Source-linked AI summary
Top++: a program for the calculation of the top-pair cross-section at hadron colliders
Michal Czakon, Alexander Mitov
TL;DR
Top++ addresses the need for a publicly available tool to calculate inclusive top-pair cross-sections at hadron colliders with advanced fixed-order and resummed QCD accuracy. It implements exact NNLO and NNLL soft-gluon resummation, offers 29 configurable options, and is designed for simple use. The program achieves sub-per-mill accuracy in realistic time, subject to numerical settings and known theoretical inputs.
Problem
A publicly available program was needed for inclusive top-pair cross-sections with soft-gluon resummation, alongside fixed-order calculations.
Method
Top++ combines exact NNLO fixed-order calculations with NNLL soft-gluon resummation and exposes 29 user-configurable options through a simple C++ interface.
Results
Sub per-mill accuracy is achievable in realistic time without Monte Carlo methods.
Takeaways & Limitations
Top++ provides flexible public access to inclusive top-pair cross-section calculations at NNLO and NNLL accuracy.
Takeaways & Limitations
The authors recommend reporting numerical results only to one-per-mill precision, while the two-loop gg matching is known only through a color-averaged combination.
Abstract
from arXiv · showhide
We present the program Top++ for the numerical evaluation of the total inclusive cross-section for producing top quark pairs at hadron colliders. The program calculates the cross-section in a) fixed order approach with exact next-to-next-to leading order (NNLO) accuracy and b) by including soft-gluon resummation for the hadronic cross-section in Mellin space with full next-to-next-to-leading logarithmic (NNLL) accuracy. The program offers the user significant flexibility through the large number (29) of available options. Top++ is written in C++. It has a very simple to use interface that is intuitive and directly reflects the physics. The running of the program requires no programing experience from the user.
Program summary
Top++ is a C++ program for top-pair cross-section calculations, offering sub-per-mill accuracy without Monte Carlo methods and running on Linux or Mac OS X.
- Top++ is distributed under the GNU Public License without warranty.
- The program was developed and tested with GNU Compiler Collection’s C++ compiler.
- Top++ runs on Linux and Mac OS X and can be adapted for other Unix systems.
- Typical memory usage is less than 200 MB, with running time depending on the selected options.
- Sub per-mill accuracy is achievable in realistic time without Monte Carlo methods.
1. Introduction: what is this program for?
Top++ calculates total inclusive top-pair production cross-sections at hadron colliders using exact NNLO fixed-order theory or NNLL soft-gluon resummation matched through NNLO. It is designed for straightforward use and customization, while numerical results should generally be reported only to per-mill precision.
- Top++ computes total inclusive top-pair production cross-sections in hadronic collisions using fixed-order NNLO or NNLL-resummed calculations matched through NNLO.
- The program is the first publicly available implementation of soft-gluon resummation for top-pair production.
- Top++ provides a simple interface for users without programming experience and is modular enough for customization.
- The manual gives only a short physics description and directs readers to cited references for detailed physics and option explanations.
- The authors recommend reporting numerical results accurate to one per-mill, because finer precision is not physically relevant for their purposes.
2. How to use the program
Top++ is configured through 29 options in top++.cfg, covering collider, PDF, scale, perturbative-order, resummation, and numerical settings. Users edit the configuration, run the program, and receive results on screen and in top++.res.
- After configuration, users save the file, execute the program, and receive progress, timing, and final results on screen and in top++.res.
- The 29 options are grouped into five subgroups, and only values differing from predefined defaults need to be specified.
- General Setup: Users configure collider type, PDF set and member, top mass, renormalization and factorization scales, and scale-variation restrictions.
- General Setup: WithResummation selects either soft-gluon resummation at configured accuracy or a fixed-order calculation, with the other option group ignored.
- Resummation: OrderFO controls fixed-order accuracy and matching order, while OrderRES controls resummation logarithmic accuracy.
- Fixed Order: The fixed-order group provides LO, NLO, and exact NNLO options for hadronic top-pair production.
3. Computation of pdf uncertainties
Top++ computes PDF uncertainties through built-in prescriptions associated with named PDF sets, while allowing users to add new prescriptions. Unsupported sets yield individual member results but no automatic uncertainty.
- Top++ includes four PDF-uncertainty prescriptions: Asymmetric, NNPDF, Symmetric, and HERA VAR.
- Each PDF set must be listed by its full name in a library that associates it with the appropriate uncertainty prescription.
- If no prescription is known for a requested PDF set, Top++ does not compute its PDF uncertainty automatically but displays all individual PDF-member results.
- Users can manually derive uncertainties from individual member results when automatic computation is unavailable.
- Adding new prescriptions: Adding a new prescription requires modifying Utilities.cpp, Utilities.h, and top++.cpp in three specified steps.
4. Once the program is running: some fine tuning
Top++ balances numerical accuracy and runtime through integration precision and PDF-grid settings, while providing diagnostics and recommendations for validating results.
- Numerical precision and speed: Precision 2 is typically adequate for per-mill accuracy, while increasing Precision slows the calculation.NaN outputs indicate numerical instability and should be addressed by rerunning with increased requested accuracy.
- Numerical precision and speed: Users should verify numerical stability by rerunning with higher Precision and a larger NPdfGrid, checking that results change beyond the required accuracy.These checks distinguish meaningful numerical uncertainty from settings that only increase runtime.
- Numerical precision and speed: The PDF-flux grid uses a second-order finite-difference scheme whose relative precision scales as ∼1/NPdfGrid^2.NPdfGrid 100 is usually accurate and fast; users are advised to compare against a larger value such as 500.
- Numerical precision and speed: Top++ treats numerical precision and speed as coupled limits that require tuning rather than maximizing a single setting.The overall uncertainty combines the Precision and NPdfGrid settings, and extreme increases can significantly slow calculations.
- Practical operation: Many calculations with varying runtimes can be parallelized using one executable, a common pdf.cfg, and calculation-specific top++.cfg files.The program also reports step timing and final results, while defaults for all 29 parameters are defined in main().
5. Contact with physics
Top++ exposes fixed-order and Mellin-space resummed calculations through physics-aligned options, combining exact NNLO results with NNLL soft-gluon resummation and matching.
- Calculation modes: Top++ computes the inclusive top-pair cross-section either at exact NNLO fixed order or with Mellin-space soft-gluon resummation through NNLL matched through NNLO.The resummed result is automatically matched through the selected fixed-order accuracy, with its perturbative expansion subtracted to avoid double counting.
- Calculation modes: WithResummation NO selects pure fixed-order perturbation theory, while LO, NLO, and NNLO independently control the known perturbative orders.NNLO calculations require setting LO, NLO, and NNLO to YES.
- Resummation controls: WithResummation YES adds soft-gluon resummation, with OrderFO selecting the fixed-order accuracy through which the result is expanded and matched.OrderFO takes LO, NLO, or NNLO values corresponding to terms through O(α_s^n) with n = 2, 3, or 4.
- Resummation controls: The resummed cross-section combines Coulomb effects, hard functions, and a Sudakov exponent containing LL, NLL, and NNLL soft logarithms.The Coulomb and hard functions have perturbative expansions controlled by OrderFO, while OrderRES selects LL, NLL, or NNLL logarithmic accuracy.
- Scope boundary: Approximate NNLO options are unavailable because exact NNLO results supersede them; earlier Top++ versions are required to use approximate NNLO calculations.
6. Summary
Top++ is a modular, speed-optimized program for top-pair cross-sections, offering extensive control over fixed-order and resummed calculations.
- Summary: Top++ provides 29 user options, giving substantial flexibility and control over the calculation.
- Summary: Version 2.0 controls resummed-cross-section matching through OrderFO, aligning with standard conventions for exact NNLO calculations.
- Summary: Top++ is optimized for speed, but resummed calculations can take longer because they integrate rapidly oscillating functions in the complex plane.Fixed-order calculation runtimes are very short.
Appendix A. Installation
Top++ uses a standard C++ build with GNU Scientific Library and LHAPDF dependencies, requiring a small set of Makefile path and compiler settings.
- Dependencies: Top++ is written in standard C++ and requires the GNU Scientific Library and Les Houches Accord PDF interface as external libraries.GSL supplies special functions and integration, while LHAPDF supplies parton distribution functions.
- Configuration: Installation requires setting CXX, GSLDIR, and LHADIR in the Makefile.These specify the compiler and the prefix directories for the two libraries.
- Configuration: The library paths are determined automatically or can be set explicitly, commonly using /usr/local.
- Compilation: After configuration, the code can be compiled with make and is ready to use as top++.Compilation can alternatively provide CXX, GSLDIR, and LHADIR directly on the command line.
Appendix B. Examples (updated for ver.2.0 )
Appendix B documents example configurations for phenomenological studies, covering default predictions, scale and PDF variations, fixed-order calculations, resummation, and top-mass scans.
- Configuration files: The configuration files range from complete option listings to focused phenomenology and best-available NNLO+NNLL applications.Examples can be copied into top++.cfg, and top++best-precision.cfg is a subset of top++pheno.cfg.
- Example configurations: The example files support default best-precision predictions for the LHC at 8 TeV and the Tevatron.The default top++.cfg is equivalent to top++pheno.cfg, while top++best-TEV.cfg computes Tevatron scale variation.
- Example configurations: Separate configurations compute PDF variations at the LHC at 7 TeV and NNLO fixed-order scale variations at 14 TeV.
- Cross-section examples: 47.6818 [pb] is obtained for an NNLO calculation at the LHC at 7 TeV with central scales µF = µR = mtop and NNLO PDFs.
- Cross-section examples: The examples also include an NNLO+LL calculation at the LHC at 8 TeV and an NLO top-mass loop at the Tevatron.The mass-loop example evaluates three top-mass values using NLO PDFs.
Appendix C. What is new in ver.2.0: changes from ver.1.4
Version 2.0 adds exact gluon-fusion NNLO information, updates PDF-uncertainty handling and NLO inputs, and refines options, defaults, matching, and output formatting.
- Physics updates: Exact NNLO corrections for gg →t¯t + X complete the NNLO QCD correction set for top-pair production at hadron colliders.
- PDF uncertainties: PDF uncertainty evaluation was revised to support prescriptions and PDF sets with different αS values for each PDF member.
- Physics updates: The NLO partonic cross-section parameterization was replaced by a fit to the exact NLO result.
- Resummation: OrderFO now sets matching of the resummed result, equivalent to OrderRES in standard LO+LL, NLO+NLL, and NNLO+NNLL cases.
- Interface and implementation: Version 2.0 adds single-partonic-channel selection, HERA PDF-variation support, revised defaults, improved output formatting, and removes approximate-NNLO options.
Appendix D. Program’s structure: a brief overview
Top++ is organized around initialized C++ classes that precompute partonic fluxes, represent fixed-order cross-sections, and implement subtraction-related flux operations, with supporting numerical utilities.
- Program organization: The main() function initializes four program classes in a defined order, with each class assigned a specific computational role.
- Core classes: PartonicFlux precomputes partonic fluxes on an NPdfGrid-point grid for each factorization scale µF.
- Core classes: FixedOrder represents the partonic fixed-order cross-section, with one object for each factorization- and renormalization-scale combination.
- Core classes: SubtrFlux implements a fake partonic flux that mirrors the actual flux within a distance O(ETA) from the partonic threshold.
- Supporting utilities: External functions provide complex special functions, PDF-uncertainty prescriptions, and the remaining numerical computation.