6,909 research outputs found

    Numerical computation of real or complex elliptic integrals

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    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

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    We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the ρ\rho-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-NN space either. The solution of the homogeneous equations is possible in terms of convergent close integer power series as 2F1_2F_1 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 qq-product and series representations implied by Jacobi's ϑi\vartheta_i functions and Dedekind's η\eta-function. The corresponding representations can be traced back to polynomials out of Lambert--Eisenstein series, having representations also as elliptic polylogarithms, a qq-factorial 1/ηk(τ)1/\eta^k(\tau), logarithms and polylogarithms of qq and their qq-integrals. Due to the specific form of the physical variable x(q)x(q) 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

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    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

    Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves

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    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

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    The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlev\'e property are explored for ODEs of order n=2,,5n =2, \dots ,5. Among the 6 ODEs identifying the Painlev\'e transcendents only PIIIP_{III}, PVP_V and PVIP_{VI} 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

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    I have evaluated up to 1100 digits of precision the contribution of the 891 4-loop Feynman diagrams contributing to the electron gg-22 in QED. The total mass-independent 4-loop contribution is ae=1.912245764926445574152647167439830054060873390658725345(απ)4 a_e = -1.912245764926445574152647167439830054060873390658725345{\ldots} \left(\frac{\alpha}{\pi}\right)^4. I have fit a semi-analytical expression to the numerical value. The expression contains harmonic polylogarithms of argument eiπ3e^{\frac{i\pi}{3}}, e2iπ3e^{\frac{2i\pi}{3}}, eiπ2e^{\frac{i\pi}{2}}, 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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