Leptogenesis in the Universe

0.82 MB·English

by  Chee Sheng Fong

Leptogenesis in the Universe
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2 Authors: Chee Sheng Fonga, Enrico Nardia, Antonio Riottob

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4 a INFN - Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044 Frascati, Italy
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b Department of Theoretical Physics and Center for Astroparticle Physics (CAP),

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h University of Geneva, 24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland
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3 Abstract
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0 Leptogenesis is a class of scenarios in which the cosmic baryon asymmetry originates from an initial
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lepton asymmetry generated in the decays of heavy sterile neutrinos in the early Universe. We ex-
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: plain why leptogenesis is an appealing mechanism for baryogenesis. We review its motivations, the
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basic ingredients, and describe subclasses of effects, like those of lepton flavours, spectator processes,
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scatterings, finite temperature corrections, the role of the heavier sterile neutrinos and quantum cor-
r rections. We then address leptogenesis in supersymmetric scenarios, as well as some other popular
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variations of the basic leptogenesis framework.
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Contents
Leptogenesis in the Universe
Authors: Chee Sheng Fonga, Enrico Nardia, Antonio Riottob 1
1.1 The Baryon Asymmetry of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 N Leptogenesis in the Single Flavour Regime . . . . . . . . . . . . . . . . . . . . . . . 5
1
1.2.1 Type-I seesaw, neutrino masses and leptogenesis . . . . . . . . . . . . . . . . . 6
1.2.2 CP asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Classical Boltzmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Baryon asymmetry from EW sphaleron . . . . . . . . . . . . . . . . . . . . . . 9
1.2.5 Davidson-Ibarra bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Lepton Flavour Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 When are lepton flavour effects relevant? . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 The effects on CP asymmetry and washout . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Classical flavoured Boltzmann equations . . . . . . . . . . . . . . . . . . . . . . 13
1.3.4 Lepton flavour equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Beyond the Basic Boltzmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Spectator processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.2 Scatterings and CP violation in scatterings . . . . . . . . . . . . . . . . . . . . 17
1.4.3 Thermal corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.4 Decays of the heavier right-handed neutrinos . . . . . . . . . . . . . . . . . . . 21
1.4.5 Quantum Boltzmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Supersymmetric Leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.1 What’s new? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.2 General constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5.3 Superequilibration regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.5.4 Non-superequilibration regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5.5 Supersymmetric Boltzmann equations . . . . . . . . . . . . . . . . . . . . . . . 37
1.6 Beyond Type-I Seesaw and Beyond the Seesaw . . . . . . . . . . . . . . . . . . . . . . 38
1.6.1 Resonant leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.6.2 Soft leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.6.3 Dirac leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.6.4 Triplet scalar (type-II) leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.6.5 Triplet fermion (type-III) leptogenesis . . . . . . . . . . . . . . . . . . . . . . . 43
1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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Neutrino Physics Leptogenesis in the Universe
1.1 The Baryon Asymmetry of the Universe
1.1.1 Observations
Up to date no traces of cosmological antimatter have been observed. The presence of a small amount
of antiprotons and positrons in cosmic rays can be consistently explained by their secondary origin
in cosmic particles collisions or in highly energetic astrophysical processes, but no antinuclei, even as
light as anti-deuterium or as tightly bounded as anti-α particles, has ever been detected.
The absence of annihilation radiation pp¯→ ...π0 → ...2γ excludes significant matter-antimatter
admixtures in objects up to the size of galactic clusters ∼ 20Mpc [1]. Observational limits on anoma-
lous contributions to the cosmic diffuse γ-ray background and the absence of distortions in the cosmic
microwave background (CMB) implies that little antimatter is to be found within ∼ 1Gpc and that
within our horizon an equal amount of matter and antimatter can be excluded [2]. Of course, at larger
super-horizon scales the vanishing of the average asymmetry cannot be ruled out, and this would
indeed be the case if the fundamental Lagrangian is C and CP symmetric and charge invariance is
broken spontaneously [3].
Quantitatively, the value of baryon asymmetry of the Universe is inferred from observations in two
independent ways. The first way is by confronting the abundances of the light elements, D, 3He, 4He,
and 7Li, with the predictions of Big Bang nucleosynthesis (BBN) [4, 5, 6, 7, 8, 9]. The crucial time for
primordial nucleosynthesis is when the thermal bath temperature falls below T <∼ 1MeV. With the
assumption of only three light neutrinos, these predictions depend on a single parameter, that is the
difference between the number of baryons and anti-baryons normalized to the number of photons:
η ≡ nB −nB¯(cid:12)(cid:12) , (1.1)
(cid:12)
nγ 0
where the subscript 0 means “at present time”. By using only the abundance of deuterium, that is
particularly sensitive to η, Ref. [4] quotes:
1010η = 5.7±0.6 (95%c.l.). (1.2)
In this same range there is also an acceptable agreement among the various abundances, once theo-
retical uncertainties as well as statistical and systematic errors are accounted for [6].
The second way is from measurements of the CMB anisotropies (for pedagogical reviews, see
Refs. [10, 11]). The crucial time for CMB is that of recombination, when the temperature dropped
down to T <∼ 1eV and neutral hydrogen can be formed. CMB observations measure the relative
baryon contribution to the energy density of the Universe multiplied by the square of the (reduced)
Hubble constant h ≡ H /(100kmsec−1Mpc−1):
0
ρ
Ω h2 ≡ h2 B , (1.3)
B
ρ
crit
thatisrelatedtoηthrough1010η = 274 Ω h2. Thephysicaleffectofthebaryonsattheonsetofmatter
B
domination, which occurs quite close to the recombination epoch, is to provide extra gravity which
enhances the compression into potential wells. The consequence is enhancement of the compressional
phases which translates into enhancement of the odd peaks in the spectrum. Thus, a measurement of
theodd/evenpeakdisparityconstrainsthebaryonenergydensity. Afittothemostrecentobservations
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Neutrino Physics Leptogenesis in the Universe
(WMAP7 data only, assuming a ΛCDM model with a scale-free power spectrum for the primordial
density fluctuations) gives at 68% c.l. [12]
102Ω h2 = 2.258+0.057. (1.4)
B −0.056
There is a third way to express the baryon asymmetry of the Universe, that is by normalizing the
baryon asymmetry to the entropy density s = g (2π2/45)T3, where g is the number of degrees of
∗ ∗
freedom in the plasma, and T is the temperature:
Y ≡ nB −nB¯(cid:12)(cid:12) . (1.5)
∆B (cid:12)
s 0
The relation with the previous definitions is given by the conversion factor s /n = 7.04. Y is a
0 γ0 ∆B
convenient quantity in theoretical studies of the generation of the baryon asymmetry from very early
times, because it is conserved throughout the thermal evolution of the Universe. In terms of Y the
∆B
BBN results eq. (1.2) and the CMB measurement eq. (1.4) (at 95%c.l.) read:
YBBN = (8.10±0.85)×10−11, YCMB = (8.79±0.44)×10−11. (1.6)
∆B ∆B
The impressive consistency between the determinations of the baryon density of the Universe from
BBN and CMB that, besides being completely independent, also refer to epochs with a six orders of
magnitude difference in temperature, provides a striking confirmation of the hot Big Bang cosmology.
1.1.2 Theory
From the theoretical point of view, the question is where the Universe baryon asymmetry comes from.
The inflationary cosmological model excludes the possibility of a fine tuned initial condition, and since
we do not know any other way to construct a consistent cosmology without inflation, this is a strong
veto.
The alternative possibility is that the Universe baryon asymmetry is generated dynamically, a
scenario that is known as baryogenesis. This requires that baryon number (B) is not conserved. More
precisely, as Sakharov pointed out [13], the ingredients required for baryogenesis are three:
1. B violation is required to evolve from an initial state with Y = 0 to a state with Y (cid:54)= 0.
∆B ∆B
2. C and CP violation: If either C or CP were conserved, then processes involving baryons would
proceed at the same rate as the C- or CP-conjugate processes involving antibaryons, with the
overall effect that no baryon asymmetry is generated.
3. Out of equilibrium dynamics: Equilibrium distribution functions n are determined solely by
eq
theparticleenergyE, chemicalpotentialµ, andbyitsmasswhich, becauseoftheCPTtheorem,
is the same for particles and antiparticles. When charges (such as B) are not conserved, the
(cid:82) d3p
corresponding chemical potentials vanish, and thus nB = (2π3)neq = nB¯.
AlthoughtheseingredientsareallpresentintheStandardModel(SM),sofarallattemptstoreproduce
quantitatively the observed baryon asymmetry have failed.
1. In the SM B is violated by the triangle anomaly. Although at zero temperature B violating
processes are too suppressed to have any observable effect [14], at high temperatures they occur
with unsuppressed rates [15]. The first condition is then quantitatively realized in the early
Universe.
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Neutrino Physics Leptogenesis in the Universe
2. SM weak interactions violate C maximally. However, the amount of CP violation from the
Kobayashi-Maskawa complex phase [16], as quantified by means of the Jarlskog invariant[17], is
only of order 10−20, and this renders impossible generating Y ∼ 10−10 [18, 19, 20].
∆B
3. Departures from thermal equilibrium occur in the SM at the electroweak phase transition
(EWPT) [21, 22]. However, the experimental lower bound on the Higgs mass implies that
this transition is not sufficiently first order as required for successful baryogenesis [23].
This shows that baryogenesis requires new physics that extends the SM in at least two ways: It must
introduce new sources of CP violation and it must either provide a departure from thermal equilib-
rium in addition to the EWPT or modify the EWPT itself. In the past thirty years or so, several
new physics mechanisms for baryogenesis have been put forth. Some among the most studied are
GUT baryogenesis [24, 25, 26, 27, 28, 29, 30, 31, 32, 33], Electroweak baryogenesis [21, 34, 35], the
Affleck-Dine mechanism [36, 37], Spontaneous Baryogenesis [38, 39]. However, soon after the discov-
ery of neutrino masses, because of its connections with the seesaw model [40, 41, 42, 43, 44] and its
deep interrelations with neutrino physics in general, the mechanism of baryogenesis via Leptogene-
sis acquired a continuously increasing popularity. Leptogenesis was first proposed by Fukugita and
Yanagida in Ref. [45]. Its simplest and theoretically best motivated realization is precisely within the
seesaw mechanism. To implement the seesaw, new Majorana SU(2) singlet neutrinos with a large
L
mass scale M are added to the SM particle spectrum. The complex Yukawa couplings of these new
particles provide new sources of CP violation, departure from thermal equilibrium can occur if their
lifetime is not much shorter than the age of the Universe when T ∼ M, and their Majorana masses
imply that lepton number is not conserved. A lepton asymmetry can then be generated dynamically,
and SM sphalerons will partially convert it into a baryon asymmetry [46]. A particularly interesting
possibility is “thermal leptogenesis” where the heavy Majorana neutrinos are produced by scatterings
in the thermal bath starting from a vanishing initial abundance, so that their number density can be
calculated solely in terms of the seesaw parameters and of the reheat temperature of the Universe.
This review is organized as follows: in Section 1.2 the basis of leptogenesis are reviewed in the
simple scenario of the one flavour regime, while the role of flavour effects is described in Section 1.3.
Theoretical improvements of the basic pictures, like spectator effects, scatterings and CP violation in
scatterings, thermal corrections, the possible role of the heavier singlet neutrinos, and quantum effects
are reviewed in Section 1.4. Leptogenesis in the supersymmetric seesaw is reviewed in Section 1.5,
while in Section 1.6 we mention possible leptogenesis realizations that go beyond the type-I seesaw.
Finally, in Section 1.7 we draw the conclusions.
1.2 N Leptogenesis in the Single Flavour Regime
1
The aim of this section is to give a pedagogical introduction to leptogenesis [45] and establish the
notations. We will consider the classic example of leptogenesis from the lightest right-handed (RH)
neutrino N (the so-called N leptogenesis) in the type-I seesaw model [40, 43, 41, 44] in the single
1 1
flavour regime. First in Section 1.2.1 we introduce the type-I seesaw Lagrangian and the relevant
parameters. In Section 1.2.2, we will review the CP violation in RH neutrino decays induced at 1-loop
level. Then in Section 1.2.3, we will write down the classical Boltzmann equations taking into account
of only decays and inverse decays of N and give a simple but rather accurate analytical estimate of
1
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Neutrino Physics Leptogenesis in the Universe
the solution. In Section 1.2.4 we will relate the lepton asymmetry generated to the baryon asymmetry
of the Universe. Finally in Section 1.2.5, we will discuss the lower bound on N mass and the upper
1
bound on light neutrino mass scale from successful leptogenesis.
1.2.1 Type-I seesaw, neutrino masses and leptogenesis
With m(m ≥ 2)1 singlet RH neutrinos N (i = 1,m), we can add the following Standard Model (SM)
Ri
gauge invariant terms to the SM Lagrangian
(cid:18)1 (cid:19)
L = L +iN ∂/N − M Nc N +(cid:15) Y N (cid:96)aHb+h.c. , (1.7)
I SM Ri Ri 2 i Ri Ri ab αi Ri α
where M are the Majorana masses of the RH neutrinos, (cid:96) = (ν ,α−) with α = e,µ,τ and H =
i α αL L
(H+,H0)arerespectivelytheleft-handed(LH)leptonandHiggsSU(2) doubletsand(cid:15) = −(cid:15) with
L ab ba
(cid:15) = 1. Withoutlossofgenerality,wehavechosenthebasiswheretheMajoranamasstermisdiagonal.
12
The physical mass eigenstates of the RH neutrinos are the Majorana neutrinos N = N +Nc . Since
i Ri Ri
N are gauge singlets, the scale of M is naturally much larger than the electroweak (EW) scale
i i
M (cid:29) (cid:104)Φ(cid:105) ≡ v = 174 GeV. Hence after EW symmetry breaking, the light neutrino mass matrix is
i
given by the famous seesaw relation [40, 43, 41, 44]
1
m (cid:39) −v2Y YT. (1.8)
ν
M
(cid:113)
Assuming Y ∼ O(1) and m (cid:39) ∆m2 (cid:39) 0.05 eV, we have M ∼ 1015 GeV not far below the GUT
ν atm
scale.
Besides giving a natural explanation of the light neutrino masses, there is another bonus: the three
Sakharov’s conditions[13] for leptogenesis are implicit in eq. (1.7) with the lepton number violation
provided by M , the CP-violation from the complexity of Y and the departure from thermal equilib-
i iα
rium condition given by an additional requirement that N decay rate Γ is not very fast compared
i Ni
to the Hubble expansion rate of the Universe H(T) at temperature T = M with
i
(cid:115)
(Y†Y) M 2 g π3 T2
ii i ∗
Γ = , H(T) = , (1.9)
Ni 8π 3 5 M
pl
where M = 1.22×1019 GeV is the Planck mass, g (=106.75 for the SM excluding RH neutrinos) is
pl ∗
the total number of relativistic degrees of freedom contributing to the energy density of the Universe.
To quantify the departure from thermal equilibrium, we define the decay parameter as follows
Γ m
K ≡ Ni = (cid:101)i, (1.10)
i
H(M ) m
i ∗
where m is the effective neutrino mass defined as[47]
(cid:101)i
(Y†Y) v2
ii
m ≡ , (1.11)
(cid:101)i
M
i
(cid:113)
withm ≡ 16π2v2 g∗π (cid:39) 1×10−3 eV.TheregimeswhereK (cid:28) 1, K ≈ 1andK (cid:29) 1arerespectively
∗ 3M 5 i i i
pl
known as weak, intermediate, and strong washout regimes.
1Neutrino oscillation data and leptogenesis both require m≥2.
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Neutrino Physics Leptogenesis in the Universe
1.2.2 CP asymmetry
The CP asymmetry in the decays of RH neutrinos N can be defined as
i
(cid:16) (cid:17)
γ(Ni → (cid:96)αH)−γ Ni → (cid:96)αH∗ ∆γα
(cid:15) = ≡ Ni, (1.12)
iα (cid:16) (cid:17)
(cid:80)αγ(Ni → (cid:96)αH)+γ Ni → (cid:96)αH∗ γNi
where γ(i → f) is the thermally averaged decay rate defined as2
(cid:90) d3p d3p
γ(i → f) ≡ i f (2π)4δ(4)(p −p )|A(i → f)|2e−Ei/T, (1.13)
(2π)32E (2π)32E i f
i f
where A(i → f) is the decay amplitude. Ignoring all thermal effects [48, 49], eq. (1.12) simplifies to
(cid:16) (cid:17)
|A (N → (cid:96) H)|2−|A N → (cid:96) H∗ |2
0 i α 0 i α
(cid:15) = , (1.14)
iα (cid:16) (cid:17)
(cid:80) |A (N → (cid:96) H)|2+|A N → (cid:96) H∗ |2
α 0 i α 0 i α
whereA (i → f)denotesthedecayamplitudeatzerotemperature. Eq.(1.14)vanishesattreelevelbut
0
is induced at 1-loop level through the interference between tree and 1-loop diagrams shown in Figure
1.1. There are two types of contributions from the 1-loop diagrams: the self-energy or wave diagram
(middle) [50] and the vertex diagram (right) [45]. At leading order, we obtain the CP asymmetry [51]:
(cid:15) = 1 1 (cid:88)Im(cid:104)(Y†Y) Y Y∗ (cid:105)g(cid:32)Mj2(cid:33)
iα 8π(Y†Y) ji αi αj M2
ii j(cid:54)=i i
+ 1 1 (cid:88)Im(cid:104)(Y†Y) Y Y∗ (cid:105) Mi2 , (1.15)
8π(Y†Y) ij αi αj M2−M2
ii j(cid:54)=i i j
where the loop function is
√ (cid:20) 1 (cid:18)1+x(cid:19)(cid:21)
g(x) = x +1−(1+x)ln . (1.16)
1−x x
The first term in eq. (1.15) comes from L-violating wave and vertex diagrams, while the second term
is from the L-conserving wave diagram. The terms of the form (M2−M2)−1 in eq. (1.15) are from the
i j
wave diagram contributions which can resonantly enhance the CP asymmetry if M ≈ M (resonant
i j
leptogenesis scenario, see Section 1.6.1)3. Let us also note that at least two RH neutrinos are needed,
otherwise the CP asymmetry vanishes because the Yukawa couplings combination becomes real.
In the one flavour regime, we sum over the flavour index α in eq. (1.15) and obtain
(cid:15) ≡ (cid:88)(cid:15) = 1 1 (cid:88)Im(cid:104)(Y†Y)2 (cid:105)g(cid:32)Mj2(cid:33), (1.17)
i iα 8π(Y†Y) ji M2
α ii j(cid:54)=i i
where the second term in eq. (1.15) vanishes because the combination of the Yukawa couplings is real.
2Here the Pauli-blocking and Bose-enhancement statistical factors have been ignored and we also assume Maxwell-
Boltzmann distribution for the particle i i.e. fi =e−Ei/T. See Refs. [48, 49] for detailed studies of their effects.
3NoticethattheresonanttermbecomessingularinthedegeneratelimitM =M . Thissingularitycanberegulated
i j
by using for example an effective field-theoretical approach based on resummation [52].
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Neutrino Physics Leptogenesis in the Universe
H ℓ,ℓ H H
ℓ
Ni + Ni Nj + Ni N
j
H
ℓ H ℓ ℓ
α α α
Figure1.1: TheCPasymmetryintype-Iseesawleptogenesisresultsfromtheinterferencebetweentree
and1-loopwaveandvertexdiagrams. Forthe1-loopwavediagram,thereisanadditionalcontribution
from L-conserving diagram to the CP asymmetry which vanishes when summing over lepton flavours.
1.2.3 Classical Boltzmann equations
WeworkintheoneflavourregimeandconsideronlythedecaysandinversedecaysofN . Ifleptogenesis
1
occurs at T >∼ 1012GeV, then the charged lepton Yukawa interactions are out of equilibrium, and this
defines the one flavour regime. The assumption that only the dynamics of N is relevant can be
1
realized if for example the reheating temperature after inflation is T (cid:28) M such that N are
RH 2,3 2,3
not produced. In order to scale out the effect of the expansion of the Universe, we will introduce the
abundances, i.e. the ratios of the particle densities n = (cid:82) d3pf to the entropy density s = 2π2g T3:
i i 45 ∗
n
i
Y ≡ . (1.18)
i
s
TheevolutionoftheN1 densityandtheleptonasymmetryY∆L = 2Y∆(cid:96) ≡ 2(Y(cid:96)−Y(cid:96)¯)4 canbedescribed
by the following classical Boltzmann equations (BE)[53]
dY
N1 = −D (Y −Yeq), (1.19)
dz 1 N1 N1
dY
∆L = (cid:15) D (Y −Yeq)−W Y , (1.20)
dz 1 1 N1 N1 1 ∆L
where z ≡ M /T and the decay and washout terms are respectively given by
1
eq
γ z K (z) 1 Y (z)
D (z) = N1 = K z 1 , W (z) = D (z) N1 , (1.21)
1 sH(M ) 1 K (z) 1 2 1 Yeq
1 2 (cid:96)
with K the n-th order modified Bessel function of second kind. Yeq and Yeq read:5
n N1 (cid:96)
45 15
Yeq(z) = z2K (z), Yeq = . (1.22)
N 2π4g 2 (cid:96) 4π2g
∗ ∗
From eq. (1.19) and eq. (1.20), the solution for Y can be written down as follows
∆L
Y∆L(z) = Y∆L(zi)e−(cid:82)zzidz(cid:48)W1(z(cid:48))−(cid:90) zdz(cid:48)(cid:15)1(z(cid:48))ddYzN(cid:48)1e−(cid:82)zz(cid:48)dz(cid:48)(cid:48)W1(z(cid:48)(cid:48)) (1.23)
zi
where z is some initial temperature when N leptogenesis begins, and we have assumed that any
i 1
preexisting lepton asymmetry vanishes Y0 (z ) = 0. Notice that ignoring thermal effects, the CP
∆L i
asymmetry is independent of the temperature (cid:15) (z) = (cid:15) (c.f. eq. (1.17)).
1 1
4The factor of 2 comes from the SU(2) degrees of freedoms.
L
5TowritedownasimpleanalyticexpressionfortheequilibriumdensityofN ,weassumeMaxwell-Boltzmanndistri-
1
bution. However, follwing [54], the normalization factor Yeq is obtained from a Fermi-Dirac distribution.
(cid:96)
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Neutrino Physics Leptogenesis in the Universe
Weak washout regime
In the weak washout regime (K (cid:28) 1), the initial condition on the N density Y (z ) is important.
1 1 N1 i
eq
If we assume thermal initial abundance of N i.e. Y (z ) = Y (0), we can ignore the washout when
1 N1 i N1
N starts decaying at z (cid:29) 1 and we have
1
(cid:90) ∞ dYeq
Yt (∞) (cid:39) −(cid:15) dz(cid:48) N1 = (cid:15) Yeq(0). (1.24)
∆L 1 0 dz(cid:48) 1 N1
On the other hand, if we have zero initial N abundance i.e. Y (z ) = 0, we have to consider the
1 N1 i
opposite sign contributions to lepton asymmetry from the inverse decays when N is being populated
1
eq eq
(Y < Y ) and from the period when N starts decaying (Y > Y ). Taking this into account the
N1 N1 1 N1 N1
term which survives the partial cancellations are given by [55] 6
27
Y0 (∞) (cid:39) (cid:15) K2Yeq(0). (1.25)
∆L 16 1 1 N1
Strong washout regime
Inthestrongwashoutregime(K (cid:29) 1)anyleptonasymmetrygeneratedduringtheN creationphase
1 1
is efficiently washed out. Here we adopt the strong washout balance approximation[56] which states
that in the strong washout regime, the lepton asymmetry at each instant takes the value that enforces
a balance between the production and the destruction rates of the asymmetry. Equating the decay
and washout terms in eq. (1.20), we have
eq
1 dY 1 dY 2
Y (z) ≈ − (cid:15) N1 (cid:39) − (cid:15) N1 = (cid:15) Yeq, (1.26)
∆L W(z) 1 dz W(z) 1 dz zK 1 (cid:96)
1
where in the second approximation, we assume Y (cid:39) Yeq. The approximation no longer holds when
N1 N1
Y freezes and this happens when the washout decouples at z i.e. W(z ) < 1. Hence, the final
∆L f f
lepton asymmetry is given by7
2 π2
Y (∞) = (cid:15) Yeq = (cid:15) Yeq(0). (1.27)
∆L z K 1 (cid:96) 6z K 1 N1
f 1 f 1
The freeze out temperature z depends mildly on K . For K = 10-100 we have for example z = 7-
f 1 1 f
10. We also see that independently of initial conditions, in the strong regime Y (∞) goes as K−1.
∆L 1
1.2.4 Baryon asymmetry from EW sphaleron
The final lepton asymmetry Y (∞) can be conveniently parametrized as follows
∆L
Y (∞) = (cid:15) η Yeq(0), (1.28)
∆L 1 1 N1
where η is known as the efficiency factor. In the weak washout regime (K (cid:28) 1) from eq. (1.24)
1 1
we have η = 1(= 27K2 < 1) for thermal (zero) initial N abundance. In the strong washout regime
1 16 1 1
(K (cid:29) 1), from eq. (1.27), we have η = π2 < 1.
1 1 6zfK1
6ThisdiffersfromtheefficiencyinRef.[55]bythefactor 12,whichisduetothedifferentnormalizationYeq eq.(1.22).
π2 (cid:96)
7Compare this to a more precise analytical approximation in Ref. [55].
9
Neutrino Physics Leptogenesis in the Universe
If leptogenesis ends before EW sphaleron processes become active (T >∼ 1012 GeV), the B − L
asymmetry Y is simply given by
∆B−L
Y = −Y . (1.29)
∆B−L ∆L
Atthelaterstage, theB−LasymmetryispartiallytransferedtoaB asymmetrybytheEWsphaleron
processes through the relation [57]
28
Y (∞) = Y (∞), (1.30)
∆B 79 ∆B−L
thatholdsifsphaleronsdecouplebeforeEWPT.ThisrelationwillchangeiftheEWsphaleronprocesses
decouple after the EWPT [57, 58] or if threshold effects for heavy particles like the top quark and
Higgs are taken into account [58, 59].
1.2.5 Davidson-Ibarra bound
Assuming a hierarchical spectrum of the RH neutrinos (M (cid:28) M , M ) and that the dominant lepton
1 2 3
asymmetry is from the N decays, from eq. (1.17) the CP asymmetry from N decays can be written
1 1
as
(cid:15) = − 3 1 (cid:88)Im(cid:104)(Y†Y)2 (cid:105) M1. (1.31)
1 16π(Y†Y) j1 M
11 j
j(cid:54)=1
Assuming three generations of RH neutrinos (n = 3) and using the Casas-Ibarra parametrization [60]
for the Yukawa couplings
1 (cid:18)(cid:113) (cid:113) (cid:19)
Y = D R D U† , (1.32)
αi v mN mν ν
αi
where D = diag(M ,M ,M ), D = diag(m ,m ,m ) and R any complex orthogonal matrix
mN 1 2 3 mν ν1 ν2 ν3
satisfying RTR = RRT = 1, eq. (1.31) becomes
(cid:88)
m Im(R2 )
νi 1i
3 M
(cid:15) = − 1 i . (1.33)
1 16π v2 (cid:88)m |R |2
νi 1i
i
(cid:88)
Using the orthogonality condition R2 = 1, we then obtain the Davidson-Ibarra (DI) bound [61]
1i
i
3 M 3 M ∆m2
|(cid:15) | ≤ (cid:15)DI = 1(m −m ) = 1 atm , (1.34)
1 16π v2 ν3 ν1 16π v2 m +m
ν1 ν3
where m (m ) is the heaviest (lightest) light neutrino mass. Applying the DI bound on eqs. (1.28)–
ν3 ν1
(1.30), and requiring that Y (∞) ≥ YCMB (cid:39) 10−10, we obtain
∆B B
(cid:18) 0.1eV (cid:19)
M1 η1max(M1) >∼ 109GeV, (1.35)
m +m
ν1 ν3
where the ηmax(M ) is the efficiency factor maximized with respect to K eq. (1.10) for a particular
1 1 1
value of M . This allows us to make a plot of region which satisfies eq. (1.35) on the (M ,m )
1 1 ν1
plane and hence obtain bounds on M and m . Many careful numerical studies have been carried
1 ν1
out and it was found that successful leptogenesis with a hierarchical spectrum of the RH neutrinos
10