A Systematic Expansion of Running Couplings and Masses

As an alternative to directly integrating their defining equations to find the running coupling $a(\mu)$ and the running mass $m(\mu)$, we expand these quantities in powers of $\ln\left(\frac{\mu}{\mu^\prime}\right)$ and their boundary values $a(\mu^\prime)$ and $m(\mu^\prime)$. Renormalization group summation is used to partially sum these logarithms. We consider this approach using both the $\overline{MS}$ and 't Hooft renormalization schemes. We also show how the couplings and masses in any two mass independent renormalization schemes are related.