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FEWZ 2.0: A code for hadronic Z production at next-to-next-to-leading order
Ryan Gavin, Ye Li, Frank Petriello, Seth Quackenbush
TL;DR
Precision LHC studies require fully differential NNLO predictions and reliable treatment of severe acceptance cuts. The paper presents an improved FEWZ code with parallelized integration, customizable multi-histogram output, and automatic PDF errors. It demonstrates the code on LHC phenomenology, including angular observables, where a background-reducing cut raises scale uncertainty from sub-1% to ±2.5%.
Problem
NLO predictions with O(10%) errors are insufficient for LHC electroweak-boson measurements expected to face 1−2% systematic uncertainties, motivating precise NNLO predictions.
Method
FEWZ computes fully differential neutral-current dilepton production at NNLO with parallelized sector integration, user-selectable histograms, automatic PDF errors, and customizable cuts.
Results
Sub-1% technical precision is obtainable under severe cuts; an angular separation cut raises scale uncertainty from sub-1% to ±2.5% because the observable begins at O(αs).
Takeaways & Limitations
NNLO QCD should be used throughout analyses, including acceptance corrections, when comparing experimental results with precise theory.
Takeaways & Limitations
Near the back-to-back lepton region, large logarithms make fixed-order predictions unstable and require resummation.
Abstract
from arXiv · showhide
We introduce an improved version of the simulation code FEWZ (Fully Exclusive W and Z Production) for hadron collider production of lepton pairs through the Drell-Yan process at next-to-next-to-leading-order (NNLO) in the strong coupling constant. The program is fully differential in the phase space of leptons and additional hadronic radiation. The new version offers users significantly more options for customization. FEWZ now bins multiple, user-selectable histograms during a single run, and produces parton distribution function (PDF) errors automatically. It also features a signifcantly improved integration routine, and can take advantage of multiple processor cores locally or on the Condor distributed computing system. We illustrate the new features of FEWZ by presenting numerous phenomenological results for LHC physics. We compare NNLO QCD with initial ATLAS and CMS results, and discuss in detail the effects of detector acceptance on the measurement of angular quantities associated with Z-boson production. We address the issue of technical precision in the presence of severe phase-space cuts.
1 Introduction
FEWZ addresses the need for precise, exclusive NNLO predictions by improving customization, histogram production, PDF-error calculation, and numerical integration for LHC studies.
- Motivation: NLO predictions with O(10%) errors are insufficient against expected 1−2% systematic uncertainties, motivating NNLO QCD calculations.Electroweak gauge-boson production is a high-statistics LHC standard candle and constrains luminosity, PDFs, and electroweak parameters.
- Motivation: The previous FEWZ version required separate runs for each cross section or kinematic histogram and was not easily customizable.It also faced technical difficulties for severe leptonic phase-space cuts.
- New features: The new code bins multiple arbitrary kinematic variables automatically during a single run.Commonly desired histograms are included in the new FEWZ distribution.
- New features: FEWZ divides the calculation into 230 parallel sectors, improving speed and reducing integration error under significant phase-space restrictions.Sub-percent integration errors are reported as readily obtainable.
- New features: PDF errors are calculated automatically for the total cross section and every histogram bin across current PDF sets.Most run parameters, including cuts and couplings, are moved to an external input file for customization.
- Applications: The paper demonstrates the updated code with LHC phenomenology, including PDF and scale uncertainties, detector acceptance, angular observables, and ATLAS/CMS comparisons.It also addresses technical precision under severe cuts.
2 Review of FEWZ
FEWZ computes fully differential neutral-current dilepton production at NNLO, combining divergent QCD contributions into finite numerical integrals with realistic cuts and resonance-aware transformations.
- Scope: The code predicts neutral-current dilepton production through photons and Z bosons at NNLO, including spin correlations, finite-width effects, and realistic acceptance cuts.Typical residual scale errors are below 1%.
- NNLO calculation: NNLO corrections combine double-virtual, real-virtual, and double-real contributions, whose separate divergences cancel only after summation.The calculation includes two-loop, one-loop-plus-emission, and tree-level two-parton-emission terms.
- NNLO calculation: Sector decomposition separates soft and collinear singular limits into independently extractable sectors.The implementation uses roughly 200 sectors and numerically checks cancellation of epsilon-pole coefficients.
- Numerical code: Finite contributions are implemented in Fortran and integrated over (4, 7, 11) dimensions for LO, NLO, and NNLO calculations.Reconstructed final-state kinematics allow cuts to be imposed by zeroing the integrand when requirements fail.
- Numerical integration: A variable transformation flattens the Z resonance and improves integration efficiency, while the new code allows this optimization to be switched on or off.The previous version could require days to reach precision better than 1% for nontrivial cuts.
3 Description of code improvements
FEWZ 2.0 improves speed, precision, flexibility, and per-run information by parallelizing NNLO sectors and automating customizable histograms and PDF uncertainties.
- Overview: The rewritten FEWZ improves numerical integration, expands physics-study options, and targets precision studies of lepton-pair production at hadron colliders.The same improvements are intended to apply to W production, which is deferred to future work.
- Parallelization: Each of the 230 NNLO sectors can run independently, enabling adaptive integration per sector and simultaneous use of multiple processor cores.Scripts support multicore local execution, Condor distribution, and result combination.
- Run parameters: Cuts, electroweak couplings, and other run parameters are set in an external input file, giving users flexible FEWZ customization.The input-file approach removes the need to modify source code for these settings.
- Histograms: User-selected kinematic distributions are produced automatically during a run by tabulating event weights with little overhead.Most distributions of interest are included by default.
- PDF errors: For PDF sets with error eigenvectors, FEWZ calculates eigenvector results for every histogram bin and reports integration and PDF errors.Included scripts combine sector outputs into a final file containing these uncertainties for total and binned cross sections.
3.2 Details of the numerical integration
FEWZ uses adaptive Vegas integration with saved intermediate state and variable transformations that smooth NNLO integrands and improve numerical stability.
- Adaptive integration: Total cross sections and kinematic distributions are computed by numerically integrating the NNLO integrand with the adaptive Vegas routine from Cuba 1.7.Vegas supports saving integration state between grid adaptations and returns sampling-point weights for histogram bins.
- Integrand transformations: Variable transformations remove the Z propagator and smooth logarithmic endpoint behavior in NNLO sector integrands.The transformations keep the integration support within the unit hypercube while removing singularities of the form ln(x).
3.3 Parallelization
FEWZ parallelizes NNLO sector calculations and assigns work to improve efficiency and convergence. Sector combinations exploit covariance, while difficult sectors can be split across independent jobs and run locally or through Condor.
- Sector organization: Approximately 260 decomposed sectors were organized into 230 NNLO sectors spanning partonic channels, diagram types, and singularity structures.
- Sector organization: Covariance-guided sector combinations reduce the resulting integration error when sectors cancel statistically, while keeping other sectors separate preserves adaptive-grid efficiency.
- Load balancing: High-variance or slow sectors are replicated with different random seeds so they do not delay cluster calculations.
- Load balancing: Sector precision targets are weighted using estimated sampling variance and evaluation time to improve CPU efficiency.
- Execution: Provided scripts run sectors on a selected number of local cores or submit one job per sector to Condor before combining results.
3.4 Run parameters
FEWZ moves relevant run controls into an external input file, allowing users to configure electroweak parameters, cuts, jet merging, and PDFs.
- Run configuration: Users can set vector-boson masses, photon and Z couplings, electroweak schemes, and Z-peak integration optimization in the input file.
- Cuts: Selectable cuts include lepton transverse momenta, dilepton transverse momentum, lepton pseudorapidities, and dilepton rapidity.
- Cuts: The input file supports cuts on dilepton invariant mass, jet transverse momenta and pseudorapidities, and the number of observable jets.
- Cuts: Users can configure jet-jet, jet-lepton, and lepton-lepton isolation requirements.
- Algorithms and PDFs: Users can choose cone or anti-kT jet merging and select the PDF set.
3.5 Histogramming
FEWZ saves weighted phase-space information during integration so one run can reconstruct multiple one-dimensional distributions and PDF-reweighted histogram results.
- Histogram construction: Each Vegas-generated point corresponds to specific kinematics, enabling detailed distributions beyond the total cross section with little overhead.
- Histogram construction: Histogram bins are duplicated and reweighted for each PDF error eigenvector so PDF uncertainties can be determined later.
- Supported distributions: The default FEWZ version includes distributions of dilepton transverse momentum, invariant mass, lepton transverse momenta, and lepton pseudorapidities.
- Supported distributions: Additional default distributions include jet transverse momenta and HT, the scalar sum of all transverse momenta.
- Angular observables: Reweighting phase-space points with trigonometric functions reconstructs Collins-Soper moments Ai binned in dilepton transverse momentum.
- Angular observables: FEWZ currently supports only one-dimensional histograms, although saved event weights make higher-dimensional extensions straightforward.
3.6 PDFs
FEWZ 2.0 substantially expands its supported PDF coverage and computes PDF uncertainties alongside central predictions through phase-space reweighting.
- PDF coverage: The supported PDF sets were drastically expanded, with all modern distributions available in the code.
- PDF coverage: Supported sets include ABKM 09 NLO and NNLO, CTEQ 6L1, 6.5, 6.6, 10 and 10W, GJR 08 LO/NLO, and JR 09 NNLO.
- PDF coverage: The code also supports MRST 2006 NNLO and MSTW 2008 LO/NLO/NNLO.
- PDF uncertainties: All supported sets except CTEQ 6L1 include PDF error eigenvectors, whose results are computed in parallel with the central PDF by reweighting phase-space points.
- PDF uncertainties: PDF uncertainties are combined according to the set type using quadrature for symmetric sets, standard deviation for neural-network sets, or the specified alternative procedure.
3.7 Additional Features
FEWZ supports parallel sector execution locally or through Condor, then combines sector results while consistently propagating integration and PDF errors.
- FEWZ provides shell scripts to farm sectors out in parallel locally or on Condor, with a finishing script to combine individual-sector results.
- The finishing script combines sector outputs and computes PDF errors while treating integration and PDF errors consistently.
- It also supports addition, subtraction, multiplication, and division across runs with consistent error treatment.
3.8 Runtime benchmarks
Runtime benchmarks show that the new FEWZ version substantially reduces computation time while simultaneously producing numerous histograms and PDF errors.
- The benchmark used standard cuts, an approximately 440 pb cross section, and 46% acceptance on an 8-core Intel Xeon machine.
- 20 hours versus 230 hours (9.8 days): the new version reached 0.84% error, while the previous version reached 0.97% relative error at 1% target precision.
- The new version outperforms the old even per core while computing over a dozen kinematic distributions and PDF errors for each.
4 Phenomenological results
The phenomenological studies quantify NNLO predictions, PDF and scale uncertainties, and the effects of acceptance and angular cuts on LHC observables. They also show that the revised FEWZ achieves sub-0.5% integration precision under severe phase-space restrictions.
- Inclusive cross sections and uncertainties: PDF uncertainties dominate many inclusive predictions, while ABKM and MSTW agree closely and the JR 2009 acceptance is 1.5% lower than MSTW.The acceptance PDF errors range from 0.4% for ABKM 2009 to approximately 0.8% for MSTW 2008.
- Technical precision: Severe forward-region cuts yield ±0.5% technical precision, compared with only ±3% using the old FEWZ implementation.The study concludes that integration errors are no longer limiting for the LHC observables examined.
- Angular observables: Inclusive normalized cos θ and Collins-Soper moments show little variation across NNLO PDF sets, but acceptance cuts dramatically reshape angular distributions and moments.The cuts invalidate the usual orthogonality extraction conditions when only a finite cos θ region is integrated experimentally.
- Angular-separation cut: Fixed-order predictions become unreliable below approximately ∆φll ≈3°, where large logarithms require resummation; the corresponding scale errors reach ±2.5%.The increased theoretical uncertainty should be included when using this angular cut to reduce experimental backgrounds.
- Standard acceptance cuts: Standard acceptance cuts leave basic transverse-momentum and rapidity shapes largely unchanged, aside from endpoint changes and a flattening of some Z-boson rapidity distributions.The reported flattening of MSTW and ABKM relative to JR suggests that their differences occur at low-x.
5 Conclusions
FEWZ 2.0 provides efficient, differential NNLO lepton-pair predictions with automated histograms and PDF uncertainties, while enabling precise studies under severe cuts. Its phenomenology shows that acceptance strongly affects angular observables and that a transverse-plane angular cut increases scale uncertainty.
- FEWZ studies differential lepton-pair production through NNLO QCD, including lepton kinematics and associated hadronic radiation.
- Histograms of most interesting kinematic variables and PDF errors for each bin are produced automatically during a single run.
- Sub-1% technical precision is obtainable with severe phase-space cuts through parallelized integration using local multi-core or Condor computing.
- Angular distributions: The transverse-plane angular cut raises scale uncertainty from sub-1% to ±2.5% because its prediction begins at O(αs).Fixed-order calculations also become unstable when leptons are nearly back-to-back because of large logarithms.
- Angular distributions: Acceptance cuts dramatically change the qualitative features of Collins-Soper angles and moments, despite their perturbative stability.
- NNLO QCD should be used when determining acceptance corrections so experimental results can be compared with the most precise available theory.