Warped Passages - Lisa Randall [120]
Before we consider the most general version of the hierarchy problem, we’ll first consider the hierarchy problem in the context of a Grand Unified Theory, where the problem was first identified and where it’s a little simpler to understand. We’ll then look at the problem in its larger (and more pervasive) context and see why it ultimately boils down to the weakness of gravity compared with all the other known forces.
The Hierarchy Problem in a GUT
Imagine that you visit a very tall friend, and discover that although he is 6? 5? tall, his fraternal twin brother is only 4? 11?. That would be surprising. You’d expect both your friend and his brother, who should have similar genetic makeups, to be similar in height. Now imagine something even more bizarre: you walk into your friend’s house and find that your friend’s brother is ten times smaller or ten times bigger. That would be very strange indeed.
We don’t think that particles should all have the same properties. But unless there is a good reason, we expect particles that experience similar forces to be somewhat similar. We expect them to have comparable masses, for example. Just as you have good reason to expect similar heights among family members, particle physicists have valid scientific reasons to expect similar masses among particles in a single theory, such as a Grand Unified Theory. But in a GUT the masses are not at all the same: even those particles that experience similar forces must have enormously different masses. And not by a mere factor of ten: the discrepancy between masses is more like a factor of ten trillion.
The problem in a Grand Unified Theory is that although the Higgs particle that breaks electroweak symmetry has to be “light”—with roughly the weak scale mass—a GUT partners the Higgs particle with another particle that interacts through the strong force. And that new particle in the GUT has to be extremely heavy—with a mass of roughly the GUT scale mass. In other words, two particles that are supposed to be related by a symmetry (the GUT force symmetry) have to have enormously different masses.
The two different but related particles must appear together in a GUT because the weak force and the strong force should be interchangeable at high energy. That’s the whole idea behind a unified theory—all forces should ultimately be the same. So when the strong and the weak forces are unified, every particle that experiences the weak force, including the Higgs particle, must be partnered with another particle that experiences the strong force and has interactions similar to those of the original Higgs particle. However, there is a big problem with the new Higgs-related particle that experiences the strong force.
The strongly charged particle that is partnered with the Higgs particle can interact simultaneously with a quark and a lepton and thereby enable the proton to decay—even more rapidly than a GUT would otherwise predict. To avoid too rapid a decay, the strongly interacting particle—which must be exchanged between two quarks and two leptons for proton decay to take place—must be extremely heavy. The current limit on the proton lifetime tells us that the strongly charged Higgs partner, if it exists in nature, has to have a mass similar in size to the GUT scale mass, about one million billion GeV. If this particle existed but was not this heavy, you and this book would decay before you finished reading this sentence.
However, we already know that the weakly charged Higgs particle has to be light (around 250 GeV) to give the weak gauge boson masses that have been measured in experiments. So experimental constraints tell us that the Higgs particl’s mass must be wildly different from the mass of the Higgs partner that experiences the strong force. The strongly charged Higgs particle, which is supposed to have very similar interactions to the weakly charged Higgs particle in a unified theory, must have a different mass,