How the Higgs Boson Gives Particles Mass 
Peter Higgs' famous boson has been in the news again lately. I've seen all sorts of "explanations" and analogies given in the press and even in popular science mags about how particles acquire mass. Many of these are just plain wrong and most don't even come close to what the theory actually says. So I thought I'd have a go at a much simplified explanation of the Higgs' mechanism. It misses out a lot of the detail, but I hope it conveys the basics.

My usual warning about when I do stuff like this: I'm not a physicist, I'm just an amateur student who reads physics textbooks for fun. I know we get a couple of proper physicists and mathematicians along here occasionally. If I've got the whole thing seriously wrong (or even a little bit wrong) then I'll be very happy to be corrected.

So how does the Higgs boson give other particles their mass? The short answer is, it doesn't! It's something called the Higgs Field that gives particles their mass. To understand what the Higgs field is, we first need to understand what physicists are talking about when they talk about fields. A good example here might be a gravitational field. This is something that fills all of space. "It surrounds us and penetrates us; it binds the galaxy together..." no, no, no that's Star Wars, although come to think of it, regarding the gravitational field that's actually true!

Near large concentrations of mass, like stars, planets or black holes, the gravitational field is very strong. Way out in the depths of inter-galactic space, it's very close to zero. The same is true of electric fields. Close to a large concentration of electric charge, the field is strong, but where the net charge is zero, the electric field will also be zero.

It probably doesn't surprise you to learn that it takes a large amount of energy to create a strong field. A strong gravitational field requires a large amount of mass (which is the same thing as energy). It takes a lot of energy to bring a lot of charge together. So you can think of a strong field as possessing a large amount of energy and a very weak field as possessing little or no energy.

The way fields are defined in physics, the energy stored in a field is always proportional to the value of the field multiplied by the same value again: the field squared in other words. So if we were to graph the energy in the field, E, against the value of the field, F, we would get something like the following.

As you can see, more field means more energy. Almost all the fields we're familiar with look something like the above.[1] The Higgs field is different. Instead of having the shape that almost all other fields have, it looks more like this.

There are a couple of things to notice about this graph.

1. The minimum energy doesn't occur when the field is at zero, it occurs at a different value that I've labelled H0.
2. The graph is symmetric and has two possible minima.

Dealing with point 2 first. The full Higgs theory has more degrees of freedom than I've shown. Imagine the second graph being rotated around the E axis, to form a shape a bit like the bottom of a champagne bottle. The resulting field actually has an infinite number of minima. At some point in the early universe, and we're talking some tiny fraction of a second after the Big Bang, the Higgs field was pushed into one of these minimum values and has been there ever since. This is an example of what physicists called spontaneous symmetry breaking. The initial conditions were symmetric but unstable. Imagine trying to balance a football at the dead centre of a hill shaped like the Higgs graph. It would quickly fall one way or the other, breaking the symmetry.

You might wonder how we might go about finding which minimum nature has chosen. Fortunately, we don't have to. All the minima have the same energy and they can all be transformed into one another using a simple rotation. In practice, we are free to choose whatever minimum we like and physicists tend to choose a value that makes the calculations as simple as possible.

We'll stick to the simple version with only two minima and arbitrarily choose the positive value of H0.

Point 1 is the property that we're really interested in. In order to make some progress, we need a particle field that interacts with the Higgs field. I'm arbitrarily going to choose the electron field, as the electron is a particle that most people are familiar with, but the same argument will apply to all particles with mass. Next I have to say what I mean by an electron field. I'm guessing that most people probably feel reasonably comfortable with the idea of gravitational and electric fields: more mass means more gravitational field, more electric charge means more electric field. The same is true of the electron field: more electrons means more electron field.

In quantum field theory, fields are taken as the most fundamental concept. Particles are the minimum excitation of those fields: their quanta. [2] The reason we're used to thinking about electric and gravitational fields is that they have classical counterparts that we're used to dealing with in the "big world" - the world on the kind of scales that we're used to. The quanta (i.e.the particles) of the electric and gravitational fields are fundamentally different from those of the electron field. The former can bunch together to form the kind of field we're used to measuring at classical scales. The quanta of the electron field like to stay aloof from one another and always look more like distinct particles to us. [3]

Let's call our electron field F and the Higgs field H. We'll call the part of the energy due to their interaction E. As the energy should increase with both these fields we might assume a relationship that is something like.

E = HF

But remember that I said that energy is proportional to field strength squared, so we really want something like.

E = (HF)2

In addition, we should add a parameter, q, that quantifies the strength of the interaction between the two fields.

E = (qHF)2

A couple of things to notice about this.

1. If the strength of the interaction, q, is zero, then there is zero energy stored due to the interaction. This allows some particles to remain massless.
2. If the value of the H field is zero then the interaction energy is also zero. But we're assuming that the value of the H field is not zero, since that value is unstable!

Next we apply a little trick. The Higgs field, H, is non-zero everwhere. If it varies at all, it varies about the value H0. What we're really interested in is not the absolute value of H, but how much it differs from H0. So we define this "difference field", little h, as:

H = H0 + h.

It's when we place this in our expression for the interaction energy that something really interesting happens.

E = (qHF)2 = (q[H0 + h]F)2 = (qH0)2F2 + ...

The first thing to say about the above equation is DON'T PANIC! All I've done is taken our simple expression for the energy due to the interaction, replaced the Higgs field, H, with the "difference field", H0 + h, and thrown away everything but one of the resulting terms (the "+ ..." just means "everything else" ).

It's that one remaining term, (qH0)2F2, that we're interested in. Remember that the energy is proportional to the field squared, F2. In this case it is multiplied by a constant value (qH0)2. One quantum of the electron field, F, results in one fixed quantum of energy (qH0)2, and this is true regardless of anything else that that quantum (i.e. that electron) is doing, even if it is just sitting there at rest, doing nothing. In other words, (qH0)2 is the rest mass (squared) of the electron!

Notice that it's the broken symmetry of the Higgs field, the fact that its minimum energy occurs when the field is non-zero, or H0 as I've called it, that is the crucial property that creates a non-zero mass. As Matt Strassler's Higgs FAQ points out, nobody has the faintest idea why the Higgs field has this property.

If you've got this far then you probably have a whole bunch of objections to the above mechanism. First and foremost, why on earth do we bother inventing this elaborate mechanism for something like the electron? Why not just stick an m2F2 term on in the first place? After all, we know what the mass of the electron is? The answer is that the electron displays a very particular type of symmetry. Every symmetry is associated with a conservation law, in this case the law of conservation of charge. Forcing this symmetry to hold locally rather than globally introduces the need for the electromagnetic field. This type of theory is called a "gauge theory" and the symmetries that underlie them are believed to be fundamental to nature. Adding interaction terms for the electron does not interfere with this symmetry, adding mass terms does. For more details on how this works, see here.

The next objection might be that there seems to be a bit of smoke and mirrors going on here. We've replaced one arbitrary constant, the mass of the electron, with two constants, the interaction strength, q, and the Higgs field constant, H0. Actually it's much worse than this. There are other constants that define the exact shape of the Higgs field graph and that determine things like the Higgs particle mass. However, many of these constants are inter-related. Particle physicists can use them to determine things like decay modes, decay rates, scattering angles and all the kind of things that they can measure at the LHC. As far as I know, all of these calculations have turned out pretty close to the values observed for the behaviour of the Higgs boson.

I think that's why the Higgs boson discovery is so important. It's existence is the final piece of confirmation that something like the Higgs field exists. I'm also assuming (I haven't waded into the detailed calculations) that the mass of the Higgs ties down the values of other parameters and that in turn allows some of these other calculations to be verified.

I should also point out another important point that Matt Strassler raises: the Higgs mechanism is only one source of particle mass. The mass of the proton, for example, doesn't just consist of the masses of the quarks whizzing about inside it. A substantial part of the mass of the proton comes from other factors, like the kinetic energy of its constituents.

Having seen just how long my simplified introduction to the Higgs field has become, I suppose I can understand why newspapers just say, "the particle that gives all other particles mass."

[1] Graphs are plotted using
[2] I've borrowed liberally from Matt Strassler's blog in some of my descriptions. He's a theoretical physicist who puts a great deal of effort into communicating these ideas as straightforwardly as possible. His blog is well worth browsing.
[3] The quanta of the electric and gravitational fields are the photon and the (hypothetical) graviton. These have integer spin values of one and two respectively. Particles with integer spin are called Bosons. Multiple Bosons can exist in the same state: their wave functions can be superimposed on one another. The Higgs particle has spin 0, an integer value, so it counts as a Boson. The quanta of the electron field, the electron, has half integer spin. It's an example of a Fermion. The fields of spin half particles obey a different dynamic equation, called the Dirac equation. In order for this to provide consistent notions of energy and particle counts the quanta cannot be allowed in the same state. This relation between a particle's spin and the states they can adopt is called the spin-statistics theorem. It's this property of fermions that forces different electrons into different energy states within atoms, giving different atoms their different chemical properties.