Axino Dark Matter in Moduli-induced Baryogenesis

We consider axino dark matter in large R-parity violation (RPV). In moduli-dominated universe, axino is produced thermally or non-thermally via saxion decay, then late-decaying moduli dilute axino density, which results in the right abundance to explain the present dark matter. At the same time baryon asymmetry is generated due to moduli-induced baryogenesis via the large RPV. Axino is cosmologically stable in spite of the large RPV since its decay rate is suppressed by the axion decay constant, heavy squark mass or kinematics.


Introduction
The discovery of the Higgs boson at the LHC confirmed the standard model of particle physics [1,2]. So far no phenomenon, which shows severe inconsistency with the standard model (SM), has been reported on the ground-based experiments (except for neutrino oscillation). In cosmology, however, it is clear that we need new physics beyond the standard model. First of all, the standard cosmology can not explain the existence of dark matter (DM). In addition, the baryon density predicted in the standard model is too small to account for the observed value. Supersymmetry (SUSY) is a promising solution to the issues. On top of that, string theory, which requires supersymmetry for the consistency, is a viable candidate for the theory of everything.
However, such an extension may cause another problem especially in cosmology.
Moduli fields, which must be stabilized to compactify the extra dimensions in string theory, may be destabilized during inflation if the inflation scale is very high, which is indicated by the recent BICEP2 observation [3]. Even if the destabilization is avoided in some ways [4,5], it is likely that moduli are displaced far from their true minima at the end of inflation. Then moduli start to oscillate, and soon dominate the energy density of the universe. The moduli-dominated universe ends when moduli decay, accompanying a huge entropy injection. This is potentially problematic because such substantial an entropy production dilutes pre-existing matter density, then it is difficult to lead to big bang nucleosynthesis (BBN) and the structure formation of the universe. Possible way out are following: production of a large amount of the matter density before moduli decay or generation of the matter density after moduli decay. As for baryonic matter, Affleck-Dine mechanism [6,7] is a typical example of the former one. On the other hand, late-decaying gravitino [8] or saxion [9] can also produce baryon asymmetry, which corresponds to the latter.
Recently another mechanism, moduli-induced baryogenesis, was proposed [10]. It was shown that Kachru-Kallosh-Linde-Trivedi scenario [11] has built-in features for baryogenesis, such as large enough CP phase and suitable mass spectrum for superparticles. Then subsequent decays of gluino and squarks from moduli produce sufficient baryon asymmetry. In those baryogenesis due to late-decaying particle, however, a large R-parity violation (RPV) is assumed, which makes lightest superparticle (LSP) unstable. This is a downside to accounting for dark matter.
In this paper we consider axino LSP in large R-parity violation. Introducing the axion supermultiplet is motivated by Peccei-Quinn (PQ) mechanism [12], which solves the strong CP problem. Assuming that the fermionic component of the axion multiplet, axino, is the LSP, axino is copiously produced by its radial component, saxion, decay and the scattering from thermal plasma. Axino can be cosmologically stable even if the RPV is O(1) because its decay rate is suppressed by the axion decay constant, squark mass or kinematics. After saxion decay, moduli decay follows to dilute axino abundance, which results in the observed relic of dark matter. At the same time, baryon asymmetry of the universe is generated in moduli-induced baryogenesis with the RPV.

Cosmological scenario
In this section we describe the basic picture of our scenario. In the scenario moduli dominate the total energy of the universe after inflation. Axino is produced thermally or by non-thermal saxion decay in the epoch of moduli domination. Eventually moduli decay and dilute the axino abundance, which gives the right value to explain DM. Here baryon asymmetry is generated from moduli decay as well due to R-parity violated interaction. The saxion decay also generates axion. Although it is diluted by moduli decay, the produced axion may give a sizable contribution to radiation as dark radiation. Finally the stability of axino under the RPV is discussed.

Moduli-dominated universe
Let us begin with the thermal history after inflation. As we mentioned in the Introduction, the modulus field tends to be displaced from its true minimum due to the deformed potential during inflation or due to the initial condition. Then after inflation, it starts to oscillate around the true minimum when the Hubble parameter H reduces to moduli mass m X . Assuming T R , the reheating temperature after inflation, is comparable to T X,osc , the temperature when modulus begins to oscillate, the energy density of modulus field X per entropy density freezes after the oscillation starts at a value of Here T is the cosmic temperature, ρ X is the energy density of modulus, s(T ) is the entropy density and δX ini is the initial amplitude of X measured from its true minimum. Typically we expect δX ini ∼ M Pl where M Pl ≃ 2.4 × 10 18 GeV is the reduced Planck mass. T X,osc is estimated from the equation H ≃ m X as T X,osc = 90 π 2 g * (T X,osc ) 1/4 M Pl m X ≃ 6.9 × 10 13 GeV m X 10 10 GeV Here g * (T ) counts degree of freedom of relativistic particles in the thermal bath.
Due to their huge energy density, moduli soon dominate the energy density of the universe. The temperature when moduli begin to dominate the universe is estimated from the relation ρ X ≃ ρ R as where we have used Eq. (2.1). It is seen moduli dominate the total energy density soon after starting to oscillate.
Since the energy density of moduli redshift as ρ X ∝ a −3 , it is given by until moduli decay. As ρ tot , the total energy density of the universe, is equal to ρ X during moduli domination, the Hubble parameter in moduli-dominated universe is given by The epoch of moduli domination terminates when moduli decay to particles in minimal supersymmetric standard model (MSSM), and it turns into radiation domination. The temperature at the beginning of this radiation-dominated universe is determined by H ≃ Γ X as (2.6) Here we have used the decay rate of moduli, which is given by [13] where c X is O(1) constant and here and hereafter we take it as unity. #1 The moduli masses have to be larger than around 100 TeV in order not to destroy BBN. Even if m X 100 TeV is satisfied, however, a huge entropy production due to moduli decay may strongly dilute primordial relics, such as baryon and DM. The effect is described by a dilution factor, which is given by a ratio of entropy density before and after the moduli decay, (2.8) The dilution is important to reduce over-produced axino (and axion), which is discussed below.

Saxion decay
Saxion is the radial component field in the axion supermultiplet. The axion supermultiplet is determined as a flat direction of the scalar potential given by the PQ fields. Here the PQ fields have non-zero PQ charges and break PQ symmetry spontaneously. We define the axion supermultiplet as Here σ, a andã are saxion, axion and axino, respectively. For later calculation we define the axion decay constant as f a = 2 i q 2 i v 2 i where q i and v i = Φ i are the PQ charge and the vacuum expectation value (VEV) of a PQ field Φ i , respectively.
If the domain wall number N DW is not unity, then f a should be 2 i q 2 i v 2 i /N DW . Similar to moduli, saxion tends to have initial amplitude around its true minimum after inflation then it starts oscillation when H ≃ m σ (m σ is saxion mass). Around this period, moduli begin to dominate the total energy. If saxion starts to oscillate before moduli domination, the temperature at the beginning of the oscillation is given by and its energy density to entropy ratio is fixed at In the numerical analysis, we use the results given in Ref. [10].
Here δσ ini is the saxion initial amplitude, which is expected to be order of f a to M Pl . The value depends on the saxion potential (see, e.g., Refs. [14,15,16]). On the other hand, saxion starts to oscillate after the universe is dominated by moduli when T σ,osc < T dom . In such a case, saxion energy density per entropy density has a fixed value (2.12) In our scenario we consider m X is relatively larger than m σ . Then the energy density of moduli is much larger than that of saxion during the period of moduli domination in either case.
In both KSVZ and DFSZ models saxion couples to axino and axion via the kinetic term, which is given as [19] Here we have usedã as a four component axino spinor. From this interaction, the partial decay widths for σ → aa and σ →ãã are computed where mã is axino mass. In KSVZ model, the process σ → aa overwhelms the other decay modes if ξ ∼ O(1). We take ξ = 1 unless otherwise noted. Then the total decay rate is given by Γ σ ≃ Γ(σ → aa). #2 On the other hand, in DFSZ model, saxion interacts with Higgs doublets in F -term potential. Then saxion can decay to the SM-like Higgs pair, whose partial decay width is In KSVZ model, the branching ratio is simply given by Br(σ →ãã) ≃ 8m 2 a /m 2 σ in the limit m σ ≫ mã. This is also true in DFSZ model when µ m σ .

Axino production
Axino is the fermionic component in the axion supermultiplet. Axino can be produced in several ways; thermal production, non-thermal saxion decay or the next-LSP (NLSP) decay. The production due to the NLSP decay is negligible because the NLSP mainly decays to the SM particles via RPV. The other two, i.e., saxion decay and thermal production, are potentially important. In terms of yield variable Yã ≡ nã/s (nã is the number density of axino), the resultant abundance of axino is expressed as, and Y TH a are contributions from saxion decay and thermal production, respectively.
Y DEC a is easily obtained. Using Eqs. (2.11) and (2.12), the present axino density due to saxion decay is given by Here we note that the produced axino is diluted due to the late-decaying moduli, which is taken into account by the dilution factor d X . There is an entropy production due to saxion decay. However, it is much smaller than the entropy production from #3 Suppose there are two PQ fields, Φ 1 and Φ 2 , and both of them get VEVs as Φ 1 = Φ 2 . If Φ 1 couples to up-and down-type Higgses (denoted as H u and H d , respectively) in superpotential, moduli. This is because saxion decays before moduli, the energy density of saxion is smaller than that of moduli and that the branching fraction to the MSSM-sector particles in saxion decay is typically suppressed. Therefore, the dilution due to saxion decay is negligible compared to moduli decay.
The thermal production, on the other hand, is highly model-dependent. It is described by Boltzmann equation, Here a dot means derivative with respect to the cosmic time and C prd is axino production rate per unit volume, which depends on the axion model. The solution of the Boltzmann equation in radiation domination is given by (usingṪ = −HT ) In KSVZ model, axino is mainly produced by thermal scattering or decay of the particles in thermal plasma via strong interaction. For example, the production rate due to scattering processes, such asqg →ãq,gg →ãg, is roughly estimated as (α s is strong coupling constant and n MSSM is the number density of the MSSM particle.) Then from Eq. (2.21) the yield variable of axino is estimated as It is seen that the axino production is the most active at the highest temperature of the universe, i.e., T R . #4 Thus the axino abundance is almost determined by the production from thermal plasma before moduli dominates the total energy, which guarantees that we have used Eq. (2.21). More precise computation of the axino production in radiation domination is done by Refs. [20,21,22,23,24]. In our later numerical calculation we adopt the result given in Ref. [22] and fit their result as where only QCD interaction is considered. #5 It is seen that the estimate given in Eq. (2.22) roughly agrees with the expression. Y eq a is the value when axion is #4 Saxion decay reheats radiation during moduli domination. If the reheating temperature exceeds squark or gluino mass, then axino is also produced at the time of saxion decay. The production is, however, negligible since the reheating temperature is much smaller than T R and that the production is suppressed by T dom (see also later discussion). #5 The fitting formula is applicable where T R 10 4 GeV and gluino or squarks are thermalized.
thermalized, and typically Y eq a ≃ 1.8 × 10 −3 using g * = 228.75. The decoupling temperature T D can be estimated by equating the scattering rate for the production process with the Hubble parameter and it is obtained as which is consistent with Ref. [25]. Then axino is thermalized when Since we assume T R ∼ T X,osc , axino is thermalized in a wide range of the parameter space.
Thermal production of axino in DFSZ model is different from the one in KSVZ model. As it is mentioned in Refs. [26,27,28,22], the scattering process via strong interaction is suppressed at high temperature. Instead, the production due to axino interaction with Higgs and Higgsino or stop and top is effective. For example, the production rate for the processes, such asHt →ãt,tt →ãh, is roughly C prd ∼ The contribution from decay gives the same order. Here we have assumed that T R > µ. It is seen that the resultant abundance is highly suppressed by T dom . In addition, it is diluted by the late moduli decay. Thus the contribution of the thermal production to axino abundance before moduli decay is negligible in a wide parameter range. #6 Axino is also produced in the era of radiation domination after moduli decay.
Here T X plays the role of T R in the above discussion. In KSVZ model, however, the axino production is negligible because T X is smaller than gluino or squark mass in a wide parameter region, i.e., processes, such as gg →ãg, qg →ãq, are kinematically suppressed and gluino and squarks are not thermalized. #7 On the other hand, in #6 Axino thermal production occurs after saxion decay if the decay reheats radiation to a temperature larger than µ. The production results in the same order of yield variable given in Eq. (2.25).
Therefore it is negligible for the same reason. #7 This fact is crucial for moduli-induced baryogenesis. Otherwise produced baryon would be washed out. Axino can be produced via RPV interaction, such as qq →ãq. We have checked that this production is negligible in the parameter region that we are interested in.
DFSZ model, axino production may be substantial since the production is effective at low temperature. The processes without external stop, such asHt →ãt orH →ãh, contribute to the production because the number density of stop is Boltzmannsuppressed. Here we assume T X > µ. Then the production due to the scattering leads to axino yield variable after moduli decay (2.26) The contribution of Higgsino decay has the same order as one from the scattering.
The above result roughly agrees with more accurate numerical calculation in the literature. Using the recent result given in Ref. [28], the yield variable of axino is read as #8 Y th a,DFSZ ≃ min Y eq a , 10 −5 µ 1 TeV (2.27) Here axino is thermalized when T DFSZ D µ, where decoupling temperature T DFSZ D is given by (2.28) Then the condition for axino thermalization becomes µ 10 4 GeV (f a /10 11 GeV) 2 .
This estimate is roughly consistent with Eq. (2.27).
In summary, axino yield variable due to the thermal production is given by . (2.29) Then the density parameter of axino at present time is obtained by where (ρ c /s) 0 ≃ 3.6 h 2 × 10 −9 GeV for h ≃ 0.67 [29]. Here we have split two contri-  (2.32) Here we have used T dom and Y eq a in the estimation of Ω DEC a and Ω TH a , respectively. The expression of Ω DEC a can be applied in DFSZ model when µ m σ .
#8 In Ref. [28], the result is given for the case relativistic stop is in thermal bath and its mass is larger than µ. In such a case the yield variable is proportional to M Pl µ 2 f 2 a mt .

Axion production
We have seen that saxion mainly decays to axion pair. The produced axion is relativistic thus behaves as radiation, which is so-called dark radiation. The additional degree of freedom in radiation is described in terms of the effective number of neutrinos N eff = N SM eff + ∆N eff . Here N SM eff = 3.046 [30] is the prediction in the standard model. The result by Planck satellite [29], combined with the measurements of the present Hubble parameter by Hubble Space Telescope [31], gives N eff = 3.83 ± 0.54 at 95% C.L.. When the data from WMAP9 [32], the Atacama Cosmology Telescope [33] and the South Pole Telescope [34,35] are included, the analysis gives N eff = 3.62 +0. 50 −0.48 at 95% C.L. [29]. Though the current observations are consistent with the SM value, the central values are slightly deviated from the SM prediction.
We will see below that ∆N eff can be O(1) in our scenario.
Referring to Refs. [36,37], ∆N eff in our scenario is given by Here ρ ν and ρ a are the energy density of neutrinos and axion, respectively, and "ν decp" means the values at neutrino decoupling. [ρ a /ρ R ] X dec is the ratio of the energy density of axion produced by saxion and radiation at the time of moduli decay. Using ρ σ ≃ ρ a at the time of saxion decay, it is straightforward to get Assuming Γ σ ≃ Γ(σ → aa) and using Eq. (2.35) In this paper we impose a conservative bound ∆N eff 1 on our scenario.
Axion is also produced by coherent oscillation when the Hubble parameter becomes comparable to axion mass. If moduli decays before the axion coherent oscillation, the abundance of the axion due to the oscillation is the conventional value given in Ref. [38], i.e., Ω c.o. a h 2 ≃ 0.2θ 2 a (f a /10 12 GeV) 1.19 . Here θ a is the initial misalignment angle of axion and it should be small in order for axion not to overclose the universe if f a 10 12 GeV. The tuning of θ a is possible when PQ symmetry is broken during or before inflation. Meanwhile, if PQ symmetry is broken after inflation, the misalignment angle should be replaced by π/ √ 3. Then the tuning is impossible and f a is severely constrained. #9 In the case where moduli decay after the coherent oscillation begins, the axion abundance is diluted by moduli decay. Similar case is discussed in Ref. [14]. Since the axion abundance is model-dependent and that we are interested in the axino DM scenario, we simply assume that the axion energy density due to the coherent oscillation is sub-dominant. It is straightforward to take into account the axion abundance from the coherent oscillation and consider mixed axion and axino DM.

Axino stability
In our model we consider RPV for moduli-induced baryogenesis. Through RPV interaction, axino decays to SM particles even if it is the LSP. The renormalizable RPV interaction in superpotential is i are chiral superfields of left-handed lepton doublet, righthanded charged lepton, left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, respectively. i, j, k are generation indices. In the present paper, we will take phenomenological approach to determine the order of each RPV couplings as follows. In our model baryon asymmetry is generated through the RPV interaction. Among the four types of interactions, U c D c D c type is the most effective for moduli-induced baryogenesis. #10 For example, λ ′′ 332 can be order of unity evading from the severe constraint from proton decay, and generate the observed baryon asymmetry [10]. The other couplings are partly constrained phenomenologically (see, e.g., Ref. [42]). Based on the facts, we simply consider a case where at least one of λ ′′ ijk is O(1) and the others are irrelevant. #11 Axino lifetime is determined by the processã → u i d j d k . Relevant interaction for this process is dimension four axino-quark-squark coupling: In this case, the domain wall number should be unity. Even if N DW = 1, axion is also produced from axionic string and axionic domain wall, which gives stringent constraint for the the decay constant, i.e., f a (2.0-3.8) × 10 10 GeV [39]. When the PQ symmetry is broken in during inflation, on the contrary, there is constraint from the isocurvature perturbation (see, e.g., Ref. [40]). #10 In Ref. [41] a simple baryogenesis is suggested in a minimal extension of the standard model by using udd type higher dimension operator, which also contains a DM candidate. #11 Axino DM with different RPV operators is studied in, e.g., Refs. [43,44,45,46]. where P R/L = (1 ± γ 5 )/2 andq L i /R i is left-/right-handed squark. Here quark and squarks are in the MSSM sector. In KSVZ model, although axino has no interaction with quark and squark in the MSSM sector at tree level, the effective interaction is induced at loop level. The effective coupling is given by [21] g (L/R) eff where mg is gluino mass. On the other hand, in DFSZ model, tree-level interaction exists in F -term potential, given by [26] g (L/R) eff ≃ ∓i m q f a cos 2 β (for up-type quark) sin 2 β (for down-type quark) . (2.39) Here m q is quark mass and tan β = H u / H d . In our model the soft SUSY breaking scale tends to be large. To get the observed Higgs mass of around 126 GeV [1,2], tan β ≃ 1 is required in the MSSM. Therefore we consider tan β = 1, which means that the axino decay is induced mainly by axino-top-stop interaction.
In the following discussion we assume that all superparticle (except Higgsino) in the MSSM sector have the same mass scale, which is characterized by the soft mass m soft , i.e., m soft ∼ mf , mg, etc. (2.40) where mf represents sfermion mass. In the calculation of axino lifetime, we use HELAS package [47].

Results
Now we are ready to give numerical results. Before showing the results, we summarize the conditions which need to be satisfied for our scenario: ii ) τã 10 26 sec, (3.2) iii ) Γ σ > Γ X .  For the determination of the axino lifetime we take λ ′′ 332 = (3 × 10 6 GeV/m X ) 1/4 /2 to explain the present baryon density [10] and the others are taken to be zero. m X 3 × 10 6 GeV should be satisfied for the baryogenesis, which is also shown in dot-dashed line. In the figure shaded regions are excluded. Since axino mass is lighter than W boson mass, the axino decay is five body. We found that the constraint τã 10 26 sec is much more stringent than the BBN constraint, which excludes the lower mass range. Finally we discuss the experimental signatures involved in the scenario. Near the region τã ∼ 10 26-27 sec, the decay of axino produces hadrons and leptons, which may be observed as cosmic rays. Among them hadronic decay products are especially constrained by cosmic-ray anti-proton observation by PAMELA [49]. #15 When axino mass is larger than order of a hundred GeV, a large amount of high energy cosmic-ray anti-protons are generated. Then the cosmic rays will be detected by AMS-02 experiment as an exotic signal, otherwise more stringent constraint will be given. If axino mass is smaller, the energy of the produced anti-proton gets smaller.
In such a low energy range, the background cosmic ray increases. Thus it would be more difficult to see the signal, depending on the lifetime. If axino is lighter than 1 GeV, then axino becomes stable because it can not decay to the SM fermions.
However, proton decays to axino via the RPV instead. As pointed out in Ref. [52], λ ′′ 332 induces uds type coupling, κ uds , which is O(10 −7 ) × λ ′′ 332 . Then proton decay to K +ã . The decay rate of proton is estimated as Γ p→K +ã ∼ mp have used m soft = m X /50 and λ ′′ 332 ∼ 0.07 and 0.2 in KSVZ and DFSZ model, respectively. The current experimental bound is τ p→K + ν ≥ 2.3 × 10 33 yr [53]. Therefore, proton decay experiment in the future could be a test of this scenario even in high moduli mass (soft mass) region.
In our scenario, lighter neutral Higgsino may be the LSP in the MSSM sector and as light as O(100 GeV-1 TeV). If the Higgsino is produced at a collider, it would decay inside the detector via the O(1) RPV. However, its decay width is suppressed by the soft mass, thus it would decay from the interaction point. Even if the decay width of Higgsino is so suppressed by the soft mass that the decay occurs far from the interaction point, the decay would be observed. Then counting the number of the decay events, the lifetime might be determined [54]. Then it may be possible to probe the validity of this scenario in the high soft mass region. #14 See, e.g., a recent work [48], which takes into account the back-reaction effect. #15 See earlier works , e.g., [50,51], which study cosmic-ray anti-proton from decaying DM.

Conclusion
In this paper we consider axino dark matter in large R-parity violation. While moduli dominate the universe after inflation, saxion also oscillates coherently and eventually decays to produce large amount of LSP axino. Axino is also produced thermally at the reheating after inflation or lower temperature, depending on axion model. Such axinos are diluted by late moduli decay. We have found that the axino relic can give the correct amount to explain the present dark matter abundance in both KSVZ and DFSZ models. Though axino is metastable due to the large Rparity violation, its decay rate is suppressed the axion decay constant, soft SUSY breaking mass or kinematics. Then the lifetime can be longer in order axino not to produce exotic cosmic rays. With the large R-parity violation, baryon asymmetry is generated in moduli-induced baryogenesis as well. Therefore the scenario explains both dark matter and baryon existing in the present universe.