Automated NRQCD and NRQED simulations of quarkonium and leptonium production with P-wave states and physical-mass effects

Technical Summary: Automated NRQCD and NRQED Simulations of Quarkonium and Leptonium Production with P-wave States and Physical-Mass Effects

Problem Statement
The production and decay of heavy quarkonia (bound states of heavy quarks) and leptonia (bound states of leptons) serve as critical laboratories for testing Quantum Chromodynamics (QCD) and Quantum Electrodynamics (QED) in the non-relativistic regime. While the Non-Relativistic QCD (NRQCD) and Non-Relativistic QED (NRQED) frameworks provide a rigorous factorization of short-distance coefficients and long-distance matrix elements (LDMEs), the automation of event generation for these processes has historically been limited. Previous tools, such as MadOnia and HELAC-Onia, primarily focused on S-wave states or lacked full automation for arbitrary numbers of P-wave states. Furthermore, conventional perturbative calculations often approximate the bound-state mass as the sum of its constituent masses ($M_B \approx m_Q + m_{\bar{Q}}$). This approximation introduces systematic uncertainties, particularly in kinematic threshold regions and processes involving multiple bound states with identical constituents but different physical masses (e.g., $J/\psi + \psi(2S)$), where the physical mass difference is significant.

Methodology
The authors present MadSONS (MadGraph Simulations Of NRQFT States), a new module for the MadGraph5_aMC@NLO (MG5_aMC) framework. This module extends the previously established S-wave implementation to include arbitrary tree-level production processes involving P-wave states and physical-mass effects.

1. P-wave Implementation via Dual Numbers:
   Unlike S-wave states, P-wave states have a vanishing wavefunction at the origin, meaning their production rates depend on the first derivative of the wavefunction. In momentum space, this corresponds to orbital angular momentum projectors involving derivatives with respect to relative momentum.

   • The authors implement these derivatives using multivariate dual numbers, a form of automatic differentiation. This approach provides exact, numerically stable derivatives without relying on symbolic differentiation or finite-difference approximations.
   • The dual number formalism allows for the propagation of derivatives through the amplitude generation chain, enabling the evaluation of orbital ($P_L$) and total angular momentum ($P_J$) projectors directly within the matrix element calculation.
   • This method is designed to be scalable for future extensions to higher orbital angular momenta (e.g., D-waves) and higher-order derivative requirements for Next-to-Leading Order (NLO) calculations.

2. Momentum Reshuffling for Physical Masses:
   To address the discrepancy between the physical bound-state mass and the sum of constituent masses, the authors introduce a momentum-reshuffling procedure.

   • Phase Space Generation: Events are generated using the physical masses of the bound states, ensuring the correct available phase-space volume and kinematic thresholds.
   • Matrix Element Evaluation: The matrix element is evaluated using the on-shell conditions of the constituent particles (sum of constituent masses), as required by the NRQFT formalism for short-distance coefficients.
   • Reshuffling Algorithms: The mapping between physical momenta and constituent-level momenta is achieved via two distinct recoil schemes:
     • Initial-state reshuffling: Modifies the incoming parton momenta to conserve four-momentum while keeping final-state momenta fixed.
     • Final-state reshuffling: Modifies the outgoing momenta (and potentially the partonic center-of-mass energy) to satisfy on-shell conditions for the physical masses.
   • This strategy approximates the inclusion of higher-order relativistic corrections in the phase-space measure, improving the description of observables near thresholds.

3. UFO Model and Automation:
   The implementation is encapsulated in an updated UFO (Universal Feynman Output) model, sm_onia. This model defines Fock states (including P-wave configurations like $^3P_J$ and $^1P_1$) and allows users to specify physical masses and LDMEs via input cards (onia_card.dat). The framework supports both quarkonia (NRQCD) and leptonia (NRQED), including color-singlet and color-octet configurations.

Key Contributions and Results

• Validation: The P-wave implementation was validated against the HELAC-Onia generator.

  • Squared Amplitudes: Comparisons at individual phase-space points show agreement at the level of double-precision arithmetic ($10^{-14}$ to $10^{-16}$ relative deviation).
  • Integrated Cross Sections: Benchmarks for single and double quarkonium production in $pp$ and $e^+e^-$ collisions show agreement within Monte Carlo statistical uncertainties.
  • Leptonium: For NRQED, the code reproduces analytic cross-section calculations for P-wave positronium and ditauonium production at the sub-permille level across a wide range of collision energies.

• Phenomenological Applications:

  • Single and Double Quarkonium Production: The authors present cross sections for various processes at the LHC ($\sqrt{s}=13$ TeV) and B-factories ($\sqrt{s}=10.58$ GeV). They quantify the impact of momentum reshuffling, noting that while effects are often small for inclusive single production, they can be significant (up to ~20-50%) for double-production channels involving states with large mass differences (e.g., $J/\psi + \psi(2S)$).
  • $J/\psi + \psi(2S)$ Case Study: A detailed study of associated $J/\psi + \psi(2S)$ production at the LHC demonstrates the utility of the framework. By incorporating physical masses and P-wave feed-down contributions (from $\psi(2S)$ and $\chi_{cJ}$), the predicted cross section is reduced compared to conventional treatments. This reduction leads to an improved description of LHCb data, leaving more room for potential Double Parton Scattering (DPS) contributions. The study highlights that P-wave contributions account for approximately 17% of the direct cross section and 25% of the total fiducial cross section via feed-down.
  • Leptonium Production: The framework enables the study of exotic atoms like positronium and ditauonium. The authors derive closed-form NRQED predictions and confirm that the automated simulation matches analytic results, validating the P-wave projectors in the QED sector.

Significance and Claims
The paper claims that MadSONS constitutes a significant step toward the fully automated simulation of bound-state production. By completing the automation of Leading Order (LO) S- and P-wave processes, the tool allows for the systematic inclusion of all phenomenologically relevant Fock states up to order $v^7$ in NRQCD.

The authors emphasize that the use of dual numbers provides a robust foundation for future NLO extensions, where the cancellation of infrared divergences requires high numerical precision. Furthermore, the momentum-reshuffling procedure offers a practical method to incorporate physical mass effects without requiring full relativistic corrections to the matrix element, thereby improving the reliability of simulations near kinematic thresholds.

The work positions MadSONS as a versatile tool for investigating quarkonium production mechanisms, testing NRQCD factorization, and exploring exotic leptonium systems, while remaining compatible with standard parton shower and hadronization tools like Pythia8. The authors modestly note that while the LO predictions improve the description of data, discrepancies may remain due to missing higher-order QCD corrections, relativistic effects, and potential DPS contributions, which are subjects for future investigation.