- #1
selfAdjoint
Staff Emeritus
Gold Member
Dearly Missed
- 6,894
- 11
The model that Lisa Randall and Raman Sundrum propose consists of two four dimensional (Minkowskian) branes, bounding a five dimensional bulk between them. In their initial model the branes are very close together; this is not a "large extra dimensions" model.
The motivation of the model is to explain the hierarchy problem. One of the branes is called the Weakbrane; it is the home of the particles of the standard model and their interactions. The other brane is the Gravitybrane and on it are defined the bosonic (non-supersymmetric) string theory gravitons; each of them is a vibration mode of a closed bosonic string.
Each brane is at a characteristic energy, the Weakbrane as its name indicates is at the Weak energy scale, around a TEV, and the Gravitybrane is at the much higher string energy scale. The graviton, in traditional Randall-Sundrum fashion, can leave its brane and roam the five dimensional bulk space between the branes,
Now in Einstein's physics, energy curves geometry, and both of these branes carry energy, so the geometry of the bulk between them is curved. This can be calculated from Einstein's equations in five dimensions. Randall in her book characterizes this curvature as the kind technically known as a warp. This means that sections taken within the bulk but parallel to the branes have the same geometry as the branes (i.e. "flat" Minkowskian), but differ by a scale factor, again having the technical name "warp factor" (hello Mr. Spock!). The definition of size actually differs from section to section! (hello, Alice in Wonderland!).
The effect of the curved geometry is to alter the probabiity of finding a graviton (from distortion of its wave function); it is high near the Gravitybrane and low near the Weakbrane. I want to emphasize that this behavior is all deduced from the specification of the model; the curvature is calculated from Einstein's equations applied to the known brane energy, and the graviton wave function is calculated within the curvature. This graviton behavior provides the explanation of the hierarchy problem.
From the book Warped Passages page 393:
There is another consequence of the warp scaling in the bulk: from page 398:
After Randall and Sundrum published this model, physicist Alex Pomerol suggested that it could account for the unification of the forces at high energies as well as the hierarchy problem. In Pomerol's version of the model the particles of the standard model travel through the bulk, and he shows the high energy behavior of the forces converge. He uses supersymmetry (as in the older Randall-Sundrum model) to address the hierarchy problem. But Randall shows you can get both unification and hierarchy without any supersymmetry if you only restrict the Higgs boson to the Weakbrane and let all the other standard model particles wander the bulk.
Now in particuar note the graviton. Each graviton in the five dimensional bulk will have a five-momentum, and some of them will have five-momenta that have zero component orthogonal to the fifth dimension. The wave function of such a particle will have four-components in the Weakbrane that will have zero three-momentum and will therefore appear there as mass. So an interacting particle with mass about the TEV scale of Weakbrane physics is predicted to be detectable at the LHC. But there's more. The momentum of the graviton can have a quantized spactrum coming from vibration modes of the closed string as wound some number of times around the topology of the compacted manifolds. So these LHC particles will have a very characteristic mass spectrum of TEV, 2TEV, 3TEV, and so on. She can also calculate that unlike the free graviton, these particles are not as suppressed by the curved geometry; their interaction probability is 16 orders of magnitude higher than the free graviton (p. 408).
How will the physics of these particle appear at LHC? On page 409 of her book Randall shows a (pop) Feynman diagram: two protons collide and a quark interacts with an antiquark at the TEV energy scale producing one of these particles (along with jets of other particles), which then decays into something else, in her diagram an electron and a positron. The key here is the energy spectrum of the interaction; it should show the characteristic TEV, 2TEV, 3TEV, etc. pattern. Also all these particles will inherit spin 2 from the graviton and that will be an earmark. So an early show up might be the lowest level of them, about a TEV of energy and spin 2.
Notice that although Randall likes to distinguish herself from some of the more fancy-dan string thorists, her whole model is built out of basic string theory. So it's not really true to say that "string theory predicts nothing", this little piece of it predicts explixit observable physics. And this model, at least is eminently falsifiable.
The motivation of the model is to explain the hierarchy problem. One of the branes is called the Weakbrane; it is the home of the particles of the standard model and their interactions. The other brane is the Gravitybrane and on it are defined the bosonic (non-supersymmetric) string theory gravitons; each of them is a vibration mode of a closed bosonic string.
Each brane is at a characteristic energy, the Weakbrane as its name indicates is at the Weak energy scale, around a TEV, and the Gravitybrane is at the much higher string energy scale. The graviton, in traditional Randall-Sundrum fashion, can leave its brane and roam the five dimensional bulk space between the branes,
Now in Einstein's physics, energy curves geometry, and both of these branes carry energy, so the geometry of the bulk between them is curved. This can be calculated from Einstein's equations in five dimensions. Randall in her book characterizes this curvature as the kind technically known as a warp. This means that sections taken within the bulk but parallel to the branes have the same geometry as the branes (i.e. "flat" Minkowskian), but differ by a scale factor, again having the technical name "warp factor" (hello Mr. Spock!). The definition of size actually differs from section to section! (hello, Alice in Wonderland!).
The effect of the curved geometry is to alter the probabiity of finding a graviton (from distortion of its wave function); it is high near the Gravitybrane and low near the Weakbrane. I want to emphasize that this behavior is all deduced from the specification of the model; the curvature is calculated from Einstein's equations applied to the known brane energy, and the graviton wave function is calculated within the curvature. This graviton behavior provides the explanation of the hierarchy problem.
From the book Warped Passages page 393:
There is another consequence of the warp scaling in the bulk: from page 398:
After Randall and Sundrum published this model, physicist Alex Pomerol suggested that it could account for the unification of the forces at high energies as well as the hierarchy problem. In Pomerol's version of the model the particles of the standard model travel through the bulk, and he shows the high energy behavior of the forces converge. He uses supersymmetry (as in the older Randall-Sundrum model) to address the hierarchy problem. But Randall shows you can get both unification and hierarchy without any supersymmetry if you only restrict the Higgs boson to the Weakbrane and let all the other standard model particles wander the bulk.
Now in particuar note the graviton. Each graviton in the five dimensional bulk will have a five-momentum, and some of them will have five-momenta that have zero component orthogonal to the fifth dimension. The wave function of such a particle will have four-components in the Weakbrane that will have zero three-momentum and will therefore appear there as mass. So an interacting particle with mass about the TEV scale of Weakbrane physics is predicted to be detectable at the LHC. But there's more. The momentum of the graviton can have a quantized spactrum coming from vibration modes of the closed string as wound some number of times around the topology of the compacted manifolds. So these LHC particles will have a very characteristic mass spectrum of TEV, 2TEV, 3TEV, and so on. She can also calculate that unlike the free graviton, these particles are not as suppressed by the curved geometry; their interaction probability is 16 orders of magnitude higher than the free graviton (p. 408).
How will the physics of these particle appear at LHC? On page 409 of her book Randall shows a (pop) Feynman diagram: two protons collide and a quark interacts with an antiquark at the TEV energy scale producing one of these particles (along with jets of other particles), which then decays into something else, in her diagram an electron and a positron. The key here is the energy spectrum of the interaction; it should show the characteristic TEV, 2TEV, 3TEV, etc. pattern. Also all these particles will inherit spin 2 from the graviton and that will be an earmark. So an early show up might be the lowest level of them, about a TEV of energy and spin 2.
Notice that although Randall likes to distinguish herself from some of the more fancy-dan string thorists, her whole model is built out of basic string theory. So it's not really true to say that "string theory predicts nothing", this little piece of it predicts explixit observable physics. And this model, at least is eminently falsifiable.
Marcus, that's too funny! These predictions are YEARS old. See eg.