E. Remiddi

The analytical value of the electron (g − 2) at order α3 in QED

Abstract 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 π 2 e(3)− 55 12 e(5)− 16 135 πM 4 + 32 3 as+ 1 24 1n 4 2 + 14 9 π 2 1n 2 2− 1 3 e(3)+ 23 3 π 2 1n2− 47 9 π 2 − 113 48 . This completes the analytical evaluation of the (g−2) at order α3, giving a e (3−cross)= α π 3 83 72 π 2 e(3)− 215 24 e(5)+ 100 3 as+ 1 24 π 2 1n 2 2 − 1 24 π 2 1n 2 2 − 239 2160 π 4 + 139 18 e(3)− 298 9 π 2 1n2+ 17101 810 π 2 + 28259 5184 = α π 3 (1.181241456…).

Two-loop master integrals for γ∗→3 jets: the planar topologies

The calculation of the two-loop corrections to the three jet production rate and to event shapes in electron-positron annihilation requires the computation of a number of up to now unknown two-loop four-point master integrals with one off-shell and three on-shell legs. In this paper, we compute those master integrals which correspond to planar topologies by solving differential equations in the external invariants which are fulfilled by the master integrals. We obtain the master integrals as expansions in

Differential equations for Feynman graph amplitudes

\e=(4-d)/2

Numerical evaluation of harmonic polylogarithms

, where

The two-loop QCD matrix element for e^+e^->3 jets.

d

Numerical evaluation of two-dimensional harmonic polylogarithms

is the space-time dimension. The results are expressed in terms of newly introduced two-dimensional harmonic polylogarithms, whose properties are shortly discussed. For all two-dimensional harmonic polylogarithms appearing in the divergent parts of the integrals, expressions in terms of Nielsens polylogarithms are given. The analytic continuation of our results to other kinematical situations is outlined.

Electron form factors up to fourth order. - II@@@Электронные форм-факторы с точностью вплоть до четвертого порядка. II

SummaryIt is by now well established that, by means of the integration by part identities, all the integrals occurring in the evaluation of a Feynman graph of given topology can be expressed in terms of a few independent master integrals. It is shown in this paper that the integration by part identities can be further used for obtaining a linear system of first-order differential equations for the master integrals themselves. The equations can then be used for the numerical evaluation of the amplitudes as well as for investigating their analytic properties, such as the asymptotic and threshold behaviours and the corresponding expansions (and for analytic integration purposes, when possible). The new method is illustrated through its somewhat detailed application to the case of the one-loop self-mass amplitude, by explicitly working out expansions and quadrature formulas, both in arbitrary continuous dimensionn and in then→4 limit. It is then shortly discussed which features of the new method are expected to work in the more general case of multi-point, multi-loop amplitudes.

Two-loop QCD helicity amplitudes for e+e−→3 jets

Harmonic polylogarithms H(a→;x), a generalization of Nielsens polylogarithms Sn,p(x), appear frequently in analytic calculations of radiative corrections in quantum field theory. We present an algorithm for the numerical evaluation of harmonic polylogarithms of arbitrary real argument. This algorithm is implemented into a FORTRAN subroutine hplog to compute harmonic polylogarithms up to weight 4.

Two-loop master integrals for γ∗→3 jets: the non-planar topologies

Abstract We compute the O (α s 3 ) virtual QCD corrections to the γ ∗ →q q g matrix element arising from the interference of the two-loop with the tree-level amplitude and from the self-interference of the one-loop amplitude. The calculation is performed by reducing all loop integrals appearing in the two-loop amplitude to a small set of known master integrals. Infrared and ultraviolet divergences are both regularized using conventional dimensional regularization, and the ultraviolet renormalization is performed in the MS scheme. The infrared pole structure of the matrix elements agrees with the prediction made by Catani using an infrared factorization formula. The analytic result for the finite terms of both matrix elements is expressed in terms of one- and two-dimensional harmonic polylogarithms.We compute the O(α s ) virtual QCD corrections to the γ∗ → qq̄g matrix element arising from the interference of the two-loop with the tree-level amplitude and from the self-interference of the one-loop amplitude. The calculation is performed by reducing all loop integrals appearing in the two-loop amplitude to a small set of known master integrals. Infrared and ultraviolet divergences are both regularized using conventional dimensional regularization, and the ultraviolet renormalization is performed in the MS scheme. The infrared pole structure of the matrix elements agrees with the prediction made by Catani using an infrared factorization formula. The analytic result for the finite terms of both matrix elements is expressed in terms of oneand two-dimensional harmonic polylogarithms.

Strong Radiative Corrections to Annihilations of Quarkonia in QCD

The two-dimensional harmonic polylogarithms G(a→(z);y), a generalization of the harmonic polylogarithms, themselves a generalization of Nielsens polylogarithms, appear in analytic calculations of multi-loop radiative corrections in quantum field theory. We present an algorithm for the numerical evaluation of two-dimensional harmonic polylogarithms, with the two arguments y,z varying in the triangle 0⩽y⩽1, 0⩽z⩽1, 0⩽(y+z)⩽1. This algorithm is implemented into a FORTRAN subroutine tdhpl to compute two-dimensional harmonic polylogarithms up to weight 4.