All-charm tetraquarks at hadron colliders: A high-precision fragmentation perspective

This paper presents the TQ4Q2.0 fragmentation functions, a high-precision, uncertainty-quantified framework for modeling the production of all-heavy $S$-wave tetraquarks in hadronic collisions, which extends previous models by incorporating nonconstituent heavy-quark contributions and advanced evolution schemes to establish a definitive baseline for future collider phenomenology.

The Gist

Imagine the universe is built out of tiny, invisible LEGO bricks called quarks. Usually, these bricks snap together in very simple, predictable ways: two bricks make a "meson" (like a proton's cousin), and three bricks make a "baryon" (like a proton or neutron). This has been the rulebook for physics for decades.

But recently, scientists have started finding "exotic" LEGO creations that break the rules: Tetraquarks. These are structures made of four bricks stuck together. Even more bizarre are the "All-Heavy" Tetraquarks, where all four bricks are the heaviest, most stubborn type of quark available (like the "charm" or "bottom" quarks). It's like trying to build a house out of four giant, dense anvils instead of lightweight foam blocks.

This paper is a high-precision instruction manual for how to find and study these heavy, four-quark monsters inside giant particle smashers like the Large Hadron Collider (LHC).

Here is the breakdown of what the authors did, using some everyday analogies:

1. The Problem: Finding a Needle in a Haystack

When protons smash together at near light speed, they create a chaotic explosion of particles. Finding a specific, rare tetraquark in this mess is like trying to find a specific, unique snowflake in a blizzard.

Previously, scientists had a rough sketch of how these particles might form (called "fragmentation functions"). But that sketch was like a hand-drawn map with missing roads and blurry details. It didn't account for all the different ways the particles could be created, and it didn't tell you how much you could trust the numbers.

2. The Solution: The "TQ4Q2.0" GPS

The authors have created a new, ultra-detailed GPS system called TQ4Q2.0. Think of this as upgrading from a sketchy paper map to a live, 3D satellite navigation system with real-time traffic updates.

• The "All-Heavy" Focus: They focused specifically on the heavy, four-quark states (all-charm and all-bottom).
• The "Replica" Strategy: Instead of giving just one answer, they generated about 100 slightly different versions of their map (called "replicas"). Imagine asking 100 different expert chefs to make the same soup. If they all agree on the taste, you know the recipe is solid. If they disagree, you know exactly where the uncertainty lies. This helps scientists know exactly how confident they can be in their predictions.
• New Roads: They added "non-constituent" paths. Previously, the map only showed how the main heavy bricks formed the tetraquark. Now, the map shows how lighter, "passenger" particles can also help build the structure, which turns out to be a much bigger deal than anyone thought.

3. The Three Shapes of the Monster

The paper studies three different "shapes" (quantum spins) these tetraquarks can take:

• Scalar (0++): The "Standard" shape.
• Axial-Vector (1+−): A tricky shape that is very hard to make and very sensitive to the rules of physics.
• Tensor (2++): A complex, spinning shape.

The Discovery: They found that the "Scalar" and "Tensor" shapes are like soft, fluffy clouds—they form easily and are common. The "Axial-Vector" shape is like a hard, dense diamond—it's much rarer and requires very specific, high-energy conditions to form. This distinction is crucial because it tells experimentalists exactly what to look for.

4. The Prediction: What Will We See?

Using their new GPS, the authors predicted what will happen at the LHC (currently running) and the future FCC (a proposed super-collider).

• The "Yield" (The Count): They calculated exactly how many of these particles we should expect to see.
  • For the Scalar and Tensor types, the numbers are huge (millions of events at the LHC). This is great news! It means these particles are within reach of current detectors.
  • For the Axial-Vector type, the numbers are tiny. It's like looking for a specific grain of sand on a beach.
• The "Jet" Connection: They realized these tetraquarks don't just appear alone; they usually come with a "jet" (a spray of other particles). It's like finding a rare coin, but it's always stuck to a specific type of gum. By looking for the coin and the gum together, scientists can filter out the noise and find the signal much faster.

5. Why This Matters

This paper is a bridge between theory (math on a chalkboard) and experiment (real data in a detector).

• For the Experimentalists: It gives them a "shopping list." It tells them, "Look here, at these specific energies, and you have a good chance of finding a tetraquark."
• For the Theorists: It provides a "stress test." If the experiments find numbers that don't match this new, high-precision map, it means our understanding of the strong force (the glue holding the universe together) is wrong, and we need to rewrite the laws of physics.

The Bottom Line

This paper is the definitive blueprint for hunting the heaviest, most exotic LEGO structures in the universe. It moves us from "guessing" where they might be to "knowing" exactly how to find them, how many to expect, and how to measure them with extreme precision. It turns the search for these exotic particles from a game of chance into a precise science.

Technical Summary

1. Problem Statement

The production mechanisms of exotic hadrons, specifically all-heavy tetraquarks ($T_{4Q}$, composed of four heavy quarks/antiquarks like $|c\bar{c}c\bar{c}\rangle$ or $|b\bar{b}b\bar{b}\rangle$), remain poorly understood compared to conventional mesons and baryons. While experimental evidence for such states (e.g., $X(6900)$, $T_{cc}^+$) has grown at the LHC, theoretical descriptions often rely on model-dependent frameworks (e.g., color evaporation, hadron-quark duality) or lack a systematic treatment of uncertainties.

A critical gap exists in the fragmentation function (FF) description of these states at high transverse momentum ($p_T$). Previous efforts (TQ4Q1.0/1.1) were limited in scope, often neglecting non-constituent quark channels and lacking a robust, data-driven uncertainty quantification strategy. The paper aims to establish a high-precision, first-principles QCD framework for the production of all-heavy tetraquarks at hadron colliders (HL-LHC and FCC), enabling precise predictions for experimental searches.

2. Methodology

The authors employ a Non-Relativistic QCD (NRQCD) factorization approach combined with collinear fragmentation and high-energy resummation.

• Theoretical Framework:

  • NRQCD Factorization: The production is factorized into perturbative Short-Distance Coefficients (SDCs) and non-perturbative Long-Distance Matrix Elements (LDMEs). The FFs are constructed at an initial scale $\mu_{F,0}$ by matching QCD amplitudes to NRQCD matrix elements for compact $|Q\bar{Q}Q\bar{Q}\rangle$ states.
  • Partonic Channels: Unlike previous generations, this work includes all partonic channels at the initial scale:
    • Gluon fragmentation ($g \to T_{4Q}$).
    • Constituent heavy-quark fragmentation ($Q \to T_{4Q}$).
    • Non-constituent channels: Light quarks ($\tilde{q} \to T_{4Q}$) and non-constituent heavy quarks ($\tilde{Q} \to T_{4Q}$).
  • Quantum Numbers: The study covers scalar ($0^{++}$), axial-vector ($1^{+-}$), and tensor ($2^{++}$) states for both all-charm ($T_{4c}$) and all-bottom ($T_{4b}$) systems.

• Evolution and Uncertainty Quantification:

  • HF-NRevo Scheme: The initial FFs are evolved to collider scales using the Heavy-flavor nonrelativistic-evolution (HF-NRevo) scheme. This handles multiple kinematic thresholds (e.g., $4m_Q$ for gluons, $5m_Q$ for heavy quarks) via a two-stage DGLAP evolution: a semi-analytical "Expanded Decoupled" (EDevo) stage up to the highest threshold, followed by a fully numerical "All-Order" (AOevo) stage.
  • Replica-Based Uncertainty (F-MHOUs): A novel multi-scale variation strategy is implemented. Instead of simple scale variations, the authors generate an ensemble of $\sim 100$ replicas by independently varying the renormalization scale ($\mu_R$) and the evolution-ready scale ($Q_0$) logarithmically in $[1/2, 2]$. This provides a dynamically correlated estimate of missing higher-order uncertainties (F-MHOUs).
  • LDMEs: Non-perturbative inputs are derived from potential models (Cornell-like potentials) and vacuum saturation approximations, with uncertainties estimated by comparing different potential models.

• Phenomenological Application:

  • The FFs are integrated into the HyF (Hybrid Factorization) formalism, which combines NLO collinear factorization with NLL (Next-to-Leading Log) high-energy resummation (BFKL dynamics).
  • Calculations are performed using the JETHAD numerical framework for inclusive $T_{4Q} + \text{jet}$ production in proton-proton collisions at $\sqrt{s} = 13$ TeV (HL-LHC) and $\sqrt{s} = 100$ TeV (FCC).

3. Key Contributions

1. TQ4Q2.0 Fragmentation Functions: The public release of the first complete, multi-channel, uncertainty-aware FF set for all-heavy tetraquarks in LHAPDF6 format. This includes contributions from non-constituent quarks, which were previously neglected.
2. Replica-Based Uncertainty Strategy: The first application of a replica-based Multi-Higher-Order Uncertainty (MHOU) framework to bound-state fragmentation. This allows for a more robust propagation of perturbative uncertainties to collider observables and facilitates future data-driven fits.
3. Threshold-Aware Evolution: Implementation of the HF-NRevo scheme to consistently handle the complex threshold structure of tetraquark fragmentation (involving 4 heavy quarks) within the Variable Flavor Number Scheme (VFNS).
4. Comprehensive Phenomenology: Detailed predictions for rapidity-differential cross-sections and event yields for scalar, axial-vector, and tensor states, including a systematic comparison between fixed-order and resummed calculations.

4. Key Results

• Fragmentation Dynamics:

  • Scalar ($0^{++}$) and Tensor ($2^{++}$): These channels are dominated by soft fragmentation (low momentum fraction $z$) driven by gluon and light-quark channels. The inclusion of non-constituent quark channels enhances cross-sections by 15–20% compared to previous TQ4Q1.1 sets.
  • Axial-Vector ($1^{+-}$): This channel is intrinsically hard (peaked at high $z \sim 0.8$) and is generated entirely radiatively via DGLAP evolution (no direct initial-scale gluon contribution due to Landau-Yang theorem). It exhibits the smallest theoretical uncertainties and is the cleanest probe of high-energy dynamics.
  • Hierarchy: Gluon fragmentation dominates, followed by non-constituent quarks, with constituent heavy quarks being subleading but significant.

• Collider Predictions (HL-LHC & FCC):

  • Event Yields: For $T_{4c}$ at $\sqrt{s}=13$ TeV (Run 2 luminosity), expected yields are $\sim 2 \times 10^6$ for scalars and $\sim 3 \times 10^6$ for tensors, while axial-vectors are suppressed ($\sim 10^4$). At FCC ($\sqrt{s}=100$ TeV), yields increase by an order of magnitude.
  • Resummation Effects: High-energy resummation (NLL) significantly impacts large rapidity intervals ($\Delta Y$). At FCC energies, resummed predictions exceed fixed-order (HE-NLO+) results at large $\Delta Y$, indicating the dominance of BFKL dynamics.
  • Uncertainties: Perturbative uncertainties (F-MHOUs) and LDME variations are comparable in magnitude. F-MHOUs primarily affect normalization, while the shape of rapidity distributions is driven by perturbative dynamics.

• Comparison with Previous Sets: The TQ4Q2.0 set shows a systematic enhancement over TQ4Q1.1 for scalar and tensor states due to the inclusion of non-constituent channels. Axial-vector yields remain unchanged as they are purely radiative.

5. Significance

• Precision Standard: The TQ4Q2.0 framework establishes a new reference standard for studying all-heavy multiquark dynamics, moving the field from exploratory modeling to high-precision phenomenology.
• Experimental Guidance: The provided event yield benchmarks and rapidity distributions offer actionable targets for current (LHC Run 2/3) and future (HL-LHC, FCC) experiments to search for all-charm tetraquarks, particularly in di-J/$\psi$ or di-$\Upsilon$ channels.
• Theoretical Robustness: By integrating NRQCD, DGLAP evolution, and BFKL resummation with a rigorous uncertainty quantification strategy, the work bridges the gap between first-principles QCD and experimental observables.
• Future Prospects: The framework is designed to be data-driven. The replica-based uncertainty sets allow for future global fits to constrain LDMEs and probe intrinsic heavy-quark components in the proton, particularly through the axial-vector channel.

In conclusion, this paper provides the most complete and rigorous theoretical toolset to date for analyzing the production of all-heavy tetraquarks at high-energy colliders, paving the way for definitive experimental discoveries and deeper insights into the non-perturbative regime of QCD.