Scattering of massless particles: scalars, gluons and gravitons
Abstract
In a recent note we presented a compact formula for the complete treelevel Smatrix of pure YangMills 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 YangMills, 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 ParkeTaylorlike terms dressed with U( N ) color structures while the second is a Pfaffian. The Smatrix 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 YangMills by a second Pfaffian, we are led to a natural colorkinematics 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 YangMills formula we find a connection to the BCJ colorkinematics 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 colorordered planar trivalent trees, which equals a Catalan number. The scattering equations simplify dramatically and are equivalent to a special Ysystem 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 Atype Dynkin diagram.
 Publication:

Journal of High Energy Physics
 Pub Date:
 July 2014
 DOI:
 10.1007/JHEP07(2014)033
 arXiv:
 arXiv:1309.0885
 Bibcode:
 2014JHEP...07..033C
 Keywords:

 Scattering Amplitudes;
 Field Theories in Higher Dimensions;
 High Energy Physics  Theory
 EPrint:
 31 pages