Warped Passages - Lisa Randall [106]
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 Higgs1 and Higgs2 fields, or the Higgs1 and Higgs2 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 Higgs1 and Higgs2 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.18
Notice that if I plot a point at the position x = 0, y = 0, this rotational symmetry is completely preserved. But if I plot a point that has one nonzero coordinate value, for example where x = 5 and y = 0, the rotational symmetry is no longer preserved. The two axes are no longer equivalent because the x value, but not the y value, of this point is not zero.19
Figure 58. When the point x = 0, y = 0 is singled out, rotational symmetry is preserved. But when x = 5, y = 0 is singled out, rotational symmetry is broken.
The Higgs mechanism spontaneously breaks weak force symmetry in a similar fashion. When the two Higgs fields are zero, the symmetry is preserved. But when one is zero and the other is not, the weak force symmetry is spontaneously broken.
The weak gauge boson masses tell us the precise value of the energy at which the weak force symmetry is spontaneously broken. That energy is 250 GeV, the weak scale energy, very close to the masses of the weak gauge bosons, the W-, the W+ and the Z. When particles have energy greater than 250 GeV, interactions occur as if the symmetry is preserved, but when their energy is less than 250 GeV, the symmetry is broken and weak gauge bosons act as if they have mass. With the correct value of the nonvanishing Higgs field, the weak force symmetry is spontaneously broken at the right energy, and the weak gauge bosons get precisely the right mass.
The symmetry transformations that act on the weak gauge bosons also act on quarks and leptons. And it turns out that these transformations won’t leave things the same unless quarks and leptons are massless. That means that weak force symmetries would be preserved only if quarks and leptons didn’t have mass. And because the weak force symmetry is essential at high energies, not only is spontaneous symmetry breaking required for the weak gauge