Scattering of massless particles: scalars, gluons and gravitons

By F. Cachazo, Song He, E. Y. Yuan. 2013

547 citations recorded by the source corpus.

A bstractIn 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.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/126439b52dd4d6869b526251357e12858bde955e

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Citations in the corpus

Black Hole Formation and Classicalization in Ultra-Planckian 2 ! N Scattering (2014, methodology): “the nal state being given by the number of black hole constituents, as suggested by classicalization. • Next, by using powerful eld and string-theoretic techniques, in particular scattering equations [9] and Kawai-Lewellen-Tye (KLT) relations [10], we estimate the perturbative part of these N-gravit”
Closed string amplitudes as single-valued open string amplitudes (2014, background): “[8] an interesting uniform description of field–theoretical gauge– and gravity amplitudes has been presented and we shall see that some of its underlying structure is carried by the gauge amplitudes in heterotic string theory.”
Extensions of theories from soft limits (2016, background): “It was shown in [7] that the KLT kernel matrix is the inverse of a matrix whose entries are biadjoint φ(3) scalar amplitudes.”
Ambitwistor strings and the scattering equations at one loop (2013, background): “More than just providing the underlying geometric explanation of the formulæ of [6], the ambitwistor string allows us to extend these formulæ in a variety of ways.”
Celestial operator products of gluons and gravitons (2019, unknown): “This resonates with similar findings in the amplitudes program [26–30].”
Scattering forms and the positive geometry of kinematics, color and the worldsheet (2017, methodology): “And of course, these forms are again obtained as the “pushforward” via the scattering equations from the familiar differential forms on the worldsheet [15, 21], in parallel to the bi-adjoint theory.”