Citation
Herrmann, Enrico (2017) Scattering in N=4 super Yang Mills and N=8 supergravity. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z94J0C49. http://resolver.caltech.edu/CaltechTHESIS:05222017201555383
Abstract
The scattering amplitudes of planar N = 4 superYangMills 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 $\N=4$ superYangMills 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 superYangMills 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 ArkaniHamed, Bourjaily, Cachazo, and Trnka, we conjecture that the same analytic structures extend beyond the planar limit of N=4 superYangMills. 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 multiloop multileg amplitudes. For the nonplanar threeloop fourpoint and twoloop five point $\N = 4$ superYangMills 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 superYangMills 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 superYangMills theory and one might hope to ultimately reformulate more general quantum field theories in a geometric language.
Using the marvelous squaring relation between YangMills and gravity theories discovered by Bern, Carrasco, and Johansson (BCJ), we relate our newly gained knowledge on the YangMills 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 superYangMills in the gravitational setting. In particular, we study onshell 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 dlogfactors, in gravity we find a nontrivial 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.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Theoretical Particle Physics  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Physics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Group:  Caltech Theory  
Thesis Committee: 
 
Defense Date:  15 May 2017  
Funders: 
 
Record Number:  CaltechTHESIS:05222017201555383  
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:05222017201555383  
DOI:  10.7907/Z94J0C49  
Related URLs: 
 
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  10193  
Collection:  CaltechTHESIS  
Deposited By:  Enrico Herrmann  
Deposited On:  30 May 2017 23:16  
Last Modified:  17 Oct 2018 18:51 
Thesis Files

PDF
 Final Version
See Usage Policy. 2653Kb 
Repository Staff Only: item control page