[Paper Review] A recursion relation for gravity amplitudes

James L. Bedford, Andreas Brandhuber+2 more|ArXiv.org|Feb 16, 2005

Black Holes and Theoretical Physics4 citations

TL;DR

This paper proposes a new recursion relation for tree-level gravity scattering amplitudes, inspired by the BCFW recursion in Yang-Mills theory, which uses complex momentum shifts to factorize amplitudes on multi-particle poles. The method yields a new compact formula for the n-graviton MHV amplitude, confirming earlier results and suggesting broader applicability to general field theories.

ABSTRACT

Britto, Cachazo and Feng have recently derived a recursion relation for tree-level scattering amplitudes in Yang-Mills. This relation has a bilinear structure inherited from factorisation on multi-particle poles of the scattering amplitudes - a rather generic feature of field theory. Motivated by this, we propose a new recursion relation for scattering amplitudes of gravitons at tree level. Using this recursion relation, we derive a new general formula for the MHV tree-level scattering amplitude for n gravitons. Finally, we comment on the existence of recursion relations in general field theories.

Motivation & Objective

• To extend the BCFW recursion relation, successful in Yang-Mills theory, to gravity amplitudes.
• To derive a new general formula for the tree-level MHV scattering amplitude of n gravitons.
• To investigate whether recursion relations can be a generic feature of massless field theories in four dimensions.
• To explore the conditions under which boundary terms vanish in the large-z limit, ensuring the validity of the recursion.

Proposed method

• Introduce a one-parameter family of amplitudes M(z) by complex momentum shifts of two external gravitons, preserving momentum conservation and on-shell conditions.
• Use contour integration in the complex z-plane to relate the original amplitude to residues at poles, analogous to the BCFW method.
• Leverage factorization properties of gravity amplitudes on multi-particle poles to compute residues in terms of lower-point amplitudes.
• Analyze the large-z behavior of M(z) to ensure the absence of boundary terms, crucial for the recursion to close.
• Apply the recursion to the MHV case, deriving a new closed-form expression for the n-graviton amplitude.
• Consider alternative shift strategies or color-ordered amplitudes in scalar theories to suppress non-vanishing terms at infinity.

Experimental results

Research questions

• RQ1 Can a BCFW-type recursion relation be formulated for gravity amplitudes, given their distinct kinematic structure?
• RQ2 What conditions ensure that the amplitude M(z) vanishes at infinity, thereby eliminating boundary terms in the contour integral?
• RQ3 Does the proposed recursion relation yield a new, compact formula for the n-point MHV gravity amplitude?
• RQ4 Can this recursion method be generalized to other massless field theories beyond Yang-Mills and gravity?

Key findings

• The paper derives a new recursion relation for tree-level gravity amplitudes based on complex momentum shifts and factorization on multi-particle poles.
• The recursion relation successfully generates a new compact formula for the n-graviton MHV amplitude, matching the earlier result of Berends, Giele, and Kuijf.
• The large-z behavior of the amplitude is analyzed, and conditions are identified under which boundary terms vanish, enabling the recursion to close.
• The method is extended to scalar φ⁴ theory, showing that similar recursion relations can be constructed under appropriate shift choices or color ordering.
• The results suggest that recursion relations may be a generic feature of four-dimensional massless field theories, provided the amplitude vanishes at infinity under suitable shifts.