9,439 research outputs found
Feynman Integrals and Intersection Theory
We introduce the tools of intersection theory to the study of Feynman
integrals, which allows for a new way of projecting integrals onto a basis. In
order to illustrate this technique, we consider the Baikov representation of
maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of
differential forms with logarithmic singularities on the boundaries of the
corresponding integration cycles. We give an algorithm for computing a basis
decomposition of an arbitrary maximal cut using so-called intersection numbers
and describe two alternative ways of computing them. Furthermore, we show how
to obtain Pfaffian systems of differential equations for the basis integrals
using the same technique. All the steps are illustrated on the example of a
two-loop non-planar triangle diagram with a massive loop.Comment: 13 pages, published versio
Densities of short uniform random walks
We study the densities of uniform random walks in the plane. A special focus
is on the case of short walks with three or four steps and less completely
those with five steps. As one of the main results, we obtain a hypergeometric
representation of the density for four steps, which complements the classical
elliptic representation in the case of three steps. It appears unrealistic to
expect similar results for more than five steps. New results are also presented
concerning the moments of uniform random walks and, in particular, their
derivatives. Relations with Mahler measures are discussed.Comment: 32 pages, 9 figure
Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle
We derive raising and lowering operators for orthogonal polynomials on the
unit circle and find second order differential and -difference equations for
these polynomials. A general functional equation is found which allows one to
relate the zeros of the orthogonal polynomials to the stationary values of an
explicit quasi-energy and implies recurrences on the orthogonal polynomial
coefficients. We also evaluate the discriminants and quantized discriminants of
polynomials orthogonal on the unit circle.Comment: 27 pages, Latex2e plus AMS packages Fix to Eqs. (2.72) and (2.74
Precise Coulomb wave functions for a wide range of complex l, eta and z
A new algorithm to calculate Coulomb wave functions with all of its arguments
complex is proposed. For that purpose, standard methods such as continued
fractions and power/asymptotic series are combined with direct integrations of
the Schrodinger equation in order to provide very stable calculations, even for
large values of |eta| or |Im(l)|. Moreover, a simple analytic continuation for
Re(z) < 0 is introduced, so that this zone of the complex z-plane does not pose
any problem. This code is particularly well suited for low-energy calculations
and the calculation of resonances with extremely small widths. Numerical
instabilities appear, however, when both |eta| and |Im(l)| are large and
|Re(l)| comparable or smaller than |Im(l)|
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