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

QED contributions to electron g-2

In this paper I briefly describe the results of the numerical evaluation of the mass-independent 4-loop contribution to the electron g-2 in QED with 1100 digits of precision. In particular I also show the semi-analytical fit to the numerical value, which contains harmonic polylogarithms of eiπ/3, e2iπ/3 and eiπ/2 one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits. I give also some information about the methods and the program used

Master integrals for the NNLO virtual corrections to qqˉ→ttˉq \bar{q} \rightarrow t \bar{t} scattering in QCD: the non-planar graphs

We complete the analytic evaluation of the master integrals for the two-loop non-planar box diagrams contributing to the top-pair production in the quark-initiated channel, at next-to-next-to-leading order in QCD. The integrals are determined from their differential equations, which are cast into a canonical form using the Magnus exponential. The analytic expressions of the Laurent series coefficients of the integrals are expressed as combinations of generalized polylogarithms, which we validate with several numerical checks. We discuss the analytic continuation of the planar and the non-planar master integrals, which contribute to $q {\bar q} \to t {\bar t}$ in QCD, as well as to the companion QED scattering processes $e e \to \mu \mu$ and $e \mu \to e \mu$.Comment: 1+26 pages, 4 figures, 1 table, 3 ancillary files. v2: references added, text partly reworded, results unmodifie

Master integrals for the NNLO virtual corrections to μe\mu e scattering in QED: the non-planar graphs

We evaluate the master integrals for the two-loop non-planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to-leading order in QED. We adopt the method of differential equations and the Magnus exponential to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as a combination of generalised polylogarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at $e^+e^-$ colliders, as well as for the QCD corrections to top-pair production at hadron colliders. In particular, our results, together with the planar master integrals recently computed, represent the complete set of functions needed for the evaluation of the photonic two-loop virtual next-to-next-to-leading order QED corrections to $\mu e \to \mu e$ and $e^+ e^-\to\mu^+\mu^-$.Comment: published version, 33 pages, 3 figures, 1 ancillary file. arXiv admin note: text overlap with arXiv:1709.0743

Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss ${}_2F_1$ hypergeometric function, and the Appell $F_1$ function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to $n$-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.Comment: 115 pages, 29 figures; references added; additional examples added; matches published versio

Gravitational scattering at the seventh order in GG: nonlocal contribution at the sixth post-Newtonian accuracy

A recently introduced approach to the classical gravitational dynamics of binary systems involves intricate integrals (linked to a combination of nonlocal-in-time interactions with iterated $\frac1r$-potential scattering) which have so far resisted attempts at their analytical evaluation. By using computing techniques developed for the evaluation of multi-loop Feynman integrals (notably Harmonic Polylogarithms and Mellin transform) we show how to analytically compute all the integrals entering the nonlocal-in-time contribution to the classical scattering angle at the sixth post-Newtonian accuracy, and at the seventh order in Newton's constant, $G$ (corresponding to six-loop graphs in the diagrammatic representation of the classical scattering angle).Comment: This paper is a significantly extended version of our previous, separate arxiv submission "Gravitational dynamics at O(G^6): perturbative gravitational scattering meets experimental mathematics" e-Print: 2008.09389 [gr-qc]. The most significant extension is that we now present results at the 7th order in G. 24 pages; revtex macros used; 1 ancillary fil

Gravitational dynamics at O(G6)O(G^6): perturbative gravitational scattering meets experimental mathematics

A recently introduced approach to the gravitational dynamics of binary systems involves intricate integrals, linked to nonlocal-in-time interactions arising at the 5-loop level of classical gravitational scattering. We complete the analytical evaluation of classical gravitational scattering at the sixth order in Newton's constant, $G$, and at the sixth post-Newtonian accuracy. We use computing techniques developed for the evaluation of multi-loop Feynman integrals to obtain our results in two ways: high-precision arithmetic, yielding reconstructed analytic expressions, and direct integration {\it via} Harmonic Polylogarithms. The analytic expression of the tail contribution to the scattering involve transcendental constants up to weight four.Comment: 9 pages, no figures, one ancillary text fil

Airy beams from a microchip laser

It is theoretically shown that an end-pumped microchip laser formed by a thin laser crystal with plane-plane but slightly tilted facets can emit, under appropriate pumping conditions and near a crystal edge, a truncated self-accelerating Airy output beam.Comment: to be published in Optics Letter