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

  Because we can set up our equipment in different directions at every point, it becomes crucial that we can somehow compare the actual setup we choose at different points. Think of surveyors, laying out the plans for a new house. They can start with one corner, which fixes the direction in which the house will be oriented. But, presuming the house has the shape of a rectangle, they’re going to want the orientation of the other corners to line up with the first one; you can’t have the bricks at the four corners of your house just pointing in random directions. In the real world, this usually isn’t too hard; we simply need to draw some straight lines, either by pulling string between the points or through the use of surveying equipment.

  Imagine, however, that the ground on which we’re building our house isn’t completely level. The terrain is bumpy, and for aesthetic reasons our client wants us to build on top of them rather than just bringing in the bulldozers and leveling the place. In that case, our problem becomes a little trickier; we need to take the variations of the ground into account when we figure out how to line up the corners of our building.

  Here’s the subtle point: The way we connect our notions of “the same direction” at different points in space requires that there is a field filling the space between those points—a field that literally tells us how to connect them together, and in the technical literature is called a “connection.” In our architectural example, the relevant field comes from the height of the ground itself. That’s a field—it’s not a fundamental field that vibrates to give particles, but it’s a number at every point along the ground, which is all a field really is. (A topographical map would be a picture of the “height field.”) The information in that field lets us relate what happens at different points in space.

  Whenever we have a symmetry that allows us to do independent transformations at different points (a gauge symmetry), it automatically comes with a connection field that lets us compare what is going on at those locations. Sometimes the field is completely innocuous and doesn’t even get noticed, like the height of the ground on a surface that is perfectly flat. But when the connection field twists and turns from place to place, it has enormous consequences.

  For example, when the height of the ground changes from place to place, you can go skiing on it (or skateboarding, depending on the conditions). If the ground is flat, you would just sit there unmoving; when the ground is sloped, there is a force that pulls you down the hill. That’s the magic formula that makes the world go, according to modern physics: Symmetries lead to connection fields, and bends and twists in the connection fields lead to forces of nature.

  Where the forces of nature come from: Local symmetries imply the existence of connection fields, which give rise to forces.

  The four forces of nature—gravity, electromagnetism, and the strong and weak nuclear forces—are all based on symmetries. (The Higgs boson also carries a force, but it’s not what gives particles masses—that’s the Higgs field in the background. And it’s not based on any symmetry.) The boson fields that carry those forces—gravitons, photons, gluons, and the W and Z bosons—are all connection fields that relate those symmetry transformations at different points in space. They are often called “gauge bosons” to drive home the point.

  The connection fields define invisible ski slopes at every point in space, leading to forces that push particles in different directions, depending on how they interact. There’s a gravitational ski slope that affects every particle in the same way, an electromagnetic ski slope that pushes positively charged particles one way and negatively charged particles in the opposite direction, a strong-interaction ski slope that is only felt by quarks and gluons, and a weak-interaction ski slope that is felt by all the fermions of the Standard Model, as well as by the Higgs boson itself.

  For gravitons, the symmetries responsible for the force are the ones we’ve already talked about—translations (changes of position) and rotations (changes in orientation)—but in four-dimensional spacetime, not just three-dimensional space. For the strong interactions, the symmetry relates the colors red, green, or blue of the different quarks. It doesn’t matter whether we describe a certain quark as red, green, blue, or any combination thereof, so that’s a symmetry.

  You might have noticed that particles with electric charge always come in matched pairs: one with a positive charge, and one with a negative charge. That’s because, to get a charged particle, you need two fields that can rotate into each other under the gauge symmetry of electromagnetism. A single field by itself can’t be electrically charged, since there’s nothing for the symmetry to act on.

  This leaves us with the W and Z bosons of the weak interactions. They are also connection fields, born out of a certain underlying symmetry of nature. But that symmetry is masked by the Higgs field, so it takes a bit more work to describe.

  The problem with symmetries

  The symmetry underlying the weak interactions was discovered in a roundabout fashion. Back in the 1950s, before the idea of quarks had even been invented, physicists had noticed that neutrons and protons were pretty similar in some ways. The neutron is a tiny bit heavier, but its mass is close to that of the proton, all things considered. Of course the proton has an electric charge and the neutron doesn’t, but the electromagnetic interaction isn’t as strong as the strong nuclear force, and as far as the strong force goes, the two particles seem indistinguishable. If we were interested in the strong interactions in particular, we could make a lot of progress by thinking of the neutron and proton as just two different versions of a unified “nucleon” particle. That’s at best an approximate symmetry—the charges and masses really are different, so the symmetry isn’t perfect—but you can still squeeze a lot of usefulness out of it.

  In 1954, Chen Ning Yang and Robert Mills came up with the idea that this symmetry should be “promoted” to a local symmetry—i.e., that we should be allowed to “rotate” neutrons and protons into each other at every point in space. They knew what this implied: the existence of a connection field and a corresponding force of nature. At face value, it might have seemed like a somewhat crazy idea; how do you make a gauge symmetry out of something that is only approximately a symmetry in the first place? But it often happens that crazy ideas are later recognized as brilliant ones as we understand more about how nature works.

  There was a bigger problem. At the time, there were two successful theories based on local symmetries: gravity and electromagnetism. You’ll notice that they are both long-range forces, and that the bosons that mediate the forces have zero mass. Neither of these facts is a coincidence. It turns out that the requirement of local symmetry demands that the associated boson be exactly massless; and when you have a massless boson, the force it carries can extend over very long ranges. The force from a massive boson peters out quickly due to the energy required to make the massive particles, but the force from a massless one can reach out indefinitely far.

  The thing about massless particles is they’re easy to make. Especially if we are talking about a field that interacts readily with neutrons and protons, and trying to understand what happens inside an atomic nucleus, where the forces are clearly very strong. From the 1954 point of view, it seemed obvious that there weren’t any new massless particles playing an important role inside the nucleus. But Yang and Mills persevered.

  It wasn’t easy. In February of that year, Yang gave a seminar at the Institute for Advanced Study at Princeton on his new work. In the audience, among other luminaries, was the famously acerbic physicist Wolfgang Pauli. Pauli knew perfectly well that the Yang-Mills theory predicted a massless boson, in part because Pauli himself had investigated a very similar model but never published. He wasn’t the only one; other physicists, including Werner Heisenberg, contemplated similar ideas before Yang and Mills put it together explicitly.

  As an audience member in a scientific seminar, it may occasionally happen that you disagree with something the speaker is saying. The usual protocol is to ask a question, perhaps
make a statement to register your disagreement, and then let the speaker continue. That wasn’t Pauli’s style. He interrupted Yang repeatedly, demanding to know, “What is the mass of these bosons?”

  Yang, who had been born in China in 1922 and had moved to the United States to study with Enrico Fermi, would share the 1957 Nobel Prize with T. D. Lee for their work on the violation of parity (left-right symmetry). But just a few years earlier he was still relatively young and not yet established. In the face of Pauli’s onslaught, Yang found himself at a loss, and eventually he simply sat down quietly in the middle of his own seminar. Robert Oppenheimer, who was chairing the proceedings, coaxed him into resuming his talk, and Pauli stewed in silence. The next day, Pauli sent a simple note to Yang: “I regret that you made it almost impossible for me to talk to you after the seminar. All good wishes. Sincerely, W. Pauli.”

  Pauli wasn’t wrong to worry about the prediction of unseen massless particles, but Yang wasn’t wrong to pursue his idea despite this apparent flaw. In their paper, Yang and Mills admitted the problem but expressed a vague hope that quantum-mechanical effects from virtual particles would give their bosons mass.

  They were almost right. Today we know that both the strong interactions and the weak interactions are based on what we call Yang-Mills theories. And the two forces use very different, but equally clever and surprising, ways of hiding their massless particles. In the strong interactions, the gluons are massless, but they’re confined inside hadrons, so we simply never see them. In the weak interactions, the W and Z bosons would be massless if it wasn’t for the interference of the Higgs field pervading space. The Higgs breaks the symmetry on which they are based, and once that symmetry is broken there’s no reason for the bosons to remain massless. Figuring all that out required quite a journey.

  Breaking symmetries

  To understand how a symmetry can be “broken,” we descend from the land of abstraction back to the everyday world. We’ve mentioned a couple of simple examples of symmetries around us: It doesn’t matter where you are, and it doesn’t matter in what direction you are pointing. The laws of physics have another symmetry, but one that’s harder to notice: It doesn’t matter at what speed you are traveling, an idea first codified by none other than Galileo himself.

  Imagine you are on a train, zipping through the countryside. Let’s make it a supermodern train, using magnetic levitation to float above the tracks rather than old-fashioned wheels. If the train is sufficiently quiet and free of bumps along the ride, there is no way we can tell what speed we’re moving at without looking out the window. Just by minding our business, doing physics experiments inside the train, the speed at which we’re moving doesn’t matter. We could be completely still or cruising along at 100 miles an hour; the effect of dropping Mentos in the Diet Coke will be exactly the same.

  This remarkable fact is hidden from us in our everyday experience, for a simple reason: We can look outside, or just stick our hand out the window. It instantly becomes clear how fast we’re moving, because we can measure (or at least estimate) our speed relative to the ground or the air.

  This is an example of symmetry breaking. The laws of physics don’t care how fast you are going, but the ground and the air definitely do. They pick out a preferred velocity, namely “at rest with respect to the ground.” The deep-down rules of the game have a symmetry, but our environment doesn’t respect it; we say that the symmetry is broken by the environment. That’s exactly what the Higgs field does to the weak interactions. The underlying laws of physics obey a certain symmetry, but the Higgs field breaks it.

  The symmetry breaking we’ve been talking about thus far is often called “spontaneous” symmetry breaking. That’s a way of saying that the symmetry is still really there, hiding in the underlying equations that govern the world, but some feature of our environment is picking out a preferred direction. Being able to stick your hand out the window of a train and measure your speed with respect to the air doesn’t change the fact that the laws of physics are invariant with respect to different velocities. Indeed, when people are careful they will sometimes talk of symmetries as simply being “hidden” rather than “spontaneously broken.” More on this notion of spontaneity in Chapter Eleven.

  Symmetries of the weak interactions

  It turns out that Yang and Mills were basically on the right track with the idea of a symmetry between neutrons and protons. These days we know about quarks, of course, so the analogous idea would be to propose a symmetry between up quarks and down quarks. The same problems appear to get in the way: the up and down quarks have different masses and different electric charges. If those features can be traced to the existence of the Higgs field, we could be in business. And indeed they can.

  Here’s where things get messy—so much so, that the details have been relegated to Appendix One. (It’s not supposed to be simple; we’re talking about a series of discoveries that resulted in multiple Nobel Prizes.) The origin of the messiness resides in the fact that elementary fermions have a property called “spin.” Massless particles, which always move at the speed of light, can spin in one of two ways: They can be left-handed or right-handed. Think “spinning clockwise/counterclockwise if the particle is moving toward you.” The secret of the weak interactions is that there is a symmetry relating all the left-handed particles, and an associated force, but no matching symmetry for the right-handed particles. The weak interactions violate parity—they discriminate between left and right. You can think of parity as the operation of looking at the world through the reflection in a mirror, where right and left are swapped. Most forces (strong, gravitation, electromagnetism) act the same whether you look at them directly or through a mirror; but the weak interactions treat right and left differently.

  The symmetry of the weak interactions relates pairs of left-handed particles, in basically the following way:

  up quark down quark

  charm quark strange quark

  top quark bottom quark

  electron electron neutrino

  muon muon neutrino

  tau tau neutrino

  The particles that we’ve joined up in pairs here seem very different to us at first glance; they have different masses and charges. That’s because the Higgs field lurking in the background breaks the symmetry between them. If it weren’t for the masquerade put on by the Higgs, the particles in each pair would be completely indistinguishable, just like we think of red/green/blue quarks as three different versions of the same thing.

  The Higgs field itself rotates under the symmetry of the weak interactions; that’s why, when it gets a nonzero value in empty space, it picks out a direction and breaks the symmetry, just like the air picks out a velocity that we can measure things with respect to when we’re traveling in our train. Back in our pendulum example, the lowest-energy state of the regular pendulum was perfectly symmetric, pointing straight down. The upside-down pendulum, like the Higgs field, breaks the symmetry by falling either left or right.

  If you were hopelessly lost in a forest in the middle of the night, all directions would seem the same to you. You could rotate how you were standing, and your situation would be just as dire. But if you had a compass, and you knew you wanted to walk north, the direction picked out by that compass would break the symmetry; now there’s a right direction to walk, and there are wrong directions. Likewise, with no Higgs field the electron and the electron neutrino (say) would be identical particles. You could rotate them into each other, and the resulting combinations would remain indistinguishable. The Higgs field, like the compass, picks out a direction. There is now one particular combination of fields that interacts most strongly with the Higgs field, which we call the “electron,” and one that doesn’t, which we call the “electron neutrino.” It’s only with respect to the Higgs field filling space that such a distinction makes sense.

  If it wasn’t for the symmetry breaking, there would actually be four Higgs bosons, rather than just one; two pairs of particles tha
t transform into each other via the weak interaction symmetry. But when the Higgs field fills space, three of those particles get “eaten” by the three gauge bosons of the weak interactions, which thereby go from being massless force-carriers to being the massive W and Z bosons. Yes, physicists really do talk that way: The weak-interaction bosons gain mass by consuming the extra Higgs bosons. You are what you eat.

  Back to the Bang

  The analogy between the Higgs field and the upside-down pendulum is actually a pretty good one. Like the Higgs, the underlying laws of physics for the pendulum are perfectly symmetric; they don’t favor either left or right. But there are only two stable configurations for the pendulum to be in: pointing left or pointing right. If we tried to balance it carefully so that it was pointing in a symmetric configuration pointing directly upward, any tiny bump would send it falling left or right.

  The Higgs field is the same way. It could be set to zero in empty space, but that’s an unstable configuration. For the pendulum, if it’s lying peacefully to the left or right, we would have to exert some energy to lift it so that it pointed directly upward. The same is true for the Higgs field. To move it from its nonzero value at every point in space back to zero would require a superhuman amount of energy—much more than the total energy in the observable universe today.

  But the universe used to be a much denser place, with a lot more energy packed into a much smaller volume. At times near the Big Bang, 13.7 billion years ago, matter and radiation were squeezed much closer together, and the temperature was enormously higher. In terms of the pendulum analogy, think of that upside-down pendulum sitting on a table rather than being bolted to the floor. “High temperature” means a lot of random motions of particles; in terms of the analogy, it’s like someone takes a hold of the table and starts shaking it. If the shaking is sufficiently energetic, we might imagine that the pendulum is pushed so hard that it flips over from left to right (or vice versa). If the shaking is really energetic, the pendulum will vibrate like crazy, flipping quickly back and forth. On average, it will spend as much time on the left as on the right. In other words, at high temperatures, the upside-down pendulum becomes symmetric again.