Charge renormalization

Charge renormalization relates a bare gauge coupling to a coupling fixed by a physical normalization condition. In QED, the Ward identity equates vertex and fermion wave-function renormalization, leaving photon wave-function renormalization to determine the charge counterterm.

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.

For a background gauge field with $A_0=Z_A^{1/2}A$ and $g_0={\mu }^{ϵ/2}{Z}_{g}g$, background gauge covariance preserves the combination $gA$ in the gauge covariant derivative. Its Ward identity therefore gives the displayed relation. This refers to background-field renormalization, not an unrestricted identity between the coupling and the ordinary quantum-gluon wave-function factor. Gauge-invariant one-loop matter determinants can be renormalized with this identity to extract their coupling running from vacuum polarization.

One adjoint Dirac spinor contributes the displayed screening term to the four-dimensional renormalization-group beta function in the modified minimal subtraction scheme. The fermion contribution to the gluon self-energy insertion has pole $-8g^2{C}_A(k^2{\delta }_{\mu \nu }-k_{\mu }{k}_{\nu })/(3ϵ(4\pi {)}^2)$. The corresponding background gauge-field counterterm is $\delta Z_A=-8g^2{C}_A/(3ϵ(4\pi {)}^2)$; the background gauge-field renormalization identity $Z_g=Z_A^{-1/2}$ gives $g_0={\mu }^{ϵ/2}(g+4C_A{g}^3/[3ϵ(4\pi {)}^2])$. Differentiation at fixed $g_0$ yields the screening coefficient. The mass-independent subtraction includes massive fields until effective-theory threshold matching is performed.

For a non-Abelian simple gauge factor with $C_A>0$, pure Yang-Mills theory contributes $-11C_A{g}^3/[3(4\pi {)}^2]$. Adding $N_D$ adjoint Dirac spinors changes this to $-C_A(11-4N_D)g^3/[3(4\pi {)}^2]$. Vanishing would require $N_D=11/4$, which is not a nonnegative integer. This excludes cancellation by these fermions alone, apart from the trivial value $g=0$; other matter representations or fields change the coefficient.