Warped Passages - Lisa Randall [177]
The lightest KK particle is therefore the one associated with this constant probability value in the extra dimension. At low energies this is the only KK particle that can be produced. Since it has neither momentum nor structure in the extra dimension, it is indistinguishable from an ordinary four-dimensional particle with the same mass and charge. With only a low energy, the higher-dimensional particle is not able to wiggle around at all in the compact rolled-up dimension. In other words, low energy won’t produce any of the additional KK particles that distinguish our universe from one with more dimensions. Low-energy processes and the lightest KK particles will therefore tell us nothing about the existence of an extra dimension, never mind its size or shape.
However, if the universe contains additional dimensions, and particle accelerators achieve sufficiently high energies, they will create heavier KK particles. These heavier KK particles, which carry nonzero extra-dimensional momenta, will be the first real evidence of extra dimensions. In our example, those heavier KK particles are associated with waves that have structure along the circular additional dimension; the waves vary as they wind around the rolled-up dimension, oscillating up and down an integer number of times along its length.
The lightest such KK particle would be the one whose probability function has the largest wavelength. And the largest wavelength for which the oscillation fits in a circle is the one that oscillates up and down exactly once as the wave winds around the rolled-up dimension. That wavelength is determined by the size of the extra dimension’s circumference (it’s approximately the same size). A larger wavelength would not fit; the wave would be mismatched when it returned to a single point along the circle. The particle with this probability wave is the lightest KK particle that “remembers” its extra-dimensional origin.
It makes sense that the wavelength of the wave associated with this lightest particle with nonzero extra-dimensional momentum would be about the same as the extra dimension’s size. After all, intuition tells us that only something sufficiently small to probe features or interactions on a tiny scale would be sensitive to a curled-up dimension’s existence. Trying to investigate an extra dimension with a bigger wavelength would be like trying to measure the location of an atom with a ruler. For example, if you were trying to detect an extra dimension with light or some other probe of a particular wavelength, the light would have to have a wavelength smaller than the size of the extra dimension. Because quantum mechanics associates probability waves with particles, the above statements about the wavelengths of probes translate into statements about particle properties. Only particles with sufficiently small wavelength and therefore (from the uncertainty principle) sufficiently high extra-dimensional momentum and mass could be sensitive to an extra dimension’s existence.
Another attractive feature of the lightest of the KK particles with nonzero extra-dimensional momentum is that its momentum (and hence its mass) is smaller when the extra dimension is bigger. A larger observable consequences because lighter particles are easier to produce and discover.
If extra dimensions do exist, this lightest KK particle would not be the sole evidence for them. Other, higher-momentum particles would leave even sharper fingerprints of extra dimensions at particle colliders. These particles would have probability waves that oscillate more than once when traversing the curled-up dimension. Because the nth such particle would correspond to a wave that oscillated n times as it wound around the rolled-up dimension, the masses of these KK particles would all be integer multiples of the lightest one. And the higher the momenta, the sharper the fingerprints of extra dimensions that the KK particles will leave at particle colliders. Figure 74 schematically shows the values