On Nielsen’s generalized polylogarithms and their numerical calculation
TL;DR: In this article, the generalized polylogarithms of Nielsen are studied, in particular their functional relations, and an Algol procedure for calculating 10 functions of lowest order is presented.
Abstract: The generalized polylogarithms of Nielsen are studied, in particular their functional relations. New integral expressions are obtained, and relations for function values of particular arguments are given. An Algol procedure for calculating 10 functions of lowest order is presented. The numerical values of the Chebyshev coefficients used in this procedure are tabulated. A table of the real zeros of these functions is also given.
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04 May 1999
TL;DR: The harmonic polylogarithms (hpl's) as mentioned in this paper are a generalization of Nielsen's poly logarithm, satisfying a product algebra (the product of two hpl's is in turn a combination of hpls) and forming a set closed under the transformation of the arguments x=1/z and x=(1-t)/(1+t)
Abstract: The harmonic polylogarithms (hpl's) are introduced They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the transformation of the arguments x=1/z and x=(1-t)/(1+t) The coefficients of their expansions and their Mellin transforms are harmonic sums
1,100 citations
TL;DR: A constructive procedure to separate overlapping infrared divergences in multi-loop integrals by implementing it into algebraic manipulation programs and applying it to calculate numerically some nontrivial 2-loop 4-point and 3-loop 3-point Feynman diagrams.
551 citations
TL;DR: Algorithms, which allow the evaluation for arbitrary complex arguments and without any restriction on the weight are provided, which are implemented with arbitrary precision arithmetic in C++ within the GiNaC framework.
515 citations
TL;DR: In this article, a subtraction scheme relevant for NNLO perturbative calculations in e+e−→ jets is presented, where antenna functions derived from the matrix elements for tree-level 1→3 and 1→4 and one-loop 1-3 processes are constructed.
Abstract: The computation of exclusive QCD jet observables at higher orders requires a method for the subtraction of infrared singular configurations arising from multiple radiation of real partons. We present a subtraction scheme relevant for NNLO perturbative calculations in e+e−→ jets. The building blocks of the scheme are antenna functions derived from the matrix elements for tree-level 1→3 and 1→4 and one-loop 1→3 processes. By construction, these building blocks have the correct infrared behaviour when one or two particles are unresolved. At the same time, their integral over the antenna phase space is straightforward. As an example of how to use the scheme we compute the NNLO contributions to the subleading colour QED-like contribution to e+e−→3 jets. To illustrate the application of NNLO antenna subtraction for different colour structures, we construct the integrated forms of the subtraction terms needed for the five-parton and four-parton contributions to e+e−→3 jets at NNLO in all colour factors, and show that their infrared poles cancel analytically with the infrared poles of the two-loop virtual correction to this observable.
496 citations
TL;DR: In this paper, the authors derived the DGLAP and BFKL evolution equations in the N = 4 supersymmetric gauge theory in the next-to-leading approximation.
423 citations
References
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01 Jan 1966
3,150 citations
TL;DR: In this article, the authors derived cross sections for several processes involving electromagnetic fields in a nonlinear manner from the electrodynamic scattering matrix and expressed in terms of the fourth-order nonlinear vacuum polarization tensor.
Abstract: Cross sections for several processes involving electromagnetic fields in a nonlinear manner are derived from the electrodynamic scattering matrix and are expressed in terms of the fourth-order nonlinear vacuum polarization tensor. The differential cross section for the scattering of light by light is calculated as a function of energy and angle. Numerical values are given for scattering at zero and at ninety degrees in the center-of-mass system. Near 1.75 Mev the forward scattering cross section has its largest value of 4.1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}31}$ ${\mathrm{cm}}^{2}$/sterad, while the maximum right-angle scattering takes place near 0.7 Mev with a cross section of 2.8\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}31}$ ${\mathrm{cm}}^{2}$/sterad, all for unpolarized radiation. Numerical results are also given for scattering at the above angles between circularly polarized states. The conclusions in this paper agree with all the results previously calculated for special cases.
253 citations
"On Nielsen’s generalized polylogari..." refers background in this paper
...Some times the Spence function for n= 2 as defined in Eq. (2.3) is called the dilogarithm [8] and viceversa [ 9 ]....
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TL;DR: In this article, a more complete account is given of an estimation, by rigourous upper and lower bounds, of the fourth order magnetic moment of the electron which has already been published by the author.
Abstract: A more complete account is given of an estimation, by rigourous upper and lower bounds, of the fourth order magnetic moment of the electron which has already been published by the author. The method is illustrated for one typical term. Then the results of a more elaborate estimation are given. In the terminology used by Karplus and Kroll, the term symbolized by the diagram IIc is found to satisfy μ(II) c =(−0.60±0.11) α 2 π 2 − 1 2 α 2 π 2 log λ 2 m 2 , in contradiction with the value −3.18 α 2 π 2 − 1 2 α 2 π 2 log λ 2 m 2 given by the previous authors. It has been found that the other terms are not in disagreement with Karplus and Kroll values. If one accepts them, then the total fourth order moment amounts to μ(4) = (−0.39±0.11) (α2/π2). One notes finally that this new value increases the Lamb shift by (0.82±0.03) Mc/s and brings the fine structure constant to 1/α = 137.0384.
167 citations
TL;DR: The covariant S matrix formalism of Dyson has been applied to the calculation of the fourth-order radiative correction to the magnetic moment of the electron in this paper, and the results for the covariant Δ-functions which describe the interaction of virtual electrons and photons with the vacuum are given to order α.
Abstract: The covariant S matrix formalism of Dyson has been applied to the calculation of the fourth-order radiative correction to the magnetic moment of the electron. Intermediate results for the covariant Δ-functions which describe the interaction of virtual electrons and photons with the vacuum are given to order α. The addition to the magnetic moment to order α2 is found to be finite after the charge of the electron is renormalized consistently. This correction amounts to −2.97α2π2 Bohr magneton so that the magnetic moment of the electron is μ=1.001147 Bohr magnetons.
142 citations
TL;DR: The covariant $S$ matrix formalism of Dyson has been applied to the calculation of the fourth-order radiative correction to the magnetic moment of the electron as discussed by the authors.
Abstract: The covariant $S$ matrix formalism of Dyson has been applied to the calculation of the fourth-order radiative correction to the magnetic moment of the electron. Intermediate results for the covariant $\ensuremath{\Delta}$-functions which describe the interaction of virtual electrons and photons with the vacuum are given to order $\ensuremath{\alpha}$. The addition to the magnetic moment to order ${\ensuremath{\alpha}}^{2}$ is found to be finite after the charge of the electron is renormalized consistently. This correction amounts to $\frac{\ensuremath{-}2.97{\ensuremath{\alpha}}^{2}}{{\ensuremath{\pi}}^{2}}$ Bohr magneton so that the magnetic moment of the electron is $\ensuremath{\mu}=1.001147$ Bohr magnetons.
133 citations
"On Nielsen’s generalized polylogari..." refers background in this paper
...In the past, special cases of S,,,~(x) were discussed and some of their typical properties (functional relations) rediscovered by people working in the field mentioned [ 3 ], [4], [5]....
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