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Problems of Naturalness: Some Lessons from String Theory

arXiv:hep-th/9207045v2 17 Jul 1992August 24, 2018Problems of Naturalness: Some Lessons from String Theory⋆†Michael DineSanta Cruz Institute for Particle PhysicsUniversity of California, Santa Cruz, CA 95064AbstractWe consider some questions of naturalness which arise when one considersconventional field theories in the presence of gravitation: the problem of globalsymmetries, the strong CP problem, and the cosmological constant problem. Usingstring theory as a model, we argue that it is reasonable to postulate weakly brokenglobal discrete symmetries.

We review the arguments that gravity is likely to spoilthe Peccei-Quinn solution of the strong CP problem, and update earlier analysesshowing that discrete symmetries can lead to axions with suitable properties. Evenif there are not suitable axions, we note that string theory is a theory in which CPis spontaneously broken and θ in principle calculable.

θ thus might turn out to besmall along lines suggested some time ago by Nelson and by Barr.⋆Invited Talk Presented at the Cincinnati Symposium in Honor of the Retirement of LouisWitten† Work supported in part by the U.S. Department of Energy.

1. IntroductionRather than deal here, as other speakers will, directly with the difficult ques-tions raised by quantum gravity, I would like to focus on some questions of natural-ness which Einstein’s theory raises.

The three which will concern us here are: thecosmological constant; the problem of symmetries (both continuous and discrete)and, related to the second, the strong CP problem. In considering these questions,we will use string theory as a guide.

In doing this, I am not assuming that stringtheory necessarily describes the real world, but rather that characteristics of stringtheory might plauisbly be shared by any ultimate theory of nature.The cosmological constant problem arises already if we consider (semi)-classical gravity coupled to quantum fields.At one loop, for example, in fieldtheory one has a contribution to the vacuum energyEo = Λ =Xi±1/2µZd3k(2π3)q⃗k2 + m2i(1.1)Here the sum runs over all physical helicity states in the theory; the ± refers tobosons and fermions, respectively. Generically, the result isEo ∼µ4(1.2)2If µ ∼MP , this corresponds to a cosmological constant more than 120 orders ofmagnitude larger than the observational limit.

In the presence of supersymmetry,the leading divergence cancels between bosons and fermions and one might hopeto find µ of order the supersymmetry breaking scale, perhaps as small as 102GeV.‡ This is a big improvement, but not nearly good enough.Of course, infield theory the cosmological constant is not calculable, and it is not clear we areasking a physically meaningful question. However, in string theory the cosmologicalconstant is calculable; whenever supersymmetry is broken one finds it is large.

[1]So, with regards to this problem, string theory seems to offer no miracles: we willstill need to search for some deeper explanation.‡Aficianados of hidden sector supergravity models will object that µ ∼1011 GeV is morereasonable; I am just describing the best one can hope for.1

Our second topic has to do with symmetries. The notion of an exact globalsymmetry is always a troubling one; it is particularly so in the presence of gravity.Global quantum numbers can disappear in black holes; wormholes, if they arerelevant, will generate symmetry-violating operators.

Gauge symmetries, on theother hand, enjoy a different status, and are expected to survive quantum gravityeffects. Both these statements apply not only to continuous symmetries, but todiscrete symmetries as well.

Krauss and Wilczek have stressed that gauged discretesymmetries should survive quantum gravity effects. [2] The simplest example of sucha symmetry is provided by a spontaneously broken U(1) gauge symmetry.

Suppose,for example, one has two scalar fields, φ, with q = 2, and χ, with q = 1. Anexpectation value for φ leaves over the symmetry χ →−χ.String theory lends weight to these views.

It is not difficult to prove that,even at tree level, the theory possesses no unbroken, continuous global symme-tries. [3] The strategy is to show that any such symmetry implies the existence ofa conserved world sheet current, which in turn implies the presence of a masslessvector particle.

Discrete symmetries do frequently arise in compactifications ofstring theory. [4] In many cases, these can be interpreted as relics of higher dimen-sional gauge and general coordinate invariance, i.e.

as gauge symmetries. It hasbeen widely speculated that all discrete symmetries in string theory of this kind;we will have more to say about this later.The third problem we have mentioned is the strong CP problem.

QCD pos-sesses an additional parameter, θ, which enters the lagrangian through the termLθ = θ g216π2Zd4xF ˜F(1.3)From the limits on the neutron electric dipole moment, one knows that θ < 10−9. [5]Two solutions to this problem have been widely considered.

Perhaps the mostpopular is the “axion.”[6] Here one postulates that the classical lagrangian possesesa global U(1) symmetry, the “Peccei-Quinn” symmetry, under which there is amassless field, the axion, which transforms non-linearlya(x) →a(x) + δ(1.4)2

The axion is assumed to couple to F ˜F asLa = Ng216π2Zd4x(θ + afa)F ˜F(1.5)fa is the axion decay constant.QCD effects can then be shown to generate apotential for the axion,V (a) ≈−m2πf2πcos( afA+ θ)(1.6)The minimum of this potential clearly occurs when θeff = afa + θ = 0.But the whole idea of the Peccei-Quinn symmetry is quite puzzling. Not onlyis one postulating a global symmetry, but a symmetry which is necessarily brokenexplicitly!

String theory offers some insight into this question. Indeed, E. Wittenpointed out early on that string theory exhibits symmetries of precisely this type.

[7]This can be understood in a number of ways. For example, if one compactifiesthe heterotic string to four dimensions, there is a two-index antisymmetric tensor,Bµν, µ, ν = 0, .

. .

3. The corresponding gauge-invariant field strength is Hµνρ =∂µBνρ + CS, where CS denotes the Chern-Simons term.

Such an antisymmetrictensor is equivalent to a scalar field;∂µa ∝ǫµνρσHνρσ(1.7)Because in perturbation theory the low energy effective lagrangian must be writtenin terms of H, no non-derivative couplings of a appear, so in perturbation theorythe lagrangian is symmetric under a →a + δ.⋆Thus one has a symmetry to all orders of perturbation theory, broken byeffects of order e−1/g2 (perhaps e−1/g [8]). This sounds like precisely what one needsto solve the strong CP problem.

So perhaps it is not so unreasonable, in general,to postulate such symmetries.⋆Alternatively, this statement can be understood in terms of string vertex operators. Axionemission is described by R d2σǫµν(k)ǫαβ∂αxµ∂βxνeik·x.

At zero momentum, this becomesthe integral of a total divergence.3

What about the possibility that CP is spontaneously violated, with vanishingbare θ? Below we will argue that in the heterotic string theory, CP is indeedconserved at a fundamental level; all observable CP violation is necessarily spon-taneous and, in principle, calculable.We now take up each of the issues raised here in more detail.2.

Discrete SymmetriesWe have argued that gauged discrete symmetries are safe, i.e. they are unbro-ken by gravity.

We have also remarked that such symmetries arise in field theoryand are quite common in string theory. We will now show that, unlike continuoussymmetries, approximate global discrete symmetries also arise in string theory.To motivate our treatment of this subject, consider the problem of anoma-lies.

We are used to the notion that continuous gauge symmetries should be freeof anomalies. What about discrete symmetries?

That anomalies can arise in dis-crete symmetries can be understood by considering instantons in an effective lowenergy theory. Instantons generally give rise to effective operators which breaksymmetries; ’t Hooft showed long ago, for example, that instantons of the elec-troweak theory generate an effective interaction which breaks both baryon andlepton numbers.

The effective interactions generated by instantons can also violatediscrete symmetries. For a gauge symmetry, such a breaking signals an inconsis-tency, and can be viewed as an anomaly.

[10,11,12] One can attempt to understanddiscrete anomalies by embedding discrete symmetries in continuous ones. [10] How-ever, this leads to constraints which depend on the quantum numbers of massivefields.

The only constraints on discrete symmetries which involve exclusively prop-erties of light fields can be understood in terms of instantons in the effective lowenergy theory. [13,10]What about string theory?If we assume that all discrete symmetries instring theory are gauge symmetries, it is natural to ask whether discrete anomaliesever arise for modular-invariant compactifications.

If one found compactificationswith such anomalies, they would be inconsistent. Such a situation would be re-meniscient of global anomalies.

It could be quite dramatic, representing a new,non-perturbative consistency condition on string compactifications.A priori, Idon’t know of an argument that this can not occur; indeed, I know of no general4

argument that other types of global anomalies (e.g. SU(2) anomlies) do not occur.In fact, study of various compactifications quickly yields numerous examplesof anomalies!

[14] However, in all the cases which have been studied to date, onecan cancel these anomalies in the following way.⋆The axion, a, couples to all ofthe low energy gauge groups:g216π2faXa(x)F (i) ˜F (i)(2.1)It turns out that one can always cancel all of the anomalies by assigning to theaxion a non-linear transformation law of the formafa→afa+ β(2.2)for some number β. To understand what is going on here, note that the instantoneffective action is typically something of the formψψ .

. .

ψeia/fae−8π2/g2(2.3)So the phase rotation of the fermions is compensated by the shift in the axion field.This result is highly non-trivial (typically several anomalies are being taken careof by one such shift). It almost surely indicates that the symmetries are not, infact, anomalous.

So far only rather special classes of models have been examined,so that while I suspect that this is a general result, it is by no means certain. Inany case, there is still no evidence for the existence of any new, (independent)consistency condition beyond those which hold in perturbation theory.However, from these studies we learn something suprising: string theory pos-sesses global discrete symmetries which are valid to any order of perturbation theoryand broken only non-perturbatively.

For while the non-anomalous symmetry in allof these cases is spontaneously broken by the non-linear transformation law of theaxion, the original, anomalous symmetry is good to all orders, being broken only⋆The possibility that anomalies in discrete symmetries might be cancelled by a Green-Schwarzmechanism was noted in ref. 10.5

non-perturbatively. If we adopt the view that phenomena which occur in stringtheory can plausibly occur in any ultimate theory, this means that it is reasonableto postulate approximate global symmetries in a low energy theory.

Such sym-metries have been suggested for many reasons, such as avoiding flavor changingneutral currents in multi-Higgs theories and proton decay in superysmmetric theo-ries, for understanding the fermion mass matrix, and (see below) for understandingthe strong CP problem.3. Strong CP3.1.

Is CP Spontaneously Broken in String Theory?In perturbation theory, CP is conserved in string theory. [15] One might askwhether this is true non-perturbatively.After all, in field theory, θ is a non-perturbative parameter which violates CP.

It has been suggested that string theorymight possess similar non-perturbative parameters. [16]If some of these are CP-violating, they might give rise to θ parameters in the low energy theory.

However,it turns out that one can argue that CP is a gauge symmetry in string theory. [17]†This means that there can be no such CP-violating parameters, since these wouldcorrespond to an explicit breaking of the symmetry.As a result, if string theory describes nature, CP must be spontaneously bro-ken and θQCD is calculable.

This breaking might arise at Mp (e.g. through ex-pectation values for CP-odd moduli) or at lower scales (e.g.through vev’s forsome matter fields).

In either case, one expects that generically θ will be large,proportional to other CP-violating phases needed to explain the features of theK-meson system. However, in field theory, it is known that one can sometimesarrange things so that θ is small.

[18] Preliminary investigation (to be described inref. 19) indicates that certain “string inspired models” can accomplish this.

Inparticular, in a class of models, discrete symmetries insure that CP is sponta-neously violated at an “intermediate scale”, MINT , of order 1011 GeV, with θ oforder MINTMp(times coupling constants). Moreover, in these models, the low energytheory is supersymmetric, but the only CP violation lies in the KM phase and θ.† When I presented this talk in Cincinnati, I was not sure of this statement, and only men-tioned it as a possibility.

I offered in addition some alternative arguments for absence of θparameters.6

3.2. Accidental Axions in String TheoryAlternatively, one can explore axion solutions to the strong CP problem instring theory.

There are, however, two potential problems with the stringy axion, a,which we have described above. First, in many compactifications of string theory,there is more than one strongly interacting gauge group; it is necessary to have atleast one axion for each group.

Second, even if QCD is the only strong group, thedecay constant, fa, is a number of order MP . This contradicts cosmological bounds,which give fa < 1012 GeV.

[20] However, one might choose to ignore these limits;there are a number of possible loopholes. For example, these analyses assume thatthere is no entropy generation after the QCD phase transition.

However, plausiblemodels exist in which there is such entropy generation, and yet an adequate baryondensity is generated. [21] These arguments also assume that the initial value of theaxion field in the observable universe is simply a random number; in that case,for such a large fa, only one universe in 103 has a sufficiently small initial θ. ButLinde has pointed out that the size of the initial θ may be correlated with primordialdensity fluctuations.

Only those regions with small enough θ, in this view, mightresemble ours. [22] Thus a rather mild application of the anthropic principle (the“weak anthropic principle”[23]) might solve the problem.You may not wish totake any of these possibilities too seriously; however, one should be aware that thecosmological axion limit rests on certain assumptions which may not be true.For now, though, let us take the cosmological limit seriously, and ask howfa ∼1011 might arise.

We could, of course, simply postulate that there is anotherfundamental scale, and the axion arises in a manner similar to the string axion.Such an assumption is certainly troubling, however, and there is no reason to thinksuch a scale should arise in string theory. Alternatively, the Peccei-Quinn symme-try might arise by accident, in the same way that baryon and lepton number arisein the standard model.

Such an accident, however, would be quite startling if onesimply assumes that gravity generates all operators consistent with the various lo-cal symmetries of the theory. The problem is that in order that the axion tune θ tothe required precision, it is necessary that the leading operators which violate thesymmetry be of very high dimension.

This point was already raised in passing byGeorgi, Glashow and Wise[24] More recently, it has been discussed in a general andquantitative fashion by several authors. [25] To gain some appreciation of the diffi-7

culty, suppose that the lowest dimension, gauge-invariant operator which violatesthe symmetry is O(4+n), of dimension 4 + n. Then the leading symmetry-violatingterm which can occur in a low-energy effective field theory isLSB =γMnPO(4+n)(3.1)where γ is a dimensionless coupling constant. On dimensional grounds, this givesrise to a linear term in the axion potential,VSB ∝γ fn+3aMnPa(x)Sincem2a ∼m2πf2πf2a(3.2)the resulting shift in θ isδθ = δafa∼γmπ2f2πfn+4aMnP< 10−9(3.3)For fa = 1011, this gives n > 6 (i.e.

the symmetry-violating operator must at leastbe of dimension 12!) If fa = 1010, things are slightly better; one needs to suppressall operators of dimension less than 9.The lesson of all this is that if one wants a Peccei-Quinn symmetry to ariseby accident, one must forbid operators up to very high dimensions.

How mightsuch a thing occur? The authors of refs.

25 noted that with a sufficiently com-plicated continuous gauge symmetry, one could indeed suppress operators of veryhigh dimension. However, by their own admission, the resulting models were notparticularly beautiful.In light of our earlier discussion, it is natural to ask how easily discrete sym-metries can accomplish the same objective.

In fact, in the framework of stringtheory, this question was asked some time ago by Lazarides et al[26] and by Rossand Casas. [27] The latter authors also attempted to estimate how large a θ would8

be induced by higher-dimension operators which violated the Peccei-Quinn sym-metry, in precisely the spirit described above (we will see, however, that they failedto consider the most dangerous class of operators). Before reviewing these models,however, it is perhaps useful to illustrate just how powerful discrete symmetriesare in this respect by considering theories in which the Peccei-Quinn symmetry isdynamically broken by fermion condensates.⋆As an example, consider a theorywith (unbroken) gauge group (in addition to the standard model gauge group)SU(4)AC (AC is for “axi-color”), with scale ΛAC ∼fa.

In addition to the usualquarks and leptons, we suppose that the theory contains additional fields Q and¯Q, transforming as (4, 3) and (¯4, ¯3) under SU(4)AC × SU(3)c, and fields Q and¯Q transforming as a (4, 1) and a (¯4, 1). Now suppose that the model possesses adiscrete symmetry (gauged or global) under whichQ →αQQ →αQwhere α = e2πiN ; all other fields are neutral.

If, for example, N = 3, the low-est dimension chirality-violating operators one can write are of the form ( ¯QQ)3,which is dimension 9; suppression of still higher dimension operators is achievedby choosing larger N. In this theory, the would-be PQ symmetry isQ →eiωQQ →e−3iωQ(3.4)This symmetry has no SU(4) anomaly, but it does have a QCD anomaly. Oneexpects that this symmetry will be broken by the condensates< ¯QQ >∼< ¯QQ >∼f3a(3.5)This gives rise to an axion with decay constant fa, which solves the strong CPproblem.Let us turn now to the ideas of Lazarides et al and of Casas and Ross.

Inparticular, we will develop a variant of the model of the latter authors. Of course,⋆This has been noted independently, and much earlier, by A. Nelson (unpublished).9

it is not presently clear how string theory might describe the real world, so wewill view this model as “string inspired,” in that it shares features common to aclass of compactifications. We will have to assume, also, some structure of softsupersymmetry breaking.Having said that, it should be stressed that modelsof this kind have a major virtue: the axion decay constant is naturally of orderMINT = √MW MP , i.e.

within the allowed axion window. Our only truly newpoint, beyond those made in ref.

27, will be that there are operators beyond thoseconsidered by these authors which one must eliminate if one is to insure sufficientlysmall θ.Consider a theory with unification in the gauge group E6, with E6 brokento a rank 6 group at the unification scale; this is the structure which emergesfrom conventional Calabi-Yau compactification. [28] Ordinary matter fields will beassumed to arise from 27’s of E6.

Casas and Ross assume that the theory possessesa Z3×Z2 symmetry. The 27 contains two standard model singlets, which we denoteby S and N. These authors suppose that there are two fields with the quantumnumbers of S, Si, and two fields with the quantum numbers of ¯S, ¯Si.UnderZ3 × Z2, these fields transform as follows:S1 →−αS1¯S1 →−α ¯S1α = e2πi3(3.6)while S2 and ¯S2 are invariant.

The leading terms allowed in the superpotential areW =aMpS22 ¯S22 +bM3pS31 ¯S31 +cM9pS61 ¯S62 +dM9p¯S61S62(3.7)This superpotential has an approximate U(1) symmetry, broken by the final twoterms:S2 →eibS2¯S2 →e−ib ¯S2(3.8)This symmetry can play the role of a Peccei-Quinn symmetry. If S2 and ¯S2 havesoft-breaking mass terms of the correct sign, these fields will acquire expectationvalues of order MINT , breaking the symmetry spontaneously (in this model, S1and ¯S1 obtain larger expectation values).10

The authors of ref. 27 estimated the θ which would arise in this model byconsidering the explicit breaking terms in the superpotential, above, as well assoft-breaking terms of the type Am3/2W.

It is easy to see that these lead to quitea small θ. However, a generic supergravity model also leads to soft supersymmetry-breaking terms for scalar fields, φ, of the type m23/2φ∗nφm.

In the present case,this allows the operator:m23/2M2pS∗1 ¯S1S2 ¯S∗2(3.9)This breaks the Peccei-Quinn symmetry. It is a huge term on the scale of axionphysics; it gives, for example, a contribution to the axion mass of order MeV ’s!Clearly we can improve the situation if we consider a different symmetry.

Forexample,S1 →−αS1¯S1 →α2 ¯S1S2 →−S2¯S2 →¯S2(3.10)again gives a lagrangian which admits a Peccei-Quinn symmetry. Now, however,the leading symmetry-violating operators are things like ¯S1S∗21S22 ¯S∗2.

This leadsto a θ which is perhaps barely small enough.We will not attempt, here, to consider all aspects of the phenomenology ofthese models; suffice it to say that it does appear to be possible to build realisticmodels along these lines. One can debate how reasonable – or contrived – thissolution appears to be.

As we will explain elsewhere, it is probably not much betteror worse than is required to obtain a Nelson-Barr type solution of the problem. [29]The main difference is that in the Nelson-Barr case, it is not necessary to suppressoperators of such high dimension.

However, as illustrated by the examples above,rather simple discrete symmetries can accomplish this.11

4. Some Wild Speculations on StrongCP and the Cosmological ConstantI would like to conclude by describing some wilder ideas about the strong CPproblem and the cosmological constant.

These are associated with what T. Banks,N. Seiberg and I have dubbed “irrational axions.”[30] Such axions do not arise inconventional theories.

It is tempting to think that they might arise in string theory,but we have not found an example of this phenomenon. The basic ideas are verysimple.

Consider first the strong CP problem. Suppose that in addition to QCD,one has an additional strongly interacting gauge group; we will refer to this asaxicolor, QAD.

In addition, suppose one has a single Peccei-Quinn symmetry, anda is the associated axion. The couplings of the axion to the two gauge groups arewritten1f1g2316π2Zd4xaF ˜F + 1f2˜g216π2Zd4xaG ˜G(4.1)where G refers to the axicolor gauge fields, and we assume ΛQAD >> ΛQCD.

Thenthe axion acquires its mass principally from ACD dynamics; one expectsm2a ∼Λ4QADf22(4.2)Ordinarily, such an axion would have nothing to do with the solution of the strongCP problem. But suppose f1f2 is irrational.

The axion potential in this theory issomething of the formV = Λ4ACDcos(a/f1) + Λ4QCDcos(a/f2 + θ)(4.3)Now since f1/f2 is irrational, one can always find integers n1 and n2 such thata ≈2πn2f2 ≈(2πn1 −θ2)f1(4.4)with arbitrary accuracy. Thus in this theory there exist ground states with ar-bitarily small θQCD.

These states are stable on cosmological scales, but they arealso rare; about 1 in 109 local minima of the potential has such small θ. Thus,in this model the θ problem is solved without a light axion; the price one pays iscosmological: why do we find ourselves in a suitable state?12

This problem can (almost) be solved if we invoke the anthropic principle, inthe “weak sense” discussed by Weinberg. [23] Suppose that the cosmological constantof the theory is adjusted so that it vanishes as θ →0.

If this is the case, formationof galaxies requires that the cosmological constant not be larger than about 1000times the present experimental limit. This gives θ far smaller than 10−9.

We stillhave a factor of 1000 in cosmological constant to explain. Actually, it is not quiteas bad as that; since the energy goes as θ2, we are really only out by a factor of30.We have already seen that there are many ways to solve the strong CP prob-lem, so one more, which we don’t (yet) know how to get from an underlying micro-scopic theory might not seem that exciting.

However, there is another fine tuningproblem which we might like to solve without a light particle: the cosmologicalconstant problem. This may also be possible with irrational axions.

The idea wewill describe here bears some resemblance to a suggestion of Abbott. [31]Suppose, in addition to QCD, the underlying theory has two strong gaugegroups, with scales Λ1 ≈Λ2.

Suppose the “irrational axion” couples to these,in the same irrational way as before.Suppose also that the bare cosmologicalconstant, Λ4o, satisfies Λ4o < Λ41. With supersymmetry, one can show that theseconditons can be natural.

Now the axion potential is a sum of three terms:V (a) = Λ4o + Λ41cos(a/f1) + Λ42cos(a/f2 + θ2)(4.5)In this potential there exist vacua with arbitrarily small values of the cosmologicalconstant. They are even more rare than in our previous example; for example,if Λ1 ∼1010, then only one in about 1088 vacua are acceptable.

Actually, thesituation is even worse because in this case a typical local minimum with smallcosmological constant is not even approximately stable.Only a small fractionare: for example, if several adjacent minima all have higher energy, the tunnelingrate will be suppressed. Again, we can suppose that in an inflationary universe,some worlds like our own were created, and try and invoke the anthropic principle.Again, we are offby a factor of 1000.However, I don’t believe this problem is so severe; for example, somewhatstronger verions of the anthropic principle might save the day.

If we had examplesof such irrational axions, they might well solve the cosmological constant problem.13

Acknowledgements I wish to thank my collaborators T. Banks, R. Leigh,D. MacIntire and N. Seiberg for their insights, and Ann Nelson for several helpfuldiscussions.REFERENCES1.

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