arXiv:hep-ph/9905212v1 3 May 1999 un P)mcaim hc noe lbl chiral scale global, energy a invokes which U Peccei- mechanism, The (PQ) scale. Planck Quinn con- the naturally around physics be with can nected scale Fermi attractive the prob- most which [2–4]. the in CP axions perhaps framework as of strong considered partners widely is the supersymmetric SUSY are to and They [1] (SUSY) lem. solution supersymmetry Peccei-Quinn low-energy the involving els Introduction 1. 95.35 14.80.Ly, 14.80.-j, PACS: de critical the because with Universe comparable the often in is density mass relic missing their fo the candidate of component natural main a are neutralinos cold lightest the of decay Z es(eoe yatle fteeetial eta gauge neutral electrically the of bosons tilde) a by (denoted ners h lgts)neutralino (lightest) The [5]. DM ally cold from that of is a component density that mass-energy dynamical total suggest the the strongly to measured spectrum as contribution power well dominant as their structures models of large Current shape of formation candidate. the (DM) of matter dark tractive iyi h nvre ftecnrbto so re the order of is contribution the If den- stable density mass critical is Universe. relic LSP the the the in to sity that substantially fact contribute may the and is presence R-parity the unbroken in conse- cosmology of important for most supersymmetry the interesting of of an quences One are LSP. the neutral for mas- color candidate and being electrically light- Axinos, and the (LSP). sive is particle cosmology supersymmetric to and est importance searches particular literature experimental Of the both partners. in SUSY attention other less than much received have they problem. CP strong the solving of way elkonta h etaiosrlcdensity relic neutralino’s the that well-known e forder of ten nvreatemlpplto fnurlnsrmisin remains neutralinos of population thermal a Universe 12 1 ymtygopsotnosyboe tsm high some at broken spontaneously group symmetry (1) eso htaio rdcdi h al nvrei the in Universe early the in produced axinos that show We ntemnmlSS oe MS) h S susu- is LSP the (MSSM), model SUSY minimal the In xnsaetu eysrnl oiae.Dsiethis, Despite motivated. strongly very thus are Axinos W f akmte.W ru htaio a elpoiethe provide well may axinos that argue We matter. dark assumed 3 + B Z and 13 H f e ρ a ob h iheto h orneutralinos. four the of lightest the be to b 0 crit W ∼ ρ + xnsaepeitdt xs nmod- in exist to predicted are Axinos . crit 3 Z 10 thg eprtrsi h early the in temperatures high At . n ig bosons Higgs and , 14 uhapril scniee nat- an considered is particle a such , (2) 11 (3) H e (1) etrfrTertclPyis eu ainlUniversity National Seoul Physics, Theoretical for Center e ean h otcompelling most the remains GeV oe nttt o dacdSuy hogynr-og S Cheongryangri-dong, Study, Advanced for Institute Korea t 0 eateto hsc,LnatrUiest,LnatrLA Lancaster University, Lancaster Physics, of Department ftersetv emoi part- fermionic respective the of χ samixture a is ar Covi Laura H xnsa odDr Matter Dark Cold as Axinos b and 1 χ inE Kim E. Jihn , = H ρ Z u χ Spebr1 2018) 1, (September tis It . 11 sof- is nsity. B e + r 2 1 , 3 n ezkRoszkowski Leszek and , ls”teUies.Ti eursstsyn h condi- the re- satisfying “over- requires by not Ω This derived tion does Universe. been abundance the have neutralino close” the models that SUSY quiring of space eter [6,7]. candidate DM cold, or non-relativistic, always nteasneo hscniinoercvr robust a recovers scale. one grand-unified condition a bound this of independent at of model partners equal absence fermionic the are (the In bosons) gauginos gauge the the of bounds masses that these assumption the that (well-motivated) the remarking on worth depend is On strongly it [11]. hand, GeV 42 other around the already is minimal it the Con- as model, the In known supergravity in also example, [9], For (CMSSM) models [8]. MSSM higher. motivated, strained much MSSM be more can the bound perhaps the in and con- GeV restrictive, 28 limit more mass about neutralino above the LSP. pushed siderably, the now searches have Experimental is LEP justified. which at well neutralino, is axino, assumption lightest the This is the it than that assumes rather one if dramatically changes searches. including DM SUSY, and of collider studies future many and often present are in DM account and LSP into the taken as Cosmological neutralino the values. of acceptable the properties to deplete to density enough number efficient LSP be decoupling. to has at annihilation section The in cross and annihilation Universe expanding their the particular in ther- LSPs a of of population evolution mal the considering from comes condition n Ω ing oe teKV oe 3) h xn ascnarise can mass breaking SUSY axino a the with examples. level [3]), axion one-loop following model at quark interested the heavy KSVZ are the (the by of we model illustrated version which supersymmetric is range the This in GeV In it of imagine [12]. tens easily in to can one few theoretically the also but tally eauei yial ml oprdt h neutralino’s the to compared tem- mass, the small “freeze-out” typically from The is decouple [5]. perature they “freeze-out” than or Universe, smaller bath, the thermal becomes of matter annihi- expansion ordinary their the When into rate bath. lation thermal the with equilibrium m [5]. bound Weinberg oteHbl parameter Hubble the to e a aybud ntenurln asadteparam- the and mass neutralino the on bounds Many nti etrw ilso htti tnadparadigm standard this that show will we Letter this In ncnrs otenurln,tems fteaxino, the of mass the neutralino, the to contrast In ean o nyvrulyucntandexperimen- unconstrained virtually only not remains , χ χ T h h f 2 2 < ∼ ∼ < ∼ etaiovrino h ocle Lee- so-called the of version neutralino a - 1 m 5,weeΩ where [5], 1 χ eu 5-4,Korea 151-742, Seoul , / 0[] ei etaio r therefore are neutralinos Relic [5]. 20 ol1002 Korea 130-012, eoul Y,England 4YB, 1 1 H m χ 0 χ = 100 = > ∼ ρ χ e 1]fo requir- from [10] GeV 3 /ρ h crit mscMc This km/sec/Mpc. A and tr insertion -term h srelated is at the intermediate heavy squark line. Then, we expect equilibrium, will subsequently decay into the axino via, 2 2 ma (fQ/8π )A where fQ is the Yukawa coupling of the e.g., the process heavy∼ quark to a singlet field containing the axion. In χ aγ. (2) ae straightforward SUSY version of the DFSZ [4] model → axino mass is typically rather small, m (keV) as a This process was alreadye considered early on in has been pointed out in Ref. [13]. However,∼ O in addi- Ref. [16] (see also [13]) in the limit of a photino NLSP tion to the above inevitable contributionse to the KSVZ and only for both the photino and axino masses assumed and DFSZ axino masses, there can be other contributions to be very low, m 1 GeV and m 300eV, the for- from superpotentials involving singlet fields. For exam- γ ≤ a ≤ ple, if the PQ symmetry is assumed to be broken by mer bound now excluded by LEP searches. In that case, the renormalizable superpotential term (KSVZ or DFSZ the photino lifetimee was typicallye much larger than 1 2 second thus normally causing destruction of primordial models) W = fZ(S1S2 f ) where f is a coupling, and − a deuterium produced during nucleosynthesis by the ener- Z,S1 and S2 are chiral fields with PQ charges of 0, +1 and 1, respectively, then the axino mass can be at the getic photon. Avoiding this led to a lower fa-dependent soft− SUSY breaking mass scale. The axino mass arises bound on the mass of the photino in the MeV range [16]. In this Letter, we show that the conclusions and from the mass matrix of S˜1, S˜2, and Z˜, bounds of Refs. [16,13] can be evaded if one considers both the axino and the neutralino in the GeV mass 0, ma, ffa range. In this regime the neutralino decays into the ax-  ma, 0, ffa  . (1) ffa,e ffa, 0 ino typically well before nucleosynthesis thus avoiding the   e problems considered in Refs. [16,13]. The resulting non- Certainly, the tree level axino mass is zero if = 0. thermally produced axino will be a cold DM candidate. However, with soft terms included, there appears a linear 2. NLSP Freeze-Out. The effective coupling of the 2 2 2 2 term in Z, V = f ( S1 + S2 ) Z +(AfS1S2Z +h.c.); neutralino with the axino is very much weaker than | | | | | | | | thus is of order A/f, and the axino mass can that of its interactions with other matter. Therefore arise at the soft mass scale. A complete knowledge of the neutralino’s decoupling is not different from the case the superpotential is necessary to pin down the axino when it is the LSP. The freeze–out temperature Tf is mass [12,14]. Therefore, generically it is not unreasonable determined by the annihilation cross section σ(χχ to consider the axino mass scale of order tens of GeV. ordinary matter) and is normally well-approximated by→ < One severe bound of ma 2keV has been derived by iteratively solving the equation for xf = Tf /mχ Rajagopal, Turner and Wilczek∼ [13]. This bound arises from requiring that the “primordial”e axinos, produced 1 mχ 45xf = ln 3 MP σvrel (xf ) , (3) along with the axions in the very early Universe when xf 4π r NF h i  the PQ symmetry becomes broken around the scale fa 11 ∼ 19 10 GeV, do not contribute too much to the total relic where MP = 1.22 10 GeV, NF is the effective num- 2 × abundance of the Universe, Ω h < 1. As was noted in ber of relativistic degrees of freedom and σvrel is the a h i < averaged product of annihilation cross section and the Ref. [13], the bound ma 2 keV (which would make the axino a warm dark matter∼ candidate)e can be evaded by annihilating neutralinos’ relative velocity. As already assuming that, at temperaturese below fa, the Universe mentioned, the scaled freeze-out temperature xf is typi- cally very small (xf = (1/20)), justifying the iterative underwent a period of inflation and that the temperature O of subsequent reheating was sufficiently below fa. These procedure [5]. Without further decay, the neutralino co- assumptions are not too radical and have now become moving number density nχ would have remained basi- part of the standard cosmological lore [5]. The same way cally constant. out is also usually used to solve the analogous problem 3. Neutralino decay into axino. In the scenario consid- with primordial gravitinos [15]. ered here, the NLSP neutralino co-moving number den- Once the number density of the primordial axinos has sity after “freeze-out” will continue to decrease because been diluted by inflation, they can be again produced in of its decay (2) into the axino LSP. This is presented in Fig. 1. At T/mχ x < xf and for Γχ H, nχ is the decays of heavier particles [16]. Since axino’s cou- ≡ ≪ plings to matter are strongly suppressed by 1/fa, all roughly given by heavier SUSY partners first cascade-decay to the next- ′ xf ′ eq dx Γχ x to-lightest SUSY partner (NLSP). A natural candidate nχ(x) nχ (xf )C(x)exp ′3 h i (4) for the NLSP is the lightest neutralino. As stated above, ≃ − Zx x H(mχ) the neutralino “freezes-out” at Tf mχ/20. If it were ∼ where C(x) takes into account the temperature differ- the LSP, its co-moving number density after freeze-out ence between the photons and the decoupled neutralinos would remain nearly constant. In the scenario considered and Γχ x is the thermally averaged decay rate for the here, the neutralino, after decoupling from the thermal h i 2 neutralino at x, while H(mχ)=2π 2πNF /45 mχ/MP . p 2 where the phase space factors from Eq. (5) have been neglected. A comment is in order regarding a plausible range of fa [19]. A somewhat model dependent lower bound 9 fa > 10 GeV comes from astrophysical considerations, most∼ notably from requiring that axions do not overly affect processes in globular clusters, red giants and in su- pernova 1987A. An upper limit of 1012 GeV is quoted in the context of cold axion energy∼ density. A range 9−10 12 10 GeV < fa < 10 GeV gives a cosmologically in- teresting values∼ for∼ the relic density of axions. Hard photons produced in the χ decay thermalize via multiple scattering from background electrons and positrons (γ + e γ + γ + e) [20,16]. The process pro- ceeds rapidly for→ electromagnetic background tempera- tures above 1 MeV at which point background e e¯ pairs annihilate. To ensure efficient thermalization and− in or- der to avoid problems with photodestruction of light ele- ments produced during nucleosynthesis, we require that FIG. 1. A schematic behavior of the co-moving number the neutralino lifetime (6) is sufficiently less than about 1 density: the thermal equilibrium (thick solid), NLSP neu- second (which also coincides with temperatures of about tralino (dash) and LSP axino (thin solid). 1 MeV). A modest requirement τ < 10−1sec leads, in the case of the neutralino with a large∼ B-ino component (a The interaction of the axino and the gaug- neutralino is never a pure B-ino state), to an upper bound ino component of the neutralino is given by the on fa which depends on mχ. Also, at larger masses ad- ∗ ˙ √ α ¯ ¯α ¯¯ ditional decay channels open up, most notably χ Za. term αY CaY Y /(4 2πfa)[(ΦBαB )θθ + (Φ Bα˙ B )θθ]+ → α2C /(4√2πf )[B W3], where Φ is the chiral su- We can see that, for large enough values of the B-ino aW W a e permultiplet containing→ the axion and the axino, while mass, the decay (2) will almost entirely take place before nucleosynthesis. the vector multiplet B (W3) corresponds to the U(1)Y The case of higgsino-dominated neutralino as the (SU(2)L) gauge group with a coupling strength αY (α2). NLSP is probably less attractive and also more model The coefficients CaY Y and CaW W are model dependent. Usually, one performs chiral transformations so that dependent. First, the lifetime (6) is now be typically µν significantly larger, easily extending into the period of there is no axion–Wµν W˜ coupling. This is equivalent to giving vanishing Peccei-Quinn charges to left-handed nucleosynthesis and beyond. This is caused by the sup- pression of the B-ino component through which the decay doublets. In this case, CaW W = 0 and CaY Y = Caγγ. In c proceeds. (See the form of Caχγ below Eq. (5)). Much the DFSZ model with (d ,e)–unification CaY Y = 8/3, lower values of fa could be considered as a remedy or and in the KSVZ model for eQ = 0, 1/3, and 2/3 − much larger higgsino masses, and/or additional (model CaY Y = 0, 2/3 and 8/3, respectively [17]. Below the dependent) decay channels involving Higgs in the final QCD chiral symmetry breaking scale, Caγγ and CaY Y are reduced by 1.92. state. We first concentrate on the dominant decay channel (2) In the MSSM, the higgsino relic abundance in the mass range allowed by LEP is typically very small, thus lead- which is always allowed as long as ma < mχ. We will 2 comment on other channels below. The decay rate for ing to even smaller Ωah . One possibility would be to consider rather obese higgsino masses, above roughly the process (2) is given by e 2 500GeV, where Ωχh e> 1 again. A resulting value of 2 2 3 2 3 2 ∼ αemN 2 mχ ma˜ Ωah would then depend on the actual size of the hig- Γ= 3 Caχγ 2 1 2 , (5) gsino component in the decaying neutralino, as well as on 16π fa  − mχ  thee axion/axino model which would determine the cou- where αem is the electromagnetic coupling strength, plings of the decay channels to the Higgs final state. One Caχγ = (CaY Y / cos θW )Z11, and N is a model depen- could also allow for a Higgs singlet and assume that its dent factor (N = 1(6) for the KSVZ (DFSZ) model). fermionic partner is mostly the NLSP. In the theoretically most favored case of a nearly pure The resulting axino relic abundance today is simply B-ino [18,9], the neutralino lifetime can be written as given by

2 3 2 m 2 −2 1 fa/N 50 GeV a τ 3.3 10 sec (6) Ωah = Ωχh (7) ≃ × C2 1011 GeV  m  mχ aY Y χ e e 3 since all the neutralinos have decayed into axinos. The ACKNOWLEDGMENTS axino will normally be produced relativistic, except when the ratio of the neutralino-axino mass difference to the One of us (JEK) is supported in part by Korea Sci- axino mass is small, but will later redshift due to the ex- ence and Engineering Foundation, Ministry of Education pansion of the Universe and become cold by the time of through BSRI 98-2468, and Korea Research Foundation. matter dominance. It is worth noting that the neutrali- LR would like to acknowledge kind hospitality of KIAS nos will not dominate the energy density of the Universe (Korea Institute for Advanced Study) where part of the before decaying; to see this we have to compare its life- project was done, and to thank A. Masiero for helpful time, given by Eq. (6), with the time when the equality conversations. ρχ = ρrel takes place. This time is easily computed ne- glecting the decay and amounts to 106 107sec; we see therefore that the neutralinos never dominate− the energy density and matter domination starts only after the pro- duced axinos become non-relativistic. −6 With the lifetime (6) significantly larger than 10 sec 38 the neutralinos escape the high energy detectors. Thus [1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440 (1977); Phys. Rev. D16, 1791 (1977). phenomenology remains basically unchanged from the [2] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978); F. Wilczek, usual case where the neutralino is the LSP. In particu- Phys. Rev. Lett. 40, 279 (1978). lar, accelerator mass bounds on supersymmetric particles [3] J. E. Kim, Phys. Rev. Lett. 43, 103 (1979); M. A. Shif- apply. But cases previously excluded by the constraint man, V. I. Vainstein, and V.I. Zakharov, Nucl. Phys. 2 Ωχh < 1 can now be allowed via Eq. (7). This leads B166, 4933 (1980). to a possibly dramatic relaxation of the parameter space [4] M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. of SUSY models. For example, in the MSSM the region 104B, 99 (1981); A. P. Zhitnitskii, Sov. J. Nucl. Phys. of large higgsino masses mentioned above has been con- 31, 260 (1980). sidered cosmologically excluded but now can be again al- [5] E. W. Kolb and M. S. Turner, The Early Universe (Addison-Wesley, Redwood City, 1990). lowed if one takes a sufficiently small ratio ma/mχ. In the gaugino region it is normally reasonable to expect that, in [6] J. Ellis, J. S. Hagelin, D. V. Nanopoulos, K. Olive, and M. Srednicki, Nucl. Phys. B238, 453 (1984). order to reduce the LSP relic abundance below one [18], e [7] For a review, see G. Jungman, M. Kamionkowski, and at least one sfermion with mass roughly below 500 GeV K. Griest, Phys. Rep. 267, 195 (1996). should exist. In the CMSSM, the same requirement of- [8] For a recent review, see, S. Katsanevas, talk at the 2nd ten provides an upper bound on the common scalar and International Workshop on the Identification of Dark gaugino masses of order one TeV [9] over a large range Matter (IDM-98), Sheffield, England, September, 7 - 11, of parameters. Both bounds which hold for a gaugino- 1998. like LSP away from annihilation resonances can now be [9] G. L. Kane, C. Kolda, L. Roszkowski, and J. Wells, readily relaxed. Phys. Rev. D49, 6173 (1994); R. G. Roberts and B309 So far we have considered the neutralino as the NLSP. L. Roszkowski, Phys. Lett. , 329 (1993). D46 This assumption can easily be relaxed to accommodate [10] K. Griest and L. Roszkowski, Phys. Rev. , 3309 any other superpartner, either neutral or carrying an (1992). [11] J. Ellis, et al., Phys. Rev. D58, 095002 (1998). electric or color charge, provided that its effective cou- [12] E. J. Chun, J. E. Kim, and H. P. Nilles, Phys. Lett B287, pling with the axino is of order 1/fa. 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B260, 689 (1985). that, with or without its non-supersymmetric partner, [18] L. Roszkowski, Phys. Lett. B262, 59 (1991). the axino could dominate the matter in the Universe. [19] See, e.g., G.G. Raffelt, hep-ph/9806506. [20] D. A. Dicus, E. W. Kolb, and V. L. Teplitz, Astrophys. J. 221, 327 (1979).

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