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Bootstrapping Multi-Parton Loop Amplitudes in QCD

Zvi Bern, Lance J. Dixon, David A. Kosower

TL;DR

This work presents a method to compute complete one-loop QCD amplitudes, including rational terms, by combining the unitarity approach for cut-containing parts with an on-shell recursion bootstrap for the rational pieces. The authors systematize a unitarity-factorization bootstrap and apply it to multi-gluon amplitudes, reproducing known five-gluon results and deriving new six- and seven-point rational terms with two adjacent negative helicities, all without explicit loop integrals for the rational parts. The approach hinges on completing cut terms to remove spurious singularities and using on-shell recursion with careful handling of overlaps between cut and rational contributions. The results demonstrate compact, factorization-consistent expressions and pave the way for applying the method to higher-point QCD processes and processes with massive particles.

Abstract

We present a new method for computing complete one-loop amplitudes, including their rational parts, in non-supersymmetric gauge theory. This method merges the unitarity method with on-shell recursion relations. It systematizes a unitarity-factorization bootstrap approach previously applied by the authors to the one-loop amplitudes required for next-to-leading order QCD corrections to the processes e^+e^- -> Z,γ^* -> 4 jets and pp -> W + 2 jets. We illustrate the method by reproducing the one-loop color-ordered five-gluon helicity amplitudes in QCD that interfere with the tree amplitude, namely A_{5;1}(1^-,2^-,3^+,4^+,5^+) and A_{5;1}(1^-,2^+,3^-,4^+,5^+). Then we describe the construction of the six- and seven-gluon amplitudes with two adjacent negative-helicity gluons, A_{6;1}(1^-,2^-,3^+,4^+,5^+,6^+) and A_{7;1}(1^-,2^-,3^+,4^+,5^+,6^+,7^+), which uses the previously-computed logarithmic parts of the amplitudes as input. We present a compact expression for the six-gluon amplitude. No loop integrals are required to obtain the rational parts.

Bootstrapping Multi-Parton Loop Amplitudes in QCD

TL;DR

This work presents a method to compute complete one-loop QCD amplitudes, including rational terms, by combining the unitarity approach for cut-containing parts with an on-shell recursion bootstrap for the rational pieces. The authors systematize a unitarity-factorization bootstrap and apply it to multi-gluon amplitudes, reproducing known five-gluon results and deriving new six- and seven-point rational terms with two adjacent negative helicities, all without explicit loop integrals for the rational parts. The approach hinges on completing cut terms to remove spurious singularities and using on-shell recursion with careful handling of overlaps between cut and rational contributions. The results demonstrate compact, factorization-consistent expressions and pave the way for applying the method to higher-point QCD processes and processes with massive particles.

Abstract

We present a new method for computing complete one-loop amplitudes, including their rational parts, in non-supersymmetric gauge theory. This method merges the unitarity method with on-shell recursion relations. It systematizes a unitarity-factorization bootstrap approach previously applied by the authors to the one-loop amplitudes required for next-to-leading order QCD corrections to the processes e^+e^- -> Z,γ^* -> 4 jets and pp -> W + 2 jets. We illustrate the method by reproducing the one-loop color-ordered five-gluon helicity amplitudes in QCD that interfere with the tree amplitude, namely A_{5;1}(1^-,2^-,3^+,4^+,5^+) and A_{5;1}(1^-,2^+,3^-,4^+,5^+). Then we describe the construction of the six- and seven-gluon amplitudes with two adjacent negative-helicity gluons, A_{6;1}(1^-,2^-,3^+,4^+,5^+,6^+) and A_{7;1}(1^-,2^-,3^+,4^+,5^+,6^+,7^+), which uses the previously-computed logarithmic parts of the amplitudes as input. We present a compact expression for the six-gluon amplitude. No loop integrals are required to obtain the rational parts.

Paper Structure

This paper contains 11 sections, 96 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. On-Shell Recursion Relations for Loop Amplitudes
  4. Analytic behavior of shifted loop amplitudes
  5. Cut-containing terms, and their 'completion'
  6. Separate factorization of pure-cut and rational terms
  7. Residues of the remaining rational pieces ${\widehat{R}}_n(z)$
  8. Review of Known Results
  9. Recomputation of Five-Gluon QCD Amplitudes
  10. Six- and Seven-Point QCD Amplitudes
  11. Conclusions

Figures (8)

  • Figure 1: A configuration of poles and branch cuts for a term in a one-loop amplitude. The contour $C$ is a circle at $\infty$.
  • Figure 2: A configuration of poles and branch cuts for a term in a one-loop amplitude, with a branch-cut-hugging contour.
  • Figure 3: The recursive diagrams for computing the rational parts of $A_{5;1}^{[0]}(1^-, 2^-, 3^+, 4^+, 5^+)$ with the shift of legs 1,2 given in eq. (\ref{['SpinorShift12']}). '$T$' signifies a tree vertex and '$L$' a loop vertex.
  • Figure 4: The five-point overlap diagrams using the (1,2) shift given in eq. (\ref{['SpinorShift12']}).
  • Figure 5: The recursive diagrams for $A_{6;1}^{[0]}(1^-, 2^-, 3^+, 4^+, 5^+,6^+)$, using the $(1,2)$ shift. Diagrams (a), (b), (d) and (e) vanish.
  • ...and 3 more figures