Deviations from scale invariance near a general conformal background

Deviations from scale invariance resulting from small perturbations of a general two-dimensional conformal field theory are studied. They are expressed in terms of β-functions for the renormalization of general couplings under a local change of scale. The β-functions for a homogeneous background are given perturbatively in terms of the data of the original conformal theory without any specific assumptions on its nature. The renormalization of couplings to primary operators and to first descendents is considered as well as that of a dilatonic type which involve explicit dependence on world sheet curvature. The first descendent couplings are interpreted as gauge degrees of freedom in the string field action and the corresponding gauge transformation is spelled out.