232 research outputs found
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
The parameter at three loops and elliptic integrals
We describe the analytic calculation of the master integrals required to
compute the two-mass three-loop corrections to the parameter. In
particular, we present the calculation of the master integrals for which the
corresponding differential equations do not factorize to first order. The
homogeneous solutions to these differential equations are obtained in terms of
hypergeometric functions at rational argument. These hypergeometric functions
can further be mapped to complete elliptic integrals, and the inhomogeneous
solutions are expressed in terms of a new class of integrals of combined
iterative non-iterative nature.Comment: 14 pages Latex, 7 figures, to appear in the Proceedings of "Loops and
Legs in Quantum Field Theory - LL 2018", 29 April - 4 May 2018, Po
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
Baikov-Lee Representations Of Cut Feynman Integrals
We develop a general framework for the evaluation of -dimensional cut
Feynman integrals based on the Baikov-Lee representation of purely-virtual
Feynman integrals. We implement the generalized Cutkosky cutting rule using
Cauchy's residue theorem and identify a set of constraints which determine the
integration domain. The method applies equally well to Feynman integrals with a
unitarity cut in a single kinematic channel and to maximally-cut Feynman
integrals. Our cut Baikov-Lee representation reproduces the expected relation
between cuts and discontinuities in a given kinematic channel and furthermore
makes the dependence on the kinematic variables manifest from the beginning. By
combining the Baikov-Lee representation of maximally-cut Feynman integrals and
the properties of periods of algebraic curves, we are able to obtain complete
solution sets for the homogeneous differential equations satisfied by Feynman
integrals which go beyond multiple polylogarithms. We apply our formalism to
the direct evaluation of a number of interesting cut Feynman integrals.Comment: 37 pages; v2 is the published version of this work with references
added relative to v
New Results on Massive 3-Loop Wilson Coefficients in Deep-Inelastic Scattering
We present recent results on newly calculated 2- and 3-loop contributions to
the heavy quark parts of the structure functions in deep-inelastic scattering
due to charm and bottom.Comment: Contribution to the Proc. of Loops and Legs 2016, PoS, in prin
Sunrise integral in non-relativistic QCD with elliptics
The main steps of the process of obtaining the result [1] in terms of
elliptic polylogarithms for a two-loop sunrise integral with two different
internal masses with pseudothreshold kinematics for all orders of the
dimensional regulator are shown.Comment: 7 pages, contribution to the proceedings of the International
Conference on Quantum Field Theory, High-Energy Physics, and Cosmology (July
18 - 21, 2022; Dubna, Russia
Two-Loop integrals for CP-even heavy quarkonium production and decays: Elliptic Sectors
By employing the differential equations, we compute analytically the elliptic
sectors of two-loop master integrals appearing in the NNLO QCD corrections to
CP-even heavy quarkonium exclusive production and decays, which turns out to be
the last and toughest part in the relevant calculation. The integrals are found
can be expressed as Goncharov polylogarithms and iterative integrals over
elliptic functions. The master integrals may be applied to some other NNLO QCD
calculations about heavy quarkonium exclusive production, like
, ,~and~, heavy quarkonium exclusive
decays, and also the CP-even heavy quarkonium inclusive production and decays.Comment: 23 pages, 3 figures, more discussions and references adde
- …