- Email: [email protected]
Leptogenesis with type II seesaw in SO(10)
Nuclear Physics B (Proc. Suppl.) 188 (2009) 332–335 www.elsevierphysics.com
Leptogenesis with type II seesaw in SO(10) Andrea Romaninoa a
SISSA/ISAS and INFN, I–34014 Trieste, Italy
We point out that what can be considered as the most natural implementation of type-II seesaw in SO(10) naturally leads to a leptogenesis scenario that is significantly more predictive than usual. The baryon asymmetry depends on the low-energy neutrino parameters, with no unknown seesaw-scale flavour parameters involved. In particular, the necessary CP violation is provided by the CP-violating phases of the lepton mixing matrix.
The main message of this presentation [1] is that what I consider to be the most natural implementation of type-II seesaw [2] in SO(10) naturally leads to a leptogenesis scenario that is significantly more predictive than usual. Let me begin by reminding that in the standard type-I seesaw, the knowledge of lepton masses and mixings is not sufficient to determine all the high-energy parameters relevant for leptogenesis. Our ignorance on the 9 unknown high energy parameters can be parameterized in several ways, for example through a complex orthogonal matrix [3]. In the case of type-II seesaw, light neutrino masses are induced by the exchange of a heavy scalar SU(2)L triplet with twice the hypercharge of the Higgs field. It is well known that this mechanism leads to a better connection to the high energy parameters and therefore to higher predictivity in lepton flavour violation [4]. However, the minimal implementation of type-II seesaw does not incorporate the necessary ingredients to give rise to viable leptogenesis. Such ingredients are provided, as anticipated, by a simple SO(10) implementation [1]. Among the different possible embeddings of the type-II scalar triplet into a SO(10) representation, the simplest possibility, and the one we adopt, is provided by the 54 of SO(10). The latter is nothing but the symmetric product of two fundamentals. Alternative possibilities would involve sets of representations with dimension 252 or larger, leading to a non-perturbative regime for the gauge couplings well before the unification scale. The triplet in the 54 has to couple to the Standard Model (SM) lepton doublets, 0920-5632/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.02.077
which constrains the SO(10) embedding of the SM fermions. Each SM family can be embedded in a ¯ 5 + 10 of SU(5). Normally, the latter are embedded into a single 16 of SO(10), together with a singlet neutrino mediating type-I seesaw. Such a choice, however, would not allow the triplet to couple to the SM leptons. We therefore choose to embed the 10 and ¯ 5 of SU(5) in a 16 and a 10 of SO(10), respectively. Overall, the SM fermions are embedded in three 16i + 10i , i = 1, 2, 3. As a consequence, i) the SO(10) prediction of the SM gauge quantum numbers in terms of the chiral content of the theory (the three 16i ’s) is maintained and ii) the matter content includes three spare (5+¯ 5) of SU(5) that need to be made heavy. We consider a supersymmetric model whose superpotential includes the terms yij 16i 16j 10 + hij 16i 10j 16 2 fij σ 10i 10j 54 + 10 10 54. + 2 2
W ⊇
(1)
The superpotential involves a 10 and a 16. The 10 contains the up Higgs field and appears in an interaction contributing to the up quark Yukawas. The 16 appears in an interaction contributing to the down quark and charged lepton Yukawas and has a double role. It contains part of the down Higgs, shared with the 10, and it gets a O (MGUT ) vev along the SM singlet direction. The latter is necessary for a proper breaking of SO(10) and gives rise to a mass term that pairs up the three spare SU(5) (5 + ¯ 5) components in 16i + 10i and makes them heavy, as needed. In eq. (1) I have
A. Romanino / Nuclear Physics B (Proc. Suppl.) 188 (2009) 332–335
omitted a mass term for the 54 (which we allow to be different for the different SU(5) components, 24 and 15 + 15), the interactions taking care of SO(10) breaking, and non-renormalizable (NR) terms necessary to obtain the correct masses of the first and second family of SM fermions. Despite its simplicity, the model has all the ingredients to give rise to successful leptogenesis through the decay of the triplets mediating type-II seesaw. Moreover, all the high energy flavour parameters relevant to leptogenesis are determined, up to the effect of NR interactions, in terms of SM fermion masses and mixings (in particular, CP violation is provided by the phases in the lepton mixing matrix), so that only a few flavour blind parameters are left to be determined. This is quite different from the usual triplet leptogenesis scenarios [5], in which two different sets of couplings are needed in order to obtain a non-vanishing CP asymmetry. We have in fact mU ij = yij vu , T mD ij = shij vd ,
mE ij = shij vd −mνij = fij
vu2 , M15
(2)
where M15 is the mass of the seesaw triplet and its SU(5) partners, which we assume to be lighter than the rest of the 54, M15 < M24 , and 0 < s < 1 measures the size of the down Higgs component in the 16. Note that the three 16i contain right-handed neutrinos, but the latter do not contribute to the light neutrino masses, as their Yukawa coupling with the light leptons is absent. We therefore have a pure type-II seesaw, with no type-I pollution. The heavy components in 16i + 10i consist in three families of heavy vectorlike lepton doublets and right-handed down ¯ i and Dc + D ¯ c . Their masses quarks, Li + L i i turn out to be proportional to the ones of the corresponding SM fermions (up to NR terms): MLi = mli 16 /(svd ). As a consequence, the masses of the heavy leptons will also be hierarchical, with the heaviest around the GUT scale and the lightest several orders of magnitude lighter. The exact one-loop expression for the CPasymmetry in the triplet decay can be found in [1]. Let us consider here the interesting limit
333
in which ML1 < M15 < ML2 , M24 . In this case the asymmetry can be approximated by ≈
1 M15 λ4l Im[m∗ee (mm∗ m)ee ] , (3) 2 10π M24 λl + |σ|2 ( i m2i )2
where mij is the light neutrino mass matrix written in the basis in flavour space in which the charged lepton mass matrix is diagonal and positive (so that mee is the matrix element entering the neutrinoless double beta decay amplitude), and λl is a mi are the light neutrino masses, flavour-independent coupling, λ2l = ij |fij |2 . In order to obtain successful leptogenesis, the product of the asymmetry times the “efficiency” parameter η must be about η ≈ 10−8 . The asymmetry grows with λl . For λl = 1, the asymmetry can reach values as large as ∼ 10−4 , depending on the value of the unknown neutrino parameters. Of course, such large values do not necessarily correspond to a large efficiency. On the contrary, they turn out to correpond to a strong washout regime. The possibility to obtain successful leptogenesis in this regime requires a detailed numerical study. Here, I would like to illustrate the complementary regime in which the measured baryon asymmetry is obtained in a weak washout regime with η ∼ 1 and 10−4 . The fact that a quasi-maximal efficiency can be achieved despite the SU(2)L triplets are kept in equilibrium by their fast gauge interactions and decays can be surprising. However, it has been shown [6] that a large efficiency can indeed be achieved provided that i) at least one decay channel is out of equilibrium, ii) the total decay rate is faster than the gauge interaction rate, and iii) the slow channel has out of equilibrium interactions with the rest of the system. As a bonus, the final baryon asymmetry ends up not to depend on the initial conditions. In our case, the three main decay channels are into two light lepton doublets, two heavy sleptons, and two up Higgses. In the regime in which eq. (3) is written, the candidate slow decay channel is the decay into two heavy sleptons. The latter is in fact suppressed compared to the decay into light leptons by |mee |2 / i m2i . The parameter space in which the efficiency is predicted to be close to maximal (and can estimated analytically
334
A. Romanino / Nuclear Physics B (Proc. Suppl.) 188 (2009) 332–335
KL 1
0.30
0.25
Ε 1010
Ε 108
Ε 107
Σ
0.20
0.15
M15 1013 GeV
0.10
0.05
M15 1012 GeV
KLc1 1 KH 1
0.00 0.00
0.05
0.10
0.15 Λl
0.20
0.25
0.30
Figure 1. Contours of constant and M15 in the λl –|σ| plane for a normal hierarchical neutrino spectrum with m1 m2 , sin2 θ13 = 0.05, maximal CP-violation, and M15 /M24 = 0.1. In the shaded regions, the conditions for a large efficiency are not satisfied.
to be not far from 4/7 [1]) is shown in Fig. 1 for a normal hierarchical neutrino spectrum with m1 m2 , sin2 θ13 = 0.05, maximal CP-violation, and M15 /M24 = 0.1. We conclude that successful leptogenesis can be achieved in a wide portion of the parameter space. The discussion above is of course oversimplified. The decay asymmetries of the SU(5) partners of the triplets should be taken into account, and the contribution of the right-handed neutrino decays to the asymmetry should be kept under control. I refer to [1] for a detailed discussion. A comment is in order. Fig. 1 shows that successful leptogenesis in the large efficiency region requires M15 1012 GeV. Such values are in conflict with the gravitino constraint, which sets an upper bound on the reheating temperature after inflation, TRH 10(9−10) GeV for a gravitino of mass m3/2 ∼ 100 GeV [7]. If the gravitino is not the LSP, a much stronger bound comes from the requirement that its decays do not spoil Big-Bang Nucleosynthesis [8]. The gravitino constraint
might be evaded in the strong washout regime or in non-supersymmetric scenarios. Otherwise, one can still invoke an extremely light gravitino [9], which can be obtained in some scenarios of gaugemediated supersymmetry breaking [10], or a very heavy gravitino [11], m3/2 100 TeV, which can be achieved in fine-tuned scenarios [12]. Another possibility is to resort to non-thermal production of the heavy triplets, e.g. during reheating [13], preheating [14], or via another mechanism [15]. The possibility to test the predictions of this supersymmetric scenarios through the measurement of lepton flavour violation is under study. REFERENCES 1. M. Frigerio, P. Hosteins, S. Lavignac and A. Romanino, Nucl. Phys. B 806 (2009) 84 [arXiv:0804.0801 [hep-ph]]. 2. M. Magg and C. Wetterich, Phys. Lett. B 94 (1980) 61; G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181 (1981) 287; R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23 (1981) 165. See also J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980) 2227. 3. J. A. Casas and A. Ibarra, Nucl. Phys. B 618, 171 (2001) [arXiv:hep-ph/0103065]. 4. A. Rossi, Phys. Rev. D 66 (2002) 075003 [arXiv:hep-ph/0207006]. 5. P. J. O’Donnell and U. Sarkar, Phys. Rev. D 49 (1994) 2118 [arXiv:hep-ph/9307279]; E. Ma and U. Sarkar, Phys. Rev. Lett. 80 (1998) 5716 [arXiv:hep-ph/9802445]; T. Hambye and G. Senjanovic, Phys. Lett. B 582 (2004) 73 [arXiv:hep-ph/0307237]. 6. T. Hambye, M. Raidal and A. Strumia, Phys. Lett. B 632 (2006) 667 [arXiv:hepph/0510008]. 7. M. Bolz, A. Brandenburg and W. Buchmuller, Nucl. Phys. B 606 (2001) 518 [Erratum-ibid. B 790 (2008) 336] [arXiv:hep-ph/0012052]; J. Pradler and F. D. Steffen, Phys. Rev. D 75 (2007) 023509 [arXiv:hep-ph/0608344]; V. S. Rychkov and A. Strumia, Phys. Rev. D 75 (2007) 075011 [arXiv:hep-ph/0701104]. 8. M. Y. Khlopov and A. D. Linde, Phys. Lett. B 138 (1984) 265; J. R. Ellis, J. E. Kim and
A. Romanino / Nuclear Physics B (Proc. Suppl.) 188 (2009) 332–335
9. 10.
11. 12.
13.
14.
15.
D. V. Nanopoulos, Phys. Lett. B 145 (1984) 181. For recent work on the BBN constraints, see e.g. M. Kawasaki, K. Kohri, T. Moroi and A. Yotsuyanagi, arXiv:0804.3745 [hep-ph]. H. Pagels and J. R. Primack, Phys. Rev. Lett. 48 (1982) 223. See e.g. M. Ibe, K. Tobe and T. Yanagida, Phys. Lett. B 615 (2005) 120 [arXiv:hepph/0503098]; M. Ibe, Y. Nakayama and T. T. Yanagida, arXiv:0804.0636 [hep-ph]. S. Weinberg, Phys. Rev. Lett. 48, 1303 (1982). N. Arkani-Hamed and S. Dimopoulos, JHEP 0506 (2005) 073 [arXiv:hep-th/0405159]; G. F. Giudice and A. Romanino, Nucl. Phys. B 699 (2004) 65 [Erratum-ibid. B 706 (2005) 65] [arXiv:hep-ph/0406088]; N. Arkani-Hamed, S. Dimopoulos, G. F. Giudice and A. Romanino, Nucl. Phys. B 709 (2005) 3 [arXiv:hep-ph/0409232]. D. J. H. Chung, E. W. Kolb and A. Riotto, Phys. Rev. D 60 (1999) 063504 [arXiv:hepph/9809453]. G. N. Felder, L. Kofman and A. D. Linde, Phys. Rev. D 59 (1999) 123523 [arXiv:hepph/9812289]; G. F. Giudice, M. Peloso, A. Riotto and I. Tkachev, JHEP 9908 (1999) 014 [arXiv:hep-ph/9905242]. G. F. Giudice, A. Riotto and A. Zaffaroni, Nucl. Phys. B 710 (2005) 511 [arXiv:hepph/0408155]; R. Allahverdi and A. Mazumdar, JCAP 0610 (2006) 008 [arXiv:hepph/0512227]; G. F. Giudice, L. Mether, A. Riotto and F. Riva, arXiv:0804.0166 [hep-ph].
335
Leptogenesis with type II seesaw in SO(10) Andrea Romaninoa a
SISSA/ISAS and INFN, I–34014 Trieste, Italy
We point out that what can be considered as the most natural implementation of type-II seesaw in SO(10) naturally leads to a leptogenesis scenario that is significantly more predictive than usual. The baryon asymmetry depends on the low-energy neutrino parameters, with no unknown seesaw-scale flavour parameters involved. In particular, the necessary CP violation is provided by the CP-violating phases of the lepton mixing matrix.
The main message of this presentation [1] is that what I consider to be the most natural implementation of type-II seesaw [2] in SO(10) naturally leads to a leptogenesis scenario that is significantly more predictive than usual. Let me begin by reminding that in the standard type-I seesaw, the knowledge of lepton masses and mixings is not sufficient to determine all the high-energy parameters relevant for leptogenesis. Our ignorance on the 9 unknown high energy parameters can be parameterized in several ways, for example through a complex orthogonal matrix [3]. In the case of type-II seesaw, light neutrino masses are induced by the exchange of a heavy scalar SU(2)L triplet with twice the hypercharge of the Higgs field. It is well known that this mechanism leads to a better connection to the high energy parameters and therefore to higher predictivity in lepton flavour violation [4]. However, the minimal implementation of type-II seesaw does not incorporate the necessary ingredients to give rise to viable leptogenesis. Such ingredients are provided, as anticipated, by a simple SO(10) implementation [1]. Among the different possible embeddings of the type-II scalar triplet into a SO(10) representation, the simplest possibility, and the one we adopt, is provided by the 54 of SO(10). The latter is nothing but the symmetric product of two fundamentals. Alternative possibilities would involve sets of representations with dimension 252 or larger, leading to a non-perturbative regime for the gauge couplings well before the unification scale. The triplet in the 54 has to couple to the Standard Model (SM) lepton doublets, 0920-5632/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.02.077
which constrains the SO(10) embedding of the SM fermions. Each SM family can be embedded in a ¯ 5 + 10 of SU(5). Normally, the latter are embedded into a single 16 of SO(10), together with a singlet neutrino mediating type-I seesaw. Such a choice, however, would not allow the triplet to couple to the SM leptons. We therefore choose to embed the 10 and ¯ 5 of SU(5) in a 16 and a 10 of SO(10), respectively. Overall, the SM fermions are embedded in three 16i + 10i , i = 1, 2, 3. As a consequence, i) the SO(10) prediction of the SM gauge quantum numbers in terms of the chiral content of the theory (the three 16i ’s) is maintained and ii) the matter content includes three spare (5+¯ 5) of SU(5) that need to be made heavy. We consider a supersymmetric model whose superpotential includes the terms yij 16i 16j 10 + hij 16i 10j 16 2 fij σ 10i 10j 54 + 10 10 54. + 2 2
W ⊇
(1)
The superpotential involves a 10 and a 16. The 10 contains the up Higgs field and appears in an interaction contributing to the up quark Yukawas. The 16 appears in an interaction contributing to the down quark and charged lepton Yukawas and has a double role. It contains part of the down Higgs, shared with the 10, and it gets a O (MGUT ) vev along the SM singlet direction. The latter is necessary for a proper breaking of SO(10) and gives rise to a mass term that pairs up the three spare SU(5) (5 + ¯ 5) components in 16i + 10i and makes them heavy, as needed. In eq. (1) I have
A. Romanino / Nuclear Physics B (Proc. Suppl.) 188 (2009) 332–335
omitted a mass term for the 54 (which we allow to be different for the different SU(5) components, 24 and 15 + 15), the interactions taking care of SO(10) breaking, and non-renormalizable (NR) terms necessary to obtain the correct masses of the first and second family of SM fermions. Despite its simplicity, the model has all the ingredients to give rise to successful leptogenesis through the decay of the triplets mediating type-II seesaw. Moreover, all the high energy flavour parameters relevant to leptogenesis are determined, up to the effect of NR interactions, in terms of SM fermion masses and mixings (in particular, CP violation is provided by the phases in the lepton mixing matrix), so that only a few flavour blind parameters are left to be determined. This is quite different from the usual triplet leptogenesis scenarios [5], in which two different sets of couplings are needed in order to obtain a non-vanishing CP asymmetry. We have in fact mU ij = yij vu , T mD ij = shij vd ,
mE ij = shij vd −mνij = fij
vu2 , M15
(2)
where M15 is the mass of the seesaw triplet and its SU(5) partners, which we assume to be lighter than the rest of the 54, M15 < M24 , and 0 < s < 1 measures the size of the down Higgs component in the 16. Note that the three 16i contain right-handed neutrinos, but the latter do not contribute to the light neutrino masses, as their Yukawa coupling with the light leptons is absent. We therefore have a pure type-II seesaw, with no type-I pollution. The heavy components in 16i + 10i consist in three families of heavy vectorlike lepton doublets and right-handed down ¯ i and Dc + D ¯ c . Their masses quarks, Li + L i i turn out to be proportional to the ones of the corresponding SM fermions (up to NR terms): MLi = mli 16 /(svd ). As a consequence, the masses of the heavy leptons will also be hierarchical, with the heaviest around the GUT scale and the lightest several orders of magnitude lighter. The exact one-loop expression for the CPasymmetry in the triplet decay can be found in [1]. Let us consider here the interesting limit
333
in which ML1 < M15 < ML2 , M24 . In this case the asymmetry can be approximated by ≈
1 M15 λ4l Im[m∗ee (mm∗ m)ee ] , (3) 2 10π M24 λl + |σ|2 ( i m2i )2
where mij is the light neutrino mass matrix written in the basis in flavour space in which the charged lepton mass matrix is diagonal and positive (so that mee is the matrix element entering the neutrinoless double beta decay amplitude), and λl is a mi are the light neutrino masses, flavour-independent coupling, λ2l = ij |fij |2 . In order to obtain successful leptogenesis, the product of the asymmetry times the “efficiency” parameter η must be about η ≈ 10−8 . The asymmetry grows with λl . For λl = 1, the asymmetry can reach values as large as ∼ 10−4 , depending on the value of the unknown neutrino parameters. Of course, such large values do not necessarily correspond to a large efficiency. On the contrary, they turn out to correpond to a strong washout regime. The possibility to obtain successful leptogenesis in this regime requires a detailed numerical study. Here, I would like to illustrate the complementary regime in which the measured baryon asymmetry is obtained in a weak washout regime with η ∼ 1 and 10−4 . The fact that a quasi-maximal efficiency can be achieved despite the SU(2)L triplets are kept in equilibrium by their fast gauge interactions and decays can be surprising. However, it has been shown [6] that a large efficiency can indeed be achieved provided that i) at least one decay channel is out of equilibrium, ii) the total decay rate is faster than the gauge interaction rate, and iii) the slow channel has out of equilibrium interactions with the rest of the system. As a bonus, the final baryon asymmetry ends up not to depend on the initial conditions. In our case, the three main decay channels are into two light lepton doublets, two heavy sleptons, and two up Higgses. In the regime in which eq. (3) is written, the candidate slow decay channel is the decay into two heavy sleptons. The latter is in fact suppressed compared to the decay into light leptons by |mee |2 / i m2i . The parameter space in which the efficiency is predicted to be close to maximal (and can estimated analytically
334
A. Romanino / Nuclear Physics B (Proc. Suppl.) 188 (2009) 332–335
KL 1
0.30
0.25
Ε 1010
Ε 108
Ε 107
Σ
0.20
0.15
M15 1013 GeV
0.10
0.05
M15 1012 GeV
KLc1 1 KH 1
0.00 0.00
0.05
0.10
0.15 Λl
0.20
0.25
0.30
Figure 1. Contours of constant and M15 in the λl –|σ| plane for a normal hierarchical neutrino spectrum with m1 m2 , sin2 θ13 = 0.05, maximal CP-violation, and M15 /M24 = 0.1. In the shaded regions, the conditions for a large efficiency are not satisfied.
to be not far from 4/7 [1]) is shown in Fig. 1 for a normal hierarchical neutrino spectrum with m1 m2 , sin2 θ13 = 0.05, maximal CP-violation, and M15 /M24 = 0.1. We conclude that successful leptogenesis can be achieved in a wide portion of the parameter space. The discussion above is of course oversimplified. The decay asymmetries of the SU(5) partners of the triplets should be taken into account, and the contribution of the right-handed neutrino decays to the asymmetry should be kept under control. I refer to [1] for a detailed discussion. A comment is in order. Fig. 1 shows that successful leptogenesis in the large efficiency region requires M15 1012 GeV. Such values are in conflict with the gravitino constraint, which sets an upper bound on the reheating temperature after inflation, TRH 10(9−10) GeV for a gravitino of mass m3/2 ∼ 100 GeV [7]. If the gravitino is not the LSP, a much stronger bound comes from the requirement that its decays do not spoil Big-Bang Nucleosynthesis [8]. The gravitino constraint
might be evaded in the strong washout regime or in non-supersymmetric scenarios. Otherwise, one can still invoke an extremely light gravitino [9], which can be obtained in some scenarios of gaugemediated supersymmetry breaking [10], or a very heavy gravitino [11], m3/2 100 TeV, which can be achieved in fine-tuned scenarios [12]. Another possibility is to resort to non-thermal production of the heavy triplets, e.g. during reheating [13], preheating [14], or via another mechanism [15]. The possibility to test the predictions of this supersymmetric scenarios through the measurement of lepton flavour violation is under study. REFERENCES 1. M. Frigerio, P. Hosteins, S. Lavignac and A. Romanino, Nucl. Phys. B 806 (2009) 84 [arXiv:0804.0801 [hep-ph]]. 2. M. Magg and C. Wetterich, Phys. Lett. B 94 (1980) 61; G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181 (1981) 287; R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23 (1981) 165. See also J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980) 2227. 3. J. A. Casas and A. Ibarra, Nucl. Phys. B 618, 171 (2001) [arXiv:hep-ph/0103065]. 4. A. Rossi, Phys. Rev. D 66 (2002) 075003 [arXiv:hep-ph/0207006]. 5. P. J. O’Donnell and U. Sarkar, Phys. Rev. D 49 (1994) 2118 [arXiv:hep-ph/9307279]; E. Ma and U. Sarkar, Phys. Rev. Lett. 80 (1998) 5716 [arXiv:hep-ph/9802445]; T. Hambye and G. Senjanovic, Phys. Lett. B 582 (2004) 73 [arXiv:hep-ph/0307237]. 6. T. Hambye, M. Raidal and A. Strumia, Phys. Lett. B 632 (2006) 667 [arXiv:hepph/0510008]. 7. M. Bolz, A. Brandenburg and W. Buchmuller, Nucl. Phys. B 606 (2001) 518 [Erratum-ibid. B 790 (2008) 336] [arXiv:hep-ph/0012052]; J. Pradler and F. D. Steffen, Phys. Rev. D 75 (2007) 023509 [arXiv:hep-ph/0608344]; V. S. Rychkov and A. Strumia, Phys. Rev. D 75 (2007) 075011 [arXiv:hep-ph/0701104]. 8. M. Y. Khlopov and A. D. Linde, Phys. Lett. B 138 (1984) 265; J. R. Ellis, J. E. Kim and
A. Romanino / Nuclear Physics B (Proc. Suppl.) 188 (2009) 332–335
9. 10.
11. 12.
13.
14.
15.
D. V. Nanopoulos, Phys. Lett. B 145 (1984) 181. For recent work on the BBN constraints, see e.g. M. Kawasaki, K. Kohri, T. Moroi and A. Yotsuyanagi, arXiv:0804.3745 [hep-ph]. H. Pagels and J. R. Primack, Phys. Rev. Lett. 48 (1982) 223. See e.g. M. Ibe, K. Tobe and T. Yanagida, Phys. Lett. B 615 (2005) 120 [arXiv:hepph/0503098]; M. Ibe, Y. Nakayama and T. T. Yanagida, arXiv:0804.0636 [hep-ph]. S. Weinberg, Phys. Rev. Lett. 48, 1303 (1982). N. Arkani-Hamed and S. Dimopoulos, JHEP 0506 (2005) 073 [arXiv:hep-th/0405159]; G. F. Giudice and A. Romanino, Nucl. Phys. B 699 (2004) 65 [Erratum-ibid. B 706 (2005) 65] [arXiv:hep-ph/0406088]; N. Arkani-Hamed, S. Dimopoulos, G. F. Giudice and A. Romanino, Nucl. Phys. B 709 (2005) 3 [arXiv:hep-ph/0409232]. D. J. H. Chung, E. W. Kolb and A. Riotto, Phys. Rev. D 60 (1999) 063504 [arXiv:hepph/9809453]. G. N. Felder, L. Kofman and A. D. Linde, Phys. Rev. D 59 (1999) 123523 [arXiv:hepph/9812289]; G. F. Giudice, M. Peloso, A. Riotto and I. Tkachev, JHEP 9908 (1999) 014 [arXiv:hep-ph/9905242]. G. F. Giudice, A. Riotto and A. Zaffaroni, Nucl. Phys. B 710 (2005) 511 [arXiv:hepph/0408155]; R. Allahverdi and A. Mazumdar, JCAP 0610 (2006) 008 [arXiv:hepph/0512227]; G. F. Giudice, L. Mether, A. Riotto and F. Riva, arXiv:0804.0166 [hep-ph].
335