Testing the $a_\mu$ anomaly in the electron sector through a precise measurement of $h/M$

The persistent $a_\mu \equiv (g-2)/2$ anomaly in the muon sector could be due to new physics visible in the electron sector through a sub-ppb measurement of the anomalous magnetic moment of the electron $a_e$. Driven by recent results on the electron mass (S. Sturm et al., Nature 506 (2014) 467), we reconsider the sources of uncertainties that limit our knowledge of $a_e$ including current advances in atom interferometry. We demonstrate that it is possible to attain the level of precision needed to test $a_\mu$ in the naive scaling hypothesis on a timescale similar to next generation $g-2$ muon experiments at Fermilab and JPARC. In order to achieve such level of precision, the knowledge of the quotient $h/M$, i.e. the ratio between the Planck constant and the mass of the atom employed in the interferometer, will play a crucial role. We identify the most favorable isotopes to achieve an overall relative precision below $10^{-10}$.