Inflation connected to the origin of $CP$ violation

We consider a simple extension of the standard model, which could give a solution for its $CP$ issues such as the origin of both CKM and PMNS phases and the strong $CP$ problem. The model is extended with singlet scalars which could cause spontaneous $CP$ violation to result in these phases at low energy regions. The singlet scalars could give a good inflaton candidate if they have a suitable nonminimal coupling with the Ricci scalar. $CP$ issues and inflation could be closely related through these singlet scalars in a natural way. In a case where inflaton is a mixture of the singlet scalars, we study reheating and leptogenesis as notable phenomena affected by the fields introduced in this extension.


Introduction
CP symmetry is a fundamental discrete symmetry which plays an important role in particle physics. In the standard model (SM), it is considered to be violated explicitly through complex Yukawa coupling constants [1] and a θ parameter in the QCD sector [2].
The former is known to explain very well CP violating phenomena in B meson systems and so on [3]. The latter is severely constrained through an experimental search of a neutron electric dipole moment [4] and causes the notorious strong CP problem [5]. Peccei-Quinn (PQ) symmetry has been proposed to solve it [6]. If we assume that the CP symmetry is an original symmetry of nature, the complex phases of Yukawa coupling constants have to be spontaneously induced through some mechanism at high energy regions. It may be compactification dynamics in string theory near at the Planck scale [7]. In that case, since nonzero θ could be caused through radiative effects after the CP violation, the PQ symmetry is required to solve the strong CP problem again.
As an alternative scenario for the realization of CP symmetry, we may consider it to be exact so that all coupling constants including Yukawa couplings are real and also θ = 0 is kept to some scale much lower than the Planck scale. In that case, the CP symmetry is supposed to be spontaneously broken and this violation can be expected to be transformed to a complex phase in the CKM matrix effectively. If nonzero θ is not brought about in this process, it is favorable for the strong CP problem. The Nelson-Barr (NB) mechanism [8] has been proposed as such a concrete example. Unfortunately, radiative effects could cause nonzero θ with a magnitude which contradicts the experimental constraints [9]. a However, the scenario is interesting since it can present an explanation for the origin of the CP violation at a much lower energy region than the Planck scale. As a realization of the NB mechanism, a simple model has been proposed in [10]. The model is extended in [11] to the lepton sector where the existence of a CP violating phase in the PMNS matrix [12] is suggested through recent neutrino oscillation experiments [13].
Observations of the CMB fluctuation [14,15] suggest the existence of the exponential expansion of the universe called inflation. Inflation is usually considered to be induced by some slowly rolling scalar field called inflaton [16]. It is a crucial problem to identify its candidate from a viewpoint of the extension of the SM. Although the Higgs scalar has been a Introduction of the PQ symmetry could solve this fault of the model. We consider such a possibility in the extension of the model. studied as an only promising candidate in the SM [17] under an assumption that it has a nonminimal coupling with the Ricci scalar curvature [18], several problems have been pointed out [19][20][21]. In this situation, it is interesting to find an alternative candidate for inflaton in a certain extension of the SM which could solve several problems in the SM.
In this sense, the model extended from a viewpoint of the CP symmetry as described above could give such a promising candidate. It contains singlet scalars which cause the spontaneous CP violation and allow the introduction of the PQ symmetry as a solution for the strong CP problem. If they couple with the scalar curvature nonminimally, it could cause slow-roll inflation successfully. In this paper, we discuss such a possibility that the inflation of the universe could be related to the CP violation in the SM. We study reheating and leptogenesis as its phenomenology caused by extra fields introduced in the model to solve the CP issues.
Remaining parts of the paper are organized as follows. In section 2, we describe the model studied in this paper and discuss both phases in the CKM and PMNS matrices which are derived as a result of the spontaneous CP violation. In section 3, we discuss the inflation brought about by the singlet scalars which are related to the CP issues and the reheating. After that, we describe leptogenesis which could show a distinguishable feature from the usual leptogenesis in the seesaw model. The paper is summarized in   Table 1. c The SM contents b Similar models with vector-like extra fermions have been considered under different symmetry structures [11,22]. c Z 4 is imposed by hand to control the couplings of the new fields to the SM contents.
are assumed to have no charge of the global symmetry. Since this global U (1) has color anomaly in the same way as the KSVZ model [23] for a strong CP problem, it can play the role of the PQ symmetry. The present charge assignment for colored fermions guarantees the domain wall number to be one (N DW = 1) so that the model can escape the domain wall problem [24,25].
The model is characterized by new Yukawa terms and scalar potential which are invariant under the imposed symmetry y αβ M * (¯ α φ)(¯ β φ) + h.c., where d R j and e R j are the SM down-type quarks and charged leptons, respectively. α is a doublet lepton and φ is an ordinary doublet Higgs scalar. Since CP invariance is assumed, parameters in Lagrangian are considered to be all real. In eq. (1), we list dominant terms up to dimension five and M * is a cut-off scale of the model. Other invariant terms are higher order and can be safely neglected in comparison with the listed ones. V b is composed of terms which are invariant under the global symmetry but violate the S number.
For a while, we focus on a part of field space where the field values of σ and S are much larger than both φ and η to study the potential composed of σ and S only. In the present study, we assume that V b takes a following form: d where we define σ =σ √ 2 e iθ and S =S √ 2 e iρ . Along the minimum of V b for ρ which is fixed by ∂V b ∂ρ = 0, the potential ofσ andS can be written as whereκ σ andκ S are defined as and w and u are the vacuum expectation values (VEVs) ofσ andS. They are supposed to be much larger than the weak scale. They keep the gauge symmetry but break down the global symmetry U (1) × Z 4 into its diagonal subgroup Z 2 . e Since the minimum of V b can be determined by using these VEVs as cos 2ρ = − β 4α w 2 u 2 as long as β 4α w 2 u 2 ≤ 1 is satisfied, the CP symmetry is spontaneously broken to result in a low energy effective model with the CP violation. On the other hand, θ = 0 is satisfied because of the global U (1) symmetry relevant to σ [26]. The stability condition for the potential (3) can be given asκ If we consider the fluctuation ofσ andS around the vacua σ and S , mass eigenstates are the mixture of them in general. If we take account of the stability condition (5), mass eigenvalues can be approximately expressed as Although they have a tiny subcomponent in these cases, a dominant component of their eigenstates isS andσ, respectively. The mass of an orthogonal component toS is found to d Imposed symmetry allows terms m 2 S (S 2 + S †2 ) in V b . However, we assume that their contribution is negligible since we focus our attention on a potential valley whereσ ∝S is satisfied and then cos 2ρ could be a constant at the minimum of V b in that case. e It guarantees the stability of the lightest Z 2 odd field, which could be a dark matter (DM) candidate as discussed later.
be m 2 S ⊥ = 8αu 2 (1 − cos 2 2ρ). Since the global U (1) symmetry works as the PQ symmetry mentioned above, and the axion decay constant is given as f a = w, the VEV w should satisfy the following condition [3,27]: The NG-boson caused by the spontaneous breaking of this U (1) becomes axion [28] which is characterized by a coupling with photons [29] g aγγ = 1.51 10 10 GeV In the next part, we show that the effective model after the symmetry breaking can have CP phases in the CKM and PMNS matrices. They are induced by the mass matrices for the down type quarks and the charged leptons through a similar mechanism which has been discussed in [10] as a simple realization of the NB mechanism [8] for the strong CP problem.

CP violating phases in CKM and PMNS matrices
Yukawa couplings of down-type quarks and charged leptons given in eq. (1) derive mass terms as where f and F represent f = d, e and F = D, E for down-type quarks and charged This mass matrix is found to have the same form proposed in [10]. Since the global U (1) symmetry works as the PQ symmetry and all parameters in the model are assumed to be real, arg(detM f ) = 0 is satisfied even if radiative effects are taken into account after the spontaneous breaking of the CP symmetry [11].
We consider the diagonalization of a matrix M f M † f by a unitary matrix wherem 2 f is a 3 × 3 diagonal matrix in which generation indices are not explicitly written. Eq. (10) requires If , which can be realized in the case u, w φ , we find that B f , C f , and D f are approximately given as These guarantee the unitarity of the matrix A f within the present experimental bound [3] since is satisfied in each component. In such a case, it is easy to find The right-hand side of the first equation corresponds to an effective squared mass matrix of the ordinary fermions f . It is derived through the mixing with the extra heavy fermions F . Since its second term can have complex phases in off-diagonal components as long as y f i =ỹ f i is satisfied, the matrix A f could be complex. Moreover, if µ 2 F < ∼ F f F † f is realized, the complex phase of A f in eq. (13) could have a substantial magnitude because the second term in the right-hand side has a comparable magnitude with the first one.
Although it can be realized for various parameter settings, we consider a rather simple Since the masses of vector-like fermions are expected to be of O(10 8 ) GeV or larger in this case, they decouple from the SM and do not contribute to phenomena at TeV regions.
The CKM matrix is determined as where O u is an orthogonal matrix used for the diagonalization of a mass matrix for up-type quarks. Thus, the CP phase of V CKM is caused by the one of A d . The same argument is applied to the leptonic sector and the PMNS matrix is derived as V PMNS = A † e U ν , where U ν is an orthogonal matrix used for the diagonalization of a neutrino mass matrix. The Dirac CP phase in the CKM matrix and the PMNS matrix can be induced from the same origin of CP violation. A concrete example of A d is given for a simple case in Appendix A.
f Leptogenesis could depend on the strength of these couplings heavily in this model as studied later.

Effective model at a lower energy region
An effective model at lower energy regions than w and u can be obtained by integrating out the heavy fields. It is reduced to the SM with a lepton sector extended as the scotogenic neutrino mass model [30], which is characterized by the terms invariant under the remaining Z 2 symmetry where neutrino Yukawa couplingsh αj are defined on the basis for which the mass matrix of the charged leptons is diagonalized as discussed in the previous part. Thus, they are complex now. After the spontaneous breaking due to the VEVs ofσ andS, coupling constants in eq. (15) are related to the ones contained in eq. (1) as These connection conditions should be imposed at a certain scaleM , which is taken to beM =M F in the present study. Stability of the potential (15) requires the following conditions to be satisfied through scales µ <M : Potential stability (5) and (17)   Ifm 2 η > 0 is satisfied and η has no VEV, Z 2 is kept as an exact symmetry of the model. In this model, we assume both |m φ | andm η have values of O(1) TeV. Since it has to be realized under the contributions from the VEVs w and u, parameter tunings are required. g Phenomenology on neutrinos and DM could be the same as the one which has been studied extensively in various studies [31,32] unless the axion is a dominant component of DM. If the lightest neutral component of η is DM which is identified as its real component η R , both DM relic abundance and DM direct search constrain the parametersλ 3 and λ 4 [22,33]. As a reference, in Fig. 1 we show their required values in the (λ + ,λ 3 ) plane for the cases M η R =0.9, 1 and 1.1 TeV where λ + =λ 3 +λ 4 +λ 5 and M 2 η R = m 2 φ +λ + φ 2 . They should be also consistent with the stability condition (17). The figure shows that these could be satisfied with rather restricted values ofλ 3 and λ 4 . A perturbativity requirement at µ >M also constrains the model strongly since DM relic abundance requiresλ 3 and g For parameter values assumed in this analysis, the order of required tuning is estimated as O(10 −10 ). |λ 4 | to take rather large values. We have to take account of it to consider the model at high energy regions. As an example, we show the result of RGE study in the right panel of Fig Neutrino mass is forbidden at tree level due to this Z 2 symmetry except for the ones generated through dimension five Weinberg operators in eq. (15). They could give a substantial contribution to the neutrino masses depending on the cut-off scale M * and coupling constants y αβ . In the present study, however, we assume that their contribution is negligible and the dominant contribution comes from one-loop diagrams with η and N j in internal lines. Its formula is given as where M 2 η =m 2 η + (λ 3 + λ 4 ) φ 2 and M N j M η is supposed. As an example, one may assume a simple flavour structure for neutrino Yukawa couplings [35] This realizes a tri-bimaximal mixing which gives a simple and rather good 0-th order approximation for the analysis of neutrino oscillation data and leptogenesis [32]. If we impose the mass eigenvalues obtained from eq. (19) for the case |h 1 | |h 2 |, |h 3 | to satisfy the squared mass differences required by the neutrino oscillation data, we find Since we have Λ 2,3 ∼ 7 × 10 5 eV for M 2,3 ∼ 10 7 GeV and M η ∼ 10 3 GeV, the neutrino oscillation data [3] can be explained by taking as an example Even if we impose the neutrino oscillation data, h 1 can take a very small value compared with h 2,3 [32]. It can play a crucial role for low scale leptogenesis as seen later.
3 Inflation due to singlet scalars

Inflation
It is well known that a scalar field which couples nonminimally with the Ricci scalar can cause inflation of the universe and the idea has been applied to the Higgs scalar in the SM [17] and its singlet scalar extensions [36,37]. If the singlet scalars S and σ, which are related to the CP issues in the SM couple with the Ricci scalar, it can play the role of inflaton in this model. The action relevant to the inflation is given in the Jordan frame as where M pl is the reduced Planck mass and the coupling of S is controlled by the Z 4 symmetry. V (σ, S) stands for the corresponding part in the potential (1). Since inflation follows very complicated dynamics if multi scalars contribute to it, we confine our study to the inflation in a potential valley. Moreover, we assume ξ S1 = −ξ S2 is satisfied for simplicity. In that case, the coupling of S with the Ricci scalar is reduced to 1 2 ξ S S 2 I R, where S = 1 √ 2 (S R + iS I ) and ξ S = ξ S1 − ξ S2 . If S is supposed to evolve along a constant ρ, which is determined as a potential minimum ∂V b ∂ρ = 0, the radial componentS couples with the Ricci scalar as 1 2ξ SS 2 R, whereξ S is defined asξ S = ξ S sin 2 ρ and the potential V (σ, S) is expressed by eq.(3). Here we consider cases such that both ξ σ andξ S are positive only.
Stability of this potential requires the condition given in eq. (5). We neglect the VEVs w and u for a while since they are much smaller than O(M pl ), which is the field values ofσ andS during the inflation. We also suppose that other scalars have much smaller values than them.
We consider the conformal transformation for a metric tensor in the Jordan framẽ g µν = Ω 2 g µν , After this transformation to the Einstein frame where the Ricci scalar term takes a canon-ical form, the action can be written as where χ σ and χ S are defined as [36] ∂χ σ ∂σ = 1 If we introduce variablesχ, ϕ to expressσ andS asσ =χ cos ϕ,S =χ sin ϕ, the potential in the Einstein frame at the large field regions such as ξ σσ 2 +ξ SS 2 M 2 pl can be written as We find that there are three types of valley along the minimum in the ϕ direction of this potential. They realize different types of inflaton. Two of them are In each case, a kinetic term mixing between χ σ and χ S disappears and inflaton can be identified with χ σ for (i) and χ S for (ii), respectively. h Another valley which is studied in this paper is realized at under the conditions where we note that these are automatically satisfied for κ σS < 0 since eq. (5) is imposed, andξ S and ξ σ are assumed to be positive. In this case the inflatonχ is a mixture ofσ and S with a constant value ofσ/S. Although the kinetic term mixing cannot be neglected for a general sin ϕ, it can be safely neglected if we restrict it to the one in which the inflaton is dominated byS (sin 2 ϕ 1) orσ (sin 2 ϕ 0). We focus our study on the former case h In different context, the inflaton dominated byσ and theS inflaton have been discussed in [38] and [11], respectively.
whereχ σ is always satisfied during inflation. If we additionally imposeξ S ξ σ on eq. (29) and assume that the relevant couplings satisfy i sin ϕ is expressed as sin 2 ϕ = 1 + κ σS 2κσ . In this case, by usingκ S =κ S − is satisfied generally, k the global U (1) is spontaneously broken during inflation and isocurvature fluctuation could be problematic [38]. However, even in that case it is escapable since the axion needs not to be a dominant component of DM in the present model. This problem is discussed later.
The canonically normalized inflaton χ can be expressed as [38] where γ can be approximated along the valley as If we use γ 1, the potential of χ obtained through V (χ) = 1 Ω 4 V (σ,S) can be derived by using the solution of eq. (32) which is given as . However, we do not consider such a case in this paper.  As references, we also plot approximated potentialκ S 6ξ 2 S M 2 pl χ 2 andκ S 4 χ 4 in eq. (35) as χ 2 and χ 4 . In these plots, a Planck unit (M pl = 1) is used.
We derive the potential of χ through numerical calculation for a typical value ofξ S by using eq. (34). Such examples of V (χ) are shown in Fig. 2. It can be approximated as The inflation ends at χ end M pl . After the end of inflation, there is a substantial region where the potential behaves as a quadratic form before it is reduced to a quartic form at low energy regions for the caseξ S 1 as in the Higgs inflation. However, such a region can be neglected for the caseξ S < 10, which is the case considered in this study. Since the inflaton oscillating in the quartic potential behaves as radiation as shown later, the radiation domination starts soon after the end of inflation in that case.
The slow-roll parameters in this model can be estimated by using eq. (32) as [16] The e-foldings number N k from the time when the scale k exits the horizon to the end of inflation is estimated by using eq. (32) as  Fig. 3 Predicted values of the scalar spectral index n s and the tensor-to-scalar ratio r in this model.
They are read off as the values at intersection points of two lines with a fixed value ofξ S or N k . The coupling constantκ S is varied in a range from 10 −7 to 10 −10 .
Taking account of these, the slow-roll parameters in this inflation scenario is found to be approximated as The field value of inflaton during the inflation is found to be expressed as χ k = √ 6 2 M pl ln(32ξ S N k ) by using eqs. (34) and (37), and its potential V k (≡ V (χ k )) takes a constant value as shown in eq. (35). On the other hand, if we use = 1 at the end of inflation, the inflaton potential is estimated as pl which is found to be a good approximation from Fig. 2. The spectrum of density perturbation predicted by the inflation is known to be expressed as [16] If we use the Planck data A s = (2.101 +0.031 −0.034 ) × 10 −9 at k * = 0.05 Mpc −1 [15], we find the Hubble parameter during the inflation to be H I = 1.4 × 10 13 60 N k * GeV and the relation which should be satisfied at the horizon exit time of the scale k * . We confine our study to the caseξ S < 10.
In Fig. 3, we plot predicted values for the scalar spectral index n s and the tensor-toscalar ratio r in the present model. Since the quartic couplingκ S is a free parameter of the model under the constraint (39), we varyκ S in the range 10 −10 ≤κ S ≤ 10 −7 for fixed values ofξ S or N k . The CMB constraint (39) is satisfied at intersection points of the lines with a fixed value ofξ S or N k . The figure shows that the constraints of the observed CMB data [15] are satisfied for the supposed parameters.
After the end of inflation, the inflaton χ starts oscillation in the potential V (χ). At this stage, the description by χ is no longer justified, especially, at the small field regions. ϕ is not constant in general there.S andσ should be treated independently. In the following study, however, we confine our study to a special inflaton trajectory and estimate the reheating phenomena by using χ to give a rough evaluation of reheating temperature under the assumption that inflaton follows a constant ϕ trajectory. l In this case, inflaton oscillation is described by the equation Since the amplitude of χ evolves approximately as Φ(t) =ξ S √ πκ S t in the quadratic potential after the end of inflation, the inflaton χ oscillates 1 2π √ 3π (ξ S − 1) times before the potential (35) changes from a quadratic form to a quartic one. This means that preheating under the quadratic potential could play no substantial role for the caseξ S < 10. In such a case, we need to consider the preheating in the quartic potential only. The model with the quartic potential V (χ) =κ S 4 χ 4 becomes conformally invariant [39]. If we introduce dimensionless conformal time τ , which is defined by using a scale factor a as adτ = √κ S χ end dt and also a rescaled field f = aχ χ end , eq. (40) can be rewritten as The solution of this equation which describes the inflaton oscillation is known to be given by a Jacobi elliptic function f (τ ) = cn τ − τ i , 1 √ 2 . m From the Friedman equation for this inflaton oscillation, we find Since H = 1/2t is satisfied, this oscillation era is radiation dominated. If we take into account such a feature of the model that radiation domination starts just after the end l During the first several oscillations, both ϕ and ρ can be numerically confirmed to take constant values. This suggests that single field treatment is rather good during the first few oscillations at least. m If we take τ i 2.44, f (τ ) can be approximated by cos 2π τ0 τ , where τ 0 is expressed by using the complete elliptic integral of the first kind K as of inflation, the e-foldings number N k can be expressed by noting a relation k = a k H k as where H 2 k = V k 3M 2 pl and the suffix 0 stands for the present value of each quantity. Reheating temperature dependence of N k is weak or lost differently from the usual case [16] where substantial matter domination is assumed to follow the inflation era. n In the next part, we discuss the reheating temperature expected to be realized in the present model.

Preheating and reheating
Before proceeding to the study of particle production under the background oscillation of inflaton, we need to know the mass of the relevant particles which is induced through the interaction with inflaton. Such interactions are given as The particles interacting with the inflaton χ have mass varying with the oscillation of χ and their mass can be read off from eq. (44) as where F = D or E should be understood for f = d or e, respectively. Since the effect of nonminimal coupling is negligible during this oscillation period, it is convenient to use the components of σ and S, which are parallel and orthogonal to the inflaton χ to describe their interactions. If we indicate each of them as σ , σ ⊥ , S , and S ⊥ , then their n If reheating occurs through a perturbative process at χ < ∼ u where matter domination is realized, its effect on N k could also be negligible as long as Γ > H is satisfied at that stage where Γ is inflaton decay width.
interactions are expressed as By combining these interactions with the composition of χ, their masses are found to be given by o The coupling constants relevant to these masses are restricted through the assumed inflaton composition and the realization of the CP phases in the CKM and PMNS matrices. The discussion in the previous sections shows that such requirements are satisfied We assume additionallŷ Since the oscillation frequency of the inflaton is ∼ √κ S χ, decays or annihilations of the inflaton are kinematically forbidden except for the one to σ ⊥ as found from eqs. (45) and (47). In σ ⊥ case, the inflaton reaction rate to it is much smaller than the Hubble parameter at this period because of the smallness of its coupling with the inflaton, energy drain from the inflaton to σ ⊥ is ineffective to be neglected. As a result, the energy transfer from the inflaton oscillation to excited particles is expected to occur at the time when the inflaton crosses the zero where the resonant particle production is possible.
Preheating under the background inflaton oscillation can generate the excitations of χ itself and other scalars ψ which couple with χ at its zero crossing [40]. In a quartic potential case [39], the model becomes conformally invariant and the time evolution equations of χ k ( S k ) and ψ k , which are the comoving modes with a momentum k, can be o It should be noted that the mass of σ could have another non-negligible contribution which is induced by explicit breaking of the global U (1) symmetry brought about by the quantum gravitational effects.
We do not take account of it in the present study.
transformed to the simple ones by rescaling them to the dimensionless quantities in the same way as eq. (41). They are given as where the rescaled variables are defined as Function f (τ ) is the solution of eq. (41) and g ψ stands for a coupling constant of the relevant particle ψ(= σ, S ⊥ , φ, η) with the inflaton χ, which can read off from eqs. (45) and (47). Amplitudes X k and F k are known to show the exponential behavior ∝ e µ k τ with a characteristic exponent µ k , which is determined by a parameter g ψ /κ S . Using the solutions of eq. (50), the number density of the produced particle ψ can be calculated as Particle production based on eq. (50) at the inflaton zero crossing has been studied in [39] and it is shown to be characterized by the parameter g ψ /κ S . We classify the relevant couplings into five groups where we note that couplings in (A)-(D) are fixed by the present inflaton composition but the ones in (E) are not constrained. Now we consider the resonant particle production in each group. A maximum value of characteristic exponent in (D) is very small so that it plays no effective role also in preheating. In (B) and (C), both the fluctuations of S and S ⊥ are produced fast, but it stops as soon as |S | 2 and |S ⊥ | 2 reach a certain value such as 0.5χ 2 end /a 2 . Although a maximum value µ max of the characteristic exponent of (B) is much smaller than the one of (C) and also the resonance band of (B) is much narrower than (C), the interaction S 2 S 2 ⊥ accelerates the production of fluctuations of S through rescattering and they reach the similar value [38,41]. Since the backreaction of these fluctuations to the inflaton oscillation restructures the resonance band, the resonant particle production stops before causing much more conversion of the inflaton oscillation energy to particle excitations. Moreover, since the decay of excitations produced through these processes are also closed kinematically, these could not play an efficient role in reheating. In (A), since σ also couples to the inflaton directly, the resonant production of its excitation stops at a certain stage due to the same reason as (B) and (C). Even if the excited particles are allowed to decay to fermions F and N j kinematically, the decay width is much smaller than the Hubble parameters to be neglected. As a result, if the process due to (E) is not effective, preheating cannot play any role for reheating and reheating proceeds through perturbative processes after the amplitude of inflaton is smaller than the VEV u.
Here, we have to note that there is a possibility in (E) where the energy transfer from the inflaton oscillation to radiation proceeds through preheating since the produced excitations can decay to relativistic particles differently from (B) and (C). In this case, φ and η are produced as excitations at the zero crossing of the inflaton where an adiabaticity conditionω k <ω 2 k could be violated for certain values ofk. By using the analytic solution of eq. (50) derived in [39], the momentum distribution n ψ k of the produced particle ψ through one zero crossing of the inflaton can be estimated as where τ 0 is an inflaton oscillation period and τ 0 = 7.416. The resonance is efficient for k <k c . Thus, the particle number density produced during one zero crossing of the inflaton isn The energy transfer from the inflaton oscillation to relativistic particles is caused through the decay of the produced particles ψ(= φ, η) and thermalization proceeds. They can decay to light fermions through φ →qt with a top Yukawa coupling h t and η →¯ N with neutrino Yukawa couplings h j , respectively. Here, we should note that η can be heavier than N j at this stage even if η is the lightest one with Z 2 odd parity at the weak scale. It is caused by the inflaton composition in the present model as found from eq. (45).
Their decay widths in the comoving frame are given by using the conformally rescaled unit asΓ where ψ = φ, η, and c ψ are internal degrees of freedom c φ = 3 and c η = 1. The Yukawa coupling y ψ represents y φ = h t and y η = h j . SinceΓ −1 ψ < τ 0 /2 is satisfied for g ψ > 4 × 10 −7 κ S 10 −8 , the produced ψ decays to the light fermions completely before the next inflaton zero crossing [42] and then it is not accumulated in such cases. We fix τ = 0 at the first inflaton zero crossing so that f (τ ) can be expressed approximately as f (τ ) = f 0 sin(cf 0 τ ). Transferred energy density through the ψ decay during a half period of oscillation can be estimated as p where γ ψ and Y (f 0 , γ ψ ) are defined by using c = 2π/τ 0 as The energy density transferred to the light particles is accumulated at each inflaton zero crossing linearly and its averaged value for τ is estimated as where the substantial change of f 0 is assumed to be negligible during τ . Since the total energy density of the inflaton oscillation energyρ χ and the transferred energyρ r to light particles is conserved, reheating temperature realized through this process can be estimated fromρ χ end =ρ r . It can be written by transferring it to the physical unit as where we useρ χ end = 1 4κ S and g * = 130. By applying eqs. (42) and (59) to this formula, we find q T R = 5.9 × 10 15 g Since h t h j is satisfied, reheating temperature is expected to be determined by the produced φ as long as φ is dominantly produced.
If preheating cannot produce relativistic particles effectively, the dominant energy is still kept in the inflaton oscillation. When the oscillation amplitude of χ decreases to be O(u), the inflaton starts decaying to the light particles through the perturbative processes.
pρ is defined as the energy density in the comoving frame by using the conformally rescaled variables. q The same result can be obtained by using the relation H = 1 2t in the radiation dominated era together with eqs. (42) and (59).
Since the mass pattern is expected under the present assumption for the coupling constants in (48) to be the inflaton decay is expected to occur mainly through χ → η † η and χ → φ † φ at tree level. The decay width of ψ(= φ, η) is estimated as where g ψ is defined in eq. (49). After the inflaton decays to η † η and φ † φ, the SM contents are expected to be thermalized through gauge interactions with η and φ immediately.
Since Γ ψ > H is satisfied for g ψ > 10 −7.1 κ S 10 −8 1/2 u 10 11 GeV 1/2 at χ u, reheating temperature in such a case can be estimated through 1 4κ S u 4 = π 2 30 g * T 4 R as r which is independent of g ψ . However, if Γ ψ > H is not satisfied because of a small g ψ , the reheating temperature is expected to be determined through Γ ψ = H and then becomes smaller proportionally to g ψ .
In the left panel of Fig. 4, for a case ψ = φ, the expected reheating temperature through both processes is plotted as a function g φ in a caseκ S = 10 −8 ,κ σ = 10 −4.5 , |κ σS |/κ σ = 10 −1.2 , and u = 10 11 GeV. It shows that the reheating temperature is determined by the perturbative process at g φ < 10 −6 . We also found from the figure that the reheating at r Although a larger value of u can make the reheating temperature much higher, its upper bound exits.
Since a larger g ψ is required in that case, it could violate the perturbativity of the model and cause the upper bound for it. For example, if we consider the case with y F = 10 −1.2 , the perturbativity is violated for g ψ > 10 −4.4 . As a result, the reheating temperature due to the perturbative process is bounded as T R < 6.3 × 10 13 GeV. s The conditionΓ −1 ψ < τ 0 /2 can be confirmed for the parameters used here. κ φσ < 10 −2.6 should be satisfied, and then the reheating temperature cannot be higher than ∼ 10 10 GeV as found from the figure. Since the decay of φ is so effective, it decays soon after their production and much before the inflaton amplitude becomes large during the oscillation. This makes the energy transfer in the preheating inefficient.
In the usual leptogenesis in the seesaw scenario, the right-handed neutrinos are supposed to be thermalized only through the neutrino Yukawa couplings h j . In the present model, neutrino mass eigenvalues obtained from eq. (19) require h 2,3 = O(10 −3 ) to explain the neutrino oscillation data as discussed at a part of eq. (22). On the other hand, reheating temperature is found to satisfy T R > ∼ 10 8 GeV from the above discussion. Since the decay width Γ N 2,3 of N 2,3 and the reheating temperature T R satisfy Γ N 2,3 > H(T R ) and T R > M N 2,3 , N 2,3 are also expected to be in the thermal equilibrium through the inverse decay simultaneously at the reheating period. In the case of N 1 , however, it depends on the magnitude of its Yukawa coupling h 1 which can be much smaller than others. We should note that N 1 could be effectively generated in the thermal bath, even  The most interesting feature of this inflation scenario is that thermal leptogenesis could generate sufficient baryon number asymmetry even for M N 1 < 10 9 GeV without relying on resonance effect. In the ordinary seesaw framework, neutrino mass is generated as through Yukawa interaction h αj¯ α φN j . Baryon number asymmetry in the universe [43] is expected to be generated by the same interaction through thermal leptogenesis [44]. If we assume the sufficient lepton asymmetry is generated through the out-of-equilibrium decay of the lightest right-handed neutrino, which has been in the thermal equilibrium, then the reheating temperature T R is required to be larger than its mass T R > M N 1 . Moreover, since it has to be produced sufficiently in the thermal bath, its Yukawa coupling h α1 should not be so small. On the other hand, the neutrino mass formula gives a severer upper bound on h α1 for a smaller M N 1 under the constraints of neutrino oscillation data. These impose a lower bound for M N 1 such as 10 9 GeV [45]. This condition for M N 1 is not changed even if T R 10 9 GeV is satisfied. The problem is caused by such a feature of the model that both the production and the out-of-equilibrium decay of the right-handed neutrino have to be caused only by the same neutrino Yukawa coupling. It does not change in the original scotogenic model either [32]. In that model, the righthanded neutrino mass can be much smaller than 10 9 GeV keeping the neutrino Yukawa couplings to be rather larger values by fixing |λ 5 | at a smaller value in a consistent way with the neutrino oscillation data. However, the washout of the generated lepton number due to the inverse decay of the right-handed neutrinos becomes so effective in that case.
As a result, successful leptogenesis cannot be realized for a lighter right-handed neutrino than 10 8 GeV. t It is a notable aspect in the present model that this situation can be changed by the particles which are introduced to explain the CP issues in the SM.
We note that the interaction between the right-handed neutrino N 1 and extra vectorlike fermions F mediated byσ could change the situation. u The lightest right-handed neutrino N 1 can be effectively produced in the thermal bath through the extra fermions t Low scale leptogenesis in the scotogenic model has been studied intensively in [46]. However, the lightest right-handed neutrino is assumed to be in the thermal equilibrium initially there. u The similar mechanism has been discussed in models with a different type of inflaton [33,47].
scatteringD L D R ,Ē L E R → N 1 N 1 mediated byσ if D L,R and/or E L,R are in the thermal equilibrium at a certain temperature T . In that case, both conditions T >M F , M N 1 and Γ F F H(T ) are required to be satisfied, where Γ F F is the reaction rate of this scattering. Mass of these fermions is determined by the VEVs u and w which should be larger than the lower bound of PQ symmetry breaking scale. Since the rough estimation of Γ F F H(T ) for relativistic F and N 1 gives we find that T >M F , M N 1 could be satisfied for suitable values of y F and y N 1 . It is crucial that this does not depend on the magnitude of the N 1 Yukawa coupling h 1 . If an extremely small value is assumed for h 1 , successful leptogenesis is allowed in a consistent way with neutrino oscillation data even for M N 1 < 10 9 GeV.
After N 1 is produced in the thermal bath through the scattering of the extra fermions mediated byσ, it is expected to decay to α η † by a strongly suppressed Yukawa coupling. Since its substantial decay occurs after the washout processes are frozen out, the generated lepton number asymmetry can be efficiently converted to the baryon number asymmetry through sphaleron processes. This scenario can be checked by solving Boltzmann equations for Y N 1 and Y L (≡ Y − Y¯ ), where Y ψ is defined as Y ψ = n ψ s by using the ψ number density n ψ and the entropy density s. Boltzmann equations analyzed here are given as and an equilibrium value of Y ψ is represented by Y eq ψ . H(T ) is the Hubble parameter at temperature T and the CP asymmetry ε for the decay of N 1 is expressed where h j = |h j |e iθ j and F (x) = √ x 1 − (1 + x) ln 1+x x . A reaction density for the decay N j → α η † and for the lepton number violating scattering mediated by N j is expressed by γ N j D and γ N j , respectively [32]. γ F represents a reaction density for the scatterinḡ D L D R ,Ē L E R → N 1 N 1 . We assume that (D L , D R ) and (E L , E R ) are in the thermal equilibrium and Now we fix the model parameters for numerical study of eq. (66) by taking account of the discussion in the previous part. We consider two cases for the VEVs of the singlet scalars such that (I) w = 10 9 GeV, u = 10 11 GeV, (II) w = 10 11 GeV, u = 10 13 GeV, where the axion could be a dominant DM in case (II). The parametersκ S ,κ σ , and κ σS , which characterize the inflaton χ, are fixed toκ S = 10 −8 ,κ σ = 10 −4.5 , and |κ σS | = 10 −6.1 .
These are used in the right panel of Fig. 4. The condition F f F † f > µ 2 F for which the CP phases in the CKM and PMNS matrices can be generated is reformulated as δ ≡ If we confine our study on a case y f = (0, 0, y) andỹ f = (0,ỹ, 0) for simplicity, v we have a relation y 2 +ỹ 2 = (δ 2 − 1) w 2 u 2 y 2 F among Yukawa couplings of the extra fermions. We fix them as δ = if the maximum CP phase is assumed. DM is determined by the couplingsλ 3 and λ 4 .
Since they are fixed so as to realize the correct DM abundance by the neutral component of η in the case (I), it cannot saturate the required DM abundance for the sameλ 3 and λ 4 in the case (II) as found from Fig. 1. The axion could be a dominant component of DM in the case (II) since w is taken to be a sufficient value for it.
We give a remark on these couplings here. It is crucial to examine whether the above parameters used in this analysis are consistent with the potential stability conditions The lightest right-handed neutrino mass in each case is M N 1 = 7 × 10 6 and 10 8 GeV. In both cases, the sufficient baryon number asymmetry is found to be produced. The figure for the case (I) shows clearly that the present scenario works well. Y N 1 reaches a value near Y eq N 1 through the scattering of the extra fermions as expected. Substantial out-ofequilibrium decay occurs at z > 10 to generate the lepton number asymmetry. The delay of the decay due to the small h 1 could make the washout of the lepton number asymmetry ineffective. On the other hand, we cannot definitely find a signature of the scenario in the case (II), where the N 1 mass is near the bound for which the usual leptogenesis can generate the required baryon number asymmetry in the original scotogenic model [32].
The figure shows that additional contribution to the N 1 production starts at z 0.1. It is considered to be brought about by the N 1 inverse decay since it is expected to become effective around z ∼ 6.3×10  In both panels, other parameters are fixed at the ones given in (69).
φ and η. Two values of y F are used in this plot. The N 1 production in the present scenario depends on the reheating temperature and the couplings y F , y N 1 . A red solid line representing Y B is expected for the parameters given in eq.(69) for the case (I).
It becomes larger and reaches an upper bound Y B 2.5 × 10 −10 when the reheating temperature increases to 10 10 GeV. This behavior can be understood if we take into account that the equilibrium number density of extra fermions are suppressed by the Boltzmann factor at lower reheating temperature and then the N 1 production due to the scattering of extra fermions is suppressed. We also plot Y B for a smaller value of y F by green crosses at some typical T R . They show that Y B takes smaller values for a smaller y F since the N 1 production cross section is proportional to y 2 F . w In the right panel, Y B is plotted by varying |λ 5 | and y N 1 . A red solid line represents it as a function of |λ 5 | for a fixed y N 1 = 5 × 10 −3 . Since the neutrino oscillation data have to be imposed on eq. (21), Yukawa couplings h 2,3 are settled by |λ 5 |, y N 2 and y N 3 . The CP asymmetry ε and the washout of the generated lepton number asymmetry are mainly determined by h 2,3 for the fixed y N 2 and y N 3 as found from eq.(67). Since a smaller |λ 5 | makes h 2,3 larger and then both ε and washout larger, Y B takes a maximum value for a certain |λ 5 |, which is found in the figure. We also plot Y B by varying y N 1 for a fixed |λ 5 | in the same panel.
A smaller y N 1 makes the N 1 production less effective for a fixed y F and then its lower w Since the effect of Boltzmann suppression caused by its massM F = δy F w could be dominant at lower T R compared with the effect on the cross section, the smaller y F gives a larger Y B at T R < 10 8 GeV in this case.
bound is expected to appear for successful leptogenesis. It gives the lower bound of M N 1 as ∼ 4 × 10 6 GeV as predicted above.
Although other parameters are fixed at the ones given in (69) in these figures, it is useful to give remarks on their dependence here. If δ takes a larger value, the mass of extra fermionsM F becomes larger to suppress the reaction density γ F due to the Boltzmann factor. As a result, the N 1 number density generated through the scattering becomes smaller and the resulting Y B also becomes smaller. If h 1 is much smaller, N 1 decay delays and the entropy produced through the decay of relic N 1 might dilute the generated lepton number asymmetry.

Dark matter and isocurvature fluctuations
This model has two DM candidates. One is the lightest neutral component of η with Z 2 odd parity which is an indispensable ingredient of the model. It is known to be a good DM candidate which does not cause any contradiction with known experimental data as long as its mass is in the TeV range where the coannihilation can be effective [32,48,49].
As found from Fig. 1, both the DM abundance and the DM direct search bound can be satisfied if the couplingsλ 3 and |λ 4 | take suitable values of O(1). Although these parameters could affect the perturbativity of the scalar quartic couplings through the radiative corrections, we can safely escape such problems in certain parameter regions.
The results obtained for the case (I) in the previous part are derived by supposing that the required DM is η R .
Axion is another promising candidate in the model. However, the axion could be a dominant component of DM only for f a ∼ 10 11 GeV although it depends on the contribution from the axion string decay [50]. We consider the case (II) as such an example.
As described before, the PQ symmetry is spontaneously broken during the inflation since the inflaton contains the radial component of σ. As a result, the axion appears as the phase θ of σ. Since the axion potential is flat during the inflation, the axion gets a quantum fluctuation δA = (H/2π) 2 and it can cause isocurvature fluctuation in the CMB amplitude [51,52]. A canonically normalized axion A is defined by noting eq. (26) as

∂A ∂θ
=σ Since the axion interacts with other fields very weakly, it causes the isocurvature fluctuation as the fluctuation of its number density n A . The amplitude of its power spectrum can be expressed as Since the axion is only a source of the isocurvature fluctuation in this model, its fraction in the power spectrum is given as where P s (k) = A s which is given in eq. (38). R a is a fraction of the axion energy density in the CDM and defined as R a = Ω a /Ω CDM . If we use a relation [53] R a = θ 2 6 × 10 −6 f a 10 16 GeV we find Since the Planck data constrain α as α ≤ 0.037 at k = 0.
where f a = w is used. In the case (I), this gives no constraint and the parameters used in the present study to estimate the reheating temperature in Fig. 4

Summary
We have proposed a model which could give an explanation for the origin of the CP After the symmetry breaking due to the singlet scalars, the leptonic sector of the model is reduced to the scotogenic model, which can explain the small neutrino masses and the DM abundance due to the remnant discrete symmetry of the imposed symmetry.
Singlet scalars introduced to explain the CP issues can play a role of inflaton if it has a nonminimal coupling with the Ricci scalar. We suppose this coupling is of order one. In that case, although it gives the similar prediction for the scalar spectral index and the tensor to scalar ratio to the one of the Higgs inflation, reheating phenomena is different from it since the radiation domination starts just after the end of inflation.
The model has a notable phenomenological feature in addition to these. The extra fermions which are introduced for the CP issues could make the thermal leptogenesis generate the sufficient baryon number asymmetry even if the lightest right-handed neutrino mass is much lower than 10 9 GeV, which is the well-known lower bound of the right-handed neutrino mass for successful leptogenesis in the ordinary seesaw scenario.