= Pure Yang-Mills one-loop beta function
{title2=$\beta(g)=-11C_Ag^3/[3(4\pi)^2]+O(g^5)$}
For pure <Yang-Mills theory> with a compact simple <gauge group> and canonically normalized <gauge field>, the one-loop <renormalization-group beta function> is $\mu\,dg/d\mu=-11C_Ag^3/[3(4\pi)^2]+O(g^5)$, 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>.
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