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Supersymmetry, Naturalness, and

arXiv:hep-ph/9205227v1 20 May 1992August 9, 2018Supersymmetry, Naturalness, andDynamical Supersymmetry Breaking⋆Michael Dine and Douglas A. MacIntireSanta Cruz Institute for Particle PhysicsUniversity of California, Santa Cruz, CA 95064AbstractModels with dynamical supersymmetry breaking have the potential to solvemany of the naturalness problems of hidden sector supergravity models. We reviewthe argument that in a generic supergravity theory in which supersymmetry isdynamically broken in the hidden sector, only tiny Majorana masses for gauginosare generated.This situation is similar to that of theories with continuous R-symmetries, for which Hall and Randall have suggested that gluino masses couldarise through mixings with an octet of chiral fields.

We note that in hidden sectormodels, such mixing can only occur if the auxiliary D field of a U(1) gauge fieldhas an expectation value. This in turn gives rise to a catastrophically large Fayet-Iliopoulos term for ordinary hypercharge.

To solve this problem it is necessary tounify hypercharge at least partially in a non-Abelian group. We consider, also,some general issues in models with continuous or discrete R symmetries, notingthat it may be necessary to include SU(2) triplet fields, and that these are subjectto various constraints.

In the course of these discussions, we consider a number ofnaturalness problems. We suggest that the so-called “µ-problem” is not a problem,and point out that in models in which the axion decay constant is directly related tothe SUSY breaking scale, squarks, sleptons and Higgs particles generically acquirehuge masses.Submitted to Physical Review D.⋆Work supported in part by the U.S. Department of Energy.

1. IntroductionHidden sector supergravity models provide a framework in which to under-stand how various types of soft supersymmetry breaking couplings might arise atlow energies.

Unfortunately, none of the models which have been constructed todate are at all compelling. None are beautiful, and all suffer from serious nat-uralness problems.

The most severe of these is the problem of the cosmologicalconstant. Others include too large flavor changing neutral currents and neutronelectric dipole moment, the existence of a large hierarchy, put in by hand, and theneed to omit from the lagrangian numerous couplings permitted by symmetries,involving both visible sector and hidden sector particles.About the cosmological constant, we will have nothing new to say here.

Wewill have to simply assume that this problem is solved by some mechanism whichdoes not too drastically alter the low energy structure of the theory. One mighthope to find conventional field-theoretic explanations for the other questions.

If thisis the case, these might have phenomenologically interesting consequences, leadingto predictions concerning the spectrum of supersymmetric particles. About flavorchanging currents and CP, for example, it has been noted elsewhere[1] that thisproblem might be resolved if the gauginos are the most massive supersymmetricparticles.The problems of obtaining a large hierarchy and of the omission ofnumerous couplings in the hidden sector might be resolved if supersymmetry isdynamically broken.

[2−4] The need to omit various visible sector couplings, on theother hand, might be resolved by the recent suggestion of Hall and Randall thatone impose an R symmetry on the theory. [5]Theories with dynamical supersymmetry breaking (DSB) have been knownfor some time.

[2−4]DSB has the potential to explain large hierarchies, and theknown examples have the virtue that it is not necessary to omit couplings allowedby symmetries. However, if one proceeds in the most straightforward way to buildmodels based on these, one runs into difficulties.

The most severe of these concernsmasses for gauginos, which turn out to be extremely small. Of course, models withR symmetries also are in danger of yielding small (zero) gaugino masses.

In thelatter case, Hall and Randall have proposed that the problem can be solved byadding an octet of chiral fields to the low energy theory; these particles combine1

with the gauginos to gain mass. This mechanism is a potential solution to theproblems of DSB as well.

We will see, however, that in either case this mechanismcan operate only if the auxiliary D field of a U(1) gauge field in the hidden sectoracquires a large expectation value. This in turn raises the danger of a large Fayet-Iliopoulos term for hypercharge.

[6] One way (possibly the only way) to forbid sucha term is to unify hypercharge in a non-Abelian group at some scale. We will seethat the natural scale for breaking this additional symmetry is the hidden sectorscale.While DSB may resolve some questions of naturalness in the hidden sector, itis still usually necessary to forbid certain visible sector couplings.

For example, inthe minimal supersymmetric standard model, one must forbid a large mass term inthe superpotential for the two Higgs doublets, but one must have a soft breakingmass term involving both doublets. One can, as in the case of the unwanted hiddensector couplings, simply suppose that the unwanted superpotential terms are notpresent at tree level and then invoke non-renormalization theorems.

Superstringtheory suggests that such a possibility might not be unreasonable. Hall and Ran-dall[5] have recently considered an alternative possibility, noting that continuous Rsymmetries can forbid such terms.The rest of this paper is organized as follows.

In the next section, we willbriefly review some features of models with dynamically broken supersymmetry.In section 3, we will consider what happens when such models are coupled to super-gravity. We will recall the general arguments that majorana masses for gauginos insuch theories must be small, and illustrate them with a one loop calculation.

Thiscalculation is rather subtle; a conventional treatment, such as has been applied tosupergravity theories in the past,[7] gives a large mass. It turns out, however, thatthe Feynman diagrams contributing to the gaugino mass require careful regulariza-tion, and that in the end these masses are small.

We comment on the implicationsof these results for more conventional theories.In section 4, we consider the effect of adding an octet to a theory in whichmajorana masses for gauginos are small. We first consider the problems associatedwith D terms, and possible solutions.

We then consider models with dynamicalsupersymmetry breaking in which either gauge interactions or supergravity is the“messenger” of supersymmetry breaking. It does not appear too difficult to build2

realistic models of this type.In section 5 we consider some aspects of models in which majorana masses forgauginos are forbidden by continuous or discrete R symmetries. We point out, first,that in supergravity theories, if one insists on cancelling the cosmological constantin the effective lagrangian, only a discrete Z2 R symmetry can survive to lowenergies.

We argue, however, that given our poor understanding of the cosmologicalconstant problem, and given all of the naturalness problems of supersymmetrictheories, such symmetries are still worthy of study. As noted by Hall and Randall,in such theories, in addition to an octet of chiral fields, it may be necessary tohave still other fields to avoid very light states in the neutralino sector.

Theseauthors considered the possible addition of a gauge singlet superfield. We showthat such a singlet is unnatural, in the sense that in almost any conceivable schemefor supersymmetry breaking, it has unacceptable properties.

We note that thecorresponding problems do not arise for SU(2) triplet fields, and consider someaspects of such models, including the spectrum and the question of the ρ parameter.We find that such schemes typically predict that there should be new particles withmasses below MZ.Our conclusions are presented in section 6. Here we comment on possible con-nections of axion physics and supersymmetry.

In particular, it is remarkable thatboth of these require a scale of around 1011 GeV, and a number of authors[8,9] havespeculated on possible connections between them. We point out that generically, ifthe Peccei-Quinn symmetry is broken by vev’s in the hidden sector, squark, sleptonand Higgs masses tend to be of order the intermediate scale.3

2. Dynamical Supersymmetry BreakingWitten was perhaps the first to appreciate the possible importance of dynam-ical supersymmetry breaking and to clearly formulate the problem.

[10] He stressedthat dynamical supersymmetry breaking was likely to give rise to large hierarchies.Because of the non-renormalization theorems,[11] supersymmetry, if unbroken at treelevel, remains unbroken to any finite order in perturbation theory. However, hepointed out that the proofs of the non-renormalization theorems are firmly basedon perturbation theory.

Thus one can hope to find effects smaller than any power ofthe coupling constant which give rise to supersymmetry breaking. Witten went onto prove that many theories do not break supersymmetry dynamically.

[12] However,chiral gauge theories did not yield to this analysis.In a series of papers, it was in fact shown that supersymmetry is some-times dynamically broken in four dimensions. [2−4] First it was observed that non-perturbative breakdown of the non-renormalization theorems is common – almostgeneric.

The basic point is illustrated by an SU(2) gauge theory with a singlemassless flavor, i.e. containing two chiral doublets, Q and Q.

At the classical level,this theory has a continuum of physically inequivalent vacuum states. Essentiallythese are the states with Q = Q = 0v!.

For non-zero v, the gauge symmetryis completely broken, and the gauge bosons are massive. The effective couplingin a given vacuum is g(v), since the gauge boson masses are of order v, and allmomentum integrals are cut offin the infrared at this scale.As the theory isasymptotically free, by choosing v large enough the theory may be made as weaklycoupled as one wishes.

In each of these states there is one massless chiral field.This field can be written in a gauge-invariant way as Φ = QQ. Expanding thefields Q and ¯Q in small fluctuations about their vacuum expectation values, theterm linear in the fluctuations is the massless state.

The problem, then, is to un-derstand the properties of the effective low energy theory containing Φ only, andin particular to determine whether this theory possesses a superpotential for Φ.Symmetry considerations restrict the superpotential to be of the formW = Λ5Φ(2.1)4

Here Λ is the scale of this SU(2) theory, and again, this expression should be un-derstood by expanding Φ in small fluctuations about a particular ground state.Itis straightforward to show that a single instanton generates the various componentinteractions implied by this superpotential.This analysis immediately general-izes to theories with gauge group SU(N) and N −1 flavors. Adding small massterms, one finds that all of these results are consistent with Witten’s analysis ofdynamical supersymmetry breaking.

Minimizing the full superpotential yields Ngauge-inequivalent ground states, in agreement with Witten’s computation of theindex. By other methods, one can show that a superpotential is generated in manyother theories.While these examples illustrate that the non-renormalization theorems do in-deed break down non-perturbatively, they do not lead to a particularly interestingphenomenology.

Without mass terms, the potential for the field Φ falls rapidlyto zero for large Φ. Thus the model has at best a cosmological interpretation.The basic problem is that for large expectation values of the fields, the theories be-come more weakly coupled and any potential which is generated non-perturbativelymust tend to zero.

Adding mass terms to the theory eliminates the “flat direc-tions” which exist classically in the potential, but in this case the full theory hassupersymmetric ground states. A general criterion for obtaining supersymmetrybreaking with a good ground state was suggested in ref.

3. Suppose a theory has,classically, no flat directions in its potential.

At the same time suppose that thetheory possesses a continuous global symmetry which is broken in the true vac-uum. In such circumstances supersymmetry is almost certainly broken.

For, if itwere not, the goldstone bosons of the broken symmetry would have scalar part-ners which would have no potential. However, in this case their expectation valueswould not be fixed and there would be flat directions, contradicting the originalassumption.This argument is heuristic, and one can imagine a variety of loopholes.

How-ever, a number of models were studied in ref. 4 satisfying these criteria, and shownto break supersymmetry.

The simplest is a theory with gauge group SU(3)×SU(2),with chiral fields Q, ¯U, ¯D and L, transforming as (3, 2), two (¯3, 1)’s and (1, 2), re-spectively under the group. In addition to the gauge interactions, to eliminate theflat directions it is necessary to include a superpotential5

W = λQ ¯QL. (2.2)If λ is small, one can first determine the superpotential generated by instantons(as in equation (2.1)), and then treat the tree level superpotential (equation (2.2))as a small correction.

Minimimizing the resulting potential, one finds that super-symmetry is broken. If the scale of the SU(3), Λ3, is larger than that of SU(2),Λ2, one finds that at the minimumQ = a000b0!¯Q = ¯U¯D!= QL = √a2 −b20!

(2.3)where a = 1.286Λ3/λ17, b = 1.249Λ3/λ17, and the vacuum energy is E =3.593λ107 Λ43.Other models can be analyzed along similar lines. Another example of interestis an SU(5) theory with a single ¯5 and 10.

In this theory, there is no classicalsuperpotential which one can write. Even so, the theory has no flat directions.Using ’t Hooft anomaly conditions one can argue that the non-anomalous globalsymmetry of the model must be broken, and that as a result supersymmetry isbroken.3.

Coupling to SupergravityWe would like to consider a theory of this type as a candidate for the hiddensector of a supergravity model.⋆As a concrete example, we take the SU(3) ×SU(2) model described in the previous section; however, our considerations belowgeneralize almost trivially to other theories. We assume that, apart from somepossible superheavy (O(MP ) or O(MGUT )) fields, no other fields transform underthe SU(3) × SU(2) gauge symmetry of the hidden sector (these groups should notbe confused with the SU(3) × SU(2) × U(1) symmetry in the visible sector; they⋆As explained in 4, breaking SUSY at low energies tends to give unwanted axions andGoldstone bosons.6

represent additional gauge interactions). Thus, taking the characteristic scale ofthe hidden sector, Mint, to be Mint ∼1011GeV , the dynamics described in theprevious section are unaffected: supersymmetry is broken in this sector at a scaleof order Mint; various fields acquire expectation values and masses of order Mint,and there are some (pseudo) Goldstone fields with decay constants of this order.The analysis of scalar masses is similar to the case of more conventional hiddensector supergravity models.

The problem comes when one attempts to computegaugino masses. There is a simple argument that any Majorana masses for gaug-inos in such theories must be extremely small.In discussing physics at scalesabove m3/2, it should be possible to integrate out Planck (and GUT) scale physics,obtaining a (locally) supersymmetric effective lagrangian.

The usual supergravitylagrangian is specified by three functions. Only the function f(φi), which describesthe coupling of the chiral fields to the gauge multiplets, is relevant to the questionof gaugino mass through a term in the lagrangian:L ∼Zd2θf(φ)W αW αwhere f is a holomorphic function of the scalar fields.

On the other hand, in allof the models of dynamical supersymmetry breaking presently known, all of thehidden sector fields, Zi, carry charges under the various gauge symmetries. Thusf is necessarily at least quadratic (and in fact is generally cubic) in fields.

Thusone expects its coefficient to be suppressed by at least two powers of MP . If this isthe case, local supersymmetry implies that gaugino masses will be extremely small(of order eV or smaller).However, if one simply computes the gaugino masses in these models usingthe naive Feynman rules, one seems to find much larger answers.

For example,suppose that, in addition to a hidden sector of the type we have described in theprevious section, the model possesses a heavy color octet, O, of chiral fields ofmass M. Then at one loop there is a diagram contributing to the gluino mass,quite similar to the types of diagrams considered in ref. 7.

In particular, there isa (non-vanishing) term in the lagrangian of the form m3/2MO2, where O is thescalar component of the octet. Then the diagram of fig.

1 gives a non-zero mass7

for the gluino of ordermλ ∼α3πM2intMP(3.1)Notice, in particular, that this expression is independent of the mass of the octet.If correct, this would be a wonderful result, since it would give rise to a gluinomass of order 100GeV of so. Not surprisingly, in view of our general argument,this result is not supersymmetric.The problem with this calculation is most easily illustrated with a well-studiedmodel, the “Polonyi model.” This theory contains a hidden sector consisting of onlyone singlet chiral field, Z, with superpotentialW = M2int(Z + β)(3.2)Assuming that the Kahler potential is simply quadratic in Z, the minimum of thepotential occurs forZ = (√3 −1)M(3.3)where M = Mp√8π; in these equations, in order that the cosmological constant vanishat the minimum of the potential, β = (2 −√3)M. Because Z is a gauge singlet,there should be no problem obtaining a gaugino mass in this model, since any fwhich is, say, a polynomial in Z will yield mλ ∼m 32.

Indeed, if one now adds tothis model the heavy octet, O, above, one generates a gluino mass at one loop;[7]proceeding as before, the diagram of fig. 1 yieldsmλ = 3g24π2(2 −√3)e(√3−1)22m 32(3.4)with m3/2 ∼M 2intMp .

Again we find a mass of the order of 100 GeV.Now we would expect that at energy scales below the mass of the heavy octet,MO, the system would still be described by a locally supersymmetric effective la-grangian, including the usual light fields and the hidden sector fields. In particular,the gluino mass could be understood as arising from the function f of this theory,8

through the term in the supergravity lagrangian:L ∼14e−G/2Gl(G−1)kl f∗αβkλαλβ.In the above, f∗αβk is the derivative of f∗αβ with respect to Z, and the existence ofa gluino mass implies that f is a function of Z. If this is the case, on the otherhand, we expect to find couplings of Z to F 2µν and F ˜F through the terms:L ∼−14RefαβF αµνF µνβ + 14iImfαβF αµν ˜F µνβ.However, at one loop, using the lagrangian of refs.

13, these couplings vanish!To see this, consider the coupling of the pseudoscalar part of Z to the octetfermions. This coupling is proportional to ∂µZ ¯Oγµ(1 −γ5)O.

One can attemptto compute the coupling of the imaginary part of Z to F ˜F at one loop arisingfrom this term. But this calculation is identical to the famous calculation of thechiral anomaly, and is subject to the same ambiguities.

For example, it is wellknown that if one uses, say, a Pauli-Villars regulator, the F ˜F coupling vanishesin this case. Indeed, the result of this computation, as in the case of the gauginomass above, is independent of the mass of the particle running in the loop, andso is canceled by the regulator diagram.

Clearly supersymmetry requires that oneuse the same sort of regulator for all of the diagrams.But we have seen thatthe gluino mass is independent of the mass of the heavy particle running in theloop, so adding the Pauli-Villars term will give zero! Correspondingly, alternativechoices of (supersymmetric) regulators will give different results for the gluinomass.

However, in the case of the hidden sector with DSB, it is clear from ouroriginal symmetry arguments that any gauge- and supersymmetric regulator willgive zero for the gaugino mass.⋆Thus simply using a theory with dynamical supersymmetry breaking as aconventional hidden sector model yields unacceptable results.In the followingsection, we consider an alternative approach.⋆We are assuming, here, that the lagrangian given in ref. 13 is the most general one consistentwith local supersymmetry, up to terms with two derivatives.9

4. Models with OctetsMajorana masses for gauginos are also forbidden in theories with an exact Rsymmetry at low energies.

Following Hall and Randall, it is natural to attempt tobuild models with light color octet chiral fields, and to allow them to mix with thegluinos. In this section, we will consider some general issues in models of this kind(with either DSB and/or exact R symmetries).As stressed by these authors, a mass term mixing the gluino and the octetfermions is one of the allowed soft breaking terms of supersymmetry.

It is inter-esting to ask, on the other hand, how such a term might arise in the framework ofhidden sector models. Consider, first, the case of hidden sector models with globalsupersymmetry.

(It is convenient to consider this case because it is easy to writedown globally supersymmetric effective actions). In such theories, above the scaleof weak interactions (the scale of supersymmetry breaking in the visible sector ofthe theory), it is possible to describe the theory by a supersymmetric effectiveaction.

[6] Then supersymmetry breaking is the statement that, below the breakingscale, the auxiliary (F) component of some chiral superfield(s), Z, is non-vanishing,as well, possibly, as the auxiliary (D) components of some gauge fields. In suchtheories, masses for the scalar components of observable fields (denoted by φ), arisethrough operators of the typeRd4θZ†Zφ†φ; Replacing Z (FZ) by its expectationvalue immediately yields scalar masses.

On the other hand, terms which mix thefermionic components of the octet, O, with the gluinos can only arise providedthe theory contains a U(1) gauge field, ˜V , whose auxiliary component, ˜D, has anexpectation value.† Then the desired mixing can arise through the operatorLλ = 1MZd2θ ˜WαW aαOa(4.1)Of course, this U(1) cannot be ordinary hypercharge. But the large vev of˜D raises the specter of a large Fayet-Iliopoulos term for DY .

The dimension four†This does not occur in the SU(3) × SU(2) model discussed earlier. There we can gaugea U(1).

However, it is is necessary to include an additional field to cancel anomalies. Itturns out that the sign of the charge of this field is such that its expectation value givesa vanishing expectation value for ˜D.

We know of no reason for this to be true in general,however.10

operatorRd2θW Yα ˜W α gives a Fayet-Iliopoulos term of orderD˜DE. Such a couplingimplies a large negative mass-squared for scalars carrying hypercharge (of orderM2int), and potentially leads to an enormous breaking of hypercharge.

One possibleway to avoid this problem is to unify hypercharge into a non-Abelian group, brokenonly at some scale well below Mp. For example, many authors, motivated by stringtheory, have considered the possibility that down to some scale there is an unbrokenSU(3)c × SU(3)L × SU(3)R symmetry.

In such a case, the Fayet-Iliopoulos termcould be highly suppressed. In fact, one can even avoid the problem if hyperchargeis a sum of a U(1) generator and a non-Abelian generator.

In such a case, it canbe natural for some scalar field to gain a large vev, breaking some of the gaugesymmetry and leaving ordinary hypercharge.‡These considerations can be immediately extended to the case of local super-symmetry. If one examines the lagrangian of ref.

13, one can see that there is onlyone term which gives rise to a Dirac mass term mixing gauginos and matter fields,and this is only non-vanishing if there is an expectation value for ˜D.In theories with dynamical supersymmetry breaking, having obtained a suf-ficiently large gluino mass, we have more or less phenomenologically acceptablemodels. One still must check the neutralino sector.

If the superpotential containsa term mH ¯H, with m ∼m3/2, the only light neutralino is the photino. We willargue later that a term of this size will arise automatically in many circumstances.The photino may gain a small mass from loops of light fields, but it may be neces-sary to add additional light fields in order to obtain an acceptable phenomenology.‡ As an example, one can consider a set of fields with the quantum numbers of a 27 of E6,and suppose that the unbroken group is SU(2)L × SU(2)R × U(1) × U(1).

Suppose that,apart from the Fayet-Iliopoulos term for the U(1), all fields have positive soft-breaking massterms, of order m23/2, except for the two SU(3)×SU(2)×U(1) singlets, which have negativemass-squared terms. Then it is easy to check that there is a local minimum of the potentialat which the surviving gauge symmetry is SU(3) × SU(2) × U(1).11

5. Models with Continuous R symmetriesOur remarks in the previous section apply to models with dynamical super-symmetry breaking and to theories with unbroken continuous R symmetries.

Inboth types of models, the desired mixing arises if an auxiliary D field has a non-zero vev, and one must insist on at least a partial unification of hypercharge ina non-Abelian group to avoid Fayet-Iliopoulos terms. In this section, we considersome further issues associated with R symmetries.

Such theories have previouslybeen carefully considered by Hall and Randall. [5] These authors assumed that thesymmetry was continuous.

The Higgs fields were assigned R = 0, while quark andlepton superfields were assigned R = 1. In order to obtain a gluino mass, theyrequired that their models contain a color octet chiral field with R = 0; they thennoted that λaψaO, where ψO is the fermionic component of O, is one of the allowedsoft breaking terms.

Hall and Randall also observed that if one does not add addi-tional fields, the model possesses, at tree level, a massless photino and a masslesshiggsino. At tree level, one can suppress the coupling of the massless Higgsino tothe Z, however one predicts too light a Higgs.

As a result, these authors consideredtheories with an additional singlet field. We will argue shortly that in almost anyscenario for supersymmetry breaking, this is likely to lead to difficulties; insteadone needs to add SU(2) triplet fields.Hall and Randall have recently pointedout that once one loop corrections are accounted for, it may not be necessary toinclude additional fields at all.

The point is that the large radiative correctionsto the Higgs mass due to top quark loops which have been discovered recently[14]can avoid the light Higgs problem, provided the top quark is heavy enough. [15] Ofcourse, dynamical supersymmetry breaking could operate in the framework of suchmodels as well.We would like to explore some aspects of models of this type.

First, thereare some questions of “philosophy” and naturalness which must be addressed. Formost particle theorists, continuous global symmetries are anathema, and this mightbe viewed as an objection to the work of ref.

5. However, in order to implement theprogram of these authors, it is not necessary that the R symmetry be continuous;it can in fact be discrete.Discrete R symmetries have a different status.Forexample, they arise frequently in string theory, where they are usually (possibly12

always) discrete gauge symmetries. For suitable ZN, a discrete ZN R symmetryhas consequences very similar to that of a continuous R symmetry.For both discrete or continuous R symmetries, however, there is a puzzle.Supersymmetry, if it exists, is a local symmetry.Thus the underlying theorymust be a supergravity theory.

In an N = 1 supergravity theory, supersymmetrybreaking with vanishing energy at the minimum of the potential requires that thesuperpotential have a non-zero expectation value. But such an expectation valuenecessarily breaks any R-invariance (apart from Z2 symmetries).

In simple models,this breaking of R invariance tends to be large, and, for example, large Majoranamass terms for gauginos are generated.⋆Still, given our lack of understanding of thecosmological constant problem, the possibility of an unbroken R-invariance seemsworthy of investigation.On the other hand, we would like to reconsider the motivations for consideringR-symmetric theories given in ref. 5.

These authors argue that such symmetrieswould improve the “naturalness” properties of supersymmetric theories. For exam-ple, they would forbid a term in the superpotential of the form µH1H2, where Hidenote the two Higgs doublets.

This argument is not particularly compelling. Fromstring theory, for example, we know that it is plausible to have massless Higgs dou-blets at tree level and to any finite order in perturbation theory.

The question, then,is how large is µ once one takes into account supersymmetry breaking. The situa-tion is most easily described in global supersymmetry.

There, if the hidden sectorcontains some fields, Zi, with non-vanishing F-components, F ∼M2int ∼mW M,then µ is generated by operators of the formLµ = 1MZd4θZ†H1H2(5.1)Replacing the chiral field Z by its vacuum expectation value Z = . .

. θ2 ⟨F⟩givesµ = M∼mW.

A number of authors have noted that these couplings can arisein loops. In supergravity theories, they generically arise at tree level.

For example,in an SU(5) theory in which a 24 couples to Higgs in the superpotential, in such⋆The theories with dynamical supersymmetry breaking often have an approximate R invari-ance in the low energy theory, even after cancelling the cosmological constant, but this doesnot help with the basic problems of naturalness.13

a way that the Higgs mass vanishes as m3/2 →0, supersymmetry breaking shiftsthe 24 vev by an amount of order m3/2, giving rise to µ ∼m3/2. Thus, in a generictheory, the “µ-problem” does not appear to be a problem.The question of motivation aside, models with R symmetry are quite interest-ing.

Singlets, however, are likely to lead to difficulty in this context. The problemis that the dimension four term in the effective lagrangian,LS =Zd4θZ†S(5.2)gives rise, effectively, to a superpotential termWS = ⟨F⟩S(5.3)Because F is generically so large, this term generally has disastrous consequences;for example, it leads to expectation values for Higgs doublets of order the inter-mediate scale.

If one has, instead of singlets, some additional triplet fields, thisproblem does not arise. The corresponding “µ” term, as for Higgs fields, is of ordermW .Actually, in models with dynamical supersymmetry breaking, in contrast tothe more general case, this problem may be somewhat ameliorated.

The point is,again, that the Zi’s are all charged under the hidden sector gauge symmetries, soit is necessary to go to higher dimension operators in order to find these µ terms.Explicit checks show that at one loop, such terms are indeed generated only withsuitably small coefficients. Thus in this framework, models with singlets may makesense.

However, it is of some interest to explore the case of models with triplets aswell. This is rather straightforward extension of the work of ref.

5, which we nowdescribe.In the case of triplets, there are a number of phenomenological concerns.One has to insure that the triplet expectation values are small enough that the ρparameter is not significantly affected. Also, one must make sure that there are noparticles so light that their effects would already have been observed at LEP.

Fordefiniteness, we will focus on the case where the R symmetry is continuous.As usual, we assign all ordinary matter fields R-charge zero; in other words,the chiral fields associated with the quarks and leptons are assigned R = 1, while14

those associated with the Higgs are assigned R charge zero. Gauginos have R = +1.We want to add an octet and a triplet field to the model.

A moment’s thoughtindicates that it is necessary to add at least two triplets to the model if one isto avoid massless particles. The problem is that in the neutralino sector, withonly one triplet (taking, for a moment, the triplet to have R = 0) there are twopositively charged, left-handed fermions with R = −1, while there is only one withR = +1; similar problems arise in the other charge sectors.

This problem can besolved if we include two triplets in the model, one with R = 2, and one with R = 0.These will be denoted by ˆT and ˆT ′, respectively. The additional terms allowed inthe superpotential are thenWT = G ˆT a ˆH2ǫτa ˆH1 + B ˆT ′a ˆT aThe scalar potential generated by this superpotential is:V =g′2 + g232(H22−H21)2+G24 (H21H22+T 2H21+T 2H22)+B22 (T 2+T ′2)+GBt′H1H22√2+Vsoft(H1, H2)Because of the term linear in T ′, T ′ acquires a vacuum expectation value:T ′= −Ga1a22B≡t′.In addition to the superpotential we can have an R-invariant soft-term in theLagrangian of the formLsoft = A ˜T ′aλawhere the λ’s are the fermionic partner of the gauge bosons, i.e.

the gauginos, and˜T ′ is the fermionic component of the superfield ˆT ′.15

The ρ parameter,ρ =M2WM2Z cos2 θWis given at tree level byρ =Pi v2i [I(I + 1) −Y 24 ]Pi v2iY 22where I is the weak isospin and Y is the hypercharge of the scalar multiplet. Theρ parameter for our case, with a triplet and two Higgs doublets, is given by:ρ = v21 + v22 + 4t′2v21 + v22= 1 +4t′2v21 + v22Since the ρ parameter is known to equal 1 to within about 1%, the vev of thetriplet must be of the order of 12 GeV or less.The fermion mass matrix, which is our principal concern, divides into twocharged matrices, each 2×2, and a neutral 3×3 matrix.

To avoid phenomenologicaldifficulties we require that the lightest eigenvalue of each of the charged massmatrices is more than half the Z mass. The lightest neutral should either havemass greater than about half the Z mass, or should couple sufficiently weakly tothe Z that it does not give too large a contribution to the Z width.

We haveexamined various ranges of parameters, and found that it is possible to satisfy allof these constraints. The constraint on the charged masses is easy to satisfy.

Itis more difficult to avoid light neutral particles. Indeed, study of the mass matrixreveals that the lightest neutral is never more massive than sin(θW )MZ; this boundis saturated for A, B ≫MW and G ≫g, g′.

For example,ABGv1/v2ml9000900114490090010.0144On the other hand, for a wide range of parameters, this light state containsnearly equal admixtures of the two Higgsinos, ˜H1 and ˜H2. Because these fieldscouple to the Z with opposite signs, the coupling of this particle to the Z issuppressed.16

6. ConclusionsThere are a number of lessons to be drawn from this work.

First, it does notseem so difficult to build models in which supersymmetry is dynamically broken.The price one pays is the introduction of light states beyond those of the minimalsupersymmetric standard model. In addition, one requires that hypercharge beunified within a larger group.

Needless to say, it is not clear whether such anapproach will fit neatly into conventional grand unification or string theory.We have also commented on some ideas of Hall and Randall for constructingtheories with unbroken R symmetries. These have the potential to solve some ofthe other naturalness problems of supersymmetric theories.

We have noted that inthe context of supergravity, it is difficult to understand both vanishing cosmolog-ical constant and the existence of R symmetries. Ignoring this question, we haveconsidered various aspects of these theories, and have noted that it may be nec-essary to add light triplets in the low energy theory to obtain phenomenologicallyviable models.

We have seen that models of this type almost always yield new,relatively light fermions with interesting properties.It is perhaps of interest to comment on one other set of naturalness issues aswe close this paper. Cosmological and astrophysical constraints suggest that theaxion decay constant is in the range 1011 −1012 GeV.

Since this scale is similarto the scale Mint, it is natural to ask whether these two scales might be related.Indeed, this possibility was suggested some time ago by Kim,[8] and its possiblecosmological significance has been considered by Rajagopal, Turner and Wilczek. [9]However, while this coincidence is tantalizing, it is also problematic.

In the modelsconsidered by Kim, for example, the axion couples to quarks with masses of orderMint.The axion, however, also couples (with dimensionless couplings) to the‘Goldstino’ (the longitudinal component of the gravitino).As a result, in thismodel, there are diagrams at three loop order which involve only dimensionlesscouplings and give mass to squarks. These masses are of orderm2˜q ∼αsπ2 λ216π2M2int(6.1)Here λ describes the coupling of the axino multiplet to the goldstino multiplet.17

Unless λ is extremely small (λ < 10−6 or so), squarks will obtain unacceptablylarge masses.This problem appears quite general. In order to link supersymmetry breakingdirectly with the axion, the axion multiplet must couple to the goldstone multiplet.On the other hand, in order to have the correct coupling to F ˜F, the axion mustcouple to fields carrying color.

But this means that the hidden sector is not reallyhidden; while gauge fields may only couple to the hidden sector through loops,these couplings are not suppressed by factors of1MP . There is, of course, no prob-lem in simply introducing the axion multiplet separately, with no (dimensionless)couplings to the hidden sector.

However, in this case one needs some other way tounderstand the coincidence of scales. [16]AcknowledgementsWe wish to thank L. Hall and L. Randall for conversations about their work.We especially wish to thank Vadim Kaplunovsky for discussions of problems asso-ciated with anomalies, and for suggesting that various non-supersymmetric resultseven in apparently finite calculations could result from a need for regularization.This work supported in part by DOE contract DE-AC02-83ER40107.REFERENCES1.

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