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© 2010 American Institute of Physics, S-0031-9228-1011-030-8 November 2010 Physics Today 39

String theory andthe real worldGordon Kane

Although string theory is formulated in 10 or 11 dimensions, specificstring theory solutions make unambiguous, testable predictions about our four-dimensional universe.

Gordy Kane is the director of the Michigan Center for Theoretical Physics and Victor Weisskopf Collegiate Professor of Physics at the University of Michigan in Ann Arbor.

Figure 1. A world of hidden, smalldimensions. String theories are for-mulated in a world of 10 or 11 space-time dimensions. According to thosetheories, a point in what appears tobe four-dimensional spacetime is ac-tually a tiny multidimensional spaceof its own. The purple blowup hints atthe complexity of those minusculespaces; it is a projection onto threedimensions of a 6D space called aCalabi–Yau space after mathemati-cians Eugenio Calabi and Shing-TungYau. (Sunflower photo courtesy ofPDPhoto.org; Calabi–Yau drawing byAndrew J. Hanson, Indiana University.)

We live in exciting times for particle physics. The LargeHadron Collider (LHC) at CERN has begun to collect data,and laboratory and satellite experiments are investigating thedark matter of the universe. Another, less appreciated fact in-creases the excitement. Physicists now have a coherent, con-sistent theoretical framework to address basic questionsabout particles, the interactions and forces between them,why they are what they are, and how numerous phenomena

are related in a broader picture. That framework is “stringtheory.” I put the term in scare quotes because there is not yeta final formulation of the theory. But the lack of a finishedpicture is not important for my purposes, so in this article Irefer to the framework as string theory or M-theory. The per-spective that string theory is the underlying framework to address many issues facing particle physics and cosmologyis different from the more standard description of it as a

consistent quantum theory of gravity. But it is a fruitful wayto think about what string theory means.

String theory’s long reachI make a distinction between a theory’s “explaining” some-thing and the weaker claim of its “addressing” an issue.String theory and the standard model of particle physicsboth explain some facts, but string theory addresses muchmore than the standard model does. To understand the dis-tinction, consider the proton. Quantum chromodynamics, apart of the standard model, is a theory of the strong inter -actions in which quarks interact via a force mediated by glu-ons. The QCD Lagrangian does not contain the proton ex-plicitly. However, study of the QCD force reveals that boundstates of quarks form. In particular, QCD requires the exis-tence of a state that has all the properties—electric charge,spin, and so forth—of a proton. Moreover, physicists haveused lattice gauge theory to calculate the mass of the protonto about 3% accuracy. So QCD explains the proton: If it hadnot been known, QCD would have predicted its existenceand properties.

On the other hand, supersymmetric extensions of thestandard model address but do not explain the dark matterof the universe. Supersymmetry is a hypothetical symmetryof nature in which the Lagrangian is symmetric under inter-change of bosons (integer-spin particles) and fermions (withhalf-integer spin). Considerable indirect evidence suggeststhat supersymmetry is a symmetry of nature at the TeV en-ergy scale. But it can’t be exact at low energy because if itwere, physicists would have seen a boson with the mass andcharge of the electron. Thus supersymmetry is posited to bea broken symmetry with superpartners having masses differ-ent from those of the known particles. If that picture is cor-rect, it is likely that some of the superpartners of the knownparticles will be observed at the LHC. The lightest of the su-perpartners is typically stable against decay into standard-model particles, and calculations confirm that it can have theright properties to be the dark matter of the universe. Thusthe supersymmetric standard model addresses the problemof dark matter: If physicists did not know about dark matter,the model would suggest it and indicate how to look for it.

It is in the dark-matter sense that string theory providesa framework to address and relate many open questions inparticle physics. We know that particles divide into threefamilies exemplified by the electron, muon, and tau, but if wedidn’t know it, string theory would suggest that families existand why. If we did not know about forces such as the strong

and electroweak forces of the standard model, or the parityviolation of the weak interactions, or supersymmetry, or in-flation, or gravity, string theory would suggest them. Com-pelling answers to many of the key questions in particlephysics have not yet been found, but we can now search forthem systematically.

What is tested by a test?Based on the presentation of string theory in some popularbooks, articles, and blogs, one might well be suspicious oftaking purported string theory explanations seriously. Suchsources often claim that string theory is not testable, and Iagree that untestable explanations are not helpful. But Iwould also argue that string theory is testable in basically thesame ways that other theories are and that string theory is or-dinary science in terms of describing nature and testing itsexplanations. Almost 14 years ago I wrote an optimistic arti-cle in PHYSICS TODAY about testing string theory (February1997, page 40). Since then, we have learned much about thesubject, though much that is confusing or misleading has alsobeen written.

To be sure, the majority of research into string theory isnot focused on how the theory connects to the real world;rather, most physicists are exploring questions at a more the-oretical level. Such formal work is necessary, because asnoted above, we need a deeper understanding to fully for-mulate the theory. Even the many theorists who are inter-ested in how string theory connects to the real world don’ttypically think much about what it means to test the theory.Fortunately, an increasingly active group of “string phenom-enologists” are focusing on formulating a string-based de-scription of the world and testing that understanding. Theyare already making testable predictions, and will increas-ingly do so.

String theory is formulated in 10 or 11 spacetime dimen-sions. The differences between the two versions are technicaland have little effect on the questions discussed in this article.In the 10D case, the dimensions must be separated into thefour spacetime dimensions that form the world of everydayexperience and six small dimensions that typically form whatis called a Calabi–Yau manifold, a space with well-studiedmathematical properties (see figure 1). Those properties de-termine, in part, the physics that emerges from string theory,in particular the particle content and forces. The jargon forthat separation of dimensions into large and small ones iscompactification, an unfortunate word choice. In the 11Dcase, M-theory, the 7D space that remains small, called a G2

F a= m

F r= −k F = −GMmr

r2

Figure 2. Testing a theory. What does it mean to test Newton’s second lawF = ma? Newton’s law is a claim—thatcould have been wrong—about the actual relation between the force F on a particle with mass m and its accelera-tion a. One tests it by calculating the acceleration with a presumed force andcomparing it to the measured value. Thetest will fail if either Newton’s law or thepresumed force is wrong. Could F = ma

be tested more generally, without re-course to positing forces and looking atactual solutions? It seems not. The situa-

tion is similar with string theory. Properties of the real world test the overall string framework and, at the same time, “compactifica-tions” analogous to the forces of Newtonian mechanics or the Hamiltonian of quantum mechanics.

40 November 2010 Physics Today www.physicstoday.org

manifold, has somewhat different mathematical propertiesfrom the Calabi–Yau space.

String theorists have found and studied a number ofwell-defined mathematical and physical steps to implementcompactification and have developed well-known proce-dures for calculating many predictions of a compactifiedstring theory. Because of the ways in which different phe-nomena are related to one another, tests of a compactifiedstring theory can also reveal whether the underlying stringtheory is 10- or 11-dimensional.

The natural scales for the multidimensional world ofstring theory are quite unlike those that are useful in normallife. The natural length scale is the Planck scale, about 10−35 m,the length that one can form from Newton’s gravitationalconstant, the speed of light, and Planck’s constant. The asso-ciated time and energy scales are about 10−43 s and 1018 GeV,respectively. String theories can be formulated at lower en-ergy scales—for example, the 10 000-GeV scale probed di-rectly by the LHC—and physicists have studied testing stringtheories at those scales. I will not consider that possibilityhere, but rather will focus on the harder-to-test theories for-mulated near the Planck scale.

Some books and popular articles have claimed that be-cause string theories are naturally formulated at such highenergies or small distances, they cannot be tested. Obviously,collisions will never probe energy scales of 1018 GeV, some 14orders of magnitude larger than that of the LHC. But equallyobviously, one does not have to be somewhere to test what’sgoing on there. Physicists have no doubt that a hot Big Bangoccurred even though no one was there to witness it: We areconvinced by the extensive evidence from relics such as theexpanding universe, helium and other light-element abun-dances, and the cosmic microwave background radiation.You do not have to travel at the speed of light to test that itis the limiting speed. You do not have to have been present65 million years ago to test that a major cause of dinosaur ex-tinction was an asteroid impact.

To understand what it means to test string theories, it iscrucial to recognize that a compactified string theory is anal-ogous to a Hamiltonian (or Lagrangian; I won’t carefully dis-tinguish between the two) of a system. All areas of physics,including string theory, have general rules for derivingphysics from a Hamiltonian, but one defines a particular the-ory only after specifying the Hamiltonian, or perhaps forces.Physical systems are described not by the Hamiltonian butby solutions to the equations calculated from it. In quantumtheory, for example, to make predictions that test whether agiven Hamiltonian for an atom is correct, one must find theground state and calculate energy levels and transitions rel-ative to it. Comparing the calculated values to reality servesto check the Hamiltonian and Schrödinger equation together.The situation is analogous for Newtonian mechanics, as illus-trated in figure 2, or for string theory, in which the universecorresponds to a stable or metastable ground state or, in thejargon of the field, the vacuum. The key point is that solutionsto the theory are the things tested. It’s a point that is often ig-nored, even by experts, in popular discussions.

Testable predictions from string theoryTo make contact with the real world, a 10D or 11D string the-ory must be compactified. String theories with stable ormetastable ground states usually also have supersymmetry,so the compactification process must break that symmetry.Theorists have realized the procedure in a number of ways.Typically, the compactification leaves the ground state withrecognizable remnants of supersymmetry—superpartners of

the standard particles but with masses different from that oftheir partner. Sometimes it both is consistent with what weknow about the world and leads to additional, testable pre-dictions about dark matter, LHC discoveries, and more.

Indeed, anyone who has worked with compactifiedstring theories knows they make a number of generic and de-tailed predictions that are not subject to qualitative changesfrom small input changes and can be falsified in a number ofways. Of course, a theory needs only to be falsifiable in oneway to be testable. Some compactifications have generatedwrong predictions. In those cases, we have successfully im-plemented a test, but the theory failed. I give specific exam-ples of tests below; interested readers can find others in talksaccessible from the website of the international String Phe-nomenology 2010 conference held in Paris this summer(http://stringpheno.cpht.polytechnique.fr) or from that of theString Vacuum Project (http://www.northeastern.edu/svp), anetwork of universities with theorists who focus on studyingthe ground states of string theories rather than the full higher-dimensional theory. It is simply wrong to say that string the-ory is not testable in basically the same way that F = ma or theSchrödinger equation is testable.

One specific test of a compactified string theory involvesneutrino masses. Theorists have devised many interestingmodels of neutrino masses, but they have not succeeded inwriting down an underlying theory in which very light but

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Physicists do not understand why the cosmological constanthas the value that it has; naive dimensional-analysis computa-tions give a result that is way too large. But that famous discrep-ancy should not change the arguments I have advanced in thisarticle, nor should it get in the way of physicists finding thestring ground state.

The cosmological constant is the value of an appropriatepotential at its minimum, and in any particular string theory, itwill scale with the product of the Planck mass and the mass ofthe gravitino, the supersymmetric partner of the massless gravi-ton associated with the gravitational force. That result is not theone obtained by naive calculation, but still it gives too large acosmological constant. In practice, one sets the minimum valueof the potential at or near zero and calculates all observables. Inreference 7, my collaborators and I followed a standard proce-dure and explicitly confirmed that for an M-theory compactifiedon a G2 manifold, implementing the potential tuning leads toonly small numerical changes in the observables. One alwaysneeds to check that the tuning doesn’t have significant conse-quences, but it appears that the physics of dark matter, super-partner masses, families, and parity violation is not closely re -lated to the potential minimum.

Solving the cosmological-constant problem does not seemlikely to help physicists comprehend the rest of what needs tobe understood, and not solving the problem does not seemlikely to be a hindrance. Quantum chromodynamics has ananalogous “strong CP problem.” One would expect QCD to breakthe symmetry of nature under the combination of charge con-jugation C and coordinate inversion, or parity P. To tame theexpected symmetry breaking, one has to set a term in the QCDLagrangian to be about 10 orders of magnitude smaller than itsnaive value. If that term were indeed equal to the naive value,every prediction of QCD would change. But for three decades,physicists have been ignoring the offending term and movingahead without encountering any hint of a problem.

A problem, but not a concern

not massless neutrinos emerge in the sense that the protonemerges and is explained by QCD. Perhaps string theorycould explain the light masses? A few years ago Joel Giedt(now at Rensselaer Polytechnic Institute), Paul Langacker(now at the Institute for Advanced Study), Brent Nelson (nowat Northeastern University), and I realized that a particularcompactified string theory had been studied so well by Mary K. Gaillard and her PhD students (including Giedt andNelson) at the University of California, Berkeley that wecould identify all the particles that could be neutrinos.1 Weshowed that in no case could the theory generate light butnot massless neutrinos. That work represents a clear exampleof a test of string theory. Although the particular compactifi-cation we studied did not yield the desired neutrino masses,different compactifications may allow for neutrino massesconsistent with experiment and offer explanations of ob-served neutrino properties.

Sometimes when people talk about testing string theory,they are referring to tests that apply to the full 10D or 11Dtheory without compactification. Quantum mechanics ad-mits a few ways to probe the general framework of the theory; most of those test the idea that the superposition principle applies to probability amplitudes. The generalframework of quantum field theory has passed tests that con-firmed the predicted connection between spin and statisticsor that all quanta of a given kind—electrons, say—are iden-tical. Perhaps string theory has such general tests eventhough no 4D universe is defined, but that is not yet clear.

Probably the ideal goal for those who want to examinestring theory is to formulate testable properties that hold forall compactified string theories with metastable or stablevacua, regardless of the form of the compactification or otherconditions such as supersymmetry breaking. Most knowl-edgeable physicists would agree that the gravitational forceis one such property; its existence is a success for string the-ory. Some theorists have emphasized that compactified stringtheories should obey certain properties of general relativisticquantum field theories;2 any putative theory that violatesthose principles cannot be right.

Naively, string theory predicts an energy of the universe,or cosmological constant, that is much greater than observed.That discrepancy is one of the outstanding puzzles of theo-retical physics. Meanwhile, as I argue in the box on page 41,it is not expected to be an obstacle to exploring testable con-sequences of string theory.

A cosmological exampleEarlier this year Bobby Acharya, Eric Kuflik, and I proposeda test that concerns the cosmological history of the universe,one that should be applicable to any compactified string the-ory.3 Before explaining it, I need to describe two additionalimportant features of compactified theories and super -symmetry breaking: moduli fields and gravitinos.

Moduli fields characterize the sizes, shapes, and metricsof the small manifolds—for example, the 6D Calabi–Yaumanifolds in a 10D string theory. The moduli fields may beunfamiliar, but they are always present in compactified stringtheories, and the quantities they represent are physical. Oncethey take on definite values in the ground state—once theyare stabilized—they determine many measurable observ-ables such as interaction strengths and the masses of thegauge bosons that mediate the standard-model interactions.A modulus has quanta analogous to the photons associatedwith the electromagnetic field, and as usual, the fluctuationsaround the value of a modulus at the minimum of its poten-tial energy determine the mass of the quantum. As theorists

often do, I will sometimes use “moduli” to designate both thefields and the associated quanta.

The gravitino is the superpartner of the graviton, themassless quantum associated with the gravitational field.When supersymmetry is broken, the gravitino becomes mas-sive. Its mass sets the scale for all the other superpartners andfor the moduli.

String theories can have many moduli, with differentstabilized values and masses. Nevertheless, string theoriesthat could describe the world we live in generically have atleast one modulus whose mass is on the order of the gravitinomass.4 The result is so universal that my collaborators and Ithink, with further study, it will be promoted to a theorem.In any case, moduli quanta have interactions and decays withimportant cosmological implications.5 Acharya, Kuflik, and Ihave generalized the argument of reference 4 and have ex-tracted those implications in the context of compactifiedstring theories.3

Moduli interact only gravitationally, so their behavior isessentially model independent. In particular, the lifetime ofa modulus is approximately calculable in terms of its mass.If even one modulus were to decay after the beginning of nu-cleosynthesis, the decay products would alter the abun-dances of helium and other nuclei that are successfully de-scribed by the usual theory that does not include moduli.Thus the moduli and gravitino masses must be at least 30 TeVor so to guarantee that the decay occurs early enough.

The moduli decay when the universe has cooled to atemperature of order 0.01 GeV, and in doing so they intro-duce large amounts of additional particles. The dark-matterrelic “density,” as the term is commonly used in this context,is the ratio of dark-matter particles to all particles. So whenthe moduli decay adds to the number of particles, it decreasesthe relic density. On the other hand, the decay products of themoduli include candidate dark-matter particles, so the de-cays can still generate the observed dark-matter density.

The essential point is that in theories with decaying mod-uli, the universe has a nonthermal history. The conventionalthermal history, in contrast, is one in which essentially theonly thing the universe does after the Big Bang is cool adia-batically. No diluting particles are produced at any stage, andno dark-matter particles enter except from decay of relatedparticles present immediately after the Big Bang; for exam-ple, superpartners present at the earliest instants of the cos-mos quickly decay into the lightest superpartner. The newsources of particles and dark matter in the nonthermal pic-ture lead to several testable predictions. Immediately afterthe inflationary epoch, the universe is matter dominated—bythe moduli—not radiation dominated as in the usual model.The dark matter itself is not cold in the nonthermal case sinceit is created with significant kinetic energy from modulidecay. And it is not present until about 0.01 s after the BigBang. Figure 3 shows a history of the universe with two land-marks indicated. First is the time when dark matter freezesout, that is, when it is cold enough and the universe has ex-panded enough that dark-matter particles are unlikely to en-counter one another. Freeze-out fixes the amount of darkmatter in the standard thermal picture. The second landmarkshows when the later decay of moduli in compactified stringtheories dilutes the dark matter on the one hand and re -generates it on the other.

Nonthermal cosmic evolution is thus a general predic-tion for compactified string theories with broken supersym-metry and stabilized moduli. It apparently does not dependon the particular forms of the compactification, supersymme-try breaking, or stabilization. As theorists further study com-

42 November 2010 Physics Today www.physicstoday.org

pactified string theories, they will presumably find addi-tional examples of such tests.

A particularly interesting result of a moduli-based non-thermal cosmic evolution follows if all or most of the darkmatter is the lightest superpartner. In that case, the dark-matter annihilation cross section must be rather large so thatthe lightest superpartners arising from moduli decay can annihilate down to a number small enough to be consistentwith observation. Such a large annihilation cross section implies that the lightest superpartner is essentially thewino—the superpartner of the W boson. The winos annihi-late into W bosons via a well-understood process that im-plies a galactic excess of positrons and antiprotons verymuch like those obtained by the PAMELA satellite-borneexperiment.6

Can the landscape be navigated?The string theory framework apparently allows an enormousnumber of solutions that could potentially describe the uni-verse. Before theoretical physicists can claim to fully under-stand string theory, they must understand the existence andimplications of the string theory landscape—that is, its manysolutions. Some theorists have argued that the existence of somany solutions will prevent them from ever finding a theorythat describes the vacuum and answers the important ques-tions they want answered. But given the success alreadyachieved with compactified string theories, I’m optimisticabout obtaining a description of our string vacuum.

Historically, physicists have understood the natural uni-verse through an interplay of experimentation and theory.String theory fits nicely into that tradition. With guidancefrom both theory and data, physicists are looking for a com-

pactified string theory that incorporates the many phenom-ena they hope to understand—a string supersymmetric stan-dard model.

Some of those who talk about testing string theory, andmost critics of theory, are assuming the 10D or 11D approachand want somehow to test the theory without applying it toa world where tests exist. That is analogous to asking a La-grangian to be falsifiable without applying it to any physicalsystem. Is 10D string theory falsifiable? That is not the rele-vant question. What matters is that the predictions of the 10Dtheory for the 4D world are demonstrably testable and falsi-fiable. If no compactified string theory emerges that describesthe real world, physicists will lose interest in string theory.But perhaps one or more will describe and explain what isobserved and relate various phenomena that previouslyseemed independent. Such a powerful success of sciencewould bring us close to an ultimate theory.

References1. J. Giedt, G. Kane, P. Langacker, B. Nelson, Phys. Rev. D 71, 115013

(2005).2. A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, R. Rat-

tazzi, J. High Energy Phys. 2006(10), 014 (2006); J. Distler, B. Grin-stein, R. Porto, I. Rothstein, Phys. Rev. Lett. 98, 041601 (2007).

3. B. Acharya, G. Kane, E. Kuflik, http://arxiv.org/abs/1006.3272.4. F. Denef, M. Douglas, J. High Energy Phys. 2005(03), 061 (2005);

M. Gómez-Reino, C. Scrucca, J. High Energy Phys. 2006(05), 015(2006).

5. G. Coughlan, W. Fischler, E. Kolb, S. Raby, G. Ross, Phys. Lett. B131, 59 (1983); T. Moroi, L. Randall, Nucl. Phys. B 570, 455 (2000).

6. O. Adriani et al., Nature 458, 607 (2009).7. B. Acharya, K. Bobkov, G. Kane, J. Shao, P. Kumar, Phys. Rev. D 78,

065038 (2008). ■

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Inflation

ACCELERATORS

LHC

Tevatron

RHICHigh-

energy

cosmic

rays

BIGBANG

BIGBANG

Moduli decay,giving entropy andrepopulating dark

matter

Thermal dark-matter freeze-out

t

T

E

1010

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s 1010

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s3 × 103000

(Kelvin)

3 × 10(GeV)

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−10

y

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Figure 3. Revisionist history. This his-tory of the early universe indicatesparticle types (including light nuclei)that were prevalent at variousepochs. The three scales give time(t), temperature in kelvin (T), andtemperature in GeV (E). The historyis changed significantly if compact-ified string theories are correct:About 0.01 s after the Big Bang, so-called moduli fields of the theorydecay, producing additional parti-cles and dark matter. The plot alsohighlights the earlier freeze-outtime of dark matter, that is, thetime after which dark-matter parti-cles no longer interact. (Adaptedfrom a more extensive history pre-pared by the Particle Data Group,Lawrence Berkeley National Labora-tory. Courtesy of the PDG.)