Warped Passages - Lisa Randall [103]
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 = mc2 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