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

TL;DR

This paper explores the relationship between the muon g-2 discrepancy and the shift in the electromagnetic coupling at the Z boson mass scale, establishing bounds and analyzing different spectral function scenarios to understand the impact on precision measurements.

Contribution

It provides a theoretical framework linking the muon g-2 discrepancy to bounds on the running of the electromagnetic coupling, considering specific spectral function models and their implications.

Findings

Total shift remains small in quark model scenarios.

In constant spectral function scenarios, the shift is within current error margins.

Bounds on the shift are derived from the muon g-2 discrepancy.

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