Loop Amplitudes in Pure Yang-Mills from Generalised Unitarity

TL;DR

This work develops and applies generalised unitarity in $D=4-2\epsilon$ to compute complete one-loop amplitudes in non-supersymmetric Yang-Mills, including rational terms, by leveraging the supersymmetric decomposition to reduce to a scalar-loop problem. Quadruple cuts fix box-function coefficients while triple cuts fix triangles and bubbles, with $\mu^2$-dependent terms capturing higher-dimensional integrals essential for rational parts. The authors re-derive all four-gluon amplitudes and the five-gluon all-plus amplitude, obtaining exact agreement with Bern et al., and demonstrate the method’s validity for both infrared-finite and infrared-divergent cases. Overall, the paper establishes a practical, diagrammatic approach to complete one-loop amplitudes in pure Yang-Mills using higher-dimensional unitarity, highlighting the role of $4-2\epsilon$ cuts in accessing cut-constructible rational contributions.

Abstract

We show how generalised unitarity cuts in D = 4 - 2 epsilon dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions.

Figures (10)

• Figure 1: One of the two quadruple-cut diagrams for the amplitude $1^+ 2^+ 3^+ 4^+$. This diagrams is obtained by sewing tree amplitudes (represented by the blue bubbles) with an external positive-helicity gluon and two internal scalars of opposite "helicities". There are two such diagrams, which are obtained one from the other by flipping all the internal helicities. These diagrams are equal so that the full result is obtained by doubling the contribution from the diagram in this Figure. The same remark applies to all the other diagrams considered in this paper.
• Figure 2: One of the possible three-particle cut diagrams for the amplitude $1^+ 2^+ 3^+ 4^+$. The others are obtained from this one by cyclic relabeling of the external particles.
• Figure 3: The quadruple cut for the amplitude $1^- 2^+ 3^+ 4^+$.
• Figure 4: The two inequivalent triple cuts for the amplitude $1^- 2^+ 3^+ 4^+$.
• Figure 5: The quadruple cut for the amplitude $1^- 2^- 3^+ 4^+$.