# Why is the Gribov ambiguity not seen in perturbation theory?

I have read that the Gribov ambiguity doesn't appear in perturbation theory. Can anyone say why? I found the following line in "Gribov copies and confinement" by Anton Ilderton [1]:

Recall it is believed that copies are not an issue in perturbation theory, essentially due to appearances of inverse powers of the coupling in their defining equations

What exactly is meant here?

In perturbation theory, Gribov copies do not pose a significant issue, and this can be understood through the work of Daniel Zwanziger. Zwanziger approached the Gribov problem differently, focusing on the Faddeev–Popov operator.

He computed the lowest eigenvalue of this operator as a perturbative series in the gluon field. This approach led to the formulation of what he termed the "horizon function". The crucial point here is that the vacuum expectation value of this horizon function should be limited to at most one to stay within the first Gribov region. Mathematically, this can be expressed as:

$$\langle H \rangle \leq 1$$

where ( H ) represents the horizon function. This constraint is crucial for ensuring that the calculations remain within the first Gribov region.

To incorporate this into the theory, Zwanziger introduced the horizon function into the path integral, much like Gribov's original approach. This requires imposing a specific gap equation on the vacuum energy of the resulting theory, which can be formulated as:

$$\Delta E_{\text{vac}} \geq 0$$

This leads to a new path integral with a modified, albeit nonlocal, action. It's important to note that in leading order, the results align with those found by Gribov.

In the perturbative regime, where the coupling is weak, the effects of Gribov copies are diminished. This is partly because the perturbative series involves expansions in powers of the coupling constant. In a weak coupling scenario, the system is close to a non-interacting state, making the complications from Gribov copies less significant.

Now having a local theory means that it is renormalizable, (see the paper section 5) without the addition of new fields. Here is the paper:

Zwanziger's Paper