On Constraints Between $\Delta\alpha_{\rm had}(M_Z^2)$ and $(g_{\mu}-2)_{\rm HVP}$

Abstract

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 $\Delta\alpha_{\rm had}^{(5)}(M_Z^2)$.