Yang-Mills beta-function from a large-distance expansion of the Schrödinger functional

For slowly varying fields the Yang-Mills Schrödinger functional can be expanded in terms of local functionals. We show how analyticity in a complex scale parameter enables the Schrödinger functional for arbitrarily varying fields to be reconstructed from this expansion. We also construct the form of the Schrödinger equation that determines the coefficients. Solving this in powers of the coupling reproduces the results of the ‘standard’ perturbative solution of the functional Schrödinger equation which we also describe. In particular the usual result for the beta-function is obtained illustrating how analyticity enables the effects of rapidly varying fields to be computed from the behaviour of slowly varying ones.