[Paper Review] Computer Algebra in Particle Physics

Stefan Weinzierl|ArXiv.org|Sep 20, 2002

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TL;DR

This paper provides a comprehensive introduction to computer algebra systems (CAS) tailored for high-energy particle physics, emphasizing their role in precision calculations such as multi-loop amplitudes and tensor integrals. It outlines core algorithms—including Gröbner bases, integration by parts, and the cut-based method—demonstrating how symbolic computation enables the systematic evaluation of complex quantum field theory amplitudes, especially with multiple scales.

ABSTRACT

These lectures given to graduate students in theoretical particle physics, provide an introduction to the ``inner workings'' of computer algebra systems. Computer algebra has become an indispensable tool for precision calculations in particle physics. A good knowledge of the basics of computer algebra systems allows one to exploit these systems more efficiently.

Motivation & Objective

• To equip graduate students and researchers with foundational knowledge of computer algebra systems (CAS) for efficient use in precision calculations in particle physics.
• To address the limitations in current computational methods, such as algorithmic inefficiency, memory constraints, and suboptimal data representation in CAS.
• To explore the role of symbolic computation in enabling higher-loop and multi-scale calculations, which are essential for modern high-energy physics predictions.
• To identify open problems in symbolic computation, such as the existence of algorithms to decide whether a given integral or sum can be expressed in terms of multiple polylogarithms.
• To highlight the mutual feedback between particle physics and CAS development, where physics problems drive algorithmic innovation in symbolic computation.

Proposed method

• Utilizes a toy computer algebra system to illustrate fundamental data structures such as lists, containers, and object-oriented design principles in C++.
• Applies classical algorithms including the Euclidean algorithm for GCD, polynomial factorization via Berlekamp’s algorithm, and Hensel lifting for p-adic lifting.
• Employs symbolic integration via Risch’s algorithm and symbolic summation via Gosper’s algorithm for hypergeometric terms.
• Introduces advanced techniques for high-energy physics, including Passarino-Veltman reduction for one-loop tensor integrals and integration-by-parts (IBP) identities for higher-loop amplitudes.
• Applies the cut-based method to reconstruct loop amplitudes from their unitarity cuts, leveraging the power-counting criterion to determine when rational pieces can be uniquely reconstructed.
• Uses the expansion of transcendental functions in ε (dimensional regularization) to extract rational terms beyond the cut-constructible part.

Experimental results

Research questions

• RQ1 Can systematic algorithms be developed to compute multi-loop amplitudes with multiple kinematical scales in quantum field theory?
• RQ2 Under what conditions is a loop amplitude uniquely determined by its unitarity cuts, and when are rational pieces missing from such reconstructions?
• RQ3 Are there constructive algorithms to decide whether a given integral or sum can be expressed in terms of multiple polylogarithms, analogous to Risch’s or Gosper’s algorithms?
• RQ4 How can symbolic computation be optimized for high-energy physics applications, particularly in terms of data representation and algorithmic efficiency?
• RQ5 What role do computer algebra systems play in enabling the calculation of higher-order corrections in QCD and QED, such as the 4-loop β-function or three-loop anomalous magnetic moment?

Key findings

• Computer algebra systems are indispensable for precision calculations in particle physics, enabling the computation of three- and four-loop amplitudes such as the anomalous magnetic moment of the electron and the QCD β-function.
• The cut-based method allows reconstruction of loop amplitudes from their unitarity cuts, with the rational piece r being the only ambiguity, which can be recovered via ε-expansion or collinear factorization constraints.
• N=4 supersymmetric Yang-Mills amplitudes satisfy the power-counting criterion and are therefore cut-constructible, while QCD amplitudes generally are not, leaving the rational piece undetermined by cuts alone.
• The class of functions appearing in loop calculations includes multiple polylogarithms, and systematic algorithms exist for computing them via differential equations and series expansion.
• Despite progress, the existence of a decision procedure for expressing arbitrary integrals or sums in terms of multiple polylogarithms remains an open problem.
• The development of computer algebra systems has been significantly driven by high-energy physics, with systems like FORM, REDUCE, and GiNaC emerging from or being strongly influenced by particle physics research needs.