On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms (original) (raw)
Related papers
2019
In [1], two of the present authors along with P. Raman attempted to extend the Amplituhedron program for scalar field theories [2] to quartic scalar interactions. In this paper we develop various aspects of this proposal. Using recent seminal results in Representation theory [3,4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction. We generalise the results of [1] and show that for any n-particle amplitude, the positive geometry associated to the projective scattering form is a convex realisation of Stokes polytope which can be naturally embedded inside one of the ABHY associahedra defined in [2,5]. For a special class of Stokes polytopes with hyper-cubic topology, we show that they have a canonical convex realisation in kinematic space as boundaries of kinematic space associahedra. We then use these kinematic space geometric constructions to write worldsheet forms for ϕ^4 theory w...
Stokes Polytopes : The positive geometry for φ^4 interactions
2019
In a remarkable recent work [1711.09102] by Arkani-Hamed et al, the amplituhedron program was extended to the realm of non supersymmetric scattering amplitudes. In particular it was shown that for tree-level planar diagrams in massless φ^3 theory (and its close cousin, bi-adjoint φ^3 theory) a polytope known as the associa-hedron sits inside the kinematic space and is the amplituhedron for the theory. Precisely as in the case of amplituhedron, it was shown that scattering amplitude is nothing but residue of the canoni-cal form associated to the associahedron. Combinatorial and geometric properties of associahedron naturally encode properties like locality and unitarity of (tree level) scattering amplitudes. In this paper we attempt to extend this program to planar amplitudes in massless φ^4 theory. We show that tree-level planar amplitudes in this theory can be obtained from geometry of objects known as the Stokes polytope which sits naturally inside the kinematic space. As in the case of associahedron we show that residues of the canonical form on these Stokes polytopes can be used to compute scattering amplitudes for quartic interactions. However unlike associahedron, Stokes polytope of a given dimension is not unique and as we show, one must sum over all of them to obtain the complete scattering amplitude. Not all Stokes polytopes contribute equally and we argue that the corresponding weights depend on purely combinatorial properties of the Stokes polytopes. As in the case of φ^3 theory, we show how factorization of Stokes polytope implies unitarity and locality of the amplitudes.
On the Positive Geometry of Quartic Interactions III : One Loop Integrands from Polytopes
2020
Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas (AHST), we explore the positive geometry encoding one loop scattering amplitude for quartic scalar interactions. We define a new class of combinatorial polytopes that we call pseudo-accordiohedra whose poset structures are associated to singularities of the one loop integrand associated to scalar quartic interactions. Pseudo-accordiohedra parametrize a family of projective forms on the abstract kinematic space defined by AHST and restriction of these forms to the type-D associahedra can be associated to one-loop integrands for quartic interactions. The restriction (of the projective form) can also be thought of as a canonical top form on certain geometric realisations of pseudo-accordiohedra. Our work explores a large class of geometric realisations of the type-D associahedra which include all the AHST realisations. These realisations are based on the pseudo-triangulation model for type-D cluster algebras discove...
On positive geometries of quartic interactions: one loop integrands from polytopes
Journal of High Energy Physics
Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas (AHST) [1] we explore the positive geometry encoding one loop scattering amplitude for quartic scalar interactions. We define a new class of combinatorial polytopes that we call pseudo-accordiohedra whose poset structures are associated to singularities of the one loop integrand associated to scalar quartic interactions. Pseudo-accordiohedra parametrize a family of projective forms on the abstract kinematic space defined by AHST and restriction of these forms to the type-D associahedra can be associated to one-loop integrands for quartic interactions. The restriction (of the projective form) can also be thought of as a canonical top form on certain geometric realisations of pseudo-accordiohedra. Our work explores a large class of geometric realisations of the type-D associahedra which include all the AHST realisations. These realisations are based on the pseudo-triangulation model for type-D cluster algebras disc...
2021
The geometric structure of S-matrix encapsulated by the “Amplituhedron program” has begun to reveal itself even in non-supersymmetric quantum field theories. Starting with the seminal work of Arkani-Hamed, Bai, He and Yan [1] it is now understood that for a wide class of scalar quantum field theories, tree-level amplitudes are canonical forms associated to polytopes known as accordiohedra. Similarly the higher loop scalar integrands are canonical forms associated to so called type-D cluster polytopes for cubic interactions or recently discovered class of polytopes termed pseudo-accordiohedron for higher order scalar interactions. In this paper, we continue to probe the universality of these structures for a wider class of scalar quantum field theories. More in detail, we discover new realisations of the associahedron in planar kinematic space whose canonical forms generate (colour-ordered) tree-level S matrix of external massless particles with n−4 massless poles and one massive pol...
Weights, recursion relations and projective triangulations for positive geometry of scalar theories
Journal of High Energy Physics
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 the...
String-generated quartic scalar interactions
2000
The cutting and sewing procedure is used for getting two-loop order Feynman diagrams of Φ 4-theory with an internal SU (N) symmetry group, starting from tachyon amplitudes of the open bosonic string theory. In a suitably defined field theory limit, we reproduce the field theory amplitudes properly normalized and expressed in the Schwinger parametrization.
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.
The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers
2021
The hypersimplex ∆k+1,n is the image of the positive Grassmannian Gr ≥0 k+1,n under the moment map. It is a polytope of dimension n− 1 in R. Meanwhile, the amplituhedron An,k,2 is the projection of the positive Grassmannian Gr≥0 k,n into the Grassmannian Grk,k+2 under the amplituhedron map Z̃. Introduced in the context of scattering amplitudes, it is not a polytope, and has full dimension 2k inside Grk,k+2. Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as was first discovered in [LPW20]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes – images of positroid cells of Gr≥0 k+1,n under the moment map – translate into sign conditions characterizing the T-dual Grasstopes – images of positroid cells of Gr≥0 k,n under the amplituhedron map. ...
Triangulations and Canonical Forms of Amplituhedra: A Fiber-Based Approach Beyond Polytopes
Communications in Mathematical Physics, 2021
Any totally positive (k + m) × n matrix induces a map π + from the positive Grassmannian Gr + (k, n) to the Grassmannian Gr(k, k + m), whose image is the amplituhedron A n,k,m and is endowed with a top-degree form called the canonical form Ω(A n,k,m). This construction was introduced by Arkani-Hamed and Trnka in [AHT14], where they showed that Ω(A n,k,4) encodes scattering amplitudes in N = 4 super Yang-Mills theory. One way to compute Ω(A n,k,m) is to subdivide A n,k,m into so-called generalized triangles and sum over their associated canonical forms. Hence, the physical computation of scattering amplitudes is reduced to finding the triangulations of A n,k,4. However, while triangulations of polytopes are fully captured by their secondary and fiber polytopes [GKZ94, BS92], the study of triangulations of objects beyond polytopes is still underdeveloped. In this work, we initiate the geometric study of subdivisions of A n,k,m in order to establish the notion of secondary amplituhedron. For this purpose, we first extend the projection π + to a rational map π : Gr(k, n) Gr(k, k + m) and provide a concrete birational parametrization of the fibers of π. We then use this to explicitly describe a rational top-degree form ω n,k,m (with simple poles) on the fibers and compute Ω(A n,k,m) as a summation of certain residues of ω n,k,m. As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when n − k − 1 = m (even). We show that, in this case, each fiber of π is parametrized by a projective space and its volume form ω n,k,m has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey-Kirwan residue computes Ω(A n,k,m) from the fiber volume form ω n,k,m. In particular, we give conceptual proofs of the statements of [FŁP18]. Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes. Contents 1 Introduction 1 2 Fibers and triangulations of polytopes 4 3 The amplituhedron 7 4 Fibers of amplituhedra and their volume forms 11 5 Linear fibers and triangulations 19 6 Jeffrey-Kirwan residue for linear fibers 25 7 Fiber positive geometries 30 8 Conclusions and outlook 32