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Section 6: Extra Dimensions and the Hierarchy Problem

The weakness of gravity is difficult to maintain in a quantum mechanical theory, much as it is difficult to balance a pencil on its tip.

Figure 17: The weakness of gravity is difficult to maintain in a quantum mechanical theory, much as it is difficult to balance a pencil on its tip.

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At least on macroscopic scales, we are already familiar with the fact that gravity, is 1040 times weaker than electromagnetism. We can trace the weakness of gravity to the large value of the Planck mass, or the smallness of Newton's universal gravitational constant relative to the characteristic strength of weak interactions, which set the energy scale of modern-day particle physics. However, this is a description of the situation, rather than an explanation of why gravity is so weak.

This disparity of the scales of particle physics and gravity is known as the hierarchy problem. One of the main challenges in theoretical physics is to explain why the hierarchy problem is there, and how it is quantum mechanically stable. Experiments at the LHC should provide some important clues in this regard. On the theory side, extra dimensions may prove useful.

A speculative example

We'll start by describing a speculative way in which we could obtain the vast ratio of scales encompassed by the hierarchy problem in the context of extra dimensional theories. We describe this here not so much because it is thought of as a likely way in which the world works, but more because it is an extreme illustration of what is possible in theories with extra spatial dimensions. Let us imagine, as in string theory, that there are several extra dimensions. How large should these dimensions be?

The gravitational field lines spread out radially from a massive particle.

Figure 18: The gravitational field lines spread out radially from a massive particle.

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First, let us think a bit about a simple explanation for Newton's law of gravitational attraction. A point mass in three spatial dimensions gives rise to a spherically symmetrical gravitational field: Lines of gravitational force emanate from the mass and spread out radially in all directions. At a given distance r from the mass, the area that these lines cross is the surface of a sphere of radius r, which grows like r2. Therefore, the density of field lines of the gravitational field, and hence the strength of the gravitational attraction, falls like 1/r2. This is the inverse square law from Unit 3.

Now, imagine there were k extra dimensions, each of size L. At a distance from the point mass that is small compared to L, the field lines of gravitation would still spread out as if they are in 3+k dimensional flat space. At a distance r, the field lines would cross the surface of a hypersphere of radius r, which grows like r2+k. Therefore the density of field lines and the strength of the field fall like 1/r2+k —more quickly than in three-dimensional space. However, at a distance large compared to L, the compact dimensions don't matter—one can't get a large distance by moving in a very small dimension—and the field lines again fall off in density like 1/r2. The extra-fast fall-off of the density of field lines between distance of order, the Planck length, and L has an important implication. The strength of gravity is diluted by this extra space that the field lines must thread.

An only slightly more sophisticated version of the argument above shows that with k extra dimensions of size L, one has a 3+1 dimensional Newton's constant that scales like L-k. This means that gravity could be as strong as other forces with which we are familiar in the underlying higher-dimensional theory of the world, if the extra dimensions that we haven't seen yet are large (in Planck units, of course; not in units of meters). Then, the relative weakness of gravity in the everyday world would be explained simply by the fact that gravity's strength is diluted by the large volume of the extra dimensions, where it is also forced to spread.

String theory and brane power

The astute reader may have noticed a problem with the above explanation for the weakness of the gravitational force. Suppose all the known forces really live in a 4+k dimensional spacetime rather than the four observed dimensions. Then the field lines of other interactions, like electromagnetism, will be diluted just like gravity, and the observed disparity between the strength of gravity and electromagnetism in 4D will simply translate into such a disparity in 4+k dimensions. Thus, we need to explain why gravity is different.

Strings can break open and end on a brane.

Figure 19: Strings can break open and end on a brane.

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In string theories, a very elegant mechanism can confine all the interactions except gravity, which is universal and is tied directly to the geometry of spacetime, to just our four dimensions. This is because string theories have not only strings, but also branes. Derived from the term "membranes," these act like dimensions on steroids. A p-brane is a p-dimensional surface that exists for all times. Thus, a string is a kind of 1-brane; for a 2-brane, you can imagine a sheet of paper extending in the x and y directions of space, and so on. In string theory, p-branes exist for various values of p as solutions of the 10D equations of motion.

So far, we have pictured strings as closed loops. However, strings can break open and end on a p-brane. The open strings that end in this manner give rise to a set of particles which live just on that p-brane. These particles are called "open string modes," and correspond to the lowest energy excitations of the open string. In common models, this set of open string modes includes analogs of the photon. So, it is easy to get toy models of the electromagnetic force, and even the weak and strong forces, confined to a 3-brane or a higher dimensional p-brane in 10D spacetime.

Gravity leaking into extra dimensions could explain the hierarchy problem.

Figure 20: Gravity leaking into extra dimensions could explain the hierarchy problem.

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In a scenario that contains a large number of extra dimensions but confines the fundamental forces other than gravity on a 3-brane, only the strength of gravity is diluted by the other dimensions. In this case, the weakness of gravity could literally be due to the large unobserved volume in extra spacetime dimensions. Then the problem we envisioned at the end of the previous section would not occur: Gravitational field lines would dilute in the extra dimensions (thereby weakening our perception of gravity), while electromagnetic field lines would not.

While most semi-realistic models of particle physics derived from string theory work in an opposite limit, with the size of the extra dimensions close to the Planck scale and the natural string length scale around 10-32 centimeters, it is worth keeping these more extreme possibilities in mind. In any case, they serve as an illustration of how one can derive hierarchies in the strengths of interactions from the geometry of extra dimensions. Indeed, examples with milder consequences abound as explanations of some of the other mysterious ratios in Standard Model couplings.