Table Of ContentUCB-PTH-15/18
IPMU16-0003
Cosmology with a Heavy Polonyi Field
6
1
0
2 Keisuke Harigayaa,b, Taku Hayakawac, Masahiro Kawasakic,d and Masaki Yamadac,d
n
a aDepartment of Physics, University of California, Berkeley, California 94720, USA
J
bTheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley,
9
California 94720, USA
] cInstitute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha,
h
p Kashiwa, Chiba 277-8582, Japan
-
p dKavli IPMU (WPI), UTIAS, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,
e
277-8583, Japan
h
[
1
v
Abstract
0
4
We consideracosmologically consistent scenariowithaheavy Polonyi field. The
1
2 Polonyi field with a mass of (100)TeV decays before the Big-Bang Nucleosynthe-
0 O
sis (BBN) and avoids the severe constraint from the BBN. However, the abundance
.
1 of the Lightest Supersymmetric Particle (LSP) produced from the decay often ex-
0
ceeds the observed dark matter density. In our scenario, the dark matter density
6
1 is obtained by the LSP abundance with an aid of entropy production, and baryon
: asymmetryisgenerated bytheAffleck-Dine mechanism. Weshowthattheobserved
v
i baryon-to-dark matter ratio of (0.1-1) is naturally explained in sequestering mod-
X O
els with a QCD axion.
r
a
1 Introduction
Cosmological observations have revealed the existence of the dark matter (DM) and the
baryon asymmetry, while their origins remain unknown for a long time. The cosmic densi-
ties of the DM and baryonic components have been precisely measured by the observation
of the Cosmic Microwave Background (CMB) radiation, and the observed baryon-to-DM
ratio is Ω /Ω = 0.18 [1]. We need a cosmologically consistent scenario explaining both
B DM
the baryon asymmetry and the DM abundance.
Since those origins cannot be explained in the framework of the Standard Model
(SM), there should be underlying physics beyond the SM. Supersymmetry (SUSY) [2]
is one of the most interesting models since it achieves the unification of the SM gauge
couplings and can relax the hierarchy problem. In addition, supersymmetric models with
a conserved R parity predict the stability of the Lightest Supersymmetric Particle (LSP),
which becomes a goodcandidate for DM. Moreover, SUSY extensions of the SM contain a
lotofflatdirectionswithB Lcharges[3], whichcanproduceB Lasymmetry [4]. Inthe
− −
early universe, one of the flat directions, which we call the Affleck-Dine field, may receive
an angular kick from SUSY breaking and R symmetry breaking effects and rotates in its
complex plane, which corresponds to the generation of the B L asymmetry. The B L
− −
asymmetry is converted into the baryon asymmetry through the sphaleron process [5, 6].
This mechanism, known as“theAffleck-Dine (AD)mechanism” [4,7], canproduce baryon
number more effectively thanmost baryogenesis scenarios. For these advantages, we focus
on supersymmetric theories in this paper.
TheSUSYmustbespontaneouslybrokensincesuperparticleshavenotbeendiscovered
yet. The simplest SUSY breaking model is the Polonyi model [8] in which an F-termof an
elementary singlet field Z breaks the SUSY in the hidden sector.1 This model is attractive
because of its simplicity. However, such a singlet field Z, called the Polonyi field, may
cause a cosmological difficulty [9]. Since the Polonyi potential has no symmetry enhanced
points,theminimumofthepotentialduringtheinflationisdeviatedfromthetruevacuum.
After inflation, the Polonyi field begins to oscillate around the true minimum with an
amplitude of the order of the Planck scale. The energy density of its coherent oscillation
immediately dominates the universe after the inflaton decays. Since the Polonyi field very
weakly interacts with particles in the observable sector, its late-time decay upsets the
standard scenario of the Big-Bang Nucleosynthesis (BBN), which is referred to as “the
Polonyi problem”.
1InthePolonyimodel,theSUSYbreakingscalecanbeeasilyobtainedbydynamicaltransmutation,by
assigningavanishingRchargetothePolonyifieldandbreakingtheRsymmetrybygauginocondensation.
1
There are mainly two ways of solving the problem. One possible way is to dilute
the Polonyi density by some mechanisms, for example, by thermal inflation [10]. In
this case, however, pre-existing baryon asymmetry is also diluted, and it is known to
be difficult to produce a sufficient amount of baryon number beforehand. Therefore,
baryon asymmetry should be produced after the dilution, and we need an intricate model
proposed in the context of the thermal inflation [11, 12, 13]. The other simple solution is
to make the Polonyi field heavy enough to decay before the BBN. Even in this case, we
need some dilution since the abundance of LSPs produced from the Polonyi decay exceeds
the observed DM abundance unless pair annihilation is very effective [14, 15]. When such
dilution is needed, the AD mechanism is the leading candidate for baryogenesis since most
baryogensis scenarios including the thermal leptogenesis [16] cannot produce a sufficient
amount of baryon asymmetry beforehand.
In this paper, we consider the case where the Polonyi field is heavy enough to decay
before the BBN, i.e., m (100)TeV, where m is a mass of the Polonyi field. We
Z Z
∼ O
construct a cosmologically consistent scenario in the presence of the heavy Polonyi field.
In our scenario, the baryon asymmetry is created by dynamics of the AD field which
takes the vacuum expectation value (VEV) of the order of the Planck scale in the early
universe. Both the Polonyi density and the baryon asymmetry are diluted by late-time
entropy production, for example, by the thermal inflation. After the entropy production,
the Polonyi field decays into LSPs which explain the observed DM abundance. We show
that the baryon-to-DM ratio is simply determined by the LSP mass and the branching
fraction of the decay of the Polonyi field into superparticles. The correct baryon-to-DM
ratio is obtained when the LSP mass is of (1)TeV and the branching fraction of the
O
decay of the Polonyi into superparticles is of (10 3).
−
O
In order to realize the branching fraction of (10 3), we consider a specific model
−
O
satisfying the following conditions. Firstly, we assume that the SUSY breaking sector
is sequestered from the visible sector in superspace Lagrangian, which is called “the
sequestering model” [17, 18]. The squarks/sleptons and gauginos have vanishing masses
at the tree level and acquire loop-suppressed masses through quantum corrections, such
as anomaly-mediated SUSY breaking effects [18, 19] (see also Refs. [20, 21]) or one loop
corrections from Planck-suppressed interactions [18, 22]. Sfermions and gauginos are
lighter than thegravitino witha mass of (100)TeV. In addition, we introduce a (pseudo-
O
)Nambu-Goldstoneboson(NGB). The NGBcanbe identified with theQCD axion[23, 24,
25, 26], which solves the strong CP problem [27, 28, 29]. In this model, the Polonyi field
mainly decays into NGBs, but their contribution to the DM abundance is negligible since
the NGB is much lighter than the LSP (as is the case with the QCD axion) or decays
2
into SM particles. On the other hand, the decay of the Polonyi into superparticles is
suppressed for the sequestered potential, and the branching ratio is of (10 3). The DM
−
O
abundance is determined by the LSP abundance produced through the Polonyi decay.
This paper is organized as follows. In Sec. 2, we briefly review the Polonyi problem
and explain why the entropy production is needed. We explain the AD mechanism in
Sec. 3. In Sec. 4, we show that the baryon-to-DM ratio is simply determined by the LSP
mass and the branching fraction. We also introduce the sequestering model asa candidate
to realize our scenario. The final section is devoted to summary and discussions.
2 The Polonyi Problem
In this section, we explain the cosmological problem of the Polonyi model.
Let us briefly introduce the Polonyi model [8]. In this model, the only ingredient in
the SUSY breaking sector is an elementary field Z which is singlet under any symmetry.
We call it as the Polonyi field. The superpotential in the hidden sector is given by2
Z
W = µ2M 1+c + , (1)
hid pl
M ···
(cid:18) pl (cid:19)
whereµisaparameter withmassdimension 1, andc isadimensionless parameter of (1).
O
Hereafter, we use M as the reduced Planck mass (M 2.4 1018GeV). Higher order
pl pl
≃ ×
terms are expressed by the ellipsis. Note that the parameter µ breaks the R symmetry
since Z has an R charge of 0. At the true minimum, the VEV of Z is assumed to be
of the order of the Planck scale. The parameter µ is related to the gravitino mass as
µ 2 W /M m M . The F-term of Z is given by F m M , which
hid pl 3/2 pl Z 3/2 pl
| | ≃ h| |i ≃ | | ≃
implies spontaneous SUSY breaking. The mass of the Polonyi field is of the order of m
3/2
for generic Ka¨hler potentials.
The Polonyi model is attractive because of its simplicity and has been studied exten-
sively so far. However, such a singlet field causes cosmological difficulties [9]. In the early
universe, non-zero vacuum energy in the inflaton sector largely breaks SUSY, which gives
the Hubble induced mass term for the Polonyi field Z [7, 30]. Since Z is singlet under any
symmetry, the minimum determined by the Hubble induced term deviates from the true
minimum. The deviation is generically expected to be of the order of the fundamental
energy scale of the supergravity, which we assume to be the Planck scale. Considering
these SUSY breaking effects, the Polonyi field evolves as follows. During the inflation,
the Polonyi field Z sits at the minimum determined by the Hubble induced term. After
2Hereafter, we use the same letter Z to denote the scalar component of the supermultiplet.
3
the end of the inflation, the Hubble parameter decreases to m , and then Z begins to
Z
oscillate around the true minimum with an amplitude of the order of the Planck scale.
The energy density of the oscillating field Z scales as a 3, where a is the scale factor, and
−
that of radiation scales as a 4. Thus, the oscillating Polonyi field immediately dominates
−
the universe after an inflaton decays into radiation.
When the Polonyi field starts its oscillation before the inflaton decays, the Polonyi
energy density-to-entropy ratio after the inflaton decay is given by
2
ρ T z
Z inf 0
, (2)
s ≃ 8 M
(cid:18) pl(cid:19)
where T denotes the temperature when the inflaton decays, s denotes the entropy den-
inf
sity, and z represents the oscillation amplitude of Z, which is expected to be of the order
0
of the Planck scale. With the decay rate of the Polonyi field,
d m3
Γ = Z Z , (3)
Z 8πM2
pl
the decay temperature is given by
1/4
90 m 3/2
1/2 Z
T = Γ M 4MeV d , (4)
Z π2g (T ) Z pl ≃ × Z 100TeV
(cid:18) Z (cid:19)
∗ p (cid:16) (cid:17)
where d is a numerical constant, and g (T ) is the effective number of degrees of freedom
Z Z
∗
at a temperature of T . Here, we used g (T ) = 10.75. One can find that if m is
Z Z Z
∗
smaller than (100)TeV, the Polonyi field decays during and after the BBN. In this case,
O
ρ /s . (10 14-10 11)GeV is required in order not to upset the success of the BBN [31].
Z − −
O
From Eq. (2), one can find that the constraint from the BBN cannot be avoided without
dilution even if T (10)MeV. The required dilution factor ∆ for ρ /s . 10 14GeV
inf Z −
≃ O
is as follows:
s a3 T
∆ f & 1.3 1022 inf , (5)
≡ s a3 × 109GeV
i (cid:18) (cid:19)
where s and s denote the entropy density after and before the entropy production,
f i
respectively. Such huge entropy production, however, also dilutes pre-existing baryon
asymmetry. We then need an intricate model in which the baryon asymmetry is produced
after the dilution [11, 12, 13].
There is another simple way to avoid the problem, which we focus on in this paper.3
When the Polonyi is as heavy as (100)TeV, it decays before the onset of the BBN,
O
3For other solutions, see Refs. [32, 33, 34, 35, 36, 37].
4
and the constraint becomes much milder [31]. Even in this case, however, there is an
incidental problem: LSPs are abundantly produced from the decay of the Polonyi, and
the LSP density tends to exceed the observed DM density. The abundance of the LSPs
is given by
n Γ N n (T )
LSP Z LSP Z Z
Y min , , (6)
LSP
≡ s ≃ σv s(T ) s(T )
(cid:20)h i Z Z (cid:21)
where n and n denote the number density of the LSP and the Polonyi field, respec-
LSP Z
tively. σv represents a thermally averaged cross section of the pair annihilation. N is
LSP
h i
the averaged number of superparticles produced by the decay of one Polonyi field. When
the first term is relevant, the pair annihilation between LSPs proceeds after the Polonyi
decay, and the relic LSP density is approximately proportional to T 1. For example,
Z−
when the neutral wino is the LSP with a mass of (0.1-1)TeV, the decay temperature
O
T is needed to be larger than (1-10)GeV [15] in order for the wino abundance not
Z
O
to exceed the observed DM density.4 This requires that m is larger than (5000)TeV.
Z
O
Assuming that the gravitino mass is generically comparable to m and that the wino
Z
mass is generated by the anomaly-mediation, such a heavy Polonyi mass is incompatible
with the wino mass of (0.1-1)TeV.
O
In this paper, we consider the case where m (100)TeV and the LSP density is
Z
≃ O
diluted by entropy production. As we will show, it is possible to generate the baryon
asymmetry before the dilution since the required dilution factor is much smaller than
Eq. (5) for m = O(100) TeV. The most probable candidate for baryogenesis is the
Z
Affleck-Dine mechanism because it cancreate huge baryon number to survive the dilution.
3 Affleck-Dine Mechanism without Superpotential
The Affleck-Dine mechanism [4] is a promising candidate for the baryogenesis in cosmo-
logical scenarios with dilution. In this section, we briefly explain the AD mechanism in
the case where the AD field does not appear in the superpotential. We show that the
resultant baryon number density is comparable to the number density of the Polonyi field.
The minimal SUSY Standard Model (MSSM) contains a lot of flat directions which
have no scalar potentials at the renormalizable level and in SUSY limit [3]. In the AD
mechanism, a flat direction with a B L charge creates the baryon asymmetry. We
−
call it “the Affleck-Dine field”. In the early universe, SUSY breaking effects and non-
renormalizable terms affect its evolution. In particular, A-term scalar potentials violating
4Reference [15] has taken into account Sommerfeld effect and coannihilation among charged and
neutral winos.
5
B L global symmetry rotate the AD field in the complex plane, and can effectively
−
generateB L number [7]. The B L asymmetry is converted into the baryonasymmetry
− −
through the sphaleron process [5, 6].
In order to generate huge baryon number comparable to the number density of the
Polonyi field, the AD field value at the onset of its oscillation should be of the order of
the Planck scale. To obtain the large field value, we assume that the AD field does not
appear in the superpotential.5 In this case, SUSY breaking effects including the A-terms
are provided by the Ka¨hler potential. We consider the following terms in the Ka¨hler
potential:6
K
= d2θd2θ¯ 3M2 exp
LAD − pl −3M2
Z " pl!#
Φn Φ 4
d2θd2θ¯ f Φ 2 + f +h.c. +f | | + , (7)
⊃ 1| | 2nMn 2 3M2 ···
Z " pl− ! pl #
where Φ denotes the AD field, and f (i = 1,2,3) is an arbitrary real function of Z and
i
Z† which satisfies fi = fi† for i = 1,3. From these terms, the potential for the AD field Φ
is given by
m2 Φ 4
V(Φ) = m2 Φ 2 3/2 (a Φn +h.c.)+c m2 | | + , (8)
φ| | − nMn 2 n 4 3/2M2 ···
pl− pl
where m and m denote the soft scalar mass of the AD field and the gravitino mass,
φ 3/2
respectively. a and c are (1) dimensionless parameters.
n 4
O
Aswe will discuss later, m isassumed to besmaller thanm . When m m and
φ 3/2 φ 3/2
≪
c = 0, there exists charge/color breaking minima smaller than the Planck scale because
4
of the relatively large A-terms [38]. The AD field needs to avoid dropping the minima
during its evolution, which makes the AD mechanism less effective [39, 40]. The quartic
term, however, lifts the potential near the Planck scale, and the global minima disappear
when its coefficient is positive.
In addition, the AD field acquires the so-called Hubble induced terms since the non-
zero vacuum energy in the early universe largely violates SUSY [7, 30]. We assume that
the AD field has a negative Hubble induced mass term so that it takes a large field value
at the onset of its oscillation:
V = c H2 Φ 2, (9)
H H
− | |
5For example, a U(1) symmetry can prohibit appearance of the AD field in the superpotential.
R
6Thesecondterminthesecondlineisequivalenttoasuperpotentialtermsuppressedbythegravitino
mass.
6
where H is the Hubble parameter, and c is a positive dimensionless parameter of (1).
H
O
Dueto this negative mass term, the AD field takes its field value of the order of the Planck
scale until H(t) m . When H(t) . m , the position of the local minimum of the AD
3/2 3/2
≃
field, which is determined by a balance between the negative Hubble induced mass term
and the positive quartic term, becomes smaller than the Planck scale. Since the position
is quickly driven towards the origin, the AD field cannot track the local minimum and
starts to roll down to the origin when H(t) m (for details, see appendix A and [41]).
3/2
≃
Let us estimate the produced baryon number density. The evolution equation for the
B L density is expressed as
−
∂V
n˙ +3Hn = 2βIm Φ , (10)
B L B L
− − ∂Φ
(cid:20) (cid:21)
where n expresses the B L density and β denotes the B L charge of the AD
B L
− − −
field. The dot denotes the time derivative. The right-hand side is the source of the
B L asymmetry. By solving the evolution equation, one can find that the asymmetry
−
is produced most effectively at the onset of the oscillation (H m ). After that, the
3/2
≃
AD field value decreases due to the expansion of the universe, and B L violating effects
−
become negligible. The produced B L density is then estimated as
−
m2 Φ n
3/2 osc
n (t ) 2β a sin[nθ +arg(a )] | |
B−L osc ∼ | n| i n Hosc Mpnl−2
ǫH Φ 2, (11)
osc osc
≡ | |
where the subscripts of show the values when the AD field starts to oscillate. θ is the
osc i
initial phase of the AD field. ǫ is estimated as
m2 Φ n 2
3/2 osc −
ǫ 2β a sin[nθ +arg(a )] | | . (12)
≃ | n| i n H2 Mn 2
osc pl−
As mentioned above, the AD field starts to roll down to the origin from the field value of
the order of the Planck scale when H(t) = H m . Therefore, the B L asymmetry
osc 3/2
≃ −
is estimated as n (t ) ǫm Φ 2, where ǫ (1) and Φ M .
B L osc 3/2 osc osc pl
− ≃ | | ≃ O | | ≃
After the oscillation, the AD field decays into radiation, and the conserved B L
−
asymmetry is converted into the baryon asymmetry through the sphaleron effect [5, 6].
The baryon asymmetry is related to the B L asymmetry as
−
8
n = n , (13)
B B L
23 −
where n expresses the baryon number density. As mentioned in Sec. 2, entropy pro-
B
duction is often necessary in order to avoid the overproduction of LSPs by the decay of
7
the Polonyi. We consider the case where the density of the Polonyi field is diluted by
the entropy production. Then, the baryon asymmetry is also diluted. Assuming that the
inflaton decays after the onset of the oscillation of the AD field, the yield of the baryon
number is estimated as
2
n 8 1 3T n 2 ǫ T Φ
B inf B L inf osc
YB = − | | , (14)
≡ s 23∆ 4ρ ≃ 23∆m M
inf (cid:12)osc 3/2 (cid:18) pl (cid:19)
(cid:12)
where ρ denotes the energy density of the(cid:12) oscillating inflaton. Note that the baryon
inf (cid:12)
number density is comparable to the density of the Polonyi field because both the AD
field and the Polonyi field simultaneously begin their oscillation with the same amplitude
of the order of the Planck scale.
Let us make a comment on Q-ball formation. When the potential for the AD field
is shallower than a quadratic potential, the AD field fragments into non-topological soli-
tons, called Q-balls [42], just after the onset of the oscillation [43, 44, 45]. Since Q-balls
absorb the produced B L charge, the formation of Q-balls could significantly affect the
−
estimation of the baryon asymmetry. In our scenario, the AD field value at the onset of
the oscillation is as large as the Planck scale. Then, the formed Q-balls may be too large
to decay before the BBN if Q-ball formation occurs, which renders the AD mechanism
ineffective. Hence, the beta function for the soft mass of the AD field may need to be
positive in order to prohibit the Q-ball formation. This requires the AD field to involve
scalar fields which have large Yukawa couplings.
4 Baryon-to-DM Ratio
In this section, we show that the baryon-to-DMratio is simply given by the LSP mass and
the branching fraction of the decay of the the Polonyi into superparticles in our scenario.
We also explain that our scenario is realized in the so-called sequestering model [17, 18]
with a (pseudo-)NGB, which can be identified with the QCD axion.
4.1 Scenario
Before we calculate the baryon-to-DMratio, let us summarize our scenario. When H(t)
≃
m m (100)TeV, both the Polonyi and the AD fields roll down to their origins
Z 3/2
≃ ≃ O
with the amplitudes of the order of the Planck scale. At that time, the AD field generates
the B L asymmetry which is later converted to the baryon asymmetry by the sphaleron
−
process. Then, the entropy production occurs and dilutes both the Polonyi density and
the baryon asymmetry. After the dilution, the Polonyi field decays into superparticles
8
whichconsequently decayintoLSPsbeforetheepochoftheBBN.Thus, theDMdensity is
determined by the abundance of the nonthermally produced LSPs, assuming that thermal
relic density of LSPs is negligible.
First, we estimate the DM abundance. From Eq. (2), the LSP-to-entropy ratio is
estimated as
2
ρ 2Br n Br T m z
LSP SUSY Z SUSY inf LSP 0
= m , (15)
LSP
s ∆ s ≃ ∆ 4m M
i Z (cid:18) pl(cid:19)
where Br denotes a branching fraction for the Polonyi decay into two superparticles,7
SUSY
and m denotes the LSP mass. The number of the produced superparticles is almost
LSP
equaltothatoftheLSPsduetotheR-parityconservation. Notethatthepairannihilation
between LSPs is not efficient since the Polonyi field decays after the dilution. We assume
that decay products other than the LSPs do not contribute to the DM abundance. Hence,
the DM abundance is obtained by the nonthermally produced LSPs.
Next, let us compare the DM abundance with the baryon asymmetry created by the
AD mechanism. The ratio of the density of the Polonyi field to the B L number remains
−
thesameaftertheybegintheir oscillationssincethedensities ofbothcomponentsdecrease
as a 3. From Eqs. (14) and (15), we obtain the following relation:
−
2
Ω 8 ǫ m m Φ
B p Z osc
= | |
Ω 23Br m m z
LSP SUSY LSP 3/2 (cid:18) 0 (cid:19)
10 3 1TeV Φ 2
− osc
0.33ǫ | | , (16)
≃ Br m z
(cid:18) SUSY(cid:19)(cid:18) LSP (cid:19)(cid:18) 0 (cid:19)
where m represents the proton mass (m 0.938GeV). Here, we assume m m .
p p Z 3/2
≃ ≃
Note that z and Φ are of the order of the Planck scale and that ǫ is of (1). One
0 osc
| | O
can find that the baryon-to-DM ratio is determined by the LSP mass and the branching
fraction of the decay of the Polonyi into superparticles. Assuming that the LSP mass is
of (1)TeV, Br is required to be of (10 3) in order to realize the observed value,
SUSY −
O O
Ω /Ω 0.18.
B DM
≃
Our scenario needs the entropy production (see appendix B for the realization by
thermal inflation). Let us estimate the required amount of the entropy production. The
ratio of the observed DM density to the entropy density is given by [1]
(obs)
ρ
DM 4.4 10 10GeV, (17)
−
s ≃ ×
0
7GravitinosarenotproducedfromthePolonyidecayassumingthatthe decayiskinematicallyforbid-
den (m <2m ). The abundance of gravitinosproduced during the reheatingbecomes negligible after
Z 3/2
the dilution.
9
