# Technical question about loop corrections

• Worldsheep
In summary, the statement is discussing the gauge invariance of the V to V' one loop correction, where V and V' are photon and Z-boson fields respectively. The photon propagator is given by a formula involving a gauge-dependent non-interacting piece, and together with the Ward-Takahashi identity, the photon is strictly 0 mass and purely 4-transverse. To make the residuum of the photon propagator 1 at q^2=0, a renormalization condition is imposed, which also applies to the gamma-Z mixing piece. However, this renormalization condition has to be carefully chosen to avoid infrared divergences.
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

$$D_{\mu \nu}(q)=-\frac{1}{q^2-q^2 \Pi(q^2)+\mathrm{i} 0^+}[g_{\mu \nu}-q_{\mu} q_{\nu}] + A(q^2) q^{\mu} q^{\nu},$$
where ##A(q^2)## is a gauge-dependent non-interacting piece, which doesn't enter any physical result.

Now the photon has strictly 0 mass. Together with the Ward-Takahashi identity of the photon polarization tensor, which makes it purely 4-transverse, this implies that
$$\Pi_{\mu \nu}=q^2 \Pi(q^2) (g_{\mu \nu}-q_{\mu} q_{\nu}).$$
##\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.

Note that the above renormalization condition is dangerous with regard to infrared divergences, which must be resummed. For this purpose it's better to choose the renormalization point in the space-like.

## 1. What are loop corrections in scientific research?

Loop corrections refer to small adjustments or modifications made to a scientific theory or model in order to account for more accurate or precise data. These corrections are often necessary to improve the accuracy and reliability of scientific findings.

## 2. How are loop corrections calculated?

Loop corrections are calculated using mathematical equations and algorithms that take into account various factors such as experimental data, theoretical predictions, and previous corrections made to the model. These calculations involve complex mathematical concepts such as perturbation theory and renormalization.

## 3. What is the purpose of loop corrections in scientific research?

The main purpose of loop corrections is to ensure that scientific theories and models accurately reflect the real world and can make reliable predictions. By making small adjustments to the model, scientists can improve its accuracy and make more precise predictions about the behavior of physical systems.

## 4. How do loop corrections impact scientific discoveries?

Loop corrections can have a significant impact on scientific discoveries by improving the accuracy and reliability of research findings. They can also lead to new insights and discoveries by revealing discrepancies or inconsistencies in previous models.

## 5. Are loop corrections always necessary in scientific research?

Not all scientific research requires loop corrections. In some cases, the initial model or theory may be accurate enough to make reliable predictions without the need for additional corrections. However, in many cases, loop corrections are necessary to improve the accuracy and precision of scientific findings.

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