(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!): https://muon-g-2.fnal.gov/
  2. More details on contributions to the anomalous magnetic dipole moment: https://pdg.lbl.gov/2019/reviews/rpp2018-rev-g-2-muon-anom-mag-moment.pdf 
  3. The Muon g-2 Theory Initiative’s results in all of its 196-page glory: https://arxiv.org/pdf/2006.04822.pdf
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Amara McCune

Amara McCune is a PhD student in theoretical physics at UC Santa Barbara focusing on phenomenology. She has bachelor’s degrees in physics and mathematics from Stanford University, where she completed research projects in AMO, cosmology, and spent a summer at CERN as part of the University of Michigan REU program. Her current projects focus on parity solutions to the strong CP problem, and she is generally interested in cosmological applications to beyond the Standard Model (BSM) physics, including dark matter, the CMB, and inflation. She is passionate about both teaching and outreach and spent part of this past summer volunteering as a teaching fellow for the Mongolian Young Scholars Program (MYSP) in Ulaanbaatar.

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