Automated event generation for S-wave quarkonium and leptonium production in NRQCD and NRQED

This paper presents an extension of the MadGraph5_aMC@NLO framework that enables automated leading-order calculations for S-wave quarkonium and leptonium production within NRQCD and NRQED formalisms, offering seamless integration with Beyond the Standard Model scenarios and highlighting the necessity of careful theoretical analysis beyond simple velocity-scaling rules.

The Gist

Imagine the universe as a giant, high-speed construction site. At the center of this site, physicists are trying to understand how tiny building blocks—quarks and leptons—stick together to form complex structures called quarkonia (like atoms made of heavy quarks) and leptonia (like atoms made of light electrons or muons).

For decades, figuring out exactly how these structures are built during high-energy collisions (like those at the Large Hadron Collider) has been like trying to predict the outcome of a chaotic demolition derby using only a pencil and paper. It's slow, prone to errors, and very difficult to do for complex scenarios.

This paper introduces a brand-new, automated construction simulator called an extension to MadGraph5_aMC@NLO. Think of this software as a "smart drone" that can instantly calculate the odds of these tiny atoms forming, no matter how complicated the crash is.

Here is a breakdown of what they did, using some everyday analogies:

1. The Problem: The "Manual Calculator" Bottleneck

Previously, if a physicist wanted to study how a specific type of heavy atom (quarkonium) forms, they had to write custom code for every single scenario. It was like trying to bake a cake by hand-measuring every grain of sugar for every new recipe. If you wanted to add a new ingredient (like a jet of particles or a Higgs boson), you had to rewrite the whole recipe.

Furthermore, while we knew how to study heavy quark atoms, studying lepton atoms (like "True Muonium" or "Ditauonium") was almost impossible with existing tools. It was like having a map for the entire world, but the map was blank for the ocean.

2. The Solution: The "Universal Lego Kit"

The authors have built a universal Lego kit (a software module) that plugs directly into the world's most popular physics simulation engine, MadGraph.

• The "Boundstate" Brick: They created a new type of digital brick called a "Boundstate." Instead of treating a heavy atom as a messy collection of particles, they treat it as a single, pre-assembled unit.
• The "Fock State" Menu: Inside this unit, there are different "flavors" or configurations (called Fock states). Imagine a car that can be a sedan, a convertible, or a truck depending on how you assemble it. The software can now automatically generate all these different versions and calculate the odds of each one appearing.
• The "Lepton" Expansion: For the first time, this kit works for leptonia too. It's like finally getting the instruction manual for building the ocean's underwater cities, not just the land ones.

3. How It Works: The "Recipe Book"

The software uses a "recipe book" (called a UFO model) that tells the computer how to assemble these atoms.

• Input: You type a simple command like "Make a J/Psi atom with a jet."
• Process: The software automatically figures out all the invisible steps (the quantum mechanics) required to make that happen. It handles the "color" and "spin" of the particles (which are like the particle's internal ID card and orientation) without the user needing to do the math.
• Output: It spits out a list of probabilities (cross-sections) and even simulates the actual collision events, which can then be fed into other programs to see what the detectors would actually "see."

4. The Surprising Discovery: "Don't Trust Your Gut"

One of the most interesting findings in the paper is a warning to physicists: Simple logic doesn't always work here.

In everyday life, if you have a small engine, you expect a small car. In particle physics, the authors found that sometimes the "small" or "subleading" contributions (the tiny, unlikely ways an atom can form) actually dominate the result because of specific rules of nature (like conservation laws).

• Analogy: Imagine you are betting on a horse race. You might think the fastest horse (the one with the strongest engine) will win. But in this quantum race, sometimes the horse with the "slowest" engine wins because the track has a hidden shortcut that only that horse knows. The software proves that you can't just guess; you have to calculate every single possibility.

5. Why This Matters

This tool is a game-changer for two main reasons:

1. Speed and Flexibility: Physicists can now test hundreds of scenarios in minutes instead of months. They can easily switch between different collision types (proton-proton, electron-positron, etc.) without rewriting code.
2. New Frontiers: It opens the door to studying "True Muonium" and "Ditauonium." These are exotic atoms that have never been seen in a collider before. Finding them would be like discovering a new element on the periodic table, potentially revealing "New Physics" beyond our current understanding of the universe.

The Bottom Line

This paper is about handing physicists a powerful, automated remote control for the subatomic world. Instead of manually wiring every circuit, they can now press a button to simulate how the universe builds its most fragile and complex atoms. It ensures that when we look at data from giant colliders, we aren't just guessing what we see—we are seeing it with crystal-clear, mathematically precise eyes.

Technical Summary

1. Problem Statement

Heavy quark-antiquark bound states (quarkonia, e.g., $J/\psi$, $\Upsilon$) and lepton-antilepton bound states (leptonia, e.g., positronium, true muonium) are crucial probes for understanding Quantum Chromodynamics (QCD) and Quantum Electrodynamics (QED). While theoretical frameworks like Non-Relativistic QCD (NRQCD) and Non-Relativistic QED (NRQED) exist to describe their production, there has been a lack of general-purpose, automated tools capable of generating Monte Carlo (MC) event samples for these processes across diverse collider environments.

Existing tools suffer from specific limitations:

• General-purpose generators (Pythia, Herwig) often lack full compatibility between matrix elements and parton showers for quarkonia or rely on simplified hadronization models.
• Dedicated generators (MadOnia, HELAC-Onia) are often process-specific, not fully integrated into modern NLO-capable frameworks, or lack support for leptonia.
• Leptonium studies are particularly scarce due to the complexity of handling bound states in QED and the lack of public tools.

The authors aim to bridge this gap by extending the widely used MadGraph5_aMC@NLO (MG5_aMC) framework to automate the generation of leading-order (LO) matrix elements and event samples for S-wave quarkonium and leptonium production.

2. Methodology

The authors implemented a new module within MG5_aMC based on the NRQCD and NRQED factorization formalisms.

Theoretical Framework

• Factorization: The cross-section is factorized into a short-distance coefficient (perturbative production of a heavy quark/lepton pair in a specific quantum state $n$) and a Long-Distance Matrix Element (LDME) describing the non-perturbative hadronization/formation of the bound state.
• Fock State Expansion: Physical bound states are treated as superpositions of Fock states characterized by quantum numbers: spin ($S$), orbital angular momentum ($L$), total angular momentum ($J$), and color ($C$).
• Projection Method: The implementation uses a covariant projection method to isolate specific Fock states.
  • Color Projection: Projects the heavy pair onto color-singlet ($C=1$) or color-octet ($C=8$) states.
  • Spin Projection: Uses spin projectors ($\Gamma_S$) to isolate singlet/triplet states.
  • Orbital Projection: For S-wave ($L=0$), the limit of relative momentum $q \to 0$ is taken. (Note: P-wave implementation is deferred to future work).
• NRQED Adaptation: For leptonia, the color projection is trivial (always singlet), and LDMEs are calculated analytically via the Schrödinger equation with a Coulomb potential, rather than fitted to data.

Implementation in MG5_aMC

• UFO Model Extension: A new Universal Feynman Output (UFO) model, sm_onia, was created. It introduces a Boundstate class analogous to the Particle class.
• Syntax: Users define bound states using a specific naming convention (e.g., Jpsi(1|3S11) for $J/\psi$ in the $^3S_1^{[1]}$ state).
• Multiparticle Handling: Physical mesons (e.g., Jpsi) are defined as collections of Fock states in boundstates_default.txt, allowing users to generate processes for the physical particle, which automatically sums over all contributing Fock states.
• LDME Handling: LDMEs are input via an onia_card.dat file. For quarkonia, values are taken from fits or wave-function calculations; for leptonia, they are fixed by theory.
• Running Modes:
  • LO Mode: Generates unweighted parton-level events (LHEF) for showering/hadronization.
  • Standalone Mode: Generates Fortran code for evaluating helicity amplitudes for specific kinematic points (useful for validation).

3. Key Contributions

1. First Automated Leptonium Generator: This work provides the first public, automated tool for generating leptonium (positronium, true muonium, true tauonium) production events in high-energy collisions.
2. Seamless Integration: The extension integrates natively with MG5_aMC, allowing for:
   • Generation of complex processes (single, double, triple quarkonia, associated production with jets/bosons).
   • Compatibility with Parton Showers (Pythia) and Hadronization.
   • Support for Beyond Standard Model (BSM) scenarios via custom UFO models (demonstrated with Higgs Effective Field Theory).
3. Flexible Fock State Management: Unlike previous tools where LDMEs were hardcoded or required manual rescaling, this implementation allows dynamic assignment of LDMEs to specific Fock states at runtime.
4. Hybrid Renormalization Scheme: For leptonium production, the authors implemented a hybrid scheme using the Thomson limit fine-structure constant $\alpha(0)$ for photon couplings to avoid artificial enhancements from the standard $G_\mu$ scheme.

4. Results and Validation

The framework was validated against HELAC-Onia and analytic expressions.

• Validation:

  • Matrix Elements: Comparison of squared amplitudes for 42 partonic processes showed agreement at the level of floating-point precision ($\Delta_{rel} \sim 10^{-7} - 10^{-15}$).
  • Cross Sections: Total cross sections for various processes (single, pair, and triple production) in $pp$ and $e^+e^-$ collisions agreed with HELAC-Onia within statistical uncertainties.
  • Analytic Checks: Leptonium cross sections were compared against analytic limits (decoupling $Z/H$ bosons), showing agreement at the permille level.

• Phenomenological Studies (Selected Examples):

  • $pp$ Collisions (LHC, 13 TeV):
    • Calculated inclusive single charmonium/bottomonium production, showing that color-octet contributions can dominate despite velocity-scaling rules due to kinematic enhancements.
    • Studied associated production with $W/Z$ bosons and jets.
    • Demonstrated Triple $J/\psi$ production (probing Triple Parton Scattering), calculating a total LO cross section of $\sim 1$ pb.
    • Analyzed $J/\psi + H$ production using Higgs Effective Field Theory (HEFT), showing distinct rapidity distributions for different Fock states.
  • $ep$ Collisions (EIC/HERA):
    • Provided cross sections for Deep Inelastic Scattering (DIS) and photoproduction of charmonia.
  • $e^+e^-$ Colliders (B-factories):
    • Calculated inclusive and exclusive production (e.g., $J/\psi + \gamma$, $J/\psi + J/\psi$).
  • Leptonium Production:
    • Provided cross sections for Positronium, True Muonium, and True Tauonium in $pp$ and $e^+e^-$ collisions.
    • Highlighted that while cross sections are small (fb to ab range), they are within reach of high-luminosity experiments (e.g., Belle II, FCC-ee, LHC).

5. Significance

• Paradigm Shift in Quarkonium Physics: The tool democratizes access to complex NRQCD calculations, allowing researchers to easily explore processes that were previously difficult to compute (e.g., multi-quarkonium production, BSM scenarios).
• New Physics Probes: By enabling automated generation for leptonia, the framework opens new avenues for testing QED, searching for CPT violation, and probing new physics via charged-lepton flavor violation.
• Precision and Flexibility: The ability to seamlessly combine fixed-order calculations with parton showers and to switch between different theoretical models (SM, HEFT, EFT) makes this a powerful tool for future global analyses and comparisons with experimental data from the LHC, EIC, and future colliders.
• Future Roadmap: The authors outline plans to extend this to P-wave states and Next-to-Leading Order (NLO) accuracy, which will further solidify the framework as the standard for bound-state phenomenology.

In summary, this paper presents a robust, automated, and versatile extension to the MG5_aMC ecosystem, significantly advancing the state-of-the-art in simulating heavy bound-state production in both QCD and QED regimes.