Locally compact abelian p-groups revisited

Even though the structure of locally compact abelian groups is generally considered to be rather thoroughly known through a wealth of publications, one keeps encountering corners that are not elucidated in up-to-date literature. In a study of a particular class of metabelian locally compact groups (see [HHR17]) we encountered some issues about noncompact locally compact abelian groups which do not appear to be discussed in the literature even though some of them were anticipated in Braconnier's article on his local product [Bra48]. Here we treat some of them, notably some aspects of totally disconnected torsion-free locally compact abelian groups which one might consider unexpected if not pathological. However, firstly we deal with some points concerning noncompact locally compact abelian torsion groups. For compact abelian groups we often refer to the monograph [HM13]. It will be convenient to use additive notation for abelian groups

Towards e+e- --> 3 jets at NNLO by sector decomposition

A method based on sector decomposition has been developed to calculate the double real radiation part of the process e+e- to 3 jets at next-to-next-to-leading order. It is shown in an example that the numerical cancellation of soft and collinear poles works well. The method is flexible to include an arbitrary measurement function in the final Monte Carlo program, such that it allows to obtain differential distributions for different kinds of observables. This is demonstrated by showing 3-, 4- and 5-jet rates at order alpha_s^3 for a subpart of the process.Comment: 14 pages, 4 figure

Master Integrals for Massless Three-Loop Form Factors: One-Loop and Two-Loop Insertions

The three-loop form factors in massless QCD can be expressed as a linear combination of master integrals. Besides a number of master integrals which factorise into products of one-loop and two-loop integrals, one finds 16 genuine three-loop integrals. Of these, six have the form of a bubble insertion inside a one-loop or two-loop vertex integral. We compute all master integrals with these insertion topologies.Comment: 12 pages, 1 figur

Master Integrals for Fermionic Contributions to Massless Three-Loop Form Factors

In this letter we continue the calculation of master integrals for massless three-loop form factors by giving analytical results for those integrals which are relevant for the fermionic contributions proportional to N_F^2, N_F*N, and N_F/N. Working in dimensional regularisation, we express one of the integrals in a closed form which is exact to all orders in epsilon, containing Gamma-functions and hypergeometric functions of unit argument. In all other cases we derive multiple Mellin-Barnes representations from which the coefficients of the Laurent expansion in epsilon are extracted in an analytical form. To obtain the finite part of the three-loop quark and gluon form factors, all coefficients through transcendentality six in the Riemann zeta-function have to be included.Comment: 12 pages, 1 figure. References added and updated. Appendix on evaluation of Mellin-Barnes integrals added. Version to appear in PL

Two-Loop Photonic Corrections to Massive Bhabha Scattering

We describe the details of the evaluation of the two-loop radiative photonic corrections to Bhabha scattering. The role of the corrections in the high-precision luminosity determination at present and future electron-positron colliders is discussed.Comment: 20 pages, Latex; discussion, references added; to appear in Nucl.Phys.

Nine-Propagator Master Integrals for Massless Three-Loop Form Factors

We complete the calculation of master integrals for massless three-loop form factors by computing the previously-unknown three diagrams with nine propagators in dimensional regularisation. Each of the integrals yields a six-fold Mellin-Barnes representation which we use to compute the coefficients of the Laurent expansion in epsilon. Using Riemann zeta functions of up to weight six, we give fully analytic results for one integral; for a second, analytic results for all but the finite term; for the third, analytic results for all but the last two coefficients in the Laurent expansion. The remaining coefficients are given numerically to sufficiently high accuracy for phenomenological applications.Comment: 15 pages, 2 figures. Minor modifications and reference added. Matches published versio

An automatized algorithm to compute infrared divergent multi-loop integrals

We describe a constructive procedure to separate overlapping infrared divergences in multi-loop integrals. Working with a parametric representation in D=4-2*epsilon dimensions, adequate subtractions lead to a Laurent series in epsilon, where the coefficients of the pole- and finite terms are sums of regular parameter integrals which can be evaluated numerically. We fully automatized this algorithm by implementing it into algebraic manipulation programs and applied it to calculate numerically some nontrivial 2-loop 4-point and 3-loop 3-point Feynman diagrams. Finally, we discuss the applicability of our method to phenomenologically relevant multi--loop calculations such as the NNLO QCD corrections for e^+e^- --> 3 jets.Comment: 14 pages, 5 eps figures. Replaced by slightly extended, published version, typos correcte