# Low Scale Leptogenesis and Dark Matter Candidates

in an Extended Seesaw Model

###### Abstract

We consider a variant of seesaw mechanism by introducing extra singlet neutrinos and singlet scalar boson, and show how low scale leptogenesis is successfully realized in this scenario. We examine if the newly introduced neutral particles, either singlet Majorana neutrino or singlet scalar boson, can be a candidate for dark matter. We also discuss the implications of the dark matter detection through the scattering off the nucleus of the detecting material on our scenarios for dark matter. In addition, we study the implications for the search of invisible Higgs decay at LHC, which may serve as a probe of our scenario for dark matter.

## I Introduction

Two unsolved important issues in particle physics and cosmology are why there is more matter than antimatter in the present Universe and what is the origin of dark matter. In this paper, we propose a model to address both of those problems and show that they can be solved by means of a common origin..

One of the most popular models to accommodate the right amount of baryon asymmetry in the present Universe is so-called leptogenesis lepto (), which is realized in the context of seesaw mechanism seesaw (), and thus the smallness of neutrino masses and the baryon asymmetry can be simultaneously achieved. However, the typical leptogenesis demands rather large scale of the right-handed Majorana neutrino mass, which makes it impossible to probe in present experimental laboratories. To lower the scale of leptogenesis, we have recently proposed a variant of seesaw mechanism kk () and showed that the required leptogenesis is allowed at a low scale even a few TeV scale without imposing the tiny mass splitting between two heavy Majorana neutrinos required in the resonant leptogenesis reso (). The main point of our model previously proposed is to introduce an equal number of gauge singlet neutrinos in addition to the heavy right-handed singlet neutrinos. In our scenario, there exist new Yukawa interactions mediated by singlet Higgs sector which is coupled with the extra singlet neutrinos and the right-handed singlet neutrinos. As shown in kk (), the new Yukawa interactions may play a crucial role in enhancing the lepton asymmetry, so that low scale leptogenesis can be achieved.

On the other hand, it is worthwhile to examine if those new particles, either the newly introduced singlet neutrinos or singlet scalar bosons, can be a dark matter candidate because any kind of neutral, stable and weakly interacting massive particles (WIMPs) can be regarded as a good candidate for dark matter. While light singlet neutrinos with mass of order MeV or keV have been considered as a warm dark matter candidate warm (), heavy singlet neutrinos with mass of order 100 GeV as a dark matter candidate have not been studied much in detail. While the cold dark matter (CDM) models supplemented by a cosmological constant are in a good agrement with the observed structure of the Universe on large scales, the cosmic microwave background anisotropies and type Ia supernovae observations for a given set of density parameters, there exists a growing wealth of observational data which are in conflict with the CDM scenarios cdm (). To remedy the difficulties of the CDM models on galactic scales, a self-interacting dark matter candidate has been proposed self (); scalarDM (); SDM1 (), and it has been discussed that a gauge singlet scalar coupled to the Higgs boson, leading to an invisible decaying Higgs, is a good candidate for the self-interacting dark matter sh (); sh2 (); sh3 ().

In this paper, we will show that either the new gauge singlet neutrinos or singlet scalar bosons, which play an important role in enhancing the lepton asymmetry so that the low scale leptogenesis is realized, can be a good candidate for dark matter. For our purpose, we will first present how the enhancement of the lepton asymmetry through the mediation of the newly introduced particles in our framework can be achieved and then will show that the new gauge singlet particles with the parameter space in consistent with low scale leptogenesis can be satisfied with the criteria on the candidate for dark matter. We will investigate possibilities of dark matter detection through the scattering off the nucleus of the detecting material. In addition, we will study how we can probe our scenarios at high energy colliders and present that the search for the invisible Higgs decay may serve as a probe of dark matter properties.

## Ii Extended Seesaw Model and Low Scale Leptogenesis

We begin by explaining what the extended seesaw model is and examine how low scale leptogenesis can be realized in this context. The Lagrangian we propose is given in the charged lepton basis as

(1) |

where stand for SU(2) doublet, right-handed singlet and newly introduced singlet neutrinos, respectively. , , represent Dirac Yukawa coupling matrix, Singlet Yukawa coupling matrix and Heavy Majorana neutrino mass matrix, respectively. And and denote the SU(2) doublet and singlet Higgs scalars. Here, we impose symmetry under which and are odd and all other particles even, which makes this model different from the extended double seesaw model proposed in kk () even though the contents of particles are the same. The immediate consequence of the exact symmetry is that the lightest or can be a candidate for the cold dark matter of the Universe.

Due to the exact symmetry, the singlet scalar field can not drive a vacuum expectation value. Thus, in this model, light neutrino masses are generated by typical seesaw mechanism, which makes this model different from the model in kk (). After integrating out the right-handed heavy neutrino sector in the above Lagrangian, the light neutrino masses are given by

(2) |

where we omitted the indices of the mass matrix and Dirac Yukawa coupling, and GeV is the Higgs vacuum expectation value. Although the absolute values of three neutrino masses are unknown, their masses are expected to be of order of eV and eV, provided that the mass spectrum of neutrinos is hierarchical. There is also a bound on neutrino masses coming from WMAP observation, which is eV. Thus it is interesting to see how the neutrino masses of order of eV can be obtained in our scenario. Such light neutrino masses can be generated through seesaw formula Eq. (2), if we take, as an example, and to be of order and GeV, respectively.

Now, let us consider how low scale leptogenesis can be achieved by the decay of the lightest right-handed Majorana neutrino in our framework. Right handed heavy Majorana neutrinos are even under symmetry, so that they can decay into a pair of the singlet particles, and , or the standard model Higgs and the lepton doublet. Since the Yukawa couplings are taken to be large, the processes involving remain in thermal equilibrium even at , and thus the decays of can not lead to the desired baryon asymmetry. However, the decay processes of the lightest right-handed Majorana neutrino depart from thermal equilibrium at , and thus lead to the desired baryon asymmetry

It will be shown that there exists a new contribution to the lepton asymmetry which is mediated by the extra singlet neutrinos and scalar boson , and successful leptogenesis can be realized with rather light right-handed Majorana neutrino masses which can escape the gravitino problem encountered in supersymmetric standard model. Without loss of generality, we can rotate and rephase the fields to make the mass matrices and real and diagonal. In this basis, the elements of and are in general complex. The lepton number asymmetry required for baryogenesis is given by

(3) |

where is the lightest right-handed neutrino and is the total decay rate. Thanks to the new Yukawa interactions, there is a new contribution of the diagram which corresponds to the self energy correction of the vertex arisen due to the new Yukawa couplings with singlet neutrinos and Higgs scalars.

Fig. 1 shows the structure of the diagrams contributing to . Assuming that the masses of the Higgs scalars and the newly introduced singlet neutrinos are much smaller compared to that of the right-handed neutrino, to leading order, we have

(4) |

so that

(5) |

where and are given in the basis where and are diagonal, , with for ,

(6) |

and

(7) |

Notice that the term proportional to comes from the interference between the tree-level diagram with the new contribution.

In particular, for , the vertex contribution to is much smaller than the contribution of the self-energy diagrams and the asymmetry is resonantly enhanced, and we do not consider this case. To see how much the new contribution may be important in this case, for simplicity, we consider a particular situation where , so that the effect of is negligibly small. In this case, the asymmetry can be written as

(8) | |||||

where is a resonance factor defined by . For successful leptogenesis, the size of the denominator of should be constrained by the out-of-equilibrium condition, with the Hubble expansion rate H, from which the corresponding upper bound on the couplings reads . Then, the first term of Eq. (8) is bounded as . So if the first term of Eq. (8) dominates over the second one, is required to achieve TeV scale leptogenesis, which implies severe fine-tuning. However, since the size of is not constrained by the out-of-equilibrium condition, large value of is allowed for which the second term of Eq. (8) can dominate over the first one and thus the size of can be enhanced. For example, if we assume that is aligned to , with constant , the upper limit of the second term of Eq. (8) is given in terms of by , and then we can achieve the successful low scale leptogenesis by taking rather large value of , instead of imposing very tiny mass splitting between and . The right amount of the asymmetry, , can be obtained for , provided that and . We emphasize that such a requirement for the hierarchy between and is much less severe than the required fine-tuning of the mass splitting between two heavy Majorana neutrinos to achieve the successful leptogenesis at low scale.

The generated B-L asymmetry is given by , where is the number density of the right-handed heavy neutrino at in thermal equilibrium given by with Boltzmann constant and the effective number of degree of freedom . The efficient factor can be computed through a set of coupled Boltzmann equations which take into account processes that create or washout the asymmetry. To a good approximation the efficiency factor depends on the effective neutrino mass given in the presence of the new Yukawa interactions with the coupling by

(9) |

In our model, the new process of type will contribute to wash-out of the produced B-L asymmetry. The process occurs mainly through virtual exchanges because the Yukawa couplings are taken to be large in our model and the rate is proportional to . Effect of the wash-out can be easily estimated from the fact that it looks similar to the case of the typical seesaw model if is replaced with . It turns out that the wash-out factor for , and GeV is similar to the case of the typical seesaw model with GeV and eV, and is estimated so that can be enough to explain the baryon asymmetry of the Universe provided that the initial lightest right-handed neutrino is thermal Buchmuller ().

## Iii Investigation of possible Dark Matter Candidates

Now, let us examine if either the newly introduced singlet Majorana neutrino or the singlet scalar boson can be a candidate for dark matter. For our purpose, in addition to the lagrangian given in Eq. (1), we allow quartic scalar interactions for the scalar sectors. Then, the lagrangian we consider is given by

(10) |

Note that the self-interacting coupling of the singlet scalar is largely unconstrained and thus can be chosen arbitrarily. But, we assume that it should not be so large that the perturbation breaks down. As mentioned before, in order to guarantee the stability of the dark matter candidate, we impose discrete symmetry under which all the standard model (SM) particles are even whereas the singlet Majorana neutrinos and singlet scalar boson are odd. In addition, we demand that the scalar potential is bounded from below so as to guarantee the existence of a vacuum and the minimum of the scalar potential must spontaneously break the electroweak gauge group, , but must not break symmetry imposed above. After breaking the electroweak gauge symmetry, the singlet scalar - dependent part of the scalar potential is given by

(11) |

where we have adopted and shifted the Higgs boson by , where GeV is the Higgs vacuum expectation value. The physical mass of , , is then given by . We assume that the spectrum of the singlet neutrinos is not degenerate and two heavier ones are so much heavier than that they could not be dark matter candidates. Here, we notice that only the lightest odd particle (LOP) under can be a candidate for dark matter in our scenario because the next lightest odd particle (NOP) can be decayed into the LOP by the cascade decay, as shown in Fig. 2.

Since there are two kinds of odd particles under in our scenario, we can classify two possible candidates for dark matter according to which particle is the LOP. One is the case that the singlet Majorana neutrinos is the LOP and the other is that is the LOP. As will be clearer later, the primordial abundance of dark matter candidate is predicted as a function of masses and coupling . Thus, imposing the preferred values of observed from WMAP, CMB1 (); CMB2 (), we can get a strong relation between mass of dark matter candidate and coupling .

### iii.1 Singlet Neutrino as a Dark Matter Candidate

The singlet neutrino can be a dark matter candidate, provided that . We omit the generation index of hereafter. To estimate the relic abundance of the singlet neutrino at a freeze-out temperature , we need to know the annihilation processesEU (). There are two possible annihilation processes of the singlet neutrino, one is happened at loop level and the other is mediated by . But, both possibilities are not relevant to fit the required present relic density of the dark matter because the annihilation cross section for the dark matter candidate is too small, therefore, these processes would predict too much relic abundance of . However, it is well known that if the mass of NOP is close to that of LOP, it would not be decoupled from thermal equilibrium at the freeze-out temperature of the LOP and thus influences the relic abundance of LOP coan (). We call this coannihilation coan () and this mechanism lowers the present day relic density of in this scenario. It turns out that in our scenario if , annihilation processes of the NOP into a pair of the SM particles through the s-channel, as shown in Fig. 3, can significantly affect the relic abundance of to be appropriately reduced.

With the help of the standard formulae for calculating relic abundance of -wave annihilation EU (),

(12) |

we can estimate the present relic density of singlet Majorana neutrino . Here is the degrees of freedom in equilibrium at annihilation, and is the inverse freeze-out temperature in units of , which can be determined by solving the equation

(13) |

where means the relevant thermal average, with the number of degree of freedom for , and is the effective cross section defined in coan (). On calculating , we have used the micrOMEGAs 2.0 program micro (). The dominant contribution to in this case is the pair annihilation cross section of the heavier particle . Since it is that the annihilation of is important, what we need to know is the non-relativistic annihilation cross section. In the non-relativistic limit, s-channel annihilation of via Higgs exchange is given by sh (),

(14) |

where is the total Higgs decay rate, and with the partial rate for decay , for a virtual Higgs . Requiring to be in the region measured from WMAP, CMB1 (); CMB2 (), we can obtain a relation between the coupling and the mass of the scalar boson .

In the case of , in Fig. 4 we represent the relation between and for . Fig. 4-(a) corresponds to the Higgs mass GeV, whereas Fig. 4-(b) corresponds to GeV. The shadowed region is forbidden due to the breakdown of perturbation, so the region GeV for GeV and is not relevant to our scenario. In the case of , it is GeV for GeV.

We notice from Fig. 4 that there exist kinematically special regions, such as the Higgs threshold () and two-particle threshold in the final states ( or , and so on). We see from Fig. 4 that for GeV for GeV and except the regions corresponding to the poles and particle thresholds, the abundance constraint arisen from WMAP results requires . This result indicates that we do not need any fine tuning or special choice of the parameters in order to achieve right amount of relic abundance of dark matter candidate. It also turns out that if is lighter than GeV or heavier than GeV for GeV and , the relic abundance of is incompatible with WMAP results for the relic density. It is worthwhile to notice that the coupling gets significantly suppressed down to the level of near the Higgs pole. This is because the Higgs resonance is quite narrow, which in turn considerably enhances the scalar boson annihilation rate, especially if is slightly smaller than coan (); sh ()

### iii.2 Singlet Scalar Boson as a Dark Matter Candidate:

The singlet Higgs scalar can be a candidate for dark matter, provided that . Since is the lightest particle involved in the interaction term , this particle can not decay into other particles, and thus the annihilation processes relevant to a successful candidate for dark matter can occur through the Higgs interaction term . In this case, the singlet scalar bosons annihilate into the SM particles mediated by the Higgs boson .

In Ref. scalarDM (), it has been proposed that a stable, strongly self-coupled scalar field can solve the problems of cold dark matter models for structure formation in the Universe, concerning galactic scales. Thus, the singlet scalar boson in our model can serve as a self-interacting scalar dark matter candidate. If the mass difference between and is substantially large, the new Yukawa interactions will negligibly affect the relic density of , so the behavior of as a dark matter candidate is the same as that of the self-interacting scalar dark matter candidate. If the mass difference between and is substantially small, the particles are thermally accessible and they are as abundant as . In this case, the formulae for the relic abundance of the particle and its freeze-out temperature are given by the same forms of Eq. (12) and Eq. (13), respectively. In fact, however, the annihilation cross sections associated with the heavier particles are turned out to be negligibly small in this scenario, so , where is the pair annihilation cross section of and is the internal degree of freedom of . Inserting into , we see that the value of for GeV and GeV in this scenario is about lower than that in the singlet Higgs model without the particles , which in turn increases the relic abundance of .

Since several important features of our results are quite similar to those of sh (), therefore, here we just present what are the differences between sh () and our model. In Fig. 5, we show the relation between and for GeV and GeV, which is generated by requiring and 0.094. We see from Fig. 5 that should be less than 551 GeV (571 GeV) for GeV and in our model, so as to be consistent with the relic abundance constraint without breaking down perturbation (i.e. ). This upper limit on is much more restrictive than what is obtained in sh (). Also, we see in Fig. 5-(a) that when lies between 80 GeV and 551 GeV, the value of in our model is much larger than that given in sh ().

## Iv Implication for Dark Matter search

To directly detect dark matter, typically proposed method is to detect the scattering of dark matter off the nucleus of the detecting material. Since the scattering cross section is expected to be very small, the energy deposited by a candidate for dark matter on the detector nucleus is also very small. In order to measure this small recoil energy, typically of order keV, of the nucleus, a very low threshold detector condition is required. Since the sensitivity of detectors to a dark matter candidate is controlled by their elastic scattering cross section with nucleus, it is instructive to examine how large the size of the elastic cross section could be. First, to estimate the elastic cross section with nucleus, we need to know the relevant matrix element for slowly moving spin-J nuclei, which is approximately given by sh ()

(15) |

where denotes nucleons and is determined to be

(16) |

by following the method given in sh () and taking the strange quark mass to be 95 MeV and .

Now, let us estimate the sizes of the elastic scattering cross sections in each case of dark matter candidates.

(i) Case for :

The Feynman diagram describing the scattering of the singlet Majorana neutrino with nucleons and nuclei is given
by -channel Higgs and heavy Majorana neutrino exchange, as shown in Fig. 6-(a).
In this case, the non-relativistic spin-independent quasi-elastic scattering cross section is approximately given by

(17) |

where is the Planck mass.

However, the size of is turned out to be less than due to small neutrino mass
as well as small Yukawa couplings and , which is much smaller than the current bound from
dark matter experiments. Thus, it is extremely difficult to detect the signal for the singlet Majorana neutrino
at dark matter detectors.

(ii) Case for :

In this case, the Feynman diagram relevant to scalar-nucleon elastic scattering is presented in Fig. 6-(b), which
has already been considered in sh ().
Then, the non-relativistic elastic scattering cross section is given by sh ()

(18) |

where is the reduced mass for the collision. Substituting Eq. (16) into Eq. (18),

where the mass of is a mass of proton. In Fig. 7, we plot the predictions for the elastic scattering cross section as a function of the scalar mass for and , respectively, and the mass difference () is taken to be . The lower line corresponds to , whereas the upper line to . On calculating , we used the relationship between and which is obtained through the constraint from WMAP result as before.

In Fig. 7, we plot the new C.L. upper bound for the WIMP-nucleon spin-independent cross section as a function of obtained from XENON10 Dark Matter Experiment xenon (). As one can see from Fig. 7, when , the region GeV for GeV and GeV is excluded by XENON10 Dark Matter Experiment xenon ().

Comparing our results with those in the model of self-interacting scalar dark matter sh (), we see that our prediction for the elastic cross section in the region for GeV, and GeV is about 1.4 (1.5) to 4.3 (4.4) times larger than that estimated in sh (). This indicates that our scenario for scalar dark matter is distinguishable from the original model of the self-interacting scalar dark matter.

## V Implication for Higgs searches at LHC

Now we investigate the implications of our scenarios for Higgs searches at collider experiments.
The singlet scalar boson will not directly couple to ordinary matters, but only to the Higgs fields.
Although the presence of the singlet scalars will not affect electroweak phenomenology in a significant way,
it will affect the phenomenology of the Higgs boson.
Due to the large values for the coupling required by relic abundance constraints,
real or virtual Higgs production may be associated with the singlet
Higgs production, as discussed in sh ().
We see from Eq. (11) that if , the real Higgs boson can decay into a pair of singlet scalars, whereas
if , the singlet scalar bosons can not be produced by real Higgs decays, but arise only via virtual Higgs exchange.
As we know that any produced singlet scalar bosons are not expected to interact inside the collider,
thus they only give rise to strong missing energy signals.

(i) Case for :

In this case, the Higgs boson can invisibly decay into a pair of the singlet scalar bosons.
The invisible decay width is given at tree level by

(19) |

Since the relic abundance constraints require the large value of the coupling , the invisible decay width to the Higgs boson gets large. It is known that if the Higgs mass is less than so that the Higgs partial width into the SM particles is very small, the Higgs will decay predominantly into the singlet scalar bosons. Then, LHC may yield a discovery signal for an invisible Higgs with enough reachable luminosity, for instance, 10 of integrated luminosity for GeV, in associated production with Z boson xx (). To quantify the signals for the invisible decay of the Higgs boson, we investigate the ratio defined as follows sh ():

The ratio indicates how the expected signal for the visible decay of the Higgs boson can decrease due to the existence of the singlet scalar bosons.

(a) Case for singlet neutrino dark matter: In Fig. 8, we plot the value of as a function of for (a) GeV and (b) GeV, respectively. Here, we fixed the value of to be 5 GeV, and used the relation between and which is obtained from the relic abundance constraints as explained before. On calculating the decay rate of Higgs particle, we have used CalcHEP 2.4.5 calcHEP (). As can be seen in Fig. 8, the prediction for the value of is totally different from that in sh (). But, in this case, we can probe the invisible Higgs decay by using only for the very narrow region when GeV. This is because the lower limit of has been determined to be GeV for GeV, as shown in Fig. 4, and upper limit of is constrained by the condition .

From Fig. 8, similar to sh (), it turns out that the invisible Higgs decay width dominates the total width everywhere except in the vicinity . This means that a tremendous suppression of the observable Higgs signal may happen at the LHC.

(b) Case for singlet scalar dark matter: For this case, we plot the prediction for in Fig. 9
and the result is turned out to be almost the same as that in sh (). Similar to the above case,
the invisible width also dominates the total width everywhere except the region sh ().
We notice that when GeV and the parameter space below GeV are excluded by
the current bound obtained from XENON10 Dark Matter experiment.
In this case, if we impose the current bound from XENON10 Dark Matter Experiment xenon (),
the mass of scalar boson should be larger than 42 (56) GeV for GeV and .
Then, the values of in our model is constrained to be
() where the upper limit is determined by the condition as it should be.

(ii) Case for :

In this case, the singlet scalar bosons can be produced only through virtual Higgs exchange.
Similar to the previous case, the produced singlet scalar particles can be detected
as missing energy above an energy threshold, .
In this case, LHC is unlikely place for discovery of a missing energy signal, whereas future
linear collider might be a good place for detecting such a signal.

## Vi Conclusion

We have considered a variant of seesaw mechanism by introducing extra singlet neutrinos and
singlet scalar boson and showed how low scale leptogenesis is realized in our scenario.
We have examined if the newly introduced neutral particles, either singlet Majorana neutrinos or singlet scalar
boson, can be a candidate for dark matter.
We have shown that the coannihilation process between dark matter candidate and the next lightest
odd particle plays a crucial role in generating the right amount of the relic density of dark matter candidate.
We have also discussed the implications of the dark matter detection through the scattering off the nucleus
of the detecting material on our scenarios for dark matter candidate. From the recent result of XENON10 Dark Matter experiment,
we could get some constraint on the mass of singlet scalar boson.
In addition, we have studied the implications for the search of invisible Higgs decay at
LHC which may serve as a probe of our scenarios for dark matter.

Acknowledgement:
SKK is supported by the KRF Grant funded by the
Korean Government(MOEHRD) (KRF-2006-003-C00069).
CSK is supported in part by CHEP-SRC and in part
by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD)
No. KRF-2005-070-C00030.

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