1 Introduction

Leptogenesis and dark matter unified in a non-SUSY model for neutrino masses

Daijiro Suematsu ***e-mail: suematsu@hep.s.kanazawa-u.ac.jp

Institute for Theoretical Physics, Kanazawa University,

Kanazawa 920-1192, Japan

Abstract
We propose a unified explanation for the origin of dark matter and baryon number asymmetry on the basis of a non-supersymmetric model for neutrino masses. Neutrino masses are generated in two distinct ways, that is, a tree-level seesaw mechanism with a single right-handed neutrino and one-loop radiative effects by a new additional doublet scalar. A spontaneously broken U(1) brings a symmetry which restricts couplings of this new scalar and controls the neutrino masses. It also guarantees the stability of a CDM candidate. We examine two possible candidates for the CDM. We also show that the decay of a heavy right-handed neutrino related to the seesaw mechanism can generate baryon number asymmetry through leptogenesis.

1 Introduction

Neutrino masses [1], cold dark matter (CDM) [2], and baryon number asymmetry in the universe [3] suggest that the standard model (SM) should be extended. Both neutrino masses and baryon number asymmetry are well known to be explained in a unified way through the leptogenesis scenario in the framework of the seesaw mechanism [4]. Extensive studies have been done on this subject during recent several years [5]. On the other hand, supersymmetry is known to play a crucial role for the explanation of CDM abundance in the universe [6], although it has been introduced originally to solve the hierarchy problem. Supersymmetric models have good candidates for CDM such as the lightest superparticle (LSP) as long as -parity is conserved. The neutralino LSP has been extensively studied as a CDM candidate in the supersymmetric SM (MSSM) and its singlet extensions [7, 8]. If we try to explain simultaneously both the leptogenesis and the CDM abundance in supersymmetric models, we have a difficulty. The out-of-equilibrium decay of thermal heavy neutrinos can generate sufficient baryon number asymmetry only if the reheating temperature is high enough such as  GeV. For such reheating temperature, however, we confront the serious gravitino problem in supersymmetric models [9, 10]. Various trials to overcome this difficulty have been done by searching scenarios to enhance the asymmetry and lower the required reheating temperature [11, 12, 13].

In these studies, the CDM and the baryon number asymmetry are separately explained based on unrelated physics. Thus, we cannot expect to obtain any hints as to why the CDM abundance is of similar order as the baryon number asymmetry in the present universe through such studies.111There are several works to relate the CDM abundance to the baryon number asymmetry. For such trials, see [14] for example. Unfortunately, at present, we have no satisfactory supersymmetric models to explain these three experimental evidences which impose us to extend the SM. In this situation it may be worth to take a different empirical view point at first and reconsider possible models which can explain these evidences simultaneously on the basis of closely related physics [15]. As the next step, the hierarchy problem may be considered in the framework where such models are embedded.

Recently, it has been suggested that neutrino masses and the CDM abundance may be related in some kind of non-supersymmetric models for neutrino masses. In such models neutrino masses are generated through one-loop radiative effects which are induced by new scalar fields [16]. A certain symmetry prohibiting large neutrino masses can also guarantee the stability of a CDM candidate like -parity in supersymmetric models [17, 18, 19]. The baryon number asymmetry has also been discussed in this model [20]. In the same type model there is also a suggestion that the hierarchy problem can be improved by considering a heavy Higgs scalar [21]. Since these models have rather simple structure at weak scale regions, it might give us some useful hints for physics beyond the SM if they can explain the above mentioned experimental evidences consistently.

In this paper, we consider the possibility that the baryon number asymmetry is closely related to the origin of both neutrino masses and CDM abundance. We show that the ordinary leptogenesis based on heavy neutrino decay can be embedded consistently in the model for neutrino masses proposed in [19]. As we discuss below, this is closely related to an extension of [19] such that (1) an additional with zero charge under U(1) is introduced and (2) the dimension five term in the scalar potential has a complex coupling . The paper also includes new contributions added to [19] such that (1) both and are studied as dark matter candidates and (2) the constraints due to neutrino oscillation data are taken into account in a more extended way than that in [19].

The remaining parts are organized as follows. In section 2 we address features of the model and discuss a parameter space consistent with neutrino oscillation data. In section 3 we study the relation between the leptogenesis and the CDM abundance in the model. We examine two possible CDM candidates taking account of the neutrino oscillation data and the conditions required by the leptogenesis. We will find that the model can give a unified picture for the explanation of the neutrino masses, the CDM abundance, and the baryon number asymmetry. In section 4 we summarize the paper with comments on the signatures of the model expected at LHC.

2 A model for neutrino masses

The present study is based on the model proposed in [19]. Ingredients of the model and U(1) charge assignments for these are given in Table 1. We suppose that U(1) is leptophobic.222We need to introduce some fields to cancel the gauge anomalies. However, it can be done without affecting the following study. We present such an example in the Appendix. The extension to general U(1) is straightforward. The fermions listed in Table 1 are assumed to be left-handed. We note that three singlet fermions are necessary for present purposes. Although only two of them are ordered to generate appropriate masses and mixing in the neutrino sector, an additional one is necessary for the leptogenesis. The invariant Lagrangian relevant to the neutrino masses can be expressed by

(1)

Yukawa couplings for charged leptons are assumed to be diagonalized already. The most general scalar potential invariant under SU(2)U(1)U(1) gauge symmetry up to dimension five is given as

(2)

where the couplings are real except for . The phase of can induce a physical one which is found to be a Majorana phase in the neutrino mass matrix. A nonrenormalizable term and bare mass terms for are added, which will be shown to play crucial roles in the present scenario. They are supposed to be effective terms generated through some dynamics at intermediate scales. We assume that and only and are related to light neutrino masses and mixings.

U(1) 0 0 0 0
+1 +1 +1 +1 +1 +1 +1 +1

Table 1.   Field contents and their charges. is the residual symmetry of U(1).

The model includes two SU(2) doublet scalars and . plays the role of the ordinary doublet Higgs scalar in the SM but is assumed to obtain no VEV. A singlet scalar is also assumed to have a real VEV at suitable scales, which breaks U(1) down to . The charge for each field can be found in Table 1. The VEV of gives masses for and as

(3)

where is assumed to be real. Since is bounded from below by the phenomenology, has also lower bounds for fixed values of . It also yields an effective coupling constant in the term. It can be small enough to make radiative neutrino masses tiny even for values of as long as is satisfied. Since the mixing between and is induced through this small coupling, the mass eigenvalues split slightly. The states have mass eigenvalues such as

(4)

The magnitude of the difference of these eigenvalues is constrained by the direct search of the CDM if either of these is the lightest odd field. Mass of the charged states is given by

(5)

and then can be much smaller than in case of . These points will be discussed in the analysis of the CDM later. Since is complex in general, the violation may be detected through this - mixing. Although this is an interesting feature of the model, we do not discuss this subject further in this paper.

We have two distinct origins for the neutrino masses in this model. One is the ordinary seesaw mass induced by a right-handed neutrino [22]. Another one is the one-loop radiative mass mediated by the exchange of and [16, 23]. Although also has contributions to the neutrino mass generation through the seesaw mechanism, its effect can be safely neglected compared with these if is large enough. However, baryogenesis caused by leptogenesis requires this contribution since is has no lepton number as discussed below. The radiative neutrino mass generation requires some lepton number violation. We can put them either in or . If we assume that and have the lepton number and , respectively, the term in brings about this required lepton number violating effect. We adopt this choice in the following arguments. are considered to have lepton number +1.

The mass matrix for three light neutrinos induced by these origins is summarized as

(6)

where is defined by

(7)

Both and are assumed to be real, for simplicity. We note that two terms in have the similar texture although they are characterized by different mass scales. If we impose commutativity between and , the condition

(8)

is needed to be satisfied. We consider this simple case in the following as an interesting example, since it allows us to study the mass matrix analytically.333If nonzero eigenvalues are dominated by different origins respectively, this will be a good approximation to describe such cases.

We introduce a matrix to diagonalize the larger term of at first, which is defined as

(9)

Then the matrix in can be diagonalized as if the angles satisfy

(10)

Eigenvalues for this matrix are found to be

(11)

Another term is also transformed by . However, if the condition (8) is satisfied, can be diagonalized by an orthogonal transformation supplemented by an additional transformation

(12)

and we have eigenvalues

(13)

Here is defined as

(14)

We note that this transformation does not affect the diagonalization of .

If we define the mass eigenvalues as where is assumed, they can be written as

(15)

Here we find that there are two possibilities for generation of and . The first case is realized by taking and in the above formulas, and then is induced by the ordinary seesaw mechanism. In this case and are defined by

(16)

The second case is obtained by taking and , and then is determined by the radiative effect. In this case and are written as

(17)

Since only two mass eigenvalues can be considered nonzero in the present setting, neutrino oscillation data require that these mass eigenvalues should satisfy and [1]. Data of the atmospheric neutrino and the K2K experiment require . We also find that should be taken as which is a mixing angle relevant to the solar neutrino. The CHOOZ experiment gives a constraint on such as [24]. If we use these conditions, the mixing matrix can be approximately written as

(18)

By imposing the experimental values on , , , and , we can constrain the values of and [19]. For simplicity, we assume .

The condition for constrains the Yukawa coupling as

(19)

If we require and to be in perturbative regions, we find that both and should be less than  GeV. Here we introduce two parameters and . The condition for selects the regions in the plane which are consistent with the neutrino oscillation data. They are shown for both cases (i) and (ii) as the regions sandwiched by the dashed lines in Fig. 1. These figures show that the model can explain the neutrino oscillation data in rather wide parameter regions. In particular, it is useful to note in relation to the CDM that we can have solutions for large values of such as as long as stays in the constrained region: (i)  and (ii) . By using these results obtained from the neutrino oscillation data, we examine the leptogenesis and the CDM abundance in this model in the next section.

3 Leptogenesis and CDM abundance

The present model contains several new neutral fields with nonzero lepton number or an odd charge. Thus, we have sufficient ingredients with the required properties for both leptogenesis and CDM candidates. Although one might consider that there are several scenarios for these explanations in this model, they seem to be constrained by the neutrino oscillation data.

The lightest neutral field with an odd charge can be stable and then a CDM candidate since an even charge is assigned to each SM content. If is satisfied, can be a CDM candidate. As in the ordinary leptogenesis scenario, related to the ordinary seesaw mechanism can be a mother field for leptogenesis. However, since two right-handed neutrinos are necessary to realize the asymmetry, we need to introduce with the lepton number as mentioned before.

On the other hand, since has both the odd charge and the lepton number, it might be considered as the origin of the CDM or the lepton number asymmetry in the case of . However, it might be difficult to contribute both of them since it has the SM gauge interactions. The situation is similar to sneutrinos in the supersymmetric models. Sneutrinos have been rejected to be a CDM candidate through the direct detection experiments. This constraint might be escapable in the case since there is the - mixing due to the term which generates the mass difference between its components. The model has to satisfy suitable conditions for this mass difference if this possibility is realized. On the other hand, this is too light to be a mother field for sufficient production of the lepton number asymmetry through the out-of-equilibrium decay, although the sector can bring the almost degenerate mass eigenstates through the violating mixing and cause the resonant decay. We examine these subjects in detail below.

3.1 Leptogenesis

If we take account of the existence of which can be neglected in the estimation of the neutrino masses, the leptogenesis is expected to occur through the decay of . In fact, it is heavy enough for the out-of-equilibrium decay and it has the lepton number violation through a Majorana mass term. By taking account of the well known relation which comes from re-processing of the asymmetry by sphaleron transitions, the generated baryon number asymmetry is given by

(20)

where is the ratio of the equilibrium number density of to the entropy density. The asymmetry in the decay and the wash-out effect are represented by and , respectively. If temperature is much larger than , we have by using and . The relativistic degrees of freedom in this model is . Thus, the asymmetry required to produce the present baryon number asymmetry is estimated as

(21)

where we use which is predicted by nucleosynthesis and CMB measurements [3]. The violation in the decay is induced through interference between the tree and one-loop amplitudes. This induced asymmetry is estimated as [5]

(22)

Now we estimate in this model. As discussed in the previous section, there are two ways for generation of the neutrino masses and . The asymmetry can also have different values for these two cases. For simplicity, we assume . This does not affect the estimation of the neutrino masses because of the assumed setting . In that case we have

(23)

where we apply the results in eq. (19) to this estimation. We use these maximum values for in the formulas of here.

In case (i), we have the relation and then can be written as

(24)

In case (ii), we note that the seesaw mechanism gives and the relation is satisfied. Thus, we find that is expressed as

(25)

These results show that a sufficient asymmetry can be generated for

(26)

Consistency with the present setting can be satisfied for  GeV in both cases, for example. It may be useful to remind that is expected to be from the numerical study of the Boltzmann equation. Such an analysis also shows that the leptogenesis is possible only for narrow ranges of [5]. In the present model this is estimated as

(27)

This suggests that is favored by leptogenesis and it could be consistent in the present settings. The values of determine which case between them is more promising. These results show that the out-of-equilibrium decay of can produce the necessary baryon number asymmetry for intermediate values of as in the usual cases. As long as we confine ourselves to the non-supersymmetric framework, the model is free from the gravitino problem.

3.2 CDM candidates and their abundance

The lightest field with an odd charge can be stable since the even charge is assigned to each SM content. If both the mass and the annihilation cross section of such a field have appropriate values, it can be a good CDM candidate as long as it is neutral. As mentioned before, we have two such candidates, that is, the lighter one of (we represent it by ) and .

At first, we consider the case in which is the CDM. Its annihilation is expected to be mediated by both the exchange of and the U(1) gauge boson. If their annihilation is mediated only by the former one through Yukawa couplings as in the model discussed in [18], we need fine tuning of coupling constants to explain both the observed value of the CDM abundance and the constraints coming from lepton flavor violating processes such as . However, in the present case the annihilation can be dominantly mediated by the U(1) gauge interaction since Yukawa coupling constants can be small enough as estimated in eq. (19). Thus, we may expect that can cause the satisfactory relic abundance as the CDM in rather wide parameter regions . We also note that the U(1) is supposed to be a generation independent gauge symmetry and then the FCNC problem can be easily escaped in this case.

In order to estimate the abundance, we consider to expand the annihilation cross section for by the relative velocity between the annihilating as . The coefficients and are expressed as

(28)

where and =3 for quarks. is the center of mass energy of collisions and is the U(1) charge of given in Table 1. The charge of the final state fermion is defined as

(29)

Using these quantities, the present relic abundance of can be estimated as [25],

(30)

where is the degrees of freedom of relativistic fields at the freeze-out temperature of . The dimensionless parameter is determined through the condition

(31)

where is the Planck mass. If we fix the U(1) charge of the relevant fields and its coupling constant , we can estimate the present abundance using these formulas. It can be compared with given by the three year WMAP [26].

Fig. 2  Regions allowed by the WMAP data in the plane. Green and blue dotted lines represent lines for and 0.7, respectively.

We numerically examine the possibility that the CDM abundance is consistently explained in this model. We use the GUT relation and as an example. The regions in the plane allowed by the WMAP data are shown in Fig. 2. They appear as two narrow bands sandwiched by both a solid line and a dashed line. The lower bounds of come from constraints for mixing and a direct search of . Since the Higgs field is assumed to have no U(1) charge, its VEV induces no mixing. Moreover, since it is assumed to be leptophobic, the constraint on obtained from its hadronic decay is rather weak. The lower bounds of may be  GeV in the present model [27]. Since the masses of and are correlated through eq. (3), we can draw a line of in the plane by fixing a value of . In Fig. 2, such lines are represented by the green and blue dotted ones for and 0.7, respectively. For these values required by the WMAP, is found to take values such as GeV and GeV for and 0.7. Using Figs. 1 and 2, we can determine the range of , if and then is fixed. We find that takes very restricted values for the case of  TeV, especially in case (i).

In Fig 2 we can observe an interesting feature of . Although we assume it is leptophobic, it can have nonhadronic decay model as long as is satisfied. Fig. 2 shows that this condition is satisfied only at the lower allowed band but not at the upper allowed band. Thus, can have nonhadronic decay mode only for .

Fig. 3  Allowed regions in the plane. A red thin dotted line and a red thin solid line corresponds to an upper and lower bound of imposed by the WMAP data. A blue thick solid line represents a line for  GeV. A blue thick dotted line represents a boundary for .

If is satisfied, the neutral scalar is the CDM. In this case we can follow the analysis given in [21]. If it is heavier than the boson, it cannot keep the relic abundance required from the WMAP data. The reason is that they can effectively annihilate to the pair through the exchange. Thus, since we have no other candidate for the CDM within the present model, we have to assume that the mass of should be smaller than 80 GeV. Even if it is lighter than the boson, direct search experiments impose a strong constraint. The difference of the mass eigenvalues of is estimated as

(32)

Since the have a vector like interaction with boson, its elastic scattering cross section with a nucleon through exchange is 8-9 orders of magnitude larger than the existing direct search limits [28]. To forbid exchange kinematically, has to be larger than a few 100 keV [29]. Following eq. (32), this constraint can be interpreted as a condition .

If we impose that the relic abundance saturates the values required by the WMAP data, a much stronger constraint can be obtained. This abundance is dominantly determined by the -wave suppressed coannihilation process . In order to realize a suitable relic abundance, we need to decrease this coannihilation rate by requiring the heavier one of is thermally suppressed. This requires that  GeV should be satisfied for GeV [21]. Thus, if we consider is the CDM taking account of this arguments, we have an another condition . Since the leptogenesis occurs successfully for  GeV as seen in the previous part, should be a larger value than and then should be larger than  GeV.

We can search favored parameter regions in the present model by estimating numerically the relic abundance of in the same way as the case. In this estimation we need to take account of the above mentioned thermal effect which modifies the relic density in the case by a factor . In Fig. 3 we plot the allowed regions in the plane for the case of  GeV, which is a favored value for leptogenesis. In the regions sandwiched by both dotted and solid thin lines, realizes the three year WMAP data. In the same figure we add two conditions. We plot a line corresponding to  GeV by a blue solid thick one. Since we now consider regions below the threshold, allowed regions are the part below this line. The width also imposes an another condition . The boundary of this condition is plotted by a blue dotted thick line. Regions above this boundary satisfy this condition. As seen from this figure, the favored part in the regions sandwiched by these thick lines gives   GeV for , which agrees with the results given in [21, 29]. This does not contradict with experimental mass bounds for charged Higgs fields as long as has suitable negative values. The constraint from can be also satisfied for which can keep Yukawa couplings small enough in eq. (19). For the required large values for , can be still satisfied and becomes very heavy so as to be out of the range reached by the LHC experiments. 444In the original models [18], required values of and for the CDM can be consistent with the neutrino oscillation data and the FCNC constraint as long as singlet fermion masses are large enough and their Yukawa couplings are small as in the present case. Thus, we could not find substantial difference between this model and the original ones in the case. In this case is confined to very restricted regions, especially in case (i). In order to realize the favorable values of and , several coupling constants are required to be finely tuned. For example, should be very small like . Although these required parameter tuning might decrease interests for this case compared with the case, it is noticeable that can be a CDM candidate consistently with the neutrino oscillation data in this model.

4 Summary

We have studied a unified explanation for both the CDM abundance and the baryon number asymmetry in a non-supersymmetric model for neutrino masses. The model is obtained from the SM by adding a U(1) gauge symmetry and several neutral fields. The neutrino masses are generated through both the seesaw mechanism with a single right-handed neutrino and the one-loop radiative effects. Both contributions induce the same texture which can realize favorable mass eigenvalues and mixing angles. New neutral fields required for this mass generation make the unified explanation for the leptogenesis and the CDM abundance in the universe possible.

Both the neutral fermion and the neutral scalar are stable due to a subgroup which remains as a residual symmetry of the spontaneously broken U(1). Thus, they can be a good CDM candidate. In the CDM case, since it has the U(1) gauge interaction, the annihilation of this CDM candidate is dominantly mediated through this interaction. If this U(1) symmetry is broken at a scale suitable for the neutrino mass generation, its estimated relic abundance can explain the WMAP result for the CDM abundance. We examined these points taking account of the neutrino oscillation data. In the CDM case, if it is lighter than boson and the difference of its mass eigenstates forbid its coannihilation due to the exchange kinematically, it can keep the suitable relic abundance. We examined the consistency of this picture with the neutrino oscillation data.

Since another introduced neutral fermion is a gauge singlet and heavy enough, it can follow the out-of-equilibrium decay which produces the baryon number asymmetry through the leptogenesis. We showed the consistency of this scenario with the neutrino oscillation data. Although the required reheating temperature for the leptogenesis is similar values to the one in the ordinary seesaw mechanism, we have no gravitino problem since we need no supersymmetry to prepare the stable CDM candidates. The present model gives an example in which three of the biggest experimental questions in the SM, that is, neutrino masses, the CDM abundance, and the baryon number asymmetry can be explained through the closely related physics in a non-supersymmetric extension of the SM. In order to solve the hierarchy problem, a supersymmetric extension of the model may be considered along the line of [30]. We would like to discuss this subject elsewhere.

Finally, we briefly comment on signatures of the model expected at LHC. The above study fixes mass spectrum of the relatively light fields in the model. We have , and as such new fields. is expected to be produced through the fusion as in the similar way to the ordinary Higgs field. Since has Yukawa couplings with leptons only, its components and can be distinguished from others such as the Higgs fields in the MSSM through the difference of the decay modes. couples with quarks, , and . However, its decay shows different feature depending on the scheme for the CDM. If the CDM is , the results shown in Fig. 2 suggest that the decay mode of is mainly hadronic. It can include nonhadronic ones only for the case of as mentioned before. In such cases, in the decay + missing energy is also included in the final states depending on the value of . On the other hand, if one component of is the CDM, the always can decay into the pair since it is very light. Thus, has a substantial invisible width. The search of with such features may be an important check of the model.

Appendix
We give an example of a set of fields which cancel gauge anomalies without affecting the discussion in the text. We consider to introduce additional fermions as the left-handed ones:

(33)

where representations and charges for SU(2)U(1)U(1) are shown in parentheses. Number of fields are also given in front of them. The SM gauge anomalies are canceled by taking account of these fields. Since these fields are vector-like for the SM gauge group, no problem is induced by them against the electroweak precision measurements. Although these fields are odd, all of them can be massive through Yukawa couplings with or . Thus, as long as their Yukawa coupling constants with or are simply larger than , remains as the lightest odd field in the model. Some discrete symmetry such as seems to be necessary to forbid the coupling between and singlet fields shown in the last line of (33). However, it can be introduced without affecting the scenario. Since no other seeds for the U(1) breaking is necessary to make these additional fermions massive, the mass formula for does not change and the discussion on the relic abundance in the text is not affected.

Acknowledgement

This work is partially supported by a Grant-in-Aid for Scientific Research (C) from Japan Society for Promotion of Science (No.17540246).

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