Scattering of Massless Particles: Scalars, Gluons and Gravitons

TL;DR

This work extends CHY-type tree-level S-matrix constructions to a massless colored cubic scalar theory, revealing a natural scalar analogue connected to Yang-Mills and gravity via color-kinematics duality. It introduces double-partial amplitudes m^{(0)}_n(alpha|beta), demonstrates their trivalent-graph expansion and their inverse-KLT relation, and establishes a CK-duality framework enabling gravity as a double copy. A special kinematic point yields Catalan-number counting of planar trees and links the scattering equations to a Y-system with Chebyshev roots. The results unify scalar, YM, and gravity amplitudes in a single formalism and point to promising directions for supersymmetric and loop generalizations, as well as connections to string theory and integrable systems.

Abstract

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)xU(N') cubic scalar theory is obtained by simply replacing the Pfaffian with a U(N') 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. 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.

Figures (4)

• Figure 1: Computing $m^{(0)}_8(I|54376218)$ by finding its polygon decomposition. $(a)$ Points are drawn on the boundary of a disk according to the $\alpha$ ordering. $(b)$ A loop of line segments is drawn connecting the points according to the $\beta$ ordering. $(c)$ External points are moved along the boundary so that a polygon decomposition is manifest. In this example all polygons can be easily exhibited in a single step.
• Figure 2: Sign of $m^{(0)}_8(I|54376218)$
• Figure 3: Trivalent diagrams $g_t,g_s,g_u$ when particle $1,n$ are contained in two different trees, e.g. $A, D$, attached to the four-particle subdiagram. Red and blue regions correspond to $O(g_s)$ and $O(g_u)$ respectively, the union of which gives $O(g_t)$.
• Figure 4: Trivalent diagrams $g_t,g_s,g_u$ when particle $1,n$ are contained in a single tree, e.g. $A$, attached to the four-particle subdiagram. Green, red and blue regions correspond to $O=O(g_s)\bigcap O(g_u)$, $O(g_s)/O$ and $O(g_u)/O$ respectively. It is easy to see that $O(g_t)=O(g_s)\bigcup O(g_u)/O$ is the union of red and blue regions.