[Paper Review] One-Loop Gauge Theory Amplitudes with an Arbitrary Number of External Legs

Zvi Bern, Lance J. Dixon+2 more|ArXiv.org|May 9, 1994

Superconducting Materials and Applications2 references3 citations

TL;DR

This paper presents a systematic method for constructing one-loop quantum chromodynamics (QCD) amplitudes with an arbitrary number of external legs using unitarity and collinear behavior constraints. By leveraging these consistency conditions and focusing on supersymmetric Yang-Mills amplitudes, the authors show that the cut-constructible parts of the amplitudes are uniquely determined, enabling efficient computation of multi-parton amplitudes in high-energy physics beyond tree level.

ABSTRACT

We review recent progress in calculations of one-loop QCD amplitudes. By imposing the consistency requirements of unitarity and correct behavior as the momenta of two legs become collinear, we construct ansatze for one-loop amplitudes with an arbitrary number of external legs. For supersymmetric amplitudes, which can be thought of as components of QCD amplitudes, the cuts uniquely specify the amplitude.

Motivation & Objective

• To develop a general framework for computing one-loop gauge theory amplitudes with an arbitrary number of external legs.
• To address the challenge of extending one-loop amplitude calculations beyond low-multiplicity processes in QCD.
• To leverage unitarity and collinear behavior as consistency constraints to construct valid amplitude ansatze.
• To demonstrate that for supersymmetric amplitudes, the cut structure uniquely determines the full amplitude.
• To provide a foundation for efficient computation of multi-parton amplitudes in high-energy collider physics.

Proposed method

• Constructing amplitude ansatze based on the physical requirements of unitarity and correct collinear behavior.
• Using generalized unitarity methods to decompose loop amplitudes into products of tree-level amplitudes across cut propagators.
• Imposing constraints from the collinear limit to fix unknown coefficients in the ansatz.
• Focusing on maximally supersymmetric Yang-Mills theory, where the cut structure fully determines the amplitude.
• Applying the method to derive compact expressions for one-loop amplitudes with arbitrary external leg numbers.
• Validating the ansatz by ensuring consistency with known results for lower-point amplitudes.

Experimental results

Research questions

• RQ1 How can one-loop amplitudes with an arbitrary number of external legs be systematically constructed using physical consistency conditions?
• RQ2 To what extent do unitarity and collinear limits constrain the form of one-loop amplitudes in gauge theories?
• RQ3 Can the cut-constructible parts of one-loop amplitudes be uniquely determined without explicit loop integration?
• RQ4 How does the supersymmetric structure simplify the determination of one-loop amplitudes?
• RQ5 What is the role of the collinear limit in fixing the coefficients of the amplitude ansatz?

Key findings

• The cut-constructible part of one-loop amplitudes in maximally supersymmetric Yang-Mills theory is uniquely determined by unitarity and collinear behavior constraints.
• For supersymmetric amplitudes, the full one-loop amplitude is completely fixed by its unitarity cuts, eliminating ambiguity in the ansatz.
• The method enables the construction of one-loop amplitudes with any number of external legs without requiring explicit loop momentum integration.
• The ansatz successfully reproduces known results for lower-point amplitudes, confirming consistency.
• The framework provides a powerful and efficient approach to computing high-multiplicity amplitudes in quantum field theory.
• The approach is generalizable to non-supersymmetric QCD amplitudes, though additional constraints may be needed.