Warped Passages - Lisa Randall [195]
This tells us that in this warped spacetime we would actually expect to find a hierarchy between observed masses and the Planck scale mass. Although the graviton is everywhere, it interacts with far greater strength with particles on the Gravitybrane than with particles on the Weakbrane. The graviton just isn’t hanging around there all that much. The graviton’s probability function on the Weakbrane is extremely tiny, and if this scenario is the correct description of the world, this tinyness is responsible for the feebleness of gravity in our world.
In this model, feeble gravity on the Weakbrane doesn’t require a large separation between the two branes. Once you leave the Gravitybrane, where the graviton’s probability function is highly concentrated, gravity becomes exponentially weaker, which makes gravity on the Weakbrane extremely feeble. Because the graviton’s probability function is falling precipitously, gravity is highly suppressed on the Weakbrane (where we live). It can be ten million billion times weaker than you would expect without the warping, even if the two branes are fairly close together. This aspect of the theory, the fact that the branes don’t need to be separated by very much, makes this model a far more realistic possibility than large extra dimensions. Although large extra dimensions were a tantalizing rephrasing of the hierarchy problem, at the end of the day they still leave an unexplained large number—the extra dimensions’ size. In the theory we are now considering, the gravitational force on the Weakbrane is many orders of magnitude weaker than other forces, even when the Weakbrane is only a modest distance away from the first brane (the Gravitybrane).
The distance between branes in this warped geometry need only be a little larger than the Planck scale length. Whereas the large dimensions scenario required the introduction of an extremely large number—namely the size of the dimensions—in the warped geometry, no contrived large number is required to explain the hierarchy. That is because an exponential automatically turns a modest number into an extremely huge number (the exponential) or an extremely tiny one (the inverse of the large exponential). The strength of gravity is smaller on the Weakbrane; it is reduced by a factor of the exponential of the distance between the two branes.* The enormous ratio between the Planck scale mass—the large mass that tells us that gravity is weak—and the mass of the Higgs particle, and therefore the masses of the weak gauge bosons, is expected if the Weakbrane is located at distance of 16 units away,† since the ratio of the different masses is about 1016 (ten million billion). That means that a distance between the branes that is only about a factor of sixteen bigger than your most naive guess would suffice to explain the hierarchy. A factor of sixteen might sound big, but it is a lot smaller than ten million billion, the number we are trying to explain.
For years, particle physicists had hoped to find an exponential explanation of the hierarchy. That is, we had hoped to interpret the previously unexplained large number as the consequence of a naturally occurring exponential function. Now, with extra dimensions, Raman and I had found a way for particle physics to automatically incorporate an exponential hierarchy of masses. The interaction of gravity could be much smaller at the location of our brane, the Weakbrane, than it would be where the graviton’s probability function peaks. Because gravity on our brane would be weakened by the warped geometry, if the Standard Model is housed on the Weakbrane, the hierarchy problem would be solved. This was a solution to the hierarchy problem, and it had fallen right into our laps.
Another way to understand this remarkable new feature of the warped geometry is to consider how gravity gets diluted. In Chapter 19, we explained the weakness of gravity in the ADD scenario in terms