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

Question

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?

[1] https://doi.org/10.1103/PhysRevLett.126.141801 [2] https://pdg.lbl.gov/2021/listings/rpp2021-list-pi-plus-minus.pdf

Answer 1

There is a brief explanation in section II.C of this paper which accompanied the results paper you linked.

Short version: an 8 GeV proton beam strikes a fixed target and a 3 GeV secondary beam is selected from the detritus of the collision; the secondary beam is then cleaned of everything except for muons with the appropriate energy, by sending things that aren’t muons or muons with the wrong energy into beam dumps and collimators.

There may be some amount of focusing of the beam in the energy coordinate, but a lot of the secondary beam gets thrown in the trash. Details are in the long version of the answer, which is the linked paper and its references, which certainly include several PhD theses and Fermilab technical papers on the development of the accelerator and storage system.