What is the Hierarchy Problem?
An important feature of nature that puzzles scientists like myself is known as the hierarchy, meaning the vast discrepancy between aspects of the weak nuclear force and gravity. There are several different ways to describe this hierarchy, each emphasizing a different feature of it. Here is one:
- The mass of the smallest possible black hole defines what is known as the Planck Mass. [Planck was the scientist who took the first step towards quantum mechanics.] (A more precise way to define this is as a combination of Newton’s gravitational constant G, Planck’s quantum constant h-bar, and the speed of light c: the Planck mass is the square root of h-bar times c divided by G.) The masses of the W and Z particles, the force carriers of the weak nuclear force, are about 10,000,000,000,000,000 times smaller than the Planck Mass. Thus there is a huge hierarchy in the mass scales of weak nuclear forces and gravity.
When faced with such a large number as 10,000,000,000,000,000, ten quadrillion, the question that physicists are naturally led to ask is: where did that number come from? It might have some sort of interesting explanation.
But while trying to figure out a possible explanation, physicists in the 1970s realized there was actually a serious problem, even a paradox, behind this number. The issue, now called the hierarchy problem, has to do with the size of the non-zero Higgs field, which in turn determines the mass of the W and Z particles.
The non-zero Higgs field has a size of about 250 GeV, and that gives us the W and Z particles with masses of about 100 GeV. But it turns out that quantum mechanics would lead us to expect that this size of a Higgs field is unstable, something like (warning: imperfect analogy ahead) a vase balanced precariously on the edge of a table. With the physics we know about so far, the tendency of quantum mechanics to jostle — those quantum fluctuations I’ve mentioned elsewhere — would seem to imply that there are two natural values for the Higgs field — in analogy to the two natural places for the vase, firmly placed on the table or smashed on the floor. Naively, the Higgs field should either be zero, or it should be as big as the Planck Energy, 10,000,000,000,000,000 times larger than it is observed to be. Why is it at a value that is non-zero and tiny, a value that seems, at least naively, so unnatural?
This is the hierarchy problem.
Many theoretical physicists have devoted significant fractions of their careers to trying to solve this problem. Some have argued that new particles and new forces are needed (and their theories go by names such as supersymmetry, technicolor , little Higgs, etc.) Some have argued that our understanding of gravity is mistaken and that there are new unknown dimensions (“extra dimensions”) of space that will become apparent to our experiments at the Large Hadron collider in the near future. Others have argued that there is nothing to explain, because of a selection effect: the universe is far larger and far more diverse than the part that we can see, and we live in an apparently unnatural part of the universe mainly because the rest of it is uninhabitable — much the way that although rocky planets are rare in the universe, we live on one because it’s the only place we could have evolved and survived. There may be other solutions to this problem that have not yet been invented.
Many of these solutions — certainly all the ones with new particles and forces or with new dimensions — predict that new phenomena should be visible at the Large Hadron Collider. Even as I write this, the Large Hadron Collider is slowly but surely excluding many of these possibilities. So far it has not seen any unexpected new phenomena. But these are still early days.
By the way, you will often read the hierarchy problem stated as a problem with the Higgs particle mass. This is incorrect. The problem is with how big the non-zero Higgs field is. (For experts — quantum mechanics corrects not the Higgs particle mass but the Higgs mass-squared parameter, changing the Higgs field potential energy and thus the field’s value, making it zero or immense. That’s a disaster because the W and Z masses are known. The Higgs mass is unknown, and therefore it could be very large — if the W and Z masses were very large too. So it is the W and Z masses — and the size of the non-zero Higgs field — that are the problem, both logically and scientifically.)