A Note on Graviton Amplitudes for New Twistor String Theories

A new class of twistor string theories were recently constructed by Abou-Zeid, Hull and Mason. The most interesting one of these theories has the particle content of N = 8 supergravity. Arguments are given for the vanishing of the (+ − −)-helicity amplitude for gravitons which suggest that this theory describes a chiral N = 8 supergravity rather than Einstein supergravity. Recently, a new set of twistor string theories has been proposed which has the potential of describing a number of interesting field theories as twistor string theories [1]. This was done by gauging the Berkovits form of the Witten-Berkovits twistor string theory with respect to a holomorphic one-form on twistor space [2, 3]. The Witten-Berkovits string theory describes N = 4 supersymmetric Yang-Mills theory coupled to N = 4 conformal supergravity [4]. There are two reasons why one might seek to modify this string theory. If the twistor string theory is to be used for calculations in Yang-Mills theory, then it is necessary to decouple (or eliminate) the conformal supergravitons which can otherwise contribute in loops. Being conformally coupled, there is no dimensional parameter which can be tuned to eliminate them. Secondly, since conformal gravity is a theory with fourth-order derivatives, the linearized equations of motion have additional solutions which do not have a positive Hilbert space norm and hence leads to loss of unitarity. If a new gauge symmetry can be implemented which eliminates these unphysical modes, then both these problems can be eliminated. Abou-Zeid, Hull and Mason introduced a new gauge symmetry which has the potential for doing just this [1]. A number of different theories, which correspond to different choices of the one-form used for gauging, were obtained; selfdual gravity and selfdual supergravity are among these. Perhaps the most intriguing one was the case of gauging with a weightless one-form which seemed to lead to N = 8 supergravity. A twistor string descriptiuon of N = 8 supergravity will have implications for the finiteness of certain loop calculations [5] and would also explain the known and suggestive formulae for graviton scattering amplitudes [6]. In this note we will focus on this special case and calculate some graviton scattering amplitudes; we will present evidence that this twistor string theory is actually N = 8 chiral supergravity and not N = 8 Einstein supergravity. We begin with the basics of Berkovits’ version of the twistor string theory [3]. The target space variables are the supertwistors ZI = (ωA, πA′ , ψ a), A,A′ = 1, 2, a = 1, ..., 4, and YI are the conjugate variables. We also have Z̃I , ỸI with Z I = Z̃I , YI = ỸI on the world-sheet boundary. For genus zero, the world-sheet may be taken as the upper hemisphere of a CP. The action is given by S = ∫ Y (∂̄ + Ā)Z + SC + tilde part (1) Here Ā is a GL(1,C) gauge field and the action (1) has the gauge symmetry, Z → λZ, Y → λY, Ā → Ā− ∂̄ log λ (2) This symmetry will help to reduce the physical set of variables to the projective supertwistor space.