Integrability and scattering amplitudes in supersymmetric theories
Final Report Abstract
Scattering amplitudes are central quantities in perturbative quantum field theories, since they can be used to construct cross-sections measured at high-energy colliders. Despite the often remarkable simplicity of the final results, the complexity of standard Feynman diagrammatic computations increases very quickly with the number of particles and loops. There is strong evidence that more fundamental, underlying principles need to be put into game. Many of the recent developments have been achieved in the domain of four-dimensional, maximally supersymmetric Yang-Mills theory in the planar limit. This model plays a special role due to its high degree of symmetry: an infinite-dimensional Yangian symmetry, which appears also in integrable spin chains and is an hallmark of integrability. Various aspects of the connection between integrability, symmetries and scattering amplitudes in this theory were analysed in this project. First, the very first attempt to systematically construct tree-level amplitudes in four-dimensional Minkowski space from integrability was made. Using Grassmannian integrals and the reality of the particle momenta in Minkowski signature, unitary integration contours were defined which guarantee the Yangian invariance of the integrals. Fully comprehending the features of these results represents the starting point in turning integrability into a powerful tool for the construction of amplitudes. Moreover, with the help of methods proper to integrable theories, we clarified the meaning of Yangian invariance for the tree-level amplituhedron. In particular, amplituhedron volume forms have been constructed from an underlying spin chain. We showed that the integrable structure is still present, although in a non-standard way. We also proposed a new on-shell diagrammatics for the amplituhedron. Furthermore, the relation between scattering amplitudes and geometry has been strengthened by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. We showed how this residue prescription allows to extract the correct amplituhedron volume functions in the specific cases of cyclic polytopes. Notably, this also naturally exposed the rich combinatorial and geometric structures of amplituhedra, such as their regular triangulations. Finally, we used the transmutation operators to link the various Cachazo-He-Yuan integrands at tree and one-loop level. These computations provided a powerful tool for computing integrands for arbitrary theories by applying the appropriate combination of differential operators.
Publications
- “Yangian Symmetry for the Tree Amplituhedron”, J. Phys. A 50, no. 29, 294005 (2017)
L. Ferro, T. Lukowski, A. Orta and M. Parisi
(See online at https://doi.org/10.1088/1751-8121/aa7594) - “Amplituhedron meets Jeffrey-Kirwan Residue” J. Phys. A 52, no. 4, 045201 (2019)
L. Ferro, T. Lukowski and M. Parisi
(See online at https://doi.org/10.1088/1751-8121/aaf3c3) - “Graßmannian integrals in Minkowski signature, amplitudes, and integrability”, JHEP 1904 (2019) 070, arXiv:1811.04949 [hep-th] M. Bollmann and L. Ferro, “Transmuting CHY formulae”, JHEP 1901 (2019) 180
N. Kanning and M. Staudacher
(See online at https://doi.org/10.1007/JHEP01(2019)180)
