Production of doubly heavy quarkonium associated with two heavy quarks via top quark decays

This paper analyzes the production of doubly heavy quarkonia ($\bar{B}_c^{(*)}$ and charmonia) via the $1 \rightarrow 4$ top-quark decay channel within the NRQCD framework, revealing significant decay widths that could yield thousands of events annually at the LHC and offering a sensitive probe for validating the narrow-width approximation.

The Gist

Imagine the universe as a giant, high-speed particle factory. At the very top of the production line sits the Top Quark. It's the "heavyweight champion" of the particle world—so massive and unstable that it doesn't just sit around; it immediately explodes (decays) into smaller pieces.

Usually, scientists study what happens when this champion breaks apart into standard pieces. But in this paper, a team of physicists (Niu, Zheng, and Ma) is looking at a very specific, rare, and complicated way the Top Quark can break apart. They are asking: "What if the Top Quark's explosion creates a rare, double-heavy 'couple' (a quarkonium) along with three other stray particles?"

Here is a simple breakdown of their discovery:

1. The Rare "Family Reunion" (The Decay)

Normally, when a Top Quark decays, it's like a parent splitting up a family. It usually gives birth to a Bottom Quark and a W Boson (which then splits into other things).

The authors are studying a "four-way split" (a 1-to-4 decay). Imagine the Top Quark as a parent who, instead of just having two kids, suddenly produces:

1. A rare, heavy "couple" made of two different heavy quarks (either a $B_c$ meson or a Charmonium like $J/\psi$).
2. Two other heavy quarks (Charm quarks).
3. One light quark (Strange antiquark).

It's like a parent giving birth to a twin pair of heavyweights, plus two other kids, all at once. This is a very crowded, chaotic, and rare event.

2. The "Heavyweight Couples" (The Products)

The paper focuses on two types of "couples" formed in this chaos:

• The $B_c$ Meson: This is a unique "mixed-race" couple made of a heavy Bottom quark and a heavy Charm quark. It's the only meson made of two different heavy flavors.
• Charmonium ($J/\psi$ and $\eta_c$): These are couples made of two identical Charm quarks.

The scientists calculated exactly how often this happens. They found that while these events are rare, they happen enough to be interesting. Specifically, they found that this specific "four-way split" is actually the main way nature produces certain types of Charmonium ($J/\psi$ and $\eta_c$) when Top Quarks are involved.

3. The "Factory Output" (What we can expect at the LHC)

The Large Hadron Collider (LHC) is like a massive particle collider that smashes protons together billions of times a second, creating millions of Top Quarks.

The authors did the math to predict how many of these rare "heavy couples" we could catch in a year:

• $B_c$ Mesons: We could see about 10,000 to 1,000,000 of these per year.
• Charmonium ($J/\psi$): We could see about 1,000 to 100,000 of these per year.

This is a goldmine for physicists! It means that by watching Top Quarks decay, we have a new, efficient "factory" to study these heavy particles, which helps us understand the strong force that holds the universe together.

4. The "Shortcut" Test (Narrow-Width Approximation)

In physics, when calculating complex explosions, scientists often use a "shortcut" called the Narrow-Width Approximation (NWA). It's like assuming that if a middleman (the W Boson) in the chain is very stable, you can treat the whole process as two separate steps:

1. Top Quark $\to$ Middleman + Stuff.
2. Middleman $\to$ Final Stuff.

The authors used this complex four-particle decay to test if that shortcut is accurate. They found that the shortcut works very well (within about 0.6% error), but only because they checked the full, messy reality first. It's like checking a map's "as the crow flies" distance against the actual winding road to make sure the shortcut is safe to use.

5. Why This Matters

• New Insights: It gives us a fresh way to study heavy particles that are hard to make otherwise.
• Precision: It helps physicists understand the "rules of the road" (Quantum Chromodynamics) for heavy particles.
• Future Hunting: The paper provides a "map" (differential distributions) showing exactly where to look and what the particles look like when they are created, helping experimentalists at the LHC spot these rare events in the noise.

In a nutshell:
This paper is about finding a hidden, rare pathway where the universe's heaviest particle (the Top Quark) breaks apart to create unique, heavy "couples." The scientists calculated exactly how often this happens, proved it's a major source of certain particles, and showed that our current mathematical shortcuts for predicting these events are surprisingly accurate. It's a new window into the heavy side of the subatomic world.

Technical Summary

1. Problem Statement

The paper addresses the production mechanisms of doubly heavy quarkonia (specifically $B_c$ mesons and charmonia like $J/\psi$ and $\eta_c$) in high-energy physics. While these particles have been studied through direct hadronic production, $e^+e^-$ collisions, and decays of $Z$, $W$, and Higgs bosons, their production via top-quark decays remains an under-explored but potentially rich channel.

Specifically, the authors investigate a rare four-body ($1 \to 4$) decay channel of the top quark:

• $t \to (b\bar{c}) + c + c + \bar{s}$ (producing $B_c$ mesons)
• $t \to (c\bar{c}) + b + c + \bar{s}$ (producing charmonia)

The challenge lies in the complexity of the multi-body phase space integration and the need to rigorously test theoretical approximations, such as the Narrow-Width Approximation (NWA), in a regime where intermediate particles (like the $W$ boson) are off-shell or where kinematic correlations are non-trivial.

2. Methodology

The study employs the Non-Relativistic QCD (NRQCD) factorization framework to calculate the decay widths.

• Factorization: The decay width $\Gamma$ is factorized into perturbative short-distance coefficients ($\hat{\Gamma}$) and non-perturbative long-distance matrix elements ($\langle \mathcal{O}^H[n] \rangle$).
  $$\Gamma = \sum_n \hat{\Gamma}(t \to \langle Q\bar{Q}' \rangle[n] + \dots) \times \langle \mathcal{O}^H[n] \rangle$$
  The authors focus on dominant color-singlet S-wave states: $[1S_0]$ and $[3S_1]$.
• Feynman Diagrams: The calculation includes tree-level diagrams where the top quark decays via $t \to bW^+$, followed by $W^+ \to c\bar{s}$. Crucially, an additional hard gluon splits into a heavy quark-antiquark pair ($c\bar{c}$ or $b\bar{c}$) to form the quarkonium.
  • For $B_c$ production, the dominant diagrams involve the $b$-quark fragmenting or the top quark radiating the gluon.
  • For charmonium ($c\bar{c}$) production, the dominant mechanism involves the $W^+$ boson decaying into a $c\bar{c}$ pair plus other quarks.
• Phase Space Integration: The authors perform a rigorous full four-body phase space integration. They utilize a recursive parameterization technique, splitting the 4-body phase space ($d\Phi_4$) into lower-dimensional sub-systems using invariant masses ($s_{ij}$) and angles. This avoids the simplifications of the NWA to obtain exact theoretical predictions.
• NWA Validation: The calculated full widths are compared against results derived using the Narrow-Width Approximation (factorizing the process into $t \to \dots W$ and $W \to \dots$) to assess the validity of NWA in this specific kinematic regime.

3. Key Contributions

• First Calculation of 1-to-4 Top Decay: This work provides the first comprehensive calculation of doubly heavy quarkonium production via the specific four-body top-quark decay channels $t \to (b\bar{c}) + c + c + \bar{s}$ and $t \to (c\bar{c}) + b + c + \bar{s}$.
• Dominant Source Identification: The authors identify that for $J/\psi$ and $\eta_c$ production, this specific four-body top-quark decay channel provides the dominant contribution among all known top-quark decay mechanisms.
• NWA Benchmark: The study serves as a rigorous quantitative benchmark for the Narrow-Width Approximation. By comparing the full phase-space calculation with the NWA result, they demonstrate that while NWA is generally robust, precise deviations (approx. 0.6% when using experimental branching fractions) arise due to finite-width effects and off-shell contributions.
• Kinematic Analysis: The paper provides detailed differential distributions (invariant masses, angles, and longitudinal momentum fractions $z$) which encode unique kinematic signatures. These are crucial for distinguishing signal from background in experimental searches.

4. Key Results

Using input parameters from the Particle Data Group (PDG) and NRQCD matrix elements, the authors obtained the following partial decay widths:

Decay Channel	Decay Width ($\Gamma$)
$t \to \bar{B}_c + c + c + \bar{s}$	0.2251 MeV
$t \to \bar{B}^*_c + c + c + \bar{s}$	0.3099 MeV
$t \to J/\psi + b + c + \bar{s}$	0.0537 MeV
$t \to \eta_c + b + c + \bar{s}$	0.0555 MeV

Phenomenological Estimates (LHC Run 3):
Based on an annual top-quark yield of $10^8$–$10^{10}$ at the LHC:

• $B_c$ and $B^*_c$ events: Estimated at $O(10^4 - 10^6)$ per year.
• Charmonium ($J/\psi, \eta_c$) events: Estimated at $O(10^3 - 10^5)$ per year.
• Future Colliders (CEPC, LHeC): Yields are significantly lower ($O(10^1 - 10^2)$) due to smaller top-quark statistics, making the LHC the primary facility for observing these events.

Theoretical Uncertainties:

• $m_b$ dependence: Weak. Varying $m_b$ by $\pm 0.3$ GeV changes widths by less than 2%.
• $m_c$ dependence: Strong. Varying $m_c$ by $\pm 0.1$ GeV causes variations of 17% to 25% in the decay widths. This highlights that the charm quark mass is the dominant source of theoretical uncertainty due to its impact on the multi-body phase space.

NWA Comparison:
The NWA prediction is slightly larger than the exact full phase-space result when using tree-level widths. However, when corrected using experimental branching fractions, the NWA underestimates the exact result by only ~0.6%, confirming its high reliability for this process (within the expected $O(\Gamma_W/M_W)$ correction level).

5. Significance

• New Discovery Channel: This work proposes a viable and potentially abundant source for detecting $B_c$ mesons and charmonia at the LHC, complementing existing direct production methods.
• Validation of Theoretical Tools: By providing exact calculations for a complex 4-body decay, the paper validates the use of NWA in top-quark physics while quantifying its limits, offering a template for future precision studies.
• Experimental Guidance: The provided differential distributions (angular and invariant mass spectra) offer specific kinematic "fingerprints" (e.g., the non-monotonic angular distribution for $B_c$ vs. monotonic for $c\bar{c}$) that experimentalists can use to isolate these rare events from background noise.
• Heavy Flavor Physics: The results deepen the understanding of QCD interactions in both perturbative (hard gluon splitting) and non-perturbative (quarkonium formation) regimes, particularly in the unique environment of top-quark decays.