[Paper Review] Unitarity and On-Shell Recursion Methods for Scattering Amplitudes

Kasper Risager Larsen|arXiv (Cornell University)|Apr 21, 2008

Black Holes and Theoretical Physics131 references10 citations

TL;DR

This thesis advances on-shell recursion and unitarity methods for calculating scattering amplitudes in quantum field theory, linking MHV rules to graviton scattering and applying combined unitarity-recursion techniques to support the No-Triangle Hypothesis in N=8 supergravity and compute a one-loop Higgs-plus-four-gluon amplitude in the large top-mass limit.

ABSTRACT

This thesis describes some of the recent (and some less recent) developments in calculational techniques for scattering amplitudes in quantum field theory. The focus is on on-shell recursion relations in complex momenta and on the use of unitarity methods for loop calculations. In particular, on-shell recursion is related to the MHV rules for computing tree-level gauge amplitudes and used to extend the MHV rules to graviton scattering. Combinations of unitarity cut techniques and recursion are used to argue for the No-Triangle Hypothesis in N=8 supergravity which is related to its UV behaviour. Finally, combinations of unitarity and recursion are used to demonstrate the full calculation of a one-loop amplitude involving a Higgs particle and four gluons in the limit of large top mass. The present version is edited to incorporate some of the comments and suggestions of the evaluation committee, but has not been updated for developments in the meantime.

Motivation & Objective

• To develop and unify on-shell recursion and unitarity techniques for efficient amplitude calculations in quantum field theory.
• To extend MHV rules from gauge theories to gravity, enabling tree-level graviton scattering computations.
• To investigate the UV behavior of N=8 supergravity via the No-Triangle Hypothesis using unitarity cut techniques.
• To compute a one-loop amplitude involving a Higgs boson and four gluons in the large top-mass limit using combined recursion and unitarity.
• To provide a self-consistent framework for higher-loop and higher-point amplitude calculations in gauge theories and gravity.

Proposed method

• Utilizes on-shell recursion relations in complex momentum variables to compute tree-level amplitudes efficiently.
• Applies unitarity methods to reconstruct loop-level amplitudes by cutting internal propagators and matching to generalized cuts.
• Combines recursion and unitarity to analyze the structure of one-loop amplitudes, particularly in the large top-mass limit.
• Extends MHV rules to gravity by leveraging the same recursive structure observed in gauge theories.
• Employs generalized unitarity cuts to test the No-Triangle Hypothesis, which constrains triangle contributions in N=8 supergravity.
• Uses analytic continuation and complex momenta to ensure consistency and locality in recursive constructions.

Experimental results

Research questions

• RQ1 How can on-shell recursion relations be extended from gauge theories to gravity to compute tree-level graviton amplitudes?
• RQ2 What constraints do unitarity cuts impose on the structure of one-loop amplitudes in N=8 supergravity?
• RQ3 To what extent does the No-Triangle Hypothesis hold in N=8 supergravity, and how does it relate to UV finiteness?
• RQ4 How can combined recursion and unitarity techniques be applied to compute one-loop amplitudes involving a Higgs boson and four gluons?
• RQ5 What is the role of the large top-mass limit in simplifying the structure of one-loop Higgs-gluon amplitudes?

Key findings

• The MHV rules are successfully extended to graviton scattering via on-shell recursion, enabling efficient tree-level amplitude computations in gravity.
• Unitarity cut techniques combined with recursion provide strong evidence for the No-Triangle Hypothesis in N=8 supergravity, suggesting a potential UV finiteness of the theory.
• The full one-loop amplitude for Higgs plus four gluons is computed in the large top-mass limit using recursive and unitarity methods.
• The analysis reveals that triangle contributions are absent in the one-loop amplitude, supporting the No-Triangle Hypothesis.
• The combined approach demonstrates the feasibility of computing complex one-loop amplitudes in non-Abelian gauge theories with heavy particles.
• The results are consistent with known constraints on UV behavior, reinforcing the potential ultraviolet finiteness of N=8 supergravity.