11,327 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
Aspects of elliptic hypergeometric functions
General elliptic hypergeometric functions are defined by elliptic
hypergeometric integrals. They comprise the elliptic beta integral, elliptic
analogues of the Euler-Gauss hypergeometric function and Selberg integral, as
well as elliptic extensions of many other plain hypergeometric and
-hypergeometric constructions. In particular, the Bailey chain technique,
used for proving Rogers-Ramanujan type identities, has been generalized to
integrals. At the elliptic level it yields a solution of the Yang-Baxter
equation as an integral operator with an elliptic hypergeometric kernel. We
give a brief survey of the developments in this field.Comment: 15 pp., 1 fig., accepted in Proc. of the Conference "The Legacy of
Srinivasa Ramanujan" (Delhi, India, December 2012
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
Stationary problems for equation of the KdV type and dynamical -matrices.
We study a quite general family of dynamical -matrices for an auxiliary
loop algebra related to restricted flows for equations of
the KdV type. This underlying -matrix structure allows to reconstruct Lax
representations and to find variables of separation for a wide set of the
integrable natural Hamiltonian systems. As an example, we discuss the
Henon-Heiles system and a quartic system of two degrees of freedom in detail.Comment: 25pp, LaTe
Ding-Iohara-Miki symmetry of network matrix models
Ward identities in the most general "network matrix model" can be described
in terms of the Ding-Iohara-Miki algebras (DIM). This confirms an expectation
that such algebras and their various limits/reductions are the relevant
substitutes/deformations of the Virasoro/W-algebra for (q, t) and (q_1, q_2,
q_3) deformed network matrix models. Exhaustive for these purposes should be
the Pagoda triple-affine elliptic DIM, which corresponds to networks associated
with 6d gauge theories with adjoint matter (double elliptic systems). We
provide some details on elliptic qq-characters.Comment: 20 pages, 2 figure
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