Scattering in N=4 Super Yang Mills and N=8 Supergravity

The scattering amplitudes of planar N = 4 super-Yang-Mills theory (sYM) exhibit a number of remarkable analytic structures, including dual conformal symmetry, logarithmic singularities of integrands, and the absence of poles at infinite loop momentum. None of these properties are apparent from our usual formulation of quantum field theory in terms of Lagrangians and Feynman rules. In the past years, the hidden features inspired a dual formulation for scattering amplitudes that is not built on the two pillars of locality and unitarity. Instead, a new geometric formulation in terms of Grassmannians and the amplituhedron emerged, which is based on the key analytic properties of scattering amplitudes in the planar sector of $\N=4$ super-Yang-Mills theory. Starting from geometric concepts, the amplituhedron geometry derives all properties of scattering amplitudes in said theory, including locality and factorization. From a practical perspective, expanding the amplitude in terms of a local diagrams, the amplituhedron construction implies that scattering amplitudes in planar N=4 super-Yang-Mills are fully specified by a surprisingly simple subset of all unitarity cuts. Concretely, integrands are uniquely (up to an overall constant) fixed by demanding their vanishing on all spurious singularities. Extending an initial proposal by Arkani-Hamed, Bourjaily, Cachazo, and Trnka, we conjecture that the same analytic structures extend beyond the planar limit of N=4 super-Yang-Mills. Furthermore we show that the $\dlog$ and \emph{no pole at infinity} constraints give the key integrand level analytic information contained in dual conformal symmetry in the planar sector. While it is presently unclear how to extend either dual conformal symmetry or the amplituhedron picture beyond the planar sector, our results suggest that related concepts might exist and await discovery. In order to support our conjectures, we have analyzed several nontrivial multi-loop multi-leg amplitudes. For the nonplanar three-loop four-point and two-loop five point $\N = 4$ super-Yang-Mills amplitudes, we explicitly construct a complete basis of diagram integrands that has only logarithmic singularities and no poles at infinity. We also give examples at three loops showing how to make the logarithmic singularity properties manifest by writing explicit dlog forms. We give additional evidence at four and five loops supporting the nonplanar logarithmic singularity conjecture. Our investigations show that the singularity structures of planar and nonplanar integrands in N = 4 super-Yang-Mills are strikingly similar. Finally, we express the complete amplitude in terms of our special basis diagrams, with the coefficients determined by the vanishing conditions on the amplitude. By successfully carrying out this procedure, we provide nontrivial evidence that the “zero conditions” also carry over into the nonplanar sector. Our analysis suggests that the concept of the amplituhedron can be extended to the nonplanar sector of N = 4 super-Yang-Mills theory and one might hope to ultimately reformulate more general quantum field theories in a geometric language. Using the marvelous squaring relation between Yang-Mills and gravity theories discovered by Bern, Carrasco, and Johansson (BCJ), we relate our newly gained knowledge on the Yang-Mills side to properties of gravity. We conjecture that to all loop orders, while N = 8 supergravity has poles at infinity, at least at four points it has only logarithmic singularities at finite locations. We provide nontrivial evidence for these conjectures. We describe the singularity structure of N = 8 supergravity at three loops and beyond. In order to approach a geometric formulation for scattering in gravitational theories, we retrace the initial steps taken in planar N=4 super-Yang-Mills in the gravitational setting. In particular, we study on-shell diagrams for gravity theories with any number of supersymmetries and find a compact Grassmannian formula in terms of edge variables of the graphs. Unlike in gauge theory where the analogous form involves only dlog-factors, in gravity we find a non-trivial numerator as well as higher degree poles in the edge variables. Based on the structure of the Grassmannian formula for $\N=8$ supergravity we conjecture that gravity loop amplitudes also possess similar properties. In particular, we find that there are only logarithmic singularities on cuts with finite loop momentum and that poles at infinity are present.