# Thermal leptogenesis in brane world cosmology

###### Abstract

The thermal leptogenesis in brane world cosmology is studied. In brane world cosmology, the expansion law is modified from the four-dimensional standard cosmological one at high temperature regime in the early universe. As a result, the well-known upper bound on the lightest light neutrino mass induced by the condition for the out-of-equilibrium decay of the lightest heavy neutrino, eV, can be moderated to be in the case of with the lightest heavy neutrino mass () and the “transition temperature” (), at which the modified expansion law in brane world cosmology is smoothly connecting with the standard one. This implies that the degenerate mass spectrum of the light neutrinos can be consistent with the thermal leptogenesis scenario. Furthermore, as recently pointed out, the gravitino problem in supersymmetric case can be solved if the transition temperature is low enough GeV. Therefore, even in the supersymmetric case, thermal leptogenesis scenario can be successfully realized in brane world cosmology.

###### pacs:

^{†}

^{†}preprint: KEK-TH-1019

## I Introduction

The origin of the cosmological baryon asymmetry is one of the prime open questions in particle physics as well as in cosmology. The asymmetry must have been generated during the evolution of the universe. In fact, such a generation is possible if three conditions, i) the existence of baryon number violating interactions, ii) C and CP violations and iii) the departure from thermal equilibrium, are satisfied SakharovCond .

Among various mechanisms of baryogenesis, leptogenesis FukugitaYanagida is attractive because of its simplicity and the connection to neutrino physics. Particularly, the simplest scenario, namely thermal leptogenesis, requires nothing but the thermal excitation of heavy Majorana neutrinos which generate tiny neutrino masses via the seesaw mechanism seesaw and provides several implications for the light neutrino mass spectrum Buchmulleretal . In leptogenesis, the first condition is satisfied by the Majorana nature of heavy neutrinos and the sphaleron effect in the standard model (SM) at the high temperature KRS , while the second condition is provided by their CP violating decay. The departure from thermal equilibrium is provided by the expansion of the universe.

The out-of-equilibrium decay is realized if the decay rate is smaller than the expansion rate of the universe,

(1) |

where and are the decay rate and the mass of the lightest heavy neutrino, respectively, and denotes the Hubble parameter. Note that the expansion law is governed by the gravitational theory. Therefore, if general relativity is replaced by another theory at a high energy scale, the universe would undergo non-standard expansion. One of such examples is the brane world cosmology braneworld . The following discussion is based on the so-called “RS II” model first proposed by Randall and Sundrum RS . In the model, the Friedmann equation for a spatially flat spacetime is given by

(2) |

where

(3) |

is the energy density of the radiation with being the effective degrees of freedom of relativistic particles,

(4) |

with being the five dimensional Planck mass
^{1}^{1}1
We define as the “reduced” five dimensional Planck mass.
In some papers, the normal five dimensional Planck mass is used.
The reduced five dimensional Planck mass is defined
as by using the normal one ().
,
and the four dimensional
cosmological constant has been tuned to be zero.
Here we have omitted the term so-called “dark radiation”,
since this term is severely constrained by
big bang nucleosynthesis ichiki and
does not affect the results in this paper.
The second term proportional to
is a new ingredient in the brane world cosmology
and leads to a non-standard expansion law.

Note that according to Eq. (2) the evolution of the early universe can be divided into two eras. At the era where the second term dominates and the expansion law is non-standard (brane world cosmology era), while at the era the first term dominates and the expansion of the universe obeys the standard expansion law (standard cosmology era). In the following, we call a temperature defined as “transition temperature”, at which the evolution of the early universe changes from the brane world cosmology era into the standard one. The transition temperature is determined as

(5) |

once is given. Using the transition temperature, we rewrite Eq. (2) into the form,

(6) |

where is the Hubble parameter in the standard cosmology.

This modification of the expansion law at a high temperature () leads some drastic changes for several cosmological issues. In fact, some interesting consequences in the brane world cosmology such as the enhancement of the dark matter relic density OS-NOS and the suppression of the overproduction of gravitino OSGravitino have been pointed out. In this paper, we investigate how the modified expansion law in the brane world cosmology affects the thermal leptogenesis. Clearly, if , we can expect some effects according to the condition of Eq. (1).

## Ii A brief overview of thermal leptogenesis

In the seesaw model, the smallness of the neutrino masses can be naturally explained by the small mixings between left-handed neutrinos and heavy right-handed Majorana neutrinos . The basic part of the Lagrangian in the SM with right-handed neutrinos is described as

(7) |

where denote the generation indices, is the Yukawa coupling, and are the lepton and the Higgs doublets, respectively, and is the lepton-number-violating mass term of the right-handed neutrino (we are working on the basis of the right-handed neutrino mass eigenstates). In this paper, we assume the hierarchical mass spectrum for the heavy neutrinos, , for simplicity as in many literature.

In the case of the hierarchical mass spectrum for the heavy neutrinos, the lepton asymmetry in the universe is generated dominantly by CP-violating out-of-equilibrium decay of the lightest heavy neutrino, and . The leading contribution is given by the interference between the tree level and the one-loop level decay amplitudes, and the CP-violating parameter is described as FandG ; SUSYFandG

(8) | |||||

(9) |

Here and correspond to the vertex and the wave function corrections,

(10) |

respectively. These functions are slightly modified in supersymmetric models SUSYFandG . In our case, both functions are reduced to for , and can be simplified as

(11) |

Through the relations of the seesaw mechanism, this formula can be roughly estimated as

(12) |

where is the heaviest light neutrino mass, GeV is the vacuum expectation value (VEV) of Higgs and is an effective CP phase. Here we have normalized by eV which is a preferable value in recent atmospheric neutrino oscillation data eV atm . Using the above , the resultant baryon asymmetry generated via thermal leptogenesis is described as

(13) |

where is the effective degrees of freedom in the universe at , and is the so-called dilution factor. This factor parameterizes how the naively expected value is reduced due to washing-out processes. To evaluate the resultant baryon asymmetry precisely, numerical calculations are necessary, and the lower bound on the lightest heavy neutrino mass in order to obtain the realistic baryon asymmetry in the present universe for fixed has been found to be GeV LowerBound .

For successful thermal leptogenesis, the reheating temperature after inflation should be higher than the lightest heavy neutrino mass. This fact causes a problem, when we consider the thermal leptogenesis in supersymmetric models. In supersymmetric models, if gravitinos produced from thermal plasma decay after big bang nucleosynthesis (BBN), high energy particles originating from the gravitino decay would destroy light nuclei successfully synthesized by the BBN. In order to maintain the success of the BBN, the number density of the produced gravitino is severely constrained to be small. The resultant number density of the produced gravitino is proportional to the reheating temperature, and then the upper bound on the reheating temperature has been found to be GeV GravitinoProblem for the gravitino mass being around the electroweak scale. The reheating temperature should be far below the lightest heavy neutrino mass. Therefore, in order to realize the successful thermal leptogenesis in supersymmetric models, new ideas are necessary, such as the gravitino as the lightest supersymmetric particle (LSP) LeptoGravino .

In the standard cosmology, the expansion of the universe is governed by

(14) |

The condition of the out-of-equilibrium decay in Eq. (1) to provide sufficient lepton asymmetry without dilution is rewritten as

(15) |

with the decay width of the lightest heavy neutrinos

(16) |

This condition can be regarded as the upper bound on the lightest neutrino mass , since the inequality can be shown Davidson ; Fujii . Considering Eq. (12), this upper bound is normally interpreted as an implication that thermal leptogenesis cannot generate sufficient baryon asymmetry in the case of the degenerate mass spectrum of light neutrinos Fujii ; DegImp .

## Iii Leptogenesis in brane world cosmology

Let us consider the case where the lightest heavy neutrinos decays in the brane world cosmology era, namely . In the era, the expansion law of the universe is nonstandard such as

(17) |

Accordingly,
the condition for the out-of-equilibrium decay
of the heavy neutrino is modified as
^{2}^{2}2
It is nontrivial whether the same condition of
the out-of-equilibrium decay is applicable
for the brane world cosmology.
After this work is finished,
a related preprint has appeared Bentoetal ,
where the same subject is addressed and
Boltzmann equations are numerically solved.
Their results justify our rough estimation here
in the same manner as the one in the standard cosmology,
though there are some discrepancies between resultant
numerical values given by their numerical calculations
and the ones we estimate in this paper.

(18) |

Now we obtain the upper bound on the lightest neutrino mass in the brane world cosmology such that

(19) |

Note that the upper bound has been moderated due to the enhancement factor for . This result implies that, if is low enough, the thermal leptogenesis scenario is successful even in the case of the degenerate light neutrino mass spectrum.

In the above discussion, we have implicitly assumed that
the lightest heavy neutrino is in thermal equilibrium
at a high temperature .
In the following, let us verify whether this situation
can be realized in the brane world cosmology.
Since the right-handed neutrinos are singlet
under the SM gauge group,
the only interaction through which the right-handed neutrino
can be in thermal equilibrium is the Yukawa coupling
in Eq. (7).
^{3}^{3}3
If we extend our model into a model
such as the left-right symmetric model,
we can consider the case where
the right-handed neutrinos can be in thermal equilibrium
through new gauge interactions Plumacher .
Consider a pair annihilation process of
the lightest heavy neutrino through the Yukawa couplings.
The thermal averaged annihilation rate is roughly estimated as

(20) |

where is the number of annihilation channels, stands for dominant Yukawa couplings, and the factor denotes the kinematical phase factor. The freeze-in temperature, , in the brane world cosmology era can be defined from the condition , and we obtain

(21) |

Here we have used a large value for the normalization of the Yukawa coupling constant. It is necessary to fine-tune each elements of large Yukawa couplings, in order to obtain masses of the light neutrinos much smaller than the scale naively obtained through the see-saw mechanism. Large Yukawa couplings might be reasonable because the thermal leptogenesis with the degenerate light neutrino mass spectrum is possible in the brane world cosmology as discussed above. Consistency of our discussion, namely , leads to the upper bound on the transition temperature such as

(22) |

The thermal leptogenesis in the brane cosmology era can take place if the lightest heavy neutrino mass is in the range

(23) |

Recall that GeV is required in order to generate the sufficient baryon asymmetry. This implies GeV GeV ( GeV GeV ) from Eqs. (21) and (23).

Finally, there is an additional interesting possibility for the thermal leptogenesis in the supersymmetric case. In the standard cosmology, the thermal leptogenesis in supersymmetric models is hard to be successful, since the reheating temperature after inflation is severely constrained to be GeV due to the gravitino problem GravitinoProblem . However, as pointed out in Ref. OSGravitino , the constraint on the reheating temperature is replaced with the one on the transition temperature in the brane world cosmology. Therefore, when the transition temperature is low enough GeV, the gravitino problem can be solved even if the reheating temperature is much higher. Interestingly, the transition temperature GeV we found above can realize the thermal leptogenesis scenario successfully and also solve the gravitino problem. However, for this scenario, one should notice that inflation need to have a very efficient reheating with GeV, because the potential energy during inflation must be less than GeV in order for inflation models to be consistently treated in the context of the five-dimensional theory. In fact, such inflation models are possible but limited CS . Further studies on inflation would provide informations for this possibility.

## Iv Conclusion

We have studied the thermal leptogenesis in the brane world cosmology. The nonstandard expansion law of the brane world cosmology affects the condition of the out-of-equilibrium decay of the lightest heavy neutrino, and moderates the upper bound on the lightest light neutrino mass. As a result, the degenerate mass spectrum for the light neutrinos can be consistent with the successful thermal leptogenesis scenario, if the transition temperature is lower than the lightest heavy neutrino mass. We have verified that the lightest heavy neutrino can be in thermal equilibrium through the Yukawa couplings among light neutrinos and Higgs bosons and found the region of the transition temperature consistent with the successful thermal leptogenesis in the brane world era. Then, we obtain the constraint on (). Furthermore, in supersymmetric case, we have noticed that the transition temperature required for the successful thermal leptogenesis can solve the gravitino problem simultaneously.

## Acknowledgments

The work of N.O. is supported in part by the Grant-in-Aid for Scientific Research in Japan (#15740164). The work of O.S. is supported by PPARC. The authors thank the Yukawa Institute for Theoretical Physics at Kyoto University, where this work was completed during the YITP Workshop (YITP-W-05-02) on ”Progress in Particle Physics 2005”.

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