In QED, one can calculate the correction to the form factor $F_2$. To the lowest order, $F_1=1$ and $F_2=0$. At one loop, it is found that $F_2(0)$ receives a non-zero finite correction which is related to the anomalous magnetic moment of the electron. However, the correction to charge form factor $F_1$ turns out to be divergent and one gets rid of the divergence by renormalization.
$\bullet$ Is there any fundamental reason to believe that $F_1$ should not receive any finite correction?
$\bullet$ The vertex correction $\delta\Gamma_\mu$ is divided into two parts (in the following, I used notation of L. H. Ryder): $$\delta\Gamma_\mu=\Gamma_\mu-\gamma_\mu=\Lambda_\mu^{(1)}+\Lambda_\mu^{(2)}$$ $\Lambda_\mu^{(1)}$ contributing to $F_1$ and is divergent while $\Lambda_\mu^{(2)}$ contributes to $F_2$ and is convergent. It turns out that, $\Lambda_\mu^{(1)}$ itself has a divergent and a finite part. Renormalization gets rid of total $\Lambda_\mu^{(1)}$ including its finite part.
Does it not mean that we don't want to have any quantum correction to $F_1$?
