Title: An Application of Rubi: Series Expansion of the Quark Mass Renormalization Group Equation,
Abstract: We highlight how Rule-based Integration (Rubi) is an enhanced method of
symbolic integration which allows for the integration of many difficult
integrals not accomplished by other computer algebra systems. Using Rubi, many
integration techniques become tractable. Integrals are approached using
step-wise simplification, hence distilling an integral (if the solution is
unknown) into composite integrals which highlight yet undiscovered integration
rules. The motivating example we use is the derivation of the updated series
expansion of the quark mass renormalization group equation (RGE) to five-loop
order. This series provides the relation between a light quark mass in the
modified minimal subtraction ($\overline{\text{MS}}$) scheme defined at some
given scale, e.g. at the tau-lepton mass scale, and another chosen energy
scale, $s$. This relation explicitly depicts the renormalization scheme
dependence of the running quark mass on the scale parameter, $s$, and is
important in accurately determining a light quark mass at a chosen scale. The
five-loop QCD $\beta(a_s)$ and $\gamma(a_s)$ functions are used in this
determination.