Even if you find the above logic mysterious, rest assured that experimenters have already seen the effects of a third polarization of a massive gauge boson and have confirmed its existence. The third polarization is called the
longitudinal polarization
. When a massive gauge boson is moving, the longitudinal polarization is the wave that oscillates along the direction of motion—the direction of sound wave oscillations, for example.
This polarization doesn’t exist in the case of massless gauge bosons such as the photon. However, for massive gauge bosons, like the weak gauge bosons, the third polarization is truly a part of nature. This third polarization must be a part of the weak gauge boson theory.
Because this third polarization is the source of the weak gauge
boson’s overly large interaction rate at high energy, its existence poses a dilemma. We already know that we need a symmetry to eliminate the bad high-energy behavior. But this symmetry gets rid of the incorrect predictions by eliminating the third polarization as well, and that polarization is essential to a massive gauge boson and therefore to the theory that describes it. Although an internal symmetry would eliminate bad predictions for high-energy behavior, it would do so at too high a price: the symmetry would get rid of the mass as well! A symmetry in the theory of massive gauge bosons seems poised to throw away the baby with the bathwater.
The impasse at first glance looks insurmountable, since the requirements for a theory of massive gauge bosons appear to be entirely contradictory. On the one hand, an internal symmetry—the one described in the previous chapter—should not be preserved, since otherwise massive gauge bosons with three physical polarizations would be forbidden. On the other hand, without an internal symmetry to eliminate two of the polarizations, the theory of forces makes incorrect predictions when the gauge bosons have high energy. We still need a symmetry to eliminate the third polarization of each massive gauge boson if we are to have any hope of eliminating the bad high-energy behavior.
The key to resolving this apparent paradox and figuring out the correct quantum field theory description of a massive gauge boson was recognizing the difference between the ones with high energy and the ones with low energy. In the theory without an internal symmetry, only predictions about the high-energy gauge bosons looked as if they would be problematic. Predictions about low-energy massive gauge bosons were sensible (and true).
These two facts together imply something fairly profound: to avoid problematic high-energy predictions, an internal symmetry is essential—the lessons of the previous chapter still apply. But when the massive gauge bosons have low energy (low compared with the energy that Einstein’s relation
E
=
mc
2
associates with its mass), the symmetry should no longer be preserved. The symmetry must be eliminated so that gauge bosons can have mass and the third polarization can participate in the low-energy interactions where the mass makes a difference.
In 1964, Peter Higgs and others discovered how theories of forces could incorporate massive gauge bosons by doing exactly what we just said: keeping an internal symmetry at high energies, but eliminating it at low energies. The Higgs mechanism, based on spontaneous symmetry breaking, breaks the internal symmetry of the weak interactions, but only at low energy. That ensures that the extra polarization will be present at low energy, where the theory needs it. But the extra polarization will not participate in high-energy processes, and the nonsensical high-energy interactions will not appear.
Let’s now consider a particular model that spontaneously breaks the weak force symmetry and implements the Higgs mechanism. With this exemplar of the Higgs mechanism, we’ll see how the elementary particles of the Standard Model acquire mass.
The Higgs Mechanism
The Higgs mechanism involves a field that physicists call the
Higgs field
. As we have seen, the fields of quantum field theory are objects that can produce particles anywhere in space. Each type of field generates its own particular type of particle. An electron field is the source of electrons, for example. Similarly, a Higgs field is the source of Higgs particles.
As with heavy quarks and leptons, Higgs particles are so heavy that they aren’t found in ordinary matter. But unlike heavy quarks and leptons, no one has ever observed the Higgs particles that the Higgs field would produce, even in experiments performed at high-energy accelerators. This doesn’t mean that Higgs particles don’t exist, just that Higgs particles are too heavy to have been produced with the energies that experiments have explored so far. Physicists expect that if Higgs particles exist, we’ll create them in only a few years’ time, when the higher-energy LHC collider comes into operation.
Nevertheless, we are fairly confident the Higgs mechanism applies to our world, since it is the only known way to give Standard Model particles their masses. It is the only known solution to the problems that were posed in the previous section. Unfortunately, because no
one has yet discovered the Higgs particle, we still don’t know precisely what the Higgs field (or fields) actually is.
The nature of the Higgs particle is one of the most hotly debated topics in particle physics. In this section, I will present the simplest of many candidate models—possible theories that contain different particles and forces—that demonstrates how the Higgs mechanism works. Whatever the true Higgs field theory turns out to be, it will implement the Higgs mechanism—spontaneously breaking the weak force symmetry and giving masses to elementary particles—in the same manner as the model I’m about to present.
In this model, a pair of fields experience the weak force. It will be useful later to think of these two Higgs fields, which are subject to the weak force, as carrying weak force charge. The Higgs mechanism terminology is sometimes sloppy, with “the Higgs” sometimes denoting the two fields together, and at other times one of the individual fields (and often the Higgs particles we hope to find). Here I will distinguish the possibilities and refer to the individual fields as Higgs
1
and Higgs
2
.
Both Higgs
1
and Higgs
2
have the potential to produce particles. But they can also take nonzero values even when no particles are present. We haven’t encountered such nonzero values for quantum fields up to this point. So far, aside from the electric and magnetic fields, we have considered only quantum fields that create or destroy particles but take zero value in the absence of particles. But quantum fields can also have nonzero values, just like the classical electric and magnetic fields. And according to the Higgs mechanism, one of the Higgs fields takes a nonzero value. We will now see that this nonzero value is ultimately the origin of particle masses.
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When a field takes a nonzero value, the best way to think about it is to imagine space manifesting the charge that the field carries, but not containing any actual particles. You should think of the charge that the field carries as being present everywhere. This is, alas, a rather abstract notion because the field itself is an abstract object. But when the field takes a nonzero value, its consequences are concrete: the charge that a nonzero field would carry exists in the real world.
A nonzero Higgs field, in particular, distributes weak charge throughout the universe. It is as if the nonzero weak-charge-carrying
Higgs field paints weak charge throughout space. A nonzero value for the Higgs fields means that the weak charge that Higgs
1
(or Higgs
2
) carries is everywhere, even when there are no particles present. The vacuum—the state of the universe with no particles present—itself carries weak charge when one of the two Higgs fields takes a nonzero value.
Weak gauge bosons interact with this weak charge of the vacuum, just as they do with all weak charge. And the charge that pervades the vacuum blocks the weak gauge bosons as they try to communicate forces over long distances. The further they try to travel, the more “paint” they encounter. (Because the charge actually spreads throughout three dimensions, you might prefer to imagine a fog of paint.)
The Higgs field plays a role very similar to that of the traffic cop in the story, restricting the weak force’s influence to very short distances. When attempting to communicate the weak force to distant particles, the force-carrying weak gauge bosons bump into the Higgs field, which gets in their way and cuts them off. Like Ike, who could travel freely only within a half-mile radius of his starting point, weak gauge bosons move unimpeded only for a very short distance, about one ten thousand trillionth of a centimeter. Both weak gauge bosons and Ike are free to travel short distances, but are intercepted at longer distances.
The weak charge in the vacuum is spread out so thinly that at short distance there is very little sign of the nonzero Higgs field and the associated charge. Quarks, leptons, and the weak gauge bosons travel freely over short distances, almost as if the charge in the vacuum didn’t exist. The weak gauge bosons can therefore communicate forces over short distances, almost as if the two Higgs fields were both zero.
However, at longer distances, particles travel further and therefore encounter a more significant amount of weak charge. How much they encounter depends on the charge density, which depends in turn on the value of the nonzero Higgs field. Long-distance travel (and communication of the weak force) is not an option for low-energy weak gauge bosons, for on long-distance excursions the weak charge in the vacuum gets in the way.
This is exactly what we needed to make sense of weak gauge bosons. Quantum field theory says that particles that travel freely over short distances, but only extremely rarely travel over longer distances, have nonzero masses. The weak gauge bosons’ interrupted travel tells us that they act as if they have mass, since massive gauge bosons just don’t get very far. The weak charge that permeates space hinders the weak gauge bosons’ travel, making them behave exactly as they should in order to agree with experiments.
The weak charges in the vacuum have a density that corresponds roughly to charges that are separated by one ten thousand trillionth of a centimeter. With this weak charge density, the masses of the weak gauge bosons—the charged Ws and the neutral Z—take their measured values of approximately 100 GeV.
And that’s not all that the Higgs mechanism accomplishes. It is also responsible for the masses of quarks and leptons, the elementary particles constituting the matter of the Standard Model. Quarks and leptons acquire mass in a very similar fashion to the weak gauge bosons. Quarks and leptons interact with the Higgs field distributed throughout space and are therefore also hindered by the universe’s weak charge. Like weak gauge bosons, quarks and leptons acquire mass by bouncing off the Higgs charge distributed everywhere throughout spacetime. Without the Higgs field, these particles would also have zero mass. But once again, the nonzero Higgs field and the vacuum’s weak charge interfere with motion and make the particles have mass. The Higgs mechanism is also necessary for quarks and leptons to acquire their masses.
Although the Higgs mechanism is a more elaborate origin of mass than you might think necessary, it is the only sensible way for the weak gauge bosons to acquire mass according to quantum field theory. The beauty of the Higgs mechanism is that it gives the weak gauge bosons mass while accomplishing precisely the task I laid out at the beginning of this chapter. The Higgs mechanism makes it look as though the weak force symmetry is preserved at short distances (which, according to quantum mechanics and special relativity, is equivalent to high energy) but is broken at long distances (equivalent to low energy). It breaks the weak force symmetry spontaneously, and this spontaneous breaking lies at the root of the solution to the
problem of massive gauge bosons. This more advanced topic is explained in the following section (but feel free to skip ahead to the following chapter if you wish).
Spontaneous Breaking of Weak Force Symmetry
We have seen that the internal symmetry transformation associated with the weak force will interchange anything that is charged under the weak force because the symmetry transformation acts on anything that interacts with weak gauge bosons. Therefore, the internal symmetry associated with the weak force must act on the Higgs
1
and Higgs
2
fields, or the Higgs
1
and Higgs
2
particles they would create, and treat them as equivalent, just as it treats up and down quarks, which also experience the weak force, as interchangeable particles.
If both of the Higgs fields were zero, they would be equivalent and interchangeable, and the full symmetry associated with the weak force would be preserved. However, when one of the two Higgs fields takes a nonzero value, the Higgs fields spontaneously break the symmetry of the weak force. If one field is zero and the other is not, the electroweak symmetry, by which Higgs
1
and Higgs
2
are interchangeable, is broken.
Just as the first person to choose his left or right glass breaks the left-right symmetry at a round table, one Higgs field taking a nonzero value breaks the weak force symmetry that interchanges the two Higgs fields. The symmetry is broken spontaneously because all that breaks it is the vacuum—the actual state of the system, the nonzero field in this case. Nonetheless, the physical laws, which are unchanged, still preserve the symmetry.
A picture might help convey how a nonzero field breaks the weak force symmetry. Figure 58 shows a graph with two axes, labeled
x
and
y
. The equivalence of the two Higgs fields is like the equivalence of the
x
and
y
axes with no points plotted. If I were to rotate the graph so that the axes switched places, the picture would still look the same. This is a consequence of ordinary rotational symmetry.
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