[Paper Review] Tree Level Recursion Relations In General Relativity

Freddy Cachazo, Peter Svrcek|ArXiv.org|Feb 17, 2005

Black Holes and Theoretical PhysicsPhysics and Astronomy20 references92 citations

TL;DR

This paper proposes tree-level recursion relations for graviton scattering amplitudes in general relativity, generalizing the on-shell recursion approach used in Yang-Mills theory. It derives compact formulas for up to six-graviton amplitudes, including the first published expression for the six-graviton non-MHV amplitude, and proves the relations for MHV, next-to-MHV, and all eight-graviton amplitudes using KLT relations and analytic properties of auxiliary functions.

ABSTRACT

Recently, tree-level recursion relations for scattering amplitudes of gluons in Yang-Mills theory have been derived. In this note we propose a generalization of the recursion relations to tree-level scattering amplitudes of gravitons. We use the relations to derive new simple formulae for all amplitudes up to six gravitons. In particular, we present an explicit formula for the six graviton non-MHV amplitude. We prove the relations for MHV and next-to-MHV n-graviton amplitudes and for all eight-graviton amplitudes.

Motivation & Objective

• To extend on-shell recursion relations, successful in Yang-Mills theory, to tree-level graviton scattering amplitudes in general relativity.
• To explain the observed simplicity and helicity selection rules in tree-level gravity amplitudes, such as the vanishing of amplitudes with more than n−2 positive-helicity gravitons.
• To derive explicit, compact formulas for n-graviton amplitudes up to n=6, including the first published result for the six-graviton non-MHV amplitude.
• To prove the validity of the proposed recursion relations for MHV and next-to-MHV amplitudes using analytic properties of auxiliary functions and the KLT relations.
• To establish a framework that may extend to higher-point amplitudes, despite challenges in proving asymptotic behavior for n≥9.

Proposed method

• Proposes a recursion relation for n-graviton amplitudes by marking two external gravitons and summing over products of lower-point subamplitudes connected by a propagator with momentum P_I.
• Uses a complex shift of the momenta of the two marked gravitons, p_i(z) and p_j(z), such that they remain on-shell for all complex z, preserving total momentum.
• Defines an auxiliary function A(z) = A(p_1, ..., p_i(z), ..., p_j(z), ..., p_n), which is a physical on-shell amplitude for all z.
• Employs the KLT relations to express gravity amplitudes as products of two Yang-Mills amplitudes weighted by kinematic invariants, enabling asymptotic analysis.
• Analyzes the behavior of A(z) as z→∞, showing it vanishes for n≤8 due to the 1/z^2 decay of Yang-Mills amplitudes and polynomial growth of KLT factors.
• Uses an auxiliary recursion relation for NMHV amplitudes to prove the vanishing of A(z) at infinity, thereby establishing the validity of the recursion for these cases.

Experimental results

Research questions

• RQ1 Can on-shell recursion relations, successful in Yang-Mills theory, be generalized to tree-level graviton amplitudes in general relativity?
• RQ2 Why do tree-level gravity amplitudes exhibit such remarkable simplicity, particularly the vanishing of amplitudes with more than n−2 positive-helicity gravitons?
• RQ3 Is the auxiliary function A(z) constructed from the amplitude vanishing at infinity for all n-graviton amplitudes, as required for the recursion to hold?
• RQ4 Can the recursion relations be rigorously proven for MHV and next-to-MHV amplitudes using the KLT relations and analytic properties of the amplitude function?
• RQ5 Do the recursion relations extend to higher-point amplitudes, particularly for n≥9, despite potential failure of A(z) to vanish at infinity?

Key findings

• The paper derives a new, explicit formula for the six-graviton non-MHV amplitude A(1⁻,2⁻,3⁻,4⁺,5⁺,6⁺), the first published result of its kind.
• The recursion relations are rigorously proven for all MHV n-graviton amplitudes by showing that the auxiliary function A(z) vanishes at infinity using the BGK formula.
• For next-to-MHV amplitudes, the recursion relations are proven via an auxiliary recursion scheme that simplifies the analytic proof, even though it yields more complex expressions.
• The recursion relations are validated for all eight-graviton amplitudes by demonstrating that A(z) vanishes at infinity using the KLT relations and the 1/z^2 decay of non-adjacent gluon amplitudes.
• For n≥9, the KLT relations suggest that A(z) may not vanish at infinity unless unexpected cancellations occur, leaving the general validity of the recursion open for higher-point amplitudes.
• The study establishes that the recursion relations are valid for all MHV amplitudes regardless of n, contrary to expectations from the KLT relations alone.