Warped Passages (55 page)

Nima and Savas assumed that other particles responsible for flavor-symmetry breaking were sequestered on other branes. As with sequestered supersymmetry breaking, flavor-symmetry breaking could then be communicated to Standard Model particles only via interactions with particles in the bulk. If there were many bulk particles interacting with the Standard Model, each of which communicated flavor-symmetry breaking from a different brane at a different distance, their model could explain the different masses of the Standard Model flavors. Symmetry breaking communicated from distant branes would induce smaller masses than symmetry breaking communicated from nearby branes. Nima and Savas named their idea
shining
to emphasize this fact. Just as light looks dimmer when its source is further away, the effect of symmetry breaking is smaller when it originates on a more distant brane. In their scenario, different flavors of quarks and leptons would be different because they each interact with a different brane at a different distance.

Extra dimensions and sequestering are novel and exciting ways to address problems in particle physics. And it doesn’t necessarily stop there. Recently we have shown that sequestering could even play an important role in cosmology, the science of the evolution of our universe. It’s clear that we have yet to discover all the merits of a universe (or multiverse) that contains sequestered particles, and new ideas are still to come.

What’s New

• Particles can be sequestered on different branes.
  
• Even tiny extra dimensions can have consequences for the properties of observable particles.
  
• Sequestered particles are not necessarily subject to the anarchic principle. Not all interactions necessarily occur, since distant particles cannot directly interact.
  
• In a model in which particles that play a role in supersymmetry breaking are sequestered from Standard Model particles, supersymmetry can be broken without introducing interactions that would change particles into other flavors.
  
• Sequestered supersymmetry breaking is testable. If gauginos are produced at high-energy colliders, we can compare gaugino masses and see whether they agree with predictions.
  
• Sequestered flavor-symmetry breaking might help to explain disparate particle masses.
  

18

Leaky Passages: Fingerprints of Extra Dimensions

I was peeking

But it hasn’t happened yet

I haven’t been given

My best souvenir

I miss you

But I haven’t met you yet.

Bjork

Athena had to admit that she missed Ike. Even though she had often found him annoying, she was pretty lonely without him. She was looking forward to spending time with K. Square, an exchange student who was planning to visit. But she was appalled by the closed-mindedness of her neighbors, who were all apprehensive about K. Square’s impending arrival. It didn’t matter that he spoke the same language and behaved the same way as everyone else. In the current climate, K. Square’s foreign origin alone was enough to make them wary.

When Athena asked her neighbors why they were so anxious, they replied, “What if he sends for his heavier relatives? What if they’re not so well behaved as he is and stick to their foreign laws? And when they all arrive together, what will happen then?”

Unfortunately, Athena heightened their suspicions by telling them that K. Square and his relatives couldn’t possibly stay long in any case, since they were all very unstable and the K. Square family could visit only during the commotion of energetic gatherings. Recognizing her unfortunate choice of words, Athena reassuringly added that the foreigners would stick to local laws during their brief and exciting visits. Convinced, her neighbors then joined Athena in welcoming the K. Square clan.

Earlier in this book I explained how extra dimensions might be hidden. They could be rolled up or hemmed in by branes so as to be imperceptibly small. But can an extra-dimensional universe really hide its nature so completely that none of its features distinguishes it from a four-dimensional world? That would be hard to believe. Even if compactified dimensions are so small that we could be lulled into believing that the world is four-dimensional, a higher-dimensional world must contain some new elements that distinguish it from a truly four-dimensional one.

If there are extra dimensions, such fingerprints of extra dimensions are sure to exist. Such fingerprints are particles called
Kaluza-Klein
(
KK
)
particles
.
^*
KK particles are the additional ingredients of an extra-dimensional universe. They are the four-dimensional imprint of the higher-dimensional world.

Should KK particles exist and be sufficiently light, high-energy colliders will produce them and they will leave their mark in experimental data. The extra-dimensional detectives—the experimenters—will piece together these clues, transforming data into forensic evidence of a higher-dimensional world. This chapter is about these Kaluza-Klein particles, and why, in a higher-dimensional world, you can be confident of their existence.

Kaluza-Klein Particles

Even if a bulk particle travels in higher-dimensional space, we still should be able to describe its properties and interactions in four-dimensional terms. After all, we don’t see extra dimensions directly, so everything should appear to us as if it is four-dimensional. Just as Flatlanders, who see only two spatial dimensions, could observe only
two-dimensional disks when a three-dimensional sphere passed through their world, we can see only particles that look like they travel in three spatial dimensions, even if those particles originated in higher-dimensional space. These new particles that originate in extra dimensions, but appear to us as extra particles in our four-dimensional spacetime,
^*
are Kaluza-Klein (KK) particles. If we could measure and study all their properties, they would tell us everything there is to know about the higher-dimensional space.

Kaluza-Klein particles are the manifestation of a higher-dimensional particle in four dimensions. Just as you can reproduce any sound a violin string could make by the superposition of many resonant modes, you can reproduce a higher-dimensional particle’s behavior by replacing it with appropriate KK particles. The KK particles fully characterize higher-dimensional particles and the higher-dimensional geometry in which they travel.

In order to mimic the behavior of higher-dimensional particles, KK particles would have to carry extra-dimensional momentum. Every bulk particle that travels through the higher-dimensional space gets replaced in our effective four-dimensional description by KK particles that have the correct momenta and interactions to mimic that particular higher-dimensional particle.
³¹
A higher-dimensional universe hosts both familiar particles and their KK relatives that carry extra-dimensional momenta that are determined by the detailed properties of the curled-up space.

However, a four-dimensional description doesn’t include information about extra-dimensional position or momentum. Therefore, the extra-dimensional momentum of the KK particles must be called something else when viewed from our four-dimensional perspective. The relationship between mass and momentum imposed by special relativity tells us that extra-dimensional momentum would be seen in the four-dimensional world as mass. KK particles are therefore particles like the ones we know, but with masses that reflect their extra-dimensional momenta.

The KK particles’ masses are determined by the higher-dimensional
geometry. However, their charges are the same as those of known four-dimensional particles. That is because if known particles originate from higher-dimensional spacetime, higher-dimensional particles have to carry the same charges as known particles. That’s also true for the KK particles that mimic the higher-dimensional particles’ behavior. So for each particle we know about, there should be many KK particles with the same charge, each with different mass. For example, if an electron travels in higher dimensions, it would have KK partners that have the same negative charge. And if a quark travels in higher dimensions, it would have KK relatives that, like the quark, experience the strong force. KK partners have identical charges to the particles we know, but masses that are determined by extra dimensions.

Determining Kaluza-Klein Masses

Understanding the origin and masses of KK particles requires taking a step beyond the intuitive picture of invisible curled-up dimensions that we looked at earlier on. For simplicity, we’ll first consider a universe without branes, in which every particle is fundamentally higher-dimensional and is free to move in all directions—including any additional ones. To be concrete, we’ll imagine a space with only one additional dimension which has been rolled up into a circle and elementary particles that travel inside that space.

Had we lived in a world where classical Newtonian physics was the final word, Kaluza-Klein particles could have had any amount of extra-dimensional momentum and therefore any mass. But because we live in a quantum mechanical universe, this is not the case. Quantum mechanics tells us that, just as only the resonant violin modes contribute to the sounds the violin strings can make, only quantized extra-dimensional momenta contribute when KK particles reproduce the motion and interactions of a higher-dimensional particle. And just as the notes of a violin string depend on its length, the quantized extra-dimensional momenta of the KK particles depend on the extra dimensions’ sizes and shapes.

The extra-dimensional momenta that the KK particles carry would appear to us in our apparently four-dimensional world as a distinctive
pattern of KK particle masses. If physicists discover KK particles, these masses will tell us about the geometry of the extra dimensions. For example, if there is a single extra dimension that is curled into a circle, these masses would tell us the extra dimension’s size.

The procedure for finding the allowed momenta (and hence masses) for KK particles in a universe with a curled-up dimension is very similar to the method you use to mathematically determine resonant violin modes, and also to the method that Bohr used to determine quantized electron orbits in an atom. Quantum mechanics associates all particles with waves, and only those waves that can oscillate an integer number of times over the extra-dimensional circle are allowed. We determine the allowed waves, and then use quantum mechanics to relate wavelength to momentum. And the extra-dimensional momenta tell us the allowed KK particles’ masses, which is what we want to know.

The constant wave—the one that doesn’t oscillate at all—is always allowed. This “wave” is like the surface of a perfectly still pond, without any visible ripples, or a violin string that has not yet been plucked. This probability wave has the same value everywhere in the extra dimension. Because of the constant value of this flat probability wave, the KK particle associated with this wave doesn’t favor any particular extra-dimensional location over any other. According to quantum mechanics, this particle carries no extra-dimensional momentum and, according to special relativity, has no additional mass.

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.