Fermilab $g-2$ muon experiment: How to obtain $a_{\mu}$?

Question

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

Answer 1

When I teach introductory students about converting units, I explain that multiplying by $\frac{\rm 1000\,m}{\rm 1\,km}$ is “multiplying by one,” and therefore something that they can do to change the appearance of their result without changing its value.

In the algebraically straightforward expression $a_\mu = \frac{\omega m}{q B}$, you have a product/ratio of four hard-to-measure quantities, each of which will have its own systematic errors. A common technique in precision measurements is to construct ratios of related quantities which have the same sets of systematic errors. Then the systematic errors cancel out in the ratio, and you can apply fewer and smaller corrections to your final result. A pioneering example of the technique is Millikan’s oil-drop experiment; see this guided tour and links.

Without having read more than a few sentences of your linked paper: mapping magnetic fields, to put a $B$ in the denominator, is really hard. The factor ${\omega_a}/{\tilde\omega_p’}$ at the front of your expression suggests to me that the $g-2$ collaborators short-circuited the magnetic-field mapping details by storing a polarized proton beam in the muon ring and measuring its precession frequency. But to store the protons in the storage ring with the same magnetic field, the proton energy would have to be different — this is the meaning of the squiggle $\tilde{\ }$ and/or the prime $’$ which decorate that frequency. Furthermore, the proton and muon magnetic moments are different, so now your muon magnetic-moment measurement is subject to the precision of the proton magnetic-moment measurement.

The other ratios in your expression are drawn from the literature and associated with citations in the paper. The citation associated with the second ratio $\mu_p/\mu_e$ is a paper measuring the proton’s magnetic moment in units of the Bohr magneton $\mu_B = e\hbar/2m_e$. Apparently that’s more precise than, or has better systematic error properties than, the dead-reckoned proton magnetic moment in macroscopic units.

Compare to astronomy. We know the gravitational parameter for the Sun extremely precisely, to about 0.1 parts per billion, because we have millenia of observations of the naked-eye planets whose orbits are determined by the product $GM$. We know the ratio of the Sun’s mass to the Earth’s mass at the part-per-million level. However we don’t actually know the Sun’s mass, and we don’t actually know the Earth’s mass, except at part-per-thousand precision. The problem is $G$. Precision astronomical measurements are made by constructing ratios where $G$ cancels out. The same thing is happening in the $g-2$ paper, but you have to know more QED to follow the details.