Gradient formula for the beta function of 2D quantum field theory

We give a non-perturbative proof of a gradient formula for beta functions of two-dimensional quantum field theories. The gradient formula has the form. partial derivative(i)c = -(g(ij) + Delta g(ij) + b(ij))beta(j) where beta(j) are the beta functions, c and g(ij) are the Zamolodchikov c-function and metric respectively, b(ij) is an antisymmetric tensor introduced by Osborn and Delta g(ij) is a certain metric correction. The formula is derived under the assumption of stress-energy conservation and certain conditions on the infrared behavior the most significant of which is the condition that the large-distance limit of the field theory does not exhibit spontaneously broken global conformal symmetry. Being specialized to nonlinear sigma models this formula implies a one-to-one correspondence between renormalization group fixed points and critical points of c.