Renormalization Scheme Dependence and the Renormalization Group Beta Function

The renormalization that relates a coupling "a" associated with a distinct renormalization group beta function in a given theory is considered. Dimensional regularization and mass independent renormalization schemes are used in this discussion. It is shown how the renormalization $a^*=a+x_2a^2$ is related to a change in the mass scale $\mu$ that is induced by renormalization. It is argued that the infrared fixed point is to be a determined in a renormalization scheme in which the series expansion for a physical quantity $R$ terminates.