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We present a complete reevaluation of the irreducible two-loop vacuum-polarization correction to the photon propagator in quantum electrodynamics, i.e. with an electron-positron pair in the fermion propagators. The integration is carried out by reducing the integrations to a limited set of master integrals, which are calculated using integration-by-parts identities. Dimensional regularization is used in D = 4-2*epsilon dimensions, and on-mass shell renormalization is employed. The one-loop effect is given to order epsilon, to be combined with the 1/epsilon divergence of the two-loop amplitude. Master integrals are given. Final evaluations of two-loop energy shifts for 1S, 2S, and 2P states are done analytically, and results are presented, with an emphasis on muonic hydrogen. For relativistic Dirac-Coulomb reference states, higher-order coefficients are obtained for the Zalpha-expansion. We compare the results obtained to the existing literature., Comment: 17 pages; RevTeX; see Physical Review D for Supplementary Material

The mini-proceedings of the STRONG2020 Virtual Workshop "Space-like and Time-like determination of the Hadronic Leading Order contribution to the Muon $g-2$", November 24--26 2021, are presented. This is the first workshop of the STRONG2020 WP21: JRA3-PrecisionSM: Precision Tests of the Standard Model (http://www.strong-2020.eu/joint-research-activity/jra3-precisionsm.html). The workshop was devoted to review of the working group activitity on: $(\it i)$ Radiative Corrections and Monte Carlo tools for low-energy hadronic cross sections in $e^+ e^-$ collisions; $(\it ii)$ Annotated database for $e^+e^-$ into hadrons processes at low energy; $(\it iii)$ Radiative Corrections and Monte Carlo tools for $\mu$-$e$ elastic scattering., Comment: 43 pages, 21 contributions. Editors: A. Kup\'s\'c and G. Venanzoni

We review the current status of the theory predictions for elastic $\mu$-$e$ scattering, describing the recent activities and future plans of the theory initiative related to the proposed MUonE experiment., Comment: 23 pages, 14 figures. Invited review for EPJ-C, identical to published version

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.cls

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 4-loop sunrise graph with two massless lines, two lines of equal mass M and a line of mass m, for external invariant timelike and equal to m^2 is considered. We write differential equations in x=m/M for the Master Integrals of the problem, which we Laurent-expand in the regularizing continuous dimension d around d=4, and then solve exactly in x up to order (d-4)^3 included; the result is expressed in terms of Harmonic PolyLogarithms of argument x and maximum weight 7. As a by product, we obtain the x=1 value, expected to be relevant in QED 4-loop static quantities like the electron (g-2). The analytic results were checked by an independent precise numerical calculation, Comment: 18 pages, 3 figures; minor changes into the Introduction; references added; accepted by Nucl. Phys. B

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 references

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., Comment: 7 pages, 1 figure, LaTex, to be published in Physics Letters B

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, LaTex

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 B

The general lines of the derivation and the main properties of the master equations for the master amplitudes associated to a given Feynman graph are recalled. Some results for the 2-loop self-mass graph with 4 propagators are then presented., Comment: 11 pages, Latex, 1 encapsulated Postscript figure with epsfig, some references added; Talk presented at the Zeuthen Workshop on Elementary Particle Physics - Loops and Legs in Gauge Theories - Rheinsberg, 19-24 April 1998

The master differential equations in the external square momentum p^2 for the master integrals of the two-loop sunrise graph, in n-continuous dimensions and for arbitrary values of the internal masses, are derived. The equations are then used for working out the values at p^2 = 0 and the expansions in p^2 at p^2 =0, in (n-4) at n to 4 limit and in 1/p^2 for large values of p^2 ., Comment: 28 pages, Latex, 2 encapsulated Postscript figures with epsfig, mistype corrected in Eq.(56)

We have evaluated in closed analytical form the contribution of the three-loop non-planar `triple-cross' diagrams contributing to the electron (g-2) in QED; its value, omitting the already known infrared divergent part, is a_e(3-cross) = 1/2 pi^2 Z(3) - 55/12 Z(5) - 16/135 pi^4 + 32/3 (a4 + 1/24 ln(2)^4) + 14/9 pi^2 ln(2)^2 - 1/3 Z(3) + 23/3 pi^2 ln(2) - 47/9 pi^2 - 113/48. This completes the analytical evaluation of the (g-2) at order alpha^3, giving a_e(3-loop) = (alpha/pi)^3 { 83/72 pi^2 Z(3) - 215/24 Z(5) + 100/3 [( a4 + 1/24 ln(2)^4 ) - 1/24 pi^2 ln(2)^2 ] - 239/2160 pi^4 + 139/18 Z(3) - 298/9 pi^2 ln(2) + 17101/810 pi^2 + 28259/5184 } = (alpha/pi)^3 (1.181241456...)., Comment: plain TeX, 16 pages, 2 figures. Submitted to Physics Letters B. Two postscript files (z2.ps, zx1gen.ps) are included

The contributions to the (g-2) of the electron from the corner-ladder graphs in sixth-order (three-loop) QED perturbation theory are evaluated in closed analytical form. The results obtained are in excellent agreement with the most precise numerical evaluations already existing in the literature. Our results allows one to reduce the numerical uncertainty of the theoretical determination of the g-2 of the electron., Comment: 10 pages + 2 figures available upon request, plain Tex, DFUB 94-18

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 94-02

We have extended the hyperspherical variables method to the analytical calculation of the angular integral of the box graph. We discuss the applications of our results to the analytical calculation of the QED contribution to the electron g-2 of the set of three-loop triple-cross vertex graphs., Comment: plain TEX, 11 pages + 2 figures (appended as postscript file), DFUB 94-01

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 93-06