a simple model for particle physics and cosmology
TRANSCRIPT
JHEP07(2010)085
Published for SISSA by Springer
Received: April 15, 2010
Accepted: July 3, 2010
Published: July 23, 2010
A simple model for particle physics and cosmology
Wan-Il Park
Department of Physics, KAIST,
373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea
E-mail: [email protected]
Abstract: We propose a simple extension of the minimal supersymmetric standard model
by introducing a gauge singlet in addition to right-handed neutrinos. The model resolves
the strong CP problem by Pecci-Quinn symmetry, explains the origin of left-handed neu-
trino masses as well as MSSM µ-parameter. It also gives rise to thermal inflation, baryoge-
nesis and dark matter in a remarkably consistent way. Interestingly, resolution of moduli
problem by thermal inflation constrains tightly axion coupling constant and flaton decay
temperature to be fa ∼ 1012 GeV and Td ∼ 100MeV, respectively. Model parameters in
this case are likely to give right amount of baryon asymmetry and dark matter at present.
The main component of dark matter is expected to be the axino whose mass is nearly fixed
to be about 1GeV.
Keywords: Cosmology of Theories beyond the SM, Supersymmetric Effective Theories
ArXiv ePrint: 1004.2326
c© SISSA 2010 doi:10.1007/JHEP07(2010)085
JHEP07(2010)085
Contents
1 Introduction 1
2 The model 2
3 Cosmology 5
3.1 Moduli problem and thermal inflation 6
3.2 Baryogenesis 10
3.3 Dark matter 11
4 Conclusion 15
A Renormalization group equations 16
1 Introduction
Although its great success in describing particle physics, the standard model (SM) of par-
ticle physics has been faced on some big questions: hierarchy problem associated with the
unnatural stability of weak scale Higgs mass against large quantum corrections induced by
the large hierarchy between electroweak (mew ∼ 103 GeV) and Planck (MPl ∼ 1019 GeV)
scales, strong CP problem [1, 2] and the origin of left-handed neutrino masses which is
implied by the neutrino oscillation [3, 4]. Any plausible theory for particle physics, which
is based on SM, should address all these questions.
The minimal supersymmetric standard model (MSSM) is a very natural and simple
supersymmetric extension of standard model, in which large hierarchy problem is absent
by the help of supersymmetry (SUSY) [5–7], hence it is quite attractive. MSSM may be
the right direction to a theory for low energy physics, but it has a mysterious parameter µ,
the dimensionful coefficient of Higgs bilinear superpotential term, whose origin need to be
explained [8]. In addition, MSSM still should be extended to resolve strong CP problem
and explain the origin of left-handed neutrino masses.
Meanwhile, string/M theory, which may be the fundamental theory at high energy,
predicts the existence of light particles with gravitationally suppressed interactions: grav-
itinos and moduli/modulinos. Typically, those particles are expected to be produced too
much due to large reheating temperature or coherent oscillation after primordial inflation,
disturbing the successful Big-Bang nucleosynthesis (BBN) or over-closing the present uni-
verse [9–13]. The most compelling solution to this gravitino/moduli problem is thermal
inflation [14, 15] which is very likely to occur in the framework of SUSY. But thermal
inflation invalidates most of known baryogenesis mechanisms, which work typically above
– 1 –
JHEP07(2010)085
or at the electroweak scale, by diluting out pre-existing baryon/lepton asymmetry.1 Since
the reheating temperature after thermal inflation is typically well below the electroweak
scale, baryogenesis mechanism is difficult to work in general.
In this paper, we propose a simple extension of MSSM by introducing just a gauge sin-
glet and right-handed neutrino chiral superfields, but without any ad-hoc mass parameter
except soft SUSY-breaking ones. In the framework of gravity-mediated SUSY-breaking,
the model realizes Peccei-Quinn symmetry [18–25] to resolve strong CP problem and ex-
plains the origin of left-handed neutrino masses through see-saw mechanism [26–28] as well
as that of MSSM µ-parameter in a very efficient way. The model also provides remarkably
consistent cosmology including thermal inflation, baryogenesis and dark matter.
This paper is organized as follows. In section 2, we propose a model and describe how
it provides proper low energy physics. In section 3, we briefly describe cosmology which
the model can realize. In section 4, we conclude.
2 The model
Motivated by the drawbacks of MSSM, strong CP problem, the origin of left-handed neu-
trino masses and µ parameter, we extend the MSSM to the following, assuming SUSY-
breaking is mediated by gravitationally suppressed interaction:2,3
W = λuQHuu + λdQHdd + λeLHde + λνLHuν + λµΦ2HuHd
MPl
+1
2λΦΦν2 (2.1)
with an assumption
m2L + m2
Hu< 0 (2.2)
where m2L and m2
Huare respectively the soft mass-squared parameters of L and Hu around
electroweak scale. In eq. (2.1), indices for gauge group and family structure has been omit-
ted for simplicity, MPl = 2.4×1018 GeV is the reduced Planck mass, Φ = φ+√
2θa+ · · · is
a gauge singlet chiral superfield and ν is the right-handed neutrino superfield. Various new
Yukawa couplings are constrained by low-energy physics which will be addressed shortly.
Note that the model has an accidental U(1) symmetry. Normalized such that Φ has charge
1One may think that Affleck-Dine baryogenesis in a scenario of gauge-mediated SUSY-breaking can
work well even in the presence of thermal inflation [16]. However, the formation of Q-balls makes it difficult
to work [17].2In the scenario of gravity-mediated SUSY-breaking, the symmetry breaking scale of thermal inflation
required to resolve moduli problem coincides accidentally with that of Peccei-Quinn symmetry. Hence, in
the scope of this paper, it is natural to consider such a mediation scenario as the framework of our argument.
We refer the reader to ref. [29] for the case of mirage-mediation of SUSY-breaking [30, 31], and will discuss
the cases of other mediation scenarios, for example gauge-mediation [32–37], in other place.3For the MSSM µ-term, we can use a renormalizable interaction λµΦHuHd instead of the non-
renormalizable interaction λµΦ2HuHd/MPl which we are using in this paper. Our model then involves
only renormalizable interactions though λµ ∼ 10−9 is hierarchically small. It may be still plausible to
consider such a hierarchically small Yukawa coupling, since Yukawa couplings of MSSM already have a
hierarchy. Compared to the case of non-renormalizable interaction for µ-term, the physics is barely affected
and can be applied to most of known mediation mechanisms of SUSY-breaking except gauge mediation.
– 2 –
JHEP07(2010)085
one, the charges assigned to fields are as follows.
U(1)
Q,L,Hu,Hd, u, d, e,Φ, ν
= 1, 3/2,−1,−1, 0, 0,−1/2, 1,−1/2 (2.3)
The key feature of our model is that if λΦ ∼ O(1) the associated Yukawa interaction
can drive the soft mass-squared parameters of φ and right-handed sneutrino (denoted as
ν except cases of confusion) to be negative through renormalization group running.4 If ν
develops vacuum expectation value (vev) before φ does, our model becomes inconsistent
with low energy phenomenology since MSSM µ-term is not reproduced. In order to avoid
this disaster, we assume at Planck scale
m2φ ≪ m2
ν + |AΦ|2 , m2ν & |AΦ|2 (2.4)
where m2φ and m2
ν are respectively the soft mass-squared parameters of φ and ν, and
AΦ is the A-parameter of the trilinear coupling associated with λΦ. In this case, only φ
can develop non-zero vev and the spontaneously broken accidental U(1) symmetry can be
identified as the Peccei-Quinn (PQ) symmetry for DFSZ axion [24, 25].
The potential along φ is of the form
V (|φ|) ≃ V0 + m2φ(Q = |λΦφ|)|φ|2 (2.5)
where m2φ(Q) is the running soft mass-squared of φ evaluated at a renormalization scale
Q. Now
dV
d|φ| =
[(
2 +d
d ln |φ|
)
m2φ(|φ|)
]
|φ| (2.6)
d2V
d|φ|2 =
[(
1 +d
d ln |φ|
)(
2 +d
d ln |φ|
)
m2φ(|φ|)
]
(2.7)
and the renormalization group equation of m2φ is
dm2φ
d ln Q=
1
8π2|λΦ|2
(
m2φ + m2
ν + |AΦ|2)
(2.8)
hence at vacuum where dV/d|φ| = 0 we find
mφ(φ0) = − 1
16π2|λΦ|2
(
m2ν + |AΦ|2
)
(2.9)
where the parameters at right-hand side are evaluated at Q = |λΦφ0| and φ0 is the vacuum
position of φ. Since the renormalization group runnings of λΦ, m2ν and |AΦ|2 are rather
slow as shown in appendix, from eq. (2.8) we may approximate m2φ(φ0) crudely as
m2φ(φ0) ∼ m2
φ(MPl) +1
8π2|λΦ|2
(
m2ν + |AΦ|2
)
ln|λΦφ0|MPl
(2.10)
Then the vacuum value φ0, which is assumed to be real without loss of generality, is
φ0 ∼ MPl
λΦ
exp
[
−1
2− 1
α
]
(2.11)
4See appendix for renormalization group equations of relevant parameters.
– 3 –
JHEP07(2010)085
where
α ≡ |λΦ|28π2
m2ν + |AΦ|2
m2φ(MPl)
(2.12)
and requiring zero cosmological constant, one finds
V0 =1
2αm2
φ(MPl)φ20 (2.13)
Decomposing φ as φ =(
φ0 + s/√
2)
exp[
ia/(√
2φ0
)]
, the physical mass of saxion (s),
denoted as mPQ, is
m2PQ ≡ 1
2
d2V
d|φ|2∣
∣
∣
∣
φ0
≃dm2
φ
d ln |φ|
∣
∣
∣
∣
∣
φ0
≃ 1
8π2|λΦ|2
(
m2ν + |AΦ|2
)
(2.14)
The masse of axon (a) at zero temperature is [38]
ma ∼ 6 × 10−5 eV
(
1012 GeV
fa
)
(2.15)
where the axion coupling constant fa is defined as
fa ≡√
2φ0
N(2.16)
with N = 6 the coefficient of U(1)PQ-QCD anomaly in our model in normalization of unit
PQ charge for PQ field (φ). fa is lower-bounded by the cooling rate of SN 1987A [39–43]
and upper-bounded by over-closure limit of cold axions from misalignment and strings [38].
If there is no dilution after axion condensation from misalignment is formed, currently al-
lowed window of fa is given by
109 GeV . fa . 1012 GeV (2.17)
The masse of axino (a) is generated at 1-loop in our model. It is given by [44]
ma =1
16π2λ2
ΦAΦ ≃ 0.6GeV λ2Φ
(
AΦ
100GeV
)
(2.18)
hence axino is the lightest supersymmetric particle(LSP) in our model.
The large vev of φ, implied in eq. (2.17), also makes ν become very heavy due to the
Yukawa coupling associated with λΦ in eq. (2.1), and integrating out ν in eq. (2.1) gives
an effective low-energy superpotential which contains a left-handed neutrino mass term:
Weff = λuQHuu + λdQHdd + λeLHde + λµφ2
0HuHd
MPl
− 1
2
λ2ν (LHu)2
λΦφ0
(2.19)
In order to provide right size of masses to left-handed neutrinos, we need
λΦ =λ2
νv2 sin2 β
mνLφ0
≃ 0.33
(
λν
10−2
)2(10−2 eV
mνL
)(
1012 GeV
φ0
)
sin2 β (2.20)
– 4 –
JHEP07(2010)085
where mνLis the left-handed neutrino mass, v = 174GeV is the vev of Higgs field and
sin β ≡ vu/v with vu the vev of up-type Higgs field (Hu). Note that in eq. (2.19) MSSM µ
parameter is given by
µ = λµφ2
0
MPl
= 4.2TeV
(
λµ
10−2
)(
φ0
1012 GeV
)2
(2.21)
Note also that using eq. (2.20), we can re-express eq. (2.11) as
mφ(MPl)√
m2ν + |AΦ|2
≃ λΦ
2√
2π
√
−1
2− ln
λΦφ0
MPl
(2.22)
≃ 0.14
(
λν
10−2
)2(10−2 eV
mνL
)(
1012 GeV
φ0
)
sin2 β (2.23)
which provides a constraint on mass parameters.
As described before this, introduction of a gauge singlet chiral superfield allows natural
realization of the PQ-symmetry to resolve the strong CP problem, and explains the origin
of small mass of left-handed neutrino and MSSM µ-parameter.
Our model is similar to the models of refs. [45–47] studied in refs. [48–52] and those
of refs. [53–56], but it differs from those models mainly by how symmetry breaking field
is stabilized and/or how a right-handed neutrino obtains large mass for a see-saw mech-
anism. In our model, instead of non-renormalizable higher order operator, the effect of
field-dependent running of a mass-squared parameter is used to stabilize the symmetry
breaking field whose vacuum expectation value provides a large mass to right-handed neu-
trinos. Although it may be thought of a rather simple variation of the models, the difference
makes our model simpler and maybe more natural with rather different cosmology which
will be described in the next section.
3 Cosmology
As our ansatz, we assume only LHu, HuHd flat directions [57, 58] and φ develop non-
zero field values temporarily or eventually.5 This will be justified shortly in subsequent
arguments. Parametrizing LHu and HuHd flat directions as
L = (0, l)T , Hu = (hu, 0)T , Hd = (0, hd)T (3.1)
with the remaining D-term constraint
D = |hu|2 − |hd|2 − |l|2 = 0 (3.2)
we find a potential from eqs. (2.1) and (2.5)
V = m2L|l|2 + m2
Hu|hu|2 + m2
Hd|hd|2 + mφ(|φ|)|φ|2 (3.3)
+
(
Aµλµφ2
MPl
huhd + c.c.
)
(3.4)
5Although HuHd may be stable near the origin as we are assuming here, dynamics of fields can lead
non-zero field value of the flat direction.
– 5 –
JHEP07(2010)085
+
∣
∣
∣
∣
λµφ2
MPl
hu
∣
∣
∣
∣
2
+
∣
∣
∣
∣
λµφ2
MPl
hd
∣
∣
∣
∣
2
+
∣
∣
∣
∣
2λµφ
MPl
huhd
∣
∣
∣
∣
2
+ |λν lhu|2 (3.5)
+1
2g2(
|hu|2 − |hd|2 − |l|2)
(3.6)
where terms in the first to forth lines are soft SUSY-breaking mass terms, A-terms, F -
terms and D-terms, respectively, and g2 = (g21 + g2
2)/4. Based on this potential, in this
section we will describe cosmology of our model including thermal inflation which may be
indispensable to resolve moduli problem, baryogenesis and dark matter.
3.1 Moduli problem and thermal inflation
In string/M theories, that might be candidates of the fundamental theory, there are flat
directions called moduli. A modulus (ϕ) has interactions suppressed by Planck scale, and
starts to oscillate coherently when expansion rate becomes comparable to the mass of the
modulus, mϕ. We assume that radiation is dominant at this time, then the abundance of
the moduli (called Big-Bang moduli) is
(nϕ
s
)
BB∼(
MPl
mϕ
)1/2
≃ 4.5 × 107
(
1TeV
mϕ
)1/2
(3.7)
where nϕ is the number density of moduli and s is the entropy density. The decay rate of
moduli is given by
Γϕ = γϕ1
8π
m3ϕ
M2Pl
≃ 6.9 × 10−30 GeV γϕ
( mϕ
1TeV
)3
(3.8)
where γϕ ∼ O(1) is a numerical coefficient, hence moduli decay after BBN. In order for
successful BBN, the abundance of moduli when they decay is constrained to be [59]
nϕ
s.
10−14 GeV
mϕ≃ 10−17
(
1TeV
mϕ
)
(3.9)
Therefore, we need a dilution more than O(1024).
Thermal inflation is the most compelling solution to this moduli problem. Moduli may
start to oscillate coherently while φ is held around the origin due to the interaction with
ν which is in very hot thermal bath. Thermal inflation begins when the potential energy
density at the origin becomes dominant over the energy density of moduli at a temperature
given by
Tb ∼ ρ1/4ϕ
(
V0
ρϕ
)1/3
∼(
V 20
mϕMPl
)1/6
(3.10)
where we have used ρϕ ∼ m2ϕM2
Pl as the initial energy density of moduli coherent oscilla-
tion. Thermal inflation ends as φ starts to roll out when temperature drops to the critical
temperature
Tc ∼√
|m2φ(|φ| ∼ 0)| ∼
∣
∣
∣
∣
lnmsoft
λΦφ0
∣
∣
∣
∣
1/2
mPQ (3.11)
where msoft ∼ 1TeV is the scale of soft SUSY-breaking.
– 6 –
JHEP07(2010)085
The coherent oscillation of φ after thermal inflation eventually decays with rate
Γφ ≃ Γφ→aa + Γφ→SM (3.12)
where
Γφ→aa =1
64π
m3PQ
φ20
(3.13)
is the partial decay rates of φ to axions. Γφ→SM is the partial decay rates of φ to SM
particles. It is mainly due to the mixing between φ and Higgs fields, induced by the term
λµΦ2HuHd/MPl. Since mPQ ∼ O(10 − 100)GeV for λΦ ∼ O(1) from eq. (2.14), Γφ→SM is
expected to be dominated by the decay to bottom quarks and given by [56]
Γφ→SM ≃ 1
4π
∣
∣
∣
∣
m2A − |B|2m2
A
∣
∣
∣
∣
2( |µ|4mPQφ2
0
)
[
f
(
m2h
m2PQ
)]
(3.14)
where mA is the mass of CP-odd neutral Higgs particle, B = Aµ in our model,
f(x) =εx
(1 − x)2
(
1 − εx
3
)3
2
(3.15)
and
ε ∼ 12m2b
m2h
∼ 0.02 (3.16)
with mb and mh the masses of bottom quark and light neutral Higgs particle, respectively.
Successful BBN limits energy contributions of relativistic non-SM particles, so it requires
Γφ→aa/Γφ→SM . 0.3 [56] which is easily satisfied for mPQ ∼ O(10 − 100)GeV. From
eqs. (3.13) and (3.14), the decay temperature of flaton (φ) is given by
Td ≡(
π2
15g∗(Td)
)−1/4
(Γφ→SMΓφ)1/4 M1/2
Pl (3.17)
≃(
5
8π4 g∗(Td)
)1
4
∣
∣
∣
∣
1 − |B|2m2
A
∣
∣
∣
∣
|µ|2
m1/2
PQφ0
[
f
(
m2h
m2PQ
)]1
2
(3.18)
≃ 26GeV
∣
∣
∣
∣
1 − |B|2m2
A
∣
∣
∣
∣
(
1012 GeV
φ0
)( |µ|1TeV
)2(30GeV
mPQ
)1
2
[
f
(
m2h
m2PQ
)]1
2
(3.19)
where we have used g∗(Td) = 100 in the last line. Figure 1 shows Td as a function of
mPQ/mh.
The e-foldings of thermal inflation is, from eqs. (3.10) and (3.11),
Nφ = lnTb
Tc
(3.20)
∼ 7.6 +1
6ln
[
(
φ0
1012 GeV
)4(30GeV
mPQ
)2(1TeV
mϕ
)
]
− 1
2ln
∣
∣
∣
∣
ln( msoft
1TeV
)
− ln
( |λΦφ0|1012 GeV
)∣
∣
∣
∣
(3.21)
– 7 –
JHEP07(2010)085
10-1
100
101
102
100
Td
/ GeV
mPQ / mh
Figure 1. Flaton decay temperature versus flaton mass: Td versus mPQ/mh for φ0 = 1012 GeV,
|µ| = 103 GeV, mh = 125 GeV and mA = 2|B|.
and the dilution factor due to the entropy release in the decay of φ is
∆φ ∼ V0
T 3c Td
(3.22)
∼ 2 × 1021
(
φ0
1012 GeV
)2(30GeV
mPQ
)(
10−1 GeV
Td
)∣
∣
∣
∣
ln
(
109msoft
|λΦφ0|
)∣
∣
∣
∣
−3/2
(3.23)
This is little bit small to dilute out moduli produced before thermal inflation to a safe level.
Fortunately, there is additional source of dilution. We assume
Tc < TLHu (3.24)
with
T 2LHu
∼ m2LHu
≡ −1
2
(
m2L + m2
Hu
)
(3.25)
Then, as temperature drops below TLHu , LHu which is expected to be held near the origin
at high temperature rolls out from the origin before flaton is destabilized. It is stabilized
by the F -term potential, |λν lhu|2 in eq. (3.5) at
|l∗| ≃ |hu∗| ∼ mLHu/|λν | (3.26)
The initial energy density of the coherent oscillation of LHu is
VLHu ∼ m2LHu
|l∗|2 ∼ m4LHu
/|λν |2 (3.27)
If this energy can be quickly dumped into radiation we would get an additional dilution to
eq. (3.22).
– 8 –
JHEP07(2010)085
The condensation of LHu decays initially through prheating [60–62] and eventually
through perturbative decays. For λν & 10−2, the perturbative decay of the condensation
is dominated by the coupling λν with a partial decay rate given by6
ΓLHu ∼ 1
8π|λν |2mLHu (3.28)
Using eqs. (2.13), (2.12) and (2.20), one finds
ΓLHu
HTI
∼ 280
sin2 β
( mνL
10−2 eV
)
(
mLHu
mν
)
(3.29)
where HTI is the Hubble expansion rate during thermal inflation, hence the coherent os-
cillation of LHu decays within a Hubble time.
It is expected that preheating is at best dominant for modes k ∼ mLHu and order one
fractional energy density of LHu is still in the form of condensation, hence the Universe is
slightly reheated to a temperature
T∗ ∼ V1/4
LHu∼ |λν |−1/2mLHu (3.30)
before flaton is destabilized. Thermal inflation is then extended by e-foldings given by
NLHu = lnT∗
TLHu
∼ −1
2ln |λν | (3.31)
and, using eq. (3.30), the dilution factor due to the entropy release in the decay of LHu is
∆LHu ∼V
3/4
LHu
m3LHu
∼ |λν |−3/2 (3.32)
Therefore, from eqs. (3.7), (3.22), (3.32) and (2.20) the abundance of Big-Bang moduli
when they decay is
(n/s)BB
∆LHu∆φ∼(
MPl
mϕ
)1/2 T 3c Td
V0
|λν |3/2
∼ 2 × 10−17
(
1TeV
mϕ
)1/2( |λν |10−2
)3/2
(3.33)
×(
1012 GeV
φ0
)2( mPQ
30GeV
)
(
Td
10−1 GeV
)∣
∣
∣
∣
ln
(
109msoft
|λΦφ0|
)∣
∣
∣
∣
3/2
(3.34)
which can manage to satisfy the bound from observation in eq. (3.9). Thus, Big-Bang
moduli can be diluted out to a safe level.
Moduli are also produced after thermal inflation due to the finite energy density of
thermal inflation. The abundance when they are produced is
(nϕ
s
)
TI∼ V 2
0
T 3c m3
ϕM2Pl
(3.35)
6If a coupling of LHu to MSSM particles is larger than λν , the particles becomes heavier than LHu,
hence the decay of LHu is kinematically forbidden.
– 9 –
JHEP07(2010)085
Using eqs. (3.22) and (2.13), the abundance when they decay is
(nϕ/s)TI
∆φ∼ V0Td
m3ϕM2
Pl
(3.36)
∼ 8 × 10−21( mPQ
30GeV
)2(
φ0
1012 GeV
)2( Td
100MeV
)(
1TeV
mϕ
)3
(3.37)
Compared to Big-Bang moduli of eq. (3.34), it is negligible, hence it is harmless as long as
Big-Bang moduli is diluted to a safe level.
Apart from the dilution of unwanted relics, thermal inflation has a very interesting
aspect. It wipes out pre-existing gravitational wave backgrounds and generates its own one
which may be detected at BBO or DECIGO type experiment [63]. Since the peak frequency
and amplitude depend on the details of model parameters and physics of phase transition,
careful investigation may be necessary to see if the wave in our model is really detectable.
3.2 Baryogenesis
As φ rolls out below Tc, thermal inflation ends and ν becomes very heavy. By integrating
out ν in eq. (2.1), one finds an effective superpotential
W = λuQHuu + λdQHdd + λeLHde + λµφ2HuHd
MPl
− 1
2
λ2ν (LHu)2
λΦφ(3.38)
which gives a potential
V = m2L|l|2 + m2
Hu|hu|2 + m2
Hd|hd|2 + m2
φ(φ)|φ|2 (3.39)
+
(
Aµλµφ2huhd
MPl
− 1
2Aν
λ2ν (lhu)2
λΦφ+ c.c.
)
(3.40)
+
∣
∣
∣
∣
λµφ2hd
MPl
− λ2ν l
2hu
λΦφ
∣
∣
∣
∣
+
∣
∣
∣
∣
λµφ2hu
MPl
∣
∣
∣
∣
+
∣
∣
∣
∣
2λµφhuhd
MPl
+1
2
λ2ν l2h2
u
λΦφ2
∣
∣
∣
∣
(3.41)
+1
2g2(
|hu|2 − |hd|2 − |l|2)2
(3.42)
for φ ≫ msoft.
As φ becomes large, LHu is shifted to a larger value
|l|2 ≃ |hu|2 ≃|λΦφ|
(√
12m2LHu
+ |Aν |2 + |Aν |)
6 |λ2ν |
(3.43)
For msoft ≪ φ ≪ φ0, the phase of LHu is determined by the Aν -term in eq. (3.40). As
φ reaches its vev, MSSM µ-parameter is reproduced, hence LHu is lifted up and brought
back into the origin.7 Simultaneously, HuHd becomes large8 due to the cross term of the
7In order to avoid possible local minimum along LHu, we need m2
L + m2
Hu+ |µ|2 > |Aν |
2/6, otherwise
LHu may be trapped there that is disastrous.8Large HuHd holds dangerous quark and lepton flat directions near the origin, hence considering only
LHu, HuHd and φ is justified.
– 10 –
JHEP07(2010)085
third term in eq. (3.41), and the cross term of the first term in the same equation provides
angular kick to LHu. This is nothing but an Affleck-Dine (AD) mechanism [64–66] to
generate charge asymmetry.
The generated lepton asymmetry is expected to be conserved due to the rapid pre-
heating of the AD fields [56]. The oscillating AD fields decay earlier than flaton, reheat
the Universe partially and activate sphaleron process [67] in which lepton asymmetry is
converted to baryon asymmetry. Eventually, flaton decays and reheats the Universe in
order for successful BBN. There is huge amount of entropy release in the flaton decay. The
baryon asymmetry at present is then given by
nB
s∼ nB
nφ
Td
mPQ
∼ nL
nAD
nAD
nφ
Td
mPQ
∼ nL
nAD
mLHu
mPQ
( |l0|φ0
)2 Td
mPQ
(3.44)
where nφ, nL and nAD are number densities of φ, lepton asymmetry and AD field respec-
tively, and l0 is the value of l when φ reaches φ0 from the origin. From the experience of a
similar model of ref. [56], we expect
nL
nAD
∼ O(10−3 to 10−2) (3.45)
and from eqs. (3.43) and (2.20)
|l0| ∼ 109 GeV
√
(mLHu
1TeV
)
(
10−2 eV
mνL
)
(3.46)
hence we expect
nB
s∼ 10−10
(
nL/nAD
10−3
)
×(mLHu
1TeV
)2(
30GeV
mPQ
)2(10−2 eV
mνL
)(
1012 GeV
φ0
)2(Td
100MeV
)
(3.47)
Note that the model parameters to resolve moduli problem give right order of the baryon
asymmetry which can match the present observation.
3.3 Dark matter
In our model, axinos and axions are expected to be the main components of dark matter
at present. These dark matter components can be cold, warm or even hot, depending on
their masses and how they are produced. In this subsection, we will derive cosmological
constraints from the dark matter.
Axinos produced in the flaton decay. As the LSP, axinos are produced mainly in
the decay of flaton with a rate given by [56]
Γφ→aa =α2
am2amφ
32πφ20
(3.48)
– 11 –
JHEP07(2010)085
where αa ≡ (d ln ma) / (d ln |φ|). In our model, from eqs. (2.18), (A.4) and (A.5),
αa ≃ 5
8π2|λΦ|2 (3.49)
The axino number density from the flaton decay is
na =2Γφ→aa
mφa(t)3
∫ t
0
a(t′)3ρφdt′ (3.50)
where a is the scale factor and ρφ is the energy density of φ. Using eqs. (3.17) and (3.50),
one find [56]na
s=
2.2TdΓφ→aa
mφΓSM
(3.51)
From eqs. (3.17) and (3.48), the current axino abundance is
Ωa ≃ 5.6 × 108( ma
1GeV
) na
s(3.52)
≃ 0.36Γ
1/2
φ
Γ1/2
SM
(
10
g1/2∗ (Td)
)
( αa
10−1
)2 ( ma
1GeV
)3(
100MeV
Td
)(
1012 GeV
φ0
)2
(3.53)
Therefore Ωa ≤ ΩCDM ≃ 0.25 requires
ma . 0.9GeV
Γ1/2
SM
Γ1/2
φ
(
g1/2∗ (Td)
10
)
(
Td
100MeV
)(
10−1
αa
)2(φ0
1012 GeV
)2
1
3
(3.54)
which can be easily satisfied for λΦ . 1 from eq. (2.18).
Axinos in the decay of thermally generated NLSP. If the next-to-lightest super-
symmetric particle (NLSP) denoted as χ is of neutralino type, the coupling λµφ2HuHd/MPl
in eq. (2.1) leads to the decay of χ to axino via various channels with rate [68, 69]
Γχ→a =Ca
16π
m3χ
ϕ20
, (3.55)
where Ca ∼ 1 may contain a factor of m2Z/m2
χ and we have neglected the masses of decay
products.
The thermal bath generates χ with the number density
nχ =1
π2
∫ ∞
0
dkk2
exp
√
k2+m2χ
T 2 + 1
, (3.56)
and they subsequently decay into axinos. The late time axino abundance is thus esti-
mated as
na =Γχ→a
a(t)3
∫ t
0
a(t′)3nχ(t′)dt′ (3.57)
– 12 –
JHEP07(2010)085
whose numerical solution results in [56]
na
s= g
−1/4∗ (Td)g
−5/4∗ (Tχ)
(
Γφ→SM
Γφ
)1/2 Γχ→aMPl
m2χ
Fa(x) (3.58)
where Fa(x) is approximated as
Fa(x) ∼
5.3x7 for x ≪ 1
5.4 for x ≫ 1(3.59)
with
x =2
3
(
g∗(Td)
g∗(Tχ)
)1/4 Td
Tχ(3.60)
and Tχ ≃ 2mχ/21 being the temperature at which the axino production rate is maximized.
From (3.55) and (3.58), the current abundance of axino dark matter is obtained
Ωa ≃ 5.6 × 108( ma
1GeV
) na
s(3.61)
≃ 2.7Ca
(
103
g1/4∗ (Td)g
5/4∗ (Tχ)
)
( mχ
102GeV
)( ma
1GeV
)
(
1012GeV
φ0
)2
Fa(x) (3.62)
where we have used Γφ ≃ Γφ→SM. Therefore, for x ≪ 1, one finds
Ωa ≃ 1.2 × 107 Ca
(
Td
mχ
)7(103g3/2∗ (Td)
g3∗(Tχ)
)
( mχ
102GeV
)( ma
1GeV
)
(
1012GeV
φ0
)2
(3.63)
which does not exceed ΩCDM if the flaton decay temperature satisfies
Td . 7.5GeV( mχ
100GeV
)
(
g∗(Tχ)
g∗(Td)
)1/4
×[
C−1a
(
g1/4∗ (Td)g
5/4∗
103
)(
102GeV
mχ
)(
1GeV
ma
)(
φ0
1012GeV
)2]1/7
(3.64)
Note that in order to resolve moduli problem (see eq. (3.34)) and explain baryon asymme-
try of the Universe (see eq. (3.47)) we need Td ∼ O(100)MeV, and in that case eq. (3.64)
is satisfied automatically. Since Td is expected to be much less than the freeze-out temper-
ature of χ, which is about mχ/20 [38], in order to avoid direct production of χ from the
flaton decay, we may require mφ < 2mχ, which seems to be typical in our model.
Axinos will also be produced by the decay of χ after they freeze out. However, the
standard Big-Bang neutralino freeze-out abundance is good match to the dark matter abun-
dance, our freeze-out abundance of χ will typically be less than the standard abundance,
and ma ≪ mχ, therefore the axino abundance generated after the freeze-out should be safe.
– 13 –
JHEP07(2010)085
Axions from misalignment and strings. The current abundance of axions produced
in these ways is given by [38]
Ωa =0.56
∆a
(
fa
1012 GeV
)1.167
(3.65)
where ∆a is the dilution factor of axion misalignment due to the late time entropy release
of thermal inflation. For Td ≪ 1.3GeV [56],
∆a ∼(
1.3GeV
Td
)1.96
(3.66)
hence for Ωcolda ≤ ΩCDM ≃ 0.25 we need
Td . 2.0GeV
(
1012 GeV
φ0
)1.167/1.96
(3.67)
Since we expect Td . O(100)MeV, these axions are likely to be a sub-dominant contribu-
tion to cold dark matter at present.
Axions in the flaton decay. Axion can be warm or hot when they are produced in
the decay of flaton. The constraint on hot dark matter comes from CMBR and structure
formation [70–73]. Currently allowed hot dark matter fractional contribution to the present
critical density is ΩHDM . 10−2 [38]. The constraint may be more stringent as suggested
from the analysis of the early re-ionization of the Universe at high redshift [74]. Taking
into account the recent analysis of WMAP 5-year data [75], the allowed warm/hot dark
matter fractional contribution to the present critical density is likely to be
ΩWHDM . 10−3. (3.68)
We will take this as the upper bound on the fractional energy density of our warm/hot
dark matter.
Axions produced by the flaton decay have a current momentum
pa =a
a0
mφ
2, (3.69)
where a is the scale factor at the time they were created and a0 is the scale factor now.
Whereas, the current momentum of an axion produced at td = Γ−1φ is
pd =ad
a0
mφ
2(3.70)
=S
1/3
d g1/3
∗S (T0)T0
S1/3
f g1/3
∗S (Td)Td
mφ
2(3.71)
≃ 2.06 × 10−2 eV
(
10
g∗(Td)
)1/3(100MeV
Td
)
( mφ
30GeV
)
(3.72)
– 14 –
JHEP07(2010)085
where Sd and Sf are respectively the total entropy at decay time td and present, so it would
be highly relativistic now. The current number density spectrum is given by
padnhot
a
dpa=
(
a
a0
)3 2ρφ
mφ
Γφ→aa
H=
16p3d
m4φ
Γφ→aap3a
Hp3d
ρφ, (3.73)
which may provide an observational test of our model in the future. The energy density of
the axions is
ρhota
ρSM
=g∗(Td)g
4/3
∗S (T )
g∗(T )g4/3
∗S (Td)
Γφ→aa
Γφ→SM
(3.74)
Therefore, assuming that the hot axions are still relativistic now, their current energy
density is estimated as9
Ωhota ≃ 4.3 × 10−5
(
10
g∗(Td)
)1
3 Γφ→aa
Γφ→SM
(3.75)
4 Conclusion
In this paper, we have proposed maybe the simplest extension of MSSM by introducing
just one additional gauge singlet chiral superfield and right-handed neutrino superfield (ν)
to MSSM without any ad-hoc mass parameter except soft SUSY-breaking ones:
W = λuQHuu + λdQHdd + λeLHde + λνLHuν + λµΦ2HuHd
MPl
+1
2λΦΦν2 (4.1)
with an assumption for the soft mas-squared parameters of L and Hu
m2L + m2
Hu< 0 (4.2)
and surely m2L + m2
Hu+ |µ|2 > 0 around electroweak scale. This model can realizes Peccei-
Quinn symmetry to resolve strong CP problem and explains the origin of left-handed
neutrino mass and µ-parameter of MSSM. In addition, the model realizes thermal inflation
followed by Affleck-Dine type leptogenesis and provides dark matter which matches well
to observation in a remarkably consistent way.
At low energy, the scalar component of the gauge singlet field, φ has negative mass-
squared around the origin due to strong Yukawa coupling to ν, hence develops non-zero
vacuum expectation value. In the absence of self-interactions, the field can be stabilized
radiatively around intermediate scale, so it can be identified as the Pecci-Quinn field which
breaks the Pecci-Quinn symmetry spontaneously. The vacuum expectation value of φ make
ν heavy, hence integrating out ν at low energy generates the mass term of left-handed
neutrino through seesaw mechanism. The vev of φ also generates MSSM µ-parameter.
Cosmologically, this model gives rise to thermal inflation while φ is held near the origin
due to large thermal effect at high temperature. Notably, the non-trivial assumption of
9The energy density of thermally produced axions is Ωa ∼ ma/131eV, hence it will be subdominant [76].
– 15 –
JHEP07(2010)085
m2L + m2
Hu< 0 allows LHu flat direction to be destabilized before the end of thermal
inflation. This causes an extension of thermal inflation by few e-foldings. Although it is
small, the additional e-foldings are crucial to dilute out moduli to a safe level. Thermal
inflation ends as φ is destabilized at a critical temperature. As φ reaches its vacuum
expectation value, MSSM µ-parameter is reproduced, leading Affleck-Dine leptogenesis
involving LHu and HuHd flat directions. The oscillation of Affleck-Dine fields is expected
to be quickly damped due to a rapid preheating induced by oscillation and preheating of
flaton field φ, so the generated lepton asymmetry can be conserved.
The eventual decay of flaton reheats the Universe with a temperature Td ∼O(100)MeV, releasing huge amount of entropy. By its own, the entropy release in the
decay of flaton is not enough to resolve moduli problem. But the decay of LHu conden-
sation before the end of thermal inflation provides an additional dilution. As the result,
moduli can be dilute out to a safe level. Interestingly axion coupling constant and flaton
decay temperature are tightly constrained to be fa ∼ 1012 GeV and Td ∼ 100MeV, respec-
tively. Remarkably, model parameters in this case are likely to give naturally right amount
of baryon asymmetry at present.
The main component of dark matter in our model is expected to be the axino whose
mass is nearly fixed to be about 1GeV to match cold dark matter abundance at present.
Axions from misalignment and strings also contribute to cold dark matter, but they are sub-
dominant. Axions from the decay of flaton are sub-dominant too, but they are expected
to be hot and highly relativistic at present in our model. The current number density
spectrum of the hot axions may provide an observational test of our model in the future.
Acknowledgments
WIP thanks to Ewan D. Stewart for valuable comments. WIP is supported by the
KRF grant (KRF-2008-314-C00064), KOSEF grants (KOSEF 303-2007-2-C000164 and No.
2009-0077503) and Brain Korea 21 projects funded by the Korean Government.
A Renormalization group equations
The relevant part of superpotential of eq. (2.1) for the running of soft SUSY-breaking
parameters associated with φ and ν is
Wpart = λνLHuν +1
2λΦΦν2 (A.1)
Then, one finds renormalization group equations (RGEs)
dm2φ
d ln Q=
1
8π2|λΦ|2
(
m2φ + m2
ν + |AΦ|2)
(A.2)
dm2ν
d ln Q=
1
8π2
[
|λΦ|2(
m2φ + 2m2
ν + |AΦ|2)
+ 2|λν |2(
m2ν + m2
L + m2Hu
+ |Aν |2)]
(A.3)
– 16 –
JHEP07(2010)085
daΦ
d ln Q=
1
16π2
(
15
2aΦ|λΦ|2 + 4aνλΦλ∗
ν + 2aΦ|λν |2)
(A.4)
dλΦ
d ln Q=
1
16π2λΦ
(
5
2|λΦ|2 + 2|λν |2
)
(A.5)
where Q is the renormalization scale, m2i is the soft mass-squared of a scalar field i, and
ai ≡ Aiλi with Ai the A-parameter of the trilinear coupling associated with λi. From
eq. (A.5), eq. (A.4) can be rewritten in terms of |Aφ| as
d|AΦ|2d ln Q
=1
8π2
[
5|AΦ|2|λΦ|2 + 2 (AΦA∗ν + A∗
ΦAν) |λν |2]
(A.6)
If λΦ ≫ λν , which should be the case for consistency of theory as described in the
text, we can ignore all the contributions with λν . Then, eqs. (A.2), (A.3), (A.6) and (A.5)
can be approximated to
dm2φ
d ln Q=
1
8π2|λΦ|2
(
m2φ + m2
ν + |AΦ|2)
(A.7)
dm2ν
d ln Q=
1
8π2|λΦ|2
(
m2φ + 2m2
ν + |AΦ|2)
(A.8)
d|AΦ|2d ln Q
=5
8π2|λΦ|2|AΦ|2 (A.9)
d ln λΦ
d ln Q=
5
32π2|λΦ|2 (A.10)
and in order for only m2φ to be negative at low renormalization scale we need
m2φ ≪ m2
ν + |Aφ|2, m2ν & |AΦ|2 (A.11)
For example, for a potential only with the soft-mass term of φ, a numerical analysis shows
that input values
mν(MPl) = 250GeV , |AΦ(MPl)| = 150GeV , mφ(MPl) = 100GeV (A.12)
with |λΦ(MPl)| = 1 generates vacuum at φ0 ≃ 6 × 1012 GeV where
mν ≃ 210GeV , |AΦ| ≃ 110GeV ,√
−m2φ = 12GeV (A.13)
with |λΦ| ≃ 0.84 and mPQ ≃ 23GeV the physical mass of φ.
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