The anomalous magnetic moment of the muon and its potential for new particle physics phenomena

In this article, I discuss a quantity that is extremely well predicted and measured, and that subsequently offers a great handle to constrain new phenomena in particle physics. This quantity is named the anomalous magnetic moment of the muon.

It is extremely precisely predicted by the Standard Model of particle physics, the value being
0.00116591828 ± 0.00000000049

On the other side, the best measurement (measured using the magnet in the picture) is given by
0.00116592089 ± 0.00000000063

Again: precision.

There is however only one chance out of about 1000 that these two results agree, when error bars are accounted for. Oho?! Isn’t it where new phenomena could occur?

In this post, I will define what is the anomalous magnetic moment of the muon and explain how those very precise predictions and measurements are made and can be used to constrain new phenomena in particle physics.

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MUONS

In the anomalous magnetic moment of the muon, there is the word muon. I know I have defined those beasts in other posts, but I would like to try to write something self-contained today.

A muon is the little guy represented by a plush on the left (particles can always be represented by plushes).

More seriously, a muon can be seen as an electron, but in a heavier version. It has the same electric charge, spin and properties.

One exception, the muon is unstable so that after some time, it decays into an electron and two invisible neutrinos.

Muons have been discovered (in the 1930s) thanks to the fact that Earth is bombarded by muons, originating from cosmic rays, on a regular basis. Thousands of them indeed reach each place on Earth every minute.

Muons are deeply studied since then, the measurement of their anomalous moment being one of the most famous study.

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MAGNETIC MOMENTS

Let us focus on the title of the post another time and the magnetic moment part in particular.

The magnetic moment of an object tells us about the magnitude and direction of the rotational force that it feels when it is plunged into an external magnetic field.

To take a simple example, we can imagine a compass.

The needle of the compass will rotate to align itself parallelly to the surrounding magnetic field.

The rotational force (or the torque) experienced by the needle is proportional, on the one hand, to the magnetic field (the more intense is the field, the larger is the torque) but also, on the other hand, to the magnetic moment.

But the relative direction between both also plays a role, so that when both the magnetic moment and the magnetic field are aligned, the torque vanishes and the needle stops.

We actually say that the magnetic moment of any object tries to get aligned with the surrounding magnetic field.

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WHERE ARE MAGNETIC MOMENTS COMING FROM?

There are two contributions to a magnetic moment (that must be summed): a first comes from the motion of electric charges, and a second is related to the intrinsic nature of the object constituent.

For instance, the magnetic moment of a hydrogen atom is given by the sum of the intrinsic moment of the constituting proton and electron, as well as the contribution stemming from the motion of the electron around the proton.

This intrinsic magnetic moment (μ) is the one we are interested in. It is determined by the spin (S), the mass (m) and the electric charge (q) of the particle under consideration,

μ = g_s q/2m S

But what is this g_s-factor in the middle of the formula? This quantity is actually called the g-factor, and it can be predicted by the theory. The predictions say g=2.

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THE ANOMALOUS MAGNETIC MOMENT OF THE MUON

However, the microscopic world is quantum… and quantum mechanics implies that there are quantum corrections to the value of the g-factor, making it slightly different from 2. The difference with 2 is called the anomalous magnetic moment.

One can calculate those quantum corrections, in the context of the Standard Model of particle physics, and one gets the number shown at the beginning of this post:
0.00116591828 ± 0.00000000049

Experimentally, muon beams are prepared by first smashing protons on a fixed target, and by then collecting the remnants of these collisions. After a while, those remnants (called pions) decay into muons that are gathered into a beam.

The experimental value has been measured at the Brookhaven National Laboratory,
0.00116592089 ± 0.00000000063

To handle this measurement, physicists prepared a muon beam, switched on a huge magnetic field and studied the behavior of the muons in the field.

The spin of the latter precesses, and the precession rate can be very precisely measured.

From the related measurements (through the observation of the properties of the muon decay products) and a very good knowledge of the magnetic field, physicists can then extract the experimental value of the anomalous magnetic moment of the muon with a very good precision.

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TOWARDS NEW PHENOMENA

There is thus a slight difference between the theoretical and experimental value of the anomalous magnetic moment of the muon.

The reason may be that the deviation between theory and experiment stems from some new particles that we have not observed yet, and that must thus be included in the calculations**.

Inverting the argument, we have here a way to constrain the presence of new particles indirectly. They should preferably contribute in reducing the gap between theory and experiment (or there are chances that the new theory can go to the bin).

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SUMMARY AND REFERENCES

In this post, I discussed a little bit what is the anomalous moment of the muon. I detailed how the theoretical predictions and the experimental measurement are made.

Although new measurement will come (data taking at the g-2 experiment at Fermilab has just started), I quickly sketched how current results can already be used as constraints on new phenomena.

As a bonus and for a little reward: where is the name “g-2” for the Fermilab experiment coming from?

References:

• The topic of this post was inspired by this article that I have recently read.
• More information on magnetic moments can be found on many lecture notes available all over the web, and even on Wikipedia.
• Experimental information can be found on the BNL website (an older experiment) here and on the Fermilab website (the new experiment) there.

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