# Technical question about loop corrections

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1. May 28, 2015

### Worldsheep

Does anyone know a simple explanation for the following statement:

Gauge invariance ⇒ $Πμνϒϒ(0) = ΠμνϒZ(0) = 0$

Where ΠVV' is the V to V' one loop correction, ϒ is the photon field and Z is the Z-boson field. The argument of Π is the incoming momentum q2 = 0

2. May 29, 2015

### vanhees71

$$D_{\mu \nu}(q)=-\frac{1}{q^2-q^2 \Pi(q^2)+\mathrm{i} 0^+}[g_{\mu \nu}-q_{\mu} q_{\nu}] + A(q^2) q^{\mu} q^{\nu},$$
where $A(q^2)$ is a gauge-dependent non-interacting piece, which doesn't enter any physical result.
$$\Pi_{\mu \nu}=q^2 \Pi(q^2) (g_{\mu \nu}-q_{\mu} q_{\nu}).$$
$\Pi$ is a logarithmically divergent scalar. Now to make the residuum of the photon propgator 1 at $q^2=0$, you impose the renormalization condition
$$\Pi(q^2=0)=0.$$
The same argument holds for the $\gamma$-Z mixing piece too.