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