From the Fledgling Physicist Archives: An Introduction to Axions

I wrote this in December 2013 and published it on my blog as a six-part series called "A Series on Axions". 



I’ve been taking a course on particle cosmology. We had to choose a topic to write a research paper on, and, being interested in particle physics, I chose to research axions. 

Axions were proposed by Peccei and Quinn in 1977 to solve the Strong CP Problem, which, in a nutshell, is that CP violation doesn’t occur in QCD but there is no reason why. The Strong CP problem is one of the greatest unsolved mysteries in physics, and, if axions are discovered, it will have been solved. It turns out that, in addition to solving the strong CP problem, discovering axions could mean that we have discovered the constituents of dark matter, as axions are a perfect candidate for the mysterious matter that makes up 80% of matter in the universe!

I’ve decided to write up a short series here on my blog about axions, covering the Strong CP Problem, the Peccei-Quinn Mechanism, and axions in cosmology. I’ve found axions incredibly interesting, and maybe someone else will find them as enthralling as I do!



Our story begins in 1975, before axions were proposed. There was this big problem in Quantum Chromodynamics (QCD), which Weinberg named “The U(1) Problem”…before I describe it, let’s back up a little big and decipher what all of this means, and we’ll get to it in the next post. 

To put it really simply, the Standard Model of particle physics is composed of several sub-theories that describe the interactions between different types of particles: quarks (which carry a “color” charge), leptons (like electrons), gauge bosons (like photons), and the Higgs boson. The way that these particles interact has to do with specific gauge symmetries in the standard model. QCD describes the interactions between quarks, and has a special particle called a gluon which is the gauge boson of the gauge group that describes QCD, which is called SU(3). For leptons, we have the Electroweak Theory, which describes electromagnetic and weak interactions; the electroweak theory deals with the leptons we all know and love (electrons, muons, taus, and their antiparticles), along with gauge bosons that come from this symmetry group called SU(2) x U(1) interactions too, but the leptons don’t interact with quarks through the strong force because they are only part of the SU(2) x U(1) group, and not of SU(3). All together, the gauge group of the standard model is all of these gauge groups combined: SU(3) x SU(2) x U(1). 

The Higgs boson plays a special role in all of this. See, there was an unexplained problem in the Standard Model: except for the W and Z bosons, all the gauge bosons were massless. The fact that the photon and gluon were massless made sense, but the W and Z not only had mass, they were very heavy, and there was no reason why.

A bunch of guys in the 1960’s came up with this crazy idea: suppose that there was some extra field in the standard model, that did something to break a symmetry that would otherwise prevent the W and Z bosons from having mass. They worked it out, and discovered that if they assumed this field was real, it would indeed solve this problem! The particle associated with this new field was called the Higgs boson, named after one of the many guys who worked on it (Peter Higgs).

The fact that a symmetry can be spontaneous broken, and give us a particle, like in the case of the Higgs mechanism, is really important to my later discussion about axions, because axions are kind of like the Higgs. At a really, really high energies, before the Higgs breaks the electroweak symmetry, there are no W bosons, Z bosons, or photons, there are just three special particles called W’s (of which there are three) and B (of which there is one), and the Higgs field is just hanging out, and not doing anything to these W’s and B’s.

At some point, the Higgs acquires what is called a “vacuum expectation value”, which is the lowest value of energy that the field can have (it’s the minimum!), and the electroweak symmetry is broken. Everything in the electroweak sector goes crazy, and the three W’s and the one B get all mixed up. The B and one of the W’s get mixed together, and out come two different particles: the photon and the neutral Z boson (which gets a mass!). The two leftover W’s get mass from this weird mixing, which is described in terms of what is called the “weak mixing angle” (which is how we quantify the particles getting mixed up), and two different, massive, charged bosons come out: the W+ and the W-. This leaves us with all of the electroweak gauge bosons in the standard model, along with the Higgs boson!

Now that I’ve mentioned the basics of the Standard Model, and little bit about symmetry breaking, we can get back to axions…


Weinberg and the U(1) Problem

Now we are back in 1975. Steven Weinberg, one of the greatest physicists of all time, realizes there is a big problem in the standard model, and he calls this problem “The U(1) Problem”.

When you look at the equation that describes the way quantum chromodynamics acts – its Lagrangian – you find this global symmetry in the limit that all the quark masses go to zero. This global symmetry is composed of  a vector symmetry (for isospin and baryon number) times what is called an “axial” symmetry (which basically just corresponds to rotating something about its axis). That is, if you are setting N flavors of quark masses to zero, this symmetry is U(N)v x U(N)a (where v = vector and a = axial).

In QCD, if we have this global symmetry, we need all of the strong interactions to be invariant under U(2)v x U(2)a. We’ve verified that strong interactions are invariant under U(2)v,  but axial symmetries behave differently. In a nutshell, we see a U(2)a symmetry, but not a U(1)a symmetry! Weinberg pointed this problem out, in his paper “The U(1) Problem”, and said that there simply must be no axial U(1) symmetry in quantum chromodynamics.

(Read: Weinberg’s original paper:


The Strong CP Problem

Clearly, this U(1)a problem that Weinberg discovered needed to be solved, and so Gerard ‘t Hooft, a brilliant Dutch physicist, looked into it, and found the solution. There was no U(1)a problem, he said, because the vacuum of QCD is so complicated that there is no true U(1) a symmetry, even though the QCD Lagrangian makes it look like there is. Unfortunately, the newly-discovered complex QCD vacuum came with an even larger problem than Weinberg’s U(1)a problem.

Discovering the complicated nature of the QCD vacuum added a new term to the QCD Lagrangian – one that contained something called  “the quark mass mixing phase”, or “theta”. Due to the kind of gauge theory that QCD is (nonabelian), such a parameter is required. Theta comes with a lot of baggage: if it is not zero or less than zero, something called “CP symmetry” is violated.

The “C” in CP stands for charge conjugation: for this to be preserved, the physical system in question must be invariant under swapping a particle for its antiparticle. Parity (the “P” in CP), on the other hand, requires that a physical system be invariant under inversion of its spatial coordinates. Therefore, CP symmetry tells us that if we were to swap out particles for antiparticles, and invert the spatial coordinates of our system, the new system with the switched particles would be physically equivalent to the old system that we had before we messed around with the particles and the coordinates!

Now CP symmetry is violated in weak interactions, but not in strong or electromagnetic interactions. It makes sense why it’s not violated in electromagnetic interactions, but since it’s preserved in QCD, this means that the theta term in the QCD Lagrangian must be zero or, well, basically zero. There isn’t any reason why this should be the case, though, because of all the values that theta can lie between (from -pi to pi), it is absurd that some how, miraculously, it found itself a value in which we get CP symmetry. This is a so-called “fine tuning problem” in the standard model, and we call it “The Strong CP Problem”.

The Strong CP Problem is one of nature’s great unsolved mysteries, and it’s up there on the list with quantum gravity and dark energy . We know so much about the standard model, and we’ve even now found the Higgs boson, but we have no clue why on earth this theta in quantum chromodynamics is zero (or very close to zero).


The Peccei-Quinn Mechanism

In 1977, two physicists named Roberto Peccei and Helen Quinn came along and proposed something radical: let’s add an additional U(1) symmetry to the standard model. This wouldn’t just be any symmetry, they said – it’ll be a global symmetry that we will force to be spontaneously broken, resulting in a new gauge boson that will take the place of the troublesome “theta” in QCD. There is a lot of jargon in these last few sentences, so let’s break it down.

Spontaneous symmetry breaking is when you have a physical system that is all physically symmetrical, and do something to “break” the symmetry, like force your physical system to change in some way. For example, the Higgs field “breaks” the electroweak symmetry when it acquires a vacuum expectation value and changes all of the masses in the electroweak sector of the standard model.

Usually, when you spontaneously break a global symmetry, you get something called a “Nambu-Goldstone boson”. These Nambu-Goldstone bosons don’t have any spin, charge, or mass. The Higgs, which I mentioned in Part 1, is almost a Nambu-Goldstone boson, with one exception: it has mass. When the symmetry is broken in a certain way, if the symmetry isn’t “exact”, the Nambu-Goldstone boson acquires mass, and becomes a pseudo-Nambu-Goldstone boson.

This is exactly what happens with the Peccei-Quinn mechanism. Basically, you add an additional U(1) symmetry to the standard model (like I mentioned earlier). This additional symmetry comes with its own field, i.e., with a gauge boson called the axion. When you force this symmetry to be spontaneously broken, the potential of the axion falls into its minimum. Like in the case of the Higgs, the symmetry isn’t exact, which is due to that crazy QCD vacuum that Gerard ‘t Hooft figured out. This means that the axion is a pseudo-Nambu-Goldstone boson – it has no spin, and no charge, but isn’t massless.

Adding the axion field changes everything in QCD, because you can make it fit into the QCD Lagrangian so that it replaces that “theta” that screwed everything up. Since the axion replaces “theta” and its sitting at the bottom of its potential (where it is zero), it gives us a perfect explanation as to why CP violation doesn’t occur in strong (QCD) interactions.

There is one problem, however: we still haven’t solved the Strong CP Problem, because, while we’ve been searching for the axions like we searched for the Higgs, we haven’t found it yet. The axion is extremely hard to find, because it doesn’t interact with many of the particles in the standard model, with the exception of photons, which it can decay to. While this makes it difficult to detect, it makes it the perfect candidate for dark matter.

(Read: Peccei and Quinn’s 1977 paper:


Axions, Dark Matter, and Cosmic Strings

In cosmology, the roles that axions play are pretty complicated, but I’m going to try to do my best to explain them.

80% of the matter in the universe is made up of “dark matter”, and we don’t really have any idea what it is. A lot of cosmologists think that, along with dark energy, dark matter may be left over from something that happened during the big bang. Cosmologists have proposed that dark matter could be composed of the LSP (the lightest supersymmetric particle), gravitinos (the supersymmetric partner of the graviton, which is the “gravity” particle that we haven’t found yet), or, interestingly enough, axions!

The reason why axions make a great dark matter candidate is because, as I briefly mentioned earlier, they basically don’t interact with any of the other particles in the standard model with the exception of photons. If axions were the constituents of dark matter, the question “why doesn’t dark matter interact with the rest of the matter we know of?” would be solved.

There are two types of dark matter – “hot” dark matter, and “cold” dark matter, and axions could possibly constitute both. To understand why, let’s take a look at a special theory in cosmology about axions (there are, of course, other theories about axions in cosmology, but this one is my favorite).

In cosmology, there are these hypothesized things called “cosmic strings”. Contrary to their name, these aren’t the same sorts of strings that are studied in string theory – rather, they are a certain kind of “topological defect” that physicist Tom Kibble hypothesized played a role in the evolution of the universe, and they form when a symmetry is broken.

Let’s suppose that these cosmic strings exist, as do axions, and look at the early universe. While the universe is evolving and the temperature of the universe is changing, different symmetries are being broken. At a certain temperature, we have the global U(1) symmetry that Peccei and Quinn proposed, along with its (at first) massless axions. Not long after, this symmetry is broken, and axions acquire mass while these massive cosmic strings are formed.

As the universe is expanding, these cosmic strings begin to radiate axions. The axions emitted by these strings could be either relativistic or non-relativistic. If they are relativistic, then they make up what is called “hot” dark matter, and would be very fast and energetic. If, on the other hand, the axions are non-relativistic, they could make up “cold” dark matter, because they would be very slow.

There is a problem with the axions being relativistic, however, because axions that were relativistic would be waaay too energetic to make dark energy: due to the way that they decay to photons, they would all decay quickly and there would be no dark matter in the universe today!

If the axions were non-relativistic, then they would be radiated by these big cosmic strings as the strings begin to loop around one another and form “domain walls”, which are two-dimensional topological defects (strings are one-dimensional topological defects). At this point, all of the axions would fall to the minimum of their potential where they remain now, and form clusters throughout the universe, becoming the dark matter we are trying to understand today.


Experimental Searches for Axions

In my earlier posts, I mentioned several times that axions don’t interact very much with the particles in the standard model, with the exception of photons. Awesomely enough, not only can axions decay to photons, but photons can decay to axions as well! This is called the Primakoff Effect (and Inverse Primakoff Effect): when axions are in a strong magnetic field, they decay into photons (and vice versa, which is the Inverse Primakoff Effect). This opens up a big opportunity for experimental searches for axions – in principle, if axions exist, we can exploit the Primakoff Effect by using large magnets to induce magnetic fields that would – hopefully – lead axions to decay to photons and vice versa.

There are three types of axion experiments going on today – all of which are trying to exploit the Primakoff Effect: experiments using helioscopes (like CAST), some experiments using lasers (like PVLAS), and RF cavity experiments (like ADMX). Let’s take a closer look at each of these:

(1) ADMX. ADMX (The Axion Dark Matter eXperiment) is searching for axions converting into microwave photons. They use a huge 8 Tesla magnet in an RF cavity, and try to induce these axion-photon conversions and “tune” into (like tuning into a radio station) the frequency of the axions that have clustered around the halo of our galaxy (and in the Milk Way!).

(2) PVLAS. The PVLAS (Polarizzazione del Vuoto con LASer) experiment has been searching for axions since 2001. They use a gigantic polarized laser beam and a magnetic field to search for weird changes in the rotations of photon polarization. Due to the Primakoff effect, when axion-photon conversions occur, you should, in principle, be able to visibly detect changes in the photon polarization.

(3) CAST. Like ADMX and PVLAS, CAST (the CERN Axion Solar Telescope) is also searching for axion-photon conversions, but does so using a large telescope pointed at the Sun. They put a huge magnet near the telescope, and try to track axions that come from the Sun and convert them into photons!

Unfortunately, right now, there are no conclusive results from these experiments, though they have started singling out certain values that the axion mass can’t be – since they don’t see any axions at a certain mass, they can “exclude” these masses as possible values the axion mass could be.

To conclude my short summary of axions, I think that the discovery of axions would be nothing less than a victory for physics. If axions exist as described by Peccei-Quinn Theory, discovering them would mean a solution to the Strong CP Problem, which would be an amazing step forward for particle physics. If it happens to be the case that axions are the constituents of dark matter, their discovery would be revolutionary not only for particle physics, but for cosmology as well!