Amplituhedra, and Beyond

© 2020 The Author(s). Published by IOP Publishing Ltd. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence (https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.This review is a primer on recently established geometric methods for observables in quantum field theories. The main emphasis is on amplituhedra, i.e. geometries encoding scattering amplitudes for a variety of theories. These pertain to a broader family of geometries called positive geometries, whose basics we review. We also describe other members of this family that are associated with different physical quantities and briefly consider the most recent developments related to positive geometries. Finally, we discuss the main open problems in the field. This is a Topical Review invited by Journal of Physics A: Mathematical and Theoretical.Peer reviewe

Bootstrapping octagons in reduced kinematics from A2A_2 cluster algebras

Multi-loop scattering amplitudes/null polygonal Wilson loops in ${\mathcal N}=4$ super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an $1+1$ dimensional subspace of Minkowski spacetime (or boundary of the $\rm AdS_3$ subspace). Since the edges of a $2n$-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of $G(2,n)/T \sim A_{n{-}3}$. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping $A_2$ functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters $v, 1+v, w, 1+w$ of $A_1 \times A_1$, there are two letters $v-w, 1- v w$ mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping $A_2$ functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to $2n$-gons in terms of $A_2$ functions and beyond.Comment: 26 pages, several figures and tables, an ancilary fil

New Aspects of Scattering Amplitudes, Higher-k Amplitudes, and Holographic Quark Gluon Plasmas

We present new results on different aspects of quantum field theory, which are divided into three main parts. In part I, we find and prove a new behavior of massless tree-level scattering amplitudes, including the biadjoint scalar theory, the U(N) non-linear sigma model, and the special Galileon, within specific subspaces of the kinematic space. We also derive new formulas for the double-ordered biadjoint scalar and $\phi^p$ amplitudes, which can be obtained as integrals over the positive tropical Grassmannian and under limiting procedures on the kinematic invariants. This reveals surprising connections with cubic amplitudes. We also present alternative versions of the formulas for $\phi^p$ amplitudes from combinatorial considerations in terms of non-crossing chord diagrams. In part II, we investigate the generalization of quantum field theory introduced by Cachazo, Early, Guevara and Mizera (CEGM) in 2019. We use soft limits to determine the number of singular solutions of the generalized scattering equations in certain cases and propose a general classification of all configurations that can support singular solutions. We also describe the generalized Feynman diagrams that compute CEGM amplitudes. These are planar arrays of Feynman diagrams satisfying certain compatibility conditions, and we propose combinatorial bootstrap methods to obtain them. Finally, in part III, we analyze different types of quark gluon plasmas in the presence of a background magnetic field using top-down holographic models. We explore conformal and nonconformal theories as consistent truncations of ${\cal N}=8$ gauged supergravity and identify a universal behavior in the ${\cal N}=2^*$ gauge theory

Tree-level amplitudes from the pure spinor superstring

We give a comprehensive review of recent developments on using the pure spinor formalism to compute massless superstring scattering amplitudes at tree level. The main results of the pure spinor computations are placed into the context of related topics including the color-kinematics duality in field theory and the mathematical structure of $\alpha'$-corrections.Comment: 196 pp. Invited review for Physics Reports. We welcome the readers' help in spotting typos or technical mistakes. Every correction that is firstly brought to our attention will be rewarded with 20 Euro Cent per numbered equation, to be paid in cash during the next in-person encounter with one of the authors. v2: version accepted for publication in Physics Report

The momentum amplituhedron

Diese Dissertation befasst sich mit einigen der jüngeren theoretischen Entwicklungen auf dem Gebiet der Streuamplituden. In den letzten Jahren wurde immer mehr der traditionelle Ansatz der Extraktion von Streuamplituden aus Feynman-Diagrammen zugunsten von Techniken, die als On-Shell-Methoden bekannt sind, aufgegeben. Diese Methoden offenbaren eine interessante Beziehung zwischen Streuamplituden und einer Geometrie, die als positive Grassmannsche Geometrie bekannt ist und zu einer radikalen Neuformulierung von Streuamplituden durch so genannte positiven Geometrien geführt hat. Positive Geometrien sind Geometrien mit Rändern aller Kodimensionen und gewissen zugehörigen \emph{kanonischen Formen}, aus denen Streuamplitude extrahiert werden können. Der zentrale Akteur dieser Dissertation ist das Impulsamplituhedron, welches durch die Positive Geometrie gegeben ist und die on-shell Amplituden auf Baumniveau in der maximal supersymmetrischen Yang-Mills-Theorie kodiert, die im Raum der Spinor-Helizitätsvariablen definiert ist. Die canonical Form das Impulsamplituhedron verfügt über eine besondere Singularitätsstruktur, die die physikalischen Singularitäten der Streuamplituden in allen Helizitätssektoren auf Baumniveau kodiert, aus denen die Streuamplituden extrahiert werden können. Dies ermöglicht es, Streuamplituden in maximal supersymmetrischen Yang-Mills Theorie zu bestimmen ohne Bezug auf Felder, Lagrangedichten, Raumzeit oder Feynman-Diagramme zu nehmen. In neueren Arbeiten über das Impulsamplituhedron konnten wir sehen, das seine kanonische Form mit der kanonischen Form - die mit einer Geometrie assoziiert ist, welche die Streuamplituden für bi-adjungierte Skalare - dem kinematischen Associahedron kodiert, in Verbindung gebracht werden kann. Die Definition des Impusamplituhedron auf dem Raum der Spinor-Helizitäts-Variablen ermöglicht einen direkten Vergleich von Geometrien, mit unterschiedlich Farb-geordneten Streuamplituden im selben Raum verbunden sind. Die wird genutzt, um die Kleiss-Kuijf-Relationen -- eine Reihe von Beziehungen zwischen Streuamplituden verschiedener Farbordnungen, wiederherzustellen, die sich aus der Farbzerlegung von Streuamplituden ergeben. Die Kleiss-Kuijf-Relationen manifestieren sich als orientierte Summen von Impulsamplituhedronen verschiedener Farbordnungen ohne Vertices in ihren Rändern. Wir leiten einen homologischen Algorithmus ab, der auf diesem Prinzip basiert, um Kleiss-Kuijf-Beziehungen für Impulsamplituhedronen zu finden.This dissertation focus on some of the modern theoretical developments in the field of scattering amplitudes. Recent years have seen a departure from the traditional approach of extracting scattering amplitudes in terms of Feynman diagrams in favor of techniques known as on-shell methods. These methods reveal a striking relationship between scattering amplitudes and a geometry known as the positive Grassmannian, leading to a radical reformulation of scattering amplitudes in terms of so-called positive geometries. Positive geometries are geometries with boundaries of all codimensions and have a certain associated canonical form. In some special cases, physical observables can be extracted from the canonical forms of positive geometries. The central player in this dissertation is the \emph{momentum amplituhedron} which is the positive geometry encoding on-shell tree-level amplitudes in maximally supersymmetric Yang-Mills theory defined on the space of spinor helicity variables. The momentum amplituhedron is equipped with a canonical form with a particular singularity structure, encoding the physical singularities of scattering amplitudes in all helicity sectors at tree-level, from which scattering amplitudes can be extracted. This allows us to determine scattering amplitudes in maximally supersymmetric Yang-Mills without reference to fields, Lagrangians, space-time, or Feynman diagrams. We will in this dissertation report on the most recent results for the momentum amplituhedron obtained in collaboration with other authors. In particular, we will see that its canonical form can be related to the canonical form associated with a geometry encoding scattering amplitudes for bi-adjoint scalars -- the kinematic associahedron. Furthermore, since we can define the momentum amplituhedron on the space of spinor helicity variables, it allows for a direct comparison of geometries associated with differently color-ordered scattering amplitudes in the same space. This ability to compare momentum amplituhedra of different color orderings will be employed to rederive the Kleiss-Kuijf relations, a set of relations between scattering amplitudes of different color orderings stemming from the color decomposition of scattering amplitudes. The Kleiss-Kuijf relations will appear as oriented sums of momentum amplituhedra of different color orderings with no vertices in their boundary stratifications. We will use this fact to derive a homological algorithm based on this principle to find Kleiss-Kuijf relations for momentum amplituhedra

ALOS-2/PALSAR-2 Calibration, Validation, Science and Applications

Twelve edited original papers on the latest and state-of-art results of topics ranging from calibration, validation, and science to a wide range of applications using ALOS-2/PALSAR-2. We hope you will find them useful for your future research

Sq and EEJ—A Review on the Daily Variation of the Geomagnetic Field Caused by Ionospheric Dynamo Currents

A record of the geomagnetic field on the ground sometimes shows smooth daily variations on the order of a few tens of nano teslas. These daily variations, commonly known as Sq, are caused by electric currents of several μA/m^2 flowing on the sunlit side of the E-region ionosphere at about 90–150 km heights. We review advances in our understanding of the geomagnetic daily variation and its source ionospheric currents during the past 75 years. Observations and existing theories are first outlined as background knowledge for the non-specialist. Data analysis methods, such as spherical harmonic analysis, are then described in detail. Various aspects of the geomagnetic daily variation are discussed and interpreted using these results. Finally, remaining issues are highlighted to provide possible directions for future work