Vector quarkonia at the LHC with JETHAD: A high-energy viewpoint

This review extends the study of inclusive vector quarkonium production at the LHC using the JETHAD method and a hybrid NLL/NLO$^+$ factorization approach, demonstrating enhanced stability in forward rapidity regions and offering a unique opportunity for precision high-energy QCD studies and resolving the quarkonium production puzzle.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle smasher. Scientists fire protons at each other at nearly the speed of light to see what happens when they collide. Usually, they look for the "big hits"—massive new particles that could change our understanding of the universe.

But sometimes, the most interesting clues are hidden in the "debris" of these collisions. Specifically, this paper focuses on a very specific type of debris: Vector Quarkonia.

Think of these quarkonia as exotic, heavy-duty marbles made of a heavy quark and its anti-quark (like a charm-anticharm or bottom-antibottom pair) glued together tightly. They are the "gold-plated" messengers of the strong force, the glue that holds the universe's matter together.

Here is the story of what this paper does, explained simply:

1. The Problem: A Messy Puzzle

For decades, physicists have been trying to figure out exactly how these heavy marbles are formed. It's like trying to understand how a specific type of cake is baked just by looking at the crumbs on the floor.

• The Old Way: Scientists used to try to calculate the formation by looking at the very first split-second of the collision (short-distance). But when the marbles are moving very fast (high energy), this method gets messy and gives unstable, unreliable answers. It's like trying to predict the weather by only looking at the clouds right above your head, ignoring the wind systems miles away.
• The "Fragmentation" Idea: The paper suggests a better way. Instead of looking at the initial crash, imagine a single piece of the collision (a gluon or a heavy quark) flying off, losing energy, and then "condensing" or "crystallizing" into the heavy marble. This is called fragmentation. It's like a steam cloud cooling down to form a perfect ice crystal.

2. The New Tool: JETHAD

To test this idea, the author uses a sophisticated computer program called JETHAD. Think of JETHAD as a super-powered simulation engine or a "digital wind tunnel."

• It doesn't just simulate one collision; it simulates millions, accounting for the complex "high-energy" rules of physics that usually break standard calculators.
• It uses a "hybrid" approach: it combines the standard rules of particle physics (collinear factorization) with a special high-energy rulebook (BFKL resummation) that handles the chaos of particles flying apart at huge speeds.

3. The Discovery: The "Natural Stability"

The biggest surprise in this paper is a phenomenon the author calls "Natural Stability."

Imagine you are trying to balance a stack of plates. If you add a little weight to the top, the whole stack might wobble and fall. This is what usually happens in these high-energy physics calculations; small changes in the math make the results jump around wildly, making them useless for precise measurements.

However, when the author looked at these heavy quark marbles (especially when they are detected in the "forward" regions of the detector, like looking through the end of a telescope), the stack of plates stopped wobbling.

• The Analogy: It's as if these heavy marbles have a built-in "gyroscope." When they are created via the fragmentation process, they naturally stabilize the chaotic high-energy calculations.
• Why it matters: This stability is so strong that it works even better when looking at the edges of the detector (forward regions) than in the center. It means scientists can finally make precision measurements of the strong force in these high-energy regimes, something that was previously thought to be too unstable to measure accurately.

4. The "ZCW19+" Recipe

To make this work, the author created a new set of "recipes" (mathematical functions called Fragmentation Functions) named ZCW19+.

• Think of these as a cookbook that tells the computer exactly how a heavy quark or a gluon turns into a quarkonium marble at different speeds.
• These recipes were built using a theory called NRQCD (Non-Relativistic QCD), which is perfect for describing how heavy particles move slowly relative to each other inside the marble, even when the marble itself is flying fast.

5. The Big Picture: Why Should We Care?

This paper is a breakthrough for two main reasons:

1. Solving the Puzzle: It offers a fresh, stable way to solve the "Quarkonium Production Puzzle," helping us finally understand how these heavy particles are made.
2. A New Window into the Universe: Because the calculations are now stable, we can use these heavy marbles as precise tools to probe the inner structure of the proton. Just as a doctor uses X-rays to see inside a body, physicists can use these stable quarkonium signals to see the "glue" (gluons) inside the proton, especially in regions where the proton is packed with many low-energy gluons.

In summary:
This paper is like finding a new, steady hand for a shaky artist. By using a new "fragmentation" recipe and a powerful simulation tool (JETHAD), the author shows that heavy quark marbles produced at the LHC are naturally stable. This stability allows physicists to finally take precise measurements of the universe's strongest force, opening the door to discovering new physics and understanding the fundamental building blocks of matter.

Technical Summary

1. Problem Statement

The paper addresses two interconnected challenges in Quantum Chromodynamics (QCD) phenomenology at the Large Hadron Collider (LHC):

• The Quarkonium Production Puzzle: Despite decades of study, the mechanism governing the production of heavy quark-antiquark bound states (quarkonia like $J/\psi$ and $\Upsilon$) remains elusive. Standard models (Color Evaporation, Color Singlet, and Non-Relativistic QCD) often struggle to simultaneously describe production rates, polarization, and kinematic distributions across different energy scales.
• Instability in High-Energy Resummation: In the semi-hard regime (large transverse momentum $p_T$ and large rapidity separation $\Delta Y$), standard fixed-order perturbative QCD calculations fail due to large logarithmic terms ($\ln s$) that spoil the convergence of the perturbative series. While the Balitsky–Fadin–Kuraev–Lipatov (BFKL) formalism resums these terms, previous applications to semi-inclusive processes (like Mueller-Navelet jets) have suffered from severe instabilities under renormalization and factorization scale variations, often yielding unphysical negative cross-sections.

The specific goal of this work is to investigate whether the hybrid high-energy and collinear factorization approach, previously successful for heavy-flavored hadrons (like $\Lambda_c$ and $B$-mesons), can provide a stable and precise description of vector quarkonium production in association with a jet or another quarkonium.

2. Methodology

The authors employ a sophisticated theoretical framework combining high-energy resummation with collinear factorization, implemented via the JETHAD computational tool.

• Hybrid Factorization Scheme:

  • High-Energy Side: The process is described using the BFKL formalism at Next-to-Leading Logarithmic (NLL) accuracy. This resums large energy logarithms ($\alpha_s \ln s$) arising from $t$-channel gluon exchanges in the large rapidity gap.
  • Collinear Side: The incoming protons are described by standard Parton Distribution Functions (PDFs). Crucially, the final-state quarkonium is described via collinear Fragmentation Functions (FFs). This is valid in the regime where the quarkonium has large transverse momentum ($p_T \gg m_Q$), allowing the use of the Variable-Flavor Number Scheme (VFNS).
  • The "Plus" ($+$) Correction: The calculation includes NLO corrections to the impact factors (the transition from partons to final states), denoted as NLL/NLO+.

• NRQCD Fragmentation Inputs (ZCW19+):

  • The authors construct a novel set of FFs, named ZCW19+, specifically for vector quarkonia ($J/\psi$ and $\Upsilon$).
  • These FFs are built using Non-Relativistic QCD (NRQCD) factorization at the initial scale. They incorporate:
    • Heavy-quark fragmentation: $c \to J/\psi$ and $b \to \Upsilon$ (using NLO inputs from Ref. [78]).
    • Gluon fragmentation: $g \to J/\psi$ and $g \to \Upsilon$ (using LO inputs from Ref. [68]).
  • These initial inputs are then evolved to higher scales using the DGLAP equations (time-like evolution).

• Computational Tool (JETHAD):

  • The analysis uses JETHAD v0.5.0, a modular Python/Fortran interface designed for calculating observables in hybrid factorization. It handles multidimensional integrations over rapidity and transverse momentum, allowing for the study of $\Delta Y$ distributions.

• Kinematics:

  • Processes studied: $pp \to \mathcal{Q} + \mathcal{Q} + X$ (double quarkonium) and $pp \to \mathcal{Q} + \text{jet} + X$ (quarkonium + jet).
  • Configurations: Both "standard" (barrel detectors) and "extended" (forward/endcap detectors) rapidity ranges to test stability at the edges of applicability.

3. Key Contributions

1. Development of ZCW19+ FFs: The paper presents the first determination of NLO collinear FFs for vector quarkonia that consistently combine NRQCD initial-scale inputs (both heavy-quark and gluon channels) with DGLAP evolution. This provides a robust tool for future phenomenological studies beyond the high-energy domain.
2. Demonstration of "Natural Stability": The central theoretical finding is the confirmation of natural stability in the high-energy series for quarkonium production. The authors show that the specific behavior of the gluon fragmentation function (which grows with the factorization scale $\mu_F$) compensates for the growth of the gluon PDF and the running of the coupling constant. This cancellation stabilizes the cross-section against scale variations, a feature previously observed in heavy-flavor hadrons but not explicitly confirmed for quarkonia in this hybrid framework.
3. Extension to Forward Regions: The study extends the analysis to forward rapidity regions (endcaps), demonstrating that the stability pattern is not only preserved but magnified in these kinematic regimes, even when approaching threshold regions where large Sudakov logarithms might typically spoil convergence.

4. Results

• Stability under Scale Variation:

  • The authors varied the renormalization ($\mu_R$) and factorization ($\mu_F$) scales by factors of $C_\mu = 1/2$ to $30$.
  • Result: The differential cross-sections ($d\sigma/d\Delta Y$) for both double quarkonium and quarkonium-plus-jet channels showed remarkable stability. The uncertainty bands were narrow, and the curves for different scale choices overlapped significantly, unlike the large instabilities seen in light-hadron or jet-only processes.
  • This stability was observed in both standard ($\Delta Y < 4.8$) and extended ($\Delta Y < 9.4$) rapidity configurations.

• Comparison with LL/LO:

  • When comparing the full NLL/NLO+ predictions with pure Leading Log/Leading Order (LL/LO) approximations, the NLL/NLO+ bands were found to be narrower and often nested within the LL/LO bands. This indicates that the higher-order corrections are under control and do not introduce large destabilizing effects.

• Cross-Section Magnitudes:

  • The predicted cross-sections for the quarkonium-plus-jet channel are substantial (values $> 10^{-2}$ pb), suggesting that these processes are experimentally accessible at the LHC with current or near-future luminosity. The double quarkonium channel has lower statistics but remains theoretically significant.

5. Significance

• Precision QCD Probe: The results establish vector quarkonia at the LHC as a "gold-plated" channel for precision studies of high-energy QCD. The natural stability of the hybrid factorization allows for reliable extraction of QCD parameters and tests of the BFKL dynamics without the need for ad-hoc scale choices to force stability.
• Insight into the Production Puzzle: By successfully describing quarkonium production using a single-parton fragmentation mechanism within a stable high-energy framework, the paper supports the hypothesis that fragmentation dominates in the high-$p_T$ regime. This offers a potential resolution to parts of the quarkonium production puzzle, specifically regarding the interplay between short-distance mechanisms and long-distance hadronization.
• Future Outlook: The work paves the way for:
  • Comparing these NRQCD-inspired FFs with global fit extractions (using neural networks).
  • Investigating Multi-Parton Interactions (MPI) and Double-Parton Scattering (DPS) effects in these stable channels.
  • Applying the hybrid factorization to future colliders (FCC, EIC) and exploring connections with gluon saturation frameworks.

In summary, the paper provides a robust theoretical and phenomenological framework that resolves previous instabilities in high-energy quarkonium calculations, validating the hybrid factorization approach and offering a new, stable window into the dynamics of strong interactions at the LHC.