  Using methods from effective field theory, we develop a novel, systematic
framework for the calculation of the cross sections for electroweak gauge-boson
production at small and very small transverse momentum q_T, in which large
logarithms of the scale ratio M_V/q_T are resummed to all orders. These cross
sections receive logarithmically enhanced corrections from two sources: the
running of the hard matching coefficient and the collinear factorization
anomaly. The anomaly leads to the dynamical generation of a non-perturbative
scale q_* ~ M_V e^{-const/\alpha_s(M_V)}, which protects the processes from
receiving large long-distance hadronic contributions. Expanding the cross
sections in either \alpha_s or q_T generates strongly divergent series, which
must be resummed. As a by-product, we obtain an explicit non-perturbative
expression for the intercept of the cross sections at q_T=0, including the
normalization and first-order \alpha_s(q_*) correction. We perform a detailed
numerical comparison of our predictions with the available data on the
transverse-momentum distribution in Z-boson production at the Tevatron and LHC.
