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 \phi axis runs from -\pi 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 S^1/\mathbb Z^2 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:

(1) ds^2 = e^{-2\sigma(\phi)}(\eta_{\mu\nu}dx^\mu dx^\nu) + r^2d\phi^2.

One can see heuristically that the warp factor e^{-2\sigma(\phi)} 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 \sigma. The cosmological constant turns out to be negative so that the bulk space is Anti deSitter (AdS_5). The heuristic form of the warp factor is depicted in Fig. 1 as a red line. Explicitly, one ends up with:

(2) \sigma = kr|\phi|.

Where k = -\sqrt{\Lambda/12M^3} is naturally on the order of the inverse `fundamental’ Planck scale. The combination kr 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 M_{Pl} is roughly the same order as the 5D `fundamental’ Planck scale. The exact relation ends up as:

(3) M_{Pl}^2 = \frac{M^3}{k}(1-e^{-2\pi kr})

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 k \sim M and kr \sim \mathcal{O}(1), the right hand side is indeed \sim M^2.

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 e^{-2\pi kr} 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:

(4) e^{-4\pi kr} \left(e^{2\pi kr}|D_\mu H|^2 + \lambda(|H|^2-v_0^2)^2\right).

The exponential factors come from the metric as discussed above. We canonically normalise the Higgs field by rescaling  H \rightarrow H' = e^{-\pi kr} H. Thus, we discover that the effective Lagrangian (properly normalised) takes the form:

(5) |D_\mu H'|^2 + \lambda(|H'|^2-e^{-2\pi kr} v_0^2)^2.

That is to say that the effective Higgs vev v=e^{-\pi kr}v_0 is equal to the `fundamental’ scale v_0 multiplied by the warp factor. The natural value for v_0 is the fundamental Planck scale, M. For values of kr \sim 30, we are able to reproduce the observed Standard Model Higgs vev. Meanwhile, the effective lagrangian (with this definition for v) 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?):

For some great background reading, I suggest looking at several TASI lectures on the subject that have been given in recent years:

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.


15 Responses to “Braneworld and the Hierarchy in RS1”

  1. 1 evankeane

    nice post there flip – well explained.

    This is a nice model – I especially like the idea of the 4D cosmological constant compensating for the negative cosmological constant of the bulk to keep the 4D brane flat.

    I will have to trust you that “this scenario comes out naturally from Horava-Witten models in M-theory”. How serious do you treat this model then? I figured it was simply a nice toy model until I read that … Could tests be thought up? Would the small differences in Planck scales be noticeable to some super experiment of the future?



  2. 2 evankeane

    oh ya forgot to say – the figure is very nice! 🙂

  3. Thanks, Evan. 🙂 Indeed the basic RS1 model is more of a toy model, but I think it has many of the necessary `moving parts’ for people to build upon it. There are a few questions that I didn’t bring up, for example radius stabilisation: 5D graviton excitations should also make the radius of the extra dimension vary (i.e. we have a “radion field”); but in the toy model we assume the radius is constant. Thus model-builders have to play with radius-stabilisation mechanisms (the standard one is called Goldberger-Wise).

    The signature of these models, as is generic for models with compactified extra dimensions, are Kaluza-Klein excitations of bulk fields. Since we only have gravity propagating in the bulk, this would mean we would expect a tower of Kaluza-Klein gravitons. Think of the particle in a box question in ordinary quantum mechanics. If we impose periodic boundary conditions (the same thing as turning the interval into a circle), then we have a standing wave solution and all higher harmonics. Thus the signature would be a family of gravitons with masses corresponding to each harmonic. As we are stuck on the visible brane, I don’t think there’s a straightforward way to probe the small difference between the effective 4D Planck scale and the fundamental 5D Planck scale… but if we see the Kaluza-Klein modes, we know that we should be thinking about extra dimensional models. (That is to say *if* we can disambiguate the KK modes from, say, supersymmetric partners… see the paper “Getting Fooled at the LHC” by Cheng et al.)

    As far as how serious should we treat the model, this is a good question. The appropriate context is that the past 30 years has been marked by the hegemony of the Standard Model in nearly all experimental tests. Model-builders had all this time to think of creative new physics, with very little experimental hints about which direction to proceed. The fact that there is some sector of string theory that accomodates braneworld scenarios is somewhat comforting from a motivational point of view… but it’s important to stress that braneworld does not imply string theory and string theory does not imply braneworld.

    The two most popular models for new TeV-scale physics are supersymmetry and extra dimensions. Supersymmetry is the unique extension of the 4D Poincare group under the Coleman-Mandula theorem. Extra dimensions extends the 4D Poincare group by turning it into the (4+N)D Poincare group. From this point of view we are trying to squeeze in new physics by circumventing well-accepted restrictions.

  4. 4 evankeane

    i could be cynical (me?) and play on your comment on string theory and braneworld –> string theory implies nothing, nothing implies string theory (oh look i did say it!)

    interesting stuff though – also i am heartened that even though in the past few decades, as you say, there has been no experimental pointers as to where the modelling should go, particle physicists have tried to find holes, ambigueities, possible problems, etc. in their theories! cf: the Cheng et al paper you mentioned! this is healthy and scientific!

    Q. what are the motivations for supersymmetry and extra dimensions, besides string theory, if any?

    NB. “because it works” IS an acceptable answer by the way. in fact this is often given as the ‘derivation’ of the Schrodinger Equation if I recall correctly and everyone is happy with that!!!



  5. Motivation for SUSY and XD! Excellent question. Maybe I’ll write a brief post on these some time… but these are rather standard in most SUSY literature. Here’s a short list: (By SUSY, by the way, I mean TeV-scale SUSY)

    1) The Hierarchy Problem (why is the electroweak scale so much smaller than the Planck scale) … SUSY and XD provide ways to avoid the fine tuning. In SUSY there are new particles that cancel the radiative corrections to the Higgs mass. In XD one either reparameterises the Hierarchy in terms of fine-tuning the size of the XD (the Large XD scenario) or one says that there’s no fine-tuning because all parameters are natural (the RS1 scenario above).

    2) Dark Matter. We know DM exists. We have good reason to believe that it is weakly interacting. SUSY with R-parity (which also prevents unfavourable interactions that lead to proton decay/flavour changing neutral currents) provides a natural DM candidate, the lightest supersymmetric particle (LSP). R-parity prevents the LSP from decaying into anything else.

    3) Grand unification. Based on results at LEP, running the renormalisation group equations in the Standard Model would not lead to gauge coupling unification. However, the minimally supersymmetric standard model provides the framework for the gauge couplings (as measured by LEP at the 100 GeV scale) to unify at a higher scale.

    4) Theoretical beauty. SUSY is the only extension of the Poincare Group under the Coleman-Mandula theorem. If SUSY did NOT exist, we would be left wondering why Nature didn’t make full use of this symmetry.

    5) Other stuff… there are some other nonperturbative effects that are relevant (BPS states), but unfortunately I don’t understand these well.

    6) It’s a necessary ingredient for String Theory. (Some people accept this, some people don’t.) Though this doesn’t mean we need to have low-energy SUSY (TeV scale).

    7) Anthropic principle: if we didn’t have low energy SUSY, then a LOT of people will have essentially wasted their time with their research in the past 30 years.

    8) Strong Anthropic principle: I’m interested in studying SUSY. So the universe should accomodate my interest.

  6. 6 Alejandro Rivero

    One should not say that string theory is a motivation for SUSY. On the contrary, SUSY plus Kaluza Klein plus other math technicalities (Leech lattice, Exceptional Groups, …) are motivations for string theory. This was an argument attributed to Polchinski: that if one follows the research on top-down particle theory, then it does not matter the road you travel (I say: GUTs, octonions, Susy, etc) you arrive to string theory at the end.

  7. 7 Alejandro Rivero

    Now, an amateur can find more colateral motivations for SUSY. One of them is that the representations you use in GUT theories are generated by fermionic creation operators. This was observed time ago in a paper from Wilczek and Zee, and you can still find their |++—> notation both in Zee book and in Wilczek opencourseware lecture notes (QFT III at MIT). Well, the case is that if we are speaking fermions here in the representation theory, we can hope a GUT setup will integrate smoothly with supersymmetry and its powers-of-two. And remember than neutrino oscillations have strongly motivated the GUT way again, as now we need the right handed neutrino.

    Another motivation I revisited recently (in but I had noticed it already in hep-ph/0512065v1 ) is a coincidence with the number of different open QCD strings (or Reggee traj.) when you terminate them wither with quarks or antiquarks. They happen to be, charge-by-charge, the same number that the number of degrees of freedom of the standard model fermions. This is because you the mass of the top is larger than the QCD scale and then you can not attach the top to a QCD string. In the spr thread I stress that if we took this equality between strings and fermions as postulate, then forcefully num generations >= 3, and if we extend it to the leptons, then =3.

    Still another motivation, of course, is the use of SUSY symmetry in other areas of physics. Generically, graded Lie superalgebras and supergroups can be useful in nuclear physics, and who knows if also in solid state physics.

  8. 8 evankeane

    hey flip – i actually looked up some SUSY books (while in Heffers) re: the motivations for SUSY and XD and i’m happy enough with them! so with TeV SUSY (ie. this is the scale it breaks at) the heirarchy problem is solved, the couplings unify and there is a dark matter candidate. fair enough! so that’s your first 3 reasons and that’s enough for me. Mars wil be happy with your 4th reason 🙂 i will take your word for it on number 5. 6 and 7 – yes well string theory and the anthropic principle … no comment. your final reason is also cool 🙂

  9. 9 evankeane

    ok i will comment on string theory …

    i agree with Alejandro Rivero – string theory is not a motivation for SUSY and XD but SUSY and XD (which are themselves motivated by what they produce as you mentiones) are motivations for string theory. this makes sense to me!


  10. 10 Alejandro Rivero

    hey evan! I like your abreviation for eXtra Dimensions. In emoticon language, it would translate as “dead while laughing” ( X for the eyes, dead shut, and D for the mouth, laughing loud). Btw, let me to apologise by the poor grammar in previous postings.

    About what motivates XD, I would think that its sources of motivation are: 1) the representation theory of Clifford Algebras 2) SuGra or Susy, and 3) The paper of Witten on Kaluza Klein. This paper actually motivates, I think, M-theory, moving the dimension up to 11. A 4th motivation can be Connes’s model, where dim=6 mod 8. The exact number of extra dimensions is burred because Clifford (n,m) = Clifford (n+1,m+1) So dimension 25+1 is as well motivated as 24+0. And here a 5th) motivation: Leech lattice.

  11. 11 Bee

    Great post! I’d just like to add that the energy of the KK-excitations in the RS scenario is not equally spaced as it is in scenarios that compactify on a torus. (if one solves the wave-equation in the above background geometry, they turn out to be at the zeros of one of the Bessel functions, forgot which). It’s not theoretically a big issue, but makes a difference for the phenomenology of that scenario.

    And thanks for the link!

    @Alejandro: When we founded our group on large extra dimensions in 2000 back in Germany, we were looking for a good abbreviation. the choice LED didn’t sound too appealing since we didn’t plan to emit very much light. It was my husband who suggested ‘LXD’, what we used from then on in our papers (and also on the homepage… gee, I just saw that it still exists!). I still like XD and LXD better than ED and LED, but most people seem to use the latter.

  12. 12 alex

    Another motivation for SUSY is REWSB (radiative electroweak symmetry breaking).

  13. 13 Alejandro Rivero

    Alex, right, but only for Minimal Susy in the orthodox way, and with the mass of the top a lot greater than the mass of the other fermions. Then the mass term of the higgs field (not to be confused with the mass of the higgs) changes sign in the adequate scale.

    Which remembers me another different view of the hierarchy: the expectation of that the electromagnetic mass of the electron, with cut-off at the Planck mass, should be of the order of the total mass of the electron. This is old tale, but for instance Polchiski revisits it when speaking of hierarchies in the second volume of his book.

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