Source-linked AI summary
vh@nnlo - Higgs Strahlung at hadron colliders
Oliver Brein, Robert V. Harlander, Tom J. E. Zirke
TL;DR
The paper addresses evaluation of inclusive associated Higgs production with a massive weak gauge boson at hadron colliders. It presents vh@nnlo, which combines NNLO QCD and NLO electroweak effects, including Drell–Yan, top, and ggHZ components. The program collects current Standard Model results and permits separate consideration of perturbative contributions.
Problem
Inclusive Higgs production with an associated W or Z boson requires combining QCD, electroweak, top, and ggHZ contributions within the Standard Model.
Method
vh@nnlo numerically integrates the inclusive cross section through NNLO QCD and applies electroweak corrections as an interpolated correction factor.
Results
vh@nnlo includes the DY, top, and ggHZ contributions together with electroweak effects for inclusive Higgs Strahlung production.
Takeaways & Limitations
Individual perturbative contributions can be separately considered, while the program collects up-to-date inclusive Standard Model results for Higgs production with a weak gauge boson.
Takeaways & Limitations
The program evaluates only total inclusive cross sections and cannot calculate kinematical distributions such as Higgs transverse momentum or rapidity.
Abstract
from arXiv · showhide
A numerical program for the evaluation of the inclusive cross section for associated Higgs production with a massive weak gauge boson at hadron colliders is described, sigma(pp/pbar p -> HV), V=W,Z. The calculation is performed in the framework of the Standard Model and includes next-to-next-to-leading order QCD as well as next-to-leading order electro-weak effects.
PROGRAM SUMMARY
vh@nnlo is a numerical program for inclusive Higgs production with a W or Z boson at hadron colliders, calculated through NNLO QCD. It uses Monte Carlo integration and supports standard desktop environments.
- PROGRAM SUMMARY: vh@nnlo calculates the inclusive total cross section for associated Higgs- and W- or Z-boson production through NNLO QCD.The calculation is performed by numerical Monte Carlo integration.
- PROGRAM SUMMARY: The program is implemented in Fortran 77 and C++ and runs on Unix/Linux and Mac OS personal computers.It requires a few 100 MB of RAM and uses LHAPDF and CUBA as external libraries.
- PROGRAM SUMMARY: The program is distributed as a tar.gz package and typically evaluates one parameter set in a few seconds.A program URL is provided in the manual.
- PROGRAM SUMMARY: Tree-level contributions include q¯q →VH and q¯q →VHg, with crossed processes including qg →VHq.The diagrams describe the partonic channels underlying the associated-production calculation.
1. Introduction
The paper describes the perturbative ingredients of Higgs Strahlung, including Drell–Yan-like, top-loop, and ggHZ contributions, together with electroweak corrections. These effects are relevant because QCD corrections alter rates and uncertainties, while electroweak corrections are of order 5%.
- 1. Introduction: Higgs Strahlung denotes associated production of a Higgs boson and a weak gauge boson W or Z at hadron colliders.At leading order, the process is represented by q¯q annihilation into a virtual gauge boson that produces VH.
- 1. Introduction: The bulk of NNLO QCD corrections is described by the same production–decay structure as Drell–Yan-like terms.These corrections are denoted DY terms.
- 1. Introduction: Top terms arise from Higgs radiation off virtual top-quark loops and, for ZH, from top-loop couplings involving the Z or H.They enter through interference with the leading-order amplitudes.
- 1. Introduction: The ggHZ contribution to ZH contains two initial-state gluons and enters as the square of a purely virtual amplitude rather than through interference with lower-order terms.It is separately finite and gauge independent.
- 1. Introduction: 31% (27%, 41%) is the NLO enhancement for WH at the LHC8 (LHC14, Tevatron), while NNLO DY adds 3% (3%, 10%).Top effects are about 1% in all three cases; for ZH, the ggHZ component adds about 5% (9%, 0.2%).
- 1. Introduction: Electroweak corrections are of order 5% and are included as a correction factor obtained by linearly interpolating published values.They do not depend on QCD parameters such as PDFs or the strong coupling.
- 1. Introduction: The program evaluates only total inclusive cross sections, not kinematical distributions such as Higgs transverse momentum or rapidity.Those distributions require other calculations.
2.1. Structure
The program combines NNLO production of an off-shell weak boson with its decay into VH, while separately implementing top-loop and electroweak contributions. Numerical integrations use VEGAS, with PDFs supplied through LHAPDF.
- 2.1. Structure: The core zwprod code evaluates off-shell weak-boson production through NNLO QCD and convolves it with the vector-boson decay rate.Partonic cross sections are integrated over parton density functions.
- 2.1. Structure: vh@nnlo performs the integrations of the production and decay expressions simultaneously using the VEGAS algorithm from CUBA.This provides the numerical Monte Carlo integration used by the program.
- 2.1. Structure: LHAPDF allows the program to switch conveniently between different PDF sets, while LHAPDF and CUBA must be installed separately.The libraries are external dependencies rather than part of the distribution.
- 2.1. Structure: Top-loop contributions are evaluated with generated routines and massive box integrals using modified LoopTools/FF functionality.The PDFs are again accessed through LHAPDF.
- 2.1. Structure: The top contribution uses the heavy-top limit, with virtual and real terms requiring three- and six-dimensional numerical integrations, respectively.Both are folded with PDFs and integrated using LHAPDF and CUBA VEGAS.
- 2.1. Structure: Electroweak corrections are represented by a two-dimensional grid over Higgs masses and collider energies, with linear interpolation between grid points.The correction factor is described as very flat.
- 2.1. Structure: The inclusive cross-section orders are assembled using the electroweak correction factor δEW.The factor is applied multiplicatively to the relevant Drell–Yan components.
2.2. Installation and compilation
vh@nnlo is distributed as a directory tree containing inputs, source code, documentation, outputs, and an executable location. Compilation builds the required libraries and calculation components, with an optional reduced build when ggHZ is unnecessary.
- Directory structure: The distribution contains directories for input files, the manual, main program, output files, source code, and the compiled executable.The root README summarizes installation, compilation, and operation; the x directory receives the executable after compilation.
- Directory structure: The main program is stored in mainfiles/main.f, while the source directory contains the vh@nnlo sources, Makefile, and installation script.
- Compilation: Full compilation builds the LoopTools/FF library and the ggHZ, DY, and top components, then copies the x.main executable into the x directory.The executable should be called from the vh@nnlo root directory.
- Compilation: When ggHZ does not contribute, such as for WH production or an NLO-only calculation, a subset of the code can be compiled without LoopTools/FF.The manual describes this as an alternative to the full installation script.
2.3. Operation
vh@nnlo is operated through a single structured input file that controls collider, process, perturbative-order, scale, mass, coupling, PDF, and correction settings. It writes requested cross-section components and correction factors to a SIGMA output block.
- Input file: A single input file, structured in custom Blocks rather than SLHA format, controls vh@nnlo operation.Comments may be added after a hash symbol, and all masses, energies, and decay widths are specified in GeV.
- Input parameters: Renormalization and factorization scales are specified relative to q^2=(pV+pH)^2, the squared partonic center-of-mass energy at LO.The final-state four-momenta pV and pH belong to the weak gauge boson and Higgs boson.
- Input parameters: The input selects pp or pbar p collisions, LO, NLO, or NNLO calculation order, WH or ZH production, PDF member, and electro-weak corrections.The requested order controls the perturbative order of the partonic cross section, while PDF evolution and αs evolution follow the selected PDF set.
- Output: For the example input, sigma(all) is 0.398313 pb and the electro-weak correction is −5.10%.The same output lists sigma(DY)=0.399203 pb, sigma(gg->HZ)=0.0154259 pb, and sigma(top)=0.00404315 pb.
- Output: The output SIGMA block reports the total inclusive cross section, αs(MZ), Drell–Yan, ggHZ, electro-weak, and top contributions.The ggHZ and top entries are zero unless NNLO is requested.
- Citations and scans: The program prints texkeys for the literature contributing to its results, and parameter scans are recommended through external scripts rather than modifying the Fortran code.
3. Conclusions
vh@nnlo collects up-to-date Standard Model results for inclusive Higgs production with an associated weak gauge boson and allows individual perturbative contributions to be examined separately.
- Conclusions: vh@nnlo collects up-to-date Standard Model results for inclusive Higgs production with an associated weak gauge boson.
- Conclusions: Individual perturbative contributions can be considered separately, and parton densities can be changed through LHAPDF.
- Conclusions: The authors state that the program is intended to be useful for Higgs physics at the LHC.