A Duality For The S Matrix (original) (raw)
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Journal of High Energy Physics, 2014
Scattering amplitudes in 4d N = 4 super Yang-Mills theory (SYM) can be described by Grassmannian contour integrals whose form depends on whether the external data is encoded in momentum space, twistor space, or momentum twistor space. After a pedagogical review, we present a new, streamlined proof of the equivalence of the three integral formulations. A similar strategy allows us to derive a new Grassmannian integral for 3d N = 6 ABJM theory amplitudes in momentum twistor space: it is a contour integral in an orthogonal Grassmannian with the novel property that the internal metric depends on the external data. The result can be viewed as a central step towards developing an amplituhedron formulation for ABJM amplitudes. Various properties of Grassmannian integrals are examined, including boundary properties, pole structure, and a homological interpretation of the global residue theorems for N = 4 SYM.
Cluster Algebraic and Grassmannian Structures in Scattering Amplitudes of N = 4 Super Yang-Mills
2023
The traditional approach of calculating scattering amplitudes results in different types of redundancies and requires the use of Feynman rules and Feynman diagrams, which becomes immensely difficult as the order of the diagrams increases. Moreover, one has to often deal with divergences that appear in the Feynman integrals. The new method of Grassmannians and the Cluster Algebraic structures followed by on-shell diagrams in mathcalN=4\mathcal{N}=4mathcalN=4 Super Yang-Mills in the planer limit proves to be an efficient and computationaly easier method as compared to the conventional approach. The method uses the on-shell functions, to be precise three point on-shell diagrams, which can be joined together to form any type of scattering amplitudes. In the planer limit of mathcalN=4\mathcal{N}=4mathcalN=4 SYM, the computation then has many symmetries which in turn make the expressions simpler. These symmetries are exploited by using Grassmannians and the cluster Algebras.
Massive scattering amplitudes in six dimensions
Journal of High Energy Physics, 2019
We show that a natural spinor-helicity formalism that can describe massive scattering amplitudes exists in D = 6 dimensions. This is arranged by having helicity spinors carry an index in the Dirac spinor 4 of the massive little group, SO(5) ∼ Sp(4). In the high energy limit, two separate kinds of massless helicity spinors emerge as required for consistency with arXiv:0902.0981, with indices in the two SU(2)’s of the massless little group SO(4). The tensors of 4 lead to particles with arbitrary spin, and using these and demanding consistent factorization, we can fix 3− and 4-point tree amplitudes of arbitrary masses and spins: we provide examples. We discuss the high energy limit of scattering amplitudes and the Higgs mechanism in this language, and make some preliminary observations about massive BCFW recursion.
2011
In this paper we discuss in detail computational methods and new results for one-loop virtual corrections to N = 4 super Yang-Mills scattering amplitudes calculated to all orders in ǫ, the dimensional regularization parameter. It is often the case that one-loop gauge theory computations are carried out to O(ǫ 0), since higher order in ǫ contributions vanish in the ǫ → 0 limit. We will show, however, that the higher order contributions are actually quite useful. In the context of maximally supersymmetric Yang-Mills, we consider two examples in detail to illustrate our point. First we will concentrate on computations with gluonic external states and argue that N = 4 supersymmetry implies a simple relation between all-orders-in-ǫ one-loop N = 4 super Yang-Mills amplitudes and the first and second stringy corrections to analogous tree-level superstring amplitudes. For our second example we will derive a new result for the all-orders-in-ǫ one-loop superamplitude for planar six-particle NMHV scattering, an object which allows one to easily obtain six-point NMHV amplitudes with arbitrary external states. We will then discuss the relevance of this computation to the evaluation of the ratio of the planar two-loop six-point NMHV superamplitude to the planar two-loop six-point MHV superamplitude, a quantity which is expected to have remarkable properties and has been the subject of much recent investigation. To make the presentation as self-contained as possible, we extensively review the prerequisites necessary to understand the main results of this work. Contents 1. Overview 2. Review of Computational Technology 2.1 Color Decompositions 2.2 Planar Limit 2.3 Spinor Helicity Formalism 2.4 BCFW Recursion 2.5 Generalized Unitarity in Four Dimensions 2.6 Generalized Unitarity in D Dimensions 3. Efficient Computation and New Results For One-Loop N = 4 Gluon Amplitudes Calculated To All Orders in ǫ
One-loop gluon scattering amplitudes in theories with supersymmetries
Physics Letters B, 2005
Generalised unitarity techniques are used to calculate the coefficients of box and triangle integral functions of one-loop gluon scattering amplitudes in gauge theories with N < 4 supersymmetries. We show that the box coefficients in N = 1 and N = 0 theories inherit the same coplanar and collinear constraints as the corresponding N = 4 coefficients. We use triple cuts to determine the coefficients of the triangle integral functions and present, as an example, the full expression for the one-loop amplitude A N =1 (
Systematics of one-loop scattering amplitudes in N= 4 super-Yang-Mills theories
Physics Letters B, 2005
One-loop scattering amplitudes in N = 4 super Yang-Mills (SYM) theories are analyzed in the paradigm of maximal helicity violating Feynman diagrams. There are very limited number of loop integrals to be evaluated. For a process with n external particles, there are only [n/2] − 1 generically independent integrals. Furthermore, the relations between leading N c amplitudes A n;1 and sub-leading amplitudes A n;c are found to be identical to those obtained from conventional field theory calculations, which can be interpreted as an indirect support for the paradigm.
Non-perturbative approaches to Scattering Amplitudes
2019
This thesis is devoted to the study of scattering amplitudes using two non-perturbative approaches. In Part I we focus on a particular theory known as N = 4 Super-Yang-Mills in four spacetime dimensions. The scattering amplitudes in this theory are dual to the expectation value of null polygonal Wilson loops which can be computed non-perturbatively using integrability. The Wilson loop is decomposed into smaller polygons and computed as an evolution of the color flux tube of the theory, summing over all intermediate flux tube states. By a suitable generalization of the building blocks called pentagons we describe how this program can describe all helicity configurations of the amplitude. We also show how the contribution from all flux tube excitations can be resummed to reproduce the general kinematics result at weak coupling. In Part II we take a different approach and study the space of Quantum Field Theories (QFTs). We focus on two-dimensional theories with a mass gap and a global...
Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case
Journal of High Energy Physics
We present a compact analytic expression for the leading colour two-loop five-gluon amplitude in Yang-Mills theory with a single negative helicity and four positive helicities. The analytic result is reconstructed from numerical evaluations over finite fields. The numerical method combines integrand reduction, integration-by-parts identities and Laurent expansion into a basis of pentagon functions to compute the coefficients directly from six-dimensional generalised unitarity cuts.
New Recursion relations for Scattering Amplitudes with Massive Particles
2021
We use the recently developed massive spinor-helicity formalism [1] of Arkani- Hamed et al. to propose a new class of recursion relations for tree-level amplitudes in gauge theories. These relations are based on a combined complex deformation of massless as well as massive external momenta. We use these relations to study tree-level amplitudes in scalar QCD as well as amplitudes involving massive vector bosons in the Higgsed phase of Yang-Mills theory. We prove the validity of our proposal by showing that in the limit of infinite momenta of two of the external particles, the amplitude once again is controlled by an enhanced Spin-Lorentz symmetry paralleling the proof of BCFW shift for massless gauge theories. Simple examples illustrate that the proposed shift may lead to an efficient computation of tree-level amplitudes.