Does electric charge $e$ become complex at some scale $mu$?

Question

begin{equation}
mu frac{partial{e}}{partial mu }=frac{{e}^3}{12pi^2}=beta({e})end{equation}
This is the equation for beta function in quantum electrodynamics, it tells us about how coupling constant scales with the scale $mu$.
The solution to this equation is
begin{equation}
{e}^2(mu)=frac{{e}^2(mu_0)}{1-frac{{e}^2(mu_0)}{6pi^2}lnfrac{mu}{mu_0}} .  
end{equation}
From this equation it is clear that running coupling constant ${e}$ increases with increasing scale (i.e., with $mu$). This equation  has a pole at
begin{equation}
mu=mu_0expBigg(frac{6pi^2}{{e}^2(mu_0)}Bigg).
end{equation} and this singularity is called Landau singularity.
I refer these things from QFT by Ryder.
My question is for
$mu gg mu_0expBigg(frac{6pi^2}{e^2(mu_0)}Bigg),$  ${e}^2(mu)$ appears to be negative and hence $e$ to be complex, is this possible?

Luke Pritchett · Accepted Answer

This equation results from the one-loop calculations. As such it is only valid for small values of $e(mu)$. All it can really tell us is that the effective charge becomes larger at high energies for a while. But once the effective charge becomes big enough you need to go to higher order in the calculations and the effective charge evolution changes. Or worse, the theory becomes non-perturbative and you can't even rely on a finite number of diagrams to tell you the evolution. So even the existence of the pole is suspect, let alone any behavior at energies beyond the pole.

Correct answer by Luke Pritchett on March 19, 2021

Does electric charge $e$ become complex at some scale $mu$?

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