[Paper Review] CompHEP - a package for evaluation of Feynman diagrams and integration over multi-particle phase space. User's manual for version 33

A. Pukhov, E. Boos+8 more|arXiv (Cornell University)|Aug 6, 1999

Particle physics theoretical and experimental studies9 references219 citations

TL;DR

CompHEP is a computational package for automated evaluation of Feynman diagrams and integration over multi-particle phase space in quantum field theory, enabling symbolic and numerical calculations of scattering amplitudes and decay rates. It supports user-defined models, generates optimized Fortran and C code for matrix elements, and integrates with Monte Carlo methods via VEGAS for precision cross-section calculations in the Standard Model and beyond.

ABSTRACT

CompHEP is a package for automatic calculations of elementary particle decay and collision properties in the lowest order of perturbation theory (the tree approximation). The main idea prescribed into the CompHEP is to make available passing on from the Lagrangian to the final distributions effectively with a high level of automation.

Motivation & Objective

• To provide a fully automated tool for computing scattering amplitudes and decay rates in quantum field theory using Feynman rules.
• To enable efficient symbolic computation of squared matrix elements and their numerical integration over multi-particle phase space.
• To support the implementation of custom particle physics models, including the Standard Model in different gauges and extensions like Majorana neutrinos and leptoquarks.
• To generate optimized Fortran and C code for numerical evaluation and event generation, facilitating high-precision phenomenological studies.

Proposed method

• The symbolic engine uses a built-in calculator to compute squared matrix elements from Feynman diagrams, with support for user-defined Lagrangians and interaction vertices.
• It generates analytical expressions in Reduce and Mathematica formats, enabling further symbolic manipulation.
• The package exports optimized Fortran and C code for the squared matrix element, which are used in numerical integration routines.
• Numerical calculations employ adaptive Monte Carlo integration via the VEGAS algorithm, with support for beam spectra, parton distribution functions, and radiation effects (ISR/Beamstrahlung).
• Phase space parameterization is achieved through decay chain decomposition and polar vector formalism, with smoothing to improve integration convergence.
• The system supports gauge invariance checks via cross-comparison in different gauges and includes tools for automatic self-validation of results.

Experimental results

Research questions

• RQ1 How can Feynman diagram amplitudes be systematically computed and squared with minimal user intervention?
• RQ2 What is the most effective way to automate the transition from Lagrangian to numerical cross-sections in multi-particle processes?
• RQ3 How can phase space integration be efficiently performed for processes with up to five final-state particles?
• RQ4 Can a unified symbolic-numeric framework be built that supports both Fortran and C code generation for high-precision calculations?
• RQ5 How can model extensions such as Majorana neutrinos and leptoquarks be consistently implemented within a single computational environment?

Key findings

• CompHEP supports the calculation of processes up to 1→5 and 2→4 particle final states, with full automation of diagram generation and amplitude computation.
• The package successfully generates optimized Fortran and C code for matrix elements, enabling high-precision numerical integration using the VEGAS algorithm.
• It provides built-in support for the Standard Model in both t’Hooft-Feynman and unitary gauges, with correct implementation of Yukawa couplings and fermion mass generation.
• The system includes tools for handling beam spectra (CTEQ, MRS), ISR, and beamstrahlung, allowing realistic collider cross-section simulations.
• Automatic checks for gauge invariance and self-consistency are implemented via cross-comparison in different gauges and via ghost field cancellation tests.
• The inclusion of the Cabibbo-Kobayashi-Maskawa matrix and Majorana neutrino models enables the study of flavor physics and neutrino masses within the same framework.