Engineering term in a dimensionally continued beta function (${\lambda }_0={\mu }^{ϵ}F(\lambda ,ϵ)⟹\hat{\beta }(\lambda )=-ϵ\lambda +\beta (\lambda )$)

In $d=4-ϵ$, the quartic scalar coupling has engineering dimension $ϵ$. Factoring out ${\mu }^{ϵ}$ makes the renormalized running coupling dimensionless. Differentiating its bare expression at fixed bare parameters generates the linear term $-ϵ\lambda$, even when the quantum contribution to the renormalization-group beta function begins quadratically. The simple-pole formulas for minimal-subtraction renormalization group functions determine the remaining contribution from the simple pole of $F$.