Abstract
A central problem in quantum field theory is the computation of scattering amplitudes. However, traditional methods are impractical to calculate high order phenomenologically relevant observables. Building on a few decades of astonishing progress in developing non-standard computational techniques, it has been recently conjectured that amplitudes in planar super Yang-Mills are given by the volume of the (dual) amplituhedron. After providing an introduction to the subject at tree-level, we discuss a special class of differential equations obeyed by the corresponding volume forms. In particular, we show how they fix completely the amplituhedron volume for next-to-maximally helicity violating scattering amplitudes.
Highlights
Scattering amplitudes are among the most fundamental quantities in quantum field theory
Much progress has been achieved in the understanding of this simplicity, especially in the context of N = 4 super Yang-Mills (SYM), a four-dimensional superconformal theory which can be thought of as a supersymmetric generalization of QCD: tree-level gluon amplitudes in the two theories coincide and it has been conjectured that N = 4
The connection to Grassmannians has inspired new geometric and even combinatoric methods for studying amplitudes: on the one hand, Hodges observed in [2] that next-to-MHV (NMHV) amplitudes are volumes of polytopes in momentum twistor space; later on, the authors of [7] showed that the residues of Pn,k are in one-to-one correspondence with on-shell diagrams, objects appearing in the so-called positroid stratification of the positive Grassmannian G+(k, n): this space is just the restriction of G(k, n) to the matrices whose k × k ordered minors are all positive
Summary
Scattering amplitudes are among the most fundamental quantities in quantum field theory. The connection to Grassmannians has inspired new geometric and even combinatoric methods for studying amplitudes: on the one hand, Hodges observed in [2] that next-to-MHV (NMHV) amplitudes are volumes of polytopes in momentum twistor space; later on, the authors of [7] showed that the residues of Pn,k are in one-to-one correspondence with on-shell diagrams, objects appearing in the so-called positroid stratification of the positive Grassmannian G+(k, n): this space is just the restriction of G(k, n) to the matrices whose k × k ordered minors are all positive Both ideas combined led to the amplituhedron proposal [8], which aims at providing a fully geometric picture of the physics of scattering, at least within planar N = 4 SYM.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
