**A bold LHC application of Randall-Sundrum models got published in PRL**

Since the middle December, I thought that the \(125\,\,\GeV\) Higgs boson is a sure thing. But of course, there's always a chance I am wrong. The signals could go away. However, I will offer you a more shocking – and more likely – reason why I could have been wrong: it could actually be a sign of a different particle than the God particle, a more divine one.

Some religious readers could ask: What in the hell and the heaven may be more divine than God? The answer is that there's perhaps nothing more divine than God "in the hell and the heaven" but there could still be more divine things "out of the hell and heaven", in extra dimensions!

Lisa Randall and Raman Sundrum improved the 1998 large-dimensions scenario by ADD1 and ADD2 and invented an even more shocking, more realistic, and more popular scenarios how the extra dimensions could be experimentally accessible even though they are not large: the Randall-Sundrum extra dimension is one dimension and it is "warped", not "large".

I will explain it momentarily but the relevant 1999 papers by Randall and Sundrum are known as

RS1: A Large Mass Hierarchy from a Small Extra DimensionBoth papers rely on one extra dimension. To confuse you (or to help you remember which paper is which), RS1 uses 2 branes while RS2 uses 1 brane. The brane or branes are localized in the extra dimension. The realistic model hiding under this blog entry is RS1.

RS2: An Alternative to Compactification

Now, we're talking about a story that was covered by Ars Technica:

Hiding in the Higgs data: hints of physics beyond the standard model by Chris LeeIt's not just this article; I consider Ars Technica a very decent source of similarly interesting and relevant semi-technical news. This particular article describes the paper

Could the excess seen at \(124\)-\(126\,\,\GeV\) be due to the Randall-Sundrum Radion?by Kingman Cheung and Tzu-Chiang Yuan which was just published in Physical Review Letters.

**Randall-Sundrum introduction**

When I first heard about the Kaluza-Klein theory more than 20 years ago and learned that the masses of the new particles are quantized because their momenta in the extra dimensions \(P_5\) are quantized, I thought: Nice. It looks completely analogous to the quantization of the angular momentum \(J_z\) which occurs because the dual angle \(\phi\) is periodic, or quantization of momentum on a compact space.

But in quantum mechanics, I had seen one more reason aside from compactness of space why the spectrum of a Hamiltonian may be discrete rather than continuous: a confining potential. A quantum harmonic oscillator is the most famous model whose spectrum is quantized for the very same reason. And I thought that the curvature of space could produce such a potential.

Of course, I didn't do anything else beyond these ideas so the scenario was waiting for bolder physicists. Their spacetime geometry looks like follows: (I know that many readers expect a picture here but I was brought up to believe and I still believe than the best picture is an equation. Nevertheless, you will see a picture if you're patient.)\[

{\Large ds^2 = e^{-2kr_c \phi} \eta_{\mu\nu}dx^\mu dx^\nu +r_c^2 d\phi^2}

\] I have used larger fonts because there are some "double subscripts" over there. This Riemannian metric has 4+1=5 dimensions. The usual 3+1 dimensions carry the vector indices \(\mu,\nu\) and the geometry within these dimensions is given by the flat Minkowski metric \(\eta_{\mu\nu}\). However, there's one more dimension \(\phi\) that contributes an extra term to \(ds^2\) whose coefficient is simply constant.

However, this fifth dimension \(\phi\) appears at one more place: it is being exponentiated and the exponential multiplies the ordinary four spacetime dimensions' metric; this rescaling of the metric by a function of another coordinate is what is referred to as the "warping". So if you move in the normal 3+1 dimensions \(x^\mu\), while keeping \(\phi\) fixed, the proper distances (or times) you will measure will actually depend on your position in the extra dimension i.e. on your value of \(\phi\). And they will depend exponentially.

Such an exponential is cool because the exponentials of reasonable numbers produce vast or tiny numbers, depending on the sign, so the very formula above is a welcome source of potential hierarchies – parameters and ratios in physics that are not of order one at all even though they could be expected to be of order one. Just place different types of particles at different values of \(\phi\) i.e. different positions in extra dimensions and some of their characteristic parameters will be exponentially larger or smaller than the parameters describing their "segregated" friends. Lisa Randall, a moderate feminist, has become one of the world's most enthusiastic advocates of segregation. To make it more PC, she actually chose a more awkward word for the same thing, sequestering, which also looks like if she fights against CO2 emissions. ;-)

Now, you probably asked: But why did the exponential appear in the formula? Didn't we just cheat? The answer is that the exponential is totally natural, too. It's exactly the exponential function for which the 5-dimensional spacetime above is nothing else than the Anti de Sitter space. Locally, it's the same space that people use in the AdS/CFT correspondence. There are many overlapping ideas between AdS/CFT and Randall-Sundrum but the ultimate motivations are different and Randall and Sundrum – model builders – made many of their insights independently of the developments in the formal theorists' AdS/CFT industry (which was running since late 1997).

It is exactly the exponential function for which we obtain the AdS space which is obviously a nice solution – a maximally symmetric one – of Einstein's equations with a negative cosmological constant in the bulk. (Our four-dimensional Universe probably has a positive cosmological constant and is very close to a de Sitter space, the "opposite" of anti de Sitter space.)

Let's now choose the units and rescale the coordinate \(\phi\) so that\[

{\Large ds^2 = e^{-2\phi} \eta_{\mu\nu}dx^\mu dx^\nu + d\phi^2}

\] and we only look at the mathematical essence, not the numerical constants. The funny thing is that if you change \(\phi\) by a moderate shift such as 35 units of length, the exponential \(\exp(-2\phi)\) will be around \(10^{-30}\) which means that the distances \(ds\) in the normal 3+1 dimensions will shrink by a factor of \(10^{15}\) or so.

It means that if you define a region "inside your room" and "now" by some particular coordinates \(x^\mu\), the proper length of your room will depend on the value of \(\phi\), the extra dimensions, in which you measure it. If \(\phi\) belongs to an interval of a certain reasonable length like 35, which is what RS1 assumes, the proper distances (and times) between the corners of your rooom measured on one boundary of the 5-dimensional spacetime will be \(10^{15}\) times longer than those measured on the opposite boundary of the room.

*Click the image to zoom it in.*

Now I must tell you where the electrons and other particles in your body sit. There are two boundaries of this braneworld located at \(\phi_{\rm min}\) and \(\phi_{\rm max}\). All the electrons and friends sit at the brane/boundary \(\phi_{\rm max}\) (note that the exponential is decreasing) where the proper distances are shorter, the "weak brane". If you think about it, it's the right side of the picture above because that's where the proper size of your room is just 2 meters because you can only squeeze one person into the room.

On the opposite side, where the proper size of the room is dozens of meters as indicated by the many small people-sized blobs in the room (but people and electrons actually can't get there because they're stuck on the right brane), we find a "Planck brane" or "gravity brane" where most of the volume of the 5D space and spacetime is concentrated because the proper distances are exponentially larger over there for fixed changes of \(x^\mu\). Gravitons may always propagate to every corner of the spacetime because gravitons are perturbations of the shape of space. And because most of the volume sits near the "Planck brane" where the proper distances are large, we may say that "most of the gravitons' wave function" is concentrated near the opposite brane than the brane we call our home. And that's the reason why the gravitons only weakly overlap with particles such as electrons and that's why gravity seems so weak to us, objects composed of leptons and quarks. But if we could turn ourselves into tourists who may escape from our "weak brane" and visit the rest of the 5D space, we would see that gravity is pretty normally strong.

*A Long White Glowing Comet, an astronomical song, an original version from 1979 (try also the newer one by Leona Machálková). Comets are confined to the weak brane, too. The song is correct because she recommends a man to look for comets in the places where there's almost no gravity (most of gravity is concentrated on the Planck brane). However, she also wants him to delay the observations because she's horny (and much younger than the comet, something that less true today!) so not everything is perfectly pro-science in the lyrics.*

That was RS1 with 2 branes. RS1 has 1 brane. You abolish the "weak brane" and locate us on the "Planck brane" instead. This RS2 model is interesting conceptually: physics looks 4-dimensional even though the extra fifth dimension is actually infinitely large! (You couldn't do it in the opposite way, to allow the space continue indefinitely behind an abolished "Planck brane", because the total volume of a unit coordinate region of such a semi-infinite space would be infinite since the proper distances would blow up behind the former "Planck brane".)

**Coupling to the trace**

This possibility had been overlooked in the literature so after 1999, model builders decided for some well-deserved overcompensation. About 5,000 papers related to the Randall-Sundrum model buildings have been written as of today.

One of them is the "Higgs is actually radion" paper that motivated this blog entry.

As we said, our "weak brane" of RS1 is localized at some value of the fifth coordinate \(\phi\). Goldberger and Wise designed a mechanism involving a bulk scalar field that stabilizes this position of our brane. I think it's not necessarily "the" right solution but it has surely served as a "proof of the concept" that implies that the exponential generator of hierarchies that Randall and Sundrum envisioned really works. The natural parameters describing the physical laws including the "stabilizers" may be moderate and the model will still produce exponential hierarchies.

The thickness of the Randall-Sundrum world or, pretty much equivalently, the location of our brane \(\phi\) in the fifth dimension behaves as a four-dimensional scalar field, the radion.

What about the new paper? Why were the Asian folks led to consider the idea that the apparent God particle is actually a radion?

The funny thing is that the LHC is observing signals that are pretty much compatible with the \(125~\GeV\) Higgs boson but there are also potentially emerging discrepancies. So far, the deviations from the Standard Model Higgs predictions are modest enough so that it's not insane to say that the deviations are due to noise. But if these early hints will continue to grow, we may face the problem that the observed bumps near \(125~\GeV\) actually "significantly disagree" with the Standard Model predictions.

What are these discrepancies?

It seems that the decay \(H\to\gamma\gamma\) seems about twice as frequent in the LHC data than what is predicted by the Standard Model. Because the whole excess is of order 3 sigma, the "excess over the Standard Model prediction" is just 1.5 sigma, if you allow me to be a bit sketchy, and it is not statistically significant. But some sensitive people may already feel nervous and we may speculate what will happen if even mentally stable people will have to feel nervous later in 2012.

How do they explain the deficit?

They say that the \(125~\GeV\) bumps are not due to the Higgs boson at all. Instead, they propose that they're caused by the radion – by the oscillating position of our brane in the fifth dimension! This position is stabilized by a mechanism, like the Goldberger-Wise mechanism, and the corresponding wave-shaped fluctuations away from the equilibrium value are quantized, much like particles in all of particle physics. In their fairy-tale, these fluctuations are the \(125~\GeV\) heavy "radion" scalar particles. The corresponding "vev" of the radion field is, in the usual conventions, about \(680~\GeV\).

**What is their evidence?**

The funny thing is that the radion with those parameters has couplings very similar to Higgs but it indeed produces a twice-as-big (or so) signal for the photon pair than the Higgs boson does, a detailed deviation that is actually pretty hard to be reproduced even in supersymmetric model building. It suppresses the channels with two bottom quarks and orders other milder corrections but the final answer is that such a radion is more compatible with the data observed at the LHC, especially because of the enhanced \(\phi\to \gamma\gamma\) channel, than the Higgs boson! It may be a coincidence, the tests that this model passed were not terribly stringent, numerous, or accurate, and the model needed to adjust one additional constant, the "radion vev" of \(680~\GeV\). But there could still be nontrivial evidence that this somewhat unexpected model is promising.

If the model were right, it would mean that whenever the LHC sees a trace of the \(125~\GeV\) particle, the collider has actually kicked our brane on which our Universe and its visible particles live into the extra dimension and created the minimal possible amount of "wiggles" of this brane-shaped space that is waving into the new dimension. This wouldn't get us terribly deep into the extra dimensions (we would still have to wait for the production of "Kaluza-Klein modes" i.e. particles whose wave function depends very differently on the extra dimensions than the wave function of the known particle species) but it could mean that we are starting to study extra-dimensional physics.

Just to be sure, I must say how the new scalar field, the radion, interacts with the known particles of the Standard Model. Because the shift in the extra dimension creates a "uniform scaling of proper distances" and because the scaling in field theories is "generated" by the trace of the stress-energy tensor (the sum of its diagonal entries), the relevant interaction is\[

\LL_{\rm int} = \frac{\phi}{\Lambda_\phi} T_\mu^\mu (SM)

\] The trace \(T_\mu^\mu\) of the Standard Model Lagrangian is multiplied by the scalar field \(\phi\) itself and divided by the \(\Lambda_\phi=680\,\,\GeV\) constant we have already mentioned. The trace of the stress-energy tensor looks much like the Lagrangian \(\LL_{SM}\) of the Standard Model but not quite. The coefficients are a bit changed (if the coupling were proportional to the Lagrangian, the field would be a kind of a "dilaton", with greetings to Dilaton, not a "radion") and you have to include terms from the "renormalization group running" into it, and so on. If you do so correctly, then – assuming that the Asian authors have done it correctly as well – you will get nice predictions that may agree with the LHC data more accurately than the Higgs. In particular, the radion's coupling to the photon pair (which is being weakly observed) and the gluon pair (which may be OK) is enhanced relatively to other couplings.

If this explanation is right, we should still ask: Where is the Higgs? It doesn't seem to me that they're answering this basic question at all.

I was trying to consider scenarios in which the radion itself would act as a Higgs at the same moment. The example of enhanced gauge symmetries at the self-dual radius in heterotic (and bosonic) string theory look like good templates for such a geometrization of the field breaking the spontaneous symmetry. I couldn't find a model that works. It seems likely that a separate Higgs has to exist besides the radion and it must be hiding somewhere. However, it's possible that the radion couplings make the usual problems justifying the existence of the Higgs boson – such as the WW scattering – less urgent if not completely non-urgent.

It seems likely to me that some people will carefully look into these matters. If the LHC continues to find deviations from the Standard Model, these seemingly far-fetched possibilities may become the daily work for many more particle physicists.

And that's the memo.

Hi Lubos,

ReplyDeleteI know you like equations but images help greatly for us lay people.

I couldn't but think that a conformal field theory approach(AdS/CFT) could not have been more implicit toward describing the abstractness of a world that classically could be moved toward our perceptions in Lagrangian. I mean on the surface you are interlining deeper underpinnings of a strange world.

I mean what possible association could trigger conceptual ideas that may only make sense to scientists qualitatively while us poor lay people have to fend for ourselves?:)

So anyway the whole time you are writing I keep seeing the Ltool. Here is an image that might help?

Please forgive me for my erroneous ways.

Best,

It looks to me that a universe with extra dimensions can explain not only our universe, but any imaginary one, so, we most be careful about it. Because something works mathematically doesn't mean it's true. I'm not an physicist and I hear those examples about try to imagine a 2D person in an 2D world, and they wouldn't notice that we - 3D persons - even exist and I get real frustrated because it is an ridicolous example. And 2D person cannot exist, it's a mathematical abstraction and yet people like Kaku use that example all the time. I can't leave my skepticism about a lot of dimension until there's a real evidence they even exist, until then bring those dimensions to explain anything sounds like cheating to me.

ReplyDeleteMaybe this will help? I know it helps me.

ReplyDeleteMeanwhile I’m continuing to develop the Extra Dimensions series of articles, and I’ve now followed up my examples of extra dimensions with a next installment, a first discussion of what scientists would look for in trying to identify that our world actually has one or more extra dimensions . The new article describes one of the key clues that would indicate their presence. But this is far from the end of the story: I owe you more articles, explaining why extra dimensions would generate this clue, outlining how we try to search for this clue experimentally, and mentioning other possible clues that might arise. All in due course…The Smoking Gun for Extra Dimensions by Theoretical Physicist Matt Strassler