[Paper Review] On-shell diagrammatics and the perturbative structure of planar gauge theories

Paolo Benincasa|arXiv (Cornell University)|Oct 13, 2015

Black Holes and Theoretical Physics71 references15 citations

TL;DR

This paper develops a decorated on-shell diagrammatics for planar $χ\mathcal{N} \leq 2$ gauge theories, including pure Yang-Mills, by introducing helicity flows that encode singularity structures. It establishes an all-loop recursion relation at the integrand level for $χ\mathcal{N} = 1,2$, and provides a detailed on-shell analysis of forward limits crucial for the proof, extending the on-shell approach beyond maximally supersymmetric theories.

ABSTRACT

We discuss the on-shell diagrammatic representation of theories less special than maximally supersymmetric Yang-Mills. In particular, we focus on planar $\mathcal{N}\,\le\,2$ gauge theories, including pure Yang-Mills. For such a class of theories, the on-shell diagrammatics is endowed with a decoration which carries the information on the helicity of the coherent states. In the first part of the paper we extensively discuss the properties of this decorated diagrammatics. Particular relevance have the helicity flows that the decoration induces on the diagrams, which allows to identify the different classes of singularities and, consequentely, the singularity structure of the on-shell processes. The second part of the paper establishes a link between the decorated on-shell diagrammatics and the scattering amplitudes for the theories under examination. We prove that an all-loop recursion relation at integrand level holds also for $\mathcal{N}\,=\,1,\,2$, while for $\mathcal{N}\,=\,0$ we are able to set up a preliminary analysis at one loop. In both supersymmetric and non-supersymmetric case, the treatment of the forward limit is subtle. We provide a fully on-shell analysis of it which is crucial for the proof of the all-loop recursion relation and for the analysis of pure Yang-Mills.

Motivation & Objective

• To extend on-shell diagrammatics beyond maximally supersymmetric Yang-Mills to less supersymmetric planar gauge theories, including pure Yang-Mills.
• To introduce a decorated diagrammatics with helicity flows that classify singularity structures of on-shell processes.
• To establish an all-loop recursion relation at the integrand level for $χ\mathcal{N} = 1,2$ theories, generalizing results from $χ\mathcal{N} = 4$.
• To provide a fully on-shell treatment of the forward limit, which is essential for the recursion and for analyzing pure Yang-Mills amplitudes.
• To connect decorated on-shell diagrams to scattering amplitudes via contour integration and double-cut analysis at one loop.

Proposed method

• Introduces decorated on-shell diagrams where helicity information is encoded via flow structures on the diagram’s edges.
• Uses BCFW bridges to build higher-point diagrams and defines helicity flows that track momentum and helicity conservation across internal lines.
• Applies Möbius transformations to relate different diagram representations and identify equivalence classes under perfect orientation.
• Derives an all-loop recursion relation at the integrand level by analyzing on-shell $1$- and $2$-forms, with singularities mapped to helicity flow structures.
• Performs contour integration over $T^4$ and $T^2$ cycles to extract leading and double singularities, linking them to integral basis coefficients.
• Analyzes the forward limit on-shell by treating it as a singular limit of on-shell processes, avoiding off-shell or virtual particle concepts.

Experimental results

Research questions

• RQ1 How can on-shell diagrammatics be generalized to $χ\mathcal{N} \leq 2$ planar gauge theories beyond $χ\mathcal{N} = 4$?
• RQ2 What role do helicity flows play in classifying the singularity structure of on-shell amplitudes?
• RQ3 Can an all-loop recursion relation at the integrand level be formulated for $χ\mathcal{N} = 1,2$ theories?
• RQ4 How is the forward limit treated within a fully on-shell framework, and why is it crucial for the recursion?
• RQ5 How do decorated on-shell diagrams encode double-cut information and sub-leading singularities (e.g., triangle and bubble structures) at one loop?

Key findings

• The decorated on-shell diagrammatics for $χ\mathcal{N} \leq 2$ theories introduces helicity flows that distinguish between different factorization channels and higher-degree singularities.
• An all-loop recursion relation at the integrand level holds for $χ\mathcal{N} = 1,2$, generalizing the $χ\mathcal{N} = 4$ result to less supersymmetric theories.
• For pure Yang-Mills ($χ\mathcal{N} = 0$), a preliminary one-loop analysis is established, showing the framework's viability beyond supersymmetry.
• The forward limit is treated fully on-shell, with no reference to virtual particles, and is shown to be essential for proving the recursion relation.
• Contour integration over $T^4$ and $T^2$ cycles extracts leading and double singularities, with the on-shell $2$-form structure making the integral basis coefficients manifest.
• The on-shell process (C.7) encodes three double cuts in the $(1,2)$, $(3,4)$, and $(5,1)$ channels, with helicity flows distinguishing the factorization channels and sub-leading singularities.