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Physik Department, Technische Universit¨at M¨unchen

arXiv:hep-ph/9205229v1 22 May 1992SLAC–PUB–5810TUM-TH-144/92SNUTP 92-26April 1992T/ASAXINO MASS⋆E. J. ChunPhysik Department, Technische Universit¨at M¨unchenD-8046 Garching, GermanyJihn E. KimCenter for Theoretical Physics and Department of PhysicsSeoul National University, Seoul 151-742, KoreaandH.

P. Nilles#Stanford Linear Accelerator CenterStanford University, Stanford, California 94309ABSTRACTThe mass of the axino is computed in realistic supersymmetric extensions ofthe standard model. It is found to be strongly model dependent and can be as smallas a few keV but also as large as the gravitino mass.

Estimates of this mass canonly be believed once a careful analysis of the scalar potential has been performed.⋆Work supported by the Department of Energy, contract DE–AC03–76SF00515.# On leave of absence from Physik Department, Technische Universit¨at M¨unchen andMax-Planck Institut f¨ur Physik, P.O. Box 40 12 12, D-8000 M¨unchen.Work partiallysupported by Deutsche Forschungsgemeinschaft.

While the standard model of electroweak interactions is in complete agreementwith all experimental findings, still much effort has been devoted to constructvarious generalizations. Supersymmetric extensions have attracted much attentionin the last decade.

[1]Realistic models of this type consider local supersymmetry(supergravity), spontaneously broken in a so-called hidden sector at a mass scale ofMS ≈1011GeV. The induced SUSY-breaking mass scale in the observable sectoris given by a value of order of the gravitino mass m3/2 ∼M2S/MP , where MPdenotes the Planck mass.The breakdown scale of the weak interaction gaugesymmetry SU(2) × U(1) is then closely related to the scale of SUSY-breakdown inthe observable sector, thus explaining the smallness of MW compared to MP , oncewe understand the mechanism for the breakdown of supersymmetry.In the minimal supersymmetric extension of the standard model, however,there is one dimensionful parameter which is not proportional to m3/2, the Higgsmass term µH ¯H in the superpotential.

Such a term is allowed by supersymmetryand it remains to be understood why also µ should take values in the desiredenergy range.One suggestion to achieve this starts with the consideration ofthe axionic generalization of the standard model. [2] The decay constant Fa of theinvisible axion should lie in the range given by MS and it was argued that sucha coincidence cannot be accidental.

In fact, one can easily construct models, inwhich µ is generated dynamically to be in the range of the gravitino mass. [3]Of course, the prime motivation to consider the axionic generalization of thestandard model is the quest for a natural solution of the strong CP-problem.

Asupersymmetric realization of this mechanism is most easily achieved in a modelwith several Higgs supermultiplets. [4] Instead of just the pseudoscalar axion, sucha model now possesses a full axion supermultiplet.

This contains the axino, thefermionic partner of the axion, as well as the saxino, the scalar partner of the axino.Although these particles are very weakly interacting, they might nonetheless leadto important astrophysical and cosmological consequences. The stability of starsand the observed energy density of the universe, as is well known, restrict thedecay constant of the invisible axion to a small window.

[5]Furthermore, a very2

light saxino might give rise to new long range interactions which are incompatiblewith present observations.In a model of broken supersymmetry, however, onewould usually expect the saxino to receive a mass of order of the gravitino in the100 GeV to TeV range, avoiding such unpleasant phenomena as a fifth force.The possible consequences of the presence of an axino have not been consideredin detail until recently. [6] Given the weakness of its interactions, of course, we wouldrather expect to find only indirect manifestations of the existence of such a particle.In fact, up to now only the effects of the (possibly) stable axino on the total energydensity of the universe have been studied.

A stable axino in a certain mass rangecould lead to overcritical energy density, and the corresponding axion models aretherefore ruled out. If, on the other hand, the axino has a mass of a few keV,it is itself an interesting candidate[6] for a source of dark matter.

In fact, such aparticle is up to now the only well motivated candidate for so-called warm darkmatter. It still remains to be seen, however, whether warm dark matter can lead toa satisfactory cosmological model including questions about large scale structureformation.In any case, the existence of a (light) axino might have important consequencesin any supersymmetric extension of the standard model.

Many of these modelscontain discrete symmetries (R-parity in the simplest case) that allow only pairproduction of the new supersymmetric particles.In these cases, there exists alightest supersymmetric particle (called LSP) which is stable.In the minimalmodel one usually considers such weakly interacting massive particles (WIMPS,an example can be found in the photino) as a possible source of cold dark matter.In the presence of an axion supermultiplet, it could very well happen, that theaxino is the LSP and thus render the WIMP unstable, at least on cosmologicaltime scales.While other properties of the axino seem not to be so important for our dis-cussion, its mass is a crucial parameter and a careful analysis is required. We shallsee in the following that this value is strongly model dependent.

Before we dis-3

cuss these questions in detail, let us remark, however, that, in general supergravitymodels, there are some natural values such a mass can have. One of them could bethe mass of the gravitino that sets the scale of SUSY breakdown in the observablesector.

But this is not the only possibility. The axino could very well be muchlighter.

In fact, models based on supergravity contain a very small dimensionlessparameterη = MSMP≈10−8. (1)The gravitino mass is then given by m3/2 ∼η2MP and the natural values for themass of the axino at the tree level are given bym˜a ∼ηkMP(2)with k ≥2.It is interesting to observe, that in the case of k = 3 this mass is in the regionof 1 to 10 keV, leading to a critical mass density of the universe.

In models ofglobal supersymmetry one obtains similar estimates. Here η ∼MW /Fa where MWdenotes the scale of weak interaction breakdown.

[6] These values coincide, since MSand Fa are so close to each other.The task of determining the mass of the axino in a given model now boils downto the question about the power k appearing in (2). For large k, of course, alsoradiative corrections to m˜a have to be taken into account.Let us start our discussion in the framework of globally supersymmetric models.Although the construction of supersymmetric generalizations nowadays exclusivelyconsiders locally supersymmetric (supergravity) models, we can still learn a lotfrom the simpler models based on global SUSY.

In the present example we can seequite easily, why it makes sense to consider the possibility of a very small axinomass. In the case of unbroken supersymmetry, the whole axion supermultiplet willremain degenerate at the mass given by the anomaly, which we shall neglect inthe following.

Thus the mass of the axino and the saxino have to be proportional4

to the scale of SUSY breakdown represented by the vacuum expectation value ofan auxiliary field FG (this is the auxiliary field of the goldstino multiplet)†. Themass splitting of the chiral supermultiplet is determined by the coupling of itsmembers[1] to FG.

The axion is protected by a symmetry and does not receive amass in the presence of SUSY breakdown. The scalar saxino couples in generalto FG and will thus obtain a mass of the order g ⟨FG⟩, where g is the coupling tothe goldstino multiplet∗.

This is the reason why one usually assumes the saxinoto be heavy. In the case of the axino the situation is similar, but different.

Againits mass is determined by the coupling to FG, but the auxiliary field has canonicaldimension two. A mass term for the axino ˜a˜aFG is of dimension five and thereare no renormalizable contributions to the mass of the axino.

In a model with an(invisible) axion we have as additional dimensionful parameter the axion decayconstant Fa of order of 1011 GeV and we therefore expect a small axino massm˜a ∼FGFa as was demonstrated in ref. 7.It remains to be seen, how these results generalize once we consider modelsbased on supergravity.

The reason why one nowadays primarily considers thesemodels is the fact that in models based on spontaneously broken global SUSY auniversal mass shift for the scalar partners of quarks and leptons is not possible.We have mentioned that already in connection with the discussion of the massof the saxino. This fact holds for a large class of models and can be succinctlysummarized by the value of STrM2, the supertrace of the square of the massmatrix.

These results suggest that in realistic models the masses of the scalars,and thus also the mass of the saxino, are pushed up to a value beyond the reachof present experiments. In the case of the axino such a general statement cannotbe made.

The authors of ref. 6 assume (in order to avoid the murky depths ofsupergravity theory as they say), that the globally supersymmetric results carry† FG is in general a combination of the auxiliary fields of gauge and chiral supermultiplets.∗Actually, in many models based on global supersymmetry this coupling can be very smalland even vanish at tree level.

These vanishing scalar masses, however, are the reason, whyglobally supersymmetric models do not lead to a realistic generalization of the standardmodel. We shall come back to this point later.5

over to the supergravity case. We shall see in the following that, in general, suchan assumption is not necessarily correct.

A similar conclusion has been obtainedby Goto and Yamaguchi. [8] Their result seems to to imply, however, that a smallmass of the axino requires a special form of the kinetic terms.

We analyze this issuein a more general way and see that, independent of the choice of the kinetic terms,small (and also large) axino masses are possible, dependent on other properties ofthe theory. We also investigate the question of the axino mass in those modelsthat might be found as the low energy limit of string theory.The scalar sector of a supergravity theory is completely specified by the K¨ahlerpotential G(Φj, Φ∗j) where Φ collectively denotes the chiral superfields.

The scalarkinetic terms are given by the second derivative Gij = ∂2G/∂Φj∂Φ∗i and one oftensplits G(Φ, Φ∗) = K(Φ, Φ∗) + log |W(Φ)|2 where the superpotential W(Φ) is aholomorphic function of Φ. The scalar potential is given by[1]V = −exp G3 −Gi(G−1)ijGj.

(3)We are interested in the mass spectrum of the theory once supersymmetry is brokenspontaneously, which leads to a nontrivial value of the gravitino mass m23/2 =exp(G)†.Masses of the scalar particles can then be read offfrom the secondderivative of the potential at the minimum. For the fermions we obtainMij = exp(G/2)Gij + 13GiGj −Gk(G−1)kl Glij,(4)where we have removed the contribution to the mass of the gravitino.

We alsohave to respect the constraint from the anomalous U(1)-symmetryXiqiΦiGi −qiΦ∗i Gi= 0,(5)where qi is the U(1)PQ-charge of Φi.† We assume vanishing vacuum energy, thus ⟨V ⟩= ⟨Vi⟩= 0 at the minimum.6

Fields in the observable (hidden) sector shall be denoted by yi (zi), respectively,and we shall assume the superpotential to split: W(Φi) = h(zi) + g(yi). In ourexamples we use the well known case h(z) = m2(z + β) for simplicity.

Let us startour discussion with a special choice Gji = δji , usually referred to as minimal kineticterms. The scalar potential then readsV = exp(K/M2)|hz + z∗WM2 |2 + |gi + y∗i WM2 |2 −3|W|2M2.

(6)Within this framework Goto and Yamaguchi[8] have argued that the axino mass isas large as the gravitino mass. Let us see how this works using their superpotentialg1 = λ(AB−f2)Y, where f is a constant and A, B and Y are fields.

Minimizing (6)we find the following (approximate) vacuum expectation values (vevs): A = B ≈fand Y ≈m2/M, while the z vev remains undisturbed∗.Actual values for fand m should lie in the range of 1011 GeV. Denoting the fermions in the chiralsupermultiplets by χi, the axino is found to be the linear combination χa = (χA −χB)/√2 and it receives a mass of order of the gravitino mass m˜a ∼m3/2 ∼m2/M.Is this now a generic property of models with minimal kinetic terms?Weshall see that the answer is no by inspecting a second example with superpotentialg2 = λ(AB −X2)Y + λ′3 (X −f)3 including a new singlet chiral superfield X.Minimization of the potential now becomes more complicated since the condition⟨V ⟩= 0 leads to a shift in the vev of the hidden sector fields.

The easiest way todiscuss the potential is by expanding it in powers of m/M. To lowest order oneobtains the globally supersymmetric result for the observable sector.

In each orderone then has to adjust the vacuum energy to zero, and in the present examplethe inclusion of the terms of order m2/M2 require a shift of ⟨z⟩. In the previousexample it was sufficient to just consider the expansion up to first order.

One stillobtains vevs of A, B similar to those of the previous example, but the presence ofthe field X has important consequences on the axino mass; in fact here one obtainsm˜a ∼m3/M2.∗Observe that in the case of global supersymmetry the minimum is found at ⟨Y ⟩= 0.7

This example shows that the mass of the axino depends strongly on the modeland the special form of the superpotential.It also shows that in models withminimal kinetic terms the mass of the axino not necessarily needs to be as large asthe gravitino mass, contrary to the impression given in ref. 8.

In particular, massesof the axino in the range of a few keV can be obtained also in this framework.Let us next consider those supergravity models that have a structure similar tothose that appear in the low-energy limit of string theories. The K¨ahler potentialis given by[9]K = −log(S + S∗) −3 log(T + T ∗−CiC∗i ),(7)where S denotes the dilaton superfield, T represents the moduli and Ci the matterfields.We shall assume the superpotential of the form W = W(S) + W(Ci),postponing a discussion of the implications of moduli dependence.

The term W(S)is assumed to appear as a result of gaugino condensation in the underlying stringmodel, and is crucial for the process of supersymmetry breakdown. For a reviewand details see ref.

10. The scalar potential of the theory defined in this wayV = exp(G)|GS|2(S + S∗) + |Wi|2(8)is positive with a minimum at ⟨V ⟩= 0; the dilaton adjusts its vev to cancel anypossible contribution to the vacuum energy.Supersymmetry is broken sponta-neously through a nontrivial vev of the auxiliary field of the dilaton supermultipletFT ∼exp(G)GT, while FS = 0.

The only problem with the potential is the factthat the vevs of the moduli are not determined and thus the vacuum is highlydegenerate. Let us nonetheless discuss this simplified example first.

The minimumof (8) is found at ⟨GS⟩= ⟨Wi⟩= 0, where Wi = ∂W/∂Ci = 0 coincides with thesolution obtained in the case of global supersymmetry, independent of the specialform of the superpotential. In our case we require nontrivial vevs ⟨Ci⟩= vi for atleast one of the charged scalar fields.

The axino is then given by ˜a = Pi(qiviv )χi,where v =qPi q2i v2i should take a value of order of m in the 1011 GeV range. The8

goldstone fermion is given by η ∼GTχT + Giχi, with GT = −3/∆, Gi = 3vi/∆and ∆= T + T ∗−CiC∗i . One thus obtains P qiviGi = 0 for the axino to be or-thogonal to the goldstino.

Fermion masses can now be computed according to (4)in a straightforward manner. This gives e.g.

MT T ∼GT T + 13GT GT + 2∆GT = 0,since GT T = 3/∆2 and GT = −3/∆. Also the terms mixing T- and i-componentsvanish as well as Maj = Pi(qiviv )Mij because of the constraint (5).

Thus all thesefermions including the axino remain massless.One could have expected such a result from the outset because of the fact thatmodels with kinetic terms of the structure (7) are very closely related to globallysupersymmetric models. We have confirmed that above finding Wi = 0 at theminimum, the globally supersymmetric solution.

Thus one might obtain a lightaxino in a natural way. [8] But this is probably not the whole story.The otherfermions remain massless as well and, more importantly, also the scalar particleslike the saxino remain massless at tree level.

At the present stage of the discussionwe can conclude that this model not only shares the desirable features of globallysupersymmetric models but also the more problematic ones. Observe that in themodels based on minimal kinetic terms the scalar particles and thus also the saxinoreceived a large mass of order of m3/2.Again the question arises whether in models with K¨ahler potential as in (7) onealways obtains a light axino.

Unfortunately this question cannot yet be answereddefinitely. One way to proceed is to compute radiative corrections and see howaxino and saxino masses are shifted.

[8] We would like to argue, however, that thisis not necessarily the correct way to attack this problem. After all the potentialgiven in (8) has still a large vacuum degeneracy and many massless scalars andthus is unstable under small changes of the parameters.

In fact, naively includingradiative corrections might destabilize the potential in such a way that it becomesunbounded from below. As long as we do not know the correct position of theminimum we can not really be sure that our estimate of the axino mass is reliable.This can be demonstrated quite easily in the framework of explicit models.

Wehave seen this in our discussion of the models with minimal kinetic terms comparing9

those with superpotentials g1 and g2. Although there the potential is less unstablethe actual value of the axino mass strongly depends on details of the potential.Similar things will happen also in models with nonminimal kinetic terms.In addition we know that the potential as given in (8) is incomplete.

In a firststep one should include the moduli-dependent contributions to the superpotential.Unfortunately the incorporation of such a dependence in W leads to enormouscomplications. The potential is no longer positive definite and nobody succeededyet to find a satisfactory minimum with broken supersymmetry and a vanishingcosmological constant.

As long as such a result is missing, any reliable computationof the axino mass in such models is impossible. Unfortunately this is also true inthose models with a composite axino that constituted our prime motivation tostudy these questions in detail.

[11,12] This does not mean that the axino cannotbe light.In fact our discussion of the models with minimal kinetic terms hasdemonstrated that light axinos could actually exist. We want to stress here thatany estimate of axino masses is unreliable as long as a detailed calculation of theunderlying potential has not been performed.Acknowledgements:The work of E. J. Chun is supported by a KOSEF-fellowship.

J. E. Kim issupported by KOSEF through the Center for Theoretical Physics at Seoul NationalUniversity.10

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