Abstract
The search for a theory of the S-Matrix over the past five decades has revealed surprising geometric structures underlying scattering amplitudes ranging from the string worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to the kinematical space where amplitudes actually live. Motivated by recent advances providing a reformulation of the amplituhedron and planar mathcal{N} = 4 SYM amplitudes directly in kinematic space, we propose a novel geometric understanding of amplitudes in more general theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint ϕ3 scalar theory, we establish a direct connection between its “scattering form” and a classic polytope — the associahedron — known to mathematicians since the 1960’s. We find an associahedron living naturally in kinematic space, and the tree level amplitude is simply the “canonical form” associated with this “positive geometry”. Fundamental physical properties such as locality and unitarity, as well as novel “soft” limits, are fully determined by the combinatorial geometry of this polytope. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between the interior of this old “worldsheet associahedron” and the new “kinematic associahedron”, providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula. We also find “scattering forms” on kinematic space for Yang-Mills theory and the Non-linear Sigma Model, which are dual to the fully color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact—“Color is Kinematics”— whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, all our scattering forms are well-defined on the projectivized kinematic space, a property which can be seen to provide a geometric origin for color-kinematics duality.
Highlights
Scattering amplitudes are arguably the most basic observables in fundamental physics
The search for a theory of the S-Matrix over the past five decades has revealed surprising geometric structures underlying scattering amplitudes ranging from the string worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to the kinematical space where amplitudes live
Motivated by recent advances providing a reformulation of the amplituhedron and planar N = 4 super Yang-Mills (SYM) amplitudes directly in kinematic space, we propose a novel geometric understanding of amplitudes in more general theories
Summary
Scattering amplitudes are arguably the most basic observables in fundamental physics. As a simple consequence of momentum conservation and on-shell conditions, the wedge product of the d(propagator) factors associated with any cubic graph satisfies exactly the same algebraic identities as the color factors associated with the same graph, as indicated in figure 4 for a n = 5 example This “Color is Kinematics” connection allows us to speak of the scattering forms for Yang-Mills theory and the Non-linear Sigma Model in a fascinating new way. The usual colored amplitudes can be obtained from these forms by replacing the wedges of the d of propagators with color factors in a completely unambiguous way These forms are rigid, god-given objects, entirely fixed (at least at tree level) by standard dimensional power-counting, gauge-invariance (for YM) or the Adler zero (for the NLSM) [20], and the requirement of projectivity. We proceed to describe all the ideas sketched above in much more detail before concluding with remarks on avenues for further work in this direction
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