84,326 research outputs found

    Differential equations for Feynman integrals beyond multiple polylogarithms

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    Differential equations are a powerful tool to tackle Feynman integrals. In this talk we discuss recent progress, where the method of differential equations has been applied to Feynman integrals which are not expressible in terms of multiple polylogarithms.Comment: 9 pages, talk given at RADCOR 201

    Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

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    Let TT be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over TT, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analogs of TT. This leads to a relation between certain multiple polylogarithm values and multiple zeta values.Comment: 19 pages, to appear in Mat Zametki. Ver 2.: Added Remark 3 on a Chen (Drinfeld-Kontsevich) iterated integral; simplified Proposition 2; gave reference for (19); corrected [16]; fixed typ

    SV-map between Type I and Heterotic Sigma Models

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    The scattering amplitudes of gauge bosons in heterotic and open superstring theories are related by the single-valued projection which yields heterotic amplitudes by selecting a subset of multiple zeta value coefficients in the α′\alpha' (string tension parameter) expansion of open string amplitudes. In the present work, we argue that this relation holds also at the level of low-energy expansions (or individual Feynman diagrams) of the respective effective actions, by investigating the beta functions of two-dimensional sigma models describing world-sheets of open and heterotic strings. We analyze the sigma model Feynman diagrams generating identical effective action terms in both theories and show that the heterotic coefficients are given by the single-valued projection of the open ones. The single-valued projection appears as a result of summing over all radial orderings of heterotic vertices on the complex plane representing string world-sheet.Comment: 28 page

    Skorohod and Stratonovich integration in the plane

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    This article gives an account on various aspects of stochastic calculus in the plane. Specifically, our aim is 3-fold: (i) Derive a pathwise change of variable formula for a path indexed by a square, satisfying some H\"older regularity conditions with a H\"older exponent greater than 1/3. (ii) Get some Skorohod change of variable formulas for a large class of Gaussian processes defined on the suqare. (iii) Compare the bidimensional integrals obtained with those two methods, computing explicit correction terms whenever possible. As a byproduct, we also give explicit forms of corrections in the respective change of variable formulas

    Rarefied elliptic hypergeometric functions

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    Two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved. They generalize evaluations of the type I and II elliptic beta integrals attached to the root system CnC_n. In a special n=1n=1 case, the simplest p→0p\to 0 limit is shown to lead to a new class of qq-hypergeometric identities. Symmetries of a rarefied elliptic analogue of the Euler-Gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed. Some extensions of the latter function to CnC_n and AnA_n root systems and corresponding symmetry transformations are considered. An application of the rarefied type II CnC_n elliptic hypergeometric function to some eigenvalue problems is briefly discussed.Comment: 41 pp., corrected numeration of formula
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