Moduli-induced Baryogenesis

We study a scenario for baryogenesis in modular cosmology and discuss its implications for the moduli stabilization mechanism and the supersymmetry (SUSY) breaking scale. If moduli fields dominate the Universe and decay into the standard model particles through diatonic couplings, the right amount of baryon asymmetry can be generated through CP violating decay of gluino into quark and squark followed by baryon-number violating squark decay. We find that, in the KKLT-type moduli stabilization, at least two non-perturbative terms are required to obtain a sizable CP phase, and that the successful baryogenesis is possible for the soft SUSY breaking mass heavier than O(1) TeV. A part of the parameter space for successful baryogenesis can be probed at the collider experiments, dinucleon decay search experiment, and the measurements of electric dipole moments of neutron and electron. It is also shown that similar baryogenesis works in the case of the gravitino- or the saxion-dominated Universe.


I. INTRODUCTION
The origin of baryon asymmetry is one of the most profound puzzles in modern cosmology and particle physics. It is well-known that baryon accounts for about 5% of the total energy density of the Universe [1]: where n B is the number density of baryon, and s is the entropy density. The standard model (SM) of particle physics and standard cosmology, however, do not provide any viable mechanism to generate the observed baryon number. To be specific, three conditions known as the Sakharov conditions [2] have to be satisfied for successful baryogenesis: (i) baryon number violation, (ii) C and CP violation, and (iii) departure from thermal equilibrium. These conditions require physics beyond the SM and set tight constraints on possible scenarios for baryogenesis.
The mechanism for baryogenesis is closely related to the thermal history of the Universe. The recent observations of cosmic microwave background as well as large scale structure of the Universe firmly support the inflationary Universe in which our Universe experienced an accelerating cosmic expansion at an early stage of the evolution [3,4].
Thus baryon asymmetry must be generated after inflation since otherwise the exponential expansion during inflation would completely dilute any pre-existing baryon asymmetry.
However, the history of the Universe after inflation is poorly known. The inflaton may directly decay into the SM particles to reheat the Universe. The right amount of baryon asymmetry can be generated through thermal leptogenesis if the reheating temperature is sufficiently high [5]. On the other hand, the evolution of the Universe could be more involved. Indeed, there are many moduli fields in supergravity and superstring theories, which may have a significant effect on the thermal history. They are known to easily dominate the energy density of the Universe because the moduli fields are copiously produced by the coherent oscillation and that their interactions are extremely weak, suppressed by the Planck scale or Grand Unified Theory (GUT) scale. Then a huge amount of entropy is released by the moduli decay, diluting pre-existing baryon asymmetry. As a consequence we have two possibilities for baryogenesis. One is to create a sufficiently large amount of baryon asymmetry before the moduli decay by, e.g., the Affleck-Dine mechanism [6,7], which has been extensively studied in a context of modular cosmology [8][9][10][11][12][13]. The other is to generate baryon asymmetry after the moduli decay. This is the main focus of the present paper.
We assume that moduli fields dominate the energy density of the Universe in the framework of supergravity. After dominating the Universe, moduli decay to lighter particles in the minimal extension of supersymmetric standard model (MSSM). The moduli decay temperature can be higher than O(1) MeV for the moduli mass heavier than about 10 TeV if their interactions are suppressed by the Planck mass, thus avoiding bounds from the big bang nucleosynthesis (BBN) [14]. In this paper we study a baryogenesis scenario in which the baryon asymmetry is generated through CP violating decay of gluino into quark and squark, followed by baryon-number violating squark decay. We will introduce renormalizable R-parity violating interactions to get the required baryon number violation. Late-time moduli decay can easily realize the out-of-equilibrium decays of gluino and squark. On the other hand, the size of the CP phase depends on how supersymmetry (SUSY) breaking is transmitted to the visible sector. As we will see shortly, the required CP violation for the baryogenesis has important implications for moduli stabilization. Successful baryogenesis is possible in moduli stabilization such that moduli-mediated SUSY breaking is sizable, and the axionic shift symmetry associated with the modulus is broken at least by two terms in the modulus superpotential. We will focus on mixed modulus-anomaly mediation in the Kachru-Kallosh-Linde-Trivedi (KKLT)-type moduli stabilization [15], as it provides a natural framework for the moduli-induced baryogenesis.
We will also show that similar baryogenesis works in other cases where the gravitino, the saxion, or the Polonyi field dominates the Universe, and decays mainly into the SM gauge sector.
Let us mention differences of the present paper from the works in the past. Baryogenesis induced by a modulus-like particle has been studied since long time ago. The gravitino-induced baryogenesis was proposed by Cline and Raby [16], followed by Moller-ach and Roulet who studied a similar scenario where the saxion plays the role of the gravitino [17]. Recently baryogenesis via such late-decaying particles was studied in a generic way by with higher dimension operators [18]. In those studies the origin of the CP phase was simply assumed to be independent of the decaying particle. In the present paper we study a baryogenesis within a concrete moduli stabilization mechanism, where the modulus field mediates SUSY breaking with a non-vanishing CP phase, and its decay provides out-of-equilibrium decays of gluino and squark. Therefore the moduli-induced baryogenesis studied in this paper offers a realistic and self-contained scenario. Later in this paper we will revisit the gravitino-induced baryogenesis. Here we will provide analytic formulae which can be applied to a more generic mass spectrum of superparticles.
Note that it was shown that the inflaton [19][20][21][22][23] as well as moduli [24] generically decays into a pair of gravitinos. Therefore the gravitino-dominated Universe can be realized in a broader scenario than originally assumed. Its cosmological and phenomenological implications have been studied recently in Ref. [25]. We will also revisit the saxion-induced baryogenesis. In Ref. [17], it was indicated that the saxion decay to gauginos is helicitysuppressed. As clearly shown in Ref. [26], however, it is not helicity-suppressed and its rate is generically comparable to that into gauge bosons. Taking into account this fact, we will show that the saxion-induced baryogenesis is more efficient than originally considered before.
The rest of this paper is organized as follows. In Sec. II we study mixed modulusanomaly mediation in the KKLT-type moduli stabilization, focusing on how to obtain the CP violation required for successful baryogenesis. We will then examine the baryogenesis by modulus decay in Sec. III, and study the implications for the SUSY breaking scale and discuss its experimental consequences. In Sec. IV we discuss similar baryogenesis by the decay of the gravitino and the saxion. Sec. V is devoted to conclusions and discussion.
In this paper we take the reduced Planck mass M P ≃ 2.4 × 10 18 GeV to be unity unless otherwise noted.

II. MODULI STABILIZATION
We consider the possibility that late-decaying moduli account for the observed baryon asymmetry of the Universe. To produce baryon asymmetry, we need couplings that violate both baryon number and CP. Baryon number violation arises from R-parity violating interactions while satisfying various experimental constraints, such as neutron-antineutron oscillation, dinucleon decay, and proton decay. The required CP violation, on the other hand, can be induced after SUSY breaking from the R-parity violating interactions. However, generating a nonzero CP phase is nontrivial. It depends on the SUSY breaking mechanism and the moduli stabilization.
As is well-known, anomaly mediation [27] (and also gauge mediation) generates flavor and CP conserving soft SUSY breaking terms because it takes place mainly through the SM gauge interactions. The soft terms induced by moduli F -terms are also considered to preserve both flavor and CP. Here the flavor conservation is a result of flavor-universal rational numbers called the modular weight, which determines the coupling between moduli and the matter fields in visible sector [28], meanwhile CP conserving soft terms are the consequence of axionic shift symmetries associated with the moduli [29]. However, this is true only for a simple KKLT-type moduli stabilization. It turns out that a sizable CP violation can be obtained from the moduli sector if • Soft SUSY breaking terms receive sizable contributions from the moduli F -terms, and • Non-perturbative corrections, which generate a modulus potential and break the axionic shift symmetry, involve at least two terms.
We will see that the KKLT scenario is a natural framework for the moduli-induced baryogenesis that works at low temperatures. It will be also shown in the next section that such a sizable CP violation for baryogenesis could have interesting experimental consequences.

A. KKLT mechanism
In the KKLT compactification [15], all the moduli are stabilized at a phenomenologically viable vacuum. There are three types of moduli; dilaton, complex structure moduli, and Kähler moduli. The dilation and complex structure moduli are fixed by fluxes while obtaining masses hierarchically larger than the gravitino mass. Incorporating nonperturbative corrections to the superpotential, one can also stabilize Kähler moduli at a supersymmetric anti-de Sitter (AdS) minimum. Since our vacuum is a SUSY breaking de where m 3/2 denotes the gravitino mass. Hence, if stabilized with mass much larger than 4π 2 m 3/2 , the modulus does not play an important role in generating the soft SUSY breaking terms because they always receive anomaly-mediated SUSY breaking contribution, m soft ∼ m 3/2 /4π 2 , in supergravity [27].
The stabilization of Kähler moduli can be examined within the effective theory after integrating out the dilation and complex structure moduli as they are much heavier than the gravitino and Kähler moduli [30]. Let us consider a simple case with a single Kähler modulus X. 1 It is straightforward to generalize to a case with multi Kähler moduli. The Kähler potential takes the no-scale form at the leading order in the α ′ (the string tension) and the string loop expansions, It should be noticed that the theory possesses the axionic shift symmetry, This shift symmetry can be explicitly broken at non-perturbative level. For example, hidden gaugino condensation or stringy instanton effects induce non-perturbative terms in superpotential. Including the non-perturbative corrections, the modulus superpotential is given by where the constant term ω 0 is originated from background fluxes, which is assumed to be much smaller than order unity in the Planck unit so as to get the soft SUSY breaking scale much lower than the Planck scale. We also note that the constant term explicitly breaks U(1) R . The above superpotential stabilizes the modulus at a SUSY AdS minimum Here ∂ X means a partial derivative with respect to the scalar field X, i.e., ∂ X ≡ ∂ ∂X . Then the modulus mass is determined by the curvature around the minimum as where the gravitino mass is given by m 3/2 = e K/2 W ∼ ω 0 , and we have used ∂ X K 0 = ∂ X * K 0 and the SUSY condition (6). After adding a sequestered uplifting potential V lift ∝ e 2K 0 /3 to cancel the negative cosmological constant, a small vacuum shift is induced. This vacuum shift results in a nonzero modulus F -term, which is order of |m 3/2 | 2 /m X .
Let us continue to examine how the SUSY breaking is mediated to the visible sector.
The soft SUSY breaking terms receive contributions from both modulus mediation and anomaly mediation [30][31][32], where φ i denotes the scalar component of the visible sector chiral superfield Φ i , λ a is the gaugino, and y ijk is the Yukawa coupling for the superpotential term Φ i Φ j Φ k . Here i, j, k are the indices of matter fields, and a = 1, 2, 3 corresponds to the U(1) Y , SU(2) L , SU(3) C gauge group, respectively. The mass of gaugino is obtained as when the gauge kinetic function is at the cut-off scale Λ, which we assume to be the GUT scale. k a is a positive constant.
Here we note that the gauge kinetic function is allowed to have only a term which is linear to X (except for constant term) due to the axionic shift symmetry, i.e., Eq. (4).
The gauge coupling reads g 2 a = 1/ Re(f a ) , and the beta-function coefficients are given by b a = (33/5, 1, −3) in the MSSM. On the other hand, m i and A ijk depend on the anomalous dimension γ i and the modular weight n i of φ i [33] that determines the modulus dependence of the matter wave function Z i . 2 For instance, the A-parameter reads where n i is a rational number of order unity, and it can have various values in the KKLTtype moduli stabilization with anomalous U(1) gauge symmetry [34]. Low energy values of the soft SUSY breaking parameters are determined by the renormalization group running, and extra gauge-charged matters if they exist at an intermediate scale [35][36][37].

B. CP violation in the moduli sector
As will be shown in the next section, in order to generate sufficient baryon asymmetry from modulus decays, the phase of the combination A ijk M * g should be sizable, i.e., where Mg is the gluino mass, and hereafter A ijk denotes the soft trilinear parameter associated with baryon-number violating superpotential terms, which will be presented soon.
In the KKLT scenario, moduli and anomaly mediations are comparable to each other, and the relative phase between the modulus F -term and the gravitino mass is not rotated away in the presence of two or more non-perturbative superpotential terms. This indeed makes it as a natural framework to implement the baryogenesis in the modulus-dominated Universe. From the relations (10) and (12), the phase is naively estimated to be omitting an order unity coefficient which depends on the renormalization group running and extra gauge-charged matter fields. The parameter α represents the ratio between moduli and anomaly mediations, where we have used that the modulus F -term is given by Eq. (8). The original KKLT scenario gives α ≃ ln(M P /m 3/2 )/4π 2 = O(1). It is obvious that the phase is suppressed if one of the mediation mechanisms dominates over the other. On the other hand, the approximate symmetry, U(1) R and U(1) X , indicates that W np should include at least two non-perturbative terms to get nonzero arg(m 3/2 F X ). This is understood as follows.
Suppose that there is only a single non-perturbative term. One can then remove the CP phases in the superpotential by redefining ω 0 and X. However, it is not generically possible to rotate away the CP phase(s) of additional non-perturbative term(s). That is why at least two non-perturbative terms are needed to get a CP phase. 3 Now let us estimate the CP phase more quantitatively. For this purpose, we consider the superpotential, with a = b and a ∼ b ∼ 4π 2 , where A and B are order unity complex constants. Here the approximate U(1) R and U(1) X (explicitly broken by the constant and non-perturbative terms) allow us to make ω 0 and A be real and positive numbers without loss of generality, which is the convention we will take hereafter. This explains why X is fixed at a CPconserving minimum with mass in the absence of the last term in Eq. (16), as was considered in the original KKLT mechanism. Here we have used m 3/2 ∼ ω 0 and A ∼ 1. Let us now turn on the Be −bX term with Im(B) = 0. Then the SUSY condition reads neglecting small terms suppressed by 1/a(X + X * ) or 1/b(X + X * ). Since the righthand side is a real number, Im(B) = 0 shifts the minimum to a CP-violating one, i.e., Im(X) = 0. Using the facts that the modulus F -term is generated according to the relation (8) after uplifting, and that the modulus is fixed at a Re(X) ≃ ln(M P /m 3/2 ), where the superpotential (16) gives Therefore, the CP phase can be sizable if aAe −aX and bBe −bX are comparable to each other in size. 4 Under the assumption that |bBe −bX | |aAe −aX |, the vacuum shift induced by Im(B) leads to for b similar to but larger than a. Here a/b is a rational number if the non-perturbative terms arise from hidden gaugino condensation. For instance, the phase is around 0.05-0.1 Let us close this section by briefly mentioning the SUSY breaking scale. We are interested in the case where the CP violation in the moduli sector gives rise to a sizable arg(A ijk M * g ) to implement the baryogenesis. Thus, to avoid the SUSY CP problems, the MSSM sparticles should be heavier than 1 TeV unless there is a cancellation among sparticle contributions in the amplitudes of physical processes. We will quantitatively discuss the constraint on the CP phase later. Heavy sparticles around over TeV scale may indicate a little hierarchy problem regarding the Higgs boson mass. However, to put it another way, such a large sparticle (mainly stop) mass is one of possible explanations for the 126 GeV Higgs boson discovered at the LHC [39,40].

III. BARYOGENESIS
In this section we will discuss the generation of baryon asymmetry by modulus decay.
During inflation the modulus is likely deviated from the true vacuum in the low energy, due to the deformation of the potential through gravitational interactions with the inflaton.
After inflation ends, it starts coherent oscillation when the Hubble parameter becomes comparable to m X with a large initial displacement of the order of the Planck scale, and then soon dominates the energy density of the Universe. Eventually the modulus field decays to lighter particles. The modulus decay releases a huge amount of entropy, which would dilute any (harmful) relic of the early Universe, and reheats the Universe at the temperature T X given by where g * is the relativistic degrees of freedom, and Γ X is the total decay width of the modulus field. Although the late modulus decay might washout pre-existing baryon asymmetry as well, we will show that sufficient baryon asymmetry is generated by the modulus decay followed by the gluino decay if the R-parity is violated. Here we will also discuss the experimental consequences of this baryogenesis scenario.

A. Baryon asymmetry
The modulus decays into the MSSM particles through its couplings to the visible sector.
For the case that the gauge couplings are determined by X , the modulus X dominantly decays into gauge boson pairs, and gaugino pairs through the interactions (written in two-component notation) [24], being the canonically normalized fluctuation about the vacuum. Here G a µν is the field strength of gauge field, and the modulus couplings are given by where the modulus coupling to gauginos has been derived by using the relation evaluated at the vacuum. Then the decay rates to a gauge boson pair and a gaugino pair are given by respectively, for both the real and imaginary components of X. Here A a denotes the gauge boson, neglecting its mass, and we have used the gauge kinetic function f a = k a X.
Under the assumption that modulus decays into hidden sector particles are suppressed, the total decay rate of the modulus is simply written as neglecting the gaugino mass in the final state. Then, from Eqs. (22) and (28), the reheating temperature is estimated as Since the decay rate is Planck-suppressed, the reheating temperature is usually lower than the typical SUSY breaking scale, which means that the produced sparticles are out of equilibrium. This is an essential ingredient, i.e., for the Sakharov condition (ii), for the baryogenesis via subsequent gluino and squark decays. It is also important that the branching fraction into a gluino pair is given by Thus gluinos, whose decay is the source of the baryon asymmetry, are abundantly produced by modulus decay.
For the baryogenesis we introduce the baryon-number violating renormalizable opera- tors in addition to the MSSM superpotential W MSSM ; 5 where U c i and D c j are the SU(2) L singlet up-type and down-type quarks, respectively, and i, j, k are flavor indices while color indices are implicit. The other renormalizable lepton number violating operators can be forbidden by generalized lepton parities or discrete R symmetries [42]. From the superpotential we obtain the following Lagrangian, Here the tilde denotes the scalar superpartner, and P R/L are projection operators defined as P R/L = (1 ± γ 5 )/2 (plus for P R and minus for P L ). The first line in the right-hand side includes SUSY couplings, while the second line is the soft SUSY breaking trilinear terms, which correspond to the second term in Eq. (9).
The gluino mainly decays to quark and squark through the tree-level diagram in Fig. 1.
The decay rate forg →q iqi reads where r i = m 2 q i /|Mg| 2 , neglecting the quark mass. Here α s = g 2 3 /4π is the strong coupling constant, and mq i denotes the mass of squark in the final state. The total decay rate of the gluino is given by the sum of all possible final sates of quark and squark, On the other hand, an asymmetry between Γ(g →q iqi ) and Γ(g →q * i q i ) may be induced at the loop level. If there exists another decay mode toq * j q j , the interference between tree and loop diagrams (Fig. 1) generates nonzero asymmetry via the baryon-number violating operator given in Eq. (32); where Here the constant c ij has a nonzero value only for the process kinematically allowed: c ij = 2 if the intermediate and final states (i.e.,q * j q j andq iqi ) are both down-type (s)quarks, and c ij = 1 otherwise. In the above, we have ignored left-right mixing in the squark sector for simplicity. It is straightforward to take into account the left-right mixing. The function f (r i , r j , r k ) behaves as f (r i , r j , r k ) ≃ 1 when r i,j,k ≪ 1. On the other hand, it is possible that r k is greater than one. If r k > 1 (and r i,j ≪ 1), then it is suppressed as f (r i , r j , r k ) ≃ 1/2r k . This behavior is expected because the loop diagram in Fig. 1 should vanish in the limit mq k → ∞. 6 Now we discuss net baryon number generated by the gluino decay. First we write the (s)quark number which is generated by the decay of a single gluino as Even though it is zero just after the gluino decay, nonzero baryon number is generated by subsequent ∆B = 0 squark decay processes. Possible decay modes of the squark are ∆B = 1 processes such asq i →q jqk , and ∆B = 0 processes such asq →χ 0 q withχ 0 being a neutralino. Here the ∆B = 1 process also includes the case whereq →q ′ W/Z and subsequentlyq ′ decays to quark pairs via the baryon-number violating operator.
Eventually the net baryon number generated by a single gluino is obtained as Here Brq i is the branching fraction for the ∆B = 1 decay process, which can be the dominant mode when the λ ijk is order unity, or it is unity whenq i is the lightest sparticle.
Then, the baryon number density from a gluino decay is where ng and n X are the number density of gluino and the modulus, respectively, and we have used Eq. (30) at the last step. Under the assumption that all the energy density of the modulus turns into radiation at its decay, i.e., ρ X = m X n X ≃ ρ R (ρ R is the energy density of radiation), the yield of baryon asymmetry is obtained as Let us estimate the baryon asymmetry in this baryogenesis scenario. For the estimation of the asymmetry, we assume that only λ 332 is nonzero for simplicity. 7 The total decay 7 Actually the case where only λ 332 is sizable is realistic in the phenomenological point of view. One of well-motivated scenarios is the minimal flavor violation [43].
width of the gluino is then given by Here a factor four counts CP conjugate, and left-handed and right-handed fields, and n f is the number of quark flavors to which gluino can decay. We have taken the limit r i ≪ 1 for the kinematically allowed decay for simplicity. In the same limit and taking n f = 6, then the asymmetry is given by which leads to 8 This is the typical scale of asymmetric parameter when the gluino is much heavier than the squarks. Furthermore taking Brt ,b,s = Brq, we get which results in n B s ≃ 4.9 × 10 −10 g * 106.75 Therefore, the observed baryon asymmetry of the Universe can be obtained with O(0.1) CP phase and O(1) baryon-number violating coupling if the modulus X has a mass around 10 6 -10 8 GeV. In the later numerical calculation, we also take into account the effect of sphaleron processes. Namely, we replace n B /s as (28/79) × n B /s when T X 4πm W /g 2 2 , where m W is the W boson mass.

B. Numerical result
In this subsection we give the numerical results. Since we are considering the KKLT moduli stabilization where the modulus and anomaly mediations give comparable contributions to the soft SUSY breaking terms, the typical mass scale for the sparticles is given by The precise sparticle mass spectrum is model-dependent, i.e., it relies on the details of For both cases we take the effective CP violation parameter to be where we have taken into account that the KKLT moduli stabilization leads to CP violation according to Eq. (21) and assumed |λ 332 | ∼ O(1). The observed baryon asymmetry given in Eq. (1) is drawn in the plot. The baryon asymmetry in the case (ii) is expected to be suppressed by M 2 g /m 2 s as we mentioned in the previous subsection. We show it as an reference of one of the possible SUSY mass spectra. Fig. 2 shows n B /s as a function of m 3/2 . The baryon asymmetry from the modulus decay is shown by the upper band for the case (i), and by the lower one for the case (ii). We have checked the out-of-equilibrium condition for the gluino and the squark is fulfilled in a wide range of the parameter space, such as m 3/2 10 14 GeV. One can see that, in the case (i), the correct amount of baryon asymmetry is generated when the gravitino mass is around 10 5 -10 7 GeV, which corresponds to m soft ∼ 10 3 -10 5 GeV. This is consistent with a rough estimation given in Eq. (45). On the other hand, in the case (ii), the suppression by the strange squark mass makes the baryon asymmetry smaller compared to the previous case by around an order of magnitude. 9 As a consequence, a larger gravitino mass is required to account for the baryon asymmetry of the Universe.
The figure shows that the gravitino mass should lie in the range about 10 7 -10 10 GeV.
The gravitino can be produced by the modulus decay with a branching fraction of O(0.01). The produced gravitino acts as radiation until it decays because it is originally produced relativistically. Thus it never dominates the Universe, and it does not cause significant entropy production. In addition, the gravitino decay does not destroy the successful BBN if m 3/2 10 TeV, which is satisfied in the parameter range of our interest.
Therefore, the asymmetry generated by modulus (and subsequent sparticle) decays is not diluted. It is also important to note that the moduli-induced gravitino problem [24] is avoided in the presence of R-parity violating operators because the lightest sparticle is not stable anymore.
Finally we comment on constraints from other experiments. A large CP phase coming from arg(A ijk M * g ) may induce sizable electric dipole moments (EDMs) of neutron and electron. The current experimental bounds are given by [44,45], for neutron and electron EDMs, respectively. We have estimated the neutron and electron EDM based on Refs. [46][47][48] (see also Refs. [49,50] and Ref. [51]), and found that both constraints give a similar bound as Here we have taken the universal soft SUSY breaking mass, and neglected the CP phase from µ/Bµ term [32]. Therefore the current bound allows O(0.1) CP phase even if sparticles are as low as around TeV, which confirms the parametrization in the numerical calculation, i.e., Eq. (47). If the CP phase from µ/Bµ term is included, the bound may 9 The suppression factor can be estimated as follows. Since the gluino cannot decay to strange (s)quark, Γg becomes (2/3) × Γg compared to the case (i), and ∆Γg becomes (1/4)(M 2 g /2m 2 s ) × ∆Γg. This gives a suppression factor 3/63, which is consistent with the numerical result given in Fig. 2. become more stringent by a factor, depending on the model. In any case, it is interesting since the parameter space we are interested in may be probed in the ongoing or future measurements of the EDMs.
respectively. Here we have set Mg = mq = m soft for simplicity.Λ is the hadronic scale used in the evaluation of the hadronic matrix element, which has theoretical uncertainty.
As it is seen, the dinucleon decay experiment gives more stringent constraint than n-n oscillation. Especially the bound from dinucleon decay is interesting. Recall that the numerical result implies that mq ∼ O(1) TeV with λ 332 ∼ O(1) leads to the right abundance of baryon (i.e., case(i)). This means that a part of the parameter region where our baryogenesis works can be probed in the current or future experiments. In addition, the squark with a mass of O(1) TeV may be discovered at the collider experiments, such as the LHC or the ILC.

IV. OTHER SCENARIOS
So far we have discussed baryogenesis by the modulus decay. As it has been shown, the KKLT scenario has important ingredients for the baryogenesis, i.e., suppressed interactions of the moduli with the visible sector, a large enough CP phase and a large branching fraction to gluinos in the modulus decay. Other than the KKLT scenario, there are several possible scenarios which possess (some of) the ingredients stated above. In this section we will discuss the gravitino-and the saxion-induced baryogenesis.

A. Gravitino-induced baryogenesis
The baryogenesis by the gravitino decay was originally discussed by Cline and Raby in Ref. [16]. In their work relatively low SUSY breaking scale, which is around several hundred GeV, was studied. Recently Ref. [41] pursued the gravitino-induced leptogenesis using lepton number violating operators. Here we revisit the original case by Cline and Raby, also considering high scale SUSY breaking region.
The gravitino is produced effectively at a high temperature. If the reheating temperature after inflation is high enough, the gravitino is copiously generated, and eventually dominates the Universe. In addition, the gravitino can also be produced by the inflaton decay [19][20][21][22][23]. In the typical mass spectra given in Eq. (46), the gravitino decays to the MSSM particles. The total decay rate is given by where ψ µ is the gravitino,φ i is the fermionic superpartner of scalar φ i and 10 Here we have neglected gauge boson masses. Since the decay rates are Planck suppressed as in moduli decay, the reheating temperature is typically the same as Eq. (29). Now let us suppose that the gravitino decays to all MSSM particles. Then taking the large gravitino mass limit, the total decay width is given by where N = 49 is the number of chiral superfields (neglecting right-handed neutrinos) and N g = 12 is the number of gauginos in the MSSM. Then the reheating temperature is estimated as T 3/2 ≃ 1.9 × 10 2 GeV g * 106.75 In the gravitino decay a fair amount of gluinos is produced if it is kinematically allowed.
The branching ratio of the decay to gluino and gluon, under the assumption above, is then This large branching fraction is important for the baryogenesis. Net baryon asymmetry from the gluino decay is then expressed as Using Eqs. (57) and (58), n B /s is estimated as n B s gluino ≃ 3.9 × 10 −10 g * 106.75 On the other hand, the decay modes to quark and squark are also important. In the mass spectrum mentioned above, the branching fraction is where n f is the number of quark flavors which can be produced by the gravitino decay. As it is seen, the branching fraction can be comparable to Br(ψ µ →gg). Note that this decay mode has the asymmetry between its CP conjugate and itself, which is given by a similar diagram depicted in Fig. 1 just by replacing the gluino with the gravitino. Therefore if physical CP phase Im(A ijk m * 3/2 ) is nonzero, then baryon asymmetry is generated. As a result of straightforward calculation, we get where Here r i = m 2 q i /|m 3/2 | 2 . Similarly to f (r i , r j , r k ), g(r i , r j , r k ) approaches to unity when r i,j,k ≪ 1, while it is suppressed as g(r i , r j , r k ) ≃ 1/2r k when r k > 1 and r i,j ≪ 1 as expected. Then net baryon number directly generated by the gravitino decay is obtained as Now, let us consider only nonzero λ 332 as in the Sec. III A. In the large gravitino mass limit, we get 11 which gives rise to Taking Brt ,b,s = Brq, net baryon number generated due to this process is given by Then baryon asymmetry directly generated by the gravitino decay to quark and squark is Therefore, it is seen that the baryon number which is resulted in the processes ψ µ →q RqR , q * R q R has the same order as gluino-mediated one. Finally let us compare those results with Ref. [16]. In their work, they took the maximum value for CP phase which is allowed by constraints from neutron EDM experiment at that time. (d n ≤ 10 −26 e cm was used there.) We have estimated the CP phase by following the way they described in their paper. Then taking Brq i = 1 and |λ 332 | = 4π × 0.1 as they did, we get This is almost consistent with Ref. [16] up to O(1) factor.

B. Saxion-induced baryogenesis
The Peccei-Quinn (PQ) mechanism is a plausible solution to the strong CP problem, and it predicts a pseudo Nambu-Goldstone boson, the axion [57,58], which acquires a small mass predominantly from the QCD anomaly. The axion is stable on cosmological time scale, thus it is a good candidate for dark matter. In the supersymmetric extension of the PQ mechanism, its scalar partner, called saxion, remains relatively light as it acquires a mass only from the SUSY breaking effects. In fact, it is known that the saxion tends to dominate the energy density of the early Universe, and it plays a similar role as the modulus. The advantages of using the saxion as a source of the baryogenesis are two folds. First, it necessarily couples to the QCD gauge sector in order to solve the strong CP problem, and therefore naturally decays into gluons and gluinos. Secondly, the baryogenesis can be more efficient because the decay temperature will be relatively high for the axion decay constant F a is smaller than the GUT scale.
Let us assume that the coherent oscillation of the saxion dominates the energy density of the Universe. The amount of baryon asymmetry generated by the saxion decay depends on the reheating temperature T s at the saxion decay as well as the branching fraction for the saxion decay into a gluino pair. In particular, the latter depends on the saxion coupling to axion, where s and a are the saxion and axion, respectively, and ξ is a model-dependent numerical coefficient. In order to have a sizable branching fraction into gluino pair, and also to avoid the overproduction of axionic dark radiation, ξ must be highly suppressed below unity since the saxion coupling to the QCD gauge sector is one-loop suppressed for a given axion decay constant F a . 12 As the precise value of ξ depends on how the PQ scalars are stabilized, let us see the axion models in some detail.
The axion models can be categorized according to whether the PQ symmetry is realized linearly or non-linearly. For the former case, which is sometimes dubbed a field-theoretic axion model, the axion decay constant F a ranges from an intermediate scale up to the GUT scale. The saxion coupling with axion depends on the stabilization mechanism. In 12 The branching fraction into the axions can also be suppressed by introducing a coupling of the saxion to the Higgs fields [59] or the right-handed neutrinos [60]. The axionic dark radiation has been extensively studied in e.g., Refs. [59][60][61][62][63][64][65].
a model with a single PQ scalar, ξ is of order unity. One way to suppress ξ is to consider a model with two PQ scalars S 1 and S 2 , which have an opposite PQ charge each other, and a U(1) PQ singlet Σ, so that the PQ symmetry is broken along the F -flat direction S 1 S 2 = µ 2 , which is lifted by SUSY breaking effects. In this model, the axion decay constant is given by µ. If S 1 and S 2 obtain the soft scalar masses of a similar size, then ξ is suppressed as ξ 2 ∼ ∆m 2 S /m 2 S , where m 2 S is the typical size of their soft masses and ∆m 2 S is the mass splitting between them [60].
The latter case, which is called a string axion model, on the other hand, corresponds to the case where the axionic shift symmetry of some moduli remains unbroken (except for the QCD anomaly). The axion decay constant is then typically around M P /8π 2 unless the Kähler metric of the saxion is hierarchically smaller than unity at the vacuum. Thus, in this case, one needs to assume a small initial misalignment of the axion in order for the axion relic energy density not to overclose the Universe. On the other hand, the saxion coupling to axion is typically suppressed as ξ ∼ 1/8π 2 , and thus the branching fraction of the saxion decay into gluino pair is naturally sizable and is comparable to that into the axions [63]. This can be understood by noting that the coupling to the QCD gauge sector arises at tree level in this case.
We note that the KKLT mechanism provides a natural framework to implement the saxion-induced baryogenesis both for the field-theoretic and the string axion models. First, the CP phase is obtained in the same way as explained in Sec. II. Note that the F -term in the axion multiplet does not induce a new CP phase in the MSSM soft terms because the soft terms generating a saxion potential depend on the moduli F -terms and the gravitino mass. Besides, the suppression for ξ can be also implemented in both models with the KKLT scenario. For the axion model described in Eq. (71), let us assume that the Yukawa couplings of S 1 and S 2 are small, and the PQ scalars S 1 and S 2 have the same modulus dependence, i.e., the same modular weight. Since the symmetry under interchanging between S 1 and S 2 is a good symmetry, then one can naturally achieve ∆m 2 S ≪ m 2 S , suppressing the saxion coupling to the axion. On the other hand, in the string axion model, the QCD axion can arise from a Kähler modulus in the KKLT scenario with multiple Kähler moduli, if the axionic shift symmetry for the Kähler modulus is not broken explicitly. [66,67]. This modulus is stabilized with a mass of about √ 2m 3/2 through Kähler mixing with other moduli, which are stabilized by the non-perturbative superpotential as in the original KKLT.
In either field-theoretic axion model or string axion model, the modulus X may dominate the Universe before the saxion-dominated Universe. This is because the saxion is much lighter than the modulus. Although the modulus decay generates the baryon asymmetry as we have described, the asymmetry is washed out by the saxion decay later.
However, the saxion decay can generate baryon asymmetry. Now let us estimate the baryon asymmetry produced by the saxion. For simplicity we consider the case where the axion production from the saxion decays is negligible, and the saxion dominantly decays to a gluon pair and a gluino pair. Including the decay into the axions does not affect our results as long as |ξ| 1/8π 2 .
The saxion couplings to gluon and gluino are given (in two-component notation) by where G µν is the gluon field strength and m s is the saxion mass, and the order unity constant κ is determined by how the saxion is stabilized. The above couplings mediate the saxion decay into a gluon pair and a gluino pair with Then the total decay rate of the saxion is given as neglecting the gluino mass in the final state. Consequently the branching ratio for the process s →gg is obtained as On the contrary, Br(s →gg) ≃ (1 + m 2 s /8M 2 g ) −1 was given in Ref. [17]. This result indicates a chiral suppression due to the gluino mass since Br(s →gg) → 0 when Mg → 0.
As it is seen, however, there is no such chiral suppression in Eq. (76), which is already pointed out by Ref. [26]. Since |κ| is order unity, the saxion can dominantly decay into a gluino pair. This makes the baryogenesis by the saxion decay more efficient than originally considered in Ref. [17].
After the gluino production, the same story follows as in the modulus case. In the string axion models, the reheating temperature is similar to the modulus case, while it can be higher in field-theoretic axion models if the axion decay constant is smaller than the GUT scale. The reheating temperature is given by where we have taken κ = 1. The resultant asymmetric yield is estimated by Thus the right amount of baryon asymmetry can be generated for the saxion mass around 10 3 − 10 5 GeV. On top of that, as mentioned above, the axion produced by the misalignment mechanism contributes to the cold dark matter. This fact is especially important because the lightest sparticle is no longer stable due to the R-parity violation.

V. CONCLUSIONS AND DISCUSSION
In this paper we have studied a baryogenesis induced by late-decaying moduli, and examined its implications for the moduli stabilization and the SUSY breaking scale. If the branching fraction of the modulus into a gluino pair is sizable, the right amount of baryon asymmetry can be generated through CP violating decay of gluino into quark and squark followed by baryon-number violating squark decays. We have shown that a natural framework realizing the baryogenesis is provided by the KKLT-type moduli stabilization since sufficient CP violation is obtained from mixed modulus-anomaly mediated SUSY breaking in the presence of two or more non-perturbative terms in the modulus superpotential. Successful baryogenesis is possible for the gravitino mass around 10 5 -10 7 GeV or equivalently the soft SUSY breaking mass m soft ∼ 10 3 -10 5 GeV (or heavier for a suppressed CP phase or effective CP violation parameter). Such low SUSY breaking scale can be probed directly at the collider experiments, dinucleon decay search, and electric dipole moments of neutron and electron. We also found that similar baryogenesis works successfully in other scenarios where the saxion or the gravitino dominates the Universe.
Lastly let us discuss baryogenesis in a couple of other moduli stabilization and SUSY breaking scenarios. We have mainly focused on the mixed modulus-anomaly mediation so far, assuming a sequestered uplifting. If the uplifting is not sequestered, sfermion masses are generically heavier than gaugino masses, then the gluino decay into quark and squark will be kinematically forbidden. Even in such case, however, the gravitino produced by the modulus decay will be able to generate baryon asymmetry.
The large volume scenario [68] can also lead to CP violating soft terms if one takes the same form of the superpotential for a small 4-cycle modulus as in Eq. (16). However, one cannot simply apply the moduli-induced baryogenesis discussed in the present paper to the large volume scenario because the overall volume modulus, which has a small mass compared to other moduli, does not appear in the visible gauge kinetic function and thus weakly couples to gluino. In addition, the modulus decay into a pair of gravitinos is kinematically forbidden and the decay into Higgsinos is suppressed due to the approximate no-scale structure even in the presence of the Giudice-Masiero term [62]. Those fact indicate much smaller amount of gluino produced by modulus decay, which suppress the resultant baryon number.
Regarding the moduli stabilization, we note that there is a general tension between moduli stabilization by non-perturbative effects and chirality [69]. To be specific, (a combination of) the moduli X L appearing in the gauge kinetic function of the visible gauge sector may not be stabilized by instantonic exponential terms. The saxion in the string axion model corresponds to this case, and as discussed before, it can be stabilized by another way [66,67] and the saxion-induced baryogenesis works. Alternatively, we may consider a modulus field which is stabilized a la KKLT, while not appearing in the SM gauge kinetic function. Such modulus field can still decay into gauge bosons and gauginos through the kinetic mixing with X L . Thus the baryon asymmetry will be generated by the subsequent gluino decay, although it may be diluted by the decay of lighter moduli X L to some extent.
Similar baryogenesis can be also realized in gravity mediation. In the gravity-mediated SUSY breaking, the MSSM gaugino masses are generated by Planck-scale suppressed interactions with an elementary gauge singlet field called the Polonyi field. Since it is a singlet under any symmetry of the theory, there is no special point in its field space.
Therefore, during inflation, the Polonyi field is generically deviated from the true minimum, as the effective potential for the Polonyi field during inflation is deformed by its gravitational couplings with the inflaton. After inflation, the Polonyi starts to oscillate about the minimum with an amplitude of order the Planck scale, and soon dominates the energy density of the Universe. The Polonyi field will decay into the MSSM gauge sector, and thus the baryogenesis by the Polonyi decay will be possible. In particular, a sizable A-term as well as large CP phases are naturally generated in the gravity mediation, which nicely fits with the current scenario. The resultant baryon asymmetry is similar to (45) if the modulus mass is replaced with the Polonyi mass, which is order of the gravitino mass.