[Paper Review] What can we learn about QCD and collider physics from N=4 super Yang-Mills?

Johannes M. Henn|arXiv (Cornell University)|May 30, 2020

Black Holes and Theoretical Physics75 references7 citations

TL;DR

This review explores how insights from $/mathcal{N}=4$ super Yang-Mills (sYM) — a maximally supersymmetric, exactly solvable theory — have advanced QCD and collider physics calculations. By leveraging dual conformal symmetry, on-shell methods, symbol calculus, and the amplituhedron geometry, researchers have developed powerful tools for computing scattering amplitudes and Feynman integrals, with direct applications to high-precision LHC phenomenology.

ABSTRACT

Tremendous ongoing theory efforts are dedicated to developing new methods for QCD calculations. Qualitative rather than incremental advances are needed to fully exploit data still to be collected at the LHC. The maximally supersymmetric Yang-Mills theory (${\mathcal N}=4$ sYM) shares with QCD the gluon sector, which contains the most complicated Feynman graphs, but at the same time has many special properties, and is believed to be solvable exactly. It is natural to ask what we can learn from advances in ${\mathcal N}=4$ sYM for addressing difficult problems in QCD. With this in mind, we review here several remarkable developments and highlights of recent results in ${\mathcal N}=4$ sYM. This includes all-order results for certain scattering amplitudes, novel symmetries, surprising geometrical structures of loop integrands, novel tools for the calculation of Feynman integrals, and bootstrap methods. While several insights and tools have already been carried over to QCD and have contributed to state-of-the-art calculations for LHC physics, we argue that there is a host of further fascinating ideas waiting to be explored.

Motivation & Objective

• To identify and systematize conceptual and computational advances in $/mathcal{N}=4$ sYM that can be transferred to QCD and collider physics.
• To address the challenge of computing high-loop-order scattering amplitudes in QCD, where traditional Feynman diagram methods become intractable.
• To explore how physical principles like unitarity, infrared finiteness, and symmetry can guide the construction of amplitudes without explicit diagram evaluation.
• To assess the potential of bootstrap and integrand-based methods for solving difficult problems in QCD, especially in infrared-divergent and high-energy limits.
• To identify open questions where $/mathcal{N}=4$ sYM offers a testing ground for novel ideas applicable to non-supersymmetric, realistic gauge theories like QCD.

Proposed method

• Utilizing on-shell recursion and unitarity methods to compute loop integrands in $/mathcal{N}=4$ sYM, revealing hidden analytic and geometric structures.
• Applying the principle of maximal transcendentality to extract exact results for the cusp anomalous dimension in the planar limit.
• Employing the duality between scattering amplitudes and polygonal Wilson loops to constrain amplitudes via conformal symmetry and near-collinear limits.
• Using the symbol calculus to decompose transcendental functions in amplitudes, enabling systematic computation of Feynman integrals in any quantum field theory.
• Leveraging cluster algebras and canonical differential equations to systematically solve Feynman integrals beyond multiple polylogarithms.
• Implementing the amplitude bootstrap program by imposing physical constraints such as unitarity, analyticity, and symmetry to reconstruct amplitudes iteratively.

Experimental results

Research questions

• RQ1 To what extent do the analytic and geometric structures observed in $/mathcal{N}=4$ sYM, such as dual conformal symmetry and the amplituhedron, carry over to QCD integrands?
• RQ2 What is the precise form of canonical differential equations for Feynman integrals in QCD that evaluate to functions beyond multiple polylogarithms?
• RQ3 How can cluster algebras describe the function space of QCD amplitudes, and what new constraints or simplifications do they provide?
• RQ4 What extensions of four-dimensional leading singularities are needed to describe the coefficients of transcendental functions in QCD scattering amplitudes?
• RQ5 To what extent are massless QCD amplitudes at the conformal fixed point determined by conformal symmetry, as in $/mathcal{N}=4$ sYM?

Key findings

• Planar four- and five-particle amplitudes in $/mathcal{N}=4$ sYM are known exactly, with results matching both weak- and strong-coupling calculations.
• A duality between scattering amplitudes and polygonal Wilson loops in $/mathcal{N}=4$ sYM provides a powerful geometric description of amplitudes, constrained by dual superconformal symmetry.
• The bootstrap program enabled the computation of planar six- and seven-gluon amplitudes in $/mathcal{N}=4$ sYM at high loop orders, up to seven loops.
• The singularity structure of loop integrands in $/mathcal{N}=4$ sYM is governed by transcendental weight, leading to a novel method for computing Feynman integrals in any QFT.
• Symbol calculus, developed in $/mathcal{N}=4$ sYM, has become a standard tool in QCD for handling transcendental functions in amplitudes and anomalous dimensions.
• The discovery of uniform weight integrals and canonical differential equations in $/mathcal{N}=4$ sYM has led to systematic, algorithmic methods for computing Feynman integrals in QCD.