The Particle at the End of the Universe: How the Hunt for the Higgs Boson Leads Us to the Edge of a New World

by Sean Carroll

  Of course we can wonder why the metaphorical Higgs pendulum is upside down rather than right-side up. The answer is, nobody really knows. There are some speculations, which rely on physics way beyond the Standard Model, but at the present state of knowledge it’s just a brute fact about the universe. There’s nothing wrong with the Higgs taking on a nonzero value in empty space; it either does or it doesn’t, and it turns out that it does. And good thing, because otherwise the world would be a lot more boring (and not just for particle physicists).

  Giving particles mass

  It wouldn’t matter that the Higgs field filled empty space—indeed, we wouldn’t even notice—if it didn’t interact with other particles. And the most obvious effect of that interaction is to “give mass” to the elementary particles of the Standard Model. But this concept is sufficiently subtle that it’s worth our time thinking about it a bit. For even more details about how this works, see Appendix One.

  First and foremost, we should say what the “mass” of an object is. Probably the best way of thinking about it is “how much resistance you feel when you push on the object,” which is another way of saying “how much energy you need to get the thing moving to a certain speed.” A car has a lot more mass than a bicycle, which we know because it takes a lot more work to push a car than to push a bike. Another definition would be the “amount of energy an object has while at rest.” That’s working backward from Einstein’s E = mc2. We usually think of this equation as telling us how much energy there is in an object with a certain mass; equivalently, we can think of it as a definition of the mass of an object that isn’t moving.

  It’s important to emphasize that mass is not directly related to gravity at all. We tend to associate the two, because the easiest way to measure the mass of something is to weigh it by putting it on a scale, and we all know that it’s gravity that’s pulling us down on the scale. Out in empty space where gravity isn’t important, things become weightless, but they still have mass. It is harder to get a massive rocket ship moving than a tiny pebble, and it would be harder still to push the moon or a planet. Gravity is something different, which affects all forms of energy, even ones that have no mass. Light, which consists of massless photons, is definitely affected by gravity, as has been vividly demonstrated by the phenomenon of gravitational lensing (bending of light rays) by galaxies and dark matter in the universe.

  If you take a look at the Particle Zoo table in Appendix Two, you’ll see that some particles have mass and some don’t. Among the force-carrying bosons, the gluons, graviton, and photon all have zero mass, but the W and Z bosons do have mass, as does the Higgs itself. Under the fermions, we see that the neutrino masses are listed as “small,” while the quarks and charged leptons have specific masses.

  This messy situation is ultimately due to the influence of the Higgs field. The rule is simple: If you don’t interact directly with the Higgs, you have zero mass; if you do interact directly with the Higgs, you have a nonzero mass, and your mass is directly proportional to how strong that interaction is. Particles like the electron and up-and-down quarks interact with the Higgs boson relatively weakly, so their masses are small; the tau lepton and the top-and-bottom quarks interact with it strongly, so their masses are relatively large. (The neutrinos are a special case; they have tiny masses, but our understanding of where those masses come from is still far from settled. For the most part we’ll be ignoring them in this book, and sticking to the parts of the Standard Model that we understand.)

  If the Higgs were like other fields, resting at zero in empty space, its interaction strength with other particles would simply measure how likely it would be for the Higgs boson to interact with that particle if they happened to pass by each other. Mostly a Higgs and an electron would pass by in peace, while a Higgs and a top quark would scatter very strongly. (I can pass by strangers on the street without being interrupted, but Angelina Jolie would be hassled at every step.) But because the expectation value is not zero, it’s like the other particles are interacting with it constantly—and it’s those persistent, inevitable interactions with the background that create the mass of the particle. When a particle interacts strongly with the Higgs, it’s as if it carries a large crowd of Higgs hangers-on everywhere it goes, contributing to its mass.

  The formula for the mass of a particle is pretty easy: It’s the value of the Higgs field in empty space, times the particular interaction strength that the particle has with the Higgs. Why do some particles, like the top quark, interact strongly with the Higgs, and others, like the electron, interact relatively weakly? And what explains the specific numbers? Nobody knows. Right now, these are unanswered questions. At the current state of the art, we treat those coupling strengths as constants of nature that we simply have to go out and measure. We’re hoping to get some clues by studying the Higgs itself, which is one reason the LHC is so important.

  A world without Higgs

  Despite all that, it’s misleadingly sloppy to say, “The Higgs is responsible for mass,” as we physicists sometimes do. Remember that we don’t see the quarks directly; they are confined, along with the gluons, inside hadrons such as protons and neutrons. The mass of a proton or neutron is much greater than the masses of its individual quarks, and for good reason; it mostly comes from the energy of the virtual particles that are binding the quarks together. If there were no Higgs, quarks would still bind together to form hadrons, whose masses would be practically unchanged. This means that most of the mass of, say, a desk, or a person, doesn’t come from the Higgs boson at all. The large majority of the mass of ordinary objects comes from their protons and neutrons, and that comes from the strong interactions, not from the Higgs field.

  Which isn’t to say that the Higgs is irrelevant to everyday physics. Imagine we got our hands on a secret control panel that governed all the laws of physics, and by slowly turning the dial labeled HIGGS we could decrease the value of the Higgs field in empty space from 246 GeV to any smaller number. (Note: There is no such secret panel.) As the background Higgs field all around us diminished in value, so would the masses of the quarks, the charged leptons, and the W and Z bosons. The changes in the masses of the quarks and W and Z bosons would lead to tiny changes in the properties of protons and neutrons, but nothing immediately dramatic. The changes to the muon and tau are basically irrelevant to everyday life. But any change in the mass of the electron would be hugely significant.

  In our usual mental cartoon picture of an atom, electrons orbit around the nucleus just like planets orbit the sun or the moon orbits the earth. This is a case where the cartoon breaks down, and we have to take quantum mechanics seriously. Unlike a planet orbiting the sun, a typical electron isn’t orbiting at some random distance; it’s actually going to be as close to the nucleus as it can possibly get. (If it’s farther away, it will tend to lose energy by giving off a photon and therefore move closer.) And how close it can get depends on its mass. Heavy particles can squeeze into small regions of space, while lighter particles are always more spread out. The size of atoms, in other words, is determined by a fundamental parameter of nature, the mass of the electron. If that mass were less, atoms would be a lot larger.

  That’s a big deal. If we made atoms bigger, it’s not as if the size of ordinary objects would grow along with them. What makes ordinary stuff hang together is chemistry—the ways in which atoms stick together in interesting combinations. And the reason they stick together is because they share electrons, at least under the right circumstances. And those circumstances would completely change if atoms had different sizes. If the mass of the electron changed just a little bit, we would still have things like “molecules” and “chemistry,” but the specific rules that we know in the real world would change in important ways. Simple molecules like water (H2O) or methane (CH4) would be basically the same, but complicated molecules like DNA or proteins or living cells would be messed up beyond repair. To bring it home: Change the mass of the electron just a little bit, and a
ll life would instantly end.

  Change the mass of the electron by a lot, and the effects would be correspondingly more dramatic. As the Higgs field got closer and closer to zero, electrons would get lighter and lighter, and atoms would get correspondingly bigger. Eventually they would reach macroscopic size, and then astronomical size. Once every atom is as big as the solar system, or the Milky Way galaxy, there’s no sense in talking about “molecules” anymore. The universe would just be a collection of individual super-enormous atoms, bumping into one another in the cosmos. If the electron mass were turned all the way down to zero, there wouldn’t be atoms at all—the electrons wouldn’t be able to stick to the nuclei. And if that happened suddenly, Hal Eisner’s leading question would be answered—the popcorn kernel would explode.

  There is something more subtle going on, as well. Think of the three charged leptons: the electron, the muon, and the tau. The only differences between these particles are their masses. If we turn off the Higgs field, those masses go to zero, and the particles become identical. (Technical aside: The strong interactions can also give fields expectation values, mimicking the effects of the Higgs but at a much lower value; we’re ignoring those in this discussion.) The same holds true for the three quarks with charge +2/3 (up, charm, and top) and for the three quarks with charge -1/3 (down, strange, and bottom). Each group of particles would be identical if it weren’t for the Higgs background. This points to perhaps the most basic role of the Higgs field: It takes a symmetric situation and breaks it.

  Defining symmetry

  When we think of the word “symmetry,” what comes to mind is a pleasing regularity. Studies have shown that symmetric faces, ones that look the same on the left and the right, are generally found to be more attractive. But physicists (and the mathematicians from whom they learn things like this) want to go deeper, studying what makes something “symmetric” in the most general sense, and how those symmetries appear in nature.

  The simple notion of “matching left and right sides” reflects a broader idea: We say that an object possesses a symmetry whenever we can do something to it and be left with exactly what we started with. For a symmetric face, we can imagine reflecting it around a line down the middle and getting back the same face. But simpler objects can have much more symmetry than that.

  A circle, a square, and a scribble. The circle has a great deal of symmetry, including rotations of any angle and reflections around any axis. The symmetries of the square are fewer: rotations by ninety degrees, reflections around vertical or horizontal axes, or combinations thereof. The scribble has no symmetry at all.

  Think of a geometric figure like a square. We can take its mirror image, reflecting both sides of the square around a vertical axis drawn precisely down the middle, and get back exactly the square we started with—that’s a symmetry. We could also do the same thing around a horizontal axis, which indicates an additional symmetry. (That wouldn’t have worked with a face; even the most beautiful person looks different when seen upside down.) For that matter, we could reflect about either diagonal axis—but not a random axis, which would move the corners of the square around. We can also rotate the square clockwise around its center by ninety degrees, or any multiple thereof.

  A circle, like a square, looks very symmetric, and in fact it’s much more so. We cannot only reflect it around any axis through the center, we can rotate it by any angle whatsoever, and it will always come back to an identical-looking circle. That’s much more freedom than we had with the square. A random scribble, by contrast, doesn’t have any symmetry at all. Any way in which we alter it will leave it looking different.

  A symmetry is a way of saying “we can alter things in some particular way and nothing important changes.” It doesn’t matter if we rotate the square by ninety degrees, or reflect it about a central axis: It ends up looking the same.

  From this perspective, the idea of symmetry might not seem that powerful. So it doesn’t matter if we rotate the circle; who cares? The reason we care is because sufficiently powerful symmetries place very strong constraints on what can possibly happen. Suppose someone tells you, “I have drawn a figure on this piece of paper, with so much symmetry that you can rotate the paper by any angle and the figure will look the same.” Then you know that the figure has to be a circle (or a single point, which is sort of a circle of zero size). That’s the only figure that has so much symmetry. Likewise, when it comes to physics, we can often figure out how experiments should behave just by understanding that there is an underlying symmetry at work.

  A classic case of symmetry in physics is the simple observation that it doesn’t matter where we do a certain experiment; if the experiment reflects basic underlying principles, we will get the same result. For example, there is a famous experiment in which a scientist (usually young, and often filmed for later YouTube consumption) introduces Mentos candies into a bottle of Diet Coke. The porous structure of the mints helps to catalyze the release of carbon dioxide from the soda, resulting in an impressive geyser of foam. The experiment doesn’t work as well with other kinds of candies, or other kinds of soda; but it works exactly the same when carried out in Los Angeles, Buenos Aires, or Hong Kong. There is no symmetry of nature under the interchange of different kinds of food or drink, but there is a symmetry of changing position. Physicists call this “translation invariance,” because they can’t resist the opportunity to give an intimidating name to a simple concept.

  When it comes to particles or fields, symmetries tell us that we can exchange different kinds of particles, or even “rotate them into each other.” (Scare quotes are useful here because we’re transforming fields into each other, not rotating directions in the honest three-dimensional space in which we live.) The most obvious example is the three kinds of colored quarks, conventionally labeled “red,” “green,” and “blue.” Which label is which is completely irrelevant—if you have three quarks in front of you, it doesn’t matter which one you call the “red quark” and which one you call the “blue quark” and which one you call the “green quark.” You can change those labels and all the important physics remains unaltered—that’s the power of the symmetry. If you had one quark and one electron, you wouldn’t want to switch their labels. A quark is very different from an electron; it has a different mass, a different charge, and it feels the strong interaction. There’s no symmetry at work there.

  If it wasn’t for the Higgs field giving masses to the elementary particles, there would be a symmetry that related the electron, muon, and tau, since those particles would be identical in every way, just as Angelina and I moved at equal speeds through the empty room. We could switch a muon in for an electron in some interaction, and the details would be the same. We could even (according to the rules of quantum mechanics) make a particle that was half-electron and half-muon, and it would also be identical, or for that matter any combination of the three particles—much like we can rotate a circle by any angle. Similar symmetries would apply to the up/charm/top quarks, as well as to the down/strange/bottom quarks. These are known as “flavor” symmetries, and even though the Higgs prevents them from being perfectly respected in nature, they remain very helpful to particle physicists analyzing different basic processes.

  But there’s another symmetry, deeper and more subtle than the flavor symmetries, that seems completely hidden at first but turns out to be of absolutely crucial importance. That’s the symmetry underlying the weak interactions.

  Connections and forces

  The real importance of symmetries—the reason why physicists can’t stop talking and thinking about them—is that sufficiently powerful symmetries give rise to forces of nature. That’s one of the most astonishing insights of twentieth-century physics, but it’s not an easy one to grasp. It’s worth going down the rabbit hole just a bit to understand how symmetries and forces are connected.

  Just as there is a symmetry of the everyday world that says “it doesn’t matter where you do your experiment,” there is a
nother one that says “it doesn’t matter in which direction your experiment is pointing.” Put the Mentos in the Diet Coke and watch the foam fly; then rotate the whole apparatus from facing north to facing east, do it again, and (within experimental uncertainties) you should get the same result. This is called “rotational invariance,” for obvious reasons.

  In fact it goes further than that. Let’s say I’m doing my experiment in the parking lot outside my office, and a friend is doing another experiment a few feet away, completely unconnected to mine. We could both rotate our equipment by some angle and expect to get the same results. But even better, I can rotate my equipment and she could keep hers just as it was, or we could both rotate by some arbitrary angle. In other words, the symmetry is not just a single rotation of the world (it doesn’t matter whether we’re all facing north, or some other direction), but separate rotations at every single point (it doesn’t matter what direction any of us is individually pointing in).

  That’s an enormously larger amount of symmetry. In the trade this kind of megasymmetry is called a “gauge invariance.” The name was given by German mathematician Hermann Weyl, who likened the choice of how to measure things at different points to the choice of gauge (distance between rails) in railroad tracks. They are also called “local” symmetries, since we can do the symmetry transformation separately at every location. A “global” symmetry, by contrast, would be based on a transformation that must be carried out uniformly everywhere at the same time. (Local doesn’t mean “only at one point”; it means “separately at every point.” Local symmetries are bigger and more powerful than global symmetries.)