Warped Passages - Lisa Randall [124]
If you are not a particle physicist, this might not sound like a very significant problem in itself, even though those numbers are dramatically big. After all, we can’t necessarily explain everything, and two masses just might be different for no very good reason. But the situation is actually far worse than it appears. Not only is there the unexplained enormous mass ratio. In the following section, we’ll see that in quantum field theory, any particle that interacts with the Higgs particle can participate in a virtual process that raises the Higgs particle mass to a value as high as the Planck scale mass, 1019 GeV.
In fact, if you asked any honest particle physicist who knew gravity’s strength but knew nothing about the measured weak gauge boson masses to estimate the Higgs particle’s mass using quantum field theory, he would predict a value for the Higgs particle—and hence the weak gauge boson masses—that is ten million billion times too large. That is, he would conclude from his calculation that the ratio between the Planck scale mass and the mass of the Higgs particle (or the weak scale mass, which is determined by the Higgs particle’s mass) should be far closer to unity than to ten million billion! His estimate of the weak scale mass would be so close to the Planck scale mass that particles would all be black holes, and particle physics as we know it would not exist. Although he would have no a priori expectation for the value of either the weak scale mass or the Planck scale mass individually, he could use quantum field theory to estimate the ratio—and he would be totally wrong. Clearly, there is an enormous discrepancy here. The next section explains why.
Virtual Energetic Particles
The reason that the Planck scale mass enters quantum field theory calculations is a subtle one. As we have seen, the Planck scale mass determines the strength of the gravitational force. According to Newton’s law, the gravitational force is inversely proportional to the value of the Planck scale mass, and the fact that gravity is so weak tells us that the Planck scale mass is huge.
Generally, we can ignore gravity when making predictions in particle physics because the gravitational effects on a particle with mass of about 250 GeV are completely negligible. If you really need to account for gravitational effects you can systematically incorporate them, but it’s not usually worth the bother. Later chapters will explain the new and very different scenarios in which higher-dimensional gravity is strong and cannot be neglected. But for the conventional, four-dimensional Standard Model, neglecting gravity is a standard and justifiable practice.
However, the Planck scale mass has another role as well: it is the maximum mass that virtual particles can take in a reliable quantum field theory calculation. If particles carried more mass than the Planck scale mass, the calculation would be untrustworthy, and general relativity would not be reliable and would have to be replaced by a more comprehensive theory, such as string theory.
But when particles (including virtual particles) have mass that is less than the Planck scale mass, conventional quantum field theory should apply and calculations based on quantum field theory should be trustworthy. That means that a calculation involving a virtual top quark (or any other virtual particle) with mass almost as big as the Planck scale mass should be reliable.
The problem for the hierarchy is that the contribution to the Higgs particle’s mass from virtual particles with extremely high mass will be about as big as the Planck scale mass, which is ten million billion times greater than the Higgs particle mass we want—the one that will give the right weak scale mass and elementary particle masses.
If we consider a path, such as the one shown in Figure 62, in which the Higgs particle turns into a virtual top quark-antitop