Scattering Amplitudes from Intersection Theory.
TL;DR: Pic Picard-Lefschetz theory is used to prove a new formula for intersection numbers of twisted cocycles associated with a given arrangement of hyperplanes, which become tree-level scattering amplitudes of quantum field theories in the Cachazo-He-Yuan formulation.
Abstract: We use Picard-Lefschetz theory to prove a new formula for intersection numbers of twisted cocycles associated with a given arrangement of hyperplanes. In a special case when this arrangement produces the moduli space of punctured Riemann spheres, intersection numbers become tree-level scattering amplitudes of quantum field theories in the Cachazo-He-Yuan formulation.
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TL;DR: In this article, the authors show that the scattering equations act as a diffeomorphism between the interior of this old worldsheet associahedron and the new kinematic associahy, providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula.
Abstract: The search for a theory of the S-Matrix over the past five decades has revealed surprising geometric structures underlying scattering amplitudes ranging from the string worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to the kinematical space where amplitudes actually live. Motivated by recent advances providing a reformulation of the amplituhedron and planar $$ \mathcal{N} $$
= 4 SYM amplitudes directly in kinematic space, we propose a novel geometric understanding of amplitudes in more general theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint ϕ3 scalar theory, we establish a direct connection between its “scattering form” and a classic polytope — the associahedron — known to mathematicians since the 1960’s. We find an associahedron living naturally in kinematic space, and the tree level amplitude is simply the “canonical form” associated with this “positive geometry”. Fundamental physical properties such as locality and unitarity, as well as novel “soft” limits, are fully determined by the combinatorial geometry of this polytope. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between the interior of this old “worldsheet associahedron” and the new “kinematic associahedron”, providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula. We also find “scattering forms” on kinematic space for Yang-Mills theory and the Non-linear Sigma Model, which are dual to the fully color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact—“Color is Kinematics”— whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, all our scattering forms are well-defined on the projectivized kinematic space, a property which can be seen to provide a geometric origin for color-kinematics duality.
213 citations
Cites background from "Scattering Amplitudes from Intersec..."
...lar to the Kawai-Lewellen-Tye relations connecting open- and closed-string amplitudes [45] (See [46, 47] for related ideas)? We have seen the YM scattering form as pushforward of the Pfaffian form on the worldsheet, which is unique gauge invariant under the assumptions provided....
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TL;DR: The precision frontier in collider physics is being pushed at impressive speed, from both the experimental and the theoretical side as discussed by the authors, and the aim of this review is to give an overview of recent developments in precision calculations within the Standard Model of particle physics, in particular in the Higgs sector.
140 citations
Cites background from "Scattering Amplitudes from Intersec..."
...nsion in the parameter of the Feynman i-prescription. From a more formal side, a very promising alternative to IBP reduction has been suggested based on the intersection theory of dierential forms [518, 519, 500, 520{523], for which the Baikov representation [524, 525] is particularly suited. In this approach, a Feynman integral written in terms of master integrals Mias I= X i ciMi (11) is considered as a vector in a ...
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TL;DR: In this article, the authors propose a geometric understanding of amplitudes for a wide range of massless theories in general spacetime dimensions, including the S-Matrix and the amplituhedron.
Abstract: The search for a theory of the S-Matrix has revealed surprising geometric structures underlying amplitudes ranging from the worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to kinematic space where amplitudes live. In this paper, we propose a novel geometric understanding of amplitudes for a large class of theories. The key is to think of amplitudes as differential forms directly on kinematic space. We explore this picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint cubic scalar, we establish a direct connection between its "scattering form" and a classic polytope--the associahedron--known to mathematicians since the 1960's. We find an associahedron living naturally in kinematic space, and the tree amplitude is simply the "canonical form" associated with this "positive geometry". Basic physical properties such as locality, unitarity and novel "soft" limits are fully determined by the geometry. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between this old "worldsheet associahedron" and the new "kinematic associahedron", providing a geometric interpretation and novel derivation of the bi-adjoint CHY formula. We also find "scattering forms" on kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual to the color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, our scattering forms are well-defined on the projectivized kinematic space, a property that provides a geometric origin for color-kinematics duality.
129 citations
Book•
23 Sep 2020
TL;DR: In this paper, intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with $n$ punctures, $\mathcal{M}_{0,n}), compute tree-level scattering amplitudes in quantum field theories with a finite spectrum of particles.
Abstract: We propose that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with $n$ punctures, $\mathcal{M}_{0,n}$, compute tree-level scattering amplitudes in quantum field theories with a finite spectrum of particles. The relevant cohomology groups are twisted by representations of the fundamental group $\pi_1(\mathcal{M}_{0,n})$ that describes how punctures braid around each other on the Riemann surface. Such a structure can be used to link the space of Riemann surfaces with the space of kinematic invariants. Intersection numbers of said cohomology classes—whose representatives we call twisted forms—can be shown to fully localize on the boundaries of $\mathcal{M}_{0,n}$, which are in a one-to-one correspondence with trivalent trees that have an interpretation as Feynman diagrams. In this work we develop systematic approaches towards accessing such boundary information. We prove that when twisted forms are logarithmic, their intersection numbers have a simple expansion in terms of trivalent Feynman diagrams weighted by residues, allowing only for massless propagators on the internal and external lines. It is also known that in the massless limit intersection numbers have a different localization formula on the support of so-called scattering equations. Nevertheless, for physical applications one also needs to study non-logarithmic forms as they are responsible for propagation of massive states. We utilize the natural fibre bundle structure of $\mathcal{M}_{0,n}$—which allows for a direct access to the boundaries—to introduce recursion relations for intersection numbers that "integrate out" puncture-by-puncture. The resulting recursion involves only linear algebra of certain matrices describing braiding properties of $\mathcal{M}_{0,n}$ and evaluating one-dimensional residues, thus paving a way for explicit analytic computations of scattering amplitudes. Together with a reformulation of the tree-level S-matrix of string theory in terms of twisted forms, the results of this work complete a unified geometric framework for studying scattering amplitudes from genus-zero Riemann surfaces. We show that a web of dualities between different homology and cohomology groups allows for deriving a host of identities among various types of amplitudes computed from the moduli space, which in this setup become a consequence of linear algebra. Throughout this work we emphasize that algebraic computations can be supplemented—or indeed replaced—by combinatorial, geometric, and topological ones.
126 citations
Cites background or methods from "Scattering Amplitudes from Intersec..."
...1 as describing different ways of inserting identity operators, 〈ua|xd〉 = 〈ua|I|xd〉 [8], for instance with...
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...The overall sign can be fixed by a more careful computations and it turns out to depend on the relative winding number w(α|β) 12An alternative—but equivalent—proof of the equality between the low-energy limit of closed-string amplitudes, the CHY formalism, and the intersection numbers in the logarithmic case was given in [8] and can be summarized as follows....
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...fact, using Morse theory one can prove that 〈φ−|φ+〉ω in the massless limit Λ→ 0 takes a simple form [8]:...
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...2Note that in several instances we will depart from the notational conventions used in [7, 8]....
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...In this work we studied a formulation of the quantum field theory S-matrix at tree-level, introduced in [8], in terms of intersection theory of twisted forms on the moduli space M0,n....
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TL;DR: In this paper, a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers is presented, where the decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.
Abstract: We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss 2F1 hypergeometric function, and the Appell F1 function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to n-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.
113 citations
Cites background or methods from "Scattering Amplitudes from Intersec..."
...The available case of intersection numbers of dlog n-forms [78, 79] is not sufficient for Feynman integrals, which belong to the wider class of generic rational n-forms....
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...This task can be achieved by applying to Feynman integrals concepts and computational tools borrowed from the intersection theory of differential forms [77–79]....
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...In the literature, the case of intersection numbers of dlog n-forms has been understood [78, 79], but Feynman integrals belong to the wider class of generic rational n-forms....
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...We refer the interested reader to [80, 83] for review of twisted (co)homologies and their intersection theory, as well as [1, 79, 84, 85] and [86, 87] for some recent applications of these ideas to physics....
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References
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TL;DR: In this paper, Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves, which is a consequence of an equivalence between the perturbative expansion of = 4 super Yang-Mills theory and the D-instanton expansion of a certain string theory.
Abstract: Perturbative scattering amplitudes in Yang-Mills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves. This in turn is apparently a consequence of an equivalence between the perturbative expansion of = 4 super Yang-Mills theory and the D-instanton expansion of a certain string theory, namely the topological B model whose target space is the Calabi-Yau supermanifold
1,626 citations
TL;DR: In this article, the authors derive a formula which expresses any closed string tree amplitude in terms of a sum of the products of appropriate open string tree amplitudes, which is applicable to the heterotic string as well as to the closed bosonic string and type II superstrings.
1,330 citations
TL;DR: A compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimensions is presented and Gauge invariance is completely manifest as it follows from a simple property of the Pfaffian.
Abstract: A new formula for the scattering of massless particles may simplify predictions and analyses of LHC experiments and shed new light on quantum gravity theories.
828 citations
TL;DR: In this paper, a natural formulation for a massless colored cubic scalar theory is presented, which is an integral over the space of n marked points on a sphere and has as integrand two factors: the first is a combination of Parke-Taylor-like terms dressed with U(N ) color structures while the second is a Pfaffian.
Abstract: In a recent note we presented a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. In this paper we show that a natural formulation also exists for a massless colored cubic scalar theory. In Yang-Mills, the formula is an integral over the space of n marked points on a sphere and has as integrand two factors. The first factor is a combination of Parke-Taylor-like terms dressed with U(N ) color structures while the second is a Pfaffian. The S-matrix of a U(N ) × U(N ) cubic scalar theory is obtained by simply replacing the Pfaffian with a U(N ) version of the previous U(N ) factor. Given that gravity amplitudes are obtained by replacing the U(N ) factor in Yang-Mills by a second Pfaffian, we are led to a natural color-kinematics correspondence. An expansion of the integrand of the scalar theory leads to sums over trivalent graphs and are directly related to the KLT matrix. Combining this and the Yang-Mills formula we find a connection to the BCJ color-kinematics duality as well as a new proof of the BCJ doubling property that gives rise to gravity amplitudes. We end by considering a special kinematic point where the partial amplitude simply counts the number of color-ordered planar trivalent trees, which equals a Catalan number. The scattering equations simplify dramatically and are equivalent to a special Y-system with solutions related to roots of Chebyshev polynomials. The sum of the integrand over the solutions gives rise to a representation of Catalan numbers in terms of eigenvectors and eigenvalues of the adjacency matrix of an A-type Dynkin diagram.
611 citations
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TL;DR: The Amplituhedron as discussed by the authors is a mathematical object that generalizes the positive Grassmannian Locality and Unitarity in the planar limit of the Grassmannians, and it is shown that locality and unitarity emerge hand-in-hand from positive geometry.
Abstract: Perturbative scattering amplitudes in gauge theories have remarkable simplicity and hidden infinite dimensional symmetries that are completely obscured in the conventional formulation of field theory using Feynman diagrams This suggests the existence of a new understanding for scattering amplitudes where locality and unitarity do not play a central role but are derived consequences from a different starting point In this note we provide such an understanding for N=4 SYM scattering amplitudes in the planar limit, which we identify as ``the volume" of a new mathematical object--the Amplituhedron--generalizing the positive Grassmannian Locality and unitarity emerge hand-in-hand from positive geometry
541 citations