The Particle at the End of the Universe: How the Hunt for the Higgs Boson Leads Us to the Edge of a New World
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Counting degrees of freedom helps explain why gauge bosons are massless without the Higgs. The reason gauge bosons exist in the first place is that there is a local symmetry—one that operates separately at every point in space—and we need to define connection fields that relate the symmetry operations at different points. It turns out that you need precisely two degrees of freedom to define this kind of field. (Trust me here. It’s hard to think of a sensible explanation that doesn’t amount to going through all the math.) When you have a spin-1 or spin-2 particle with just two degrees of freedom, that particle is necessarily massless. The Higgs field is a completely separate degree of freedom; when it gets eaten by the gauge bosons, they now become massive. If there were no extra degrees of freedom lying around, the gauge bosons would have had to remain massless, as they do for the other known forces.
Hopefully this helps explain why physicists were so confident that something like the Higgs must exist, even before it had been discovered. In some sense, it had been discovered—three of the four scalar bosons were already there, as the zero-spin parts of the massive W and Z bosons. All we needed to do was find the fourth.
Why fermions are massless without the Higgs
Here is why the fact that fermions have mass is something that demands an explanation in the first place. Notice that the degrees-of-freedom argument we used for the gauge bosons isn’t relevant in this case; a spin-1/2 fermion has two possible spin values whether it’s massive or massless.
Start by thinking of a massive spin-1/2 particle like the electron. Imagine that it’s moving directly away from us, and we measure its spin to be +1/2 along an axis pointing in the direction of its motion. But we can imagine accelerating our own velocity so that we start catching up to the electron—now we’re moving toward it. Nothing intrinsic to the electron has changed, including its spin, but its velocity with respect to us has. We define a quantity called the “helicity” of a particle, which is the spin as measured along the axis defined by its motion. The helicity of the electron goes from being +1/2 to -1/2, and all we did is change our own motion—we didn’t touch the electron at all. Clearly the helicity isn’t an intrinsic feature of the particle; it depends on how we look at it.
Now consider a massless spin-1/2 fermion (like the electron would be without spontaneous symmetry breaking). Let it be moving away from us, and we measure its spin to be +1/2 along an axis defined by its direction of motion, so its helicity is +1/2. In this case, the fermion is necessarily moving at the speed of light (because that’s what massless particles always do). Therefore we can’t catch up to it and change its apparent direction of motion just by accelerating ourselves. Every observer in the universe will observe this massless particle to have a unique value for its helicity. For massless particles, in other words, the helicity is a well-defined quantity no matter who is measuring it, unlike the case for massive particles. A particle with positive helicity is “right-handed” (spinning counterclockwise as it comes toward us), while a negative-helicity particle is “left-handed” (spinning clockwise as it comes toward us).
And the reason why all this matters is because the weak interactions couple to fermions of one helicity but not the other. In particular, before the Higgs comes along to break the symmetry, the massless gauge bosons of the weak interactions couple to left-handed fermions and not right-handed ones, and they also couple to right-handed antifermions and not left-handed ones. Don’t ask why that’s the way nature works, except that it’s what we need to fit the data. The strong force, gravity, and electromagnetism all couple equally well to left- and right-handed particles; but the weak force couples to one but not the other. That also explains why the weak interactions violate parity: Looking at the world through a mirror switches right with left.
Having a force couple to one helicity but not the other clearly doesn’t make sense if the helicity is different to observers moving at different speeds. Either the weak force couples to a certain particle, or it doesn’t. If the weak force couples only to left-handed particles and right-handed antiparticles, it must be true that such particles have one helicity or the other once and for all. And that can happen only if they move at the speed of light. Which implies, at last, that they must have zero mass.
If you can swallow that, it helps make sense of some of the dancing around we did while first defining the Standard Model. We said that the known fermions come in pairs, which would be symmetric if it weren’t for the Higgs lurking in empty space. Up and down quarks form a pair, electrons and electron neutrinos form a pair, and so on. But really it’s only the left-handed up and down quarks that form a symmetric pair; there is no local symmetry connecting right-handed up quarks to right-handed down quarks, and likewise for the electron and its neutrino. (In the original version of the Standard Model, neutrinos were thought to be massless, and right-handed neutrinos didn’t even exist. Now we know that neutrinos have a small mass, but the status of right-handed neutrinos remains murky.) Once the Higgs fills space, the weak symmetry is broken, and the observed quarks and charged leptons are all massive, with both right- and left-handed helicities allowed.
Now we see why the Higgs is needed in order for Standard Model fermions to have mass. If the weak interaction symmetry were unbroken, helicity would be a fixed property of each fermion, which means that they all would be massless particles moving at the speed of light. It’s all because the weak interactions can tell left from right. If that weren’t true, there would be no obstacle to fermions simply having mass, with or without the Higgs. Indeed, the Higgs itself is a scalar field with mass, but it’s not as if the Higgs gives mass to itself; it simply has mass, since there’s no reason for it not to.
APPENDIX TWO
STANDARD MODEL PARTICLES
Throughout the book we’ve talked about the various particles of the Standard Model, but not always in a systematic way. Here we provide a summary of the particles and their properties.
There are two types of elementary particles: fermions and bosons. Fermions take up space; that is, you cannot put two identical fermions right on top of each other in precisely the same configuration. They therefore serve as the basis for solid objects, from neutron stars to tables. Bosons can be piled on top of one another as much as you like. They therefore are able to create macroscopic force fields, such as the electromagnetic field and the gravitational field.
The fermions
Let’s consider the fermions first. There are twelve fermions in the Standard Model that fall into strict patterns. Fermions that feel the strong nuclear force are quarks, while those that don’t are leptons. There are six types each of quarks and leptons, arranged into three pairs, each pair forming a generation. It’s a rule that the spin of a fermion must equal an integer plus one half; all the known elementary fermions are spin-1/2 particles.
The elementary fermions, with their electric charges and approximate masses. The masses of neutrinos haven’t yet been accurately measured, but they are all lighter than the electron. Quark masses are also approximate; they are hard to measure because quarks are confined inside hadrons.
There are three up-type quarks, with an electrical charge of +2/3 each. In order of increasing mass, they are the up quark, the charm quark, and the top quark. There are also three down-type quarks, with charge -1/3 each: the down quark, the strange quark, and the bottom quark.
Each type of quark comes in three colors. It would be perfectly legitimate to count each color as a separate kind of particle (in which case there would be eighteen types of quarks, not just six), but because the colors are all related by the unbroken symmetry of the strong interactions, we usually don’t bother. All particles with color are confined into colorless combinations known as “hadrons.” There are two simple types of hadrons: mesons, which consist of a quark and an antiquark, and baryons, consisting of three quarks, one each of the three colors red, green, and blue. Protons (two ups and one down) and neutrons (two downs and one up) are both baryons. An example of a meson is th
e pion, which comes in three types: one with positive charge (up plus antidown), one with negative charge (down plus antiup), and one that is neutral (a mixture of up-antiup and down-antidown).
Unlike quarks, leptons are not confined; each one can move by itself through space. The six leptons also come in three generations, each with one neutral particle and one with charge -1. The charged leptons are the electron, muon, and tau. The neutral leptons are the neutrinos: the electron neutrino, the muon neutrino, and the tau neutrino. Neutrino masses are not well understood and don’t arise in the same way as those of other Standard Model fermions, so we have essentially ignored them in this book. They’re known to be small (less than one electron volt) but not zero.
The twelve different fermions should really be thought of as six different matched pairs of particles. Each charged lepton comes in a pair with its associated neutrino, while the up and down quarks form a pair, as do the charm and strange quarks, and the top and bottom. As an example of this pairing in action, when a W- boson decays into an electron and an antineutrino, it’s always an electron antineutrino. Likewise, when a W- decays into a muon, it’s always accompanied by a muon antineutrino, and so on. (You would like to say the same thing about the quarks, but they actually mix together in subtle ways.) The particles within each pair would actually have identical properties, if it weren’t for one sneaky influence lurking in the background: the Higgs field. In the world we see, the particles within each pair have different masses and different electrical charges, but that’s only because the Higgs is hiding their underlying symmetric nature.
Is it possible that the quarks and leptons aren’t really elementary, and that they are actually made of an even smaller level of particles? Sure, it’s possible. Physicists don’t have a vested interest in the current particles being truly elementary; they would love to find yet more mysteries hidden within them, and they have spent a great deal of time inventing models along those lines and testing them experimentally. The hypothetical particles that could make up quarks and leptons even have a name: “preons.” What they don’t have is any experimental evidence, or, for that matter, any compelling theory. The consensus these days is that quarks and leptons seem to be truly elementary, rather than being composites of some other kind of particle, but we can always be surprised.
The bosons
Now we turn to the bosons, which always have integer spins. The Standard Model includes four types of gauge bosons, each arising from local symmetries of nature, and corresponding to a certain force.
Photons, which carry the electromagnetic force, are massless, neutral, spin-1 particles. Gluons, which carry the strong nuclear force, are also massless, neutral, and spin-1. A major difference is that gluons do carry color, so they are confined inside hadrons just like quarks are. Because of these colors there are actually eight different kinds of gluons, but once again they are related by an unbroken symmetry, so we don’t even bother to give them specific labels.
Force-carrying particles, the bosons. Masses are measured in giga electron volts (GeV).
Gravitons, which carry gravity, are also massless and neutral but have spin-2. Gravitons do interact with gravity themselves—because everything interacts with gravity—but for the most part gravity is so weak that you wouldn’t notice. (Things change when you collect a large amount of mass to create a strong gravitational field, of course.) Indeed, the weakness of gravity means that the graviton is mostly irrelevant for particle physics, at least within the Standard Model. Because we don’t have a full theory of quantum gravity, and because individual gravitons are almost impossible to detect, people often don’t include the graviton as a particle, but there’s every reason to believe that it’s real.
The weak force is carried by the charged W bosons and the neutral Z bosons. All three are spin-1 but massive, and they decay quickly when they are produced. It’s the broken symmetry due to the Higgs field that is responsible for the weak bosons becoming massive and differentiating from one another; if it weren’t for the Higgs, the W and Z bosons would be more like gluons, but with only three varieties instead of eight.
Unlike the three forces previously mentioned, the weak force is so feeble that it isn’t able to hold any two particles together all by itself. When other particles interact via the weak force, there are essentially only two ways to do it: Two particles can scatter off each other by exchanging a W or Z, or one massive fermion can decay into a lighter fermion by emitting a W, which then decays into other particles itself. Those processes play a crucial role when it comes to looking for new particles at the LHC.
The Higgs itself is a scalar boson, which is to say that it is spin-0. Unlike the gauge bosons, it doesn’t arise from a symmetry, and there’s no reason to expect that its mass should be zero (or even small). We can talk about a Higgs “force,” and it might even be relevant to detecting dark matter in deep underground experiments. But the major interest in the Higgs comes from the fact that the field on which it is based is nonzero in empty space, and its presence influences other particles by giving them mass.
If you’ve read this far, you’re probably pretty familiar with the Higgs boson.
Table summarizing which particles (bosons and fermions) interact with which forces. Photons carry the electromagnetic force, but they don’t interact directly with themselves, since they are electrically neutral. The origin of neutrino mass is still mysterious, so their interaction with the Higgs is unknown.
APPENDIX THREE
PARTICLES AND THEIR INTERACTIONS
This appendix, in which we talk about Feynman diagrams, is also more technical than the main body of the book. Feel free to skip it, or to just look at the pictures. Richard Feynman himself, when he first invented the diagrams, thought it would be hilarious if someday these little scribbles were all over the place in the physics research journals. That hilarity has come to pass.
Feynman diagrams are a simple way to figure out what can happen when elementary particles get together to interact. Let’s say you want to ask whether a Higgs boson can decay into two photons. You know that photons are massless, and that the Higgs interacts only with particles that have mass, so your first guess might be that it doesn’t happen. But by concatenating Feynman diagrams, you can discover processes in which virtual particles can connect the Higgs to photons. A physicist will then go further, using those diagrams to calculate the actual probability that such an event will occur; each diagram is associated with a specific number, and we add up all the different diagrams to get the final answer. We’re not playing the role of professional physicists, but it’s still helpful to see the various allowed interactions portrayed in Feynman-diagram language. There are many rules that go along with these diagrams; we’ll delve into them just enough to get an idea of what’s going on, but if you want to be precise, it will behoove you to consult a textbook on particle physics or quantum field theory.
Some basic principles: Each diagram is a cartoon of particles interacting with one another and changing identity, with time running from left to right. The incoming particles at the far left of a diagram, and the outgoing particles at the far right, are “real”—they have the mass that we’ve listed in the Particle Zoo tables in Appendix Two. Particles that exist only inside a diagram, not sneaking out to either side, are “virtual”—their mass can be anything at all. That’s worth emphasizing: Virtual particles aren’t real particles, they’re just bookkeeping devices that indicate how quantum fields are vibrating in the course of a particle interaction.
We’ll portray fermions with solid lines, gauge bosons with wavy lines, and scalar bosons (such as the Higgs) with dashed lines. Fermion lines never end—they either travel in closed loops, or they stretch to the beginning and/or end of the diagram. Boson lines, in contrast, can easily come to an end, either on fermion lines or on other boson lines. A place where lines come together is called a “vertex.” At each vertex, electric charge is conserved; so if an electron emits a W boson to turn into a neutrino, we
know it must be a W-. The total number of quarks and the total number of leptons (where antiparticles count as -1) are also conserved at each vertex. We can take any line and turn it backward if we exchange particles with antiparticles. So if an up quark can convert to a down quark by emitting a W+, an antidown can convert to an antiup by the same means.
We’ll start by writing down the basic diagrams of the Standard Model. More complicated diagrams can be constructed by combining these fundamental building blocks in various ways. We’re not going to be completely comprehensive, but hopefully enough that the basic pattern becomes clear.
First, consider what can happen to a single fermion coming in from the left. Fermion lines can’t end, so some sort of fermion has to come out the other side. But we can spit out a boson. Essentially, if a fermion feels a certain force, it can emit the boson that carries that force. Here are some examples.
Every particle feels gravity, so every particle can emit a graviton. (Or absorb a graviton, if we run the diagram backward; like photons and the Higgs, gravitons are their own antiparticles.) Even though we’re drawing a straight line as if the particle is a fermion, there are equivalent diagrams for all the bosons as well.