## Knocking on Heaven's Door: How Physics and Scientific Thinking Illuminate the Universe and the Modern World - Lisa Randall (2011)

### Part IV. MODELING, PREDICTING, AND ANTICIPATING RESULTS

### Chapter 17. THE WORLD’S NEXT TOP MODEL

In January 2010, colleagues gathered at a conference in Southern California to discuss particle physics and dark matter searches in the LHC era. The organizer, Maria Spiropulu, a CMS experimenter and member of the Caltech physics department, asked me to give the first talk and outline the LHC’s major issues and physics goals for the near future.

Maria wanted a dynamic conference, so she suggested we start with a “duel” among the three opening speakers. As if the term “duel” applied to three people wasn’t confusing enough, the audience of invited guests posed an even greater challenge since it ranged from experts in the field to interested observers from the California technology world. Maria asked me to dig deep and look into subtle and overlooked features of current theories and experiments, while one of the attendees, Danny Hillis—a brilliant nonphysicist from the company Applied Minds—suggested I make everything as basic as possible so the nonexperts could follow.

I did what any rational person would do in the face of such contradictory and impossible-to-satisfy advice: procrastinate. The result of my web surfing was my first slide (see Figure 56), which ended up in Dennis Overbye’s *New York Times* article on the subject—typo and all.

The topics referred to the subject matter that the subsequent speakers and I were scheduled to cover. But the humor in the sound effects I inserted to accompany the entrance of each of the dueling cats (which I can’t reproduce here) was meant to reflect both the enthusiasm and the uncertainty associated with each of these models. Everyone at the conference, no matter how strongly convinced of an idea he or she had worked on, knew that data were coming soon. And data would be the final arbiter of who had the last laugh (or a Nobel Prize).

**[ FIGURE 56 ]** Candidate models, as I presented on a slide at a conference.

The LHC presents us with a unique opportunity to create new understanding and new knowledge. Particle physicists hope to soon know the answers to the deep questions we have been thinking about: Why do particles have the masses they do? What is dark matter composed of? Do extra dimensions solve the hierarchy problem? Are extra symmetries of spacetime involved? Or is there something completely unforeseen at work?

Proposed answers include models with names like supersymmetry, technicolor, and extra dimensions. The answers could turn out to be different from anything anticipated, but models give us concrete targets of what to look for. This chapter presents a few of the candidate models that address the hierarchy problem and gives a flavor of the type of explorations that the LHC will perform. Searches for these and other models happen concurrently and will provide valuable insights no matter what turns out to be the true theory of nature.

**SUPERSYMMETRY**

We’ll begin with the bizarre symmetry called supersymmetry and the models that incorporate it. If you did a survey among theoretical particle physicists, a good fraction of them would likely say that supersymmetry solves the hierarchy problem. And if you asked experimenters what they wanted to look for, a large fraction of them would suggest supersymmetry as well.

Since the 1970s, many physicists have considered the existence of supersymmetric theories so beautiful and surprising that they believe it has to exist in nature. They have furthermore calculated that forces should have the same strength at high energy in a supersymmetric model—improving on the near-convergence that happens in the Standard Model, allowing the possibility of unification. Many theorists also find supersymmetry to be the most compelling solution to the hierarchy problem, despite the difficulty in making all the details agree with what we know.

Supersymmetric models posit that every fundamental particle of the Standard Model—electrons, quarks, and so on—has a partner in the form of a particle with similar interactions but different quantum mechanical properties. If the world is supersymmetric, then there exist many unknown particles that could soon be found—a supersymmetric partner for every known particle. (See Figure 57.)

Supersymmetric models could help solve the hierarchy problem and, if so, would do it in a remarkable fashion. In an exactly supersymmetric model, the virtual contributions from particles and their superpartners cancel exactly. That is, if you add together all the quantum mechanical contributions from every particle in the supersymmetric model and tally their effect on the Higgs boson mass, you would find they all add up to zero. In a supersymmetric model, the Higgs boson would be massless or light, even in the presence of quantum mechanical virtual corrections. In a true supersymmetric theory, the sum of the contributions of both types of particles exactly cancel. (See Figure 58.)

**[ FIGURE 57 ]** In a supersymmetric theory, every Standard Model particle would have a supersymmetric partner. The Higgs sector is also enhanced beyond that of the Standard Model.

This sounds miraculous perhaps but is guaranteed because supersymmetry is a very special type of symmetry. It’s a symmetry of space and time—like the symmetries you are familiar with such as rotations and translations—but it extends them into the quantum regime.

**[ FIGURE 58 ]** In a supersymmetric model, contributions from virtual supersymmetric particles exactly cancel the Standard Model particles’ contributions to the Higgs boson mass. For example, the sum of the contributions from the two diagrams above is zero.

Quantum mechanics divides matter into two very different categories—bosons and fermions. Fermions are particles that have half-integer *spin*, where spin is a quantum number that essentially tells us something like how much the particle acts as if it is spinning. Half-integer means values like 1/2, 3/2, 5/2, and so on. The quarks and leptons of the Standard Model are examples of fermions and have spin -1/2. Bosons are particles such as the force-carrying gauge bosons or perhaps the yet-to-be-discovered Higgs boson that have integral spin, indicated by whole numbers such as 0, 1, 2, and so on.

Fermions and bosons are distinguished not only by their spins. They behave very differently when there are two or more of them of the same type. For example, identical fermions with the same properties can never be found in the same place. This is what the *Pauli exclusion principle*, named after the Austrian physicist Wolfgang Pauli, tells us. This fact about fermions accounts for the structure of the periodic table that tells us that electrons, unless distinguished by some quantum number, have to orbit around the nucleus differently from each other. It is also the reason why my chair isn’t falling to the center of the Earth, since the fermions in my chair can’t be in the same place as the material of the Earth.

Bosons, on the other hand, behave in exactly the opposite manner. They are actually more likely to be found in the same place. Bosons can pile on top of each other—kind of like crocodiles, which is why phenomena like Bose condensates that require many particles to pile up in the same quantum mechanical state can exist. Lasers, too, rely on bosonic photons’ affinity for each other. The intense beam is created by the many identical photons that shoot off together.

Remarkably, in a supersymmetric model, particles that we take to be very different—bosons and fermions—can be exchanged in such a way that the result in the end is the same as the theory you started with. Each particle has a partner particle of the opposite quantum mechanical type, but with exactly the same mass and charges. The nomenclature for the new particles is a bit funny—it never fails to elicit giggles when I speak on this topic in public. For example, the fermionic electron is paired with a bosonic *selectron*. A bosonic photon is paired with a fermionic *photino*, and a W is paired with a *Wino*. The new particles have related interactions to the Standard Model particles with which they are paired. But they have opposite quantum mechanical properties.

In a supersymmetric theory, the properties of each boson are related to the properties of its superpartnered fermion and vice versa. Since each particle has a partner and the interactions are carefully aligned, the theory permits this bizarre symmetry that interchanges fermions and bosons.

One way to understand the apparently miraculous cancellation of virtual contributions to the Higgs mass is that supersymmetry relates any boson to a partner fermion. In particular, supersymmetry partners the Higgs boson with a Higgs fermion, the Higgsino. Even though quantum mechanical contributions radically influence the mass of a boson, the mass of a fermion will never be much bigger than the classical mass, which is the mass before you account for quantum contributions you started out with—even when quantum mechanical corrections are included.

The logic is subtle, but large corrections don’t occur because fermion masses involve both left-handed and right-handed particles. Mass terms allow them to convert back and forth into each other. If there were no classical mass term and they couldn’t convert into each other before quantum mechanical virtual effects were included, they couldn’t do so even with quantum mechanical effects taken into account. If a fermion has no mass to begin with (no classical mass), it will still have zero mass after quantum mechanical contributions are included.

No such argument applies to bosons. The Higgs boson, for example, has zero spin. So there is no sense in which we can talk about a Higgs boson spinning to the left or to the right. But supersymmetry tells us that boson masses are the same as fermion masses. So if the Higgsino mass is zero (or small), so too must be the mass of the partnered Higgs boson in a supersymmetric theory—even when quantum mechanical corrections are taken into account.

We don’t yet know if this rather elegant explanation for the stability of the hierarchy and cancellation of large corrections to the Higgs mass is correct. But if supersymmetry does address the hierarchy problem, then we know a lot about what we would expect to find at the LHC. That’s because we know what new particles should exist, since every known particle should have a partner. On top of that, we can estimate what the masses of the new supersymmetric particles should be.

Of course, if supersymmetry were exactly preserved in nature, we would know precisely the masses for all the superpartners. They would be identical to the mass of the particle they were paired with. However, none of the superpartners have been observed. That tells us that even if supersymmetry applies in nature, it cannot be exact. If it were, we would have already discovered the selectron and the squark and all the other supersymmetric particles a supersymmetric theory would predict.

So supersymmetry has to be *broken*, meaning the relationships that are predicted in a supersymmetric theory—though possibly approximate—cannot be exact. In a broken supersymmetric theory, every particle would still have a superpartner, but those superpartners would have different masses than their partner Standard Model particles.

However, if supersymmetry were too badly broken, it wouldn’t help with the hierarchy problem, since the world would then look as if supersymmetry didn’t apply to nature at all. Supersymmetry has to be broken in just such a way that we wouldn’t have yet discovered evidence of supersymmetry, while the Higgs mass is nonetheless protected from large quantum mechanical contributions that would give it too big a mass.

This tells us that supersymmetric particles should have weak scale masses. Any lighter and they would have been seen, and any heavier and we would expect the Higgs mass to be heavier as well. We don’t know precisely the masses since we only know the Higgs mass approximately. But we do know that if the masses were overly heavy, the hierarchy problem would persist.

So we conclude that if supersymmetry exists in nature and addresses the hierarchy problem, lots of new particles with masses in the range of a few hundred GeV to a few TeV should exist. This is precisely the range of masses the LHC is positioned to search for. The LHC, with 14 TeV of energy, should be able to produce these particles even if only a fraction of the protons’ energy goes into quarks and gluons colliding together and making new particles.

The easiest particles to produce at the LHC would be the supersymmetric particles that are charged under the strong nuclear force. These particles could be made in abundance when protons collide (or more specifically the quarks and gluons within them). When these collisions happen, new supersymmetric particles that interact via the strong force can be produced. If so, they will leave very distinctive and characteristic evidence in the detectors.

These *signatures*—the experimental pieces of evidence they leave—depend on what happens to the particle after its creation. Most supersymmetric particles will decay. That’s because, in general, lighter particles (such as those in the Standard Model) exist for which the total charge is the same as the heavy supersymmetric particle. If that’s the case, the heavy supersymmetric particle will decay into lighter Standard Model particles in a way that conserves the initial charge. Experiments will then detect the Standard Model particles.

That’s probably not sufficient to identify supersymmetry. But in almost all supersymmetric models, a supersymmetric particle won’t decay solely into Standard Model particles. Another (lighter) supersymmetric particle remains at the end of the decay. That’s because supersymmetric particles appear (or disappear) only in pairs. Therefore, a supersymmetric particle has to remain at the end after a supersymmetric particle has decayed—one supersymmetric particle cannot turn into none. Consequently, the lightest such particle must be stable. This lightest particle, which has nothing to decay into, is known to physicists as the lightest supersymmetric particle, the LSP.

Supersymmetric particle decays are distinctive from an experimental vantage point in that the lightest of the neutral supersymmetric particles will remain, even after the decay is complete. Cosmological constraints tell us that the LSP carries no charges, so it won’t interact with any elements of the detector. This means that whenever a supersymmetric particle is produced and decays, momentum and energy will appear to be lost. The LSP will disappear from the detector and carry away momentum and energy to where it can’t be recorded, leaving as its signature missing energy. Missing energy isn’t specific to supersymmetry alone, but since we already know a good deal about the supersymmetric spectrum, we know both what we should and shouldn’t see.

For example, suppose a squark, the supersymmetric partner of a quark, is produced. Which particles it can decay into will depend on which of the particles are lighter. One possible mode of decay will always be a squark turning into a quark and the lightest supersymmetric particle. (See Figure 59.) Recall that because decays can occur essentially immediately, the detector records only the decay products. If such a squark decay occurred, detectors would record the passage of the quark in the tracker and in the hadronic calorimeter that measures energy deposited by a strongly interacting particle. But the experiment will also measure that energy and momentum are missing. Experimenters should be able to tell that momentum is missing in the same way they can when neutrinos are produced. They would measure momentum perpendicular to the beam and find that it doesn’t add to zero. One of the biggest challenges the experimenters face will be to unambiguously identify this missing momentum. After all, anything that is not detected appears to be missing. If something is wrong or mismeasured and even small amounts of energy go undetected, the missing momentum could add up to mimic an escaping supersymmetric particle’s signal, even though nothing exotic was produced.

**[ FIGURE 59 ]** A squark can decay into a quark and the lightest super-symmetric particle.

In fact, because the squark is never created on its own, but only in conjunction with another strongly interacting object (such as another squark or an antisquark), the experimenters will measure at least two jets (see Figure 60 for an example). If two squarks are created by a proton collision, they would give rise to two quarks that detectors would record. The net missing energy and momentum would escape undetected, but their absence would be noted and provide evidence for new particles.

**[ FIGURE 60 ]** The LHC might produce two squarks together, both of which decay into quarks and LSPs, leaving a missing energy signature.

One major advantage of all the delays in the LHC schedule was that experimenters had time to fully understand their detectors. They calibrated them so that measurements were very precise from the day the machine went on line, so missing energy measurements should be robust. Theorists, on the other hand, had time to think about alternative search strategies for supersymmetric and other models. For example, together with a theorist from Williams College, Dave Tucker-Smith, I found a different—but related—way to search for the squark decay just described. Our method relies on measuring only the momentum and energy of the quarks emerging from the event, with no need to explicitly measure missing momentum, which can be tricky. The great thing about the recent LHC excitement was that a number of CMS experimenters immediately ran with the idea and not only showed that it worked, but generalized and improved it within a few months. It’s now part of the standard supersymmetry search strategy and the first supersymmetry search from CMS used the technique we had so recently suggested.^{62}

Down the road, even if supersymmetry is discovered, experimenters won’t stop there. They will try their best to determine the entire supersymmetric spectrum, and theorists will work to interpret what the results could mean. A lot of interesting theory underlies supersymmetry and the particles that could spontaneously break it. We know which supersymmetric particles should exist if supersymmetry is relevant to the hierarchy problem, but we don’t yet know the precise masses they should have or how those masses arise.

Different mass spectra will make an enormous difference to what the LHC should see. Particles can only decay into other particles that are lighter. The decay chain, the sequence of possible decays of supersymmetric particles, depends on the masses—what is heavier and what is lighter. The rates of various processes also depend on particle masses. Heavier particles in general decay more quickly. And they are usually more difficult to produce since only collisions with a good deal of energy can create them. Combining all the results together could give us important insights into what underlies the Standard Model and what awaits at the next energy scales. This will be true of any analysis of new physical theories that we might find.

Nonetheless, one should keep in mind that despite supersymmetry’s popularity among physicists, there are several reasons for concern about whether it truly applies to the hierarchy problem and the real world.

The first, and perhaps the most worrisome, is that we have not yet seen any experimental evidence. If supersymmetry exists, the only explanation for why we haven’t yet seen evidence is that the superpartners are heavy. But a natural solution to the hierarchy problem would require that superpartners be reasonably light. The heavier the superpartners are, the more inadequate supersymmetry appears as a solution to the hierarchy problem. The fudge required is determined by the ratio of the mass of the Higgs boson to the supersymmetry breaking scale. The bigger this is, the more “fine-tuned” the theory.

Not yet having seen the Higgs boson either compounds the problem. It turns out that in a supersymmetric model, the only way to make the Higgs heavy enough to have eluded detection is to have big quantum mechanical contributions that can come only from heavy superpartners. But again, those masses need to be so heavy that the hierarchy becomes a little unnatural, even with supersymmetry.

The other problem with supersymmetry is the challenge of finding a fully consistent model that includes supersymmetry breaking and agrees with all experimental data to date. Supersymmetry is a very specific symmetry that relates many interactions and prohibits interactions that quantum mechanics would otherwise permit. Once supersymmetry is broken, the “anarchic principle” takes over. Anything that can happen will. Most models would predict decays that have either never been seen in nature or are seen only much too infrequently to agree with predictions. Because of quantum mechanics, a whole can of worms is opened once supersymmetry is broken.

Physicists might simply be missing the right answers. We certainly cannot say definitively that no good models exist or that a little fine tuning doesn’t happen. Certainly, if supersymmetry is the correct resolution of the hierarchy problem, we should find evidence for it soon at the LHC. So it is certainly worth pursuing. A discovery of supersymmetry would mean that this exotic new spacetime symmetry applies not just in a theoretical formulation on a piece of paper, but also in the real world. However, in the absence of discovery, it is also worth considering alternatives. The first we’ll consider is known as *technicolor*.

**TECHNICOLOR**

Back in the 1970s, physicists also first considered an alternative potential solution to the hierarchy problem known as *technicolor*. Models under this rubric involve particles that interact strongly via a new force, playfully named the *technicolor force*. The proposal was that technicolor acts similarly to the strong nuclear force (which is also known as the color force among physicists), but binds particles together at the weak energy scale—not the proton mass scale.

If technicolor is indeed the answer to the hierarchy problem, the LHC wouldn’t produce a single fundamental Higgs boson. Instead it would produce a bound state, something like a hadron, that would play the role of the Higgs particle. The experimental evidence in support of technicolor would be lots of bound state particles and many strong interactions—very much like the hadrons we are familiar with, but that appear only at much higher energy—at or above the weak scale.

Not yet having seen any evidence poses a significant constraint on technicolor models. If technicolor is truly the solution to the hierarchy problem, we would expect to have already found evidence—though of course we could be missing something subtle.

On top of that, model building with technicolor is even more challenging than with supersymmetry. Finding models that agree with everything we have observed in nature has posed significant challenges, and no entirely suitable model has been found.

Experimenters will nonetheless keep an open mind and search for technicolor and any other evidence of new strong forces. But hopes are not overly high. If, however, technicolor turns out to be the underlying theory of the world, maybe Microsoft Word will stop automatically spellcorrecting and inserting a capital “T” whenever I write about it.

**EXTRA DIMENSIONS**

Neither supersymmetry nor technicolor are obviously perfect solutions to the hierarchy problem. Supersymmetric theories don’t readily accommodate experimentally consistent supersymmetry breaking and deriving technicolor theories that predict the correct quark and lepton masses is even more difficult. So physicists decided to look further afield and considered ideas that are superficially even more speculative alternatives. Remember, even if an idea seems ugly or not obvious at first, only after we fully understand all the implications can we decide which idea is most beautiful—and, more importantly, correct.

The better understanding of string theory and its components that physicists gained in the 1990s led to new suggestions for addressing the hierarchy problem. These ideas were motivated by elements of string theory—though not necessarily directly derived from its very constrained structure—and involved extra dimensions of space. If extra dimensions exist—and we have reason to think they might—they could hold the key to solving the hierarchy problem. If that is indeed the case, they would give rise to experimental evidence of their existence at the LHC.

Additional spatial dimensions is an exotic concept. If the universe has such dimensions, space would be very different from what we observe in our everyday lives. In addition to the three directions—left-right, up-down, forward-backward, or alternatively longitude, latitude, and altitude—space would extend in directions no one has ever observed.

Clearly, since we don’t see them, these new dimensions of space must be hidden. That could be because they are too small to directly influence anything we could possibly see, as physicist Oskar Klein suggested back in 1926. The idea is that as much that owing to our limited resolution, the dimensions might be too small to discern. We might not notice a curled-up dimension that we cannot travel through—much as a tightrope walker would view his path as one-dimensional, whereas a tiny ant on the wire might experience two, as illustrated in Figure 61.^{63}

**Man on tightrope**

**Ant on tightrope**

**[ FIGURE 61 ]** A person and a tiny ant experience a tightrope very differently. For the person, it appears to have one dimension, whereas the ant experiences two.

Another possibility is that dimensions can be hidden because space-time is curved or warped, as Einstein taught us will happen in the presence of energy. If the curving is sufficiently dramatic, the effects of the additional dimensions are obscured, as Raman Sundrum and I determined in 1999. This meant that warped geometry might also provide a way in which a dimension might hide.^{64}

But why would we even think extra dimensions could be out there if we have never seen them? The history of physics holds many examples of finding things no one could see. No one could “see” atoms and no one could “see” quarks. Yet we now have strong experimental evidence of the existence of both.

No law of physics tells us that only three dimensions of space can exist. Einstein’s theory of general relativity works for any number of dimensions. In fact, soon after Einstein completed his theory of gravity, Theodor Kaluza extended Einstein’s ideas to suggest the existence of a fourth spatial dimension, and, five years later, Oskar Klein suggested how it might be curled up and differ from the familiar three.

String theory, a leading proposal for a theory combining quantum mechanics and gravity, is another reason physicists currently entertain the notion of extra dimensions. String theory does not obviously lead to the theory of gravity we are familiar with. String theory necessarily involves additional dimensions of space.

People often ask me the number of dimensions that exist in the universe. We don’t know. String theory suggests six or seven extra ones. But model builders keep an open mind. It’s conceivable that different versions of string theory will lead to other possibilities. In any case, dimensions model builders care about in the following discussions are only the ones that are sufficiently warped or so large that they can affect physical predictions. Other dimensions even smaller than the ones relevant to particle physics phenomena might exist, but we will ignore anything so super-tiny. We again take the effective theory approach and ignore anything too small or invisible to ever make any measurable differences.

String theory also introduces other elements—notably branes—that make for richer possibilities for the geometry of the universe, if indeed it contains extra dimensions. In the 1990s, the string theorist Joe Polchinski established that string theory was not just a theory of one-dimensional objects called strings. He, along with many others, demonstrated that higher-dimensional objects known as branes were also essential to the theory.

The word “brane” derives from “membrane.” Like membranes, which are two-dimensional surfaces in three-dimensional space, branes are lower-dimensional surfaces in higher-dimensional space. These branes can trap particles and forces so that they don’t travel through the full higher-dimensional space. Branes in higher-dimensional space are like a shower curtain in your bathroom, which is a two-dimensional surface in a three-dimensional room. (See Figure 62.) Water droplets might travel only over the two-dimensional surface of the curtain, much as particles and forces might be stuck on the lower-dimensional “surface” of a brane.

**[ FIGURE 62 ]** A brane traps particles and forces, which can move along it but not off—much like water droplets that can move on a shower curtain but don’t travel away.

Broadly speaking, two types of strings exist: *open strings* that have ends and *closed strings* that form loops like rubber bands. (See Figure 63.) String theorists in the 1990s realized that the ends of open strings can’t be just anywhere—they have to end on branes. When particles arise from the oscillations of the open strings that are anchored to a brane, they too are confined there. Particles, the oscillations of those strings, are then stuck. As with water drops on a shower curtain, they can travel along the dimensions of the branes, but they can’t travel off them.

**[ FIGURE 63 ]** An open string with two ends, and closed string with none.

String theory suggests the existence of many types of branes, but the ones that will be of most interest for models addressing the hierarchy problem involve those that extend over three dimensions—the three physical dimensions of space that we know. Particles and forces can be trapped on these branes, even when gravity and space extend through more dimensions. (Figure 64 presents a schematic of a braneworld showing a person and a magnet on a brane, with gravity spreading both on and off it.)

String theory’s extra dimensions might have physical import for the observable world and so too might three-dimensional branes. Perhaps the most important reason to consider extra dimensions is that they might affect visible phenomena, and, in particular, address outstanding puzzles such as the hierarchy problem of particle physics. Extra dimensions and branes could be the key to resolving this question—addressing the issue of why gravity is so weak.

**[ FIGURE 64 ]** Standard Model particles and forces can be stuck on a braneworld that lives in higher-dimensional space. In that case, my cousin Matt, the matter and stars we know, forces such as electromagnetism, and our galaxy and universe all live in its three spatial dimensions. Gravity, on the other hand, can always spread throughout all of space. (Photo courtesy of Marty Rosenberg)

Which brings us to what is perhaps the best reason right now to think about extra dimensions of space. They can have consequences for phenomena we are now trying to understand, and if so, we might see evidence in the imminent future.

Recall that we can phrase the hierarchy problem in two different ways. We can say it is the question of why the Higgs mass—and hence the weak scale—is so much smaller than the Planck mass. This is the question we considered when thinking about supersymmetry and technicolor. But we can also ask an equivalent question: Why is gravity so weak compared to the other known fundamental forces? The strength of gravity depends on the Planck mass scale, the enormous mass ten thousand trillion times greater than the weak scale. The bigger the Planck mass, the weaker the force of gravity. Only when masses are at or near the Planck scale is gravity strong. As long as particles are a good deal lighter than the scale set by the Planck mass, as they are in our world, the force of gravity is extremely weak.

The puzzle of why gravity is so weak is in fact equivalent to the hierarchy problem—the solution of one solves the other. But even though the problems are equivalent, phrasing the hierarchy problem in terms of gravity helps guide our thinking toward extra-dimensional solutions. We’ll now delve into a couple of the leading suggestions.

**LARGE EXTRA DIMENSIONS AND THE HIERARCHY**

Ever since people first started thinking about the hierarchy problem, physicists thought the resolution must involve modified particle interactions at the weak energy scale of about a TeV. With only Standard Model particles, the quantum contributions to the Higgs particle mass are simply too enormous. Something has to kick in to tame the large quantum mechanical contributions to the Higgs particle mass.

Supersymmetry and technicolor are two examples in which new heavy particles might participate in high-energy interactions and cancel the contributions or prevent them from arising in the first place. Until the 1990s, all proposed solutions to the hierarchy problem could be categorized similarly, with new particles and forces and even new symmetries emerging at the weak energy scale.

In 1998, Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali__ ^{65}__ proposed an alternative way of addressing the problem. They pointed out that since the problem involves not just the weak energy scale alone, but its ratio to the Planck energy scale associated with gravity, perhaps the problem lay in an incorrect understanding of the basic nature of gravity itself.

They suggested that there is in fact no hierarchy in masses at all—at least with respect to the fundamental scale of gravity compared to the weak scale. Maybe gravity is instead much stronger in the extra-dimensional universe, but is only measured to be so feeble in our three-plus-one-dimensional world because it is diluted throughout all the dimensions that we don’t see. Their hypothesis was that the mass scale at which gravity becomes strong in the extra-dimensional universe is in fact the weak mass scale. In that case, we measure gravity to be minuscule in strength not because it is fundamentally weak but rather because it spreads throughout large unseen dimensions.

One way to understand this is to imagine an analogous situation with a water sprinkler. Think about the water that emerges from this sprinkler. If the water spread only in our dimensions, its impact would depend on the amount of water emerging from the hose and how far it had to travel. But if there were additional dimensions to space, the water would spread throughout those dimensions as well after emerging from the end of the hose. We would experience much less water than we would otherwise at a given distance from the source because water would also spread throughout the dimensions we don’t observe. (This is illustrated schematically in Figure 65.)

**[ FIGURE 65 ]** The strengths of forces weaken more quickly with distance in a higher-dimensional space than in a lower-dimensional one. This is analogous to a higher-dimensional water sprinkler for which the water dilutes much more quickly with distance. The water spreads more in three dimensions than it spreads in two—in the picture, only the flower receiving water from the lower-dimensional sprinkler is adequately maintained.

If the extra dimensions were of finite size, the water would reach the boundaries of the extra dimensions and no longer spread out. But the amount of water anything would receive at any given place in the extra-dimensional space would be far less than if it had never spread out in those dimensions in the first place.

Similarly, gravity could spread into other dimensions. Even though the force wouldn’t spread out forever if the dimensions have finite size, large dimensions would dilute the gravitational force we would experience in our three-dimensional world. If the dimensions were sufficiently large, we would experience very weak gravity, even though the fundamental strength of higher-dimensional gravity could be quite big. Keep in mind, however, that for this idea to work, the extra dimensions have to be enormous compared to what theoretical considerations lead us to expect, since gravity indeed appears so weakly in a three-dimensional world.

Nonetheless, the LHC will subject this idea to experimental tests. Even though the idea now seems improbable, reality and not our ease in finding models is the final arbiter of what is right. If realized in the world, these models would lead to a distinctive characteristic signature. Because higher-dimensional gravity is strong at energies of about the weak scale—the energies that the LHC will generate—p articles would collide together and produce a higher-dimensional graviton—the particle that communicates the force of higher-dimensional gravity. But this graviton travels into the extra dimensions. The gravity we are familiar with is extremely weak—far too weak to produce a graviton if there are only three dimensions of space. But in this new scenario, higher-dimensional gravity would be sufficiently strong to produce a graviton at the energies reached by the LHC.

The consequence would be the production of particles known as Kaluza-Klein (KK) modes, which are the manifestation of the higher-dimensional gravitation in three-dimensional space. They are named after Theodor Kaluza and Oskar Klein, who first thought about extra dimensions in our universe. KK particles have interactions similar to those of the particles we know, but with heavier masses. These heavier masses are the result of their additional momentum in the direction of the extra dimension. If the KK mode is associated with the graviton—as the large extra dimensional scenario predicts—once produced, it would disappear from the detector. The evidence of its ephemeral visit would be the energy that would go missing. (See Figure 66, in which a KK particle is produced and takes away unseen energy and momentum.)

**[ FIGURE 66 ]** In the large extra-dimensional scenario, a Kaluza-Klein partner of the graviton with momentum in the extra dimensions can be produced. If so, it will disappear from the detector, leaving as evidence missing energy and momentum.

Of course, missing energy is also characteristic of supersymmetric models. The signals could even appear so similar that even if a discovery is made, people from both extra-dimensional and supersymmetry camps are likely to interpret the data as supporting their expectations—at least initially. But with detailed understanding of the consequences and predictions of both types of models, we will be able to determine which idea—if either—is correct. One of our goals in building models is to match experimental signatures and details to their true implications. Once we have characterized different possibilities, we know the rate and features of the signatures that follow, and we can use subtle features to distinguish among them.

In any case, at this point, along with most of my colleagues, I doubt that the large-extra-dimensional scenario is truly the solution to the hierarchy problem, though we will soon see a very different extra-dimensional example that seems much more promising. For one thing, we don’t expect extra dimensions to be so large. It turns out that the extra dimensions would have to be enormous relative to the other scales posed in the problem. Even though the hierarchy between the weak scale and the gravity scale is in principle eliminated, a new hierarchy involving the new dimensions’ size gets introduced in this scenario.

Even more worrisome is that in this scenario, we would expect the evolution of the universe to be very different from what has been observed. The problem is that these very large dimensions would expand along with the rest of the universe until the temperatures are very low. For a model to be a potential candidate for reality, the evolution of the universe it predicts would have to mimic that which has been observed that is consistent with only three dimensions of space. That poses a difficult challenge for scenarios with such large additional dimensions.

These challenges are not enough to definitively rule out the idea. Clever enough model builders can find solutions to most problems. But the models tend to become overly complicated and convoluted in order to agree with all observations. Most physicists are skeptical about such ideas on aesthetic grounds. Many have therefore turned to more promising extra-dimensional ideas such as the ones described in the following section. Even so, only experiments will tell us for certain whether models with large extra dimensions apply to the real world or not.

**A WARPED EXTRA DIMENSION**

Large extra dimensions are not the only potential solution to the hierarchy problem, even in the context of an extra-dimensional universe. Once the door was opened to extra-dimensional ideas, Raman Sundrum and I identified what seems to be a better solution__ ^{66}__—one that most physicists would agree is much more likely to exist in nature. Mind you, that doesn’t mean that most physicists think it is likely to be true. Many suspect that anyone would be lucky to correctly predict what the LHC will reveal or to get a model completely correct without further experimental clues. But it’s an idea that probably stands as good a chance as any of being right, and—like most good models—presents clear search strategies so that theorists and experimenters can more fully exploit all the LHC’s capabilities—and maybe even discover evidence that the proposal is true.

The solution that Raman and I proposed involves only a single extra dimension, and that dimension need not be large. No new hierarchy involving the dimension’s size is necessary. And—as opposed to large-extra-dimensional scenarios—the universe’s evolution automatically agrees with late time cosmological observations.

Although our focus is the single new dimension, additional dimensions of space might exist as well—but in this scenario they won’t play any discernible role in explaining particle properties. Therefore, we can justifiably ignore them when investigating the hierarchy solution—in accordance with the effective theory approach—and concentrate on the consequences of the single extra dimension.

**[ FIGURE 67 ]** The Randall-Sundrum setup contains two branes that bound a fourth dimension of space (a fifth dimension of spacetime). In this space, the graviton wavefunction (which tells the probability of finding the graviton at any point in space) decreases exponentially from the Gravitybrane to the Weakbrane.

If the idea that Raman and I had is right, the LHC will soon teach us fascinating properties about the nature of space. It turns out that the universe we suggested is dramatically curved, in accordance with what Einstein taught us about spacetime in the presence of matter and energy. In technical terminology, the geometry we derived from Einstein’s equations is “warped” (that really was the pre-existing technical term). What that means is that space and time vary along the single additional dimension of interest. It does so in such a way that space and time, as well as masses and energy, are all rescaled as you move from one place in extra-dimensional space to another, as we will soon get to and is illustrated in Figure 68.

One important consequence of this warped spacetime geometry is that whereas the Higgs particle would have been heavy in some other location in extra-dimensional space, it will have weak scale mass—exactly as should be the case—in the location where we reside. This might sound somewhat arbitrary, but it is not. According to our scenario, there is a brane on which we live—the Weakbrane—and a second brane where gravity is concentrated, known as the Gravitybrane—or among physicists, the Planck brane. This brane would contain another universe that is separated from us in an extra dimension. (See Figure 67.) In this scenario, the second brane would in fact be right next door—separated by an infinitesimal distance, a million trillion trillion times smaller than a centimeter.

The remarkable property that follows from the warped geometry (illustrated in the Figure 67), is that the *graviton*, the particle that communicates the force of gravity, is far more heavily weighted on the other brane than on ours. That would make gravity strong elsewhere in the other dimension, but very weak where we live. In fact, Raman and I found that gravity should be exponentially weaker in our vicinity than on the other brane, thereby giving a natural explanation for the weakness of gravity.

An alternative way of interpreting the consequences of this setup is through the geometry of spacetime, schematically illustrated in Figure 68. The scale of spacetime depends on location in the fourth spatial dimension. Masses get exponentially rescaled too—and they do so in a way that the Higgs boson mass is what it needs to be. Although one can debate the assumptions our model relies on—namely, two large flat branes bounding an extra-dimensional universe—the geometry itself follows directly from Einstein’s theory of gravity once you postulate the energy carried by the branes and by the extra-dimensional space known as the bulk. Raman and I solved the equations of general relativity. And when we did, we found the geometry I just described—namely, the curved warped space in which masses get rescaled in the way required to solve the hierarchy problem.

**[ FIGURE 68 ]** Another way to understand why warped geometry solves the hierarchy problem is in terms of the geometry itself. Space, time, energy, and mass all are rescaled exponentially as you go from one brane to the next. In this scenario, it would be very natural to find that the Higgs mass is exponentially smaller than the Planck mass.

Unlike the large extra-dimensional models, the models based on the warped geometry don’t replace the old enigma of the hierarchy problem by a new one (why are the extra dimensions so large?). In the warped geometry, the extra dimension is not large. The large numbers arise from an exponential rescaling of space and time. The exponential rescaling makes the ratio of sizes—and masses—of objects enormous, even when those objects are separated only modestly in extra-dimensional space.

The exponential function isn’t made up. It arises from the unique solution to Einstein’s equations in the scenario we proposed. Raman and I calculated that in the warped geometry, the ratio of the strength of gravity and the weak force is the exponential of the distance between the two branes. If the separation between the two branes has a reasonable value—a few dozens or so in terms of the scale set by gravity—the right hierarchy between masses and the strength of forces naturally emerges.

In the warped geometry, the gravity we experience is weak—not because it is diluted throughout large extra dimensions—but instead because it is concentrated somewhere else: on the other brane. Our gravity arises only as the tail end of what in other regions of the extra-dimensional world feels like a very intense force.

We don’t see the other universe on the other brane because the lone shared force is gravity, and gravity is too weak in our vicinity to communicate readily observable signals. In fact, this scenario can be thought of as one example of a *multiverse*, in which the stuff and elements of our world interact very weakly, or in some cases not at all, with the stuff in another world. Most such speculations cannot be tested and will be left to the realm of imagination. After all, if matter is so far distant that light couldn’t reach us in the lifetime of the universe, we can’t detect it. The “multiverse” scenario that Raman and I proposed is unusual in that the shared gravitational force leads to experimentally testable consequences. We don’t directly access the other universe. But particles that travel in the higher-dimensional bulk can come to us.

The most obvious effect of the extra-dimensional world—in the absence of detailed searches such as those at the LHC—would be the explanation for the hierarchy of mass scales that particle physics theories need in order to successfully explain observed phenomena. This of course is not sufficient for us to know if the explanation is the one operational in the world, since it doesn’t distinguish among proposed solutions.

However, the higher energy that will be achieved at the LHC should help us discover whether an extra dimension of space is just an outlandish idea or an actual fact about the universe. If our theory is correct, we would expect the LHC to produce Kaluza-Klein modes. Because of the connection to the hierarchy problem, the right energy scale to look for KK modes in this scenario is the one that will be probed at the LHC. They should have mass of about a TeV—the weak mass scale. Once the energy achieved is high enough, these heavy particles might be produced. The discovery of these KK particles would provide the key confirmation that gives us insight into a greatly expanded world.

In fact, the KK modes of the warped geometry have an important and distinctive feature. Whereas the graviton itself has extraordinarily feeble interaction strength—after all, it communicates the extremely weak gravitational force—the KK modes of the graviton interact far more strongly, almost as strongly as the force called the weak force, which is in actuality trillions of times stronger than gravity.

The reason for the KK gravitons’ surprisingly strong interaction strength is the warped geometry they travel in. Owing to spacetime’s dramatic curvature, the interactions of KK gravitons have far greater strength than those of the graviton that communicates the gravitational force we experience. In the warped geometry, not only do masses get rescaled, but gravitational interactions do as well. Calculations demonstrate that in the warped geometry, KK gravitons have interactions comparable to that of weak scale particles.

This means that unlike supersymmetric models, and unlike large extra-dimensional ones, the experimental evidence for this scenario will not be missing energy where the interesting particle escapes unseen. Instead, it will be a much cleaner, and easier to identify, signature, consisting of the particle decaying inside the detector into Standard Model particles that leave visible tracks. (See Figure 69, in which a KK particle is produced and decays into an electron and positron for example.)

**[ FIGURE 69 ]** In Randall-Sundrum models, a KK graviton can be produced and decay inside the detector into visible particles, such as an electron and a positron.

This is in fact how experimenters have discovered all new heavy particles so far. They don’t see the particles directly. But they observe the particles that they decay into. That’s a lot more information in principle than would be provided by missing energy. By studying the properties of these decay products, experimenters can figure out the properties of the particle that was initially present.

If the warped geometry scenario is correct, we will soon see particle pairs originating from the decay of KK graviton modes. By measuring the energies and charges and other properties of the final state particles, experimenters will be able to deduce the mass and other properties of the KK particles. These identifying features, along with the relative frequency with which the particle decays into various final states, should help experimenters determine whether they have discovered a KK graviton or some other new and exotic entity. The model tells us the nature of the particle that should be found so that physicists can make predictions to distinguish among the possibilities.

A friend of mine (a screenwriter who both extols and satirizes the excesses of human nature) doesn’t understand how, given the potential implications of the discoveries that might happen, I’m not sitting on the edge of my seat waiting for results. Whenever I see him, he insistently asks me, “Won’t the results be life-changing? Might they not confirm your theories?” He also wants to know, “Why aren’t you over there (in Geneva) talking to people all the time?”

Of course, in some sense his instincts are right. But experimenters already know what to look for, so much of the job of theorists is already done. When we have new ideas about what to look for, we communicate them. We don’t necessarily have to be at CERN or even in the same room to do that. Experimenters can be found all over the United States and almost anywhere on the globe for that matter. And remote communication works pretty well—in part due to the initial Internet insight that Tim Berners-Lee had many years ago at CERN.

I also know enough to know what a challenge these searches might be, even once the LHC is fully operational. So I know we might have a bit of a wait. Fortunately for us, the KK modes just described are one of the most straightforward things experimenters can look for. The KK gravitons decay into all particles—after all, every particle experiences gravity—so experimenters can focus on the final states that they find easiest to identify.

However, there are two cautionary notes—two reasons that the searches might be more challenging than initially anticipated and why we might have to wait awhile for discovery, even if the underlying idea is correct.

One is that other candidate models of warped geometry could lead to messier experimental signatures that will be more difficult to find. Models describe the underlying framework—which here involve an extra dimension and branes. They also suggest specific implementations of the general principles the framework embodies. Our original scenario suggested that only gravity was spread throughout the higher-dimensional space known as the *bulk*. But some of us later worked on alternative implementations. In these alternative scenarios, not all particles are on branes. This would mean more KK particles since each bulk particle would have its own KK modes. But it also turns out that these KK particles would be considerably harder to find. This challenge has prompted a great deal of research into how to discover these more elusive scenarios. The investigations that followed will prove useful not only in the search for KK particles, but also for any energetic massive particles that might be present in any new model.

The other reason that searches might prove to be difficult is that KK particles could be heavier than we hope. We know the range of masses we might anticipate for KK particles, but we don’t yet know the precise values. If KK particles are nice and light, the LHC will readily produce them in abundance and discovery will be easy. But if the particles are heavier, the LHC might create only a few of them. And if they turn out to be heavier still, the LHC might not produce any at all. In other words, the new particles and new interactions might only be produced or occur at higher energies than the LHC will achieve. This was always a concern for the LHC with its fixed tunnel size and constrained energy reach.

As a theorist, I can only do so much about that. The LHC energy is what it is. But we can try to find subtle clues about the existence of extra dimensions, even if the KK modes turn out to be too heavy. When Patrick Meade and I did our calculations about the production rate of possible higher-dimensional black holes, we focused not only on the negative result—the much lower black hole production rate than had originally been claimed—but also thought about what would happen if higher-dimensional gravity was strong, even if no black holes were produced. We asked whether the LHC might produce any interesting signals of higher-dimensional gravity at all. We found that even without discovering new particles or exotic objects like black holes, experimenters should be able to observe deviations from Standard Model predictions. Discovery is not guaranteed, but experimenters will do everything they can with the existing machine and detectors. In other more advanced research, colleagues have thought about improved methods for finding KK modes, even if Standard Model particles reside in the bulk.

There is also a chance that we could be lucky and that the scales for new particle masses and interactions might turn out to be lower than we anticipate. If that turns out to be the case, we would not only find KK modes sooner than expected, but we would also see other new phenomena. If string theory is the underlying theory of nature and the scale of new physics is low, the LHC could even produce—in addition to KK particles and new interactions—additional particles associated with oscillating underlying strings. These particles would be much too heavy to create under more conventional assumptions. But with warping, there is hope that some string modes will be much lighter than anticipated and could thus appear at the weak energy scale.

Clearly there are several interesting possibilities for warped geometry and we eagerly await experimental results. If the consequences of this geometry are discovered, they will change our view of the nature of the universe. But we will only know which—if any—of these possibilities is realized in nature after the LHC has done its search.

**REDUX**

Experiments at the LHC are currently testing all the ideas in this chapter. We hope that if any of these models are right, hints will soon appear. There might be solid evidence like KK modes, or there might be subtle changes to Standard Model processes. Either way, both theorists and experimenters are alert and waiting. Every time the LHC does or does not see something, it constrains the possibilities further. And if we’re lucky, one of the ideas that have been discussed might prove right. As we learn more about what the LHC will produce and how detectors work, we will hopefully also learn more about how to extend the LHC’s reach to test as large a range of possibilities as possible. And as data become available, theorists will incorporate that data into their proposals.

We don’t know how long it will be before we start getting answers since we don’t know what is there or what the masses and interactions might be. Some discoveries may happen within a year or two. Others could take more than a decade. Some might even require higher energies than the LHC will ever achieve. The wait is a little anxiety provoking, but the results will be mind-blowing. That should make the nail-biting worth it. They could change our view of the underlying nature of reality, or at least the matter of which we are composed. When the results are in, whole new worlds could emerge. Within our lifetimes, we just might see the universe very differently.