QCD two-loop amplitudes for e^{+} e^{−} → 3 jets: the fermionic contribution

S. Moch^{a}^{∗}, P. Uwer^{b}^{†}, and S. Weinzierl^{c}^{‡}

aDeutsches Elektronensynchrotron DESY Platanenallee 6, D–15738 Zeuthen, Germany

bInstitut für Theoretische Teilchenphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany

cDipartimento di Fisica, Università di Parma, INFN Gruppo Collegato di Parma, 43100 Parma, Italy

We review a new technique for the calculation of two-loop amplitudes and discuss as an example the fermionic contributions to e^{+} e^{−} → qq̄g.

1. Introduction

The construction of fully differential next-to-next-to-leading order (NNLO) programs is a challenging problem and urgently needed to match the accuracy reached in today’s collider experiments. While for inclusive observables like for example the total hadronic cross section in e^{+} e^{−} annihilation the step from NLO to NNLO is far less demanding – and has been taken a long time ago – the contrary is true for less inclusive quantities as for example jet rates. Essentially one has to adress two problems in jet physics at NNLO accuracy. The first one is related to the fact that one needs to treat two-loop integrales with more than one scale. These integrals are much harder to solve than one-scale integrals. The second problem which needs to be solved is the combination of virtual and real corrections – which are separately infrarot (IR) divergent – into a IR finite cross section. Given these difficulties it is clear that only for specific reactions NNLO calculations can be envisaged. Important reactions for which one should go beyond the NLO approximation are for example: Bhabha scattering, pp,pp̄→ 2 jets, and e^{+} e^{−} → 3 jets. Among these examples the e^{+} e^{−} annihilation into 3 jets is of particular interest. Historically this reaction was one of the first clean tests of Quantum Chromodynamics (QCD) as the theory of strong interaction. Today the 3-jet production in e^{+} e^{−} annihilation is still an excellent laboratory for precise tests of QCD and jet physics. In particular e^{+} e^{−} → 3 jets is also a perfect reaction for measureing the QCD coupling α_{s} with high accuracy. It is exactly here where the need for NNLO predictions for 3-jet production becomes most prominent: at present the measurements of α_{s} are plagued by scale uncertainties of the theoretical predictions [?]. These uncertainties arise from uncalculated higher order contributions. Although formally of higher order these uncertainties can still be large. They can be reduced only by a NNLO calculation. The residual scale dependence of the 3-jet rate f_{3} can be studied using the following formula

f_{3}( y, \frac{μ^{2}}{s}) = \frac{α_{s}(μ)}{2π} A(y)

+( \frac{α_{s}(μ)}{2π} ) ^{2}( B(y)+ \frac{1}{2}β_{0}ln( \frac{μ^{2}}{s}) A(y))

+( \frac{α_{s}(μ)}{2π} ) ^{3}( C(y)+ β_{0}ln( \frac{μ^{2}}{s}) B(y)

+\frac{1}{4} [β_{0}^{2}ln( \frac{μ^{2}}{s}) ^{2}+ β_{1}ln( \frac{μ^{2}}{s}) ] A(y)) . (1)

Here β_{0},β_{1} denote the coefficients of the QCD βfunction, y is the resolution of the jet algorithm and A, B, and C are the LO, NLO, NNLO coefficients at μ^{2} = s. The residual scale dependence is illustrated in fig. ?? for the Durham algorithm at y = 0.1. For the NNLO curves three different scenarios have been assumed: one scenario in which the NNLO corrections vanish for μ^{2} = s, and two scenarios where a ± 5 % correction at μ^{2} = s have been assumed. It is clearly visible that the NNLO corrections improves the residual scale dependence. For a 3-jet prediction

∗moch@ifh.de †uwer@particle.uni-karlsruhe.de (speaker)

‡stefanw@fis.unipr.it