Abstract
As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints for any M\"obius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.
Highlights
Cachazo, He and Yuan have generalized the construction to φ4-theory and various related scalar theories coupled to gauge fields and gravitons [15]
We give integration rules for the CHY integrands that contain a Pfaffian and appear in the CHY φ4-theory [15], and we examine the connection to the dual formulation in string theory
We are mostly interested in string theory integrands H(z) that can lead to convergent integrals for finite α in a neighborhood around the origin
Summary
In order to develop integration rules for CHY integrands it is useful to first describe the link between superstring theory and the CHY prescription [10]. This separation of terms is equivalent to a complete cancellation of tachyon poles in the superstring integrand At this point, one can insert the δ-function constraint (2.6) to recover the CHY prescription [1,2,3] for Yang-Mills theory. We are mostly interested in string theory integrands H(z) that can lead to convergent integrals for finite α in a neighborhood around the origin As mentioned above, this can be achieved in the superstring case by combining terms, perhaps after integration by parts. The final integral is the sum over these products for each of the collections τ It follows from these rules that if there are no (n − 3)-element collections of mutually compatible subsets, the integral (combined with the α n−3 prefactor) will vanish in the α → 0 limit.
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