6,909 research outputs found
Numerical computation of real or complex elliptic integrals
Algorithms for numerical computation of symmetric elliptic integrals of all
three kinds are improved in several ways and extended to complex values of the
variables (with some restrictions in the case of the integral of the third
kind). Numerical check values, consistency checks, and relations to Legendre's
integrals and Bulirsch's integrals are included
Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
We calculate 3-loop master integrals for heavy quark correlators and the
3-loop QCD corrections to the -parameter. They obey non-factorizing
differential equations of second order with more than three singularities,
which cannot be factorized in Mellin- space either. The solution of the
homogeneous equations is possible in terms of convergent close integer power
series as Gau\ss{} hypergeometric functions at rational argument. In
some cases, integrals of this type can be mapped to complete elliptic integrals
at rational argument. This class of functions appears to be the next one
arising in the calculation of more complicated Feynman integrals following the
harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic
polylogarithms, square-root valued iterated integrals, and combinations
thereof, which appear in simpler cases. The inhomogeneous solution of the
corresponding differential equations can be given in terms of iterative
integrals, where the new innermost letter itself is not an iterative integral.
A new class of iterative integrals is introduced containing letters in which
(multiple) definite integrals appear as factors. For the elliptic case, we also
derive the solution in terms of integrals over modular functions and also
modular forms, using -product and series representations implied by Jacobi's
functions and Dedekind's -function. The corresponding
representations can be traced back to polynomials out of Lambert--Eisenstein
series, having representations also as elliptic polylogarithms, a -factorial
, logarithms and polylogarithms of and their -integrals.
Due to the specific form of the physical variable for different
processes, different representations do usually appear. Numerical results are
also presented.Comment: 68 pages LATEX, 10 Figure
On a computer-aided approach to the computation of Abelian integrals
An accurate method to compute enclosures of Abelian integrals is developed.
This allows for an accurate description of the phase portraits of planar
polynomial systems that are perturbations of Hamiltonian systems. As an
example, it is applied to the study of bifurcations of limit cycles arising
from a cubic perturbation of an elliptic Hamiltonian of degree four
Large scale analytic calculations in quantum field theories
We present a survey on the mathematical structure of zero- and single scale
quantities and the associated calculation methods and function spaces in higher
order perturbative calculations in relativistic renormalizable quantum field
theories.Comment: 25 pages Latex, 1 style fil
Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
We show that for a class of two-loop diagrams, the on-shell part of the
integration-by-parts (IBP) relations correspond to exact meromorphic one-forms
on algebraic curves. Since it is easy to find such exact meromorphic one-forms
from algebraic geometry, this idea provides a new highly efficient algorithm
for integral reduction. We demonstrate the power of this method via several
complicated two-loop diagrams with internal massive legs. No explicit elliptic
or hyperelliptic integral computation is needed for our method.Comment: minor changes: more references adde
Lie point symmetries and ODEs passing the Painlev\'e test
The Lie point symmetries of ordinary differential equations (ODEs) that are
candidates for having the Painlev\'e property are explored for ODEs of order . Among the 6 ODEs identifying the Painlev\'e transcendents only
, and have nontrivial symmetry algebras and that only
for very special values of the parameters. In those cases the transcendents can
be expressed in terms of simpler functions, i.e. elementary functions,
solutions of linear equations, elliptic functions or Painlev\'e transcendents
occurring at lower order. For higher order or higher degree ODEs that pass the
Painlev\'e test only very partial classifications have been published. We
consider many examples that exist in the literature and show how their symmetry
groups help to identify those that may define genuinely new transcendents
High-precision calculation of the 4-loop contribution to the electron g-2 in QED
I have evaluated up to 1100 digits of precision the contribution of the 891
4-loop Feynman diagrams contributing to the electron - in QED. The total
mass-independent 4-loop contribution is . I have fit a semi-analytical expression to
the numerical value. The expression contains harmonic polylogarithms of
argument , , ,
one-dimensional integrals of products of complete elliptic integrals and six
finite parts of master integrals, evaluated up to 4800 digits.Comment: 14 pages, 3 figures, 3 tables v2: version published in PRL (specified
"mass-independent contribution", figure 2 reformatted
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