Nonanalyticity of the perturbation series for a physical quantity

Using the obvious constraint that jet or particle fractions add up to one, we derive a simple relation between the scheme invariants characterizing these quantities in a general renormalizable field theory. This relation provides surprisingly direct evidence that the corresponding perturbation series are nonanalytic. Approximating this relation can generate corrections to all orders, given one-loop and tree-level coefficients. This is applied to estimate the as-yet-uncalculated $O\left({\alpha }_s^3\right)$ corrections to $e^{+}{e}^{-}$ two-, three-, and four-jet fractions.