Baryon asymmetry and dark matter from soft leptogenesis

0.26 MB·English

by  Heidi Kuismanen

Baryon asymmetry and dark matter from soft
leptogenesis
Heidi Kuismanen Iiro Vilja
and
Department of Physics and Astronomy, University of Turku, 20014 Turku, Finland
3 January 4, 2013
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n Abstract
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3 The framework for soft leptogenesis minimally extended with a DM sector is studied.

A heavy singlet neutrino superfield acts as the source for (s)lepton asymmetry and by
]
h
coupling to the singlet DM superfield it produces a DM particle density through decays.
p
-
The nature of DM generated is twofold depending on whether the Yukawa and DM
p
e
couplings areeithersmallorlarge. Withsufficiently smallYukawa andDMcouplings DM
h
[ annihilations into MSSM particles are slow and as a consequence all DM particles form

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the DM component. The solutions to Boltzmann equations are given and the dependence
v
5 between the DM masses and coupling are presented in this weak coupling regime. Also,
3
3 the behavior of the efficiency of producing asymmetric DM is determined with weak
4
couplings. We note that a different outcome arises if the couplings are larger because
.
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1 then the ADM component is dominant due to the effectiveness of DM decays into the
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1 MSSM sector.
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X
1 Introduction
r
a
Our current understanding of the universe is that around 5 % of the energy density belongs to
ordinary baryonic matter while the rest stems from dark matter (DM) and dark energy with
proportions 22 % and 73 %, respectively [1]. The existence of all three components is more
or less a mystery and points towards physics beyond the Standard Model (SM). The fact that
there seems to be almost exclusively matter and no antimatter in the universe suggests there
is matter-antimatter asymmetry. Since the SM cannot accommodate sufficient baryogenesis
on its own, baryogenesis mechanisms have been constructed in various Grand Unified Theory
1
(GUT) scenarios. Alongside GUT models, baryogenesis via leptogenesis [2] has been studied
extensively over the last twenty years or so.
As for DM, a viable candidate arises in the Minimal Supersymmetric Standard Model
(MSSM), namely the Lightest Supersymmetric Particle (LSP) [3, 4]. Usually, the lightest
neutralino (i.e. a combination of the superpartners of the neutral gauge bosons and higgs
sparticles), or in supergravity theories the gravitino, assumes the role of the LSP, and the
success of the LSP in producing the correct DM abundance depends on the adjustment of
parameters within the MSSM. However, since the DM and baryonic energy densities are close
to each other, Ω /Ω 5, it is natural to assume that DM and baryogenesis are somehow
DM b
∼
related. The connection between baryogenesis and DM has also been widely explored [5].
Moreover, the possibility of the DM energy density being the product of an asymmetry has
led to the concept of asymmetric DM (ADM) where DM and DM antiparticles are distinct
from another and produce an asymmetry analogous to baryons. Note, however, that as ADM
scenarios incorporate MSSM, also LSP is always included in the ADM models but is not
necessarily good candidate for DM.
Over the years various realizations of leptogenesis have been devised [6]-[28]. They are
usually motivated by SO(10) scenarios where B L is a gauge symmetry, and it is natural to
−
have a heavy right-handed SM singlet neutrino that can also be responsible for light neutrino
masses via the seesaw mechanism [24, 25]. Adding a heavy singlet neutrino superfield into the
Minimal Supersymmetric SM (MSSM) gives rise to soft leptogenesis where the soft breaking of
supersymmetry allows the scalar superpartner of the heavy neutrino to generate the (s)lepton
asymmetry [26, 27]. In contrast to non-SUSY models, only one family of heavy singlets is
required.
Furthermore, various scenarios incorporating leptogenesis and DM have been introduced
[29]-[37]. In this article, we investigate a model that encompasses soft leptogenesis and DM
generation from the same heavy singlet superfield. Previously a model with SU(2) triplet
L
DM and supersymmetric leptogenesis has been studied [38]. Our model is a minimal extension
to the Minimal Supersymmetric Standard Model (MSSM) and it consists of MSSM augmented
with a single SM singlet DM superfield and a heavy singlet neutrino superfield that enables
soft leptogenesis after soft SUSY breaking. In section 2 we introduce our model and present
the CP violation parameters, in section 3 we solve the Boltzmann equations and the resulting
baryon and DM abundances, and in section 4 we present our conclusions.
2
2 The model
Models incorporating leptogenesis and DM generation have the potential to produce the ob-
served matter and DM energy densities and offer a possible explanation as to why the abun-
dances are similar in magnitude. So far the MSSM has been considered as one of the most
appealing extension to the SM, and with the addition of a heavy singlet (s)neutrino, soft lep-
togenesis can arise. To account for DM, the soft leptogenesis framework can be extended with
an additional DM sector. We have augmented the MSSM with a singlet superfield consisting of
a heavy neutrino and sneutrino and a singlet DM superfield. The heavy (s)neutrino couples to
the MSSM (s)leptons, Higgses and higgsinos, which gives rise to the soft leptogenesis scenario.
The DM sector couples to the heavy (s)neutrino as well and this provides additional decay
channels for the singlet (s)neutrino and a source for DM production through these decays. The
superpotential is
1 1
W = W +y NL H + MNN +m ΦΦc + λNΦΦ. (2.1)
MSSM α α u Φ
2 2
The discrete symmetry Z imposed on Φ forbids terms Φ3 and NNΦ . After adding
2
∼ ∼
soft supersymmetry breaking terms we get the part of the Lagrangian that includes the soft
leptogenesis and darkgenesis interactions
= λφNχ+y H L N +y L H N +h.c.
α u α α α u
L
1 1 1
+N λe−iθb/2χχ+ y e−iθb/2L H + λM∗eiθb/2φ2
+ √2 √e2 eα α u 2√2
(cid:20)
1 1 1
+ e y M∗eiθb/2L H + c e−iθb/2φ2 +e c e−iθb/2L H +h.c.
α α u 1 2 α u
√2 2 2
(cid:21)
i i i
+N λe−iθb/e2χχ+ y e−iθb/2L H λeM∗eiθb/2φ2
− α α u
√2 √2 − 2√2
(cid:20)
i i i
e y M∗eiθb/2L H + c e−iθb/2φ2e+ c e−iθb/2L H +h.c. , (2.2)
α α u 1 2 α u
−√2 √2 √2
(cid:21)
e e
where the sneutrino mass eigenstates are
1
N = eiθb/2N +e−iθb/2N∗ , (2.3)
+
√2
(cid:16) (cid:17)
1
Ne = eiθb/2eN e−iθb/2eN∗ . (2.4)
−
√2i −
(cid:16) (cid:17)
e e e
The DM particles χ and φ represent the DM fermion and scalar, respectively.
The lepton and DM asymmetries arise from the self-energy loop diagrams shown in Figs.
1(a)-1(f). While the process N χχc does not produce a DM asymmetry, i.e. an asymmetry
±
→
e 3
between χ and χc, the decays N φφ and N φ∗φ∗ can create an asymmetry between
± ±
→ →
φ and φ∗. This is the source for possible ADM in our model and the asymmetric component
e e
consists of scalar DM. The CP violation parameter for (s)lepton production is defined as
¯ ¯
Γ(N f) Γ(N f)+Γ(N f) Γ(N f)
f + → − + → − → − − →
ε = , (2.5)
L P hΓ(N f)+Γ(N f¯)+Γ(N f)+Γ(N f¯)i
f e+ → e+ → e− → e− →
h i
P
where f stands for the feermionic (l Hc)eand bosonic (e˜l H ) final staetes. The DM asymmetry
α u α u
is defined in a similar manner:
e
Γ N φφ Γ N φ∗φ∗ +Γ N φφ Γ N φ∗φ∗
+ + − −
→ − → → − →
ε = (2.6)
DM (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Γ N φφ +Γ N φ∗φ∗ +Γ N φφ +Γ N φ∗φ∗
e+ e+ e− e−
→ → → →
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
e e e e
4
The temperature-dependent CP violation parameter for the (s)leptons is
M2 M2
4 e e M
ε (T) = N+ − N− | |m
L − 16π(M2 M2 )2 +Π2 v2 eff
" Ne+ − Ne− −− u
M 2 A 2 π M
| | | | 2 A M cos θ + +θ θ | |m
× 2 − 2 | || | A 2 M − b v2 eff
(cid:18) (cid:19)(cid:18) u
(cid:16) (cid:17)
π c
B
+ λ M c cos θ + +θ θ +θ
| || || 1| − λ 2 M − b c1 16πMe
(cid:16) (cid:17)(cid:19) N+
4 M2e M2e M 2 A 2 M
N− − N+ | | + | | | |m
−16π(M2 M2 )2 +Π2 − 2 2 v2 eff
Ne− − Ne+ ++ (cid:18) (cid:19) u
π M
2 M A cos θ +θ θ + | |m
× | || | M A − b 2 v2 eff
(cid:18) u
(cid:16) (cid:17)
π c
B
+ c λ M cos θ θ +θ θ +
| 1|| || | c1 − λ M − b 2 16πMe
(cid:16) (cid:17)(cid:19) N−
M2 M2
4 e e M m M
N− − N+ | | eff 2 M A sin( θ θ +θ ) | |m
−16π(M2 M2 )2 +Π2 v2 2 | || | − M − A b v2 eff
Ne− − Ne+ ++ u (cid:18) u
c
F
+|c1||λ||M|sin(θλ −θc1 −θM +θb) 16πMNe−
(cid:19)
M2 M2
4 e e M m M
N+ − N− | | eff 2 A M sin(θ θ θ ) | |m
−16π(M2 M2 )2 +Π2 v2 2 | || | b − A − M v2 eff
Ne+ − Ne− −− u (cid:18) u
c
F
+|c1||M||λ|sin(θb −θc1 +θλ −θM) MNe+16π
#
(cid:19)
M c
M 2 + A 2 +2 M A cos(θ θ +θ ) | |m B
×" | | | | | || | M − b A vu2 eff8πMNe+
(cid:0) (cid:1)
M c
+ M 2 + A 2 2 M A cos(θ θ +θ ) | |m B
| | | | − | || | M − b A v2 eff8πMe
u N−
(cid:0) −1 (cid:1)
M c M c
F F
+MNe−|v2|meff8π +MNe+|v2|meff8π , (2.7)
u u #
where we see the influence of the DM sector couplings in the N self-energies and N N
± ± ∓
→
e e e
5
transitions. We have used c A y and m = v2 y 2/M. The net φ asymmetry is
2 ≡ α α eff u α| α|
4 M2eP M2e M P2 λ 2 c 2 c
ε (T) = N+ − N− | | | | | 1| φ
DM "− 16π(MN2e+ −MN2e−)2 +Π2−− (cid:18) 8 − 2 (cid:19) 16πMNe+
π M
2 A M cos θ + +θ θ | |m
× | || | A 2 M − b v2 eff
(cid:18) u
(cid:16) (cid:17)
+ c M λsin(θ θ +θ θ )
| 1|| | b − c1 λ − M
(cid:19)
4 M2e M2e c 2 λ 2 M 2 c
N− − N+ | 1| | | | | φ
−16π(MN2e− −MN2e+)2 +Π2++ (cid:18) 2 − 8 (cid:19) 16πMNe−
π M
2 M A cos θ +θ θ + | |m
| || | M A − b 2 v2 eff
(cid:18) u
(cid:16) (cid:17)
+ c λ M sin(θ +θ θ θ )
| 1|| || | λ b − c1 − M
#
(cid:19)
c c 2 M 2 λ 2 1
φ 1
| | + | | | | + c M λ cos(θ θ +θ θ )
×"8πMNe+ (cid:18) 2 8 2| 1|| || | c1 − b M − λ (cid:19)
−1
c M 2 λ 2 c 2 1
φ 1
+ | | | | + | | M λ c cos(θ θ +θ θ ) (2.8)
8πMNe− (cid:18) 8 2 − 2| || || 1| λ − M b − c1 (cid:19)#
with the factors [12]
2 eq eq
cF = 1−xl −xHe 1+xl −xHe −4xl(1−fl ) 1−fHe , (2.9)
q (cid:16) (cid:17)
(cid:0) (cid:1) 2(cid:0) (cid:1)eq eq
cB = 1+xH −xel −4xH (1+fH ) 1+fel , (2.10)
c = q(cid:0)1 4x 1+(cid:1)feq 1+feq , (cid:16) (cid:17) (2.11)
φ − φ φ φ
m (T)2 1
x = pa , f(cid:0)eq = (cid:1)(cid:0) , (cid:1) (2.12)
a M2 a eEa/T 1
±
where the thermal masses are
3 g2 3
m (T)2 = T2 g2 + Y + 4πα , (2.13)
H 8 2 8 4 t
(cid:18) (cid:19)
1 3 g2 3
mHe(T)2 = 2T2 8g22 + 8Y + 44παt , (2.14)
(cid:18) (cid:19)
3 g2
me(T)2 = T2 g2 + Y , (2.15)
l 8 2 8
(cid:18) (cid:19)
1 3 g2
m (T)2 = T2 g2 + Y , (2.16)
l 2 8 2 8
(cid:18) (cid:19)
m (T)2 = m2 +m2 +λ2T2. (2.17)
φ Φ Φsoft
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The phases θ and θ can be rotated away and are set to zero and so M is real and we can
M b
replace M M. The factors c reflect the production thresholds of MSSM particles
F,B,φ
| | →
and DM boson φ imposed by energy conservation. Since we normalize the DM asymmetry
to the DM boson production, the factors c cancel and consequently there is no temperature
φ
dependence in ε . We have also kept the soft terms at the vertices for completeness, usually
DM
these are neglected [27].
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3 Boltzmann equations
We move on to solve the relic baryon and DM abundances this scenario generates. We assume
the Yukawa couplings m and DM coupling λ are sufficiently small and consider only pro-
eff
| |
cesses with cross sections (y2) and (λ2). Due to this, DM annihilation processes are left
O O
out, which rules out ADM as the primary source of the DM energy density. Thus, we write
the Boltzmann equations also for the total abundances of χ and φ. The particle abundances
are expressed in terms of the number densities normalized to entropy density Y n /s and
i i
≡
their evolution with changing z M/T is to be determined. Our Boltzmann equations are
≡
dY
N eq (0) (1) (2) (3) (4)
= (Y Y ) D +D +4S +4S +4S +2S +4S , (3.1)
dz − N − N N Nχφ t t t t t
(cid:16) (cid:17)
ddYzNe = − YNe −2YNeeq D2Ne +DN(e3) + DN2eφφ + DN2eχχ +3S22
"
(cid:16) (cid:17)
(5) (6) (7) (9) (8)
+2 S +S +S +S +S , (3.2)
t t t t t
#
(cid:16) (cid:17)
dYd∆zLtot = εL(T) YNe −2YNeeq D2Ne
−YY∆Leqt(cid:16)ot D2NYNeq +(cid:17) D2NeYNeeq +YNeeqDN(e3) +YNe St(5) + S2t(8)
l " !
(0) (3) eq (1) (2) (4)
+Y 2S +S +2Y S +S +S
N t t N t t t
(cid:16) (cid:17) (cid:16) (cid:17)
+2Yeq S(6) +S(7) +S(9) +S 2Yeq + YNe , (3.3)
Ne t t t 22 Ne 2
#
(cid:18) (cid:19)
(cid:16) (cid:17)
dY De Y De +De D
∆φ eq Nφφ ∆φ Nφφ Nχχ eq Nχφ eq
dz = εDM YNe −2YNe 2 − Yeq 2 YNe + 2 YN , (3.4)
φ (cid:18) (cid:19)
(cid:16) (cid:17)
dY 4m2 m2 m2 2 m2 +m2
dzχ = − 1− M2χDNeχχ +s1+ χM−4 φ −2 φM2 χDNχφ
r (cid:0) (cid:1)
 
eq
× YNe −2YNe , (3.5)
(cid:16) (cid:17)
dY 4m2 m2 m2 2 m2 +m2
dzφ = −s1− M2φDNeφφ +s1+ χM−4 φ −2 φM2 χDNχφ
(cid:0) (cid:1)
 
eq
× YNe −2YNe . (3.6)
(cid:16) (cid:17)
with YNeeq ≈ YNeeq± and YNe/2 ≈ YNe± [39]. Recall that we take into account O(y2) and O(λ2)
decay rates and scattering diagrams involving the (s)top. The expressions for the decay D
and (s)top scattering rates S can be found in appendix A. The solutions to these Boltzmann
8
equations are shown in Figs. 2-4. The baryon abundance is related to the (s)lepton abundance
through
8
Y = Y . (3.7)
B −15 ∆Ltot
We have chosen m = 10−14 TeV, M = 105 TeV, M =125 GeV, A = 1 TeV, λ = 10−8,
eff h
| | | |
c = 0.001 TeV and m = 0.1 TeV1 The evolution of the Y abundance is similar to those
1 Φ B
| |
found in [27, 39].
The DM abundances Y and Y are not sensitive to changes in the DM mass parameter m
φ χ Φ
as long as it is a lot smaller than the heavy scale M associated with the singlet (s)neutrino,
m M. It is the coupling λ and the positive mass dimension soft coupling c that largely
Φ 1
≪
determine the order of magnitude in the DM abundances. With suitable choices of these
couplings, the abundances Y and Y attain final values 10−11 suggesting that both the
χ φ
∼
scalar and fermionic DM particles can be viable DM candidates.
At small couplings λ 10−8 and c 0.001 TeV, the asymmetric component is extremely
1
∼ ∼
tiny with abundance 10−17 and does not play a role in DM energy density. However, an
∼
increase in both DM sector couplings causes the magnitude of Y to rise and also compels to
∆φ
consider scatterings where DM particles are destroyed and MSSM particles are produced. So,
in order to have ADM as the primary source of all DM, it is crucial to find a balance between
sufficient DM production through asymmetric decays and scattering processes
φφ L˜ H , φφ L Hc,
→ α u → α u
χχc L˜ H , χχc L Hc,
→ α u → eα u
χφ L H ,χφ L˜ Hc,
→ α u → α ue
where χc denotes the right-handed fermion obtained frome χ through χ¯α˙ = ǫα˙β˙ (χ )∗, where
β
ǫα˙β˙ = ( iσ2)α˙β˙. WebelieveitispossibletocreateconditionswheretheDMparticlesannihilate
−
and the asymmetric component remains and forms a part of final DM abundance although
this scenario entails a different parameter space in terms of DM couplings and other parameter
inputs. We are studying this alternative in an upcoming paper.
We have explored the allowed parameter space and made scatter plots of the DM masses
versus the DM coupling λ , Figs. 5-6. The fixed parameters are m = 10−14 TeV, M = 105
eff
| |
1If the heavy (s)neutrinos are supposed to be produced thermally, large M implies reheating temperature
TR >M. In many inflatory supergravity models this high TR leads to overproduction of gravitinos. However,
recently it has been proposed that with the present value of the Higgs mass about 125 GeV [42] the reheating
temperature constraint could be relaxed, TR 109 1010 GeV, not in conflict with our scenario [43].
∼ −
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Table 1: The DM sector parameters and their scanned values.
Parameters Scanned values
λ 10−10 10−8
| | −
c 10−5 10−3 TeV
1
| | −
m 0.01-1 TeV
Φ
m2 10−4-1 TeV2
Φsoft
TeV and A = 1 TeV while λ , c , m and m are varied as shown in Table 1. The
1 Φ Φsoft
| | | |
range for the soft parameter c has been chosen so that c λ M. The DM coupling
1 1
| | | | ∼ | |
10−10 < λ < 10−8 is the most favorable region of producing the correct DM abundance which
| |
is why this particular region is studied. The DM mass parameters are chosen to be in the
weak scale and even in the 10 GeV range which brings the typical ADM regime closer. The
required DM abundances come from the relation
3Ω
DM
m Y +m Y . T (3.8)
χ χ φ φ
| | | | 4 Ω
γ
and the baryon abundance has to satisfy
Y . 8.8 10−11 (3.9)
B
| | ×
whichwehaveusedasconstraintsinextractingtheallowedparameterspace. Knownparameter
values from observations are T = 2.725 K, Ω = 0.11/h2 and Ω = 2.47 10−5/h2. We have
DM γ
×
taken the lower limit in (3.9) to be 30 % of the observed baryon abundance while in (3.8) the
lower limit is 90 % of the observed value. Figs. 5-6 reflect the hyperbola-like boundary set by
Eq. (3.8), and there is an indication of slight preference of λ values in the 10−9 ballpark.
| | ∼
A similar observation of the favorable magnitude of λ can be made from a plot with the
| |
left-hand side of (3.8) plotted against λ .
| |
We have also determined the dependence of efficiency factors related to baryon and ADM
production on the Yukawa and DM couplings. The efficiency factors η and η are shown in
B DM
Figs. 7-8 and they are given by
Y
η = ∆Ltot,∆φ . (3.10)
B,DM eq
2ε Y (z 0)
(cid:12) L,DM e (cid:12)
(cid:12) N ≈ (cid:12)
(cid:12) (cid:12)
The ADM production efficiency falls w(cid:12)ith rising m whi(cid:12)le with increasing λ η increases
(cid:12) eff (cid:12) DM
| |
and after reaching a maximum a steep decline ensues. The fall in η is due to the fact that
DM
10