Pure Yang-Mills one-loop beta function ($\beta (g)=-11C_A{g}^3/[3(4\pi {)}^2]+O(g^5)$)

For pure Yang-Mills theory with a compact simple gauge group and canonically normalized gauge field, the one-loop renormalization-group beta function is $\mu dg/d\mu =-11C_A{g}^3/[3(4\pi {)}^2]+O(g^5)$, where $C_A>0$ is the Casimir operator eigenvalue in the Adjoint representation. This coefficient includes the gauge-field and Faddeev-Popov ghost field loops; it is the standard pure-gauge input when adding matter-loop contributions. Its negative sign gives asymptotic freedom. Integrating the leading equation gives $g(\mu {)}^{-2}=g({\mu }_0{)}^{-2}+11C_A\log (\mu /{\mu }_0)/[24{\pi }^2]$, while perturbation theory is valid. Combining this term with the adjoint Dirac contribution to the Yang-Mills beta function gives the integer obstruction to one-loop marginality with adjoint Dirac fermions.