Nonfactorizable QCD and Electroweak Corrections to the Hadronic Z Boson Decay Rate.

We present an analysis of two-loop mixed QCD and electroweak corrections to the decay of the Z boson into light quarks. We find that the naive factorization of QCD and electroweak corrections does not describe correctly the two-loop effects. The nonfactorizable corrections shift the width of the Z boson by approximately −0.59(3) MeV and increase the central value of the strong coupling constant determined at LEP by 0.001. With a total number of about four million hadronic Z decays collected at each of the four LEP experiments a final precision in the Z boson decay rate of about 2 MeV is expected. At the same time the ratio Rhad = Γ(Z → hadrons)/Γ(Z → μ μ) will be measured with an accuracy of about 0.1 per cent. These measurements will determine αs with a remaining uncertainty of about δαs = ±3× 10 . This puts not only severe constraints on the experimental analysis but also on the theoretical understanding of subtle higher order effects. QCD corrections have been calculated up to order α s [1] and leading as well as subleading electroweak corrections have been calculated up to two loops [2]. Another important class of effects is provided by the interplay between electroweak interactions and QCD. The relation between MW , MZ , GF , and α is affected by the large top quark mass and again two and even three loop self-energies have been evaluated to arrive at accurate predictions. In addition there is of course the large class of “reducible” corrections which originate from one-loop electroweak diagrams, multiplied by the QCD correction factor (1+αs/π). The remaining two-loop effects are induced by nonfactorizable vertex corrections. With an order of magnitude of αweak π αs π ≈ 0.4× 10 and an unknown coefficient, they are not a priori negligible for an analysis at the 0.1 per cent level. In particular it is evident that an ansatz assuming factorization between QCD and weak effects cannot be valid for the irreducible vertex diagrams. For the Z decay rate into bottom quarks this has been confirmed by the calculation of the leading contributions which are enhanced by a factor mt/m 2 W [3–6] and by logarithms ln(m 2 t/m 2 W ) [7,8]. In both cases factorization is invalidated and nontrivial additional terms are obtained. These results were derived by expanding in the “small” parameter mW/m 2 t . In contrast,