RITS-PP-005
Reheat Temperature and the Right-handed Neutrino Mass
Takeshi Fukuyama,? Tatsuru Kikuchi,? and Wade Naylor?
Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan (Dated: February 2, 2008) We discuss the reheating temperature in the instant preheating scenario. In this scenario, at the last stage of in?ation, the in?aton ?eld ?rst decays into another scalar ?eld with an enormous number density via the instant preheating mechanism. Subsequently, the produced scalar ?eld decays into normal matter accompanied by the usual reheating mechanism. As an in?ationary model, we identify the in?aton as a ?eld which gives rise to a mass for the right-handed (s)neutrino. One of the interesting consequences of the instant preheating mechanism is the fact that the reheating temperature is proportional to the mass of the decayed particle, the right-handed sneutrino, TR ∝ MR . This is very di?erent from the ordinary perturbative reheating scenario in which the reheating temperature is proportional to the mass of the in?aton.
Keywords: Beyond the Standard Model, In?ation
arXiv:hep-ph/0510003v2 26 Oct 2005
In?ation is a well-motivated scenario for solving many problems in the standard Big Bang cosmology: the ?atness problem, monopole problem and so on [1]. The basic framework is constructed by using a single scalar ?eld with a monomial potential. Although such a simple ‘toy’ model may be attractive, it has serious di?culties from the particle physics point of view. This is the so called gauge hierarchy problem in the standard model: When we take into account the radiative corrections of the scalar mass, it receives quadratically divergent contributions from UV physics. The most promising way to solve the gauge hierarchy problem is to introduce supersymmetry (SUSY) [2]. In models with SUSY it also gives a basic tool for constructing in?ationary potentials in a rather natural way, rather than non-SUSY models, due to the enhanced symmetry and the fact that radiative corrections can be kept under control. In making an in?ationary model in the non-SUSY set up we put a scalar ?eld in by hand; however, in SUSY models we are fortunate to have many candidates for such scalar ?elds representing ?at directions in the ?eld con?guration space. Indeed, there are many ?at directions even in the MSSM [3]. From the low energy phenomenological point of view, supersymmetric grand uni?ed theory (GUT) provides an attractive framework for the understanding of low-energy physics. In fact, for instance, the anomaly cancellation between the several matter multiplets present is automatic in GUT, since the matter multiplets are uni?ed into a few multiplets, and the experimental data supports the fact of uni?cation of three gauge couplings at the GUT scale, MGUT = 2 × 1016 GeV, assuming the particle content of the minimal supersymmetric standard
model (MSSM) [4, 5]. The right-handed neutrino, which appears naturally in the SO(10) GUT, provides a natural explanation for the smallness of the neutrino masses through the see-saw mechanism [6]. However, there is no clear connection between the reheating temperature and GUT scale physics, like for the masses of the right-handed neutrinos. Hence, we shall discuss the reheating process using the instant preheating mechanism and show that the reheating temperature is given by a mass of the right-handed (s)neutrino. Consider ?rst, the following superpotential relevant for in?ation [7] W = MI I 2 + MRi Nic Nic + λi INic Nic + Yνij Nic Lj Hu , (1) where Nic and Lj are the right-handed neutrino and lepton doublet super?elds and I is a complete Standard Model singlet super?eld; later the scalar component of the singlet will be identi?ed with the in?aton ?eld. From the superpotential (1), we obtain the Lagrangian relevant for the preheating as follows: 1 2 2 1 2 c2 ij c 2 c2 L = ? MI I ? MRi Ni ? λ2 i I Ni + Yν Ni Lj Hu . 2 2 (2) In such a model the right-handed sneutrino is coupled to the in?aton, and after developing a VEV the righthanded neutrinos obtain their masses at the order of about 1013 [GeV]. Furthermore, because Yν ? Yu (Yu : up-type quark Yukawa coupling) is naturally expected in models with an underlying SU(4) ? SO(10) Pati-Salam symmetry; hence, we can naturally expect there to be many large couplings between the scalar ?eld Nic and the fermionic ?elds Lj and Hu , which are required in order to obtain a viable instant preheating: I → Nic → Lj Hu . First, let us brie?y consider the perturbative treatment of reheating. When the in?aton potential is given as above the in?aton decay rate is found to be Γ(I → Nic Nic ) ? |λi |2 MI 4π (3)
? Email: ? Email:
fukuyama@se.ritsumei.ac.jp rp009979@se.ritsumei.ac.jp ? Email: naylor@se.ritsumei.ac.jp
2 and thus, within the perturbative treatment of reheating, the reheating temperature is obtained in terms of the decay rate as TR = 45 2π 2 g?
1/4
it is possible to show that, see [8, 9], nk = exp ?
2 π (k 2 /a2 + MR ) i λi MRi I
(8)
(ΓMPL )1/2 ? 0.1 × |λi | MI · MPL (4)
? 1.3 × 1015 GeV (for |λi | ? 1) .
˙ and as discussed in [9] MRi I can be replaced by |I| which leads to nk = exp ?
2 π (k 2 /a2 + MR ) i ˙ λi |I|
Here the mass of the in?aton, MI , has been determined from the CMB anisotropy constraint, 4 MI = 5×10?5 ? MI ? 1.4×1013 GeV , 3π MPL (5) where we have taken the number of e-foldings to be N ≈ 60. In the chaotic in?ation scenario, the value of the in?aton at the time √ of terminating in?ation is given by Iend = MPL /(2 π ), and the corresponding energy density is therefore δρ =N ρ ρend
2 2 3MI MPL 3 = (6.5 × 1015 GeV)4 . (6) = V (Iend ) = 2 16π
.
(9)
This can then be integrated to give the number density for the right-handed sneutrinos, Nic , nN c
i
1 = 2π 2
∞
dk k 2 nk =
0
2 ˙ )3/2 πMR (λi I i exp ? ˙ 8π 3 λi |I|
3 Ri MR (MRi λi I )3/2 ? πM i λi I = e e ?π . ? 8π 3 8π 3
(10)
and so we expect several oscillations per Hubble time. Therefore, we would expect many oscillations of the in?aton ?eld before it decays, which in general leads to broad parametric resonance [8]. In the usual broad parametric resonance for a hyperbolic potential, it is assumed that there is a succession of scatterings by the potential every time the ?eld oscillates about the origin. However, there are cases when the ?eld only needs to oscillate about the origin once (before rolling back down the potential again it decays into other particles by the standard reheating mechanism). This model is known as instant preheating [9], which in many ways is far simpler than general parametric resonance theory. Indeed, as mentioned in [9] under certain conditions one does not even need a parabolic potential, provided that the in?aton is coupled to another ?eld quadratically. Also, recently, it has been pointed out in [10] that the thermalization process is very slow in SUSY models due to the presence of ?at directions in the SUSY potential. However, in this letter, we adopt a model where the thermalization process occurs quickly by taking a suitable choice of parameters in the model. Given any in?ationary models, we would like to investigate the e?ects of preheating to generate a large decay rate for the in?aton. This can be achieved by using the instant preheating mechanism [9]. Thus, if the in?aton oscillates about the minimum of the potential only once
Let us now return to non-perturbative reheating, i.e. preheating. During reheating there are in general three time scales: ? ?1 tosc ? MI ? 10?36 s ? ?1 ? tosc ? texp ? tdec (7) texp Hend ? 10?35 s ? ?1 tdec ? ΓI ? 10?25 s
As argued in [9], if the couplings are of order λi ? 1 then there need not be an exponential suppression of the number density. This fact has recently been used in an interesting model of non-thermal leptogenesis in [11]. The resultant reheating temperature from instant preheating is given by TR = 30 · mN c · n N c i i g? π 2 ? = 0.05 × MRi .
1/4
?
15 4π 5 g?
1/4
MRi e?π/4 (11)
It should be stressed that the reheating temperature in equation (11), obtained from the preheating mechanism, is proportional to the mass of the decayed particle, the right-handed sneutrino, i.e. TR ∝ MR , and does not depend on the in?aton mass. This is very di?erent from the ordinary perturbative reheating scenario in which the reheating temperature is proportional to the mass of the in?aton ?eld, see Eq. (4). This characteristic of proportionality is applicable to all the models using preheating. A nice example is in the next to minimal supersymmetric standard model (NMSSM) [12], where we can identify a singlet in this model as the in?aton [13]. In such a case, very interestingly, the reheating temperature is determined by the Higgs mass: TR ∝ mH . To summarise, in this letter we have discussed the connection between the reheating temperature and the masses of the right-handed (s)neutrinos. The reheating process has been described as follows: At the last stage of in?ation, the in?aton ?eld ?rst decays into another scalar ?eld with an enormous number density, via the instant preheating mechanism. Subsequently, the produced scalar ?eld decays into normal matter accompanied by the usual reheating mechanism. Interestingly, the reheating temperature is proportional to the mass of the decayed particle, the right-handed sneutrino. We emphasise that this is very di?erent from the ordinary perturbative reheating scenario in which the reheating temperature is proportional to the mass of the in?aton.
3 The work of T.F. is supported in part by the Grantin-Aid for Scienti?c Research from the Ministry of Education, Science and Culture of Japan (#16540269). The work of T.K. is supported by the Research Fellowship of the Japan Society for the Promotion of Science (#7336). We thank the referee for useful comments.
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