Mass and Coupling Constant Limits from Convergence Conditions on the Effective Charge in QED

The massless fermion limit of QED is discussed. For on-shell renormalisation the high energy behaviour fixes no lower limit on the mass of the lightest fermion if the fine structure constant $\alpha$ is allowed to vary. The choice of an arbitary (space-like) subtraction point does, however, fix a lower limit on the mass of the lightest fermion, for any subtraction scale $\mu$, if the effective charge $\aefmu$ respects both quantum mechanical superposition and renormalisation scale invariance. Limits on the values of $\alpha$ or the electron mass are obtained within the Standard Electroweak Model by requiring convergence of $\aefmz$.