KANAZAWA-12-06

July, 2012

Baryon number asymmetry and dark matter in the neutrino mass model with an inert doublet

Shoichi Kashiwase^{1}^{1}1e-mail: shoichi@hep.s.kanazawa-u.ac.jp
and Daijiro Suematsu^{2}^{2}2e-mail: suematsu@hep.s.kanazawa-u.ac.jp

Institute for Theoretical Physics, Kanazawa University,

Kanazawa 920-1192, Japan

Abstract

The radiative neutrino mass model with an inert doublet scalar has been
considered as a promising candidate which can explain neutrino masses,
dark matter abundance and baryon number asymmetry if dark matter
is identified with the lightest neutral component of the inert doublet.
We reexamine these properties by imposing all the data of the neutrino
oscillation, which are recently suggested by the reactor experiments.
We find that the sufficient baryon number asymmetry seems not to be
easily generated in a consistent way with all the data of the neutrino
masses and mixing as long as the right-handed neutrinos are kept in TeV
regions. Two possible modifications of the model are examined.

## 1 Introduction

The standard model (SM) is now considered to be extended on the basis of several evidences clarified by recent experiments and observations, that is, the neutrino masses and mixing [1], the existence of dark matter (DM) [2] and also the baryon number asymmetry in the Universe [3]. Although there are a lot of models which are proposed to explain these independently, it is not so easy to construct a model which can explain all of them simultaneously without causing any other phenomenological problems. If we can find such a model, it would give us crucial hints for the new physics beyond the SM. The study along this line might play a crucial role for the search of physics beyond the SM prior to the study based on purely theoretical motivation such as the gauge hierarchy problem.

The radiative neutrino mass model with an inert doublet [4] could be such a promising candidate. It is a very simple extension of the SM by an inert doublet scalar and three right-handed neutrinos only. An imposed symmetry controls the scalar potential and forbids the tree-level neutrino masses since its odd parity is assigned to these new fields and the even parity is assigned to the SM fields. It also guarantees the stability of the lightest field with its odd parity. Thus, the lightest neutral component of the inert doublet scalar [5, 6, 7, 8, 9] or the lightest right-handed neutrino [10, 11] could play a role of DM. This feature opens a possibility for the model such that it can explain all the above-mentioned three problems on the basis of closely related physics, simultaneously. However, the model could explain only two of the three problems if the right-handed neutrino is identified with DM [12]. In this case the model is required to be extended in some way for the explanation of all three issues [13]. On the other hand, it is noticeable that the above-mentioned three problems are suggested to be consistently explained by the new fields with TeV-scale masses as long as the lightest neutral component of the inert doublet scalar is identified with DM [6]. This latter case seems very interesting and worthy for further quantitative study since the signature of the model could be seen in various ongoing or future experiments.

In this paper we reexamine this radiative neutrino mass model by fixing the parameters relevant to the neutrino masses and mixing on the basis of neutrino oscillation data including the recent results for given by T2K, Double Chooz, RENO and Daya Bay [14]. We proceed this study by imposing the conditions required by the DM relic abundance and its direct detection. Based on these results, we analyze what amount of the baryon number asymmetry can be generated via thermal leptogenesis. We show that the sufficient baryon number asymmetry seems difficult to be generated consistently for the parameters that are favored by the presently known phenomenological requirements.

The remaining parts of paper are organized as follows. In section 2, we briefly review the scalar sector of the model and then discuss the constraints brought about by the DM relic abundance and the DM direct detection. Next, we fix the parameters relevant to the neutrino mass matrix to realize the neutrino oscillation data, which are also closely related to leptogenesis. In section 3, we apply them to the study of leptogenesis and estimate the baryon number asymmetry via the out-of-thermal-equilibrium decay of the lightest right-handed neutrino by solving the Boltzmann equations numerically. Conditions to generate the suitable baryon number asymmetry are discussed. Summary of the paper is given in section 4.

## 2 DM abundance and neutrino masses

### 2.1 The model and nature of its scalar sector

We consider the radiative neutrino mass model with an inert doublet scalar [4]. The model is a very simple extension of the standard model (SM) with three right-handed neutrinos , and a scalar doublet which is called the inert doublet and assumed to have no vacuum expectation value. Although both and are supposed to have odd parity of an assumed symmetry, all SM contents are assigned by its even parity. Invariant Yukawa couplings and scalar potential related to these new fields are summarized as

(1) | |||||

where is a left-handed lepton doublet and is an ordinary Higgs doublet. All the quartic coupling constants are assumed to be real, for simplicity. We also assume that neutrino Yukawa couplings are written by using the basis under which both matrices for Yukawa couplings of charged leptons and for masses of the right-handed neutrinos are real and diagonal. These neutrino Yukawa couplings are constrained by the neutrino oscillation data and also the lepton flavor-violating processes such as .

In the following study, we assume the mass spectrum of the right-handed neutrinos to satisfy

(2) |

and also the flavor structure of the neutrino Yukawa couplings to be

(3) |

where are real constants. This assumption for the neutrino Yukawa couplings could reduce free parameters of the model substantially. Moreover, it can cause the favorable lepton flavor mixing as found later. We note that there remains a freedom, that is, which type structure represented by the suffix and in eq. (3) should be assigned to each right-handed neutrino. In the following part, we adopt two typical cases for it as follows,

(4) |

Now, we briefly review the scalar sector of the model [5, 6]. If we take unitary gauge and put and where , the scalar potential in eq. (1) can be written as

(5) | |||||

where we use the definition and

(6) |

The expression of in eq. (5) shows that the assumed vacuum is stable for

(7) |

We also require these quartic couplings to satisfy so that the perturbativity of the model is guaranteed.

Since the new doublet scalar is assumed to have no vacuum expectation value, the symmetry is kept as the unbroken symmetry of the model. Thus, the lightest field with the odd parity of this is stable and then its thermal relic behaves as DM in the Universe. If it is identified with here, the following condition should be satisfied

(8) |

These are easily found from eq. (6). The value of might be estimated by using GeV, which is suggested through the recent LHC experiments. If we apply it to the tree-level formula in eq. (6), we have . Using this value of and the conditions given in eqs. (7) and (8), we can roughly estimate the allowed range of as

(9) |

for the sufficiently small values of . The lower bound of is settled by the requirement for the perturbativity of the model.

The mass difference among the components of is estimated as

(10) |

which could be a good approximation for the large value of
such as TeV.^{3}^{3}3Such a large value of is favored
from the analysis of the parameter for the precise measurements in
the electroweak interaction [5, 6].
In that case, the model has no constraint from it.
These formulas show that coannihilation among the components of
could play an important role in the estimation of the relic
abundance of [6].

### 2.2 Inert doublet dark matter

In several articles [5], the DM abundance is found to be well explained if the lightest neutral component of is identified with DM. In the high-mass case, in particular, it is suggested that the relic abundance could be a suitable value if one of the quartic couplings in eq. (1) has magnitude of [6].

The relic abundance is known to be estimated as [15]

(11) |

where the freeze-out temperature and are defined as

(12) |

The effective annihilation cross section
and the effective degrees of
freedom are expressed
by using the thermally averaged (co)annihilation cross section
and the equilibrium number density
as^{4}^{4}4We may use the notation such as
for
convenience in the following discussion.

(13) |

The thermally averaged (co)annihilation cross section may be expanded by the thermally averaged relative velocity of the annihilating fields as . Since is satisfied for cold DM candidates and then gives the dominant role for determining the relic abundance of , we take account of it only neglecting the contribution in this analysis. In the present model, the corresponding cross section is caused by the weak gauge interactions and also the quartic couplings . It is approximately calculated as [6]

(14) | |||||

where is defined as

(15) |

Using these formulas, we examine the condition on the relevant parameters of the model to realize the relic abundance which is required from the WMAP data [2]. In Fig. 1, we plot as a function of for and some typical values of . We should note that the allowed region of is restricted by eq. (9). This figure shows that the required value for could be obtained for a wide range value of if has a value of . Since we consider the high-mass region such as , the coannihilation among the components of could be effective to reduce the relic abundance of .

The above analysis shows that the DM relic abundance gives only a weak condition on some of the quartic couplings but no conditions on the neutrino Yukawa couplings. This is completely different from the case in which the lightest is identified with DM [10, 11, 12]. On the other hand, the direct search of DM could give a severe constraint on the value of , which plays a crucial role in this radiative neutrino mass generation. Elastic scatterings between and nucleus could be mediated by the Higgs exchange at tree level and also by the gauge boson exchange at one-loop level. However, their effects are much smaller than the present upper bounds of the sensitivity for the direct detection. Thus, we can neglect their effects in this discussion. On the other hand, inelastic scattering of with nucleus mediated by the exchange could bring about an important effect to the direct search experiments [8, 9], since the masses of and are almost degenerate for small values of as found from eq. (10).

If we note that the interaction of relevant to this process is given by

(16) |

it is found that the inelastic nucleus-DM scattering can occur for the DM with velocity larger than a minimum value given by [16]

(17) |

where is the mass difference between and
defined in eq.(10). is the nucleus recoil
energy, and and are the mass of the target nucleus and the
reduced mass of the nucleus-DM system.
Thus, the mass difference is constrained by the fact that
no DM signal is found in the direct
DM search yet [17, 18].^{5}^{5}5The DAMA data have
been suggested
to be explained by the DM inelastic scattering [8, 9].
However, we do not consider it here.
This condition might be estimated as keV [9].
Since is related to through eq.(10) ,
this constrains the allowed value of
such as^{6}^{6}6We should note that the bound of largely
depends on the DM velocity in the neighborhood of the Earth. If we take
MeV, this inelastic scattering effect can
be completely neglected even for the maximally estimated DM velocity.
In that case, the lower bound of becomes one order of
magnitude larger.

(18) |

We take account of this constraint in the following analysis of the neutrino masses and the baryon number asymmetry.

### 2.3 Neutrino masses and mixing

Neutrino masses are generated through one-loop diagrams with the contribution of new odd fields. They can be expressed as [4, 10]

(19) |

where . Since we consider the high-mass region such as , the mass difference among caused by nonzero is negligible in the neutrino mass analysis. Thus, we treat their masses as . Both neutrino masses and mixing are determined by the couplings and , the right-handed neutrino masses ’s and the inert doublet mass . Here we note that the neutrino Yukawa couplings could take rather large values even for the light right-handed neutrinos with masses of TeV as long as takes a small value in the range given by eq. (18). This freedom is crucial when we consider leptogenesis in this model as seen in the next section.

Now we have a lot of information on the feature of lepton flavor mixing on the basis of the neutrino oscillation data including the recent results for [19]. We can use it to restrict the neutrino Yukawa couplings. The flavor structure of the neutrino Yukawa couplings assumed in eq. (3) makes the neutrino mass matrix take a simple form such as

(20) |

where should be understood to stand for (i) and (ii) following eq. (4). If we put in both cases, the PMNS mixing matrix is easily found to have a tribimaximal form

(21) |

where Majorana phases are determined by the phases and . If we put and , they are expressed as

(22) |

In this case, one of mass eigenvalues is zero. Thus, if is assumed to take a sufficiently small value compared with others, we find that the mass eigenvalues should satisfy

(23) |

where and stand for
the squared mass differences required by the neutrino oscillation analysis
for both atmospheric and solar neutrinos [1, 19].^{7}^{7}7We
confine our study to the normal hierarchy here.

Using the formulas (21) and (23), we can examine whether parameters obtained in the previous part could be consistent with the neutrino oscillation data. This gives a useful starting point for the analysis. However, if we take account of the fact that is found to have a nonzero value now, we cannot use them in the analysis directly. Here, we numerically diagonalize the mass matrix (20) and impose both the neutrino oscillation data and the constraint on given in eq. (18) to restrict the neutrino Yukawa couplings. By fixing to typical values, in Fig. 2, we plot contours in the plane which correspond to 2 bounds of the neutrino oscillation parameters given in [20]. This figure shows the mass matrix (20) can explain all the neutrino oscillation data consistently at the regions in the plane, which are the region including in the left panel () and the region including in the right panel (). These regions in two panels are obtained for each set of parameters listed in Table 1. We also give the predicted values for in each case of the same Table. These examples show that neutrino Yukawa couplings of can explain the neutrino oscillation data for and the right-handed neutrinos with the mass of TeV.

Lepton flavor-violating processes such as are also induced through one-loop diagrams which have and in the internal lines [10]. Their present experimental bounds could impose severe constraints on the model depending on the values of neutrino Yukawa couplings . However, since the small of can realize the appropriate values for neutrino masses even for the TeV-scale values of and as discussed above, the new contributions to the lepton flavor-violating processes are sufficiently suppressed such as and . These values show that the lepton flavor-violating processes bring about no substantial constraints on the model. We should note that the freedom of in this mass generation scheme makes it possible.

max | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

(ia) | 0.85 | 1 | 2 | 6 | 10 | 3.41 | 1.50 | 0.085 | ||

(ib) | 0.85 | 1 | 2 | 20 | 200 | 4.62 | 4.16 | 0.085 | ||

(ic) | 1 | 1 | 2 | 6 | 10 | 3.41 | 1.50 | 0.053 | ||

(iia) | 0.85 | 1 | 2 | 6 | 10 | 1.34 | 3.81 | 0.085 | ||

(iib) | 0.85 | 1 | 2 | 20 | 200 | 1.82 | 10.6 | 0.085 |

Table 1 The predicted value of and for the model parameters that can satisfy the neutrino oscillation data. Cases (i) and (ii) correspond to the ones defined in eq. (4). In all cases, and are fixed to and , respectively. The value of is evaluated at for and for , where all other neutrino oscillation data are satisfied. A TeV unit is used as the mass scale.

## 3 Baryon number asymmetry

### 3.1 Leptogenesis via the decay of the TeV-scale right-handed neutrino

We consider the thermal leptogenesis [21] in this model with the mass spectrum given in eqs. (2) and (8). In this case, the lepton number asymmetry is expected to be generated through the out-of-thermal-equilibrium decay of the right-handed neutrino . The dominant contribution to the asymmetry in this decay is brought about by the interference between the tree diagram and the one-loop vertex diagram as usual. However, we should note that the mass is not negligible compared with the one of in this model. Taking account of this feature, can be calculated as [22]

(24) | |||||

where is defined by

(25) |

and is represented as

(26) |

If we use the flavor structure of neutrino Yukawa couplings (i) and (ii) given in eq. (4), are expressed for each case as

(27) | |||||

where .

The decay of should be out of equilibrium so that the lepton number asymmetry is generated through it. If we express the Hubble parameter and the decay width of by and respectively, this condition is given as at where the lepton number asymmetry is considered to be dominantly generated. Since is expressed as , we find that the Yukawa coupling should be sufficiently small such as

(28) |

We note that this constraint could be weaker since both and the Boltzmann suppression factor are neglected in this estimation. As found in the numerical calculation, can be somewhat larger than this bound.

The generated lepton number asymmetry could be washed out by both the lepton number violating 2-2 scattering such as and and also the inverse decay of . If the relevant Yukawa couplings are much smaller than , these processes are expected to decouple before the temperature of the thermal plasma decreases to . In order to study this quantitatively, we numerically solve the coupled Boltzmann equations for the number density of and the lepton number asymmetry which are expressed by and here, respectively. The Boltzmann equations for these quantities are written as [23]

(29) |

where and . and are defined as and by using the entropy density . Their equilibrium values are expressed as and , where is the number of relativistic degrees of freedom and is the modified Bessel function of the second kind. In these equations we omit terms whose contributions are considered to be negligible compared with others. The formulas of the relevant reaction density contained in these equations are given in the Appendix. If we use the relation which is derived with the chemical equilibrium condition in this model, the baryon number asymmetry in the present Universe is found to be estimated as

(30) |

by using the solution of the coupled equations in eq. (29). Here is related to the sphaleron decoupling temperature as .

The solutions of eq. (29) are shown in the upper panels
of the left and the middle columns in Fig. 3 for cases (ia)
and (ib) in Table 1.
The generated lepton number asymmetry is smaller than the value
required for the explanation of the baryon number asymmetry
at least by one order of magnitude.^{8}^{8}8The parameters used in
case (ia) are almost equivalent to the one which is presented as the
promising one for the generation of the sufficient baryon number
asymmetry in [6].
In case (ia), the asymmetry parameter takes
rather large value such as .
This suggests that the washout of the generated lepton number
asymmetry is effective.
In the figure of the right column we use the same parameters as
the one of case (ia) except for
which is fixed to the larger value .
From this figure, we can see the role of this coupling which is
discussed above.
Although this change does not affect the values of which explain
the neutrino oscillation data, we expect that
the deviation of the number density of from the equilibrium value
becomes smaller than other cases with the smaller value of .
This is shown in the figures.

In the lower panel we plot the behavior of the relevant reaction rates for each case. These processes are crucial for the leptogenesis in this model. The figures show that the lepton number-violating scatterings induced by the -channel exchange are kept in the thermal equilibrium until rather late period and the large part of the generated lepton number asymmetry is washed out. This situation is common for all cases in Table 1. We give the predicted value of for each case in the last column of Table 1. These examples show that the sufficient amount of baryon number asymmetry seems difficult to be generated through the thermal leptogenesis in the present neutrino mass generation scheme at least as long as we impose the full neutrino oscillation data and the DM direct search constraint.

In order to confirm this statement in case (ia), we plot the
generated baryon number asymmetry for various values of
in Fig. 4.
The maximum value of is found to be realized at a certain value of
and it moves to the smaller
region for the smaller .
This may be explained as follows. At the region with a larger
value, the neutrino Yukawa couplings have smaller values to
give a smaller value for . On the other hand, at the
region with a smaller value, the neutrino Yukawa couplings
have larger values to bring about the large washout.
For the TeV case, the requited can be obtained
at a rather small value such as due to this nature.
However, it is excluded by the direct search
experiments as shown in eq. (18).^{9}^{9}9It is useful to
note that the smaller brings about the
smaller value for the lower bound of . However, its effect
is only a change of the factor in case of the high-mass .
Although we do not search the whole parameter space,
we could say that the above-mentioned result does not change so easily
for the values of which satisfy the condition
(18).
This is suggested by the fact that
the value of becomes smaller for the larger
which makes the neutrino Yukawa couplings smaller under the
constraints of the neutrino oscillation data.

Finally, we give some comments on the leptogenesis in the case where
the right-handed neutrinos have large masses comparable to the ones in
the ordinary seesaw case.^{10}^{10}10This is considered in [7].
However, the neutrino oscillation data are not imposed
in a quantitative way there.
In this case, we find that the DM relic abundance and the
neutrino masses and mixing could be explained consistently by
setting and the neutrino Yukawa couplings appropriately.
The right-handed neutrino masses could be smaller than the ones
in the ordinary tree-level seesaw case for the same neutrino Yukawa
couplings since the neutrino masses are generated through the
one-loop effect.
In the ordinary type I seesaw scenario with the hierarchical
right-handed neutrino masses, there is an upper bound
for the asymmetry which is known as the Davidson-Ibarra (DI)
bound [25] and may be written as
.
In the present case with the assumed flavour structure,
the asymmetry is related to the DI bound as
.
This shows that the DI bound could be evaded depending on the value of
. However, this feature does not
mean that the model causes
more efficient leptogenesis than the ordinary seesaw model.
Since the smaller requires the larger neutrino Yukawa
couplings under the constraints of the neutrino oscillation data,
the washout of the lepton number asymmetry is expected to become large.

We examine this aspect under the assumed lepton flavor structure by imposing all the neutrino oscillation data quantitatively. We assume the hierarchical right-handed neutrino mass spectrum and fix the parameters as follows,

(31) |

where is determined by taking account of the condition (28).
We find that the asymmetry can be written for these
parameters as where the DI bound
is given as .
This relation shows that the asymmetry could escape the DI bound
without causing the contradiction with the neutrino oscillation data if
and take suitable values.
In order to fix the values of neutrino Yukawa couplings,
we impose the neutrino oscillation data in the same way as in the previous
examples with , , and .
For such neutrino Yukawa couplings, we obtain
independently on the value of .
The baryon number asymmetry obtained at through the analysis of
the Boltzmann equations is plotted for some typical values
of and in the left panel of Fig. 5.
Since this is much smaller than ,
could be much smaller than the plotted value if the
washout effects do not decouple.
However, we can confirm that the plotted is recognized as
, at least for , where
gives the maximum value of .
In the region of , the neutrino
Yukawa couplings become large enough to continue reducing the generated lepton
number asymmetry through the lepton number-violating scatterings
which do not decouple at completely.
Since is required to be larger for the smaller values of
from the neutrino oscillation data,
the large part of the generated lepton number asymmetry
is considered to be washed out effectively although a larger
value is expected.
The figure shows that the required value of is generated for
GeV, which is somewhat smaller than the one required
in the ordinary seesaw case.^{11}^{11}11The required baryon number
asymmetry could be generated even in the case with GeV
if special texture for the neutrino mass matrix is assumed even in the
ordinary seesaw case [26].

### 3.2 Improvement by suppressing the washout

In the previous part we found that it is difficult to generate the sufficient baryon number asymmetry in consistent with all the neutrino oscillation data and the DM direct search as long as the mass of the lightest right-handed neutrino is assumed in a TeV range. This result is considered to be brought about by the large washout effect of the generated lepton number asymmetry. Here, we consider a possible improvement of this situation by making the neutrino Yukawa couplings small enough to suppress the washout. In this improvement, the asymmetry should be kept to a suitable value such as or more, simultaneously. Resonant leptogenesis can realize it.

As such an example, we suppose that the right-handed neutrino masses and are nearly degenerate in case (ia). is fixed to 2 TeV and is replaced with . In this case, we can make the neutrino Yukawa couplings much smaller than the ones in case (ia) by assuming a larger value for . For instance, if we put , all the neutrino oscillation data can be satisfied for and . Although the smaller neutrino Yukawa couplings tend to make the asymmetry smaller, we can enhance by supposing the degenerate right-handed neutrino masses. On the other hand, in that case the washout effect could be suppressed sufficiently. In the usual resonant leptogenesis with the right-handed neutrino masses at TeV regions [22, 27, 28], the generation of the sufficient lepton number asymmetry requires rather strict mass degeneracy. We examine whether this situation can be changed in this neutrino mass generation scenario.

In the nearly degenerate right-handed neutrino case, the dominant
contribution to the asymmetry in the decay of comes
from the interference between the tree and the self-energy
diagrams.^{12}^{12}12Since the Yukawa couplings of
are required to be much larger than for ,
are considered to be in the thermal equilibrium.
Thus, we can not expect a substantial contribution to the lepton number
asymmetry from the decays of .
In the present case, the asymmetry can be expressed
as [22, 27, 28]

(32) | |||||

where . For case (ia) with , can have a value of for . Thus, we can expect that the mass degeneracy required to bring about the sufficient baryon number asymmetry could be much milder than the usually assumed value in the TeV-scale resonant leptogenesis [28].

We use the formula for given above and estimate by solving the Boltzmann equations in eq. (29) for some typical values of by varying the value of . In the right panel of Fig. 5, we plot the numerical results of the obtained baryon number asymmetry. The required baryon number asymmetry is found to be generated for the right-handed neutrinos with the mass degeneracy for each value of . This degeneracy is much milder in comparison with the ordinary resonant leptogenesis at TeV scales. The result is caused by the nature of the model such that the neutrino Yukawa couplings can take sufficiently small values to suppress the washout effect keeping the neutrino masses in the appropriate range for the explanation of the neutrino oscillation data. The freedom of makes it possible. The present neutrino mass generation scheme can give the consistent explanation for the three phenomenological problems in the SM if only the rather mild mass degeneracy between two light right-handed neutrinos is assumed without large extension of the model.

## 4 Summary

The inert doublet model extended with the right-handed neutrinos is a simple and interesting framework for both neutrino masses and dark matter. In this scenario, the lightest right-handed neutrino or the lightest neutral component of the inert doublet can be a dark matter candidate. Since the neutrino Yukawa couplings should be to reduce the relic abundance of dark matter in the former case, the thermal leptogenesis is difficult due to the strong washout effect. On the other hand, the neutrino Yukawa couplings can be irrelevant to the dark matter abundance in the latter case. Thus, the latter scenario has been considered to give the consistent explanation for the origin of the small neutrino masses, the existence of dark matter and the baryon number asymmetry in the Universe.

In this paper, we reexamined the possibility for the simultaneous explanation of these three problems in the latter case. We took account of the quantitative explanation for all the neutrino oscillation data including the recent results for in the analysis. The results of our study suggest that the sufficient amount of baryon number asymmetry seems not to be generated in a consistent way with the full neutrino oscillation data. The neutrino Yukawa couplings could be large enough to enhance the asymmetry even for the TeV right-handed neutrinos in the consistent way with the neutrino oscillation data by using the freedom of . However, the same Yukawa couplings induce the lepton number-violating scattering processes to wash out the generated lepton number asymmetry. We checked this point through the numerical study. Although we do not study the whole parameter space, this feature seems to be rather general and then it seems not so easy to find the parameters to escape this situation without any modification.

We also examined the same problem in the modified situation such as the case with the heavy right-handed neutrinos like the ordinary type I seesaw and also the case with the degenerate light right-handed neutrinos. In the former case, we found that the right-handed neutrino could be somewhat lighter than the ordinary seesaw one but needs to be heavy enough to the similar level to it. In the latter case, even if the right-handed neutrino masses are in a TeV range, the resonant effect can enhance the asymmetry to generate the sufficient amount of baryon number asymmetry even for the small neutrino Yukawa couplings which can suppress the washout sufficiently. An interesting point is that the required mass degeneracy in this case is much milder than the one in the ordinary seesaw case. This possibility is brought about by the characteristic feature in this inert doublet model with the radiative neutrino mass generation. More complete study of the parameter regions which are not searched in this paper will be presented in future publication.

## Acknowledgement

This work is partially supported by Grant-in-Aid for Scientific Research (C) from Japan Society for Promotion of Science (No. 21540262 and No. 24540263), and also a Grant-in-Aid for Scientific Research on Priority Areas from The Ministry of Education, Culture, Sports, Science and Technology (No. 22011003).

## Appendix

In this Appendix, we give the formulas of the reaction density
contributing to the Boltzmann equations for the number density
of and the lepton number asymmetry.
For the processes relevant to their evolution, we could refer to
the reaction density given in [24].
In the present model, however, interaction terms of and
are restricted by the symmetry and also the masses of
and take the similar order values, which cause large difference
from the ordinary seesaw leptogenesis.
Thus, we need to modify these formulas by taking account of the features
of the present model.^{13}^{13}13Although the modified ones are
given in Appendix of [12], the mass spectrum assumed there is
different from the present one. The following formulas are arranged to
applicable to the scenario in this paper.

In order to give the expression for the reaction density of the relevant processes, we introduce dimensionless variables

(33) |

where is the squared center of mass energy. The reaction density for the decay of can be expressed as