New detectors on the block

Article title: “Toward Machine Learning Optimization of Experimental Design”

Authors: MODE Collaboration

Reference: (pdf)

In a previous post we wondered if (machine learning) algorithms can replace the entire simulation of detectors and reconstruction of particles. But meanwhile some experimentalists have gone one step further – and wondered if algorithms can design detectors.

Indeed, the MODE collaboration stands for Machine-learning Optimized Design of Experiments and in its first paper promises nothing less than that.

The idea here is that the choice of characteristics that an experiment can have is vast (think number of units, materials, geometry, dimensions and so on), but its ultimate goal can still be described by a single “utility function”. For instance, the precision of the measurement on specific data can be thought of as a utility function.

Then, the whole process that leads to obtaining that function can be decomposed into a number of conceptual blocks: normally there are incoming particles, which move through and interact with detectors, resulting in measurements; from them, the characteristics of the particles are reconstructed; these are eventually analyzed to get relevant useful quantities, the utility function among them. Ultimately, chaining together these blocks creates a pipeline that models the experiment from one end to the other.

Now, another central notion is differentiation or, rather, the ability to be differentiated; if all the components of this model are differentiable, then the gradient of the utility function can be calculated. This leads to the holy grail: finding its extreme values, i.e. optimize the experiment’s design as a function of its numerous components.

Before we see whether the components are indeed differentiable and how the gradient gets calculated, here is an example of this pipeline concept for a muon radiography detector.

Discovering a hidden space in the Great Pyramid by using muons. ( Financial Times)

Muons are not just the trendy star of particle physics (as of April 2021), but they also find application in scanning closed volumes and revealing details about the objects in them. And yes, the Great Pyramid has been muographed successfully.

In terms of the pipeline described above, a muon radiography device could be modeled in the following way: Muons from cosmic rays are generated in the form of 4-vectors. Those are fed to a fast-simulation of the scanned volume and the detector. The interactions of the particles with the materials and the resulting signals on the electronics are simulated. This output goes into a reconstruction module, which recreates muon tracks. From them, an information-extraction module calculates the density of the scanned material. It can also produce a loss function for the measurement, which here would be the target quantity.

Conceptual layout of the optimization pipeline. (MODE collaboration)

This whole ritual is a standard process in experimental work, although the steps are usually quite separate from one another. In the MODE concept, however, not only are they linked together but also run iteratively. The optimization of the detector design proceeds in steps and in each of them the parameters of the device are changed in the simulation. This affects directly the detector module and indirectly the downstream modules of the pipeline. The loop of modification and validation can be constrained appropriately to keep everything within realistic values, and also to make the most important consideration of all enter the game – that is of course cost and the constraints that it brings along.

Descending towards the minimum. (Dezhi Yu)

As mentioned above, the proposed optimization proceeds in steps by optimizing the parameters along the gradient of the utility function. The most famous incarnation of gradient-based optimization is gradient descent which is customarily used in neural networks. Gradient descent guides the network towards the minimum value of the error that it produces, through the possible “paths” of its parameters.

In the MODE proposal the optimization is achieved through automatic differentiation (AD), the latest word in the calculation of derivatives in computer programs. To shamefully paraphrase Wikipedia, AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations and functions. By applying the chain rule repeatedly to these operations, derivatives can be computed automatically, accurately and efficiently.

Also, something was mentioned above about whether the components of the pipeline are “indeed differentiable”. It turns out that one isn’t. This is the simulation of the processes during the passage of particles through the detector, which is stochastic by nature. However, machine learning can learn how to mimic it, take its place, and provide perfectly fine and differentiable modules. (The brave of heart can follow the link at the end to find out about local generative surrogates.)

This method of designing detectors might sound like a thought experiment on steroids. But the point of MODE is that it’s the realistic way to take full advantage of the current developments in computation. And maybe to feel like we have really entered the third century of particle experiments.

Further reading:

The MODE website:

A Beginner’s Guide to Differentiable Programming:

Black-Box Optimization with Local Generative Surrogates:

Sinusoidal dark matter: ANAIS Edition

Title: Annual Modulation Results from Three Years Exposure of ANAIS-112.


This is an exciting couple of months to be a particle physicist. The much-awaited results from Fermilab’s Muon g-2 experiment delivered all the excitement we had hoped for. (Don’t miss our excellent theoretical and experimental refreshers by A. McCune and A. Frankenthal, and the post-announcement follow-up.) Not long before that, the LHCb collaboration confirmed the R_K flavor anomaly, a possible sign of violation of lepton universality, and set the needle at 3.1 standard deviations off the Standard Model (SM). That same month the ANAIS dark matter experiment took on the mighty DAMA/LIBRA, the subject of this post.

In its quest to confirm or refute its 20 year-old predecessor at Brookhaven National Lab, the Fermilab Muon g-2 experiment used the same storage ring magnet — though refurbished — and the same measurement technique. As the April 7 result is consistent with the BNL measurement, this removes much doubt from the experimental end of the discrepancy, although of course, unthought-of correlated systematics may lurk. A similar philosophy is at work with the ANAIS experiment, which uses the same material, technique and location (on the continental scale) as DAMA/LIBRA.

As my colleague M. Talia covers here and I touch upon here, an isotropic distribution of dark matter velocities in the Galactic frame would turn into an anisotropic “wind” in the solar frame as the Solar System orbits around the center of the Milky Way. Furthermore, in the Earth’s frame the wind would reverse direction every half-year as we go around the Sun. If we set up a “sail” in the form of a particle detector, this annual modulation could be observed — if dark matter interacts with SM states. The amplitude of this modulation S_m is given by

R(t) = S_m \cos(\omega (t - t_0)) + R_0 \phi_{\rm bg}(t)~,


R(t) is the rate of event collection per unit mass of detector per unit energy of recoil at some time t,

\omega = 2\pi/(365 \ {\rm days}),

R_0 captures any unmodulated rate in the detector with \phi_{\rm bg} its probability distribution in time, and

t_0 is fixed by the start date of the experiment so that the event rate is highest when we move maximally upwind on June 02.

The DAMA/LIBRA experiment in Italy’s Gran Sasso National Laboratory, using 250 kg of radiopure thallium-doped sodium-iodide [NaI(Tl)] crystals, claims to observe a modulation every year over the last 20 years, with S_m = 0.0103 \pm 0.0008 /day/kg/keV in the 2–6 keV energy range at the level of 12.9 \sigma.

It is against this serious claim that the experiments ANAIS, COSINE, SABRE and COSINUS have mounted a cross-verification campaign. Sure, the DAMA/LIBRA result is disfavored by conventional searches counting unmodulated dark matter events (see, e.g. Figure 3 here or this recent COSINE-100 paper). But it cannot get cleaner than a like-by-like comparison independent of assumptions about dark matter pertaining either to its microscopic behavior or to its phase space distribution in the Earth’s vicinity. Doing just that, ANAIS (for Annual Modulation with NaI Scintillators) in Spain’s Canfranc Underground Laboratory, using 112.5 kg of radiopure NaI(Tl) over 3 years, has a striking counter-claim summed up in this figure:

Figure caption: The ANAIS 3 year dataset is consistent with no annual modulation, in tension with DAMA’s observation of significant modulation over 20 years.

ANAIS’ error bars are unsurprisingly larger than DAMA/LIBRA’s given their smaller dataset, but the modulation amplitude they measure is unmistakably consistent with zero and far out from DAMA/LIBRA. The plot below is visual confirmation of non-modulation with the label indicating the best-fit S_m under the modulation hypothesis.

Figure caption: ANAIS-112 event rate data in two energy ranges, fitted to a null hypothesis and a modulation hypothesis; the latter gives a fit consistent with zero modulation. The event rate here falls with time as the time-distribution of the background \phi_{\rm bkg} (t) is modeled as an exponential function accounting for isotopic decays in the target material.

The ANAIS experimenters carry out a few neat checks of their result. The detector is split into 9 pieces, and just to be sure of no differences in systematics and backgrounds among them, every piece is analyzed for a modulation signal. Next they treat t_0 as a free parameter, equivalent to making no assumptions about the direction of the dark matter wind. Finally they vary the time bin size in analyzing the event rate such as in the figure above. In every case the measurement is consistent with the null hypothesis.

Exactly how far away is the ANAIS result from DAMA/LIBRA? There are two ways to quantify it. In the first, ANAIS take their central values and uncertainties to compute a 3.3 \sigma (2.6 \sigma) deviation from DAMA/LIBRA’s central values S_m^{\rm D/L} in the 1–6 keV (2–6 keV) bin. In the second way, the ANAIS uncertainty \sigma^{\rm AN}_m is directly compared to DAMA using the ratio S_m^{\rm D/L}/\sigma^{\rm AN}_m, giving 2.5 \sigma and 2.7 \sigma in those energy bins. With 5 years of data — as scheduled for now — this latter sensitivity is expected to grow to 3 \sigma. And with 10 years, it could get to 5 \sigma — and we can all go home.

Further reading.

[1] Watch out for the imminent results of the KDK experiment set out to study the electron capture decay of potassium-40, a contaminant in NaI; the rate of this background has been predicted but never measured.

[2] The COSINE-100 experiment in Yangyang National Lab, South Korea (note: same hemisphere as DAMA/LIBRA and ANAIS) published results in 2019 using a small dataset that couldn’t make a decisive statement about DAMA/LIBRA, but they are scheduled to improve on that with an announcement some time this year. Their detector material, too, is NaI(Tl).

[3] The SABRE experiment, also with NaI(Tl), will be located in both hemispheres to rule out direction-related systematics. One will be right next to DAMA/LIBRA at the Gran Sasso Laboratory in Italy, the other at Stawell Underground Physics Laboratory in Australia. ParticleBites’ M. Talia is excited about the latter.

[4] The COSINUS experiment, using undoped NaI crystals in Gran Sasso, aims to improve on DAMA/LIBRA by lowering the nuclear recoil energy threshold and with better background discrimination.

[5] Testing DAMA, article in Symmetry Magazine.

An Anomalous Anomaly: The New Fermilab Muon g-2 Results

By Andre Sterenberg Frankenthal and Amara McCune

This is the final post of a three-part series on the Muon g-2 experiment. Check out posts 1 and 2 on the theoretical and experimental aspects of g-2 physics. 

The last couple of weeks have been exciting in the world of precision physics and stress tests of the Standard Model (SM). The Muon g-2 Collaboration at Fermilab released their very first results with a measurement of the anomalous magnetic moment of the muon to an accuracy of 462 parts per billion (ppb), which largely agrees with previous experimental results and amplifies the tension with the accepted theoretical prediction to a 4.2\sigma discrepancy. These first results feature less than 10% of the total data planned to be collected, so even more precise measurements are foreseen in the next few years.

But on the very same day that Muon g-2 announced their results and published their main paper on PRL and supporting papers on Phys. Rev. A and Phys. Rev. D, Nature published a new lattice QCD calculation which seems to contradict previous theoretical predictions of the g-2 of the muon and moves the theory value much closer to the experimental one. There will certainly be hot debate in the coming months and years regarding the validity of this new calculation, but it does not stop from muddying the waters in the g-2 sphere. We cover both the new experimental and theoretical results in more detail below.

Experimental announcement

The main paper in Physical Review Letters summarizes the experimental method and reports the measured numbers and associated uncertainties. The new Fermilab measurement of the muon g-2 is 3.3 standard deviations (\sigma) away from the predicted SM value. This means that, assuming all systematic effects are accounted for, the probability that the null hypothesis (i.e. that the true muon g-2 number is actually the one predicted by the SM) could result in such a discrepant measurement is less than 1 in 1,000. Combining this latest measurement with the previous iteration of the experiment at Brookhaven in the early 2000s, the discrepancy grows to 4.2\sigma, or smaller than 1 in 300,000 probability that it is just a statistical fluke. This is not yet the 5\sigma threshold that seems to be the golden standard in particle physics to claim a discovery, but it is a tantalizing result. The figure below from the paper illustrates well the tension between experiment and theory.

Comparison between experimental measurements of the anomalous magnetic moment of the muon (right, top to bottom: Brookhaven, Fermilab, combination) and the theoretical prediction by the Standard Model (left). The discrepancy has grown to 4.2 sigma. Source:

This first publication is just the first round of results planned by the Collaboration, and corresponds to less than 10% of the data that will be collected throughout the total runtime of the experiment. With this limited dataset, the statistical uncertainty (434 ppb) dominates over the systematic uncertainty (157 pbb), but that is expected to change as more data is acquired and analyzed. When the statistical uncertainty eventually dips below, it will be critically important to control the systematics as much as possible, to attain the ultimate target goal of a 140 ppb total uncertainty measurement. The table below shows the actual measurements performed by the Collaboration.

Table of measurements for each sub-run of the Run-1 period, from left to right: precession frequency, equivalent precession frequency of a proton (i.e. measures the magnetic field), and the ratio between the two quantities. Source:

The largest sources of systematic uncertainties stem from the electrostatic quadrupoles (ESQ) in the experiment. While the uniform magnetic field ensures the centripetal motion of muons in the storage ring, it is also necessary to keep them confined to the horizontal plane. Four sets of ESQ uniformly spaced in azimuth provide vertical focusing of the muon beam. However, after data-taking, two resistors in the ESQ system were found to be damaged. This means that the time profile of ESQ activation was not perfectly matched to the time profile of the muon beam. In particular, during the first 100 microseconds after each muon bunch injection, muons were not getting the correct focusing momentum, which affected the expected phase of the “wiggle plot” measurement. All told, this issue added 75 ppb of systematic uncertainty to the budget. Nevertheless, because statistical uncertainties dominate in this first stage of the experiment, the unexpected ESQ damage was not a showstopper. The Collaboration expects this problem to be fully mitigated in subsequent data-taking runs.

To guard against any possible human bias, an interesting blinding policy was implemented: the master clock of the entire experiment was shifted by an unknown value, chosen by two people outside the Collaboration and kept in a vault for the duration of the data-taking and processing. Without knowing this shift, it is impossible to deduce the correct value of the g-2. At the same time, this still allows experimenters to carry out the analysis through the end, and only then remove the clock shift to reveal the unblinded measurement. In a way this is like a key unlocking the final result. (This was not the only protection against bias, only the more salient and curious one.)

Lattice QCD + BMW results

On the same day that Fermilab announced the Muon g-2 experimental results, a group known as the BMW (Budapest-Marseille-Wuppertal) Collaboration published its own results on the theoretical value of muon g-2 using new techniques in lattice QCD. The group’s results can be found in Nature (the journal, jury’s still out on whether they’re in actual Nature), or at the preprint here. In short, their calculations bring them much closer to the experimental value than previous collaborations, bringing their methods into tension with the findings of previous lattice QCD groups. What’s different? To this end, let’s dive a little deeper into the details of lattice QCD. 

The BMW group published this plot, showing their results (top line) compared with the results of other groups using lattice QCD (remaining green squares) and the R-ratio (the usual data-driven techniques, red circles). The purple bar gives the range of values for the anomalous magnetic moment that would signal no new physics – we can see that the BMW results are closer to this region than the R-ratio results, which have similarly small error bars. Source: BMW Collaboration

As outlined in the first post of this series, the main tool of high-energy particle physics rests in perturbation theory, which we can think of graphically via Feynman diagrams, starting with tree-level diagrams and going to higher orders via loops. Equivalently, this corresponds to calculations in which terms are proportional to some coupling parameter that describes the strength of the force in question. Each higher order term comes with one more factor of the relevant coupling, and our errors in these calculations are generally attributable to either uncertainties in the coupling measurements themselves or the neglecting of higher order terms.

These coupling parameters are secretly functions of the energy scale being studied, and so at each energy scale, we need to recalculate these couplings. This makes sense intuitively because forces have different strengths at different energy scales — e.g. gravity is much weaker on a particle scale than a planetary one. In quantum electrodynamics (QED), for example, these couplings are fairly small when in the energy scale of the electron. This means that we really don’t need to go to higher orders in perturbation theory, since these terms quickly become irrelevant with higher powers of this coupling. This is the beauty of perturbation theory: typically, we need only consider the first few orders, vastly simplifying the process.

However, QCD does not share this convenience, as it comes with a coupling parameter that decreases with increasing energy scale. At high enough energies, we can indeed employ the wonders of perturbation theory to make calculations in QCD (this high-energy behavior is known as asymptotic freedom). But at lower energies, at length scales around that of a proton, the coupling constant is greater than one, which means that the first-order term in the perturbative expansion is the least relevant term, with higher and higher orders making greater contributions. In fact, this signals the breakdown of the perturbative technique. Because the mass of the muon is in this same energy regime, we cannot use perturbation theory in quantum field theory to calculate g-2. We then turn to simulations, and since cannot entirely simulate spacetime (because it consists of infinite points), we must instead break it up into a discretized set of points dubbed the lattice. 

A visualization of the lattice used in simulations. Particles like quarks are placed on the points of the lattice, with force-carrying gauge bosons (in this case, gluons) forming the links between them. Source: Lawrence Livermore National Laboratory

This naturally introduces new sources of uncertainty into our calculations. To employ lattice QCD, we need to first consider which lattice spacing to use — the distance between each spacetime point — where a smaller lattice spacing is preferable in order to come closer to a description of spacetime. Introducing this lattice spacing comes with its own systematic uncertainties. Further, this discretization can be computationally challenging, as larger numbers of points quickly eat up computing power. Standard numerical techniques become too computationally expensive to employ, and so statistical techniques as well as Monte Carlo integration are used instead, which again introduces sources of error.

Difficulties are also introduced by the fact that a discretized space does not respect the same symmetries that a continuous space does, and some symmetries simply cannot be kept simultaneously with others. This leads to a challenge in which groups using lattice QCD must pick which symmetries to preserve as well as consider the implications of ignoring the ones they choose not to simulate. All of this adds up to mean that lattice QCD calculations of g-2 have historically been accompanied by very large error bars — that is, until the much smaller error bars from the BMW group’s recent findings. 

These results are not without controversy. The group employs a “staggered fermion” approach to discretizing the lattice, in which a single type of fermion known as a Dirac fermion is put on each lattice point, with additional structure described by neighboring points. Upon taking the “continuum limit,” or the limit that the spacing between points on the lattice goes to zero (hence simulating a continuous space), this results in a theory with four fermions, rather than the sixteen that live in the Standard Model. There are a few advantages to this method, both in terms of reducing computational time and having smaller discretization errors. However, it is still unclear if this approach is valid, and the lattice community is then questioning if these results are not computing observables in some other quantum field theory, rather than the SM quantum field theory. 

The future of g-2

Overall, while a 4.2\sigma discrepancy is certainly more alluring than the previous 3.7\sigma, the conflict between the experimental results and the Standard Model is still somewhat murky. It is crucial to note that the new 4.2\sigma benchmark does not include the BMW group’s calculations, and further incorporation of these values could shift the benchmark around. A consensus from the lattice community on the acceptability of the BMW group’s results is needed, as well as values from other lattice groups utilizing similar methods (which should be steadily rolling out as the months go on). It seems that the future of muon g-2 now rests in the hands of lattice QCD.

At the same time, more and more precise measurements should be coming out of the Muon g-2 Collaboration in the next few years, which will hopefully guide theorists in their quest to accurately predict the anomalous magnetic moment of the muon and help us reach a verdict on this tantalizing evidence of new boundaries in our understanding of elementary particle physics.

Further Reading

BMW paper:

Muon g-2 Collaboration papers:

  1. Main result (PRL): Phys. Rev. Lett. 126, 141801 (2021)
  2. Precession frequency measurement (Phys. Rev. D): Phys. Rev. D 103, 072002 (2021)
  3. Magnetic field measurement (Phys. Rev. A): Phys. Rev. A 103, 042208 (2021)
  4. Beam dynamics (to be published in Phys. Rev. Accel. Beams):

(Almost) Everything You’ve Ever Wanted to Know About Muon g-2 – Experiment edition

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, a_\mu, which is given by

a_\mu = \frac{g-2}{2}.

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 \omega_c). 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 \omega_s).

If the cyclotron and spin frequencies of the muon were exactly the same, then it would have an anomalous magnetic moment a_\mu 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

\omega_a = \omega_s - \omega_c = -a_\mu \frac{eB}{m_\mu}

where m_\mu is the muon mass, e is the muon charge, and B 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 a_\mu.

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

a_\mu = \left( \frac{g_e}{2} \right) \left( \frac{\mu_p}{\mu_e} \right) \left( \frac{m_\mu}{m_e} \right) \left( \frac{\omega_a}{\langle \omega_p \rangle} \right)

where \mu_p/\mu_e is the proton-to-electron magnetic moment ratio, g_e is the electon g-factor, and \langle \omega_p \rangle 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 a_\mu written in this form has the considerable advantage that all of the quantities have been independently and very accurately measured: to 0.00028 ppb (g_e), to 3 ppb (\mu_p/\mu_e), and to 22 ppb (m_\mu/m_e). 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 \omega_a and \omega_p 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: \omega_a, the anomalous precession frequency, and \omega_p, 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:

Sketch of muon production and injection into the storage ring starting from a beam of protons at Fermilab. Source: David Sweigart’s thesis.

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.

Photo of the Muon g-2 storage ring superconducting coil before assembly at Fermilab (2014). Source: personal photo. Link to the full assembled storage ring.

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.

Muon decay and positron trajectories into the calorimeters of the Muon g-2 experiment at Fermilab. Different positron energies correspond to different bends under the magnetic field and different impact positions on the calorimeters. Source: Aaron Fienberg’s thesis.

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).

\omega_a measurement

The key aspect of the \omega_a 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 \omega_s frequency, while the muons themselves travel around the ring with \omega_c 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, \omega_s and \omega_c, 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.

An example of a “wiggle plot” showing the number of positrons recorded by the calorimeters as a function of time. The oscillations are due to the precession of the spin motion compared to the cyclotron motion of the muon. Note: these are “blinded” results, and do not correspond to the real measurement yet (see text). Source: David Sweigart’s thesis.

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:

N(t) = N_0 \; e^{-t/\tau} \left[ 1 + A_0 \; \text{cos}(\omega_a t + \phi_0) \right]

where the five parameters are: N_0, the initial number of positrons; \tau, the time-dilated muon lifetime; A_0, the amplitude of the oscillation which is related to the asymmetry in the positron’s transverse impact position; \omega_a, the sought-after spin precession frequency; and \phi_0, 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 \omega_a measurement.

CategoryBrookhaven [ppb]Fermilab [ppb]Improvements
Gain changes12020Better laser calibration; low-energy threshold
Pileup8040Low-energy samples recorded; calorimeter transverse segmentation
Lost muons9020Better collimation in ring
Coherent Betatron Oscillation70< 30Higher n value (frequency); better match of beam line to storage ring
Electric field and pitch5030Improved tracker; precise storage ring simulations
Estimated systematic uncertainties for the \omega_a measurement, compared to the previous iteration of the experiment at Brookhaven. The total is added in quadrature. Adapted from the Muon g-2 Technical Design Report (TDR).

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

\omega_a(R) = 2 \pi \left(0.2291 \text{MHz} \right) \left[ 1 + (R + \Delta R) / 10^6 \right] .

Here R is the blinded parameter that is used instead of \omega_a, and \Delta R 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 \omega_a analyses) in the announcement on April 7th, except that the blinded R modification will be removed and the true number unveiled.

\omega_p measurement

The measurement of the Larmor frequency \omega_p (and of the magnetic field B) is equally important to the determination of a_\mu and proceeds separately from the \omega_a 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 \langle \omega_p \rangle 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 \omega_p measurement is shown in the table below.

CategoryBrookhaven [ppb]Fermilab [ppb]Improvements
Absolute probe calibration5035More uniform field for calibration
Trolley probe calibration9030Better alignment between trolley and the plunging probe
Trolley measurement5030More uniform field, less position uncertainty
Fixed probe interpolation7030More stable temperature
Muon distribution3010More uniform field, better understanding of muon distribution
Time-dependent external magnetic field5Direct measurement of external field, active feedback
Trolley temperature, others10030Trolley temperature monitor, etc.
Estimated systematic uncertainties for the \langle \omega_p \rangle measurement, compared to the previous iteration of the experiment at Brookhaven. The total is added in quadrature. Adapted from the Muon g-2 Technical Design Report (TDR) and from arxiv:1909.13742.


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 a_\mu 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:

(Almost) Everything You’ve Ever Wanted to Know About Muon g-2, Theoretically

This is post #1 of a three-part series on the Muon g-2 experiment.

April 7th is an eagerly anticipated day. It recalls eagerly anticipated days of years past, which, just like the spring Wednesday one week from today, are marked with an announcement. It harkens back to the discovery of the top quark, the premier observation of tau neutrinos, or the first Higgs boson signal. There have been more than a few misfires along the way, like BICEP2’s purported gravitational wave background, but these days always beget something interesting for the future of physics, even if only an impetus to keep searching. In this case, all the hype surrounds one number: muon g-2. 

This quantity describes the anomalous magnetic dipole moment of the muon, the second-heaviest lepton after the electron, and has been the object of questioning ever since the first measured value was published at CERN in December 1961. Nearly sixty years later, the experiment has gone through a series of iterations, each seeking greater precision on its measured value in order to ascertain its difference from the theoretically-predicted value. New versions of the experiment, at CERN, Brookhaven National Laboratory, and Fermilab, seemed to point toward something unexpected: a discrepancy between the values calculated using the formalism of quantum field theory and the Muon g-2 experimental value. April 7th is an eagerly anticipated day precisely because it could confirm this suspicion. 

It would be a welcome confirmation, although certain to let loose a flock of ambulance-chasers eager to puzzle out the origins of the discrepancy (indeed, many papers are already appearing on the arXiv to hedge their bets on the announcement). Tensions between our theoretical and measured values are, one could argue, exactly what physicists are on the prowl for. We know the Standard Model (SM) is incomplete, and our job is to fill in the missing pieces, tweak the inconsistencies, and extend the model where necessary. This task prerequisites some notion of where we’re going wrong and where to look next. Where better to start than a tension between theory and experiment? Let’s dig in. 

What’s so special about the muon?

The muon is roughly 207 times heavier than the electron, but shares most of its other properties. Like the electron, it has a negative charge which we denote e, and like the other leptons it is not a composite particle, meaning there are no known constituents that make up a muon. Its larger mass proves auspicious in probing physics, as this makes it particularly sensitive to the effects of virtual particles. These are not particles per se — as the name suggests, they are not strictly real — but are instead intermediate players that mediate interactions, and are represented by internal lines in Feynman diagrams like this:

Figure 1: The tree-level channel for muon decay. Source: Imperial College London

Above, we can see one of the main decay channels for the muon: first the muon decays into a muon neutrino \nu_{\mu} and a W^{-} boson, which is one of the three bosons that mediates weak force interactions. Then, the  W^{-} boson decays into an electron e^{-} and electron neutrino \nu_{e}. However, we can’t “stop” this process and observe the W^{-} boson, only the final states of \nu_{\mu}, \nu_{e}, and e^{-}. More precisely, this virtual particle is an excitation of the W^{-} quantum field; they conserve both energy and momentum, but do not necessarily have the same mass as their real counterparts, and are essentially temporary fields.

Given the mass dependence, you could then ask why we don’t instead carry out these experiments using the tau, the even heavier cousin of the muon, and the reason for this has to do with lifetime. The muon is a short-lived particle, meaning it cannot travel long distances without decaying, but the roughly 64 microseconds of life that the accelerator gives it turns out to be enough to measure its decay products. Those products are exactly what our experiments are probing, as we would like to observe the muon’s interactions with other particles. The tau could actually be a similarly useful probe, especially as it could couple more strongly to beyond the Standard Model (BSM) physics due to its heavier mass, but we currently lack the detection capabilities for such an experiment (a few ideas are in the works).

What exactly is the anomalous magnetic dipole moment?

The “g” in “g-2” refers to a quantity called the g-factor, also known as the dimensionless magnetic moment due to its proportionality to the (dimension-ful) magnetic moment \mu, which describes the strength of a magnetic source. This relationship for the muon can be expressed mathematically as

\mu = g \frac{e}{2m_{\mu}} \textbf{S},

where \textbf{S} gives the particle’s spin, e is the charge of an electron, and m_{\mu} is the muon’s mass. Since the “anomalous” part of the anomalous magnetic dipole moment is the muon’s difference from g = 2, we further parametrize this difference by defining the anomalous magnetic dipole moment directly as

a_{\mu} = \frac{g-2}{2}

Where does this difference come from?

The calculation of the anomalous magnetic dipole moment proceeds mostly through quantum electrodynamics (QED), the quantum theory of electromagnetism (which includes photon and lepton interactions), but it also gets contributions from the electroweak sector (W^{-}, W^{+}, Z, and Higgs boson interactions) and the hadronic sector (quark and gluon interactions). We can explicitly split up the SM value of a_{\mu} according to each of these contributions,

a_{\mu}^{SM} = a_{\mu}^{QED} + a_{\mu}^{EW} + a_{\mu}^{Had}.

We classify the interactions of muons with SM particles (or, more generally, between any particles) according to their order in perturbation theory. Tree-level diagrams are interactions like the decay channel in Figure 1, which involve only three-point interactions between particles and can be drawn graphically in a tree-like fashion. The next level of diagrams that contribute are at loop-level, which include an additional leg and usually, as the name suggests, contain some loop-like shape (further orders up involve multiple loops). Calculating the total probability amplitude for a given process necessitates a sum over all possible diagrams, although higher-order diagrams usually do not contribute as much and can generally (but not always) be ignored. In the case of the anomalous magnetic dipole moment, the difference between the tree-level value of g = 2 comes from including the loop-level processes from fields in all the sectors outlined above. We can visualize these effects through the following loop diagrams,

Figure 2: The loop contributions from each of QED, electroweak, and hadronic processes. Source: Particle Data Group

In each of these diagrams, two muons decay to a photon with an internal loop of interactions in some combination of particles . From left to right: the loop is comprised of 1) two muons and a photon \gamma, 2) two muons and a Z boson, 3) two W bosons and a neutrino \nu, and 4) two muons and a photon \gamma, which has some interactions involving hadrons.

Why does this value matter?

In calculating the anomalous magnetic dipole moment, we sum over all of the Feynman loop diagrams that come from known interactions, and these can be directly related to terms in our theory (formally, operators in the Lagrangian) that give rise to a magnetic moment. Working in an SM framework, this means summing over the muon’s quantum interactions with all relevant SM fields, which show up as both external and internal Feynamn diagram lines.

The current accepted experimental value is 116,592,091 \times 10^{-11}, while the SM makes a prediction of 116,591,830 \times 10^{-11} (both come with various error bars on the last 1-2 digits). Although they seem close, they differ by a factor of 3.7 \sigma (standard deviation), which is not quite the 5 \sigma threshold that physicists require to signal a discovery. Of course, this could change with next week’s announcement. Given the increased precision of the latest run of Muon g-2, these values could be confirmed up to 4 \sigma or greater, which would certainly give credence to a mismatch.

Why do the values not agree?

You’ve landed on the key question. There could be several possible explanations for the discrepancy, lying at the roots of both theory and experiment. Historically, it has not been uncommon for anomalies to ultimately be tied back to some experimental or systematic error, either having to do with instrument calibration or some statistical fluctuations. Fermilab’s latest run of Muon g-2 aims to deliver a value with a precision of 1 in 140 parts per billion, while the SM calculation yields a precision of 1 in 400 parts per billion. This means that next week, the Fermilab Muon g-2 collaboration should be able to tell us if these values agree.

Figure 3: The SM contributions to the anomalous magnetic dipole moment are detailed, with values given \times 10^11. HVP is the hadronic vacuum polarization (a process in which the virtual photon loop contains a quark-antiquark pair), while HLbL is hadronic light by light scattering (a process that is similar but involves more virtual photons). These two are main source of uncertainty in the SM theory prediction. Source: Muon g-2 Theory Initiative.

On the theory side, the majority of the SM contribution to the anomalous magnetic dipole moment comes from QED, which is probably the most well-understood and well-tested sector of the SM. But there are also contributions from the electroweak and hadronic sectors — the former can also be calculated precisely, but the latter is much less understood and cannot be computed from first principles. This is due to the fact that the muon’s mass scale is also at the scale of a phenomenon known as confinement, in which quarks cannot be isolated from the hadrons that they form. This has the effect of making calculations in perturbation theory (the prescription outlined above) much more difficult. These calculations can proceed from phenomenology (having some input from experimental parameters) or from a technique called lattice QCD, in which processes in quantum chromodynamics (QCD, the theory of quarks and gluons) are done on a discretized space using various computational methods.

Lattice QCD is an active area of research and the computations are accompanied in turn by large error bars, although the last 20 years of progress in this field has refined the calculations from where they were the last time a Muon g-2 collaboration announced its results. The question as to how much wiggle room theory can provide was addressed as part of the Muon g-2 Theory Initiative, which published its results last summer and used two different techniques to calculate and verify its value for the SM theory prediction. Their methods significantly improved upon previous uncertainty estimations, meaning that although we could argue that the theory should be more understood before pursuing further avenues for an explanation of the anomaly, this holds less weight in the light these advancements.

These further avenues would be, of course, the most exciting and third possible answer to this question: that this difference signals new physics. If particles beyond the SM interacted with the muon in such a way that generated loop diagrams like the ones above, these could very well contribute to the anomalous magnetic dipole moment. Perhaps adding these contributions to the SM value would land us closer to the experimental value. In this way, we can see the incredible power of Muon g-2 as a probe: by measuring the muon’s anomalous magnetic dipole moment to a precision comparable to the SM calculation, we essentially test the completeness of the SM itself.

What could this new physics be?

There are several places we can begin to look. The first and perhaps most natural is within the realm of supersymmetry, which predicts, via a symmetry between fermions (spin-½ particles) and bosons (spin-1 particles), further particle interactions for the muon that would contribute to the value of a_{\mu}. However, this idea probably ultimately falls short: any significant addition to a_{\mu} would have to come from particles in the mass range of 100-500 GeV, which we have been ardently searching for at CERN, to no avail. Some still hold out hope that supersymmetry may prevail in the end, but for now, there’s simply no evidence for its existence.

Another popular alternative has to do with the “dark photon”, which is a hypothetical particle that would mix with the SM photon (the ordinary photon) and couple to charged SM particles, including the muon.  Direct searches are underway for such dark photons, although this scenario is currently disfavored, as it is conjectured that dark photons primarily decay into pairs of charged leptons. The parameter space of possibilities for its existence has been continually whittled down by experiments at BaBar and CERN.

In general, generating new physics involves inserting new degrees of freedom (fields, and hence particles) into our models. There is a vast array of BSM physics that is continually being studied. Although we have a few motivating factors for what new particles that contribute to  a_{\mu} could be,  without sufficient underlying principles and evidence to make our case, it’s anyone’s game. A confirmation of the anomaly on April 7th would surely set off a furious search for potential solutions — however, the precision required to even quash the anomaly would in itself be a wondrous and interesting result.

How do we make these measurements?

Great question! For this I defer to our resident Muon g-2 experimental expert, Andre Sterenberg-Frankenthal, who will be posting a comprehensive answer to this question in the next few days. Stay tuned.

Further Resources:

  1. Fermilab’s Muon g-2 website (where the results will be announced!):
  2. More details on contributions to the anomalous magnetic dipole moment: 
  3. The Muon g-2 Theory Initiative’s results in all of its 196-page glory: