Warped Passages - Lisa Randall [211]
On top of that, since the KK particles are extremely light, they might be easy to produce. Colliders already operate at sufficiently high energy to make them. Even ordinary physical processes, such as chemical reactions, would generate enough energy to create graviton KK partners. If the KK particles carried too much energy to the five-dimensional bulk, the theory would be ruled out.
Fortunately, neither of these concerns turns out to be a problem. When we calculated the probability functions for the KK particles, we found that the graviton KK partners interact extremely weakly on or near the Gravitybrane. Despite the large number of graviton KK partners, they all interact so feebly that there is no danger of producing too many of them or of changing the form of the gravitational force law anywhere. If there is any problem at all, it is that this theory so closely mimics four-dimensional gravity that we don’t yet know any way to distinguish it experimentally from a truly four-dimensional world! The graviton KK partners would have such a negligible impact on anything observable that we do not yet know how to tell the difference between four flat dimensions and four flat dimensions supplemented by a fifth, warped one.
You can understand the weakness of the graviton KK partners’ interactions from the shape of their probability functions. As with the graviton, these tell us the likelihood of any particle being found at any position along the fifth dimension. Raman and I followed the more or less standard procedure for finding the masses and probability functions of each graviton KK partner in our warped geometry. This involved solving a quantum mechanics problem.
For a flat fifth dimension, the quantum mechanics problem, described in Chapter 6, was to find the waves that fit around the rolled-up dimension and thereby quantize the allowed energies.* For our warped, infinite fifth-dimensional geometry, the quantum mechanics problem looked rather different, since we needed to take into account the energy on the brane and in the bulk that warped spacetime. But we were able to modify the standard procedure to suit our setup. The results were fascinating.
The first KK particle we found was the one with no momentum in the fifth dimension. The probability function of this particle is heavily concentrated on the Gravitybrane and decreases exponentially away from it. This shape should sound familiar: it is the probability function for the same four-dimensional graviton we have already discussed. This massless KK mode is the four-dimensional graviton that communicates Newton’s four-dimensional force law.
The remaining KK particles are very different, however. None of these other KK particles are likely to be found near the Gravitybrane. Instead, what you find is that for any value of mass between zero and the Planck scale mass, there exists a KK particle with that particular mass, and the probability function for each of those particles peaks at a different place along the fifth dimension.
In fact, there is an interesting interpretation for the locations of the different peaks. We saw in Chapter 20 that in warped spacetime, in order to put all particles on the same footing in the four-dimensional effective theory so that they all interact with gravity in the same way, we rescaled all distances, times, energies, and momenta differently along the fifth dimension. As one travels out away from