scholarly journals Momentum amplituhedron meets kinematic associahedron

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
David Damgaard ◽  
Livia Ferro ◽  
Tomasz Łukowski ◽  
Robert Moerman
Keyword(s):  
Canonical Form ◽  
Tree Level ◽  
Canonical Forms ◽  
Scalar Theory ◽  
Yang Mills ◽  
Kinematic Space ◽  
The Common ◽  
Scalar Theories ◽  
Reduced Forms ◽  
Gram Determinant

Abstract In this paper we study a relation between two positive geometries: the momen- tum amplituhedron, relevant for tree-level scattering amplitudes in $$ \mathcal{N} $$ N = 4 super Yang-Mills theory, and the kinematic associahedron, encoding tree-level amplitudes in bi-adjoint scalar φ3 theory. We study the implications of restricting the latter to four spacetime dimensions and give a direct link between its canonical form and the canonical form for the momentum amplituhedron. After removing the little group scaling dependence of the gauge theory, we find that we can compare the resulting reduced forms with the pull-back of the associahedron form. In particular, the associahedron form is the sum over all helicity sectors of the reduced momentum amplituhedron forms. This relation highlights the common sin- gularity structure of the respective amplitudes; in particular, the factorization channels, corresponding to vanishing planar Mandelstam variables, are the same. Additionally, we also find a relation between these canonical forms directly on the kinematic space of the scalar theory when reduced to four spacetime dimensions by Gram determinant constraints. As a by-product of our work we provide a detailed analysis of the kinematic spaces relevant for the four-dimensional gauge and scalar theories, and provide direct links between them.

scholarly journals Weights, recursion relations and projective triangulations for positive geometry of scalar theories

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Renjan Rajan John ◽  
Ryota Kojima ◽  
Sujoy Mahato
Keyword(s):  
Scattering Amplitudes ◽  
Detailed Analysis ◽  
Tree Level ◽  
Massless Scalar ◽  
Weighted Sum ◽  
Canonical Forms ◽  
Factorization Property ◽  
Scalar Theory ◽  
Recursion Relations ◽  
Scalar Theories

Abstract The story of positive geometry of massless scalar theories was pioneered in [1] in the context of bi-adjoint ϕ3 theories. Further study proposed that the positive geometry for a generic massless scalar theory with polynomial interaction is a class of polytopes called accordiohedra [2]. Tree-level planar scattering amplitudes of the theory can be obtained from a weighted sum of the canonical forms of the accordiohedra. In this paper, using results of the recent work [3], we show that in theories with polynomial interactions all the weights can be determined from the factorization property of the accordiohedron. We also extend the projective recursion relations introduced in [4, 5] to these theories. We then give a detailed analysis of how the recursion relations in ϕp theories and theories with polynomial interaction correspond to projective triangulations of accordiohedra. Following the very recent development [6] we also extend our analysis to one-loop integrands in the quartic theory.

scholarly journals Stringy canonical forms

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Nima Arkani-Hamed ◽  
Song He ◽  
Thomas Lam
Keyword(s):  
Canonical Form ◽  
A Priori ◽  
Saddle Points ◽  
Cluster Algebras ◽  
Tree Level ◽  
Minkowski Sum ◽  
Canonical Forms ◽  
Newton Polytopes ◽  
Kinematic Space ◽  
String Amplitudes

Abstract Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce “stringy canonical forms”, which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter α′. They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes. As α′→ 0, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalently the volume of the dual of this polytope, naturally determined by tropical functions. At finite α′, they have simple poles corresponding to the facets of the polytope, with the residue on the pole given by the stringy canonical form of the facet. There is the remarkable connection between the α′→ 0 limit of tree-level string amplitudes, and scattering equations that appear when studying the α′→ ∞ limit. We show that there is a simple conceptual understanding of this phenomenon for any stringy canonical form: the saddle-point equations provide a diffeomorphism from the integration domain to the interior of the polytope, and thus the canonical form can be obtained as a pushforward via summing over saddle points. When the stringy canonical form is applied to the ABHY associahedron in kinematic space, it produces the usual Koba-Nielsen string integral, giving a direct path from particle to string amplitudes without an a priori reference to the string worldsheet. We also discuss a number of other examples, including stringy canonical forms for finite-type cluster algebras (with type A corresponding to usual string amplitudes), and other natural integrals over the positive Grassmannian.

scholarly journals On differential operators and unifying relations for 1-loop Feynman integrands

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Kang Zhou
Keyword(s):  
Sigma Model ◽  
Differential Operators ◽  
Linear Sigma Model ◽  
Tree Level ◽  
Maxwell Theory ◽  
Scalar Theory ◽  
Yang Mills ◽  
Loop Level ◽  
Wide Range ◽  
Non Linear

Abstract We generalize the unifying relations for tree amplitudes to the 1-loop Feynman integrands. By employing the 1-loop CHY formula, we construct differential operators which transmute the 1-loop gravitational Feynman integrand to Feynman integrands for a wide range of theories, including Einstein-Yang-Mills theory, Einstein-Maxwell theory, pure Yang-Mills theory, Yang-Mills-scalar theory, Born-Infeld theory, Dirac-Born-Infeld theory, bi-adjoint scalar theory, non-linear sigma model, as well as special Galileon theory. The unified web at 1-loop level is established. Under the well known unitarity cut, the 1-loop level operators will factorize into two tree level operators. Such factorization is also discussed.

scholarly journals MULTIPARTICLE THRESHOLD AMPLITUDES EXPONENTIATE IN ARBITRARY SCALAR THEORIES

1996 ◽  
Vol 11 (31) ◽  
pp. 2539-2546 ◽  
Author(s):  
M.V. LIBANOV
Keyword(s):  
Particle Production ◽  
Tree Level ◽  
Exponential Behavior ◽  
Scalar Theory ◽  
Scalar Theories

Threshold amplitudes are considered for n-particle production in arbitrary scalar theory. It is found that, like in Φ4, leading-n corrections to the tree level amplitudes, being summed over all loops, exponentiate. This result provides more evidence in favor of the conjecture on the exponential behavior of the multiparticle amplitudes.

scholarly journals Using probability distributions to account for recognition of canonical and reduced word forms

2010 ◽  
Vol 1 ◽  
pp. 49
Author(s):  
Meghan Clayards
Keyword(s):  
Canonical Form ◽  
Probability Distributions ◽  
Reduced Form ◽  
Canonical Forms ◽  
Word Form ◽  
Reduced Word ◽  
Word Forms ◽  
Lexical Items ◽  
Reduced Forms

The frequency of a word form influences how efficiently it is processed, but canonical forms often show an advantage over reduced forms even when the reduced form is more frequent. This paper addresses this paradox by considering a model in which representations of lexical items consist of a distribution over forms. Optimal inference given these distributions accounts for item based differences in recognition of phonological variants and canonical form advantage.

scholarly journals Evaluating EYM amplitudes in four dimensions by refined graphic expansion

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Hongxiang Tian ◽  
Enze Gong ◽  
Chongsi Xie ◽  
Yi-Jian Du
Keyword(s):  
Tree Level ◽  
Tree Amplitude ◽  
Four Dimensions ◽  
Yang Mills ◽  
Negative Helicity ◽  
Spanning Forests ◽  
Single Trace

Abstract The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes induces a refined graphic expansion, by which any tree-level EYM amplitude can be expressed as a summation over all possible refined graphs. Each graph contributes a unique coefficient as well as a proper combination of color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to evaluate EYM amplitudes through YM amplitudes, the latter have much simpler structures in four dimensions than the former. In this paper, we classify the refined graphs for the expansion of EYM amplitudes into N k MHV sectors. Amplitudes in four dimensions, which involve k + 2 negative-helicity particles, at most get non-vanishing contribution from graphs in N k′ (k′ ≤ k) MHV sectors. By the help of this classification, we evaluate the non-vanishing amplitudes with two negative-helicity particles in four dimensions. We establish a correspondence between the refined graphs for single-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − or $$ \left({h}_i^{-},{g}_j^{-}\right) $$ h i − g j − configuration and the spanning forests of the known Hodges determinant form. Inspired by this correspondence, we further propose a symmetric formula of double-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − configuration. By analyzing the cancellation between refined graphs in four dimensions, we prove that any other tree amplitude with two negative-helicity particles has to vanish.

scholarly journals Anthroponyms of Jewish Women in the 16th Century Grand Duchy of Lithuania

2012 ◽  
Vol 21 (26) ◽  
pp. 200-207
Author(s):  
Jūratė Čirūnaitė
Keyword(s):  
Average Length ◽  
Family Members ◽  
16Th Century ◽  
Canonical Forms ◽  
Family Status ◽  
Social Origin ◽  
Jewish Women ◽  
The Common ◽  
Grand Duchy Of Lithuania ◽  
Derivatives Of

The most popular names among Jewish women in 16th century Lithuania were Simcha, Marjam, Anna, Debora. The names were most frequently recorded as diminutives (63.3%), with only 36.4% appearing in canonical forms. The smallest group comprises names formed using only anthroponyms that were derived from those of (male) family members (29.6%). 35.2% of the namings are recorded as mixed type. The same number of women are recorded using only names in the documents.Personal names are included in 70.4% of recorded women’s namings. Andronyms (anthroponyms formed from the spouse’s name) were found in 64.8% of all the records. 9.3% of women’s namings include anthroponyms formed using the spouse’s patronymic. Only 1.9% of namings had a female patronymic (the derivative of the suffix -owna/-ewna).One-member female namings prevail (59.3%). Two-member namings comprise 33.3%. Three members are found in 5.6% of the namings, while four-membered ones comprise 1.9%. The average length of the namings is 1.5 times that of the anthroponyms.Common words explaining anthroponyms were found in 68.5% of the namings. Common words related to religion prevail (51.4%). 29.7% of the common words characterize relationships or family status, and only 10.8% describe occupation, post or trade (vocation). Common words describing descent (social origin) comprise only 8.1% of all the women’s namings.Namings consisting only of anthroponyms of family members can be subdivided into the following subgroups: 1) derivatives of the suffix -owaja/-ewaja; 2) derivatives of the suffix ‑owaja/-ewaja; 3) derivatives of the suffix -owaja/-ewaja + the genitive of a male patronymic; 4) derivatives of the suffix -owaja/-ewaja + a male patronymic + the genitive of a male patronymic. Namings without anthroponyms consisting of family members included names and names with common words. Mixed namings consisted of: 1) a name + a derivative of the suffix -owaja/-ewaja; 2) a derivative of the suffix -owaja/-ewaja + the genitive of a male patronymic + a name; 3) a derivative of the suffix -owaja/-ewaja + the genitive of a male patronymic + a name + a female patronymic.The most popular type of naming is a recorded name.

scholarly journals Reduced Canonical Forms of Stoppers

10.37236/1083 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Aaron N. Siegel
Keyword(s):  
Canonical Form ◽  
Canonical Forms ◽  
Alternate Proof

The reduced canonical form of a loopfree game $G$ is the simplest game infinitesimally close to $G$. Reduced canonical forms were introduced by Calistrate, and Grossman and Siegel provided an alternate proof of their existence. In this paper, we show that the Grossman–Siegel construction generalizes to find reduced canonical forms of certain loopy games.

Canonical forms for definable subsets of algebraically closed and real closed valued fields

1995 ◽  
Vol 60 (3) ◽  
pp. 843-860 ◽  
Author(s):  
Jan E. Holly
Keyword(s):  
Canonical Form ◽  
Simple Form ◽  
Finite Union ◽  
Canonical Forms ◽  
Boolean Combination ◽  
Valued Fields ◽  
The Real

AbstractWe present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming “Swiss cheeses” in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in “most” valued fields F, if f(x), g(x) ∈ F[x] and v is the valuation map, then the set {x: v(f(x)) ≤ v(g(x))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of “valued trees”, which we define formally.

VI.—On Singular Pencils of Zehfuss, Compound, and Schläflian Matrices

1937 ◽  
Vol 56 ◽  
pp. 50-89 ◽  
Author(s):  
W. Ledermann
Keyword(s):  
Canonical Form ◽  
Direct Product ◽  
Canonical Forms ◽  
Singular Case ◽  
Elementary Divisors ◽  
Compound Matrices ◽  
Matrix Pencils ◽  
The Subject

In this paper the canonical form of matrix pencils will be discussed which are based on a pair of direct product matrices (Zehfuss matrices), compound matrices, or Schläflian matrices derived from given pencils whose canonical forms are known.When all pencils concerned are non-singular (i.e. when their determinants do not vanish identically), the problem is equivalent to finding the elementary divisors of the pencil. This has been solved by Aitken (1935), Littlewood (1935), and Roth (1934). In the singular case, however, the so-called minimal indices or Kronecker Invariants have to be determined in addition to the elementary divisors (Turnbull and Aitken, 1932, chap. ix). The solution of this problem is the subject of the following investigation.

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