T. Gehrmann

Differential equations for two loop four point functions

At variance with fully inclusive quantities, which have been computed already at the two- or three-loop level, most exclusive observables are still known only at one loop, as further progress was hampered so far by the greater computational problems encountered in the study of multi-leg amplitudes beyond one loop. We show in this paper how the use of tools already employed in inclusive calculations can be suitably extended to the computation of loop integrals appearing in the virtual corrections to exclusive observables, namely two-loop four-point functions with massless propagators and up to one off-shell leg. We find that multi-leg integrals, in addition to integration-by-parts identities, obey also identities resulting from Lorentz-invariance. The combined set of these identities can be used to reduce the large number of integrals appearing in an actual calculation to a small number of master integrals. We then write down explicitly the differential equations in the external invariants fulfilled by these master integrals, and point out that the equations can be used as an efficient method of evaluating the master integrals themselves. We outline strategies for the solution of the differential equations, and demonstrate the application of the method on several examples.

Antenna subtraction at NNLO

The computation of exclusive QCD jet observables at higher orders requires a method for the subtraction of infrared singular configurations arising from multiple radiation of real partons. We present a subtraction scheme relevant for NNLO perturbative calculations in e+e−→ jets. The building blocks of the scheme are antenna functions derived from the matrix elements for tree-level 1→3 and 1→4 and one-loop 1→3 processes. By construction, these building blocks have the correct infrared behaviour when one or two particles are unresolved. At the same time, their integral over the antenna phase space is straightforward. As an example of how to use the scheme we compute the NNLO contributions to the subleading colour QED-like contribution to e+e−→3 jets. To illustrate the application of NNLO antenna subtraction for different colour structures, we construct the integrated forms of the subtraction terms needed for the five-parton and four-parton contributions to e+e−→3 jets at NNLO in all colour factors, and show that their infrared poles cancel analytically with the infrared poles of the two-loop virtual correction to this observable.

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

NNLO corrections to event shapes in e+ e- annihilation

\e=(4-d)/2

Infrared structure of e+ e- ---> 2 jets at NNLO

, where

Calculation of the quark and gluon form factors to three loops in QCD

d

W+ W- production at hadron colliders in next to next to leading order QCD.

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.

Antenna subtraction with hadronic initial states

We compute the next-to-next-to-leading order (NNLO) QCD corrections to the six most important event shape variables related to three-particle final states in electron- positron annihilation. The corrections are sizeable for all variables, however their magni- tude is substantially different for different observables. We observe that the NNLO correc- tions yield a considerably better agreement between theory and experimental data both in shape and normalisation of the event shape distributions. The renormalisation scale de- pendence of the theoretical prediction is substantially reduced compared to the previously existing NLO results. Our results will allow a precise determination of the strong coupling constant from event shape data collected at LEP.

High precision determination of the gluon fusion Higgs boson cross-section at the LHC

Abstract The production of two jets is the simplest exclusive quantum chromodynamics process in electron–positron annihilation. Using this process, we examine the structure of next-to-next-to-leading order (NNLO) corrections to jet production observables. We derive a subtraction formalism including double real radiation at tree level and single real radiation at one loop. For two-jet production, these subtraction terms coincide with the full matrix elements, thus highlighting the phase space structure of the subtraction procedure. We then analytically compute the infrared singularities arising from each partonic channel. For the purely virtual (two-parton) NNLO corrections, these take the well known form predicted by Catanis infrared factorization formula. We demonstrate that individual terms in the infrared factorization formula can be identified with infrared singular terms from three- and four-parton final states, leaving only single poles and a contribution from the one-loop soft gluon current, which subsequently cancels between the three- and four-parton final states. Summing over all different final states, we observe an explicit cancellation of all infrared poles and recover the known two-loop correction to the hadronic R -ratio.

Second-order QCD corrections to the thrust distribution in electron-positron annihilation.

We describe the calculation of the three-loop QCD corrections to quark and gluon form factors. The relevant three-loop Feynman diagrams are evaluated and the resulting three-loop Feynman integrals are reduced to a small set of known master integrals by using integration-by-parts relations. Our calculation confirms the recent results by Baikov et al. for the three-loop form factors. In addition, we derive the subleading