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 Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Niklas Henke ◽  
Georgios Papathanasiou
Keyword(s):  
Scattering Amplitudes ◽  
Finite Subset ◽  
Tropical Geometry ◽  
Cluster Algebra ◽  
Cluster Algebras ◽  
Square Root ◽  
Mathematical Foundation ◽  
Yang Mills ◽  
Square Roots ◽  
Related Approach

Abstract We further exploit the relation between tropical Grassmannians and Gr(4, n) cluster algebras in order to make and refine predictions for the singularities of scattering amplitudes in planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory at higher multiplicity n ≥ 8. As a mathematical foundation that provides access to square-root symbol letters in principle for any n, we analyse infinite mutation sequences in cluster algebras with general coefficients. First specialising our analysis to the eight-particle amplitude, and comparing it with a recent, closely related approach based on scattering diagrams, we find that the only additional letters the latter provides are the two square roots associated to the four-mass box. In combination with a tropical rule for selecting a finite subset of variables of the infinite Gr(4, 9) cluster algebra, we then apply our results to obtain a collection of 3, 078 rational and 2, 349 square-root letters expected to appear in the nine-particle amplitude. In particular these contain the alphabet found in an explicit 2-loop NMHV symbol calculation at this multiplicity.

scholarly journals The symbol and alphabet of two-loop NMHV amplitudes from $$ \overline{Q} $$ equations

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Explicit Formula ◽  
First Principle ◽  
Square Root ◽  
Algebraic Functions ◽  
Yang Mills ◽  
Square Roots ◽  
Total Differential ◽  
Gram Determinant ◽  
Number Of Particles

Abstract We study the symbol and the alphabet for two-loop NMHV amplitudes in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills from the $$ \overline{Q} $$ Q ¯ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N2MHV ratio functions, we explain in detail how to use $$ \overline{Q} $$ Q ¯ equations to obtain the total differential of two-loop n-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for n ≥ 8. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find 17 − 2m multiplicative-independent letters for a given square root of Gram determinant, with 0 ≤ m ≤ 4 depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and free of square roots. As an example, we present the complete symbol for n = 9 whose alphabet contains 59 × 9 rational letters, in addition to the 11 × 9 independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.

scholarly journals Bootstrapping octagons in reduced kinematics from A2 cluster algebras

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Yichao Tang ◽  
Qinglin Yang
Keyword(s):  
Scattering Amplitudes ◽  
Minkowski Spacetime ◽  
Dimensional Subspace ◽  
Cluster Algebras ◽  
Wilson Loops ◽  
Two Dimensional ◽  
Feynman Integrals ◽  
Yang Mills ◽  
Two Sectors ◽  
Special Case

Abstract Multi-loop scattering amplitudes/null polygonal Wilson loops in $$ \mathcal{N} $$ 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 AdS3 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 ∼ An−3. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping A2 functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters v, 1 + v, w, 1 + w of A1 × A1, there are two letters v − w, 1 − vw 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 A2 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 A2 functions and beyond.

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 Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

scholarly journals Symbol alphabets from tensor diagrams

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Lecheng Ren ◽  
Marcus Spradlin ◽  
Anastasia Volovich
Keyword(s):  
Yang Mills ◽  
Square Roots ◽  
Web Series

Abstract We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar $$ \mathcal{N} $$ N = 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope $$ {\mathcal{C}}^{\dagger}\left(4,9\right) $$ C † 4 9 encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.

scholarly journals Applications of Square Roots of Diffeomorphisms

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 43
Author(s):  
Yoshihiro Sugimoto
Keyword(s):  
Smooth Manifold ◽  
Contact Manifold ◽  
Diffeomorphism Group ◽  
Square Root ◽  
Square Roots ◽  
Group Diff

In this paper, we prove that on any contact manifold ( M , ξ ) there exists an arbitrary C ∞ -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C ∞ -small contactomorphism which is not “autonomous”. This paper is the first step to study the topology of C o n t 0 ( M , ξ ) ∖ Aut ( M , ξ ) . As an application, we also prove a similar result for the diffeomorphism group Diff ( M ) for any smooth manifold M.

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 Vector logic allows counterfactual virtualization by the square root of NOT

2020 ◽  
Author(s):  
Eduardo Mizraji
Keyword(s):  
Square Root ◽  
Mathematical Approach ◽  
Complex Matrix ◽  
Virtual Character ◽  
Truth Values ◽  
Square Roots ◽  
Counterfactual Conditionals ◽  
Basic Representation ◽  
Logical Functions ◽  
Matrix Vector

Abstract In this work, we investigate the representation of counterfactual conditionals using the vector logic, a matrix-vector formalism for logical functions and truth values. Inside this formalism, the counterfactuals can be transformed in complex matrices preprocessing an implication matrix with one of the square roots of NOT, a complex matrix. This mathematical approach puts in evidence the virtual character of the counterfactuals. This happens because this representation produces a valuation of a counterfactual that is the superposition of the two opposite truth values weighted, respectively, by two complex conjugated coefficients. This result shows that this procedure gives an uncertain evaluation projected on the complex domain. After this basic representation, the judgement of the plausibility of a given counterfactual allows us to shift the decision towards an acceptance or a refusal. This shift is the result of applying for a second time one of the two square roots of NOT.

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