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Axinos as dark matter particles

Laura Covi1,3 and Jihn E Kim2,3 1 Deutsches Elektronen SYnchrotron DESY, Notkestrasse 85, D-22603 Hamburg, Germany 2 Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea E-mail: [email protected] and [email protected]

New Journal of Physics 11 (2009) 105003 (15pp) Received 5 February 2009 Published 16 October 2009 Online at http://www.njp.org/ doi:10.1088/1367-2630/11/10/105003

Abstract. The identification of dark matter in our physics model is still an open question. Here, we argue that axinos can be successful dark matter candidates in models with supersymmetry and the axion solution of the strong CP problem. Axinos can be the lightest supersymmetric particle (LSP) or can be heavier than the LSP. Axinos can be produced at the right abundance by thermal scattering and, if they are the LSP, also by out of equilibrium decays of the lightest of SM fields (LSPSMs). On the other hand, heavier (not LSP) axinos can generate a part of the neutralino LSP dark matter. Depending on the nature of the supersymmetric spectrum, and if R-parity is strictly conserved or slightly broken, very different signals of the LSP axino scenario can arise in colliders and in astrophysics.

3 Authors to whom any correspondence should be addressed.

New Journal of Physics 11 (2009) 105003 1367-2630/09/105003+15$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2

Contents

1. Introduction 2 2. Axino models and axino mass 3 2.1. Axion models ...... 4 2.2. Axino, SUSY breaking and axino mass ...... 5

3. Axino cosmology with ma˜ < Mχ 6 3.1. Producing axinos in the early universe ...... 6

4. Axino cosmology with ma˜ > Mχ 8 5. Axino DM and R-parity breaking 10 6. The LSPSM and colliders 12 6.1. How to distinguish the LSP from LSPSM decay? ...... 12 7. Conclusions 13 Acknowledgments 13 References 13

1. Introduction

The nature of dark matter (DM) as a particle is still unknown today, since its main evidence relies only on the gravitational interaction and is universal. On the other hand, from some of the present data and numerical simulations of structure formation, we do know that it must be a neutral, cold, probably collisionless (i.e. quite weakly interacting) and very long- lived particle [1]. Unfortunately, a particle with these characteristics is not contained in the (SM) of : the only neutral stable and massive candidates, the electroweakly active , are so light that they are, at most, hot DM and therefore only a subdominant component. DM has therefore to be part of a larger picture and of any physics beyond the SM. Probably the best motivated models of this kind rely on supersymmetry (SUSY), which is a unique extension of the Poincaré algebra and calls for a doubling of all degrees of freedom with 1 spin difference 1s 2 [2]. In this context, it is then clear that more particles can be suitable DM candidates, if they= ± are the lightest ones and sufficiently long-lived, in particular the very well studied cases of the neutralino or the gravitino. But if we invoke the Peccei–Quinn (PQ) solution to the strong CP problem in supersymmetric models, a new multiplet has to be introduced, namely the axion multiplet [3, 4]. Such a multiplet must by its very nature interact with the SM particles, but the scale of its interaction is suppressed by the scale at which the PQ symmetry is broken, Fa. Therefore, the fermionic component of the multiplet is naturally a very weakly interacting particle and can easily be the lightest state of the spectrum, but it can also be heavy. We will present in this paper a summary of the axino cold DM (CDM) scenario [5]–[11] and explore the implications of axino DM for the phenomenology of supersymmetric models and for future indirect detection of DM.

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 3

2. Axino models and axino mass

In discussing axino models, one should refer to the corresponding axion models. So, let us start with the axion shift symmetry and the reparametrization invariance as discussed in [12]. The PQ solution of the strong CP problem [13] requires the introduction of the axion a, which renders the θ parameter4 dynamical and allows it to relax to zero after the QCD phase transition. An axion a is a pseudoscalar coupling to the anomaly as αsa µν Lθ Gµν G , (1) = 8π Fa ˜ µν where the dual field strength is G 1 µνρσ G , without any other interaction term in the ˜ 2 ρσ potential V . Below the QCD chiral symmetry= breaking scale, an axion potential is developed, which arises purely from integrating out the strongly interacting fields with that anomaly term. If the original potential contains other axion-dependent terms, they should be extremely small and not affect the position of the minimum, such that the axion vaccuum expectation value 11 (VEV), θ a /Fa, should be extremely small, <10− . Let| us| ∼ now h i focus on the QCD interactions containing the axionic degree of freedom. The axion effective Lagrangian below the chiral symmetry breaking scale reads 1 1 L f 2∂µθ∂ θ Ga Gaµν + (q iD/q + q iD/q ) + c (∂ θ)qγ µγ q θ,eff S µ 2 µν L L R R 1 µ 5 = 2 − 4gs ¯ ¯ ¯ θ aµν ic2θ  a qL mqR e + h.c. + c G G (or L ) − ¯ 3 32π 2 µν ˜ det θ iµν + c F i F + L (c`, c`), (2) θγ γ 32π 2 em,µν ˜ em ,θ 1 2 where θ a/fS with the axion decay constant fS defined up to the domain wall number = ( fS N Fa) and q are the SU(3)c charge-carrying fields. The c term is the derivative = DW 1 coupling with respecting the PQ shift symmetry, the c2 term is related to the phase in the quark mass matrix, and the c3 term is the anomalous coupling or the determinental interaction Ldet. The Lagrangian term Lleptons,θ is the axion interaction with leptons, which in principle can ` ` contain other constants c1, c2. The coupling constants c1, c2 and c3 are obtained below the axion scale fS after integrating out the heavy degrees of freedom responsible for the PQ symmetry breaking. The mass parameter m is defined to be real and positive below the electroweak (EW) scale. The Lagrangian (2) has a shift symmetry a a+ (constant), which reparametrizes the → iαa(x) couplings c1, c2 and c3. Explicitly, changing the phases of the quark fields qL e qL iαa(x) → and qR e− qR, we obtain the following reparametrization, where the effective one point irreducible→ action 0 [a(x), Aa (x) c , c , c , m, 3 ] changes to 1PI µ ; 1 2 3 QCD 0 [a(x), Aa (x) c α, c 2α, c + 2α, m, 3 ]. (3) 1PI µ ; 1 − 2 − 3 QCD So we see immediately, from this transformation for a single quark, that if it is massless, the corresponding c1, c2 parameters disappear and we can shift away the anomaly term completely with no physical effect. This is in fact one alternative solution to the strong CP problem; see [12] for a detailed discussion. For determining the axion mass, all c1, c2 and c3 terms may be relevant, but only the combination c2 + c3 actually appears [12]. Usually, in the field theoretic axion models, we start

4 Below, θ denotes the conventional θ θ0 + Arg Det mq . ¯ = New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 4 with c1 0. In any case, note that the c1 term can be reabsorbed in the c2 term using integration by parts= and the quark equations of motion. So, in the next sections we just start with the couplings c2 and c3. Usually, F is defined by transferring all couplings of the axion to the coefficient of GG and a ˜ rescaling c3 to one. On the other hand, fS is defined to be the VEV of the singlet field σ breaking the PQ symmetry. It turns out that c2 + c3 is an integer, not necessarily one in the pseudoscalar field space, and it determines the number of minima in the axion periodic potential. Thus, this integer is called the domain wall number NDW [14]: N c + c Tr Q (ψ )`(ψ ), (4) DW = | 2 3| = PQ colored colored where the trace is taken over all colored ψcolored, ` is the index of their SU(3)c representation and the PQ charge QPQ is given for the left-handed chiral representations. The scale Fa is constrained by astrophysical and cosmological bounds to lie in the narrow 10 12 axion window 10 GeV < Fa < 10 GeV [12]. Note that above the EW symmetry breaking scale, in principle, couplings to the EW gauge and the Higgs fields may also arise and we have to write then the effective Lagrangian as 1 1 L f 2∂µθ∂ θ Ga Gaµν + (q iD/q + q iD/q ) θ,eff > EW S µ 2 µν L L R R = 2 − 4gs ¯ ¯ µ µ µ + ∂ h∗∂ h I + V (h I , θ) + c h (∂ θ)(h∗∂ h I ∂ h∗h I ) µ I 1, I µ I − I µ q ic2θ  + c (∂ θ)qγ γ q Y qL h I qR e + h.c. 1 µ ¯ 5 − I ¯ θ aµν θ µν + c Ga G + c B B 3 32π 2 µν ˜ 3,Y 32π 2 Y,µν ˜ Y θ iµν + c W i W + L (c`, c`). (5) 3,EW 32π 2 SU(2),µν ˜ SU(2) leptons,θ 1 2 Then we can define an extended shift symmetry, including also transformations of the EW charged fields; these can be changed independently to the colored degrees of freedom, in such a way as to have c3,EW 0 and leave only the anomalous coupling to the hypercharge gauge bosons. As in the case of= QCD, such a coupling could be shifted away completely if one of the leptons were massless; since the mass is quite small, the residual effects, contained in ` the c1,2 terms, is negligible for many practical purposes.

2.1. Axion models

There are several types of c2 and c3 couplings that define different axion models. If c2 0 and c 0 due to the existence of PQ charge-carrying heavy quarks, the model is called= 3 6= the Kim–Shifman–Vainshtein–Zakharov (KSVZ) model [15]. If the coupling c2 is provided by the EW-scale Higgs doublets, while c 0, it is the Peccei–Quinn–Weinberg–Wilczek 3 = (PQWW) model [16]. If the phase c2 is provided by an EW-singlet with c3 0, it is the Dine–Fischler–Srednicki–Zhitnitski (DFSZ) model [17]. The model-independent= axion in superstring models gives instead c2 0 and c3 1. These values enable us to write down the axion––nucleon couplings unambiguously= = for each model [12]. But in general axion models may contain both c2 and c3 with c2 + c3 0 [12] and may have the family dependences of the variant axion [18] or invisible axion [19].6=

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 5 2.2. Axino, SUSY breaking and axino mass In the case of a supersymmetric model, the axion field is the pseudoscalar part of a whole chiral multiplet 8. Note, however, that the reparametrization invariance equation (2) still holds and represents a freedom in choosing the c1, c2 and c3 terms. We choose here the basis where the c2 term is transferred to the c3 term, and hence the 8 interaction is Z 2 αs α d ϑ 8aW Wα + h.c., (6) 4√2π Fa where now 8a (s + ia)/√2 + ϑa + (F term) is the chiral multiplet containing the s and = ˜ axion a and their fermionic partner the axino a, while Wα is the vector multiplet containing the α ˜ a m a 1 a aµν i a aµν and the gluon field strength, and W Wα ϑϑ 2iλ σ ∂mλ 2 Gµν G + 2 Gµν G + 2 | = − ¯ − ˜ D . Here, αs is the QCD coupling constant. An analogous interaction is present for the hypercharge gauge multiplet with the additional coupling c3,Y as discussed previously. As long as SUSY is unbroken, the axion multiplet remains light, since it is protected by the U(1)PQ symmetry [20]–[22]. This symmetry implies that no supersymmetric mass parameter is allowed for the axion multiplet since, as discussed above, the axion does not have a potential V (i.e. terms in the superpotential W with SUSY). Both saxion and axino masses are split from the almost vanishing axion mass if SUSY is broken, either at tree level via the VEV of some scalar field in the model and mixing with the other or via loop diagrams involving multiplets with split masses. The precise value of the axino mass depends on the model, specified by the SUSY breaking sector and the mediation sector to the axion . Most probably, the saxion mass is around the soft mass scale MSUSY. The axino mass should also be near this scale as well. But the axino mass can also be much smaller than MSUSY [4, 20, 21] or much larger than MSUSY [22]. Therefore, we take the axino mass as a free parameter. If R-parity is not conserved, the lightest supersymmetric partner of the SM particles can decay to ordinary particles. If R-parity is conserved, it cannot decay to ordinary SM particles, but it can decay to axino and/or gravitino if they are lighter. Thus, the axino cosmology depends crucially on the R-parity realization. Here, we consider first models with R-parity conservation and the thermal history of the universe can be very different depending on the hierarchy between the axino mass and the mass of the lightest supersymmetric particle (LSP) (of SM multiplets) Mχ . Firstly, we consider the case ma < Mχ and next ma > Mχ . The cosmology of a weakly interacting˜ massive˜ particle (WIMP) and an extra WIMP depends on several temperatures. For example, the neutralino cosmology depends on the neutralino freeze-out temperature [23, 24] and the gravitino/axino cosmology depends on the reheating temperature after inflation [25, 26]. We therefore define the following temperatures relevant for the axino cosmology:

Ta dcp axino decoupling temperature, ˜− = T reheating temperature after inflation, R = T neutralino freeze-out temperature, (7) fr = Ta rad axino-radiation equality temperature, ˜ = = T radiation temperature at χ or a decay, D = ˜ where note that TD corresponds to a different temperature for ma < Mχ and ma > Mχ . ˜ ˜

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 6

3. Axino cosmology with ma˜ < Mχ

Let us consider the axion supermultiplet together with the minimal supersymmetric SM (MSSM) fields. Then the LSP in the MSSM (LSPSM) χ plays an important role. The decoupling temperature of the axino supermultiplet is of the order [27]  F 2 0.13 T 1011 GeV a . (8) a dcp 12 ˜− = 10 GeV αs Cosmology with the saxion s is a simple extension of the standard cosmology if the saxion mass is around the SUSY breaking scale [28] or larger [29], but its effect is not so dramatic as the effect of the axino. If the axion interaction ever was in thermal equilibrium, e.g. TR > Ta dcp, a substantial axino number density survives to the present day and axinos have to be very˜− light. Axinos with mass in the eV range from this epoch have been considered as hot DM [20] or warm DM for masses in the keV range [27]. In the gauge-mediated SUSY breaking (GMSB) scenario, the gravitino is probably the LSP and the possibility of primordial axinos decaying to has been considered as well [30]. Here let us focus on the CDM axino LSP scenario.

3.1. Producing axinos in the early universe We briefly review here the two main mechanisms that produce axinos in the early universe. In principle, also other sources could be present like Q-balls decay [31]. We concentrate here on the hadronic type of axion models and define the axion supermultiplet in the basis where the c2 term is zero. Then the axino does not interact directly with the MSSM multiplets apart from the gluon and hypercharge vector multiplets and does not mix substantially with the standard neutralinos or other fermions. So we neglect any interaction with the leptons or the EW gauge bosons that may appear in the DFSZ type of models, and that can only increase the production ` cross-section. It is worth recalling here that the shift between c2, c2 and c3, c3,EW couplings is simply a matter of definition of the axion interactions as long as it is nearly a mass eigenstate ` and if we choose c2, c3,EW 0, the other coupling c2 is suppressed by the leptons Yukawa and therefore negligible for axino= production. We also do not consider here more complicated PQ sectors with many fields [32] or giving rise as well to thermal inflation [33].

3.1.1. Thermal scatterings. Any particle, even very weakly coupled, is produced in the thermal plasma by scattering of the particles that are in thermal equilibrium. As we have seen, axinos couple directly to the and via the ‘anomaly’ coupling in equation (6), i.e. in components

αs µν b b Lagg aγ5σ λ Gµν, (9) ˜ ˜ = 8π Fa ¯˜ where λb is the gluino field. So many scatterings in the primordial plasma involving colored particles produce axinos5. The axino number density is given by solving a Boltzmann equation of the type

dna X X ˜ + 3Hna σ (i + j a + )vrel ni n j + 0(i a + ) ni , (10) dt ˜ = h → ˜ ··· i h → ˜ ··· i i j i

5 The same happens in the case of the gravitino, but with a different vertex structure and scale [34].

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 7

109 108 EXCLUDED 7 10 TP 2 (Ωa˜ h > 1) 106

5 (GeV) 10 TP 2 (Ωa˜ h =1) R 104 T 103 102 ΩNT P h2 1 Hot Warm Cold a˜ 101 ≈ TF

8 6 4 2 2 10− 10− 10− 10− 1 10 ma˜ (GeV)

Figure 1. Maximal reheating temperature as a function of the axino mass obtained by requiring that the axino energy density is below the present DM density [6]. The difference between solid and dashed lines is due to the inclusion of the decay term in the Boltzmann equation (10). In the yellow area we expect the non-thermal production via out of equilibrium decays to be also substantial. where we are neglecting back-reactions, which are suppressed by na ni . At high temperature the two-body scatterings dominate the RHS, since they contain a vertex˜  given by the dimension 5 operator in equation (6) and show a characteristic linear dependence on T . So most of the axinos are produced at the highest temperature, and the axino number density is proportional to that temperature, which we take to be TR. Some of the two-body scatterings are IR divergent due to the massless gluon propagator; in the thermal bath such a divergence is screened by the presence of a thermal gluon mass gT . In our computation we introduced such an IR cut-off by hand [6]. A self-consistent procedure' is instead to perform a full resummation of the hard thermal loops as done in [35]. In general we expect O(1) corrections from higher order terms in αs, especially at low temperature [35]. There as well, the decay terms start dominating and the number density is no more proportional to the reheating temperature, but depends instead on the supersymmetric spectrum, in particular the gluino and squark masses [7]. Using the expression for the present axino energy density as  2  na(T ) ah ma ˜ 0.72 eV ˜ , (11) ˜ s(T ) = 0.2 3 T  3 where s(T ) 2.89 10 2.726 K cm− is the present entropy density, we can then obtain a bound on the= reheating× temperature in figure 1. 3.1.2. Out of equilibrium decays. An axino population is also generated by the LSPSM 6 decay after it freezes out from the thermal bath. The heavier cascade-decay quickly

6 In passing, we point out that the gravitino problem [25] is absent if the gravitino is the NLSP, ma < m3/2 < mχ , ˜ since a thermally produced gravitino would decay into an axino and an axion, which do not affect the Big Bang Nucleosynthesis (BBN)-produced light elements [36].

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 8 into the LSPSM (or very rarely to the axino LSP itself as we discussed above) while still in equilibrium, but the LSPSM has a lifetime much longer than its freeze-out time: since the axino 11 couplings are suppressed by the PQ scale Fa 10 GeV, the LSPSM lifetime for 100 GeV mass is of the order of seconds. Then the freeze-out' process is unaffected since the decay takes place only much later. In this case, thanks to R-parity conservation, the axino energy density can be directly computed from the LSPSM would-be-relic density as

NT ma a ˜ LSPSM. (12) ˜ = mLSPSM If the mass ratio is not too small, we still have a connection with the classical WIMP mechanism in case the LSPSM is a neutralino. On the other hand, in this scenario, the LSPSM can be more generally any superpartner that may freeze out with a sufficiently large number density. A couple of problems can arise if the LSPSM decay happens too late: 1. BBN can be spoiled by the energetic ‘active’ particles produced in the decay along with the axino: the strong limits on the injection of energetic particles depend on the electromagnetic/hadronic nature of the produced showers, the LSPSM number density and its decay time [26]. In general such limits are weak for the axino case since the LSPSM lifetime (excluding a strong mass degeneracy) is below 102 s, but they can affect the region of small mass for both the neutralino and stau LSPSM [6, 7]. These constraints disappear easily for colored LSPSM like the stop since the number density is reduced also by the Sommerfeld enhancement and the lifetime is very short as long as the decay to top is allowed [11]. Therefore a stop LSPSM is perfectly viable, but a very large stop mass and axino mass of a few TeV is needed to produce the whole DM density. 2. Are axinos from the decay cold enough to be CDM? They may be relativistic at production even if the LSPSM is not, and in general they have a non-thermal spectrum. Their velocity can be estimated as the following:  1/3 p(T ) mLSPSM g (T ) T v(T ) ∗ , (13) = ma ' 2ma g (TD) TD ˜ ˜ ∗ where TD is the temperature of the LSPSM decay time. Axinos must therefore have sufficient time to cool down before structure formation begins. In [37], such constraints have been studied and the conclusion is that an axino mass of at least 1 GeV is probably needed if the whole DM population is produced by out of equilibrium decay of an LSPSM of 100 GeV mass.

Depending on the parameters and TR, either production mechanism can give sufficient axinos to explain the present DM density. Once further information about the SUSY spectrum is available from the Large Collider (LHC), it may be possible to determine which contribution dominates and restrict the range of TR [38]. Of course another possibility is that the axino is so light as to be a subdominant (warm or hot) DM component. In the last case in our scenario, the axion [12] could be the DM.

4. Axino cosmology with ma˜ > Mχ

Now, let us consider the axino mass region, ma > Mχ . Here, we are interested in the case where the CDM density is determined by the axino˜ and in particular the axino energy density

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 9

~

11 Figure 2. The TR versus ma plot for mχ 100 GeV and Fa 10 GeV. ˜ = = is dominating the evolution history of the Universe. This is possible not only in the near past if the axino has not decayed yet and is DM as discussed in section 3, but also if a heavy axino decayed into the DM at an earlier epoch as will be considered below. Also for heavy axinos, the axino density before decay can be estimated from Ta dcp or TR as discussed above. Even in the second case, when the axion coupling never was in equilibrium,˜− an early cold axino DM domination may have happened if the number density was sufficiently min large, i.e. if TR was larger than TR defined by the equality of axino and radiation energy density at decay:

4 min maYa(TR ) TD. (14) 3 ˜ ˜ = min So for any TR > TR , axinos dominate the evolution of the universe before they decay and produce a non-negligible amount of entropy diluting the existing number densities. We recall here that in SUSY theories, we must always consider a relatively small reheating temperature of 107–108 GeV due to the gravitino problem [25, 26]. The heavy axino cosmology must also satisfy this upper bound on the reheating temperature. Heavy axinos cannot be the LSP and decay to the LSP plus light SM particles. This possibility was considered briefly in studying cosmological effects of the saxion in [29], and a more complete cosmological analysis has been presented in [10], which will be sketched here. Here the axino or the axino-decay produced neutralino is supposed to constitute the CDM fraction  0.23 of the universe. CDM ' For a heavy axino decaying to a neutralino, we present a TR versus ma plot for Fa 11 ˜ = 10 GeV in figure 2. For other parameters, we refer the reader to [10]. The region TR > Ta dcp is above the dashed blue line. The axino lifetime is greater than 0.1 s in the red shaded region˜− in

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 10 the LHS and there BBN constraints may apply. The blue shaded region in the RHS is where the axino decays before the neutralino decouples (TD > Tfr). The magenta lines (horizontal) are the contours of the entropy increase due to the axino decay, r S f /S . Above the r 1 line, axinos ≡ 0 = dominate the universe before they decay. The green lines (vertical) show the values of σannvrel , 2 h i where σann is the neutralino annihilation cross section (in units of GeV− ), which are needed to give the right amount of neutralino relic density after Tfr. In figure 2, we use neutralino and gluino masses as mχ 100 GeV and mg 2 TeV, respectively. For a larger Fa and a heavier neutralino mass, the green= lines move to˜ the= right [10].

5. Axino DM and R-parity breaking

The axino is a good DM candidate even if R-parity is not exactly conserved. In fact its lifetime can be very long, thanks to the PQ scale suppression and a small R-parity breaking. Axino CDM with R-parity breaking has been considered in [39] and subsequently in the context of the Integral anomaly in [40, 41]. Different decay channels and lifetimes are possible for the axino CDM depending on the R-parity breaking model and the axion model. If we consider bilinear R-parity breaking of the form [42]7

WR i µLi Hu, (15) / = where µ is the Higgs bilinear term, and the corresponding SUSY breaking soft term

L R Bi µLi Hu, (16) soft / = ˜ and restrict ourselves to hadronic axion models, the axino decay arises from the anomaly coupling with the hypercharge vector multiplet and the neutralino– mixing generated by the sneutrino VEV, νi . The decay rate then reads h ˜ i C 2 α2 m3 aγ γ em a 2 2 ˜ 0a γ νi ξi Uγ Z , (17) 3 2 ˜ ˜→ = 2(4π) Fa | ˜ | where ξi νi /v with the Higgs VEV given by v 174 GeV and = h ˜ i = SZα S∗ X ˜ γ α Uγ Z MZ ˜ (18) ˜ ˜ = m α χα is the –Zino mixing parameter, containing the neutralino mixing matrix S and the mass eigenvalues mχ α as in the case of the gravitino [44]. The lifetime is then given by  U  2 3  2 25 ξi γ Z −  ma − Fa τ 4 10 s | ˜ ˜ | ˜ . (19) a γ ν 10 11 ˜→ = × 10− 1 GeV 10 GeV For larger axino masses, also the decay into Z bosons opens up, and quickly dominates since it is proportional to the UZ Z mixing, which can be as large as one. The decay into W instead does not appear since we have˜ ˜ shifted away the SU(2) anomaly vertex. Note that the constraints from the diffuse gamma-ray background require a very small value for the parameters ξi and therefore a very small R-parity breaking, smaller than in the case of the gravitino DM [45]. In fact the EGRET diffuse flux already limits the lifetime of a DM particle decaying into a single gamma line to be larger than τ > 7 1026 s [46], between 0.1 and 10 GeV, with very weak dependence × 7 For a review see [43].

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 11 on the mass scale, while for lower masses bounds of the order of 1027 s are obtained from x-ray data [47]. If the axion model is instead of the DFSZ type, direct couplings with the leptons and Higgses arise from the µ or i µ terms. Then a direct mixing between axino–neutrino and axino– appears and other channels open up, in particular the three-body decay into + 8 neutrino and ` `− via intermediate Z, which could contribute to the electron flux . The mixing of the axino with leptons may arise actually also in the KSVZ models at the one-loop level [8]. Taking the mixing between the axino and Higgs multiplet typically as v/Fa and the Higgsino mixing to neutrino from the sneutrino VEV as above, we obtain for this channel

2 5 2 2 2 G F ma UH Z ξi v 0 + ˜ ˜ ˜ a e e−νi 3 | |2 ˜→ = 192π Fa 5   2  2 26 1  ma  Fa − UH Z ξi (10 s)− ˜ ˜ ˜ , (20) 11 | 12| ∼ 10 GeV 10 GeV 10− where GF is the Fermi constant and UH Z the mixing between Higgsino and Zino neutralino ˜ ˜ defined, in an analogous way, as Uγ Z . The branching ratios in the different leptons and quarks are determined by the axino mass˜ and˜ the Z couplings. In general the same decay can also arise via the R-parity breaking leptonic trilinear coupling λLLE c, and then the decay rate is given by

2 2 5 ζe λ ma 0a e+e ν | | ˜ ˜→ − = 24(8π)2 F 2m2 a eR ˜ 5 2   2  2 25 1  ma   meR − Fa − ζe λ (10 s)− ˜ ˜ , (21) 11 | 12| = 10 GeV 100 GeV 10 GeV 10− in the limit meR ma, where we have taken ζemeR /Fa as the effective coupling of the axino to the electron multiplet.˜  ˜ Which of the two leptonic˜ diagrams dominate depends on the R-parity breaking and axion model parameters. Recently, the PAMELA satellite experiment reported a significant excess [48], but no excess [49]. If this result is confirmed by another independent experiment, a vast unknown realm of the CDM cosmology will open up, because it is generally very contrived to build SUSY models producing excess , but no excess . It is clear from the above discussion, however, that the decaying axino could be a possibility, since in that case the radiative or leptonic decay channels may be preferred over the hadronic ones9. In fact, in the bilinear R-parity violating case, the two-body decays into We+ is either not open (for the hadronic axion models) or may be suppressed and therefore the antiproton flux from + W fragmentations disappears leaving only the Z contribution. The direct channel into e e−ν may be dominant for models where the sneutrino VEV, i.e. the bilinear R-parity breaking, is suppressed and the trilinear R-parity violating couplings give the dominant decay. + The model-independent case of a decaying into e e−ν has been recently studied in the context of the PAMELA anomaly in [51], and there it was shown that such a decaying DM particle may be a good fit to the data for a lifetime of about 1026 s and a mass above 300 GeV.

8 Note that also the diagram with off-shell splitting into an electron position pair generate this channel, but it is then subleading in comparison with the γ ν two-body channel due to αem and phase-space suppression. 9 The leptophilic coupling for the axino was also observed in flipped SU(5) models [19, 50].

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 12

The axino CDM could be a realization of this scenario with the appropriate choice of parameters. In that case, however, the axino has to be pretty heavy and therefore the reheating temperature very low. Even larger masses are probably needed for trying to accommodate the ATIC anomaly as well [52]. Note, however, that the PAMELA excess could be also due to astrophysical sources like pulsars [53] and then the PAMELA data give only a bound on the axino lifetime and R-parity breaking parameters. In that case also the possibility of a heavy axino ma > Mχ with a neutralino DM, which cannot fit very easily the PAMELA anomaly, is not ruled out˜ either.

6. The LSPSM and colliders

The signal of axino DM at colliders depends strongly on the nature of the LSPSM, which in turn depends on the SUSY breaking mechanism. In the constrained MSSM, where all the SUSY breaking parameters are derived by two common mass parameters, m0, m1/2, and a common trilinear coupling A at the unification scale, the value of tan β and the sign of µ, the only allowed LSPSMs are the lightest neutralino and the stau. In more general SUSY breaking models, of course other LSPSMs are allowed, in particular in the case of nonuniversal Higgs scalar masses, the stop or the sneutrino. If the neutralino or the sneutrino are the LSPSM, it will be difficult to disentangle the two and prove that they are not DM. In both cases it would be necessary to measure their mass and couplings and realize that those parameters either give a too large DM energy density or are already excluded by direct DM searches. Then we would have good reasons to imply that the neutralino or sneutrino must be unstable on cosmological timescales, but it will be very difficult to determine what they are decaying into and if that includes the axino. Other, more indirect, collider signatures may arise in models with axino DM and SO(10) Yukawa unification [54]. If the stau (or another charged sparticle) is the LSPSM instead, we will have the striking signal of an apparently stable charged heavy particle in the detector. In that case it will be clear that the LSP must be a very weakly interacting particle or that R-parity is violated, but we will need to measure and study the LSPSM decay to distinguish the two possibilities and identify if there is a DM candidate and what kind of particle it is. Unfortunately the astrophysical constraints on the R-parity violation scenarios discussed in the previous section seem to point to quite a long LSPSM lifetime, if the axinos are DM, and the decay would mostly happen outside the detector.

6.1. How to distinguish the LSP from LSPSM decay? The LSPSM decay can give information on the scenario and on the nature of the LSP, even if the LSP is not detected. In fact, the decay time and the branching ratios are model-dependent and vary substantially e.g. between R-parity-conserving and R-parity-violating scenarios. In the first case, we expect that the dominant decay is the two-body channel into the LSPSM partner and the axino, while the next open channel the subleading radiative decay with an additional photon in the final state. If instead R-parity is violated, the LSPSM decays completely into SM particles with no missing energy apart for the light neutrinos. So for the case of a stau LSPSM, we have τ τ a, τ a γ R-parity conserved (22) ˜ → ˜ ˜ ; c c τ τ ν , µ ν , b t (b b W −) R-parity violated (23) ˜ → µ τ ;

New Journal of Physics 11 (2009) 105003 (http://www.njp.org/) 13 therefore the R-parity violation case should clearly be visible via the large number breaking, since e.g. the τ and µ final states arise from the same trilinear coupling, as well as from the hadronic channel [55]. Moreover, the angular distribution of the radiative decays into a photon, an SM particle and missing energy contains, in general, information on the spin of the LSP and the interaction vertex structure. This quantity can indeed play a key role, in particular, in distinguishing between the axino and gravitino LSP, which can give rise to similar NLSP lifetimes and similar ‘visible’ decay channels [9, 56]. In that particular case, we will need to measure the branching ratio and the angular dependence of the radiative decay in order to reach a definitive identification [9].

7. Conclusions

We have discussed here different cosmological scenarios where the axinos play an important role in the DM question. If they are light, with masses in the MeV–GeV range, they can be the CDM if the reheating temperature is low and they are the LSP. In that case they can remain DM even if R-parity is broken, but the breaking has to be suppressed. If instead axinos are heavy and not the LSP, they can still produce the necessary neutralino LSP abundance in their decay and dilute dangerous relics. In general, the presence of an axino LSP and DM relaxes many of the bounds on the supersymmetric parameters, since the right number density of axinos can be obtained in a wider region of parameter space. Moreover, the possibility of different LSPSMs and therefore very different collider signatures arises. We expect that the LHC will soon clarify the situation. In the case of R-parity violation also astrophysical signatures could arise, but they are unfortunately strongly dependent on the axion model realization.

Acknowledgments

We thank C Berger, A Brandenburg, K-Y Choi, K Hamaguchi, HB Kim, S Kraml, B Kyae, F Palorini, R Ruiz de Austri, M Small, F D Steffen and in particular L Roszkowski for several years of fruitful and exciting collaboration. LC acknowledges support from the ‘Impuls- und Vernetzungsfond’ of the Helmholtz Association under contract number VH-NG-006 and from the European Network of Theoretical Astroparticle Physics ILIAS/N6 under contract number RII3-CT-2004-506222. JEK acknowledges the Korea Research Foundation under grant no. KRF-2005-084-C00001 of the Ministry of Education, Science and Technology (MEST) of the Republic of Korea for financial support.

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