Scattering of massless particles: scalars, gluons and gravitons

• Published in: The journal of high energy physics Vol. 2014; no. 7; p. 1
• Main Authors: Cachazo, Freddy, He, Song, Yuan, Ellis Ye
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-07-2014; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Elementary Particles; High energy physics; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Relativity Theory; String Theory; Scattering Amplitudes; Field Theories in Higher Dimensions
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP07(2014)033

A
bstract
In a recent note we presented a compact formula for the complete tree-level S-matrix 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.