Source-linked AI summary
Helac-nlo
G. Bevilacqua, M. Czakon, M. V. Garzelli, A. van Hameren, A. Kardos, C. G. Papadopoulos, R. Pittau, M. Worek
TL;DR
NLO calculations require reliable virtual corrections for increasingly complex scattering processes. HELAC-1LOOP combines OPP reduction with HELAC recursion to evaluate these corrections numerically and generate LHE information for HELAC-NLO. The implementation includes benchmark tests and handles numerical instabilities through gauge testing and higher precision.
Problem
Reliable virtual QCD corrections are needed for automated NLO calculations of multi-particle scattering amplitudes.
Method
HELAC-1LOOP uses OPP integrand-level reduction and HELAC tree-recursive building blocks to construct one-loop amplitudes numerically.
Results
HELAC-1LOOP evaluates virtual QCD corrections and produces LHE files containing tree-order and virtual one-loop information.
Takeaways & Limitations
The program forms part of HELAC-NLO, which supports complete NLO QCD evaluations when combined with real corrections from HELAC-DIPOLES.
Takeaways & Limitations
Real corrections currently require weighted event samples, and especially subtracted real-emission contributions converge slowly in Monte Carlo integration.
Abstract
from arXiv · showhide
Based on the OPP technique and the HELAC framework, HELAC-1LOOP is a program that is capable of numerically evaluating QCD virtual corrections to scattering amplitudes. A detailed presentation of the algorithm is given, along with instructions to run the code and benchmark results. The program is part of the HELAC-NLO framework that allows for a complete evaluation of QCD NLO corrections.
PROGRAM SUMMARY
HELAC-1LOOP is a Fortran program within HELAC-NLO for numerically evaluating QCD virtual corrections using HELAC and CutTools. It constructs one-loop amplitudes from tree-order amplitudes and produces LHE files combining tree-order and virtual contributions.
- HELAC-1LOOP evaluates QCD virtual corrections using the HELAC framework and CutTools.
- The program constructs one-loop n-particle amplitudes as components of n + 2 tree-order amplitudes.
- HELAC-1LOOP produces LHE files combining complete tree-order and virtual one-loop information.
- The program is implemented in Fortran and supports Linux, Unix, and Mac OS.
1. Introduction
HELAC-NLO is a computational framework assembled from several software components to automate numerical NLO QCD calculations. HELAC-1LOOP supplies fully numerical virtual corrections for scattering amplitudes with up to seven strongly interacting particles attached to the loop.
- The framework targets increasingly precise theoretical predictions for multi-particle processes at the LHC.
- HELAC-NLO incorporates HELAC-PHEGAS, CutTools, HELAC-DIPOLES, OneLoop, and HELAC-1LOOP.
- HELAC-1LOOP evaluates fully numerical virtual QCD corrections for amplitudes involving up to seven particles directly attached to the loop.
- The paper presents the underlying algorithm, running instructions, benchmark results, and a review of current software limitations and improvements.
2. The HELAC-1LOOP algorithm
HELAC-1LOOP uses integrand-level OPP reduction and HELAC tree-recursion to construct and numerically evaluate one-loop amplitudes. Its workflow organizes loop structures, samples loop-momentum solutions, and addresses numerical instabilities with gauge tests.
- HELAC-1LOOP constructs a tree-order skeleton containing the information needed to evaluate amplitudes from external-particle flavor assignments and physical constants.
- The program enumerates color connections and recursively constructs currents for the Dyson-Schwinger representation of the amplitude.The number of currents required is not known a priori and is estimated through a user-defined parameter.
- The algorithm reduces one-loop amplitudes at the integrand level using the OPP approach.The reduction expresses numerator functions through loop-momentum-independent coefficients and spurious terms that integrate to zero.
- Numerator contributions are evaluated as parts of n + 2 tree amplitudes after cutting a loop line and omitting loop-dependent propagator denominators.
- One-loop contributions are organized by topologically inequivalent partitions, loop assignments, and internal flavors and colors, including hexagon through bubble topologies.
- Numerical instabilities, often associated with small Gram determinants and higher-rank tensor integrals, are detected with a gauge test and may be treated in quadruple precision.The effect is reported as appreciable for very complicated processes such as gg → t¯tgg.
3. How to run the code
HELAC-1LOOP is distributed as Fortran 90 code with scripts and configurable input files for compilation, momentum handling, amplitude evaluation, and LHE reweighting. Users can run testing, gauge-check, and reweighting workflows through selectable precision, kinematic, color, and phase-space options.
- Installation and compilation: The code is written in Fortran 90, requires no additional software, and includes scripts guiding testing, reweighting, and gauge-check runs.The distribution creates HELAC1L_OFFICIAL with examples, run, source, and tree directories.
- Input configuration: HELAC-1LOOP accepts momenta from files, random RAMBO generation, or HELAC-generated LHE files, with user-defined physical parameters in constants.h.The reweighting mode reads Les Houches Event input generated by HELAC.
- Installation and compilation: Compilation uses scriptmake, with options for double or quadruple precision, full or incremental rebuilding, compiler flags, and gauge-check mode.The gauge-check command produces main_onep_dpGC.exe, while standard compilation produces main_onep_dp.exe.
- Input configuration: The input file controls reweighting versus other modes, color summation, verbosity, repetition, helicity treatment, particle content, and QCD or electroweak couplings.Momentum settings separately select user-provided, RAMBO-generated, or LHE-file momenta, alongside energy, phase-space count, scale, and output-file parameters.
- LHE generation and reweighting: The tree-order stage generates an LHE file for later reweighting, with optional color Monte Carlo, PHEGAS or KALEU phase-space generation, and an oneloop_rewgt switch.For gg → t¯t, the generated file is sampleG0G0TqTa.lhe.
4. Results and Benchmarks
The paper illustrates HELAC-1LOOP through testing and reweighting workflows, including amplitude construction, numerical output, gauge checks, and integration with HELAC-DIPOLES. Testing reports tree and loop quantities together with pole-comparison diagnostics, while reweighting produces weighted LHE events for full NLO calculations.
- Testing mode: The examples cover u ¯d →W +ng with n = 1, 2, 3 and provide distributed input, output, and amplitude-construction files.The n = 1 example includes tree-order and virtual-amplitude generation information.
- Testing mode: The output records process, color-connection, subamplitude, and loop-topology information, including 33 contributions in the illustrated example.The first contribution is identified as a box graph, while the final six are R2 tree-order-like contributions with unique special vertices.
- Testing mode: Six non-zero helicity configurations are reported, with LO = 0.73261294162118751 and ratio = 6.67299738352457872E-002 for the displayed test point.The output also lists U0, T0, U1, T1, T2, I1, I2, R0, and R1 matrix-element quantities.
- Testing mode: Agreement between T2 and I2 and between T1 and I1 measures the precision achieved by comparing renormalized loop poles with I-operator predictions.The displayed values give T1 = -2.26466317876591848E-002 and I1 = -2.26466317876590842E-002.
- Reweighting mode: For gg → t¯t, reweighting produces sampleG0G0TqTa_WEIGHTED.lhe, which can be used with HELAC-DIPOLES to obtain full NLO corrections.Failed gauge-test events can be reprocessed in higher precision and combined with passed-event results.
- HELAC-DIPOLES: HELAC-DIPOLES provides configuration for integrated dipoles and supports real-only, last-particle, divergent-dipole, hybrid, sign, and summation options.Its setup includes process, optimization, renormalization, factorization-scale, and parton-number parameters.
5. Outlook
The outlook identifies unresolved efficiency, parton-shower interfacing, broader-theory, and higher-loop challenges for automated NLO calculations. HELAC-1LOOP has already been interfaced with POWHEG-BOX for selected hadron-level predictions, while further automation and reimplementation are planned.
- Open efficiency issues: Current LHE generation covers tree-order plus virtual corrections, whereas real corrections rely on weighted events and relatively slow Monte Carlo convergence.The authors propose investigating alternative phase-space algorithms, color sampling, and subtraction methods to improve efficiency.
- Parton-shower interfacing: HELAC-1LOOP has been interfaced with POWHEG-BOX for pp →ttH and pp →ttj studies including NLO QCD, parton showers, hadronization, and hadron decay.The resulting predictions reach the hadron level for comparison with LHC and Tevatron data.
- Broader theories and efficiency: A reimplementation of HELAC is proposed to incorporate full electroweak corrections and theories beyond the Standard Model.The planned redesign also targets reduced computational time and resource use by constraining redundancy.
- Beyond one loop: The OPP reduction method and recursive scattering-amplitude computation may support highly efficient calculations beyond one loop, including the two-loop level.This is presented as a future direction rather than a current capability.