Integration rules for scattering equations

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 Möbius-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.

Sudakov representation of the Cachazo-He-Yuan scattering equations formalism

“…for the n-point amplitude. There has been steady progress in the understanding of the solutions to the SE and the calculation of amplitudes obtained from them (see, for example, [20][21][22][23][24][25][26][27]). …”

Derivation of Feynman rules for higher order poles using cross-ratio identities in CHY construction

“…Applying this method, one can extract all the correct pole structures from a given integrand, and directly obtain the result via the corresponding Feynmann diagrams, rather than solving the scattering equations. However, one shortcomings of the original integration rules [17][18][19] is that it requires the CHY integrand under consideration containing simple poles only, therefore cannot be applied to arbitrary physically acceptable integrands in general.…”

“…Above analysis shows that the first term has been reduced to the case without any higher order poles. Thus the integration rules given in [17][18][19] can be applied straightforwardly. Putting all considerations together, the first part of (5.4) gives…”

The CHY representation of tree-level primitive QCD amplitudes

“…In the gluon case, the permutation invariant function E(z, p, ε) can be written as a (reduced) Pfaffian. The CHY representation has triggered significant interest in the community [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In addition, there are interesting connections with string theory [23][24][25][26][27][28][29][30] and gravity [31][32][33][34][35][36].…”