Gradient flows for β functions via multi-scale renormalization group equations

Renormalization schemes and cutoff schemes allow for the introduction of various distinct renormalization scales for distinct couplings. We consider the coupled renormalization group flow of several marginal couplings which depend on just as many renormalization scales. The usual $\beta$ functions describing the flow with respect to a common global scale are assumed to be given. Within this framework one can always construct a metric and a potential in the space of couplings such that the $\beta$ functions can be expressed as gradients of the potential. Moreover the potential itself can be derived explicitly from a prepotential which, in turn, determines the metric. Some examples of renormalization group flows are considered, and the metric and the potential are compared to expressions obtained elsewhere.