A systematic expansion of running couplings and masses

As an alternative to directly integrating their defining equations to find the running coupling a(μ) and the running mass m(μ), we expand these quantities in powers of ln( μ μ′) and their boundary values a(μ′) and m(μ′). Renormalization group summation is used to partially sum these logarithms. We consider this approach using both the MS¯ and ’t Hooft renormalization schemes. We also show how the couplings and masses in any two mass independent renormalization schemes are related.