- Email: [email protected]
The Seesaw Scale vs Cosmology
Available online at www.sciencedirect.com
Nuclear and Particle Physics Proceedings 265–266 (2015) 307–310 www.elsevier.com/locate/nppp
The Seesaw Scale vs Cosmology P. Hern´andez,a , M. Kekica , J. L´opez-Pav´onb a IFIC
(CSIC) and Dpto. F´ısica Te´orica, Universidad de Valencia, Edificio Institutos Investigaci´on, Apt. 22085, E-46071 Valencia, Spain. b SISSA, via Bonomea 265, 34136 Trieste, Italy. INFN, sezione di Trieste, 34136 Trieste, Italy.
Abstract We will study the simplest extension of the Standard Model that can account for neutrino masses: the Type-I seesaw. The model introduces a New Physics scale, M, which is often assumed to be much larger than the electroweak scale. However, it is presently unconstrained and the light neutrino masses and mixing can be generated for any value of M above O(eV). Paying special attention to the contribution of the sterile states to Neff as a function of M, we will show that a large part of the M parameter space (8 orders of magnitude) can be excluded thanks to cosmological measurements. The implications for neutrinoless double beta decay will be discussed too. Keywords: Seesaw, Cosmology, Neutrino Mass, Sterile neutrino.
The simplest extension of the Standard Model (SM) that can account for the light neutrino masses is the Type-I seesaw model [1] with N ≥ 2 extra singlet Majorana fermions. We will dub it Minimal Seesaw Model, since it contains the minimum number of extra degrees of freedom required to generate the observed neutrino masses and mixing. The model introduces a New Physics scale, M, associated to the mass of the singlet fermions (sterile neutrinos). The Yukawa couplings are usually assumed to be of the order one because of naturalness arguments, which drives M close to the GUT scale through the seesaw mechanism (mν ∼ Y 2 v2 /M). The drawback of this assumption is that such a high scale would require an important level of fine tunning in order to stabilize the Higgs mass, since it is quadratically sensitive to M at one loop. In any case, regardless theoretical discussions, M is experimentally unconstrained and the light neutrino masses and mixing can be generated for any value of M above O(eV) [2]. Considering low M scales requires accordingly small Yukawa couplings which can be recognized unnatural. Email address: [email protected] (J. L´opez-Pav´on)
http://dx.doi.org/10.1016/j.nuclphysbps.2015.06.077 2405-6014/© 2015 Elsevier B.V. All rights reserved.
However, low Majorana scales may be also considered technically natural since the Lagrangian gains a U(1) global symmetry in the limit M → 0. In any case, we are going to explore the full parameter space of the model without any theoretical prejudice. The main goal of our analysis is to understand if a general bound on the seesaw scale can be extracted without assuming anything a priori about the parameters of the model. If lepton number conservation is not imposed, the most general renormalizable Lagrangian when N extra singlet Weyl fermions, νRi , are included is given by
L = LS M −
α,i
˜ Ri − L¯ α Y αi Φν
N 1 ic i j j ν¯ M ν + h.c., 2 R N R i, j=1
where Y is a 3 × N complex matrix and MN a diagonal N × N real matrix. This Lagrangian defines the Type-I seesaw model. We will study the minimal model, which consists in the addition of N = 2 singlet Weyl fermions, and the more popular next to minimal model with N = 3. The detailed analysis of both models in the context of the cosmological bounds studied here can be
308
P. Hernández et al. / Nuclear and Particle Physics Proceedings 265–266 (2015) 307–310
where ν0 is the energy density of one SM massless neutrino with a thermal distribution. One thermal extra sterile state that freezes out from the thermal bath being relativistic contributes ΔNeff 1 when it decouples. Indeed, the sterile neutrinos should both achieve thermalization and decouple from the thermal bath when they are still relativistic in order to have any impact on Neff . SM + ΔNeff , where the contribuRecall that Neff = Neff SM = 3.046 [7, 8], and tion of the active neutrinos is Neff Big Bang Nucleosynthesis (BBN) [9] and Cosmic Microwave Background (CMB) [10] data give us Neff < 4 at more than 2σ while Neff = 5 is fairly excluded. If Mi 100 MeV, the sterile states are always relativistic when they decouple from the thermal bath regardless the value of the free parameters [3]. For Mi 100 MeV, no general bound can be extracted on Mi since the sterile neutrinos can become non-relativistic before freezing out and the contribution to Neff would be suppressed by the Boltzmann factor. The key quantity in order to understand if the sterile neutrinos are thermal is the thermalization ratio f s j (T ): f s j (T ) ≡
Γ s j (T ) . H(T )
(2)
which measures the sterile production rate of the species s j in units of the Hubble expansion rate. Γ s j is the sterile neutrino collision rate [11]: 1 Γs j P(νa → ν s j ) × Γνα 2 a ⎛ ⎞2 M 2j ⎜⎜⎜ ⎟⎟⎟ Γνα ⎜⎜⎝ ⎟⎟ |Uαs j |2 (3) 2⎠ 2pV (T ) − M α j α=e,μ,τ where P(να → ν s j ) is the time-averaged probability να → ν s j , Γνα is the active neutrino collision rate, Vα is the effective potential induced by the coherent scattering, and Uαs j is the mixing between the active neutrinos να and the sterile state ν s j . In [3], we also derived this results from the Boltzmann equations in the assumption of no primordial large lepton asymmetries1 . In any case, 1 The details of the derivation and the expressions for Γ (T ) and να Vα (T ) can be found in [4].
1 Log 10 m 1 eV
found in [3, 4] respectively. In both cases, we have considered an extension of the Casas-Ibarra parameterization [5], which is valid at all orders in the seesaw expansion and guaranties the generation of the right pattern of light neutrino masses and mixing [6]. The energy density of the sterile neutrino species, s , is usually quantified in terms of ΔNeff defined by s (1) ΔNeff ≡ 0 , ν
10
2
1
3
0.1
4 5 6
4
2
0
2
Lo g 10 M 1 M eV Figure 1: Contours of Min[ f s1 (T max )]=0.1, 1, 10 on the plane (M1 , m1 ).
we have numerically solved the Boltzmann equations in order to compute Neff and extract the bounds on M. f s j (T ) reaches a maximum at the temperature T max , which depends on the free parameters of the model. If it is larger than one, thermalization will be achieved at early times. In the N = 2 model, both sterile neutrinos achieve thermalization f s j (T max ) > 1 in the full parameter space [3]. In Fig. 1 we show the contour plots of the minimum of f s j (T max ) (varying the unconstrained parameters of the model in the full range), as a function of m1 (the lightest neutrino mass) and M1 in the N = 3 model. The three contours correspond to Min[ f s1 (T max )] = 10−1 , 1, 10. The results are the same in the (M2 , m1 ) and (M3 , m1 ) planes. As in the N = 2 case, the Min[ f s1 (T max )] is roughly independent of the scale M. The figure shows that m1 ultimately controls the thermalization of the sterile neutrinos. In fact, there is a thermalization threshold at m1 10−3 eV. This opens two possible scenarios: (i) m1 10−3 eV. The three sterile states achieve thermalization. Therefore, for Mi 100 MeV, each ( j) 1 when sterile neutrino will contribute with ΔNeff freezing out from the thermal bath. However, there are two possible effects that can modify the contribution to Neff later on and before the active neutrino decoupling, when BBN starts: the entropy dilution and the decay of the sterile states. The effect of the entropy dilution is only relevant for Mi 10 KeV [3, 4] and sterile states of this range of masses give a contribution to Neff at BBN in agreement with data. However, these sterile neutrinos would give a huge contribution to the energy density when they become non-relativistic later, modifying in a drastic way CMB and structure formation. In fact, CMB measurements close the window all the
309
P. Hernández et al. / Nuclear and Particle Physics Proceedings 265–266 (2015) 307–310 Log10 Mi eV 1
10 101
8 m ΒΒ eV
6 4 2
NH
10 3 eV
0 Log10 Mi eV
102
IH
104
1
2
3
1 0 0 eV k eV
105 105
10
104
10 3
102
101
1
m lig h t eV
8 6 4 2 0 1
2
3
Figure 3: mββ as a function of the lightest neutrino mass: contribution from the active neutrinos (red and blue regions) and the maximum contribution of the lightest sterile neutrino, for M1 = 1 eV (solid), 100 eV (dashed), 1 KeV (dotted), for NH (blue) and IH (red) restricting Ω s1 h2 ≤ 0.12 and f s1 (T max ) ≤ 1, for M2,3 100 MeV, as a function of the lightest neutrino mass. The shaded region is ruled out for M1 ∈ [1eV-100MeV] by the thermalization bound on the lightest neutrino mass, m1 ≤ 10−3 eV.
Figure 2: Allowed spectra of the heavy states Mi for m1 ≥ 10−3 eV (upper panel) and m1 ≤ 10−3 eV (lower panel).
way down to the O(eV) [12]. The only way to scape from the CMB and BBN bounds is if the sterile neutrinos decay before BBN. Nevertheless, taking into account the bounds on the active-sterile mixing from direct searches, this possibility is excluded for Mi 100 MeV [13]. In summary, the allowed region of the M parameter space is shown in the upper panel of Fig. 2. The constraints from cosmology essentially exclude the spectra of heavy states in the range 1 eV - 100 MeV. (ii) m1 10−3 eV. One and only one of the sterile neutrinos can never thermalize and it depends on the unknown parameters [4]. In this case, one of the sterile neutrino masses can not be bounded in general. The other two states always thermalize and again the range 1 eV- 100 MeV is severely restricted [4]. If one of the sterile neutrinos never thermalizes, the contribution of the two thermal states reduces to the N = 2 case since the other is essentially decoupled from them and the mass generation. The results are qualitatively summarized in the lower panel of Fig. 2, where we show the allowed heavy neutrino spectra in the m1 10−3 eV case. Finally, the information from cosmology is highly complementary to that coming from neutrino oscillations and neutrinoless double beta decay (0νββ decay). As an example, in Fig. 3 we show the impact of the cosmological bounds when a sterile neutrino in the range 1 eV - 100 MeV is present. The 0νββ decay rate is sen-
sitive to the effective mass, mββ , and Fig. 3 shows the separate contribution to mββ from the active and sterile neutrinos as a function of the lightest neutrino mass m1 . The well-known colored bands correspond to the active neutrino contribution while the maximum contribution of the lightest sterile state is given by the solid (M1 = 1 eV), dashed (M1 = 100 eV) and dotted (M1 = 1 KeV) lines. The other two neutrinos are assumed to be well above 100 MeV. A sterile state with M1 ∈ [1 eV, 100 MeV] gives a long range contribution to the 0νββ decay which could in principle be at the reach of the future experiments. However, once the constraints from cosmology are included, it becomes subdominant with respect to the active one and well below the future sensitivity O(10−2 eV). Moreover, if M1 ∈ [1 eV, 100 MeV], the quasi-degenerate light neutrino spectrum and the region of the parameter space in which a cancellation can occur in the active neutrino contribution would be excluded. This is because in such a case m1 should be below the thermalization threshold, at m1 = 10−3 eV, in order to make the lightest sterile neutrino non-thermal. In summary, we have found that the BBN and CMB data essentially excludes the region of the seesaw scale between 1 eV and 100 MeV in both the N = 2 and the N = 3 models. Only in the N = 3 case and provided that m1 10−3 eV, one of the sterile states can be nonthermal and its mass remains unbounded. This has an important impact in the 0νββ decay phenomenology.
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P. Hernández et al. / Nuclear and Particle Physics Proceedings 265–266 (2015) 307–310
References [1] P. Minkowski, Phys. Lett. B 67, 421 (1977); M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, edited by P. van Nieuwenhuizen and D. Freedman (North-Holland, 1979), p. 315; T. Yanagida in Proceedings of the Workshop on the Unified Theory and the Baryon Number in the Universe, edited by O. Sawada and A. Sugamoto (KEK Report No. 79-18, Tsukuba, 1979), p. 95; R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980). [2] A. Donini, P. Hernandez, J. Lopez-Pavon and M. Maltoni, JHEP 1107 (2011) 105 [arXiv:1106.0064 [hep-ph]]. [3] P. Hernandez, M. Kekic and J. Lopez-Pavon, Phys. Rev. D 89 (2014) 7, 073009 [arXiv:1311.2614 [hep-ph]]. [4] P. Hernandez, M. Kekic and J. Lopez-Pavon, Phys. Rev. D 90 (2014) 6, 065033 [arXiv:1406.2961 [hep-ph]]. [5] J. A. Casas and A. Ibarra, Nucl. Phys. B 618, 171 (2001). [6] A. Donini, P. Hernandez, J. Lopez-Pavon, M. Maltoni and T. Schwetz, JHEP 1207 (2012) 161 [arXiv:1205.5230 [hep-ph]]. [7] G. Mangano, G. Miele, S. Pastor and M. Peloso, Phys. Lett. B 534 (2002) 8 [astro-ph/0111408]. [8] G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti and P. D. Serpico, Nucl. Phys. B 729 (2005) 221 [hep-ph/0506164]. [9] R. Cooke, M. Pettini, R. A. Jorgenson, M. T. Murphy and C. C. Steidel, arXiv:1308.3240 [astro-ph.CO]. [10] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 571 (2014) A16 [arXiv:1303.5076 [astro-ph.CO]]. [11] R. Barbieri and A. Dolgov, Phys. Lett. B 237, 440 (1990) and Nucl. Phys. B349 (1991) 743. K. Kainulainen, Phys. Lett. B 244, 191 (1990). [12] E. Di Valentino, A. Melchiorri and O. Mena, JCAP 1311 (2013) 018 [arXiv:1304.5981]. [13] O. Ruchayskiy and A. Ivashko, JCAP 1210, 014 (2012).
Nuclear and Particle Physics Proceedings 265–266 (2015) 307–310 www.elsevier.com/locate/nppp
The Seesaw Scale vs Cosmology P. Hern´andez,a , M. Kekica , J. L´opez-Pav´onb a IFIC
(CSIC) and Dpto. F´ısica Te´orica, Universidad de Valencia, Edificio Institutos Investigaci´on, Apt. 22085, E-46071 Valencia, Spain. b SISSA, via Bonomea 265, 34136 Trieste, Italy. INFN, sezione di Trieste, 34136 Trieste, Italy.
Abstract We will study the simplest extension of the Standard Model that can account for neutrino masses: the Type-I seesaw. The model introduces a New Physics scale, M, which is often assumed to be much larger than the electroweak scale. However, it is presently unconstrained and the light neutrino masses and mixing can be generated for any value of M above O(eV). Paying special attention to the contribution of the sterile states to Neff as a function of M, we will show that a large part of the M parameter space (8 orders of magnitude) can be excluded thanks to cosmological measurements. The implications for neutrinoless double beta decay will be discussed too. Keywords: Seesaw, Cosmology, Neutrino Mass, Sterile neutrino.
The simplest extension of the Standard Model (SM) that can account for the light neutrino masses is the Type-I seesaw model [1] with N ≥ 2 extra singlet Majorana fermions. We will dub it Minimal Seesaw Model, since it contains the minimum number of extra degrees of freedom required to generate the observed neutrino masses and mixing. The model introduces a New Physics scale, M, associated to the mass of the singlet fermions (sterile neutrinos). The Yukawa couplings are usually assumed to be of the order one because of naturalness arguments, which drives M close to the GUT scale through the seesaw mechanism (mν ∼ Y 2 v2 /M). The drawback of this assumption is that such a high scale would require an important level of fine tunning in order to stabilize the Higgs mass, since it is quadratically sensitive to M at one loop. In any case, regardless theoretical discussions, M is experimentally unconstrained and the light neutrino masses and mixing can be generated for any value of M above O(eV) [2]. Considering low M scales requires accordingly small Yukawa couplings which can be recognized unnatural. Email address: [email protected] (J. L´opez-Pav´on)
http://dx.doi.org/10.1016/j.nuclphysbps.2015.06.077 2405-6014/© 2015 Elsevier B.V. All rights reserved.
However, low Majorana scales may be also considered technically natural since the Lagrangian gains a U(1) global symmetry in the limit M → 0. In any case, we are going to explore the full parameter space of the model without any theoretical prejudice. The main goal of our analysis is to understand if a general bound on the seesaw scale can be extracted without assuming anything a priori about the parameters of the model. If lepton number conservation is not imposed, the most general renormalizable Lagrangian when N extra singlet Weyl fermions, νRi , are included is given by
L = LS M −
α,i
˜ Ri − L¯ α Y αi Φν
N 1 ic i j j ν¯ M ν + h.c., 2 R N R i, j=1
where Y is a 3 × N complex matrix and MN a diagonal N × N real matrix. This Lagrangian defines the Type-I seesaw model. We will study the minimal model, which consists in the addition of N = 2 singlet Weyl fermions, and the more popular next to minimal model with N = 3. The detailed analysis of both models in the context of the cosmological bounds studied here can be
308
P. Hernández et al. / Nuclear and Particle Physics Proceedings 265–266 (2015) 307–310
where ν0 is the energy density of one SM massless neutrino with a thermal distribution. One thermal extra sterile state that freezes out from the thermal bath being relativistic contributes ΔNeff 1 when it decouples. Indeed, the sterile neutrinos should both achieve thermalization and decouple from the thermal bath when they are still relativistic in order to have any impact on Neff . SM + ΔNeff , where the contribuRecall that Neff = Neff SM = 3.046 [7, 8], and tion of the active neutrinos is Neff Big Bang Nucleosynthesis (BBN) [9] and Cosmic Microwave Background (CMB) [10] data give us Neff < 4 at more than 2σ while Neff = 5 is fairly excluded. If Mi 100 MeV, the sterile states are always relativistic when they decouple from the thermal bath regardless the value of the free parameters [3]. For Mi 100 MeV, no general bound can be extracted on Mi since the sterile neutrinos can become non-relativistic before freezing out and the contribution to Neff would be suppressed by the Boltzmann factor. The key quantity in order to understand if the sterile neutrinos are thermal is the thermalization ratio f s j (T ): f s j (T ) ≡
Γ s j (T ) . H(T )
(2)
which measures the sterile production rate of the species s j in units of the Hubble expansion rate. Γ s j is the sterile neutrino collision rate [11]: 1 Γs j P(νa → ν s j ) × Γνα 2 a ⎛ ⎞2 M 2j ⎜⎜⎜ ⎟⎟⎟ Γνα ⎜⎜⎝ ⎟⎟ |Uαs j |2 (3) 2⎠ 2pV (T ) − M α j α=e,μ,τ where P(να → ν s j ) is the time-averaged probability να → ν s j , Γνα is the active neutrino collision rate, Vα is the effective potential induced by the coherent scattering, and Uαs j is the mixing between the active neutrinos να and the sterile state ν s j . In [3], we also derived this results from the Boltzmann equations in the assumption of no primordial large lepton asymmetries1 . In any case, 1 The details of the derivation and the expressions for Γ (T ) and να Vα (T ) can be found in [4].
1 Log 10 m 1 eV
found in [3, 4] respectively. In both cases, we have considered an extension of the Casas-Ibarra parameterization [5], which is valid at all orders in the seesaw expansion and guaranties the generation of the right pattern of light neutrino masses and mixing [6]. The energy density of the sterile neutrino species, s , is usually quantified in terms of ΔNeff defined by s (1) ΔNeff ≡ 0 , ν
10
2
1
3
0.1
4 5 6
4
2
0
2
Lo g 10 M 1 M eV Figure 1: Contours of Min[ f s1 (T max )]=0.1, 1, 10 on the plane (M1 , m1 ).
we have numerically solved the Boltzmann equations in order to compute Neff and extract the bounds on M. f s j (T ) reaches a maximum at the temperature T max , which depends on the free parameters of the model. If it is larger than one, thermalization will be achieved at early times. In the N = 2 model, both sterile neutrinos achieve thermalization f s j (T max ) > 1 in the full parameter space [3]. In Fig. 1 we show the contour plots of the minimum of f s j (T max ) (varying the unconstrained parameters of the model in the full range), as a function of m1 (the lightest neutrino mass) and M1 in the N = 3 model. The three contours correspond to Min[ f s1 (T max )] = 10−1 , 1, 10. The results are the same in the (M2 , m1 ) and (M3 , m1 ) planes. As in the N = 2 case, the Min[ f s1 (T max )] is roughly independent of the scale M. The figure shows that m1 ultimately controls the thermalization of the sterile neutrinos. In fact, there is a thermalization threshold at m1 10−3 eV. This opens two possible scenarios: (i) m1 10−3 eV. The three sterile states achieve thermalization. Therefore, for Mi 100 MeV, each ( j) 1 when sterile neutrino will contribute with ΔNeff freezing out from the thermal bath. However, there are two possible effects that can modify the contribution to Neff later on and before the active neutrino decoupling, when BBN starts: the entropy dilution and the decay of the sterile states. The effect of the entropy dilution is only relevant for Mi 10 KeV [3, 4] and sterile states of this range of masses give a contribution to Neff at BBN in agreement with data. However, these sterile neutrinos would give a huge contribution to the energy density when they become non-relativistic later, modifying in a drastic way CMB and structure formation. In fact, CMB measurements close the window all the
309
P. Hernández et al. / Nuclear and Particle Physics Proceedings 265–266 (2015) 307–310 Log10 Mi eV 1
10 101
8 m ΒΒ eV
6 4 2
NH
10 3 eV
0 Log10 Mi eV
102
IH
104
1
2
3
1 0 0 eV k eV
105 105
10
104
10 3
102
101
1
m lig h t eV
8 6 4 2 0 1
2
3
Figure 3: mββ as a function of the lightest neutrino mass: contribution from the active neutrinos (red and blue regions) and the maximum contribution of the lightest sterile neutrino, for M1 = 1 eV (solid), 100 eV (dashed), 1 KeV (dotted), for NH (blue) and IH (red) restricting Ω s1 h2 ≤ 0.12 and f s1 (T max ) ≤ 1, for M2,3 100 MeV, as a function of the lightest neutrino mass. The shaded region is ruled out for M1 ∈ [1eV-100MeV] by the thermalization bound on the lightest neutrino mass, m1 ≤ 10−3 eV.
Figure 2: Allowed spectra of the heavy states Mi for m1 ≥ 10−3 eV (upper panel) and m1 ≤ 10−3 eV (lower panel).
way down to the O(eV) [12]. The only way to scape from the CMB and BBN bounds is if the sterile neutrinos decay before BBN. Nevertheless, taking into account the bounds on the active-sterile mixing from direct searches, this possibility is excluded for Mi 100 MeV [13]. In summary, the allowed region of the M parameter space is shown in the upper panel of Fig. 2. The constraints from cosmology essentially exclude the spectra of heavy states in the range 1 eV - 100 MeV. (ii) m1 10−3 eV. One and only one of the sterile neutrinos can never thermalize and it depends on the unknown parameters [4]. In this case, one of the sterile neutrino masses can not be bounded in general. The other two states always thermalize and again the range 1 eV- 100 MeV is severely restricted [4]. If one of the sterile neutrinos never thermalizes, the contribution of the two thermal states reduces to the N = 2 case since the other is essentially decoupled from them and the mass generation. The results are qualitatively summarized in the lower panel of Fig. 2, where we show the allowed heavy neutrino spectra in the m1 10−3 eV case. Finally, the information from cosmology is highly complementary to that coming from neutrino oscillations and neutrinoless double beta decay (0νββ decay). As an example, in Fig. 3 we show the impact of the cosmological bounds when a sterile neutrino in the range 1 eV - 100 MeV is present. The 0νββ decay rate is sen-
sitive to the effective mass, mββ , and Fig. 3 shows the separate contribution to mββ from the active and sterile neutrinos as a function of the lightest neutrino mass m1 . The well-known colored bands correspond to the active neutrino contribution while the maximum contribution of the lightest sterile state is given by the solid (M1 = 1 eV), dashed (M1 = 100 eV) and dotted (M1 = 1 KeV) lines. The other two neutrinos are assumed to be well above 100 MeV. A sterile state with M1 ∈ [1 eV, 100 MeV] gives a long range contribution to the 0νββ decay which could in principle be at the reach of the future experiments. However, once the constraints from cosmology are included, it becomes subdominant with respect to the active one and well below the future sensitivity O(10−2 eV). Moreover, if M1 ∈ [1 eV, 100 MeV], the quasi-degenerate light neutrino spectrum and the region of the parameter space in which a cancellation can occur in the active neutrino contribution would be excluded. This is because in such a case m1 should be below the thermalization threshold, at m1 = 10−3 eV, in order to make the lightest sterile neutrino non-thermal. In summary, we have found that the BBN and CMB data essentially excludes the region of the seesaw scale between 1 eV and 100 MeV in both the N = 2 and the N = 3 models. Only in the N = 3 case and provided that m1 10−3 eV, one of the sterile states can be nonthermal and its mass remains unbounded. This has an important impact in the 0νββ decay phenomenology.
310
P. Hernández et al. / Nuclear and Particle Physics Proceedings 265–266 (2015) 307–310
References [1] P. Minkowski, Phys. Lett. B 67, 421 (1977); M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, edited by P. van Nieuwenhuizen and D. Freedman (North-Holland, 1979), p. 315; T. Yanagida in Proceedings of the Workshop on the Unified Theory and the Baryon Number in the Universe, edited by O. Sawada and A. Sugamoto (KEK Report No. 79-18, Tsukuba, 1979), p. 95; R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980). [2] A. Donini, P. Hernandez, J. Lopez-Pavon and M. Maltoni, JHEP 1107 (2011) 105 [arXiv:1106.0064 [hep-ph]]. [3] P. Hernandez, M. Kekic and J. Lopez-Pavon, Phys. Rev. D 89 (2014) 7, 073009 [arXiv:1311.2614 [hep-ph]]. [4] P. Hernandez, M. Kekic and J. Lopez-Pavon, Phys. Rev. D 90 (2014) 6, 065033 [arXiv:1406.2961 [hep-ph]]. [5] J. A. Casas and A. Ibarra, Nucl. Phys. B 618, 171 (2001). [6] A. Donini, P. Hernandez, J. Lopez-Pavon, M. Maltoni and T. Schwetz, JHEP 1207 (2012) 161 [arXiv:1205.5230 [hep-ph]]. [7] G. Mangano, G. Miele, S. Pastor and M. Peloso, Phys. Lett. B 534 (2002) 8 [astro-ph/0111408]. [8] G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti and P. D. Serpico, Nucl. Phys. B 729 (2005) 221 [hep-ph/0506164]. [9] R. Cooke, M. Pettini, R. A. Jorgenson, M. T. Murphy and C. C. Steidel, arXiv:1308.3240 [astro-ph.CO]. [10] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 571 (2014) A16 [arXiv:1303.5076 [astro-ph.CO]]. [11] R. Barbieri and A. Dolgov, Phys. Lett. B 237, 440 (1990) and Nucl. Phys. B349 (1991) 743. K. Kainulainen, Phys. Lett. B 244, 191 (1990). [12] E. Di Valentino, A. Melchiorri and O. Mena, JCAP 1311 (2013) 018 [arXiv:1304.5981]. [13] O. Ruchayskiy and A. Ivashko, JCAP 1210, 014 (2012).