Scattering of massless particles: scalars, gluons and gravitons

Abstract

In a recent note we presented a compact formula for the complete tree-level Smatrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. In this paper we show that a natural formulation also exists for a massless colored cubic scalar theory. In Yang-Mills, the formula is an integral over the space of n marked points on a sphere and has as integrand two factors. The first factor is a combination of Parke-Taylor-like terms dressed with U (N ) color structures while the second is a Pfaffian. The S-matrix of a U (N ) × U (Ñ ) cubic scalar theory is obtained by simply replacing the Pfaffian with a U (Ñ ) version of the previous U (N ) factor. Given that gravity amplitudes are obtained by replacing the U (N ) factor in Yang-Mills by a second Pfaffian, we are led to a natural color-kinematics correspondence. An expansion of the integrand of the scalar theory leads to sums over trivalent graphs and are directly related to the KLT matrix. Combining this and the Yang-Mills formula we find a connection to the BCJ color-kinematics duality as well as a new proof of the BCJ doubling property that gives rise to gravity amplitudes. We end by considering a special kinematic point where the partial amplitude simply counts the number of color-ordered planar trivalent trees, which equals a Catalan number. The scattering equations simplify dramatically and are equivalent to a special Y-system with solutions related to roots of Chebyshev polynomials. The sum of the integrand over the solutions gives rise to a representation of Catalan numbers in terms of eigenvectors and eigenvalues of the adjacency matrix of an A-type Dynkin diagram. 1 Note that the formulas above differ from those in [5] by some overall constant factors that can be absorbed into the definition of the coupling constants. More explicitly, M YM,here n = 1 2 M YM,there n and M gravity,here n = 2 n−1 M gravity,there n. The convention we use in this paper (which coincides with that in [6]) is more standard, and we will see that it is convenient for connecting formulas with different s.

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

“…The departing point of the CHY formalism [2,3] is a mapping from these momenta into an internal space on the n-punctured Riemann sphere…”

“…Remarkably, as shown in [2,3], all tree-level n-point Yang-Mills amplitudes in D dimensions can be obtained from the following integral representation with support on the solutions to the SE:…”

“…In the CHY formalism [2,3], the momenta {p, q, p , q } are mapped into the moduli space of spheres with four punctures, located respectively at the points {σ p , σ q , σ p , σ q } ∈ CP 1 . This is implemented by the identities…”

“…To illustrate this, we focus now on the calculation of the four-point amplitude in a ϕ 3 scalar theory. According to the general prescription given in [2,3], the amplitude can be written as the following integral supported on the solution to the SE…”

“…Usual Feynman diagram techniques become too complicated as the number of external legs or loops increases and any alternative procedures are always desirable. The Cachazo-He-Yuan (CHY) formalism [2,3] represents a very promising route in this direction. It presents n-point amplitudes as an (n − 3)-dimensional integral over the moduli space of n-punctured spheres, fully localized on the solutions to the so-called scattering equations (SE) [2,[4][5][6][7].…”