scholarly journals Positive Configuration Space

2021 ◽  
Author(s):  
Nima Arkani-Hamed ◽  
Thomas Lam ◽  
Marcus Spradlin
Keyword(s):  
Scattering Amplitudes ◽  
Configuration Space ◽  
Yang Mills ◽  
Stratified Space

AbstractWe define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel.

scholarly journals Algebraic singularities of scattering amplitudes from tropical geometry

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Drummond ◽  
Jack Foster ◽  
Ömer Gürdoğan ◽  
Chrysostomos Kalousios
Keyword(s):  
Scattering Amplitudes ◽  
Natural Language ◽  
Tropical Geometry ◽  
Cluster Algebras ◽  
Finite Alphabet ◽  
Square Root ◽  
Finite Sets ◽  
Yang Mills ◽  
Square Roots ◽  
Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

scholarly journals Two-loop rational terms in Yang-Mills theories

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Jean-Nicolas Lang ◽  
Stefano Pozzorini ◽  
Hantian Zhang ◽  
Max F. Zoller
Keyword(s):  
Scattering Amplitudes ◽  
Gauge Theories ◽  
A Posteriori ◽  
Four Dimensions ◽  
Yang Mills ◽  
General Properties ◽  
Finite Set ◽  
Numerical Tools ◽  
Loop Amplitudes

Abstract Scattering amplitudes in D dimensions involve particular terms that originate from the interplay of UV poles with the (D − 4)-dimensional parts of loop numerators. Such contributions can be controlled through a finite set of process-independent rational counterterms, which make it possible to compute loop amplitudes with numerical tools that construct the loop numerators in four dimensions. Building on a recent study [1] of the general properties of two-loop rational counterterms, in this paper we investigate their dependence on the choice of renormalisation scheme. We identify a nontrivial form of scheme dependence, which originates from the interplay of mass and field renormalisation with the (D−4)-dimensional parts of loop numerators, and we show that it can be controlled through a new kind of one-loop counterterms. This guarantees that the two-loop rational counterterms for a given renormalisable theory can be derived once and for all in terms of generic renormalisation constants, which can be adapted a posteriori to any scheme. Using this approach, we present the first calculation of the full set of two-loop rational counterterms in Yang-Mills theories. The results are applicable to SU(N) and U(1) gauge theories coupled to nf fermions with arbitrary masses.

scholarly journals Celestial superamplitude in $$ \mathcal{N} $$ = 4 SYM theory

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Hongliang Jiang
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Scattering Amplitudes ◽  
Momentum Space ◽  
Ward Identity ◽  
Yang Mills ◽  
Conformal Field ◽  
New Perspective ◽  
Celestial Sphere

Abstract Celestial amplitude is a new reformulation of momentum space scattering amplitudes and offers a promising way for flat holography. In this paper, we study the celestial amplitudes in $$ \mathcal{N} $$ N = 4 Super-Yang-Mills (SYM) theory aiming at understanding the role of superconformal symmetry in celestial holography. We first construct the superconformal generators acting on the celestial superfield which assembles all the on-shell fields in the multiplet together in terms of celestial variables and Grassmann parameters. These generators satisfy the superconformal algebra of $$ \mathcal{N} $$ N = 4 SYM theory. We also compute the three-point and four-point celestial super-amplitudes explicitly. They can be identified as the conformal correlation functions of the celestial superfields living at the celestial sphere. We further study the soft and collinear limits which give rise to the super-Ward identity and super-OPE on the celestial sphere, respectively. Our results initiate a new perspective of understanding the well-studied $$ \mathcal{N} $$ N = 4 SYM amplitudes via 2D celestial conformal field theory.

scholarly journals Spectral parameters for scattering amplitudes in $ \mathcal{N} $ =4 super Yang-Mills theory

2014 ◽  
Vol 2014 (1) ◽  
Author(s):  
Livia Ferro ◽  
Tomasz Lukowski ◽  
Carlo Meneghelli ◽  
Jan Plefka ◽  
Matthias Staudacher
Keyword(s):  
Scattering Amplitudes ◽  
Spectral Parameters ◽  
Yang Mills ◽  
Super Yang Mills Theory

scholarly journals Asymptotic dynamics on the worldline for spinning particles

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Domenico Bonocore
Keyword(s):  
Scattering Amplitudes ◽  
Gauge Boson ◽  
Classical Limit ◽  
Wilson Line ◽  
Background Field ◽  
Spinning Particle ◽  
Spinning Particles ◽  
Yang Mills ◽  
Asymptotic Dynamics ◽  
Gravity Theories

Abstract There has been a renewed interest in the description of dressed asymptotic states à la Faddeev-Kulish. In this regard, a worldline representation for asymptotic states dressed by radiation at subleading power in the soft expansion, known as the Generalized Wilson Line (GWL) in the literature, has been available for some time, and it recently found applications in the derivation of factorization theorems for scattering processes of phenomenological relevance. In this paper we revisit the derivation of the GWL in the light of the well-known supersymmetric wordline formalism for the relativistic spinning particle. In particular, we discuss the importance of wordline supersymmetry to understand the contribution of the soft background field to the asymptotic dynamics. We also provide a derivation of the GWL for the gluon case, which was not previously available in the literature, thus extending the exponentiation of next-to-soft gauge boson corrections to Yang-Mills theory. Finally, we comment about possible applications in the current research about asymptotic states in scattering amplitudes for gauge and gravity theories and their classical limit.

NON-PERTURBAITVE GREEN FUNCTIONS IN QUANTUM GAUGE THEORIES

1991 ◽  
Vol 06 (10) ◽  
pp. 909-921 ◽  
Author(s):  
S.V. SHABANOV
Keyword(s):  
Configuration Space ◽  
Gauge Theories ◽  
Gauge Condition ◽  
Green Functions ◽  
Spatial Infinity ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Global Gauge

Non-perturbative Green functions for gauge invariant variables are considered. The Green functions are found to be modified as compared with the usual ones in a definite gauge because of a physical configuration space (PCS) reduction. In the Yang-Mills theory with fermions, this phenomenon follows from the Singer theorem about the absence of a global gauge condition for the fields tending to zero at spatial infinity.

COCYCLES IN TOPOLOGICAL YANG-MILLS THEORIES

1988 ◽  
Vol 03 (01) ◽  
pp. 11-17 ◽  
Author(s):  
IZUMI TSUTSUI
Keyword(s):  
Configuration Space ◽  
Orbit Space ◽  
Translation Group ◽  
Vector Representation ◽  
Functional Connection ◽  
Yang Mills ◽  
Topological Mass

We show that the cocycles arise in the Yang-Mills theories with topological terms added. They are found in the representation of the translation group in the configuration space or gauge orbit space, and are shown to be directly related to the U(1) functional connection in these spaces. In 3+1 dimensions, if we adopt a vector representation (i.e. vanishing 2-cocycle), we find the θ-parameter in the Pontryagin term to be 2π×integers. While, in 2+1 dimensions, insisting that finite translations be associative (i.e. vanishing 3-cocycle) leads to the known quantization of the topological mass. Some related aspects with anomalies are also discussed.

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