# Fermilab $g-2$ muon experiment: How to choose the momentum of the muons?

In the $$g-2$$ muon experiment, we measure $$\omega_a = \omega_s - \omega_c$$, where $$\omega_s$$ is the spin precession frequency and $$\omega_c$$ the cyclotron frequency. Since we can relate $$\omega_a$$ with $$a_{\mu}$$, the quantity that we are really interested in, we are good.

However, according to Eq. (1) of [1], the dependence of $$\omega_a$$ on $$a_{\mu}$$ is somewhat complicated:

$$\vec\omega_a = \vec\omega_s - \vec\omega_c = -\frac{q}{m_{\mu}}\left[ a_{\mu}\vec{B} - a_{\mu}\left( \frac{\gamma_{\mu}}{\gamma_{\mu} + 1}\right)\left(\vec \beta \cdot \vec B\right)\vec \beta - \left( a_{\mu}-\frac{1}{\gamma_{\mu}^2 - 1}\right)\frac{\vec \beta \times \vec E}{c}\right].$$

By choosing a vertical magnetic field for the horizontally moving muons we have $$\left( \vec\beta\cdot \vec B\right)$$ and thus the second term vanishes. By choosing $$\gamma = \sqrt{1 + \frac{1}{a_{\mu}}} \approx 29.3$$, the last term vanishes. The value $$\gamma \approx 29.3$$ corresponds to a momentum of the muons of $$p_{\mu} \approx 3.09$$ GeV.

Question: In the $$g-2$$ muon experiment at FermiLab, how is it ensured that the muons have approximately this momentum? We should get the muons from pion decay (as pions decay with a branching ratio of $$99.98770 \%$$ into a muon and muon-neutrino [2]), but how can we control their momentum?