In the $g−2$ muon experiment, we measure the anomalous precession frequency $$\omega_a = \omega_s − \omega_c,$$ where $$\omega_s = \frac{g_{\mu}\left\vert q\right\vert B}{2m} + \frac{\left( 1-\gamma\right)\left\vert q\right\vert B}{\gamma m}$$ is the spin precession frequency and $$\omega_c = \frac{\left\vert q\right\vert B}{\gamma m}$$ the cyclotron frequency.
Now, when putting in the spin precessian and cyclotron frequency into the first equation, we obtain: $$\omega_a = \frac{a_{\mu}\left\vert q\right\vert B}{m} \Leftrightarrow a_{\mu} = \frac{\omega_a m}{\left\vert q\right\vert B}.$$
While this is certainly an equation that is given in one of the Fermilab papers [1], Eq. (3), there is also another one that is way more complicated, cf. [2], Eq. (2):
$$ a_{\mu} = \frac{\omega_a}{\tilde{\omega}_{p}^{'}(T_r)}\frac{\mu_{p}'(T_r)}{\mu_e(H)}\frac{\mu_e(H)}{\mu_e}\frac{m_{\mu}}{m_{e}}\frac{g_e}{2}.$$
Question: I think I do not understand where the last equation comes from; I though that we measure $\omega_a$ and thus obtain $a_{\mu}$ via $a_{\mu} = \frac{\omega_a m}{\left\vert q\right\vert B}$, since we precisely know the magnetic field $B$ as well, but clearly, there is more to it...
[1] https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.044002 [2] https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.126.141801