Warped Passages - Lisa Randall [123]
With a Grand Unification Theory, on the other hand, a small change in a parameter is enough to completely ruin the theory’s predictions, both quantitative and qualitative. The physical consequences of the value of the Higgs particle’s mass that breaks electroweak symmetry depend extraordinarily sensitively on a parameter. For practically all the values of that parameter, the hierarchy between the GUT mass and the weak scale mass wouldn’t exist, and structure and life, which rely on this hierarchy, would be impossible. If that parameter were off by as little as 1%, the Higgs particle’s mass would be far too large. The weak gauge boson masses, and other particle masses as well, would then all be much larger, and the consequences of the Standard Model would be nothing like what we see.
The Hierarchy Problem of Particle Physics
The last section presented an enormous mystery, the hierarchy problem in a GUT. But the true hierarchy problem is even worse. Although GUTs first alerted physicists to the hierarchy problem, virtual particles will generate overly large contributions to the Higgs particle’s mass, even in a theory without GUT-mass particles. Even the Standard Model is suspect.
The problem is that a theory consisting of the Standard Model combined with gravity contains two enormously different energy scales. One is the weak scale energy, the energy at which electroweak symmetry is broken, which is 250 GeV. When particles have energies below that scale, the effects of electroweak symmetry breaking are manifest, and weak gauge bosons and elementary particles have mass.
The other energy is the Planck scale energy, which is sixteen orders of magnitude, or ten million billion times, greater than the weak scale energy: a whopping 1019 GeV. The Planck scale energy determines the strength of gravitational interactions: Newton’s law says that the strength is inversely proportional to the second power of that energy. And because the strength of gravity is small, the Planck scale mass (related to the Planck scale energy by E = mc2) is big. A huge Planck scale mass is equivalent to extremely feeble gravity.
So far, the Planck scale mass hasn’t come up in our particle physics discussions because gravity is so weak that for most particle physics calculations it can safely be ignored. But that is precisely the question particle physicists want answered: why is gravity so weak that it can be ignored in particle physics calculations? Another way of phrasing the hierarchy problem is to ask why the Planck scale mass is so huge—why is it ten million billion times higher than the masses relevant to particle physics scales, all of which are less than a few hundred GeV?
To give you a basis for comparison, consider the gravitational attraction between two low-mass particles, such as a pair of electrons. This gravitational attraction is about a hundred million trillion trillion trillion times weaker than the electric repulsion between the electron pair. The two kinds of forces would be comparable only if electrons were heavier than they really are by a factor of ten billion trillion. That’s an enormous number—it’s comparable to the number of times you could lay the island of Manhattan end to end in the extent of the observable universe.
The Planck scale mass is enormously bigger than the electron’s mass and all other particle masses we know of, and that signifies that gravity is very much weaker than the other known forces. But why should there be such a huge discrepancy between the strengths of most forces—or, equivalently, why should the Planck scale mass be so enormous compared with known particle masses?
To particle physicists, the enormous ratio between the Planck scale mass and the weak scale mass, a factor of about ten million billion, is hard to stomach. This ratio is greater than the number of minutes that have passed since the Big Bang; it’s about a thousand times the number of marbles you can line up from the Earth