  I discuss the possibility of estimating the shift $\delta\Delta\alpha(M_Z^2)$
on the running of the EM-coupling to the $M_Z$ scale, induced by the
discrepancy $\Delta a_{\mu}$ between two precise determinations of the hadronic
vacuum polarization contribution to $g_{\mu}-2$. It is shown that the size of
$\Delta a_{\mu}$ implies rigorous bounds on the
$\delta\Delta\alpha(M_Z^2)$-shift. Any extra contribution to this minimal shift
necessarily depends on the specific shape of the underlying spectral function
$\frac{1}{\pi}\Imm\Delta(t)$ responsible for the $\Delta a_{\mu}$-discrepancy.
I show that, in the case of a quark model, the total
$\Delta\alpha(M_Z^2)$-shift remains small. I also discuss the scenario where
$\frac{1}{\pi}\Imm\Delta(t)$ is a constant in a finite $t$-region and show that
in this case, up to a $t_{\rm max}\precsim 5~\GeV^2$, the size of the total
$\delta\Delta\alpha_{\rm had}^{(5)}(M_Z^2)$-shift, remains below or of the
order of the present error value on $\Delta\alpha_{\rm had}^{(5)}(M_Z^2)$.

High Energy Physics - Phenomenology;High Energy Physics - Lattice