= Simple-pole formulas for minimal-subtraction renormalization group functions
{title2=$\beta(\lambda)=(\lambda\partial_\lambda-1)f_1(\lambda),\quad\gamma_{m^2}(\lambda)=\lambda b_1'(\lambda)$}
In the <minimal subtraction scheme>, let $\lambda_B=\mu^\epsilon F$, $F=\lambda+\sum_{k\geq1}f_k(\lambda)/\epsilon^k$, and $m_B^2=m^2H$, $H=1+\sum_{k\geq1}b_k(\lambda)/\epsilon^k$. Define the <renormalization-group beta function> at fixed bare parameters by $\widehat\beta=\mu d\lambda/d\mu=-\epsilon\lambda+\beta(\lambda)$, and the <running mass> function by $\gamma_{m^2}=\mu d\log m^2/d\mu$.
The bare coupling is scale independent, so $0=\epsilon F+\widehat\beta F'$. Its finite coefficient is $f_1+\beta-\lambda f_1'$, giving the first formula. Similarly $0=\gamma_{m^2}H+\widehat\beta H'$ has finite coefficient $\gamma_{m^2}-\lambda b_1'$. The remaining poles cancel recursively: $(\lambda\partial_\lambda-1)f_{k+1}=\beta f_k'$ and $\lambda b_{k+1}'=\gamma_{m^2}b_k+\beta b_k'$. Thus the simple poles determine the finite <renormalization group> functions, while higher poles obey consistency conditions.
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