Feynman integral
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Lectures on differential equations for Feynman integrals Open
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of …
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AMFlow: A Mathematica package for Feynman integrals computation via auxiliary mass flow Open
AMFlow is a Mathematica package to numerically compute dimensionally regularized Feynman integrals via the recently proposed auxiliary mass flow method. In this framework, integrals are treated as functions of an auxiliary mass parameter a…
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Bi-local holography in the SYK model Open
We discuss large N rules of the Sachdev-Ye-Kitaev model and the bi-local representation of holography of this theory. This is done by establishing 1/N Feynman rules in terms of bi-local propagators and vertices, which can be evaluated foll…
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Vector Space of Feynman Integrals and Multivariate Intersection Numbers Open
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 inte…
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Elliptic polylogarithms and iterated integrals on elliptic curves. II. An application to the sunrise integral Open
We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure mathematics and string theory. We then f…
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Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms Open
We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a fourfold, rational (Feynman-)parametric representation …
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Elliptic Feynman integrals and pure functions Open
A bstract We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-l…
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Bounded Collection of Feynman Integral Calabi-Yau Geometries Open
We define the rigidity of a Feynman integral to be the smallest dimension over which it is nonpolylogarithmic. We prove that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L-1) at L loops provided they are in th…
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Feynman integrals and hyperlogarithms Open
We study Feynman integrals in the representation with Schwinger parameters and derive recursive integral formulas for massless 3- and 4-point functions. Properties of analytic (including dimensional) regularization are summarized and we pr…
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Gluing Ladder Feynman Diagrams into Fishnets Open
We use integrability at weak coupling to compute fishnet diagrams for four-point correlation functions in planar ϕ^{4} theory. The results are always multilinear combinations of ladder integrals, which are in turn built out of classical po…
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Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals Open
We consider Feynman integrals with algebraic leading singularities and total\ndifferentials in $\\epsilon\\,\\mathrm{d}\\ln$ form. We show for the first time\nthat it is possible to evaluate integrals with singularities involving\nunration…
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Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction Open
We study the algebraic and analytic structure of Feynman integrals by proposing an operation that maps an integral into pairs of integrals obtained from a master integrand and a corresponding master contour. This operation is a coaction. I…
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Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms Open
In this paper we exploit factorisation properties of Picard-Fuchs operators to decouple differential equations for multi-scale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the ir…
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Tempered fractional Feynman-Kac equation: Theory and examples Open
Functionals of Brownian and non-Brownian motions have diverse applications and attracted a lot of interest among scientists. This paper focuses on deriving the forward and backward fractional Feynman-Kac equations describing the distributi…
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Causality and Loop-Tree Duality at Higher Loops Open
We relate an l-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original l-loop graph. Causality requires that the propagators of the trees have a modified iδ prescri…
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Determining Feynman Integrals with Only Input from Linear Algebra Open
We find that all Feynman integrals (FIs), having any number of loops, can be completely determined once linear relations between FIs are provided. Therefore, FI computation is conceptually changed to a linear algebraic problem. Examples up…
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Determining arbitrary Feynman integrals by vacuum integrals Open
By introducing an auxiliary parameter, we find a new representation for\nFeynman integrals, which defines a Feynman integral by analytical continuation\nof a series containing only vacuum integrals. The new representation therefore\nconcep…
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Yangian bootstrap for conformal Feynman integrals Open
We explore the idea to bootstrap Feynman integrals using integrability. In\nparticular, we put the recently discovered Yangian symmetry of conformal\nFeynman integrals to work. As a prototypical example we demonstrate that the\nD-dimension…
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Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers Open
The reduction of a large number of scalar multi-loop integrals to the smaller set of Master Integrals is an integral part of the computation of any multi-loop amplitudes. The reduction is usually achieved by employing the traditional Integ…
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Tropical Monte Carlo quadrature for Feynman integrals Open
We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The me…
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Baikov representations, intersection theory, and canonical Feynman integrals Open
A bstract The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to d -dimensional d log-form…
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On critical exponents without Feynman diagrams Open
In order to achieve a better analytic handle on the modern conformal\nbootstrap program, we re-examine and extend the pioneering 1974 work of\nPolyakov's, which was based on consistency between the operator product\nexpansion and unitarity…
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Evaluation of the general three-loop vacuum Feynman integral Open
We discuss the systematic evaluation of 3-loop vacuum integrals with arbitrary masses. Using integration by parts, the general integral of this type can be reduced algebraically to a few basis integrals. We define a set of modified finite …
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AI Feynman: A physics-inspired method for symbolic regression Open
Our physics-inspired algorithm for symbolic regression is able to discover complex physics equations from mere tables of numbers.
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Asymptotics of Wide Networks from Feynman Diagrams Open
Understanding the asymptotic behavior of wide networks is of considerable interest. In this work, we present a general method for analyzing this large width behavior. The method is an adaptation of Feynman diagrams, a standard tool for com…
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Taming Calabi-Yau Feynman Integrals: The Four-Loop Equal-Mass Banana Integral Open
Certain Feynman integrals are associated to Calabi-Yau geometries. We demonstrate how these integrals can be computed with the method of differential equations. The four-loop equal-mass banana integral is the simplest Feynman integral whos…
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The SAGEX review on scattering amplitudes Chapter 3: Mathematical structures in Feynman integrals Open
Dimensionally-regulated Feynman integrals are a cornerstone of all perturbative computations in quantum field theory. They are known to exhibit a rich mathematical structure, which has led to the development of powerful new techniques for …
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On a procedure to derive ϵ-factorised differential equations beyond polylogarithms Open
A bstract In this manuscript, we elaborate on a procedure to derive ϵ -factorised differential equations for multi-scale, multi-loop classes of Feynman integrals that evaluate to special functions beyond multiple polylogarithms. We demonst…
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Elliptic polylogarithms and Feynman parameter integrals Open
In this paper we study the calculation of multiloop Feynman integrals that cannot be expressed in terms of multiple polylogarithms. We show in detail how certain types of two- and three-point functions at two loops, which appear in the cal…
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All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering Open
We compute the complete set of two-loop master integrals for the scattering of four massless particles and a massive one. Our results are ready for phenomenological applications, removing a major obstacle to the computation of complete nex…