Vector Space of Feynman Integrals and Multivariate Intersection Numbers
TL;DR: In this article, a general algorithm for the construction of multivariate intersection numbers relevant to Feynman integrals is presented, which can be used to solve the problem of integral reduction to a basis of master integrals by projections, and directly derive functional equations fulfilled by the latter.
Abstract: Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for the construction of multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the decomposition of a few Feynman integrals at one and two loops, as first steps toward potential applications to generic multiloop integrals. The proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations.
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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
TL;DR: Kira 2.0 as discussed by the authors is the most recent version of Kira, which is a state-of-the-art integral reduction algorithm for C++ programs with support for user-provided systems of equations.
114 citations
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TL;DR: The proceedings of the 2019 Les Houches workshop on physics at TeV colliders as discussed by the authors dealt with new developments for high precision Standard Model calculations, the sensitivity of parton distribution functions to the experimental inputs, new developments in jet substructure techniques and a detailed examination of gluon fragmentation at the LHC, issues in the theoretical description of the production of Standard Model Higgs bosons and how to relate experimental measurements.
Abstract: This Report summarizes the proceedings of the 2019 Les Houches workshop on Physics at TeV Colliders. Session 1 dealt with (I) new developments for high precision Standard Model calculations, (II) the sensitivity of parton distribution functions to the experimental inputs, (III) new developments in jet substructure techniques and a detailed examination of gluon fragmentation at the LHC, (IV) issues in the theoretical description of the production of Standard Model Higgs bosons and how to relate experimental measurements, and (V) Monte Carlo event generator studies relating to PDF evolution and comparisons of important processes at the LHC.
87 citations
TL;DR: The stringy canonical form of as mentioned in this paper is a natural extension of the Minkowski sum of the Newton polytopes of the regulating polynomials of a polytope.
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.
74 citations
TL;DR: In this paper, a contour deformation based on loop-tree duality is proposed for numerical computation of loop integrals featuring threshold singularities in momentum space, without the need for fine-tuning.
Abstract: We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space. The functional form of our contour deformation automatically satisfies all constraints without the need for fine-tuning. We demonstrate that our construction is systematic and efficient by applying it to more than 100 examples of finite scalar integrals featuring up to six loops. We also showcase a first step towards handling non-integrable singularities by applying our work to one-loop infrared divergent scalar integrals and to the one-loop amplitude for the ordered production of two and three photons. This requires the combination of our contour deformation with local counterterms that regulate soft, collinear and ultraviolet divergences. This work is an important step towards computing higher-order corrections to relevant scattering cross-sections in a fully numerical fashion.
55 citations
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TL;DR: In this paper, it was proved that the counterterm for an arbitrary 4-loop Feynman diagram in an arbitrary model is calculable within the minimal subtraction scheme in terms of rational numbers and the Riemann ζ-function in a finite number of steps via a systematic "algebraic" procedure involving neither integration of elementary, special, or any other functions, nor expansions in and summation of infinite series of any kind.
1,928 citations
TL;DR: In this article, the authors presented a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory, and proved that their N = 4 ansatz is correct.
1,222 citations
TL;DR: In this paper, a large class of one-loop amplitudes for massless particles that can be constructed via unitarity from tree amplitudes, without any ambiguities, is identified.
1,164 citations
TL;DR: In this paper, the authors presented a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory, and proved that their $N=4$ ansatz is correct using general properties of the relevant one-loops $n$-point integrals.
Abstract: We present a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory. As an example, we obtain the one-loop contribution to amplitudes for $n$ gluon scattering in $N=4$ supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes. We prove that our $N=4$ ansatz is correct using general properties of the relevant one-loop $n$-point integrals. We also give the ``splitting amplitudes'' which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.
1,043 citations
TL;DR: Algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace transformation and procedures for generating and solving systems of differential equations in masses and momenta for master integrals are shown.
Abstract: We describe a new method of calculation of generic multiloop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace transformation. We also describe new algorithms for the identification of master integrals and the reduction of generic Feynman integrals to master integrals, and procedures for generating and solving systems of differential equations in masses and momenta for master integrals. We apply our method to the calculation of the master integrals of massive vacuum and self-energy diagrams up to three loops and of massive vertex and box diagrams up to two loops. Implementation in a computer program of our approach is described. Important features of the implementation are: the ability to deal with hundreds of master integrals and the ability to obtain very high precision results expanded at will in the number of dimensions.
1,023 citations