Warped Passages (63 page)

Alex Pomarol, a Spanish physicist now at the University of Barcelona, observed that unification of forces can also occur in warped geometry. However, the setup he considered is slightly different; the electromagnetic, weak, and strong forces are not confined to a brane, but are instead present in the full five-dimensional bulk. The gauge bosons of the Standard Model—the gluons, the Ws, the Z, and the photon—are not stuck on a three-plus-one-dimensional brane.

According to string theory, gauge bosons could be stuck on a higher-dimensional brane or, along with gravity, they too could be in the bulk. Unlike the graviton, which must arise from a closed string, gauge bosons and charged fermions can correspond to either open or closed strings—it depends on the model. And according to whether they arise from open or closed strings, gauge bosons and fermions will be either stuck on a brane or free to move in the bulk.

In the large extra-dimensional scenario, had nongravitational forces been in the bulk, they would have been far too weak to agree with observations. Bulk forces would have spread throughout an enormous bulk space. Therefore, as with gravity, they too would have been extremely diluted. This would be unacceptable because we have measured the forces’ strengths to be much larger than this theory would have predicted.

But if additional dimensions are not large, as is the case in the warped geometry, there is no problem with the nongravitational forces in the five-dimensional bulk. The only thing that can dilute them is the extra dimensions’ size, not the warping—and in the warped scenario that size is rather small. This means that the true theory of the world might have all four forces experienced throughout the bulk. In that case, not only particles on the brane, but also particles throughout the bulk, could then feel the electric force, the weak force, and the strong force, as well as gravity.

If gauge bosons in the warped scenario are present in the bulk, they could have energy much bigger than a TeV. The gauge bosons, which would hang out in the bulk, would experience the entire energy range. No longer tethered to the Weakbrane, they could travel anywhere in the bulk, and have energies as high as the Planck scale energy. Only on the Weakbrane does energy have to be less than a TeV. Because the forces would be in the bulk and could therefore operate at high energies, unification of forces would be a possibility. This is exciting because it means that the forces can unify at high energy, even in a theory with an extra dimension. And Pomarol found the very interesting result that unification did indeed occur, almost as if the theory were truly four-dimensional.

But it gets even better. Unification and the warped hierarchy mechanism can be combined. Pomarol showed that forces unify, but he
assumed that supersymmetry addressed the hierarchy problem. But the hierarchy problem’s solution in warped geometry requires only that the Higgs particle be on the Weakbrane, so that its mass will be about the same as the weak scale energy, between 100 GeV and a TeV. The gauge bosons need not be stuck there.

In the warped geometry, all you need in order to solve the hierarchy problem is that the Higgs particle’s mass be low. That is because the Higgs field is responsible for the spontaneous symmetry breaking that is the source of all elementary particle masses. Gauge bosons and fermions won’t have a mass unless the weak force symmetry is broken. So long as the Higgs particle has a weak scale mass, the weak gauge boson masses will turn out correct. The warped gravity solution to the hierarchy really only requires the Higgs particle to be on the Weakbrane.

What this all means is that if the Higgs particle is on the Weakbrane, but quarks, leptons, and gauge bosons are in the bulk (see Figure 83), you can have your cake and eat it too. The weak scale would be protected and would be about a TeV, but unification could still occur at very high energies—on the GUT scale. My former student Matthew Schwartz and I showed that supersymmetry isn’t the only theory that
can be consistent with unification—a warped extra-dimensions theory can be, too!

Figure 83.
Nongravitational forces can also be in the bulk. In that case, forces can unify at high energies.

Experimental Implications

The natural scale on the Weakbrane is about a TeV. Should this warped geometry scenario prove to be a true description of our world, the experimental consequences at the Large Hadron Collider at CERN in Switzerland will be tremendous. Signatures of the warped five-dimensional spacetime could include Kaluza-Klein particles, five-dimensional black holes of anti de Sitter space, and TeV-mass strings.

The KK particles of the warped spacetime are likely to be the most accessible experimental herald of this geometry. As always, KK particles are particles with momentum in the extra dimension. But the new wrinkle in this model is that because the space is curved—not flat—the masses of the KK particles would reflect the idiosyncrasies of the warped geometry.

Since the only particle that we know for certain traverses the bulk is the four-dimensional graviton, let’s concentrate on its KK partners. As was true in flat space, the lightest of the KK partners of the graviton will be the one with no momentum at all in the fourth dimension. This particle would be indistinguishable from a particle of genuine four-dimensional origin: it’s the graviton that would communicate gravity in what looks like a four-dimensional world and it is the graviton whose probability function we have studied in detail in this chapter. If there were no additional KK particles, the gravitational force would behave in exactly the same way as in a true four-dimensional universe. In this scenario, the universe is secretly five-dimensional, but the particle that acts like a four-dimensional graviton does not reveal this fact. In the absence of heavier KK particles, Athena’s world would indeed appear to her to be four-dimensional.

Only the more massive KK particles could communicate the secrets of the five-dimensional theory. But they have to be light enough to be produced. Calculating the KK particles’ masses in this theory is a little tricky, however. Because of the distinctive geometry, the KK particles
would not have masses proportional to the inverse size of the dimension, as was the case for rolled-up dimensions of flat space. A mass proportional to the inverse size would have been extremely surprising, since, for the small extra dimension we are considering, that would be the Planck scale mass. On the Weakbrane nothing much heavier than a TeV can exist; one certainly wouldn’t ever find anything there with the Planck scale mass.

Since a TeV is the mass associated with the Weakbrane, it shouldn’t come as too big a surprise that when you do the calculations correctly taking into account warped spacetime, the KK particles turn out to have masses of about a TeV. Both the lightest KK particle, and the difference between the masses of the successively heavier KK particles, turn out to be about a TeV when the fifth dimension ends at the Weakbrane, as we have been assuming. KK particles pile up on the Weakbrane (because their probability function peaks there) and they have all the properties of Weakbrane particles.

This means that there are heavy KK partners of the graviton that are about 1 TeV, 2 TeV, 3 TeV,…in mass. And, depending on the ultimate energy reach of the LHC, there is a good chance of finding one or more of them. Unlike the KK partners in the large extra dimensions scenario, these KK partners interact much more strongly than gravity.

These KK particles are not nearly as feebly interacting as the graviton of four dimensions—they have an interaction strength sixteen orders of magnitude bigger. The graviton KK partners interact so strongly in our theory that any KK partner produced at the collider will not simply disappear out of sight, carrying away energy but leaving no visible signal. Instead, they will decay inside the detector into detectable particles, perhaps muons or electrons, which can be used to reconstruct the KK particle from which they originated (see Figure 84).

This is the conventional recipe for discovering new particles: study all the decay products and deduce the properties of what they came from. If what you find isn’t something you already know about, it must be something new. If the KK particles decay in the detector, the signal of extra dimensions should be very clean. In our model, rather than simply a missing energy signature, which has no significant labels
that would definitively identify the missing energy’s origin and let us distinguish the model from other possibilities, the reconstructed masses and spins of the KK particles should be enormously helpful clues that will tell us quite a lot about the new particles’ identities. The spin value of the KK particles—spin-2—will be a virtual ID tag that will tell us that the new particles have something to do with gravity. A spin-2 particle with a mass of about a TeV would be extremely strong evidence for an extra warped dimension. Few other models give rise to such heavy spin-2 particles, and the ones that do would have other distinguishing features.

Figure 84.
Two protons collide, and a quark and an antiquark annihilate and produce a KK partner of the graviton. The KK particle can then decay into visible particles, such as an electron and a positron. The gray lines are sprays of particles from the protons.

If we’re lucky, in addition to the KK partners of the graviton, experiments might also produce an even richer set of KK particles. In a theory in which most Standard Model particles reside in the bulk, we might also see charged KK partners of quarks and leptons and gauge bosons. Those particles would be both charged and heavy. And they could ultimately give us even more information about the higher-dimensional world.
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In fact, the model builders Csaba Csaki, Christophe Grojean, Luigi Pilo, and John Terning have shown that in extra-dimensional warped spacetime with Standard Model particles in the bulk, electroweak symmetry might be broken even without a
Higgs particle, and the charged particles that experimenters might then detect could tell us whether this alternative model is true for the world in which we live.

An Even More Bizarre Possibility

I’ve described quite a few weird properties of extra dimensions. But the most extraordinary possibility is yet to come. We will shortly see that a warped extra dimension can actually stretch infinitely far, yet still be invisible, unlike a flat dimension, which always has to have a finite size to agree with observations.

This result was truly shocking. In Chapter 22, when we discuss this infinite extra dimension, we will focus on the geometry of space, not the hierarchy problem. But I’ll briefly mention here how you can solve the hierarchy problem in the infinite-extra-dimensional case as well.

So far, we have considered a model with two branes: the Gravitybrane and the Weakbrane, both of which bound a fifth dimension. However, the Weakbrane doesn’t have to be the end of the world (that is, the boundary of the fifth dimension). If the Higgs particle is confined to a second brane placed in the middle of an infinite extra dimension, such a model could also solve the hierarchy problem. The graviton’s probability function would be very small on the Weakbrane, gravity would be weak, and the hierarchy problem would be solved just as before when the Weakbrane bounded the extra dimension. The graviton’s probability function in the model with an infinite warped dimension would continue beyond the Weakbrane, but that wouldn’t affect the solution to the hierarchy problem, which relied only on the small graviton probability function on the Weakbrane.

However, because the dimension is infinite, the KK particles would have different masses and interactions, so the experimental implications of this model would be different than the ones I just described. When Joe Lykken and I first discussed this possibility at the Aspen Center for Physics (an inspirational venue if there ever was one, and also one of the reasons why many theoretical physicists like to hike), we weren’t sure whether this idea would actually work. If the fifth dimension didn’t end on the Weakbrane, not all the KK particles
would be heavy (and have mass of about a TeV). Some KK particles would have very tiny masses. If these particles were detectable but experiments hadn’t yet discovered them, the model would be ruled out.