*This is post #2 of a three-part series on the Muon g-2 experiment. Check out Amara McCune’s post on the theory of g-2 physics for an excellent introduction to the topic.*

As we all eagerly await the latest announcement from the Muon g-2 Collaboration on April 7th, it is a good time to think about the experimental aspects of the measurement and to appreciate just how difficult it is and the persistent and collaborative effort that has gone into obtaining one of the most precise results in particle physics to date.

The main “output” of the experiment (after all data-taking runs are complete) is a single number: the g-factor of the muon, measured to an unprecedented accuracy of 140 parts per billion (ppb) at Fermilab’s Muon Campus, a four-fold improvement over the previous iteration of the experiment that took place at Brookhaven National Lab in the early 2000s. But to arrive at this seemingly simple result, a painstaking measurement effort is required. As a reminder (see Amara’s post for more details), what is actually measured is the anomalous deviation from 2 of the magnetic moment of the muon, , which is given by

.

**Experimental method**

The core tenet of the experimental approach relies on the behavior of muons when subjected to a uniform magnetic field. If muons can be placed in a uniform circular trajectory around a storage ring with uniform magnetic field, then they will travel around this ring with a characteristic frequency, referred to as its *cyclotron* *frequency* (symbol ). At the same time, if the muons are polarized, meaning that their spin vector points along a particular direction when first injected into the storage ring, then this spin vector will also rotate when subjected to a uniform magnetic field. The frequency of the spin vector rotation is called the *spin* *frequency* (symbol ).

If the cyclotron and spin frequencies of the muon were exactly the same, then it would have an anomalous magnetic moment of zero. In other words, the anomalous magnetic moment measures the discrepancy between the behavior of the muon itself and its spin vector when under a magnetic field. As Amara discussed at length in the previous post in this series, such discrepancy arises because of specific quantum-mechanical contributions to the muon’s magnetic moment from several higher-order interactions with other particles. We refer to the differing frequencies as the *precession* of the muon’s spin motion compared to its cyclotron motion.

If the anomalous magnetic moment is not zero, then one way to measure it is to directly record the cyclotron and spin frequencies and subtract them. In a way, this is what is done in the experiment: the anomalous precession frequency can be measured as

where is the muon mass, is the muon charge, and is the (ideally) uniform magnetic field. Once the precession frequency and the exact magnetic field are measured, one can immediately invert this equation to obtain .

In practice, the best way to measure is to rewrite the equation above into more experimentally amenable quantities:

where is the proton-to-electron magnetic moment ratio, is the electon g-factor, and is the free proton’s Larmor frequency averaged over the muon beam spatial transverse distribution. The Larmor frequency measures the proton’s magnetic moment precession about the magnetic field and is directly proportional to B. The written in this form has the considerable advantage that all of the quantities have been independently and very accurately measured: to 0.00028 ppb (), to 3 ppb (), and to 22 ppb (). Recalling that the final desired accuracy for the left-hand side of the equation above is 140 ppb leads to a budget of 70 ppb for each of the and measurements. This is perhaps a good point to stop and appreciate just how small these uncertainty budgets are: 1 ppb is a 1/1000000000 level of accuracy!

We have now distilled the measurement into two numbers: , the anomalous precession frequency, and , the free proton Larmor frequency which is directly proportional to the magnetic field (the quantity we’re *actually* interested in). Their uncertainty budgets are roughly 70 ppb for each, so let’s take a look at how they are able to measure these two numbers to such an accuracy. First, we’ll introduce the experimental setup, and then describe the two measurements.

**Experimental setup**

The polarized muons in the experiment are produced by a beam of pions, which are themselves produced when a beam of 8 GeV protons created by Fermilab’s linear accelerator strikes a nickel-iron target. The pions are selected to have a momentum close to the required for the experiment: 3.11 GeV/c. Each pion then decays to a muon and a muon-neutrino (more than 99% of the time), and a very particular momentum is selected for the muons: 3.094 GeV/c. Only muons with this specific momentum (or very close) are allowed to enter the storage ring. This momentum has a special significance in the experimental design and is colloquially referred to as the “magic momentum” (and muons, upon entering the storage ring, travel along a circular trajectory with a “magic radius” which corresponds to the magic momentum). The reason for this special momentum is, very simplistically, the fortuitous cancelation of some electric and magnetic field effects that would need to be accounted for otherwise and that would therefore reduce the accuracy of the measurement. Here’s a sketch of the injection pipeline:

Muons with the magic momentum are injected into the muon storage ring, pictured below. The storage ring (the same one from Brookhaven which was moved to Fermilab in 2013) is responsible for keeping muons circulating in orbit until they decay, with a vertical magnetic field of 1.45 T, uniform within 25 ppm (quite a feat and made possible via a painstaking effort called magnet “shimming”). The muon lifetime is 2 microseconds in its own frame of reference, but in the laboratory frame and with a 3.094 GeV/c momentum this increases to 64 microseconds. The storage ring has a roughly 45 m circumference, so muons can travel up to hundreds of times around the ring before decaying.

When they do eventually decay, the most likely decay products are positrons (or electrons, depending on the muon charge), electron-antineutrinos, and muon-neutrinos. The latter two are neutral particles and essentially invisible, but the positrons are charged and therefore bend under the magnetic field in the ring. The magic momentum and magic radius only apply to muons – positrons will bend inwards and eventually hit one of the 24 calorimeters placed strategically around the ring. A sketch of the situation is shown below.

Calorimeters are detectors that can precisely measure the total energy of a particle. Furthermore, with transversal segmentation, they can also measure the incident position of the positrons. The calorimeters used in the experiment are made of lead fluoride (PbF2) crystals, which are Cherenkov radiators and therefore have an extremely fast response (Cherenkov radiation is emitted instantaneously when an incident particle travels faster than light in a medium – not in vacuum though since that’s not possible!). Very precise timing information about decay positrons is essential to infer the position of the decaying muon along the storage ring, and the experiment manages to achieve a remarkable sub-100 ps precision on the positron arrival time (which is then compared to the muon injection time for an absolute time calibration).

** measurement **

The key aspect of the measurement is that the **direction and energy distributions of the decay positrons are correlated with the direction of the spin of the decaying muons**. So, by measuring the energy and arrival time of each positron with one of the 24 calorimeters, one can deduce (to some degree of confidence) the spin direction of the parent muon.

But recall that the spin direction itself is not constant in time — it oscillates with frequency, while the muons themselves travel around the ring with frequency. By measuring the energy of the most energetic positrons (the degree of correlation between muon spin and positron energy is highest for more energetic positrons), one should find an oscillation that is roughly proportional to the spin oscillation, “corrected” by the fact that muons themselves are moving around the ring. Since the position of each calorimeter is known, accurately measuring the arrival time of the positron relative to the injection of the muon beam into the storage ring, combined with its energy information, gives an idea of how far along in its cyclotron motion the muon was when it decayed. These are the crucial bits of information needed to measure the difference in the two frequencies, and , which is proportional to the anomalous magnetic moment of the muon.

All things considered, with the 24 calorimeters in the experiment one can count the number of positrons with some minimum energy (the threshold used is roughly 1.7 GeV) arriving as a function of time (remember, the most energetic positrons are more relevant since their energy and position have the strongest correlation to the muon spin). Plotting a histogram of these positrons, one arrives at the famous “wiggle plot”, shown below.

This histogram of the number of positrons versus time is plotted modulo some time constant, otherwise it would be too long to show in a single page. But the characteristic features are very visible: 1) the overall number of positrons decreases as muons decay away and there are fewer of them around; and 2) the oscillation in the number of energetic positrons is due to the precession of the muon spin relative to its cyclotron motion — whenever muon spin and muon momentum are aligned, we see a greater number of energetic positrons, and vice-versa when the two vectors are anti-aligned. In this way, the oscillation visible in the plot is directly proportional to the precession frequency, i.e. how much ahead the spin vector oscillates compared to the momentum vector itself.

In its simplest formulation, this wiggle plot can be fitted to a basic five-parameter model:

where the five parameters are: , the initial number of positrons; , the time-dilated muon lifetime; , the amplitude of the oscillation which is related to the asymmetry in the positron’s transverse impact position; , the sought-after spin precession frequency; and , the phase of the oscillation.

The five-parameter model captures the essence of the measurement, but in practice, to arrive at the highest possible accuracy many additional effects need to be considered. Just to highlight a few: Muons do not all have exactly the right magic momentum, leading to orbital deviations from the magic radius and a different decay positron trajectory to the calorimeter. And because muons are injected in bunches into the storage ring and not one by one, sometimes decay positrons from more than one muon arrive simultaneously at a calorimeter — such *pileup* positrons need to be carefully separated and accounted for. A third major systematic effect is the presence of non-ideal electric and/or magnetic fields, which can introduce important deviations in the expected motion of the muons and their subsequent decay positrons. In the end, to correct for all these effects, the five-parameter model is augmented to an astounding 22-parameter model! Such is the level of detail that a precision measurement requires. The table below illustrates the expected systematic uncertainty budget for the measurement.

Category | Brookhaven [ppb] | Fermilab [ppb] | Improvements |

Gain changes | 120 | 20 | Better laser calibration; low-energy threshold |

Pileup | 80 | 40 | Low-energy samples recorded; calorimeter transverse segmentation |

Lost muons | 90 | 20 | Better collimation in ring |

Coherent Betatron Oscillation | 70 | < 30 | Higher n value (frequency); better match of beam line to storage ring |

Electric field and pitch | 50 | 30 | Improved tracker; precise storage ring simulations |

Total | 180 | 70 |

Note: the wiggle plot above was taken from David Sweigart’s thesis, which features a *blinded* analysis of the data, where is replaced by , and the two are related by:

.

Here is the blinded parameter that is used instead of , and is an arbitrary offset that is independently chosen by each different analysis group. This ensures that results from one group do not influence the others and allows all analysis to have the same (unknown) reference. We can expect a similar analysis (and probably several different types of analyses) in the announcement on April 7th, except that the blinded modification will be removed and the true number unveiled.

** measurement**

The measurement of the Larmor frequency (and of the magnetic field B) is equally important to the determination of and proceeds separately from the measurement. The key ingredient here is an extremely accurate mapping of the magnetic field with a two-prong approach: removable proton Nuclear Magnetic Resonance (NMR) probes and fixed NMR probes inside the ring.

The 17 removable probes sit inside a motorized trolley and circle around the ring periodically (every 3 days) to get a very clear and detailed picture of the magnetic field inside the storage ring (the operating principle is that the measured free proton precession frequency is proportional to the magnitude of the external magnetic field). The trolley cannot be run concurrently with the muon beam and so the experiment must be paused for these precise measurements. To complement these probes, 378 fixed probes are installed inside the ring to continuously monitor the magnetic field, albeit with less detail. The removable probes are therefore used to calibrate the measurements made by the fixed probes, or conversely the fixed probes serve as a sort of “interpolation” data between the NMR probe runs.

In addition to the magnetic field, an understanding of the muon beam transverse spatial distribution is also important. The term that enters the anomalous magnetic moment equation above is given by the average magnetic field (measured with the probes) *weighted* by the transverse spatial distribution of muons when going around the ring. This distribution is accurately measured with a set of three trackers placed immediately upstream of calorimeters at three strategic locations around the storage ring.

The trackers feature pairs of straw wires at stereo angles to each other that can accurately reconstruct the trajectory of decay positrons. The charged positrons ionize some of the gas molecules inside the straws, and the released charge gets swept up to electrodes at the straw end by an electric field inside the straw. The amount and location of the charge yield information on position of the positron, and the 8 layers of a tracker together give precise information on the positron trajectory. With this approach, the magnetic field can be measured and then corrected via a set of 200 concentric coils with independent current settings to an accuracy of a few ppm when averaged azimuthally. The expected systematic uncertainty budget for the measurement is shown in the table below.

Category | Brookhaven [ppb] | Fermilab [ppb] | Improvements |

Absolute probe calibration | 50 | 35 | More uniform field for calibration |

Trolley probe calibration | 90 | 30 | Better alignment between trolley and the plunging probe |

Trolley measurement | 50 | 30 | More uniform field, less position uncertainty |

Fixed probe interpolation | 70 | 30 | More stable temperature |

Muon distribution | 30 | 10 | More uniform field, better understanding of muon distribution |

Time-dependent external magnetic field | – | 5 | Direct measurement of external field, active feedback |

Trolley temperature, others | 100 | 30 | Trolley temperature monitor, etc. |

Total | 170 | 70 |

**Conclusions**

The announcement on April 7th of the first Muon g-2 results at Fermilab (E989) is very exciting for those following along over the past few years. Since the full data-taking has not been completed yet, it’s likely that these results are not the ultimate ones produced by the Collaboration. But even if they manage to match the accuracy of the previous iteration of the experiment at Brookhaven (E821), we can already learn something about whether the central value of shifts up or down or stays roughly constant. If it stays the same even after a decade of intense effort to make an entire new measurement, this could be a strong sign of new physics lurking around! But let’s wait and see what the Collaboration has in store for us. Here’s a link to the event on April 7th.

Amara and I will conclude this series with a 3rd post after the announcement discussing the things we learn from it. Stay tuned!

Further Reading:

- Muon g-2 Technical Design Report (TDR): https://arxiv.org/pdf/1501.06858.pdf
- David Sweigart’s thesis has an accessible yet comprehensive introduction to the experiment: https://lss.fnal.gov/archive/thesis/2000/fermilab-thesis-2020-02.pdf
- Aaron Fienberg’s thesis has some nice discussion on the g-2 calorimeter and analysis methods: https://lss.fnal.gov/archive/thesis/2000/fermilab-thesis-2019-07.pdf
*The calorimeter system of the new muon g-2 experiment at Fermilab*(L. P. Alonzi et al): https://doi.org/10.1016/j.nima.2015.11.041*Current Status of Muon g-2 Experiment at Fermilab*(Esra Barlas-Yucel @ FPCP 2020): https://indico.cern.ch/event/838862/contributions/3609622/attachments/2054740/3445104/FPCP-2020-muon_g-2_Experiment.pdf

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Is the magic momentum related to the equation that appeared in the fermilab presentation [1]?

How can such equation be derived?

what terms does the magic momentum cancel in particular?

[ 1] youtube link at the specific time: https://www.youtube.com/watch?v=81PfYnpuOPA&t=2222s

Hi Daniel, indeed the magic momentum cancels the last term in the equation you point to. With the magic momentum of 3.094 GeV/c, the Lorentz factor gamma is 29.3. Assuming some average a_mu (the ultimate precision for this purpose is not so important since the realistic momentum distribution, although centered at the magic momentum, has some experimental width), then the terms in parenthesis cancel out. The equation is derived from Larmor precession, which deals with the precession of magnetic moments about an external magnetic field. For the full equation, however, Thomas precession must also be included, which is a relativistic correction needed to account for the spin motion of the muon as well.