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In a recent note we presented a compact formula for the complete tree-levelS-matrix of pure Yang-Mills and gravity theories in arbitrary spacetimedimension. In this paper we show that a natural formulation also exists for amassless colored cubic scalar theory. In Yang-Mills, the formula is an integralover 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 obtainedby replacing the U(N) factor in Yang-Mills by a second Pfaffian, we are led toa natural color-kinematics correspondence. An expansion of the integrand of thescalar theory leads to sums over trivalent graphs and are directly related tothe KLT matrix. We find a connection to the BCJ color-kinematics duality aswell as a new proof of the BCJ doubling property that gives rise to gravityamplitudes. We end by considering a special kinematic point where the partialamplitude simply counts the number of color-ordered planar trivalent trees,which equals a Catalan number. The scattering equations simplify dramaticallyand are equivalent to a special Y-system with solutions related to roots ofChebyshev polynomials.

Scattering of Massless Particles: Scalars, Gluons and Gravitons