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 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

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

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

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

High-precision calculation of the 4-loop contribution to the electron g-2 in QED

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

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

Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D

We show that dimensional recurrence relation and analytical properties of the loop integrals as functions of complex variable $\mathcal{D}$ (space-time dimensionality) provide a regular way to derive analytical representations of loop integrals. The representations derived have a form of exponentially converging sums. Several examples of the developed technique are given.Comment: Several misprints correcte

Analytic evaluation of Feynman graph integrals

We review the main steps of the differential equation approach to the analytic evaluation of Feynman graphs, showing at the same time its application to the 3-loop sunrise graph in a particular kinematical configuration.Comment: 5 pages, 1 figure, uses npb.sty. Presented at RADCOR 2002 and Loops and Legs in Quantum Field Theory, 8-13 September 2002, Kloster Banz, Germany. Revised version: minor typos corrected, one reference adde

The muon anomalous magnetic moment in QED: three-loop electron and tau contributions

We present an analytic calculation of electron and tau O(alpha^3) loop effects on the muon anomalous magnetic moment. Computation of such three-loop diagrams with three mass scales is possible using asymptotic and eikonal expansions. An evaluation of a new type of eikonal integrals is presented in some detail.Comment: 9 pages, late