Upper Limits on the Peccei-Quinn Scale and on the Reheating Temperature in Axino Dark Matter Scenarios

Considering axino cold dark matter scenarios with a long-lived charged slepton, we study constraints on the Peccei-Quinn scale f_a and on the reheating temperature T_R imposed by the dark matter density and by big bang nucleosynthesis (BBN). For an axino mass compatible with large-scale structure, m_axino \gtrsim 100 keV, temperatures above 10^9 GeV become viable for f_a>3x10^12 GeV. We calculate the slepton lifetime in hadronic axion models. With the dominant decay mode being two-loop suppressed, this lifetime can be sufficiently large to allow for primordial bound states leading to catalyzed BBN of Lithium-6 and Beryllium-9. This implies new upper limits on f_a and on T_R that depend on quantities which will be probed at the Large Hadron Collider.


I. INTRODUCTION
In supersymmetric (SUSY) extensions of the Standard Model with conserved R-parity, the lightest supersymmetric particle (LSP) is stable and thus a compelling dark matter candidate. While the lightest neutralino χ 0 1 or the gravitino G are often considered to be the LSP, the axino a is also a well-motivated LSP candidate and hence an equally compelling dark matter candidate [1][2][3][4][5][6][7] beyond the minimal supersymmetric Standard Model (MSSM).
The axino a is the fermionic partner of the axion in SUSY extensions of the Standard Model in which the Peccei-Quinn (PQ) mechanism is embedded as a solution of the strong CP problem. Because its interactions are suppressed by the PQ scale f a 6 × 10 8 GeV [8][9][10][11], the axino can be classified as an extremely weakly interacting particle (EWIP). With the axino being the LSP, the lightest Standard Model superpartner or lightest ordinary superpartner (LOSP) is unstable and can thus be an electrically charged particle such as a charged sleptoñ l 1 . For example, the lighter stau τ 1 -which is the superpartner of the tau lepton τ -is the LOSP in a large part of the parameter space of the constrained MSSM (CMSSM). Due to the extremely suppressed axino interaction strength, such an LOSP would be long-lived and would appear as a quasi-stable charged particle in the collider detectors. Its ultimate decay into the a LSP will often occur outside of those detectors. Some decays however may be accessible experimentally and may allow one to probe the PQ scale at colliders [12]. While an axino LSP identification [12] will require challenging * Electronic address: afreitas@pitt.edu † Electronic address: steffen@mppmu.mpg.de ‡ Electronic address: nurhana@physik.uzh.ch § Electronic address: wyler@physik.unizh.ch experimental setups [13], quasi-stablel 1 's can appear as a first hint for the existence of SUSY and of the axino LSP at the Large Hadron Collider (LHC) already within the next three years.
In this Letter we focus on the axino LSP case with a long-livedl 1 LOSP and in particular on scenarios in which the axino provides the dominant contribution to the dark matter density [14] Ω 3σ dm h 2 = 0.105 +0.021 −0.030 (1) with h = 0.73 +0.04 −0.03 denoting the Hubble constant in units of 100 km Mpc −1 s −1 . The 3σ range indicated rests on a representative six-parameter "vanilla" model.
The thermally produced (TP) axino density Ω TP e a must not exceed Ω dm . This puts upper limits on the postinflationary reheating temperature T R [3,5,7,15,16]. These T R limits-which depend on the axino mass m e a and on the PQ scale f a -can be very restrictive for models of inflation and of baryogenesis. For example, T R 10 6 GeV is found for f a = 10 11 GeV and m e a = 100 keV [5]. Indeed, for m e a 100 keV, temperatures above 10 9 GeV can become viable only for larger values of the PQ scale, f a 3 × 10 12 GeV, if a standard thermal history is assumed. 1 While T R 10 9 GeV is required, e.g., by standard thermal leptogenesis with hierarchical right-handed neutrinos [21][22][23][24][25], we show in this work that f a 3 × 10 12 GeV can be associated with restrictive BBN constraints due to the long-livedl 1 LOSP and its potential to form primordial bound states. In fact, we find that those BBN constraints imply upper limits on f a and thereby new upper limits on T R . We consider hadronic (or KSVZ) axion models [26,27] in a SUSY setting [28]. In this class of models, the axino couples to the MSSM particles only indirectly through loops of heavy KSVZ (s)quarks. Thereby, the dominant 2-body decay of thel 1 LOSP into the associated lepton and the axino is described in leading order by 2-loop diagrams [4,12]. Using a heavy mass expansion, we evaluate the 2-loop diagrams explicitly and obtain the decay width that governs thel 1 lifetime τl 1 . For a given thermal freeze-out yield of negatively chargedl − 1 's, Yl− 1 , our τl 1 result allows us to infer the BBN constraints associated with primordial 6 Li and 9 Be production that can be catalyzed byl − 1 -nucleus-bound-state formation [29][30][31]. While BBN constraints were often assumed to play only a minor role in the axino LSP case, we explore the ones from bound-state effects explicitly and show that they impose new restrictive limits on f a and T R .
Before proceeding, let us comment on axion physics. We assume a cosmological scenario in which the spontaneous breaking of the PQ symmetry occurs before inflation leading to T R < f a so that no PQ symmetry restoration takes place during inflation or in the course of reheating. Since axions are never in thermal equilibrium for the large f a values considered, their relic density Ω a is governed by the initial misalignment angle Θ i of the axion field with respect to the CP-conserving position; cf. [6,9,32] and references therein. With a sufficiently small Θ i being an option, Ω a ≪ Ω dm is possible even for f a as large as 10 14 GeV. We assume Ω a ≪ Ω dm to keep the presented constraints conservative.
The remainder of this Letter is organized as follows. In the next section we review the upper limits on T R imposed by Ω TP e a ≤ Ω dm which provide our motivation to consider f a 3 × 10 12 GeV. Section III presents the results for thel 1 NLSP lifetime obtained from our 2-loop calculation. Section IV explores the BBN constraints froml 1 -nucleus-bound-state formation. In Sect. V, we show that those BBN constraints imply new T R limits if the considered axino LSP scenario is realized in nature. Analytic expressions that approximate the obtained limits in a conservative way are derived in Sect. VI.

II. CONSTRAINTS ON TR
Because of their extremely weak interactions, the temperature T f at which axinos decouple from the thermal plasma in the early Universe can be very high, e.g., T f 10 9 GeV for f a 10 11 GeV [5,33] or T f 10 11 GeV for f a 10 12 GeV [5]. Accordingly, axinos decouple as a relativistic species in scenarios with T R > T f . The resulting relic density is then insensitive to the precise value of T R [33]: Ω therm ≃ Ω dm , this is in conflict with large-scale structure which requires a smaller present free-streaming velocity of axino dark matter and thereby m e a 1 keV; cf. Sect. 5.2 and Table 1 of Ref. [34]. Focussing on scenarios in which axinos provide the dominant component of cold dark matter with a negligible present free-streaming velocity, m e a 100 keV, we thus assume T R < T f in the remainder of this work.
In scenarios with T R < T f , axino dark matter can be produced efficiently in scattering processes of particles that are in thermal equilibrium within the hot MSSM plasma [3,5,35,36]. The efficiency of this thermal axino production is sensitive to T R and f a and the associated relic density reads [5] 2 Ω TP e a h 2 ≃ 5.5 g 6 s (T R ) ln with the strong coupling g s and the axion-modeldependent color anomaly of the PQ symmetry absorbed into f a . 3 Using hard thermal loop (HTL) resummation together with the Braaten-Yuan prescription [38], this expression has been derived within SUSY QCD in a consistent gauge-invariant treatment that requires weak couplings g s (T R ) ≪ 1 and thus high temperatures. Accordingly, (2) is most reliable for T ≫ 10 4 GeV [5]. 4 Note that we evaluate g s (T R ) = 4πα s (T R ) according to its 1-loop renormalization group running within the MSSM from α s (m Z ) = 0.1176 at m Z = 91.1876 GeV.
In Fig. 1, (m e a , T R ) regions in which the thermally produced axino density (2) is within the nominal 3σ range (1) are indicated for f a values between 10 11 GeV and 10 14 GeV by gray bands (as labeled). For given values of m e a and f a , T R values above the corresponding band are disfavored by Ω TP e a > Ω dm ; see also [3,5,7,15,16]. From (2) and Fig. 1, one can see that the viability of temperatures above 10 9 GeV points to f a > 3 × 10 12 GeV if one insists on cold axino dark matter, m e a 100 keV, providing the dominant component of Ω dm . Those f a values and m e a 1 GeV are thereby favored by the viability of standard thermal leptogenesis with hierarchical right-handed neutrinos [21][22][23][24][25].
FIG. 1: Upper limits on the reheating temperature TR as a function of the axino mass m e a in scenarios with axino cold dark matter for fa = 10 11 , 10 12 , 10 13 , and 10 14 GeV (as labeled). For (m e a , TR) combinations within the gray bands, the thermally produced axino density Ω TP e a h 2 is within the nominal 3σ range (1). For given fa, the region above the associated band is disfavored by Ω TP e a h 2 > 0.126.

III. THE CHARGED SLEPTON LOSP CASE
While the T R limits discussed above are independent of the LOSP, we turn now to the phenomenologically attractive case in which the LOSP is a charged sleptoñ l 1 . To be specific, we focus on the τ 1 LOSP case under the simplifying assumption that the lighter stau is purely 'right-handed,' τ 1 = τ R , which is a good approximation at least for small tan β. The χ 0 1 -τ 1 coupling is then dominated by the bino coupling. For further simplicity, we also assume that the lightest neutralino is a pure bino: We consider SUSY hadronic axion models in which the interaction of the axion multiplet Φ with the heavy KSVZ quark multiplets Q 1 and Q 2 is described by the superpotential with the quantum numbers given in Table I For the heavy KSVZ (s)quark masses, we use the SUSY limit M e Q1,2 = M Q = y φ = yf a / √ 2 with both y and f a taken to be real by field redefinitions. The phenomenological constraint f a 6 × 10 8 GeV [8][9][10][11] thus implies a large mass hierarchy between the KSVZ (s)quarks and the weak and the soft SUSY mass scales for y = O(1), Before proceeding, let us recall axion and axino interactions to clarify the definition of f a = √ 2 φ in the considered models. By integrating out the heavy KSVZ (s)quarks, axion-gluon and axion-photon interactions are obtained as described by the effective Lagrangians where G a µν and F µν are the gluon and electromagnetic field strength tensors, respectively, whose duals are given by G a µν = ǫ µνρσ G aρσ /2 and F µν = ǫ µνρσ F ρσ /2; e 2 = 4πα. After chiral symmetry breaking, for the models described by (3) and Table I, where z = m u /m d ≃ 0.56 denotes the ratio of the up and down quark masses. The corresponding interactions of axinos with gluons and gluinos g are obtained as described by and as used in the derivation of (2). In R-parity conserving settings in which the τ R LOSP is the NLSP, its lifetime τ e τ is governed by the decay τ R → τ a. For the models given by (3) and Table I, the Feynman diagrams of the dominant contributions to the 2-body stau NLSP decay τ R → τ a are shown in Fig. 2.
Since m τ ≪ m e τ , we work in the limit m τ → 0. The decay amplitude depends on the parameters of the heavy (s)quark sector through their masses M Q = yf a / √ 2, the Yukawa coupling y, and the gauge couplings ee Q . In fact, in the calculation of the 2-loop diagrams, the hierarchy (5) allows us to make use of a heavy mass expansion in powers of 1/f a [39]. In this asymptotic expansion, it is sufficient to calculate the leading term of the amplitude ∝ 1/f a since the sub-leading terms (∝ 1/f 2 a ) are suppressed by many orders of magnitude. Details of this calculation and the full result of the leading term will be presented in a forthcoming publication [40]. The dominant leading logarithmic (LL) part of the partial width is given by where α denotes the fine structure constant, m e B the bino mass, and θ W the weak mixing angle. 5 However, all numerical results shown in the plots below rest on the full calculation. 6 Feynman diagrams of the dominant contributions to the stau NLSP decay e τR → τ e a in a SUSY hadronic axion model with one KSVZ quark Q = (q1,q2) T and the associated squarks e Q1,2. The considered quantum numbers are given in Table I. For simplicity, the lightest neutralino is assumed to be a pure bino e χ 0 1 = e B and the tau mass is neglected. 5 We use α = α MS (m Z ) = 1/129 [41] and sin 2 θ W = 1 − m 2 W /m 2 Z = 0.2221. 6 Note that the 3-body decay e τ R → τ e aγ occurs already at the 1loop level. The corresponding amplitude however is not enhanced by ln(yfa/ √ 2m e τ ) which can be as large as 20.4-27.3 for m e τ /y = 100 GeV and fa = 10 11 -10 14 GeV. In fact, the branching ratio It is interesting to note that the τ R τ a vertex-governed by 2-loop diagrams-is sensitive to the two large scales f a and M Q ; cf. (11). In contrast, there appears only the scale f a in the vertices-governed by 1-loop diagramsthat describe the interactions of axions/axinos with photons, gluons, and gluinos mentioned above.
In Fig. 3 our result of the full leading term for 1/Γ( τ R → τ a) ≈ τ e τ and its relation to m e τ is illustrated for m 2 e a /m 2 e τ ≪ 1, m e B = 1.1 m e τ , |e Q | = 1/3, and y = 1. The considered f a values are between 10 10 and 10 14 GeV.
The results show that Γ( τ R → τ a) is largely governed by the LL part (11). Comparing equation (11) with the full expression [40] (see also Fig. 3), we estimate that it gives the total width Γ e τR tot and thereby the τ R lifetime τ e τ = 1/Γ e τR tot to within 10% to maximally 15%, depending on the values of f a and m e τ . One can see that f a 10 12 GeV is associated with τ e τ > 1 s for m e τ 1 TeV, i.e., for the m e τ range that would be accessible at the LHC. Accordingly, BBN constraints on axino LSP scenarios with the stau NLSP can become important as will be discussed explicitly below. Note that not only the LL part (11) but the full leading term is strongly sensitive to the electric charge of the heavy KSVZ fields: With respect to the case in Fig. 3, τ e τ is thus reduced by a factor of 81 (16) for |e Q | = 1 (2/3). On the other hand, if e Q = 0, the decay of the τ NLSP will require 4-loop diagrams involving gluons, gluinos, and ordinary (s)quarks, which would thus lead to significantly larger lifetimes than in Fig. 3.
Let us compare our result with the one for Γ( τ R → τ a) that had been obtained in [12] with an effective theory in which the heavy KSVZ (s)quark loop was integrated out, i.e., by using the method described in [42]. There, the logarithmic divergences were regulated with the cut-off f a , and only dominant contributions were kept. While the dependence on the quantum numbers of the KSVZ (s)quarks was absorbed into the constant C aYY , the uncertainty associated with this cut-off procedure was expressed in terms of a mass scale m and a factor ξ in Ref. [12]. Our 2-loop calculation allows us to make direct connection with the parameters of the underlying model. In particular, we find from (11) that one must set C aYY = 6e 2 Q , ξ = 1, and m = √ 2m e τ /y. Assuming y 1, to avoid non-perturbative heavy (s)quark dynamics, this implies that the scale m cannot be significantly smaller than m e τ , which is an important result of the full 2-loop calculation. Furthermore, the non-LL part can account, as mentioned, for up to 15% of the decay rate.
In the early Universe, the stau LOSP decouples as a of e τ R → τ e aγ stays below about 3% once both the energy of the photon Eγ and its opening angle θ with respect to the τ direction are required to be not too small. Those cuts are needed because of an infrared and a collinear divergence for Eγ → 0 and θ → 0, respectively, which would be canceled by the virtual 3loop correction to the 2-body decay channel [40]. For a stau yield Y e τ given by (12), τ e τ values to the right of the nearly vertical solid and dash-dotted (red) lines are disfavored by the constraints (18) and (17) on catalyzed BBN (CBBN) of 9 Be and 6 Li, respectively [31]; see Sect. IV for details.
WIMP before its decay into the axino LSP. The thermal relic stau abundance prior to decay then depends on details of the SUSY model such as the mass splitting among the lightest Standard Model superpartners [43] or the left-right mixing of the stau LOSP [44,45]. However, focussing on the τ R LOSP setting, we work with the typical thermal freeze out yield described by where s denotes the entropy density and n e τR the total τ R number density for an equal number density of positively and negatively charged τ R 's. This approximation (12) agrees with the curve in Fig. 1 of Ref. [43] derived for m e B = 1.1 m e τ and for m e τ significantly below the masses of the lighter selectron and the lighter smuon.

IV. CBBN CONSTRAINTS
The presence of negatively charged τ − R 's at cosmic times of t > 10 3 s can allow for primordial 6 Li and 9 Be production via the formation of ( 4 He τ − R ) and ( 8 Be τ − R ) bound states. Indeed, depending on the lifetime τ e τ and the abundance Y e τ − R = Y e τ /2, the following catalyzed BBN (CBBN) reactions can become efficient [29][30][31] 7 The large 9 Be-production cross section reported and used in Refs. [30,31] has recently been questioned by Ref. [46], in which a study based on a four-body model is announced as work in progress to clarify the efficiency of 9 Be production.
Confronting the τ e τ -dependent Y e τ − R bounds with (12), we obtain the CBBN constraints shown in Figs. 3 and 4 by the solid ( 9 Be) lines and by pairs of dash-dotted ( 6 Li, red) lines associated, respectively, with (18) and the range in (17). The regions to the right of the corresponding lines in Fig. 3 and the ones below the corresponding lines in Fig. 4 are disfavored by CBBN due to an excess of 9 Be and 6 Li over the given limits.
In will be able to measure m e τ and m e B at the LHC. Moreover, with further experimental insights into the SUSY model, Y e τ can be calculated for a standard cosmological history with T R above the temperature at which the stau decouples from the primordial plasma. For concreteness, let us assume that m e B = 1.1 m e τ and that the resulting yield agrees with (12). The measured m e τ value can then be confronted with the CBBN constraints shown in Figs. 3 and 4. For m e τ = 500 GeV, for example, the CBBN constraints imply f a 10 13 GeV for m 2 e a /m 2 e τ ≪ 1, |e Q | = 1/3, and y = 1. Then T R 10 9 GeV-as required by standard thermal leptogenesis-will only be viable for m e a 1 MeV; cf. Fig. 1. While τ e τ is practically independent of such a small m e a , one could in principle test this m e a limit from the kinematics of the 2-body decay τ R → τ a [12], i.e., from a measurement of the energy of the emitted tau E τ , At present, however, this seems to be a realistic option only for 0.1m e τ m e a < m e τ in light of the expected experimental uncertainties. Indeed, for m e a 1 GeV, an experimental determination of m e a along (19) will be extremely challenging. Nevertheless, for a given hadronic axion model (i.e., given e Q and y), the CBBN constraints together with experimental insights into m e τ , m e B , Y e τ , and Ω dm imply new m e a -dependent upper limits on the reheating temperature T R . 8 In Fig. 5, we present upper limits on T R imposed by Ω TP e a h 2 ≤ 0.126 and by the 9 Be CBBN limit on f a given in Fig. 4, i.e., for |e Q | = 1/3, m e B = 1.1 m e τ , Y e τ given by (12), and y = 1. The shown limits range from T max R = 10 5 GeV up to 10 10 GeV (as labeled). Once m e τ is determined at colliders, this figure allows one to infer (m e a , T R ) combinations that are disfavored by CBBN and Ω dm . The 6 Li CBBN limits on f a are in close vicinity to the 9 Be limit, as can be seen in Fig. 4. Thus, we do not show the associated T max R lines since they agree basically with the ones shown in Fig. 5. For |e Q | = 1, T max R becomes less restrictive by almost exactly two orders of magnitude. For example, the T max R = 10 9 GeV line for |e Q | = 1 is in close vicinity to the T max R = 10 7 GeV line in Fig. 5.
The obtained upper limits on f a and T R are conservative ones. For instance, BBN constraints from hadronic energy emitted in 4-body decays τ R → τ aqq can become relevant already for τ e τ 100 s. These additional constraints-imposed mainly by observationally inferred limits on primordial deuterium-may imply more restrictive f a limits than obtained here, and thereby T max R values that are more restrictive than the ones in Fig. 5. Effects of late energy injection on 6 Li from CBBN have been included in the gravitino LSP case, e.g., in Refs. [49][50][51][52]. The resulting constraints differ only marginally from the ones obtained without taking this effect into account [31,53,54]. 9 We expect a similar outcome for our CBBN limits and refer the study of constraints from energy injection to a future publication.

VI. DISCUSSION
It has already been realized in Ref. [12] that collider measurements of τ e τ , m e τ , and m e B will probe the PQ scale f a in the considered axino LSP scenarios. This is also evident from the results of our 2-loop calculation shown in Fig. 3 and from the associated LL part (11). The f a value inferred for given e Q and y can then be used in (2) to extract the m e a -dependent limit T max R imposed by Ω TP e a ≤ Ω dm ; cf. Fig. 1. However, a τ e τ measurement will be challenging from the experimental point of view. In fact, while there are proposals for planned detectors at the International Linear Collider (ILC) [58,59], new detector concepts may be necessary to stop and collect long-lived τ 1 s for an analysis of their decays [13,[60][61][62]. ≤ Ω dm and on Ω NTP e a to be inferred from collider data and without considering BBN constraints in the e a LSP case with al 1 NLSP, which are the main results of our Letter. 9 At t 10 3 s when CBBN is not efficient, injection of energy may have a noticeable effect on the 6 Li abundance and could even allow for a solution of the 7 Li problem that is consistent with 6 Li in the observationally inferred range (17) [50,[55][56][57].
The limits on f a and T R presented in Figs. 4 and 5 do not rely on a measurement of τ e τ . They result from upper limits τ max e τ imposed by the CBBN constraints, which show only a very mild dependence on m e τ for typical yields such as (12); see Fig. 3. In fact, based on (20), it is possible to derive analytic expressions for the upper limits on f a and T R in a conservative way.
Aiming at an instructive derivation, we work with the LL part (11) which describes τ e τ to within 15% accuracy, where (22)  A comparison with the numerically obtained 9 Be limits at m 2 e a /m 2 e τ ≪ 1 shows a good overall agreement for τ max e τ ≈ 5 × 10 3 s. The associated analytical expression however is less restrictive (i.e., more conservative) than the numerically obtained limits towards larger m e τ . In fact, there the actual τ max e τ value imposed by CBBN becomes more restrictive as can be seen in Fig. 3.
Let us now turn to T R on which a conservative limit Here the constant "conservative" prefactor 0.6 accounts for the T R -dependent prefactor in (2), which stays in the range 0.6 < 5.5 g 6 s (T R ) ln[1.211/g s (T R )] < 1.06 for 10 4 GeV ≤ T R ≤ 10 12 GeV if the MSSM 1-loop renormalization group running of g s is considered. Using the upper limit (24) in (25), one arrives immediately at an analytic expression for the CBBN-imposed limit, which is conservative. For τ max e τ ≈ 5 × 10 3 s, we find again a good overall agreement with the limits obtained numerically. However, as expected from its derivation, the associated analytic expression can be by a factor of O(1) less restrictive than the numerical results shown in Fig. 5.
Since τ e τ depends on the ratio f a /e 2 Q , the limits (24) and (28) depend on e Q and thus on the specific axion model. It would therefore be particularly valuable to discover the axion and its mass since the relation between m a and f a does not depend on e Q ; in the models given by (3) and Table I, m a = [ √ z/(1 + z)] f π m π /f a with f π ≈ 92 MeV and m π = 135 MeV. If f a can thus be determined, T max R would be given by (25) directly. In addition, one could find e Q in a τ e τ measurement or derive a lower limit on it from the CBBN constraints (20).
In this respect we note that most axion searches probe the axion-photon-coupling g aγγ = αC aγγ /(2πf a ) in certain ranges of the axion mass m a ; cf. [6] and references therein. In the models considered, C aγγ is given by (8) so that g aγγ does also depend on f a and e Q [63]. An axion discovery at an (m a , g aγγ ) combination would thus be associated with an (f a , e Q ) combination in the considered models. The e Q value from axion searches could then be compared to the one inferred from a τ e τ measurement at colliders or, if this is not possible, to its lower limit imposed by CBBN.
The region in which the presented BBN constraints are expected to become relevant is explored by the axion dark matter experiment (ADMX) which searches for resonant conversion of dark matter axions into photons in a microwave cavity [64,65]. Axion searches of this type are sensitive to g aγγ only in the combination g 2 aγγ ρ a , where ρ a denotes the local halo density of axions. If axinos provide the dominant component of cold dark matter, ρ a can be very small so that no signals will appear at the expected g aγγ values. An axion signal in such a direct search would in turn imply a sizeable axion density, Ω a ∼ Ω dm , and thereby a restrictive T R limit in the considered m e a range, m e a 0.1 MeV, given by (25) or (28) with Ω dm → Ω dm − Ω a . Alternatively, evidence for solar axions could appear in the Tokyo Axion Helioscope or the CERN Axion Solar Telescope (CAST) [66][67][68]. This would imply Ω a ≪ Ω dm , f a 10 9 GeV and thus T R ≪ 10 6 GeV in the considered axino cold dark matter scenarios; cf. (25) with m e a 0.1 MeV. Here the CBBN constraints will be relevant only in the exceptional cases with e Q → 0 and/or m e a → m e τ .

VII. SUMMARY AND CONCLUSIONS
We have explored BBN constraints in axino cold dark matter scenarios with a long-lived charged sleptonl 1 . Calculating the lifetime τl 1 , which is governed by 2-loop diagrams in hadronic axion models, we find thatl 1 can be sufficiently long lived to allow for an efficient catalysis of 6 Li and 9 Be via bound-state formation with primordial nuclei. Observationally inferred abundances of 6 Li and 9 Be thus impose upper limits on τl 1 for typical thermal relic abundances of the long-livedl 1 . These limits have allowed us to derive upper limits on the PQ scale f a that depend mainly on the masses of the slepton, ml 1 , and the lightest neutralino, m e χ 0 1 , and on the electric charge of the heavy (s)quarks e Q . The obtained f a constraints imply new upper limits on the reheating temperature T R since f a governs not only τl 1 but also the efficiency of thermal axino production and thereby the T R constraints imposed by Ω TP e a ≤ Ω dm . We have presented both numerical results and analytical approximations for those new BBNimposed limits and have discussed their dependence on m e a , ml 1 , m e χ 0 1 , and e Q . For example, for ml 1 = 500 GeV, m e χ 0 1 = 1.1 ml 1 , and |e Q | = 1/3, we find f a 10 13 GeV and that T R 10 9 GeV is viable only for m e a 1 MeV.
We have addressed the extent to which the BBNimposed limits on f a and T R can be probed experimentally if the considered axino LSP scenario is realized. With not too heavy Standard Model superpartners, LHC experiments will allow us to measure ml 1 and m e χ 0 1 and to infer the thermal relicl 1 abundance prior to decay under the assumption of a standard cosmological history. With the ILC and/or new detector concepts, even a measurement of τl 1 is conceivable, and our τl 1 result shows that this could give insights into f a /e 2 Q . A determination of m e a however seems possible only for relatively heavy axinos 0.1ml 1 m e a < ml 1 and hopeless for m 2 e a /m 2 l1 ≪ 1 [12]. Moreover, insights into e Q -or, more generally, into the axion model-seem to require not only an axion discovery but a determination of its mass m a (and thereby of f a ) in axion search experiments.
A simple form of the superpotential has been considered that is generic for SUSY hadronic axion models in which the axion multiplet interacts with the MSSM multiplets through loops of heavy (s)quarks. While we have explored the case with a minimum number of SU(2) Lsinglet KSVZ multiplets and withl 1 being a purely righthanded stau τ R , our study can be generalized to more complicated settings in a straightforward way.
Without specifying the SUSY breaking mechanism or other details of the PQ sector, we have assumed saxion effects to be negligible and a spectrum with the a LSP and thel 1 NLSP. Our results depend crucially on these assumptions. In situations in which the saxion dominates the energy density before its decay, the entropy per comoving volume can be enhanced by a factor ∆ > 1. If this additional entropy production takes place beforel 1 decoupling, the BBN constraint on f a will not be affected but the thermally produced axino density can be diluted so that Ω TP e a → Ω TP e a /∆ and T max R → ∆T max R . If entropy increases by a large factor of ∆ > 10 3 afterl 1 decoupling and before BBN, thel 1 abundance can be diluted such that catalyzed BBN (CBBN) of 6 Li and 9 Be cannot become efficient. Then the CBBN-imposed constraints on f a and T R would not exist. Nevertheless, Ω TP e a → Ω TP e a /∆ so that the Ω dm -imposed limit on T R would be relaxed by a factor of ∆. However note that the baryon asymmetry would also be diluted by a factor of ∆ and therefore a larger asymmetry would be needed before its dilution; see Ref. [37] for a related discussion in the G LSP case.
The cosmological constraints presented in this work can also be affected by the presence of the gravitinoG even for a standard thermal history. Its mass mG-which depends on the SUSY breaking mechanism and the SUSY breaking scale-governs the strength of its interactions. The gravitino can be produced thermally in the early Universe, with the resulting abundance depending on mG and T R [69,70]. In the scenario studied in this Letter, mã < ml 1 < mG, the heavy gravitino is typically longlived and its decays may affect BBN. Thereby additional constraints on T R can be incurred [52,71].
If mG < ml 1 and Γ(l 1 → lã) ≪ Γ(l 1 → lG), τl 1 is governed byl 1 → l G. Then our f a limit can be evaded while the CBBN constraints discussed in [31, 37, 50, 52-54, 72, 73] and their implications for thermally produced gravitino abundance become relevant. On the other hand, if Γ(l 1 → lã) ≫ Γ(l 1 → lG), the CBBN limits discussed in this Letter also apply. However, the gravitinos lead to an increase of the LSP density, thus leading to more restrictive T R limits. In this case our results remain as conservative upper limits. Our investigations show that for the interesting case of new long-lived charged particles, BBN constraints play an important role and can be used to restrict the models considerably. These constraints will become particularly important if such particles are produced and detected at the upcoming LHC experiments.