Analytical and Numerical Contributions of Some Tenth-Order Graphs Containing Vacuum Polarization Insertions to the Muon (G-2) in QED

The contributions to the g-2 of the muon from some tenth-order (five-loop) graphs containing one-loop and two-loop vacuum polarization insertions have been evaluated analytically in QED perturbation theory, expanding the results in the ratio of the electron to muon mass (m_e /m_\mu). Some results contain also terms known only in numerical form. Our results agree with the renormalization group results already existing in the literature.Comment: 13 pages + 2 figures appended as 2 postscript files, plain TeX, DFUB 94-0

High-precision calculation of multi-loop Feynman integrals by difference equations

We describe a new method of calculation of generic multi-loop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace's transformation. We also describe new algorithms for the identification of master integrals and the reduction of generic Feynman integrals to master integrals, and procedures for generating and solving systems of differential equations in masses and momenta for master integrals. We apply our method to the calculation of the master integrals of massive vacuum and self-energy diagrams up to three loops and of massive vertex and box diagrams up to two loops. Implementation in a computer program of our approach is described. Important features of the implementation are: the ability to deal with hundreds of master integrals and the ability to obtain very high precision results expanded at will in the number of dimensions.Comment: 55 pages, 5 figures, LaTe

Calculation of master integrals by difference equations

In this paper we describe a new method of calculation of master integrals based on the solution of systems of difference equations in one variable. An explicit example is given, and the generalization to arbitrary diagrams is described. As example of application of the method, we have calculated the values of master integrals for single-scale massive three-loop vacuum diagrams, three-loop self-energy diagrams, two-loop vertex diagrams and two-loop box diagrams.Comment: 7 pages, 1 figure, LaTex, to be published in Physics Letters

Calculation of Feynman integrals by difference equations

In this paper we describe a method of calculation of master integrals based on the solution of systems of difference equations in one variable. Various explicit examples are given, as well as the generalization to arbitrary diagrams.Comment: LaTex, 10 pages, uses appolb.cls. Presented at the XXVII International Conference of Theoretical Physics "Matter to the Deepest", Ustron, Poland, 15-21 September 2003. To appear in Acta Physica Polonica. v2:added reference

High-precision e-expansions of massive four-loop vacuum bubbles

In this paper we calculate at high-precision the expansions in e=(4-D)/2 of the master integrals of 4-loop vacuum bubble diagrams with equal masses, using a method based on the solution of systems of difference equations. We also show that the analytical expression of a related on-shell 3-loop self-mass master integral contains new transcendental constants made up of complete elliptic integrals of first and second kind.Comment: 7 pages, 2 figures, LaTex, to be published in Physics Letters

The Analytical Contribution of Some Eighth-Order Graphs Containing Vacuum Polarization Insertions to the Muon (G-2) in QED

The contributions to the $g-2$ of the muon from some eighth-order (four-loop) graphs containing one-loop and two-loop vacuum polarization insertions have been evaluated analytically in QED perturbation theory, expanding the results in the ratio of the electron to muon mass ${(m_e / m_\mu)}$. The results agree with the numerical evaluations and the asymptotic analytical results already existing in the literature.Comment: plain TEX, 10 pages + 3 figures (figures are available upon request), DFUB 93-0

Analytic treatment of the two loop equal mass sunrise graph

The two loop equal mass sunrise graph is considered in the continuous d-dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d=2 and d=4, the second order differential equation for the scalar Master Integral is expanded in (d-2) and solved by the variation of the constants method of Euler up to first order in (d-2) included. That requires the knowledge of the two independent solutions of the associated homogeneous equation, which are found to be related to the complete elliptic integrals of the first kind of suitable arguments. The behaviour and expansions of all the solutions at all the singular points of the equation are exhaustively discussed and written down explicitly.Comment: 33 pages, LaTeX, v2: +1 figure; v3: changes in the conclusions; simplifications in the recurrences (6.3) and (6.9

The Export Trading Company Act of 1982 and the photovoltaics industry: An assessment

The potential advantages of recent export promotion legislation for the U.S. photovoltaics industry were assessed. The provisions of the Export Trading Company Act of 1982 were reviewed and the export trade sector was surveyed to determine what impact the Act is haviang on export company activity. The photovoltaics industry was then studied to determine whether the Act offers particular advantages for promoting its product overseas

Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals

In this paper we continue the work begun in 2002 on the identification of the analytical expressions of Feynman integrals which require the evaluation of multiple elliptic integrals. We rewrite and simplify the analytical expression of the 3-loop self-mass integral with three equal masses and on-shell external momentum. We collect and analyze a number of results on double and triple elliptic integrals. By using very high-precision numerical fits, for the first time we are able to identify a very compact analytical expression for the 4-loop on-shell self-mass integral with 4 equal masses, that is one of the master integrals of the 4-loop electron g-2. Moreover, we fit the analytical expressions of some integrals which appear in lattice perturbation theory, and in particular the 4-dimensional generalized Watson integral.Comment: 13 pages, 1 figure, LaTex; v2: some rephrasing of text; v3: reference added, minor modifications; v4: checks of lattice integrals up to 2400 digits; some modifications of text; version accepted for publication in IJMPA, needs the document class ws-ijmpa.cl

High-precision e-expansions of three-loop master integrals contributing to the electron g-2 in QED

In this paper we calculate at high-precision the Laurent expansions in e=(4-D)/2 of the 17 master integrals which appeared in the analytical calculation of 3-loop QED contribution to the electron g-2, using difference and differential equations. The coefficients of the expansions so obtained are in perfect agreement with all the analytical expressions already known. The values of coefficients not previously known will be used in the high-precision calculation of the 4-loop QED contribution to the electron g-2