Fully charmed tetraquark production in forward rapidity $pp$ collisions at LHC and FCC energies

This paper investigates the production of fully charmed tetraquarks in forward rapidity $pp$ collisions at LHC and FCC energies using the Color Glass Condensate formalism, revealing that the tensor state $T_{4c}(2^{++})$ is predominantly produced via gluon-initiated processes while the axial-vector state $T_{4c}(1^{+-})$ is dominated by charm-initiated processes and is highly sensitive to intrinsic charm.

The Gist

Imagine the universe's building blocks not just as tiny, solitary marbles (quarks), but as complex, swirling dance troupes. For decades, physicists believed these troupes could only have two or three members (like a proton, which is three quarks). But recently, they discovered "exotic" troupes with four or five members. One of the most exciting new discoveries is a troupe made entirely of four heavy "charm" quarks, called a fully charmed tetraquark (or T4c).

This paper is like a set of blueprints and a weather forecast for finding these rare, heavy dancers in the world's biggest particle accelerators: the LHC (Large Hadron Collider) in Europe and the proposed FCC (Future Circular Collider).

Here is the breakdown of their research, explained simply:

1. The Setting: A High-Speed Collision Course

Think of the LHC and FCC as massive, high-speed train stations where two proton trains crash into each other.

• The Goal: When these trains smash, they create a chaotic explosion of energy. The scientists want to know: "How often does this explosion create a T4c tetraquark?"
• The Location: Most previous studies looked at the "center" of the crash (where debris flies straight out). This paper looks at the forward rapidity—the debris flying off at a sharp angle, almost parallel to the train tracks. This is a tricky, high-speed region where the rules of physics get a little weird.

2. The Two Ways to Build the Tetraquark

The paper suggests there are two main "construction crews" that can build this T4c tetraquark during the crash:

• Crew A: The Gluon Team (The Glue)

  • Gluons are the "glue" that holds quarks together. In this scenario, two gluons smash together and magically fuse into the T4c.
  • The Analogy: Imagine two strong winds colliding and suddenly forming a solid tornado.
  • The Finding: This crew is the most productive. They build the "Tensor" version of the T4c (a specific shape/spin of the particle) the most often. This happens mostly at the center of the crash and at very high energies.

• Crew B: The Charm Team (The Heavy Hitters)

  • Sometimes, a "charm" quark (one of the heavy ingredients) is already hiding inside the incoming proton. It gets knocked out and fuses with another charm quark to build the T4c.
  • The Analogy: Imagine a hidden passenger on the train who jumps out and immediately grabs another passenger to form a duo.
  • The Twist: This crew is usually very small and quiet. However, the paper suggests that protons might have a secret stash of "intrinsic charm" (hidden heavy passengers) that we didn't know about. If this secret stash exists, the Charm Team suddenly becomes much more active, especially when looking at the debris flying off to the side (forward rapidity).

3. The "Intrinsic Charm" Mystery

This is the paper's biggest detective story.

• The Theory: For a long time, physicists thought protons were just three light quarks with a sea of virtual particles popping in and out. But a theory called Intrinsic Charm (IC) suggests that sometimes, a real, heavy charm quark is permanently part of the proton's structure, like a permanent resident rather than a tourist.
• The Test: The authors calculated what would happen if this "permanent resident" exists.
  • Result: If the T4c is built by the Charm Team (specifically the "Axial-Vector" shape), the production rate skyrockets if Intrinsic Charm is real.
  • Why it matters: If we see a lot of these specific T4c particles flying off to the side, it's a smoking gun that proves protons have this hidden heavy charm inside them!

4. The Tools: The "Color Glass Condensate"

To predict this, the authors used a sophisticated mathematical framework called the Color Glass Condensate (CGC).

• The Metaphor: Imagine the proton as a dense, chaotic crowd of people (gluons) packed into a small room. At high speeds, this crowd acts like a thick, sticky glass. The CGC is the set of rules used to describe how this "sticky glass" behaves when hit by a projectile. It helps the scientists calculate how the particles scatter in those tricky forward angles.

5. The Results: What Should We Expect?

The authors ran the numbers for the current LHC and the future FCC:

• The Tensor State (2++): This is the "easy" one to make. It's mostly built by the Gluon Team. We should see plenty of these (billions of events if we run the collider long enough).
• The Axial-Vector State (1+−): This is the "hard" one. It relies on the Charm Team.
  • Without Intrinsic Charm: We will see very few of these.
  • With Intrinsic Charm: We will see significantly more, especially at the FCC.
• The Verdict: The paper concludes that looking for the Axial-Vector T4c at forward angles is the best way to test if protons have this hidden "Intrinsic Charm."

Summary in a Nutshell

This paper is a guide for future experiments. It tells us:

1. Look at the edges of the collision debris, not just the center.
2. Watch for the "Tensor" T4c to confirm our general understanding of particle physics.
3. Keep a sharp eye on the "Axial-Vector" T4c. If we find more of these than expected, it proves that protons have a secret, heavy "intrinsic charm" component inside them, solving a decades-old mystery about what makes up our universe's most common matter.

It's like saying, "We know how to build a house with bricks (gluons), but if we find a house built with a specific type of rare gold brick (intrinsic charm), we'll know the architect had a secret stash we never knew about."

Technical Summary

1. Problem Statement

The production mechanism of fully charmed tetraquark states ($T_{4c}$, composed of $cc\bar{c}\bar{c}$) remains an open question in high-energy physics, particularly regarding their dynamical production in hadronic collisions.

• Context: While the $T_{4c}$ candidate was observed by LHCb (and confirmed by ATLAS/CMS) in the double $J/\psi$ mass spectrum, most theoretical studies have focused on central rapidities using collinear factorization and Non-Relativistic QCD (NRQCD).
• The Gap: At forward rapidities, the kinematics involve asymmetric parton momentum fractions ($x_1 \gg x_2$). In this regime, standard collinear approaches fail to capture:
  1. Small-$x$ effects: Non-linear QCD dynamics (saturation) in the target wave function.
  2. Large-$x$ effects: The potential presence of an Intrinsic Charm (IC) component in the projectile proton wave function, which could significantly enhance charm-initiated processes at large $x$.
• Objective: To investigate the impact of small-$x$ saturation and large-$x$ intrinsic charm on $T_{4c}$ production at forward rapidities for both the Large Hadron Collider (LHC, $\sqrt{s}=13$ TeV) and the Future Circular Collider (FCC, $\sqrt{s}=100$ TeV).

2. Methodology

The authors employ a Hybrid Factorization Formalism combined with the Color Glass Condensate (CGC) framework. This approach treats the scattering of a dilute projectile (proton) off a dense target (proton) at forward rapidities.

• Factorization Scheme:
  The cross-section is calculated as a convolution of projectile Parton Distribution Functions (PDFs), target scattering amplitudes, and Fragmentation Functions (FFs):
  $$\sigma(pp \to T_{4c}X) \propto g(x_1) \otimes N_A(x_2) \otimes D_{g/T_{4c}} + c(x_1) \otimes N_F(x_2) \otimes D_{c/T_{4c}}$$

  • Gluon-Initiated (GI): $g \to T_{4c}$ transition.
  • Charm-Initiated (CI): $c \to T_{4c}$ transition.

• Target Dynamics (Small-$x$):

  • The target scattering amplitudes ($N_A$ and $N_F$) are derived by solving the running coupling Balitsky-Kovchegov (rcBK) equation.
  • Initial conditions are based on the McLerran-Venugopalan (MV) model, fitted to HERA data.
  • This accounts for gluon saturation effects in the target.

• Projectile Dynamics (Large-$x$ & Intrinsic Charm):

  • The study utilizes PDF sets from CT18 and NNPDF.
  • Crucially, it compares scenarios with and without an Intrinsic Charm (IC) component (specifically the BHPS model for CT18 and IC models for NNPDF).
  • This allows the authors to isolate the contribution of a pre-existing $c\bar{c}$ pair in the proton wave function.

• Fragmentation Functions (FFs):

  • The transition probabilities ($g/c \to T_{4c}$) are modeled using the TQ4Q1.1 parameterization.
  • These FFs are derived within NRQCD and evolved via DGLAP evolution.
  • Three quantum number configurations are analyzed:
    1. Scalar: $J^{PC} = 0^{++}$
    2. Axial-vector: $J^{PC} = 1^{+-}$
    3. Tensor: $J^{PC} = 2^{++}$

3. Key Contributions

1. First Forward Rapidity Analysis: This is the first study to apply the CGC formalism to $T_{4c}$ production, specifically addressing the interplay between small-$x$ saturation and large-$x$ intrinsic charm.
2. Quantum Number Dependence: The paper distinguishes production mechanisms based on the quantum numbers of the tetraquark, revealing a stark dichotomy between tensor and axial-vector states.
3. Intrinsic Charm Probe: It establishes the $T_{4c}(1^{+-})$ state as a unique probe for the Intrinsic Charm hypothesis, as its production is dominated by charm-initiated fragmentation.
4. FCC Predictions: Provides the first phenomenological predictions for $T_{4c}$ production at $\sqrt{s} = 100$ TeV, a key energy for future discovery potential.

4. Key Results

• Dominance of Tensor States:

  • The Tensor state ($2^{++}$) exhibits the largest production cross-section.
  • Its production is dominated by the gluon-initiated (GI) process.
  • Consequently, the cross-section for $T_{4c}(2^{++})$ is relatively insensitive to the presence of Intrinsic Charm in the PDFs.
  • Predicted cross-sections at LHC ($p_T \ge 20$ GeV, $2.0 \le y \le 4.5$) are $\approx 1.2 - 2.9$ nb (depending on PDF/FF uncertainties). At FCC, this rises to $\approx 8 - 17$ nb.

• Sensitivity of Axial-Vector States:

  • The Axial-vector state ($1^{+-}$) is suppressed by two orders of magnitude compared to the tensor state.
  • Its production is dominated by the charm-initiated (CI) process.
  • Crucial Finding: The cross-section for $T_{4c}(1^{+-})$ is highly sensitive to the Intrinsic Charm component.
    • At LHC energies, including IC enhances the cross-section significantly (by factors depending on the PDF set, e.g., NNPDF IC shows a larger enhancement).
    • At FCC energies, the enhancement is less pronounced in the integrated cross-section because the IC effect is concentrated at very high $p_T$, while the integral is dominated by lower $p_T$ regions. However, differential distributions at high $p_T$ show strong IC dependence.

• Kinematic Dependencies:

  • Cross-sections increase with center-of-mass energy ($\sqrt{s}$) and decrease with increasing rapidity ($y$).
  • The uncertainty in predictions is dominated by the non-perturbative Long-Distance Matrix Elements (LDMEs) in the fragmentation functions, rather than the choice of PDFs (except for the CI channel).

• Experimental Feasibility:

  • The estimated event yield for $T_{4c}(2^{++})$ at the LHC with an integrated luminosity of $3000 \text{ fb}^{-1}$ is $\gtrsim 10^9$ events, suggesting that a future experimental analysis of this channel is highly feasible.

5. Significance

• Theoretical Validation: The work validates the hybrid factorization approach for heavy exotic hadron production in the forward region, bridging the gap between collinear NRQCD and small-$x$ saturation physics.
• Phenomenological Tool: It provides specific, testable predictions for the LHCb, ATLAS, and CMS collaborations, as well as future FCC experiments.
• Intrinsic Charm Discovery: The paper argues that measuring the production of the $T_{4c}(1^{+-})$ state at forward rapidities offers a robust method to confirm or constrain the existence of Intrinsic Charm in the proton, a long-standing debate in QCD.
• Quantum Number Discrimination: By predicting distinct production rates and mechanisms for different spin-parity states, the study aids in the experimental identification of the quantum numbers of observed $T_{4c}$ candidates.

In summary, the paper demonstrates that while tensor $T_{4c}$ states are the most abundant and dominated by gluon fusion, the rare axial-vector states serve as a critical "smoking gun" for intrinsic charm physics in the forward kinematic regime.