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.