Scattering of massless particles: scalars, gluons and gravitons

  • Freddy Cachazo
  • Song HeEmail author
  • Ellis Ye Yuan
Open Access


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.


Scattering Amplitudes Field Theories in Higher Dimensions 


Open Access

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Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  3. 3.Department of Physics & AstronomyUniversity of WaterlooWaterlooCanada

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