A flow invariant is a quantity depending only on the ultraviolet (UV) and infrared (IR) conformal fixed points and not on the flow connecting them. Typically, its value is related to the central charges a and c. In classically conformal field theories, scale invariance is broken by quantum effects and the flow invariant aUV-aIR is measured by the area of the graph of the beta function between the fixed points. There exists a theoretical explanation of this non-trivial fact. On the other hand, when scale invariance is broken at the classical level, it is known empirically that the flow invariant equals cUV-cIR in massive free-field theories, but a theoretical argument explaining why it is so is still missing. A number of related open questions are answered here. A general formula of the flow invariant is found, which also holds when the stress tensor has improvement terms. The conditions under which the flow invariant equals cUV-cIR are identified. Several non-unitary theories are used as a laboratory, but the conclusions are general and an application to the Standard Model is addressed. The analysis of the results suggests some new minimum principles, which might point towards a new understanding of quantum field theory.