Scattering equations and a new factorization for amplitudes. Part I. Gauge theories

Abstract

In this work we show how a double-cover (DC) extension of the Cachazo, He and Yuan formalism (CHY) can be used to provide a new realization for the factorization of the amplitudes involving gluons and scalar fields. First, we propose a graphic representation for a color-ordered Yang-Mills (YM) and special Yang-Mills-Scalar (YMS) amplitudes within the scattering equation formalism. Using the DC prescription, we are able to obtain an algorithm (integration-rules) which decomposes amplitudes in terms of three-point building-blocks. It is important to remark that the pole structure of this method is totally different to ordinary factorization (which is a consequence of the scattering equations). Finally, as a byproduct, we show that the soft limit in the CHY approach, at leading order, becomes trivial by using the technology described in this paper. arXiv:1810.05407v2 [hep-th]

Scattering equations and a new factorization for amplitudes. Part II. Effective field theories

“…The integration rules share a strong resemblance to the Yang-Mills integration rules given in Ref. [21]. The integration of the double-cover variables y a localizes the integrand to the curves C a = 0, with the solutions y a = ± σ 2 a − Λ 2 , ∀ a.…”

“…As shown in Refs. [17,21], to obtain the usual CHY matrices in the double-cover prescription we use the identification 1 z ab → T ab = 1 (ya+σa)−(y b +σ b ) (see the above section), which gives us the naive identification z a → (y a + σ a ). However, we need all elements of Π β 1 ,...,βm to transform in the same way under a global scaling (y 1 , σ 1 , ..., y n , σ n , Λ) → ρ (y 1 , σ 1 , ..., y n , σ n , Λ), ρ ∈ C * .…”

“…The graphical representation for effective field theory amplitudes in the double-cover prescription is analogous to one presented in Ref. [21]. The only difference is that we are going to work with determinants instead of Pfaffians.…”

“…, we recall how the Pfaffian in Yang-Mills theory was represented [21]. In YM, the half-integrand (−1) i+j n a=1…”

“…The notation for the fixed punctures by yellow, green and red vertices is the same as in Ref. [21]. When all particles are on-shell, the expression is independent of the choice of fixed punctures and reduced determinant.…”