Axino LSP Baryogenesis and Dark Matter

We discuss a new mechanism for baryogenesis, in which the baryon asymmetry is generated by the lightest particle in another sector, for example the supersymmetric particle (LSP), decaying to quarks via baryonic-number-violating interactions. As a specific example, we use a supersymmetric axion model with an axino LSP and baryonic $R$-parity violation. This scenario predicts large $R$-parity violation for the stop, and an upper limit on the squark masses between {15 and 130 TeV}, for different choices of the Peccei-Quinn scale and the soft $X_t$ terms. We discuss the implications for the nature of dark matter in light of the axino baryogenesis mechanism, and find that both the axion and a metastable gravitino can provide the correct dark matter density. In the axion dark matter scenario, the initial misalignment angle is restricted to be ${\cal O}(1)$. On the other hand, the reheating temperature is linked to the PQ scale and should be higher than $10^4-10^5$ GeV in the gravitino dark matter scenario.


Introduction
Two fundamental observations about our universe, namely the presence of dark matter and the prevalence of matter over antimatter, can individually find an explanation in theories that go beyond the Standard Model. However, it is harder to explain both in a unified framework. A simple argument based on symmetries can justify this: a stable dark matter candidate would be protected by a symmetry, while baryogenesis requires the violation of another symmetry, namely baryon number (or lepton number in high-energy leptogenesis). At the same time, flavor physics is sensitive to baryon-number violation, and proton decay is mediated by the simultaneous breaking of baryon and lepton number.
Although the symmetries involved in the two mechanisms need not to be related, they are in supersymmetric theories, where R-parity forbids baryon-number-violating operators and at the same time provides a stable dark matter candidate, the Lightest Supersymmetric Particle (LSP). The LSP is also the source of the missing energy signature of supersymmetric events at colliders. Given the stringent LHC limits for light R-parity conserving supersymmetry (SUSY) and the null results in dark matter direct detection experiments, it is appealing to investigate phenomenological consequences of R-parity violation (RPV). Besides collider studies, its cosmological implication is very interesting because one can relate the baryogenesis mechanism and the nature of dark matter.
Sizable lepton and proton number violation are not consistent with bounds on the proton lifetime, thus we will focus on models allowing only for baryonic R-parity violation. In addition to the Minimal Supersymmetric Standard Model (MSSM) superpotential, we include the operator where the RPV couplings λ ijk are antisymmetric under the exchange j ↔ k and the sum over color indices is understood. Neutralinos decay away in the early universe. Without an additional symmetry to warrant dark matter stability, a natural candidate for dark matter should be super-weakly interacting: for example, the gravitino, which can be metastable and have a lifetime longer than the age of the universe, and the axion, which was originally introduced [1] to explain the absence of CP-violation in strong interactions (the strong CP problem). Introducing the axion supermultiplet gives a further implication that relates dark matter and baryogenesis; there is a fermionic superpartner, the axino, whose out-of equilibrium decay through the operator (1.1) can generate a baryon asymmetry. The idea of using the operator (1.1) for baryogenesis is not new. Refs. [2][3][4] discussed a baryon asymmetry generated at low temperatures (down to the MeV scale) from late decays of inflaton, gravitinos and axinos into superpartners, respectively 1 . It is important that baryon asymmetry is generated at a temperature well below the superpartner mass scale so that it is not washed out by baryon-number-violating processes in the thermal bath [9]. For large couplings, baryogenesis should happen at a temperature T mq/20. Otherwise, if the asymmetry is generated at or above the superpartner scale, all the R-parity violating couplings have to be smaller than O(10 −7 ) for baryons to survive.
We shortly review these baryogenesis mechanisms, which all involve R-parity violating Aterms. In [2], out-of-equilibrium decays of the inflaton generate a non-thermal population of squarks, which later decay to a quark and a gluino, provided that mq > mg +m q . Interference between the tree level and two-loop diagrams gives a baryon asymmetry. In [3], the decay of a gravitino to a quark and a squark (or to a gluino and a gluon, with the subsequent gluino decay to a quark and a squark) was used to generate the asymmetry (given the hierarchies m 3/2 > mq + m q or m 3/2 > mg > mq + m q , respectively), through a one-loop diagram involving A-terms. In [4], the same diagrams were used to discuss the asymmetry generated by the decay of the axino (or saxion) to a gluino and finally to a quark and a squark.
For the cases described above, the parent particle decays to some on-shell superpartner via R-parity conserving interactions, and interference with a R-parity violating loop diagram generates the baryon asymmetry. A specific hierarchy is required in each case: the particle decaying out of equilibrium is always heavier than the superpartners that are on-shell in the interfering diagram. This automatically excludes the case of an LSP decay (in the following the LSP is defined as in R-parity conserving SUSY, and it is stable only when the RPV interactions are turned off). In fact, this reflects a more general statement: Nanopoulos and Weinberg proved in Ref. [10] that an LSP defined in such a way cannot give a baryon asymmetry at first order in the baryon-number-violating interactions.
In this work, we will investigate how to generate both a baryon asymmetry at low energies and a correct dark matter density. We focus on models with baryonic RPV and show a new baryogenesis mechanism in which a late-decaying axino LSP gives rise to the observed baryon asymmetry (at second order in the R-parity violating interactions). Baryogenesis takes place below the weak scale and before Big Bang Nucleosynthesis. Both gravitino and axion dark matter are discussed, with the nature of dark matter constraining the parameter space for baryogenesis and vice versa.
This paper is organized as follows: we recall the Nanopoulos-Weinberg theorem in section 2, and discuss the implications for baryogenesis through an LSP decay. In section 3 we derive a new mechanism that generates substantial baryon asymmetry through late decays of an axino LSP. In section 4, we investigate the possibilities for dark matter candidates and discuss the range of parameter space in which both a baryon asymmetry and a correct dark matter density exist. We discuss the collider bounds on our model and conclude in section 5.

The Nanopoulos-Weinberg theorem and LSP baryogenesis
In Ref. [10] it was shown that, given a particle X which is stable in the limit of no baryonnumber-violating interactions, the decay rate of X into all final states with a given baryon number B equals the decay rate of its antiparticle X into all states with baryon number −B, at first order in the baryon-number-violating interactions. Because the net number of baryons is proportional to the difference of the two decay rates, no baryon asymmetry can be generated by an LSP decay at first order.
The Nanopoulos-Weinberg theorem was further investigated in Ref. [11], where it was generally argued that a difference in the two decay rates exists only if the on-shell intermediate particles and the final particles have a different baryon number. In other words, the process to the right of the "cut" must violate baryon number. This results holds at all orders in the baryon-number-violating interactions. For an LSP decay, baryon-number-violating operators must also appear on the left of the "cut" to have on-shell intermediate particles. If the parent particle is not the LSP (it has baryon-number-conserving decay channels), it is possible to have an asymmetry at first order, with a baryon-number-conserving interaction to the left of the "cut". The models of [2][3][4] fall in this last category, as they used decays of heavier particles, and were indeed able to get an asymmetry at first order in the RPV couplings.
For the LSP case, we find it useful to consider a low-energy effective theory containing the LSP χ and the SM fields, where all the heavier degrees of freedom have been integrated out. Baryon number violation is present in non-renormalizable operators which are suppressed by the heavier particles masses (in RPV SUSY, the squarks). The effective operators involving the quarks q and the LSP will be of the schematic form where the gauge indices are contracted to form gauge singlets and Λ is the mass scale of the heavier particles. Then the decay rate of χ becomes An asymmetry between Γ χ→qqq and Γχ →qqq will come from the interference between the tree level ∆B = 1 decay and a two-loop decay diagram involving both L |∆B|=1 and L |∆B|=2 : It is worth to note that the asymmetry is generated not only at second order in the baryonnumber-violating interactions (as expected by the Nanopoulos-Weinberg theorem), but also at two-loops: a 1-loop diagram would require a dimension six, ∆B = 2 operator in the effective theory, which is not allowed. Finally, the present baryon asymmetry depends on the LSP abundance at decay time, This can be compared to its experimental value [12], (Y ∆B ) obs = (0.80 ± 0.018) × 10 −10 .
In the next section, we will see the axino LSP is a good example to realize this mechanism.

Baryogenesis from axino decays
In this section we present a concrete example of the LSP baryogenesis scenario just outlined. First, the particle has to decay out of thermal equilibrium. A typical candidate would be an LSP gravitino, produced in the reheating epoch [13,14]: without R-parity, gravitinos decay to three quarks via the RPV operator of eq. (1.1). Compared a non-LSP gravitino, this decay is much slower, as it is suppressed by the intermediate squark mass, by the RPV coupling λ and by the three-body kinematic factor. In order for the decay products not to interfere with Big Bang Nucleosynthesis (BBN), the gravitino LSP would be extremely heavy, m 3/2 10 3 TeV, with the other superpartners being even heavier. Even so, without additional sectors there would be no dark matter candidate, as all superpartners have decayed in the early universe.
Thus, we examine a supersymmetric QCD axion model in which the axion (a), the pseudo-Goldstone boson associated with the spontaneous breaking of an anomalous Peccei-Quinn (PQ) U (1) symmetry at a scale v P Q , solves the strong CP problem of QCD [15] and is a good dark matter candidate. Assuming that the axino (ã), the fermionic superpartner of the axion, is the LSP, a baryon asymmetry can be generated by its decays in a more natural region of the parameter space than in the gravitino LSP case. The large value of v P Q ensures that axinos are out-of-equilibrium when they decay, and the lifetime is much longer than the period in which RPV interactions are in thermal equilibrium (thus, the asymmetry is not washed out). The chiral axion superfield can be written as where the saxion, the scalar superpartner of the axion, is denoted as s. We consider a following superpotential terms for A to give interactions with the MSSM particles: where H u , H d are the MSSM Higgs doublets, and Φ, Φ c are SM charged matter fields with This is a hybrid of the DFSZ [18,19] and KSVZ [16,17] axion models, in which the axino decay is dominated by the first term as in the DFSZ case while at high temperature its thermal production is mostly given by that of the KSVZ model [20]. Because all the sparticles are heavier, we can consider a low-energy effective theory with SM quarks supplemented by the axino, a Majorana particle with a mass mã. Nonrenormalizable interactions for the quarks remain after integrating out the squarks in the diagrams of Fig. 2. The following effective axino interactions are given by the mixing of the axino with the higgsino, after integrating out the squarks: where q α (respectively, q c α ) are the left-handed (right-handed) quarks. The holomorphic term in the second line comes from the LR squark mixing, X uα ≡ A uα + µ cot β. κ α is an O(1) coefficient given by the charges of the SM fields under the PQ symmetry. 2 The six-fermion holomorphic ∆B = 2 Lagrangian is obtained with the contribution of soft SUSY breaking A-terms, ∆L soft = λ ijk A ijkũ RidRjdRk + h.c., after integrating out the right-handed squarks: Because the effective axino coupling to quarks is proportional to the up-type quark masses, the dominant one involves the top quark. The relevant R-parity violating couplings are λ 312 , λ 313 , λ 323 . For simplicity, we take the assumption that λ 323 is the only dominant coupling. 3 Assuming mã m t , the tree-level decay rate of the axino LSP (Fig. 2) is given by Thus the total decay rate is Γã = Γã →tbs + Γã →tbs ≈ 2Γã →tbs . Corrections to this result are proportional to m 2 t /m 2 a and m 2 a /m 2 t R and are shown in the Appendix.

Axino cosmology
Even if the interactions between the axinos and the MSSM particles are quite suppressed, axinos can be generated from the thermal bath in the early Universe. For a reheating temperature (T R ) much higher than the MSSM sparticle masses and much lower than the PQ breaking scale (µ , axinos are mainly produced by scattering processes mediated by gluinos [24][25][26][27][28][29]. The axino yield from this thermal production is , where nã is the axino number density, s the entropy density of the Universe given by (2π 2 /45)g * s T 3 , g * (T ) g * s (T ) the number of relativistic degrees of freedom at T . If the reheating temperature is high enough (T R > T dec. ), the scattering processes can be in chemical equilibrium. In such a case, the axino number density is nã = (3ζ(3)/2π 2 )T 3 before it decouples, and the corresponding yield after decoupling (T < T dec. ) is given by the second term of the RHS of Eq. (3.6).
In our cosmological consideration, axinos should decay before BBN, as the baryon asymmetry is generated by their decay. We denote by T D the axino decay temperature, defined by Γã = H(T D ): The condition that the axino decays before BBN corresponds to T D 10 MeV. In the absence of large axino-squark hierarchy, 4 it gives a upper bound on v P Q of about 10 12 GeV. Because T D mã, axinos become non-relativistic when the temperature drops below mã. Unless they decay beforehand, they will eventually dominate the energy density of the universe. The temperature at which the axino density equals the radiation density is For T D > T eq , axinos decay before dominating the energy density; for T D < T eq , they decay after, injecting a non-negligible amount of high-energy decay products in the thermal plasma. This has the effect of increasing the entropy, and the axino yield at decay is One can check that, for T D > T eq , the axino yield is given by Eq. (3.6), while for lower decay temperatures it is given by (3/4)(T D /mã).
In the axino decay, a difference in the decay rate to quarks vs. antiquarks is needed to generate a baryon asymmetry: the parameter = (Γ(ã → qqq) − Γ(ã →qqq))/(Γ(ã → qqq) + Γ(ã →qqq)) gives the net asymmetry per axino decay, and the net yield is On the other hand, the saxion is also produced from the thermal plasma in a similar amount as the axino. Its decay rate is much larger than that of the axino, because the saxion can decay through the R-parity conserving interactions. The saxion always decays to axions with Γ s→aa = m 3 s /(64πv 2 P Q ), where m s is the saxion mass. If kinematically allowed, the dominant decay mode of the saxion would be s → hh with (3.11) Thus saxions decay much earlier than the axinos. Furthermore, because their decays do not produce any baryon asymmetry, the role of the saxion cosmology is really negligible.

Axino baryogenesis
As discussed in section 2, no contribution to comes at one-loop. The interference between the tree-level decay and the two-loop decay (involving the ∆B = 2 interaction in Fig. 2(c)) gives a non-zero asymmetry, Here, all mass squared terms represent real values. Again, this is computed in the limit of massless final states and heavy intermediate squarks, and the full expression is discussed in the Appendix. In the following we will denote by Φ the relative phase between the axino mass and the RPV A-term A 323 , Φ ≡ Im[mãA * 323 ]/|mãA 323 |, and by mq the average squark mass scale, mq ≡ (m 2 s R ) 1/6 . As an example, we give two benchmark points that reproduce the correct baryon asymmetry: fixing |λ 323 | = 1, Φ = 1 and mã = 500 GeV, the other parameters are: when the axino yield at decay is maximal, (nã/s) D = (135ζ(3)/4π 4 )(1/g * ). We will discuss these benchmark points in relation to the nature of dark matter in the next section. But first, let us discuss our result, Eq. (3.12) in more details: • The asymmetry is proportional to the fourth power of the RPV coupling. This is expected from our discussion of the Nanopoulos-Weinberg theorem in section 2 and it differs from other baryogenesis mechanisms, such as in [2][3][4], where the coupling is only squared. Thus, a large λ 323 is strongly preferred. How large can it be? In [30], the RG running of the R-parity violating couplings was considered, and an absolute upper limit of λ 323 = 1.07 was set from the condition that perturbativity is valid up to the GUT scale. Although from the point of view of a low-energy effective theory this is not a problem, and one can just expect that new degrees of freedom appear around the Landau pole, we will assume λ max = 1 in the rest of this paper.
• The baryon asymmetry is proportional to (mã/mq) 3 (m t /mq) 2 ; it is suppressed for a large hierarchy between the squark mass and either the axino mass or the weak scale. Even for mã ∼ mq, there is a suppression by m 2 t /m 2 q , which points to an upper limit on the squark scale. For λ 1, we find an absolute upper bound on the squark masses at 35 TeV for large soft terms A 323 X t 3mq; larger values for A-terms are potentially dangerous in that they can generate color-breaking vacua [31].
• Additionally, the baryon asymmetry is proportional to the relative phase between the axino mass mã and the soft SUSY breaking parameter A 323 . There is no direct constraint on the CP phase of A 323 but an indirect constraint is provided by the null results in the measurement of the Electric Dipole Moment (EDM). CP phases for the MSSM A-terms A U,D (in particular, the phase φ A Qg ≡ Im[mgA Q ]/m 2 g , where mg is the gluino mass and Q = U, D) contribute to the neutron EDM [32], TeV mq 2 (3.14) while experimentally the upper limit is |d n | < 2.9 × 10 −26 e cm [33]. This implies either a small phase φ A Qg or superpartners in the multi-TeV range. In our model, we can have a large baryon asymmetry and a small contribution to the neutron EDM in two ways. First, unlike the models of [3] in which gluino decays contribute to the baryon asymmetry, even with a common CP phase for all the A-terms the baryon asymmetry and the neutron EDM are proportional to different phases. Thus Im[mãA 323 ] = Im[mãA Q ] could be maximal, while Im[mgA Q ] could be small. Second, the phase of A Q and A 323 might be independent at the messenger scale so that A 323 has a large phase while the MSSM A-terms could have small ones. The RG running can generates a non-zero (but small) A Q , phase at low energies, that does not contribute too much to the neutron EDM.
Summarizing, the contributions to the neutron EDM depends on the SUSY breaking sector and on the phases generated at that scale. The CP phase needed for baryogenesis is not the same as the one contributing to neutron EDM.
To conclude this section, we have a mechanism for generating the right baryon asymmetry that points to large R-parity violation, not too large squark masses, and can be safe from the null experimental results for neutron EDM. Large R-parity violation is not a problem if it is confined in interactions involving heavy quarks, otherwise there are many potentially large baryon-number-violating contributions to low-energy flavor physics (see [34] for a review). Even if the only non-zero coupling is λ 323 , couplings involving light quarks are generated at 1-loop level [2]: for λ 323 1, we find λ 112 10 −8 , λ 223 10 −5 , which are too small to significantly contribute to KK mixing or n-n oscillation. We can also revisit the assumption of single-coupling dominance in the decay of the axino and see if the presence of other couplings is consistent with flavor physics. An important bound for the case with a non-negligible λ 313 coupling comes from contributions to ∆m K [35], which for λ 323 1 and TeV-scale squarks, implies |λ 313 | 3×10 −2 . Then, the single-coupling dominance assumption was justified and the λ 313 -mediated contribution to the axino decay is negligible.

Dark Matter
We now turn our attention to the presence of dark matter. We first recall the dark matter density from the Planck satellite's CMB measurements (combined with WMAP9 polarization maps) [12], Without R-parity, no supersymmetric particle is stable and indeed the axino, which can be a viable dark matter candidate in R-parity-conserving models [24], decays and generates the baryon asymmetry. There are two natural candidates for dark matter that are already in the model: axions and gravitinos. Coherent oscillation of the axions can give rise to cold dark matter if the PQ symmetry breaking scale is properly taken. If the gravitinos are the LSP, their lifetime can be long enough that they are the dark matter at present times. The abundance of the gravitino can be sizable by taking proper values of its mass and the reheating temperature T R .

Heavy gravitino scenario
When the gravitino is heavy enough to decay through the R-parity conserving interactions, the only possible candidate for dark matter is the axion. The axion cold dark matter is generated when the axion starts to oscillate coherently at the QCD phase transition. Its abundance is given as [36] Ω a h 2 = 1 ∆ a k a θ 2 a v P Q 10 12 GeV where k a is a numerical factor of O(1), θ a is the axion misalignment angle, and ∆ a is the possible dilution factor from entropy release when axinos decay after the axion coherent oscillation has started. In viable parameter regions, we find that ∆ a is just O(1). The initial angle θ a is not averaged out because we assume the PQ symmetry is broken from the inflation epoch. There is no dark matter contribution from the axionic string decays for the same reason. With the natural value of the angle θ 2 a = θ 2 a ∼ 3, v P Q ∼ 10 11 GeV explains the present density of dark matter. From the dark matter constraint, a larger value of v P Q is allowed if we take a small value of θ a . However, v P Q cannot be too large, otherwise axinos will decay after the BBN era. For a low T D , using Eqs. (3.7) and (3.12), v P Q can be represented as v P Q = 10 12 GeV mã 0.5 TeV Thus, we get v P Q 10 12 GeV for reasonable parameter values. The allowed range is rather small, 10 11 GeV v P Q 10 12 GeV. (4.4) In order to produce sizable baryon asymmetry, the reheating temperature should be high enough, but it is notable that T R need not be as large as v P Q . This is consistent with the assumption that the PQ symmetry is not restored in the reheating epoch. As an example, the observed dark matter abundance and baryon asymmetry are generated for v P Q = 10 11 GeV, T R = 10 7 GeV at the benchmark point BP1, where mã = 500 GeV, mq = 750 GeV, A 323 X t = √ 6mq, λ 323 = 1, Φ = 1. Because the baryon asymmetry is inversely proportional to the squark mass mq and the PQ scale v P Q , with such a high value of v P Q we find an absolute upper bound of mq 7 TeV. Note that this upper limit is found taking mã mq and large A-terms, A 323 X t 3mq, so that it corresponds to a rather tuned region of the parameter space. For a more natural choice of parameters, the squark mass has to be below 1 TeV.
On the other hand, although the gravitino is not a present dark matter candidate, its lifetime can be long enough to cause problems. The decay rate of the gravitino is where n V (n C ) is the number of vector (chiral) supermultiplets whose masses are smaller than the gravitino mass. When the gravitino is heavier than the MSSM sparticles, its decay products and their amounts are strongly constrained by successful prediction of the standard Big Bang nucleosynthesis [37]. For reheating temperatures around 10 7 GeV (needed to generate enough axinos), the gravitinos have to decay before the BBN era, i.e. τ 3/2 < O(0.1) sec. This requires m 3/2 50 TeV, corresponding to a spectrum typical of anomaly-mediation of SUSY breaking.
The late time decay of heavy gravitinos could also contribute to the baryon asymmetry, as in [3]. However, in [3] the gravitinos were dominating the energy density of the universe at decay time, implying T R ∼ 10 15 GeV, while in our case the yield of the gravitino is too small to contribute to a baryon asymmetry of n B /s ∼ 10 −10 .

Light gravitino scenario
When the gravitino is the true LSP, the axino is the NLSP and can decay to the gravitino and the axion with a decay rate [38] Γã →aψ 3/2 = 1 96π Because the branching ratio Br(ã → aψ 3/2 ) is quite small, the baryogenesis mechanism is effectively the same as for an axino LSP, and our previous discussion holds. However the non-thermal production of gravitinos by axino decays can provide a sizable abundance of dark matter as (4.7) The second line is evaluated for T D > T eq . For T D < T eq , there is a further dilution by the factor T D /T eq . On the other hand, the thermal production at reheating reads [14,39,40] Here the gluino mass mg is assumed to be of the same order of magnitude as the squark mass. We note that for given m 3/2 and mq mg, the thermal production is always the dominant contribution. For relatively low T R , the non-thermal production also can be important.
If light gravitinos are produced in the right amount, they can give the correct relic density, provided that their lifetime is longer than the age of the universe (they decay via RPV interactions, ψ 3/2 → qqq). As a matter of fact, the condition on the gravitino lifetime is stronger, as the hadronic decay products would contribute to the cosmic ray antiparticle population, which is looked at in experiments such as PAMELA or AMS-02 [41][42][43]. For example, in [44] it was shown that the lifetime of a vanilla DM candidate decaying to bb is constrained to be bigger than about 5 × 10 27 sec from the non-observation by PAMELA of an excess in thep/p fraction, for 80 GeV m DM 500 GeV (future antideuterons experiments will do better in the lower mass range). To translate these results to the case of a gravitino decaying to three quarks, it is necessary to find how many antiprotons are generated and compare it to the case of a bb final state, for each value of the DM mass. This effort is being tackled by one of the authors in [45], and it is generally found that the number of antiprotons in the experimental energy range produced in the ψ 3/2 → qqq case is approximately the same as in the χ → bb case, with variations of around ±30%, depending on the particle mass and the specific flavor structure of the final three-quarks state. It is then reasonable to take the lower bound τ exp ψ 3/2 10 27 sec on the gravitino lifetime, when it makes up all of the dark matter. This bound is conservative enough to not be sensitive to the uncertainties in the precise number of antiparticles arising from the gravitino decay.
The gravitino lifetime can be computed as [43] τ ψ 3/2 →u i d j d k = 1.28 × 10 26 sec 1 λ ijk q ). Finally, there is also a lower bound on the gravitino mass, coming from the one-loop proton decay channel p → K + ψ 3/2 setting a limit on λ 323 [46]: For λ 323 = 1, mq = 1 TeV, the corresponding lower limit on the gravitino mass is m 3/2 2 MeV. In Fig. 3, we fix the RPV coupling to λ 323 = 1, the axino and squark masses to mã = 0.5 TeV and mq = 1 TeV, and vary the remaining parameters in the v P Q − T R plane. The black horizontal dashed lines are contours of different values of m 3/2 that give the correct dark matter relic abundance in the range 2 MeV m 3/2 4 GeV. The ranges excluded by proton decay and cosmic ray observations are respectively shown at the bottom in green and at the top in red (taking λ 223 1). The non-thermal production from axino decays contributes to a dip in the lines, more easily seen in the lower part of the plot. For higher gravitino masses (and higher reheating temperatures) a spike can be seen, corresponding to the parameter region in which axinos decay when they dominate the energy density of the universe (T D < T eq : at higher reheating temperatures axinos are produced more efficiently); because the gravitinos produced at reheating are diluted by axino decays, a higher reheating temperature than naively thought is allowed, resulting in a higher number of produced gravitinos. The diagonal lines (becoming vertical around the center of the plot) are contours that give the correct baryon asymmetry, with different values of the soft terms and the CP phase Φ = Im[mãA * 323 ]/|mãA 323 |. Their behavior can be understood in the following way: for low reheating temperatures, the axino yield depends on both T R and on the PQ scale v P Q , see Eq. (3.6). For high T R , the yield is just given by the thermal scattering expression, independent of v P Q . As we increase the variables Φ, A , X t that determine the asymmetry parameter , the correct baryon asymmetry can be generated at higher values of v P Q , that is, with weaker axino interactions. In the shaded region to the left, not enough asymmetry can be generated, because the axino yield is too small to start with and cannot be too large. In the light-blue region on the right, a correct baryon asymmetry can be generated only by taking large values of the soft A-terms, which is dangerous from the point of view of color-breaking vacua (the stop squarks might acquire a vev). We excluded the region with A 323 X t 3mq (see [31] for a more detailed discussion). On the right, in the light yellow region the axion can be dark matter (depending on the precise value of the misalignment angle) and gravitinos can either have decayed already or be a sub-dominant dark matter component, while in the gray rightmost region the axino decays at 0.1 − 1 sec, compromising the observed abundances for light nuclei produced during BBN. We note that both baryogenesis and dark matter can be accounted for in most of the "axion window", 10 9 GeV < v P Q < 10 12 GeV, for reheating temperatures as low as 10 TeV and as high as 10 7 −10 8 GeV. It is interesting to point out that the choice of fixed parameters in Fig. 3 is in some way optimal: for heavier squark masses (for fixed mã/mq) the asymmetry parameter becomes smaller. For an almost degenerate axino LSP, mã ≈ mq, more parameter space opens up, as is bigger, and smaller A-terms (and phases) are allowed; in this case, the higher squark mass allowed is 35 TeV.
We finish this sub-section with a comment on the Higgs mass: large A-terms are needed to achieve a 125 GeV Higgs boson with light stops in the MSSM, and at the same time large A-terms increase the asymmetry parameter . A 125 GeV Higgs with maximal mixing 6mq) allows non-maximal values for the relative CP-violating phase Φ, such as Φ = 1/3.

Conclusions
We have discussed a new mechanism for baryogenesis through the R-parity violating decay of an axino LSP, at the two-loop level and at the second order in the baryon-number-violating couplings. A suitable dark matter density is also generated by related processes, namely by the coherent oscillation mechanism for axions and by thermal scatterings and the axino decays for gravitinos. The scenarios described are very predictive: for the case of axion dark matter, the allowed range for the squarks extends to about 7 TeV; additionally, the initial axion misalignment angle is large. For the case of gravitino dark matter, the gravitino mass is between a few MeV and a few GeV, with proton decay and cosmic rays experiments capable of narrowing this interval; in this case the upper limit on squark masses is higher, of order 35 TeV. The cited limits on the squark masses correspond to tuned regions of the parameter space, where the axino mass is very close to the squarks masses; requiring that the axino and the squarks masses are different by at least 10% brings down the upper squark mass limits to 5 TeV and 25 TeV. In both cases, the axino should be close to the squark mass, up to a factor of a few, and R-parity violation should be maximal, corresponding to prompt decays of superpartners.
At the LHC, the most important signatures of light RPV squarks would be multijets, with at least two jets from each squark, and three jets from the decay of a gluino. The most relevant experimental searches are [47] from CMS and [48,49] from ATLAS. In particular, Ref. [48] studied the decay of pair-produced gluinos to six quarks, and used b-tagging to probe the flavor structure of the RPV couplings λ ijk . Gluino masses below 874 GeV are excluded for gluinos whose decay products include a top and a bottom (as it is the case for large λ 323 coupling). Unfortunately these are limits on the gluino masses and, apart from the matter of naturalness, are of little importance for our baryogenesis model, which is mediated by squarks. In fact, because the cross section for pair-produced stops is smaller than for pair-produced gluinos, RPV squarks are relatively unprobed at the LHC; for example, LSP squarks are best probed at the Tevatron by the CDF experiment, excluding squark masses up to about 100 GeV [50]. With dedicated searches, the LHC at 14 TeV has the potential to exclude RPV squarks up to about a TeV [51,52]. For our axion dark matter scenario, most of the natural region of the parameter space can be probed.  Exact numerical results Even in the limit of massless final states, we could not find a simple analytical expression for the asymmetry parameter . We can integrate the fourdimensional phase space integrals numerically and check that the simple expressions given in the main part of the article do not introduce a large error. We show the full numerical results for Γã and in the mt mã − mã mq plane in Figs. 4 . The benchmark points of eq. (3.13), used in Sec. 4, are indicated by white star markers. We see that Γ and can decrease by a factor of about three. This is mainly due to relaxing the approximation m t = 0, reducing the phase space available for the decay. The exact numerical results have been used for Fig. 3.