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 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 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 Plahte diagrams for string scattering amplitudes

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Pongwit Srisangyingcharoen ◽  
Paul Mansfield
Keyword(s):  
Scattering Amplitudes ◽  
Analytic Continuation ◽  
Complex Plane ◽  
Open String ◽  
Tree Level ◽  
Recursion Relations ◽  
Sign Changes ◽  
String Amplitudes

Abstract Plahte identities are monodromy relations between open string scattering amplitudes at tree level derived from the Koba-Nielsen formula. We represent these identities by polygons in the complex plane. These diagrams make manifest the appearance of sign changes and singularities in the analytic continuation of amplitudes. They provide a geometric expression of the KLT relations between closed and open string amplitudes. We also connect the diagrams to the BCFW on-shell recursion relations and generalise them to complex momenta resulting in a relation between the complex phases of partial amplitudes.

scholarly journals Gravity loop integrands from the ultraviolet

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Alex Edison ◽  
Enrico Herrmann ◽  
Julio Parra-Martinez ◽  
Jaroslav Trnka
Keyword(s):  
Scattering Amplitudes ◽  
Momentum Space ◽  
Indirect Evidence ◽  
Tree Level ◽  
Large Momentum ◽  
Recursion Relations ◽  
Yang Mills ◽  
The One ◽  
Geometric Picture ◽  
Super Yang Mills Theory

We demonstrate that loop integrands of (super-)gravity scattering amplitudes possess surprising properties in the ultraviolet (UV) region. In particular, we study the scaling of multi-particle unitarity cuts for asymptotically large momenta and expose an improved UV behavior of four-dimensional cuts through seven loops as compared to standard expectations. For N=8 supergravity, we show that the improved large momentum scaling combined with the behavior of the integrand under BCFW deformations of external kinematics uniquely fixes the loop integrands in a number of non-trivial cases. In the integrand construction, all scaling conditions are homogeneous. Therefore, the only required information about the amplitude is its vanishing at particular points in momentum space. This homogeneous construction gives indirect evidence for a new geometric picture for graviton amplitudes similar to the one found for planar N=4 super Yang-Mills theory. We also show how the behavior at infinity is related to the scaling of tree-level amplitudes under certain multi-line chiral shifts which can be used to construct new recursion relations.

scholarly journals Constraining the weights of Stokes polytopes using BCFW recursions for ϕ4

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Ishan Srivastava
Keyword(s):  
Scattering Amplitudes ◽  
Great Depth ◽  
Tree Level ◽  
Recursion Relations ◽  
3 Dimensional ◽  
Loop Level ◽  
Boundary Terms ◽  
Geometric Objects ◽  
The Relationship ◽  
Geometric Formulation

Abstract The relationship between certain geometric objects called polytopes and scattering amplitudes has revealed deep structures in QFTs. It has been developed in great depth at the tree- and loop-level amplitudes in $$ \mathcal{N} $$ N = 4 SYM theory and has been extended to the scalar ϕ3 and ϕ4 theories at tree-level. In this paper, we use the generalized BCFW recursion relations for massless planar ϕ4 theory to constrain the weights of a class of geometric objects called Stokes polytopes, which manifest in the geometric formulation of ϕ4 amplitudes. We see that the weights of the Stokes polytopes are intricately tied to the boundary terms in ϕ4 theories. We compute the weights of N = 1, 2, and 3 dimensional Stokes polytopes corresponding to six-, eight- and ten-point amplitudes respectively. We generalize our results to higher-point amplitudes and show that the generalized BCFW recursions uniquely fix the weights for an n-point amplitude.

scholarly journals Open associahedra and scattering forms

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Aidan Herderschee ◽  
Fei Teng
Keyword(s):  
Scattering Amplitudes ◽  
Canonical Form ◽  
Recursion Relations ◽  
Fiber Product ◽  
New Phenomena

Abstract We continue the study of open associahedra associated with bi-color scattering amplitudes initiated in ref. [1]. We focus on the facet geometries of the open associahedra, uncovering many new phenomena such as fiber-product geometries. We then provide novel recursion procedures for calculating the canonical form of open associahedra, generalizing recursion relations for bounded polytopes to unbounded polytopes.

scholarly journals On polytopes and generalizations of the KLT relations

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Nikhil Kalyanapuram
Keyword(s):  
Scattering Amplitudes ◽  
Large Class ◽  
Moduli Space ◽  
Differential Forms ◽  
Natural Generalization ◽  
Intersection Theory ◽  
Hyperplane Arrangements ◽  
Number Of Solutions ◽  
Scalar Theories ◽  
Double Copy

Abstract We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.

TREE DIAGRAMS IN WITTEN’S SUPERSTRING FIELD THEORY

1989 ◽  
Vol 04 (21) ◽  
pp. 2063-2071
Author(s):  
GEORGE SIOPSIS
Keyword(s):  
Field Theory ◽  
Scattering Amplitudes ◽  
Invariant Theory ◽  
Higher Order ◽  
Contact Term ◽  
Tree Level ◽  
Gauge Invariant

It is shown that the contact term discovered by Wendt is sufficient to ensure finiteness of all tree-level scattering amplitudes in Witten’s field theory of open superstrings. Its inclusion in the action also leads to a gauge-invariant theory. Thus, no additional higher-order counterterms in the action are needed.

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 Wormholes in Wyman's solution

2014 ◽  
Vol 23 (11) ◽  
pp. 1450086 ◽  
Author(s):  
J. B. Formiga ◽  
T. S. Almeida
Keyword(s):  
Scalar Field ◽  
General Solution ◽  
Detailed Analysis ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Human Beings ◽  
Einstein Field Equations ◽  
Field Equations

The most general solution of the Einstein field equations coupled with a massless scalar field is known as Wyman's solution. This solution is also present in the Brans–Dicke theory and, due to its importance, it has been studied in detail by many authors. However, this solutions has not been studied from the perspective of a possible wormhole. In this paper, we perform a detailed analysis of this issue. It turns out that there is a wormhole. Although we prove that the so-called throat cannot be traversed by human beings, it can be traversed by particles and bodies that can last long enough.

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