84,326 research outputs found
Differential equations for Feynman integrals beyond multiple polylogarithms
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
Let be the triangle with vertices (1,0), (0,1), (1,1). We study certain
integrals over , 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 . 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
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
(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
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
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 . In
a special case, the simplest limit is shown to lead to a new
class of -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 and root systems and
corresponding symmetry transformations are considered. An application of the
rarefied type II elliptic hypergeometric function to some eigenvalue
problems is briefly discussed.Comment: 41 pp., corrected numeration of formula
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