Warped Passages (33 page)
The Origin of Elementary Particle Masses: Spontaneous Symmetry Breaking and the Higgs Mechanism
One of these mornings the chain is gonna break.
Aretha Franklin
The stricter enforcement of speed limits made long-distance driving a nightmare for Icarus III. He longed to race as fast as he pleased, but police pulled him over nearly every half-mile. The cops never bothered with dull, neutral cars, but harassed only the lively, turbo-charged vehicles, like his own.
Ike resigned himself to driving only short distances, since that way he could avoid the police altogether. Within the half-mile-wide region around where he started, police never interfered and he could always drive impressively fast. Though the Porsche engine’s force was unknown outside his neighborhood, closer to home it became legendary.
Symmetries are important, but the universe usually doesn’t manifest perfect symmetry. Slightly imperfect symmetries are what makes the world interesting (but organized). For me, one of the most intriguing aspects of physics research is the quest for connections that make symmetry meaningful in an unsymmetric world.
When a symmetry is not exact, physicists say the symmetry is
broken
. Although broken symmetry is often interesting, it isn’t always aesthetically appealing: the beauty and economy of the underlying system or theory can be lost (or lessened). Even the very symmetric Taj Mahal doesn’t have perfect symmetry, since the builder’s parsimonious heir decided not to build a planned second monument,
adding instead an off-center tomb to the original. This second tomb destroys the Taj Mahal’s otherwise perfect fourfold rotational symmetry, detracting slightly from its underlying beauty.
But fortunately for aesthetically minded physicists, broken symmetries can be even more beautiful and interesting than things that are perfectly symmetrical. Perfect symmetry is often boring. The
Mona Lisa
with a symmetric smile just wouldn’t be the same.
In physics, as in art, simplicity alone is not necessarily the highest goal. Life and the universe are rarely perfect, and almost all symmetries you care to name are broken. Although we physicists value and admire symmetry, we still have to find a connection between a symmetric theory and an asymmetric world. The best theories respect the elegance of symmetric theories while incorporating the symmetry breaking necessary to make predictions that agree with phenomena in our world. The goal is to make theories that are richer and sometimes even more beautiful without compromising their elegance.
The concept of the
Higgs mechanism
, which relies on the phenomenon of
spontaneous symmetry breaking
(which we will consider in the following section), is an example of such a sophisticated, elegant theoretical idea. This mechanism, named after the Scottish physicist Peter Higgs, lets the Standard Model particles—quarks, leptons, and weak gauge bosons—acquire mass.
Without the Higgs mechanism, all elementary particles would have to be massless; the Standard Model with massive particles but without the Higgs mechanism would make nonsensical predictions at high energies. The magical property of the Higgs mechanism is that it lets you have your cake and eat it too: particles get mass, but they act as if they are massless when they have energies at which massive particles would otherwise cause problems. We will see that the Higgs mechanism allows particles to have mass but travel freely over a restricted range, in much the same way that Ike’s car, which was stopped by policemen after half a mile, traveled undisturbed over limited distances, and that this suffices to solve high-energy problems.
Although the Higgs mechanism is one of the nicest ideas in quantum field theory and underlies all fundamental particle masses, it is also somewhat abstract. For this reason it is not well known by most people aside from specialists. While you can understand many features
of ideas I discuss later in the book without knowing the details of the Higgs mechanism (and you can skip now to the summary bullets if you like), this chapter does provide an opportunity to delve a bit deeper into particle physics and into the ideas, such as spontaneously symmetry breaking, that buttress theoretical developments in particle physics today. As an added bonus, some familiarity with the Higgs mechanism will let you in on an amazing insight into electromagnetism that was discovered only in the 1960s, once the weak force and the Higgs mechanism were properly understood. And later on, when we come to explore extra-dimensional models, some understanding of the Higgs mechanism will make the potential merits of those recent ideas meaningful.
Spontaneously Broken Symmetry
Before describing the Higgs mechanism, we need first to investigate spontaneous symmetry breaking, a special type of symmetry breaking that is central to the Higgs mechanism. Spontaneous symmetry breaking plays a big role in many of the properties of the universe that we already understand and is likely to play a role in whatever we have yet to discover.
Spontaneous symmetry breaking is not only ubiquitous in physics, but is a prevalent feature of everyday life. A spontaneously broken symmetry is a symmetry that is preserved by physical laws but not by the way things are actually arranged in the world. Spontaneous symmetry breaking takes place when a system cannot preserve a symmetry that would otherwise be present. Perhaps the best way to explain how this works is to give a few examples.
Let’s first consider a dinner arrangement in which a number of people are seated around a circular table with water glasses placed between them. Which glass should someone use, the one on the right or the one on the left? There is no good answer. I am told that Miss Manners says the one on the right, but aside from arbitrary rules of etiquette, left and right serve equally well.
However, as soon as someone chooses a glass, the symmetry is broken. The impetus to choose would not necessarily be part of the
system; in this case it would be another factor—thirst. Nonetheless, if one person spontaneously drank from the glass on their left, so would that person’s neighbors, and in the end everyone would have drunk from the glass on the left.
The symmetry exists until the moment someone picks up a glass. At that moment the left-right symmetry is spontaneously broken. No law of physics dictates that anyone has to choose left or right. But one has to be chosen, and after that, left and right are no longer the same in that there is no longer a symmetry that interchanges the two.
Here’s another example. Imagine a pencil standing on end at the center of a circle. For the split second in which it rests on its tip and is exactly vertical, all directions are equivalent and a rotational symmetry exists. But a pencil standing on end won’t just stay there: it will spontaneously fall in some direction. As soon as the pencil topples over, the original rotational symmetry is broken.
Notice that it would not be the physical laws themselves that determined the direction. The physics of the pencil falling over would be exactly the same no matter the direction in which it fell. What would break the symmetry would be the pencil itself, the state of the system. The pencil simply cannot fall in all directions at once. It has to fall in one particular direction.
A wall that is infinitely long and high would also look the same everywhere and in all directions along it. But because an actual wall has boundaries, if you are to see the symmetries you will have to get close enough to it that the boundaries are out of your field of vision. The wall’s ends tell you that not everywhere along the wall is the same, but if you were to press your nose up against it so that you could see only a short distance away, the symmetry would appear to be preserved. You might want to briefly reflect on this example, which shows that a symmetry can appear to be preserved when viewed on one distance scale, even though it appears to be broken on another—a concept whose importance will become apparent very soon.
Almost any symmetry you care to name is not preserved in the world. For example, there are many symmetries that would be present in empty space, such as rotational or translational invariance, which tell us that all directions and positions are the same. But space is not empty: it is punctuated by structures such as stars and the solar system,
which occupy particular positions and are oriented in particular ways that no longer preserve the underlying symmetry. They could have been anywhere, but they can’t be everywhere. The underlying symmetries must be broken, although they remain implicit in the physical laws describing the world.
The symmetry associated with the weak force is also spontaneously broken. In the rest of this chapter I’ll explain how we know this and discuss some of the consequences. We’ll see that spontaneously breaking the weak force symmetry is the only way to explain massive particles while avoiding incorrect predictions for high-energy particles that cannot be avoided in any other candidate theory. The Higgs mechanism acknowledges both the requirement of an internal symmetry associated with the weak force and the necessity for it being broken.
The Problem
The weak force has one especially bizarre property. Unlike the electromagnetic force, which travels over large distances—which you benefit from each time you turn on the radio—the weak force affects only matter that is within extremely close range. Two particles must be within one ten thousand trillionth of a centimeter to influence each other via the weak force.
For the physicists who studied quantum field theory and quantum electrodynamics (QED, the quantum field theory of electromagnetism) in its earliest days, this restricted range was a mystery. QED made it look as if forces, such as the well-understood electromagnetic force, should be transmitted arbitrarily far away from a charged source. Why wasn’t the weak force also communicated to particles at any distance and not just to those nearby?
Quantum field theory, which combines the principles of quantum mechanics and special relativity, dictates that if low-energy particles communicate forces only a short distance, they must have mass; and the heavier the particle, the shorter the particle’s range. As explained in Chapter 6, this is a consequence of the uncertainty principle and special relativity. The uncertainty principle tells us that you need
high-momentum particles to probe or influence physical processes at short distances, and special relativity relates that momentum to a mass. Although this is a qualitative statement, quantum field theory makes this relationship precise. It tells how far a massive particle will travel: the smaller the mass, the bigger the distance.
Therefore, according to quantum field theory, the short range of the weak force could mean only one thing: the weak gauge bosons communicating the force had to have nonzero mass. However, the theory of forces I described in the previous chapter works only for gauge bosons such as the photon, which communicates a force over large distances and has zero mass. According to the original theory of forces, the existence of nonzero masses was strange and problematic—the theory’s high-energy predictions when gauge bosons have mass make no sense. For example, the theory would predict that very energetic, massive gauge bosons would interact much too strongly—so strongly in fact that particles would appear to be interacting more than 100% of the time. This naive theory is clearly wrong.
Furthermore, the masses for weak gauge bosons, quarks, and leptons (all of which we know to have nonzero mass) do not preserve the internal symmetry which, as we saw in the previous chapter, is a key ingredient in the theory of forces. Physicists who hoped to construct a theory with massive particles clearly needed a new idea.
Physicists have shown that the only way to make a theory that avoids nonsensical predictions about energetic, massive gauge bosons is to have the weak force symmetry break spontaneously through the process known as the Higgs mechanism. Here’s why.
You might recall from the previous chapter that one of the reasons we wanted to include an internal symmetry that eliminates one of the three possible polarizations of a gauge boson was that a theory without the symmetry makes the same sort of nonsensical predictions I’ve just mentioned. The simplest theory of forces without an internal symmetry predicts that any energetic gauge boson, with or without a mass, interacts with other gauge bosons far too often.
The successful theory of forces eliminates this bad high-energy behavior by forbidding the polarization that is responsible for the incorrect predictions and doesn’t actually exist in nature. Spurious polarizations are the source of the problematic predictions for
high-energy scattering, so the symmetry allows only physical polarizations—the ones that really exist and are consistent with the symmetry—to remain. The symmetry, which rids the theory of nonexistent polarizations, also eliminates the incorrect predictions they would otherwise induce.
Although I didn’t say so explicitly at the time, this idea works as stated only for massless gauge bosons. The weak gauge bosons, unlike the photon, have nonzero masses. Weak gauge bosons travel at less than the speed of light. And that puts a wrench in the works.
Whereas massless gauge bosons have only two polarizations that exist in nature, massive gauge bosons have three. One way to understand this distinction is that massless gauge bosons always travel at the speed of light, which tells us that they are never at rest. They therefore always single out their direction of motion, so you can always distinguish the perpendicular directions from the remaining polarization along the direction of travel. And it turns out that for massless gauge bosons, physical polarizations oscillate only in the two perpendicular directions
Massive gauge bosons, on the other hand, are different. Like all familiar objects, they can sit still. But when a massive gauge boson isn’t moving, it doesn’t single out a direction of motion. To a massive gauge boson sitting at rest, all three directions should be equivalent. But if all three directions are equivalent, then all three possible polarizations would have to exist in nature. And they do.
