Bootstrapping One-Loop QCD Amplitudes with General Helicities

TL;DR

The paper extends the on-shell unitarity-bootstrap framework to compute complete one-loop QCD amplitudes with general gluon helicities by pairing recursion relations to handle non-standard complex factorization and large-shift behavior. It introduces an auxiliary shift technique to determine large-z contributions when the primary shift alone yields ambiguous or divergent results, and it provides detailed constructions for NMHV-like three-negative-helcity sequences, including explicit six-, seven-, and eight-gluon results. The work decomposes amplitudes into completed-cut, recursive, and overlap pieces, and demonstrates consistency through real-momentum factorization checks and symmetry verifications, with numerical phase-space tests. Collectively, these methods enable analytic and semi-analytic access to high-multiplicity one-loop QCD amplitudes, with clear potential for collider phenomenology and automation in multi-jet processes.

Abstract

The recently developed on-shell bootstrap for computing one-loop amplitudes in non-supersymmetric theories such as QCD combines the unitarity method with loop-level on-shell recursion. For generic helicity configurations, the recursion relations may involve undetermined contributions from non-standard complex singularities or from large values of the shift parameter. Here we develop a strategy for sidestepping difficulties through use of pairs of recursion relations. To illustrate the strategy, we present sets of recursion relations needed for obtaining n-gluon amplitudes in QCD. We give a recursive solution for the one-loop n-gluon QCD amplitudes with three or four color-adjacent gluons of negative helicity and the remaining ones of positive helicity. We provide an explicit analytic formula for the QCD amplitude A_{6;1}(1^-,2^-,3^-,4^+,5^+,6^+), as well as numerical results for A_{7;1}(1^-,2^-,3^-,4^+,5^+,6^+,7^+), A_{8;1}(1^-,2^-,3^-,4^+,5^+,6^+,7^+,8^+), and A_{8;1}(1^-,2^-,3^-,4^-,5^+,6^+,7^+,8^+). We expect the on-shell bootstrap approach to have widespread applications to phenomenological studies at colliders.

Figures (17)

• Figure 1: Schematic representation of recursive contributions. The labels '$T$' and '$L$' refer to tree and loop vertices. The multi-particle factorization-function contribution (c) does not appear for MHV amplitudes.
• Figure 2: The recursive diagrams arising from a ${[3,4\rangle}$ shift in $A^{{\cal N}=0}_{5;1}(1^-,2^-,3^-,4^+,5^+)$. Diagram (b) has a non-standard complex singularity.
• Figure 3: The recursive diagrams arising from a ${[1,2\rangle}$ shift in $A^{{\cal N}=0}_{5;1}(1^-,2^-,3^-,4^+,5^+)$. As discussed in the text, only diagram (d) is nonvanishing.
• Figure 4: The overlap diagrams arising from an auxiliary ${[3,4\rangle}$ shift in $A^{{\cal N}=0}_{5;1}(1^-,2^-,3^-,4^+,5^+)$.
• Figure 5: The recursive diagrams arising from an auxiliary ${[3,4\rangle}$ shift in $A^{{\cal N}=0}_{6;1}(1^-,2^-,3^-,4^+,5^+,6^+)$. Diagram (c) has non-standard complex singularities.