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Write $d=4-ϵ$ and $C_A=C_2(G)$. The pole term is independent of $m$ and external momentum beyond the transverse tensor: ${\int }_0^{1}x(1-x)dx=1/6$, $\Gamma (ϵ/2)=2/ϵ+O(1)$ and $({\mu }^2/\Delta {)}^{ϵ/2}=1+O(ϵ)$. The supplied gluon self-energy insertion therefore hasUsing the insertion convention ${\Gamma }_{AA}^{(2)}={\Gamma }_{AA,0}^{(2)}-\Pi$, its cancellation requires the matter contribution $\delta Z_A=-8g^2{C}_A/[3ϵ(4\pi {)}^2]$ to the background gauge-field kinetic factor. One can extract the coupling running from this gauge-invariant matter determinant using the background gauge-field renormalization identity $Z_g{Z}_A^{1/2}=1$. This is a background-field Ward identity, not a claim that the ordinary quantum-gluon wave-function factor alone always determines the full coupling renormalization. For this one adjoint Dirac spinor,Differentiate at fixed $g_0$. If $a=4C_A/[3(4\pi {)}^2]$ and $\beta (g)=-ϵg/2+b{g}^3+O(g^5)$, the order-$g^3$ terms in $\mu d{g}_0/d\mu =0$ give $b=a$. Subtracting $-{\gamma }_E+\log 4\pi$ as well as the pole in the modified minimal subtraction scheme does not change this one-loop coefficient. Thus the adjoint Dirac contribution to the Yang-Mills beta function isThis is screening. The pure Yang-Mills one-loop beta function coefficient is $-11C_A{g}^3/[3(4\pi {)}^2]$. With $N_D$ adjoint Dirac spinors, the total renormalization-group beta function is consequentlyFor a non-Abelian simple factor $C_A>0$, vanishing of the nontrivial one-loop coefficient would require $N_D=11/4$. It cannot be a nonnegative integer, proving the integer obstruction to one-loop marginality with adjoint Dirac fermions. For a product group the same argument applies to every non-Abelian factor; the statement concerns the beta-function coefficient, not the trivial fixed point $g=0$. The modified minimal subtraction scheme is mass independent, so a massive spinor has this ultraviolet contribution; decoupling below its mass instead requires effective-theory threshold matching.

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.