolved with a Gaussian detector response function (mass resolution $\sigma_{\text{res}} = 6$ MeV). Signal samples are generated for mass hypotheses of 7167 MeV and 7229 MeV, with widths ranging from 0.5 to 40 MeV.
• Parameter Scanning: The study scans the parameter space of the $T_{bc}$ mass, width, production cross section ($\sigma(T_{bc})$), and $\sigma_{\text{eff}}$. Three representative production cross section scenarios are tested: an optimistic estimate of 103 nb, an intermediate value of 18 nb (scaled from $T_{cc}^+$ and baryon production ratios), and a conservative lower bound of 0.3 nb.
• Significance Calculation: The statistical significance ($Z$) is calculated using the standard formula $Z = N_{\text{Sig}} / \sqrt{N_{\text{Sig}} + N_{\text{Bkg}}}$ within a mass window of $\pm 3\sigma_{\text{eff res}}$. Systematic uncertainties, dominated by detector efficiency (50% relative uncertainty) and background normalization, are propagated to the significance estimate.

Key Contributions

• Data-Driven Background Calibration: Unlike previous theoretical studies focusing solely on mass predictions, this work provides a realistic background model calibrated to LHCb measurements, specifically addressing the DPS contribution which dominates the $B^\mp D^\pm$ background in the relevant kinematic region.
• Quantitative Discovery Benchmarks: The paper establishes specific integrated luminosity requirements for a $5\sigma$ discovery under various theoretical assumptions, moving beyond qualitative feasibility to quantitative thresholds.
• Comprehensive Parameter Scan: The study evaluates the interplay between $T_{bc}$ properties (mass, width, cross section) and experimental conditions (luminosity, $\sigma_{\text{eff}}$), providing a map of the discovery landscape for future searches.

Results

• Optimistic Scenario ($\sigma(T_{bc}) = 103$ nb): A $5\sigma$ discovery is achievable with approximately 20–70 fb$^{-1}$ of integrated luminosity, depending on the $T_{bc}$ mass, width, and $\sigma_{\text{eff}}$. This is well within the reach of the combined Run 2, 3, and 4 dataset (expected $\sim$50 fb$^{-1}$).
• Intermediate Scenario ($\sigma(T_{bc}) = 18$ nb): For this more realistic estimate, a $5\sigma$ discovery is only achievable under the most favorable parameter choices (narrow width, lower mass, high $\sigma_{\text{eff}}$) using the full Run 5 dataset (300 fb$^{-1}$). Most parameter sets in this scenario yield $3\sigma$ evidence with the full dataset.
• Conservative Scenario ($\sigma(T_{bc}) = 0.3$ nb): No significant signal is expected to be observable even with 300 fb$^{-1}$ of data.
• Minimum Observable Cross Section: With 50 fb$^{-1}$ (Runs 2–4), a $5\sigma$ discovery requires $\sigma(T_{bc}) \times \mathcal{B}(T_{bc} \to B^- D^+)$ to be in the range of 20–60 nb. With 300 fb$^{-1}$ (Runs 2–5), the sensitivity improves, allowing observation of values in the 5–25 nb range.

Significance
The paper concludes that a systematic search for the $T_{bc}$ tetraquark at LHCb is feasible. While the discovery potential is highly dependent on the production cross section, the study provides a quantitative benchmark for experimental searches. Even in the absence of a signal, the results offer valuable theoretical input by setting upper limits on $\sigma(T_{bc}) \times \mathcal{B}(T_{bc} \to B^- D^+)$, thereby providing experimental constraints to guide hadron models. The work addresses the critical challenge of background modeling for prompt $\bar{B}D$ production, establishing a foundation for further exploration of exotic hadrons containing different heavy quarks.