Non-Supersymmetric Loop Amplitudes and MHV Vertices

TL;DR

This work extends the MHV-diagram framework to non-supersymmetric Yang–Mills by computing the cut-constructible part of the one-loop MHV multi-gluon amplitude with a scalar running in the loop. The authors derive a compact expression that reproduces the adjacent-negative-helicity and five-gluon non-adjacent results, and they present a general formula featuring box and triangle contributions with a detailed dispersion-integral treatment. They also discuss twistor-space localization of the building blocks and demonstrate consistency through checks of adjacent and five-gluon cases, plus the correct infrared pole structure. Overall, the paper provides a robust, gauge-invariant method to obtain cut-constructible portions of non-supersymmetric loop amplitudes using MHV diagrams, and it offers insights into their twistor-space interpretation.

Abstract

We show how the MHV diagram description of Yang-Mills theories can be used to study non-supersymmetric loop amplitudes. In particular, we derive a compact expression for the cut-constructible part of the general one-loop MHV multi-gluon scattering amplitude in pure Yang-Mills theory. We show that in special cases this expression reduces to known amplitudes - the amplitude with adjacent negative-helicity gluons, and the five gluon non-adjacent amplitude. Finally, we briefly discuss the twistor space interpretation of our result.

Figures (5)

• Figure 1: A one-loop MHV diagram with a complex scalar running in the loop, computed in Eq. \ref{['loopint']}. We have indicated the possible helicity assignments for the scalar particle.
• Figure 2: A triangle function contributing to the amplitude in the case of adjacent negative helicity gluons. Here we have defined $P := q_{j,m-1}$, $Q := q_{m+1, i}= - q_{j,m}$ (in the text we set $i=1$, $j=2$ for definiteness).
• Figure 3: A box function contributing to the amplitude in the general case. The negative-helicity gluons, $i$ and $j$, cannot be in adjacent positions, as the figure shows.
• Figure 4: One type of triangle function contributing to the amplitude in the general case, where $i \in Q$, and $j \in P$.
• Figure 5: Another type of triangle function contributing to the amplitude in the general case. By first shifting $m_1-1 \to m_1$, and then swapping $m_1 \leftrightarrow m_2$, we convert this into a triangle function as in Figure 5 -- but with $i$ and $j$ swapped. These are the triangle functions responsible for the $i\leftrightarrow j$ swapped terms in \ref{['final']} -- or \ref{['final2']}.