Scattering of gravitons and spinning massive states from compact numerators

N Emil J Bjerrum-Bohr,Taro V Brown,Humberto Gomez

doi:10.1007/jhep04(2021)234

Abstract

We provide a new efficient diagrammatic tool, in the context of the scattering equations, for computation of covariant D-dimensional tree-level n-point amplitudes with pairs of spinning massive particles using compact exponential numerators. We discuss how this framework allows non-integer spin extensions of recurrence relations for amplitudes developed for integer spin. Our results facilitate the on-going program for generating observables in classical general relativity from on-shell tree amplitudes through the Kawai-Lewellen-Tye relations and generalized unitarity.

Highlights

Extension in the scattering equation framework and allows an expansion of the reduced Pfaffian as noticed by [82] provided by Kawai-Lewellen-Tye orthogonality [83–86], Pf (Ψ) =
We have in this paper presented a new computational tool for numerators with spinning massive particles that require only (n − 2)! diagrams
We have shown through several explicit examples and analytic computations how to write compact expressions for numerators in terms of exponential numerators

Summary

Amplitudes from scattering equations

The scattering equation formalism provides D-dimensional scattering amplitudes for a large number of theories, refs. [82, 95–98], through integration over a moduli space. For which we identify the following algebraic numerator contributions ( 1 · f2 · f3 · 4), ( 1 · 4)( 2 · k1)( 3 · k1), ( 1 · 4)( 3 · f2 · k1) In those expressions, fi denotes the linearized field strength tensor for particle i with fiμν. [72, 73] was the extension of the above diagrammatic rules for spinors We treat these cases through the same diagrams by exchanging the polarization vectors with spinors and the gluon field strength tensor coming from the baseline by the following spinor replacement. Χ1 is a ten-dimensional Majorana-Weyl spinor, obeying χ1k/ = 0 and ξn is related to χn through χn = k/nξn Based on these definitions, we have tested up to six points that the following replacements are valid for D-dimensional massive Dirac fermions. Considering the expressions (2.11), we find the corresponding contributions for two scalars to be

Exponential color-kinematic numerators

Further improvement to the algorithm

Recurrence relations with massive spinning particles-version

Conclusion

C All seven-point diagrams