Source-linked AI summary
HELAC-Onia: an automatic matrix element generator for heavy quarkonium physics
Hua-Sheng Shao
TL;DR
Heavy-quarkonium production mechanisms remain insufficiently understood, while automatic tools for quarkonium helicity amplitudes are rare. The paper extends HELAC with Dyson–Schwinger recursion, NRQCD projections, and P-wave off-shell currents to create HELAC-Onia. The resulting program automatically computes efficient quarkonium helicity amplitudes for varied collider-production studies, with applications reported for multiple collision types.
Problem
Heavy-quarkonium production mechanisms remain unclear, and automatic tools for calculating quarkonium helicity amplitudes are still rare.
Method
HELAC-Onia extends HELAC’s Dyson–Schwinger-based off-shell recursion with NRQCD quarkonium projections and new P-wave off-shell currents.
Results
HELAC-Onia provides an automatic, high-efficiency computation tool for heavy-quarkonium helicity amplitudes and has been applied to production in pp, p̄p, and e+e− collisions.
Takeaways & Limitations
The program enables automated studies of varied heavy-quarkonium production processes, including multi-leg and multi-quarkonium applications.
Takeaways & Limitations
The current implementation is restricted to pp, p̄p, and e−e+ collisions, and next-to-leading-order helicity-amplitude automation remains future work.
Abstract
from arXiv · showhide
By the virtues of the Dyson-Schwinger equations, we upgrade the published code \mtt{HELAC} to be capable to calculate the heavy quarkonium helicity amplitudes in the framework of NRQCD factorization, which we dub \mtt{HELAC-Onia}. We rewrote the original \mtt{HELAC} to make the new program be able to calculate helicity amplitudes of multi P-wave quarkonium states production at hadron colliders and electron-positron colliders by including new P-wave off-shell currents. Therefore, besides the high efficiencies in computation of multi-leg processes within the Standard Model, \mtt{HELAC-Onia} is also sufficiently numerical stable in dealing with P-wave quarkonia (e.g. $h_{c,b},χ_{c,b}$) and P-wave color-octet intermediate states. To the best of our knowledge, it is a first general-purpose automatic quarkonium matrix elements generator based on recursion relations on the market.
PROGRAM SUMMARY
HELAC-Onia is an automatic heavy-quarkonium helicity-amplitude tool based on Dyson–Schwinger recursion within NRQCD. It targets efficient multi-leg collider calculations, including P-wave states, with typical tested runtimes of several to tens of minutes.
- The program is implemented in FORTRAN 90 and tested on Windows and Unix systems.
- HELAC-Onia calculates heavy-quarkonium helicity amplitudes automatically in the NRQCD framework using a recursive algorithm derived from Dyson–Schwinger equations.
- The program supports multi-leg processes with or without multiple quarkonia, including states up to P-wave at hadron and electron–positron colliders.
- Its recursive approach has lower computational cost than traditional Feynman-diagram methods.
- Typical tested processes require several minutes to tens of minutes of runtime, depending on the process.
1 Introduction
The paper motivates an automatic heavy-quarkonium tool because production mechanisms remain unclear and multi-particle processes are increasingly important. HELAC-Onia extends recursive HELAC methods while trading some flexibility against improved speed and multi-quarkonium support.
- Heavy-quarkonium production mechanisms remain unclear despite the usefulness of quarkonia for studying QCD and collider physics.
- HELAC-Onia uses HELAC’s recursive algorithm to reduce the computational cost of multi-particle amplitude calculations.
- The recursive approach is expected to calculate matrix elements faster than the Feynman-diagram-based MADONIA approach.
- HELAC-Onia supports multi-quarkonium production, whereas MADONIA restricts the number of quarkonia to one.
- HELAC-Onia is currently restricted to proton, antiproton, and electron–positron collisions, while MADONIA supports more colliders and subsequent decays.
2 The recursive algorithm
The recursive algorithm builds higher-level off-shell currents from lower-level currents, combines them with propagators and interaction rules, and assembles the final amplitude while organizing color and helicity sums efficiently.
- Recursive current construction: HELAC initializes level-1 currents from external-leg wavefunctions and constructs higher-level currents recursively from lower-level currents.
- Recursive current construction: The recursion exhausts allowed partitions of external legs into lower-level currents, multiplies each off-shell current by its propagator, and forms the level-n amplitude.
- Computational efficiency: Summing subgraphs into shared currents avoids repeated computation of identical subgraphs and reduces the number of objects passed to later recursion levels.
- Color treatment: HELAC represents color using the color-flow basis, constructs a color matrix from permutation-based color structures, and combines it with color-stripped amplitudes.
- Helicity summation: Monte Carlo sampling over helicity configurations is used to perform helicity summation and improve computational efficiency.
3 Quarkonium amplitudes in NRQCD
HELAC-Onia implements NRQCD quarkonium production amplitudes by combining short-distance calculations with state projectors and specialized P-wave currents. Its P-wave construction is designed to remain numerically stable, especially for multiple P-wave states.
- NRQCD factorization: NRQCD factorization separates quarkonium production into perturbative short-distance parts and non-perturbative long-distance matrix elements.
- NRQCD factorization: NRQCD power counting limits the Fock states needed at a specified order in the heavy-quark relative velocity v.
- Projection method: HELAC-Onia projects heavy-quark pairs onto specified color and spin states to evaluate process-dependent short-distance coefficients.
- Projection method: The projectors use spectroscopic states 2S+1L_J^[c], with c=1 or 8 denoting color-singlet or color-octet intermediate states.
- P-wave currents: New P-wave off-shell currents handle P-wave quarkonia and color-octet P-wave states, whose fragmentation topologies can dominate at medium and high transverse momentum.
- P-wave currents: The P-wave current construction keeps only amplitudes differentiated with respect to every P-wave relative momentum, avoiding large numerical cancellations.
4 Benchmark processes
HELAC-Onia is validated against published results across B_c, charmonium, double-quarkonium, and associated heavy-quark production processes at hadron and electron-positron colliders. The calculations generally agree with the literature and extend to polarization observables and new higher-order channels.
- B_c meson production at the LHC: At the LHC, HELAC-Onia B_c production results agree with published calculations under matched input parameters.The comparison uses gluon-fusion and quark-antiquark annihilation at a center-of-mass energy of 14 TeV.
- Charmonia production at the B factory: At the B factory, inclusive η_c and J/ψ color-singlet results agree with earlier calculations.The calculation uses single- and double-photon exchange contributions at a center-of-mass energy of 10.6 GeV.
- Charmonia production at the B factory: Exclusive double-charmonia cross sections and J/ψJ/ψ and J/ψh_c production results agree with published references.The exclusive calculations use the same input parameters as the cited studies, while the J/ψJ/ψ and J/ψh_c results include the stated perturbative orders.
- Double quarkonia production at the Tevatron and the LHC: Double-quarkonium production is used to investigate the color-octet mechanism, with comparisons performed at the Tevatron and LHC.The benchmark program includes calculations at 1.96 TeV and 14 TeV, alongside associated J/ψc¯c and Υb¯b production studies.
- Spin density matrix and polarization: HELAC-Onia also computes spin density matrices and polarization observables for heavy quarkonia in multiple polarization frames.The code has been applied to inclusive J/ψ polarization, inclusive χ_c hadroproduction, and polarized χ_c production with a charm-quark pair.
5 Running the program
HELAC-Onia is run through initialization and computation phases, with user-specified process, collider, generator, physics, sampling, scale, polarization, cut, and matrix-element inputs. It then produces numerical results and, through PHEGAS, event files for selected processes.
- Initialization selects relevant sub-amplitudes and evaluates the color matrix, while computation evaluates amplitudes at phase-space points.
- Input files and particle identifiers: Users provide process.inp with the external-particle count and particle IDs, using six-digit identifiers for heavy quarkonia.The quarkonium IDs encode hadron family and intermediate Fock-state information, including examples such as 3P[8].
- Runtime configuration: The program supports pp, p¯p, and e+e− collisions, with selectable PHEGAS, RAMBO, DURHAM, or VEGAS Monte Carlo generation.A negative generator value requests one phase-space point calculation.
- Runtime configuration: Helicity handling can use explicit summation or Monte Carlo sampling over elementary-particle and quarkonium polarization states.The ranhel setting controls progressively broader sampling choices, including all heavy-quarkonium polarization vectors.
- Physics and analysis parameters: Users can configure QCD and electroweak content, running αS, gauge and width schemes, Monte Carlo iterations, PDFs, scales, cuts, and long-distance matrix elements.Additional flags select pT distributions, P-wave-state summation, and polarized observables with specified spin-density-matrix elements and polarization frames.
- Outputs: The program writes result files containing cross sections and numerical errors, while PHEGAS can generate standard Les Houches Event files for event information.The documented e−e+ →ηc+ggg example illustrates both output types.
6 Summary and outlooks
The paper presents HELAC-Onia as an automatic, efficient tool for heavy-quarkonium helicity amplitudes and applications across collider processes. Its current scope motivates improving Monte Carlo reliability and automating next-to-leading-order computations.
- Heavy-quarkonium production remains theoretically unresolved, while radiative corrections are described as indispensable even at the qualitative level.
- Les Houches Event files can currently be generated only through PHEGAS in HELAC-Onia.
- HELAC-Onia addresses the scarcity of automatic quarkonium helicity-amplitude tools by extending HELAC with recursive calculations for pp, p¯p, and e+e− collisions.The program is presented as an automatic Monte Carlo generator for heavy-quarkonium studies.
- Automating next-to-leading-order quarkonium helicity-amplitude computations is identified as the next step toward full next-to-leading-order production analyses.The paper describes such analyses as more reliable and useful, especially at the LHC.