Braneworld and the Hierarchy in RS1
Earlier this year I wrote a Part III essay on bulk neutrinos in the Randall-Sundrum I (RS1) model. After scouring the introductory braneworld literature, I noticed that there seems to be set of standard figures used. Somewhat uninspired by these, I went ahead and made my own graphic. The resulting image, below, was nifty enough that I feel compelled to write a post around it.🙂
Fig 1. Scientists are searching for nature’s answers while stuck on the brane, represented by Gauguin’s Tahitian panting, “Where Do We Come From? What Are We? Where Are We Going?” (1897, image from the Artchive). The red line depicts the warp factor, by which mass scales are exponentially shrank on our brane. The axis runs from to 0.
So what’s going on in the funky picture above (fig 1)? What’s heuristically represented is the RS1 `toy model’ of the universe in which our observed (3+1)-dimensional spacetime is embedded as a hypersurface (brane) in a 4+1 dimensional bulk spacetime. The fifth dimension is an interval, which is actually an orbifold. Our brane, the visible brane, is located at one endpoint of the interval while another brane, the hidden brane, is located at the other end. In the figure above, this hidden brane is represented by the yellow translucent sheet. All Standard Model particles are confined to live on the brane while gravity is allowed to propagate freely in the bulk.
As an ansatz, we can write our bulk metric as:
One can see heuristically that the warp factor conformally transforms 3+1-dimensional cross sections of the bulk spacetime relative to one another. This will be the mechanism by which the RS1 model can naturally generate large hierarchies (more on this shortly).
Solving the 55-component of the Einstein equation does two things for us: first it fixes the bulk cosmological constant, and then it fixes . The cosmological constant turns out to be negative so that the bulk space is Anti deSitter (AdS). The heuristic form of the warp factor is depicted in Fig. 1 as a red line. Explicitly, one ends up with:
Where is naturally on the order of the inverse `fundamental’ Planck scale. The combination is dimensionless and naturally order 1.
Just as a rubber sheet can be stretched, the branes are allowed to have tension, i.e. a constant energy density localized on the brane. To sentient life confined to live on the brane, this constant energy density looks like a 4D cosmological constant. These brane tensions compensate for the curvature of the bulk space so that the 4D branes are flat. (Sanity check: the cosmological constant indeed supplies the `missing’ energy density so that the universe is flat or very nearly so.)
After some manipulations (see references below), this construction yields something fantastic. First of all, the 4D Planck scale is roughly the same order as the 5D `fundamental’ Planck scale. The exact relation ends up as:
This comes from perturbing the 4D part of the metric (i.e. introducing 4D gravitons) and playing with the corresponding gravitaton terms in the action. One can see that with and , the right hand side is indeed .
But what about the Standard Model (SM)? The SM `lives’ on the visible brane. Effectively, this means that in the 5D action the SM lagrangian terms are accompanied by delta functions which fix these fields to the visible brane. But the extra dimension still influences the SM lagrangian in two ways: (1) since we are dealing with gravity, there is an overall factor of the square root of the metric multiplying every term in the action, and (2) contractions of vector indices depend on the metric. Since the metric is nonfactorisable (i.e. the 4D block depends explicitly on the 5th coordinate), the metric is different depending on where in the 5th dimension it is being evaluated. In particular, at the visible brane we get factors of floating around — the very same factors that were floating around in the previous paragraph.
However, we are effectively 4D beings that only care about 4D effective actions. We want to massage the Standard Model terms in the action into a form we are used to. Namely, we perform the trivial integration over the extra dimension (using the brane delta function) and we want to canonically normalise our fields relative to the 4D theory. In the Higgs sector, the effective Lagrangian looks like:
The exponential factors come from the metric as discussed above. We canonically normalise the Higgs field by rescaling . Thus, we discover that the effective Lagrangian (properly normalised) takes the form:
That is to say that the effective Higgs vev is equal to the `fundamental’ scale multiplied by the warp factor. The natural value for is the fundamental Planck scale, . For values of , we are able to reproduce the observed Standard Model Higgs vev. Meanwhile, the effective lagrangian (with this definition for ) is exactly the same as the usual Standard Model lagrangian.
Aha! Did you catch it? Something fantastic has happened. Because of the warp factor, we have been able to explain the hierarchy between the electroweak and Planck scales using only natural parameters, namely the warping of spacetime turns naturally Planck-scale quantities into [naturally] electroweak scale quantities. That is to say that this scenario `solves’ the hierarchy problem (if it really is a problem)! Ok, so the cost is that we’ve had to introduce a new parameter here and there, but the point is that these parameters all take natural values.
A good analogy (by Rattazzi in his Cargese lectures, see references) is that of redshifting. One can think of the mass (energy) of a particle in terms of its frequency. Just as photons are redshifted near a strong gravitational field, the idea of warped extra dimensions is to redshift the mass-scale of our visible brane such that the fundamental Planck scale is `warped’ to the electroweak scale.
Now back to Gauguin. Where does the brane come from? What is it? Where is it going? One may feel awkward that we’ve made a rather arbitrary set up to get a nice model. To some extent, this is true — but this is model-building, after all. It turns, though, out that this scenario comes out naturally from Horava-Witten models in M-theory. However, we shall take the stance that the mechanism that generated the brane is related to some high-energy theory that is otherwise decoupled from our low-energy universe. All that is relevant is that it is reasonable that a high-energy theory (which would turn out to have extra dimensions) may have, a Randall-Sundrum braneworld set up as a lower-energy limit.
Extra Dimensions Review Articles I Liked
An excellent starting point is Bee’s literature review. Seriously, if more bloggers wrote similar lit reviews with lots of useful and beginner-friendly references, then the world would be a better place for grad students.
Because I hope to cater to colleagues thoroughly rooted in the 21st century, here are some nice multimedia references… they’re great for serious studying, background noise, or even for an iPod while study-jogging (does anyone else do this?):
- Rizzo’s SSI2004 Talk (write up)
- Rubakov lectures at CERN (related review)
- Lykken’s lecture
- Hill’s lecture
- Joanne Hewett’s ASTI lectures
For some great background reading, I suggest looking at several TASI lectures on the subject that have been given in recent years:
- Sundrum: To the Fifth Dimension and Back
- Csaki: Lectures on😄 and Branes
- Dienes: New Directions for New Dimensions (not readily available on the arXiv… but it’s online if you look for it, otherwise you can find it in the 2002 TASI book)
- Kribs: Lectures on the Pheno. of😄
For the RS1 set up, Csaki’s reference above is very nice, as well as the original paper by Randall and Sundrum. Some good background can be found in an earlier papers by Sundrum: EFT for a 3-Brane Universe, Compactification for a 3-Brane Universe.
I’ll specialise even a bit deeper and suggest a very nice paper by Grossman and Neubert regarding bulk neutrinos in a warped extra dimension. This paper provides the mathematical framework for placing fermions in the bulk, something which became a bit of a model-building trend.
Hopefully these will provide enough of a framework to make the subject easily accessible to a first year postgraduate.
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