Massive Neutrinos and Leptogenesis
David de Sousa Seixas
Dissertacao para a obtencao de Grau de Mestre em
Engenharia Fısica Tecnologica
Juri
Presidente: Jorge Romao
Orientador: Gustavo Castelo Branco
Vogais: David-Emmanuel Costa
Setembro 2009
Abstract
We review here the recent developments in the lepton sector since neutrino masses and leptonic mixings
have been found. We will give a brief account of neutrino oscillations which are a consequence of
neutrino mass and we will present ways to give mass to the neutrinos in the Standard Model and the
Left-Right Symmetric Model. We will then address the basics of generating the baryon asymmetry of the
universe through these new couplings in the neutrino sector, a mechanism which is called leptogenesis.
Contents
1 Introduction 3
2 Massive Neutrinos 4
2.1 Fermions in the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Gauge Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Masses and Mixings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Massive Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Majorana Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Properties under discrete space-time symmetries . . . . . . . . . . . . . . . . . . . 13
2.2.3 Majorana and Dirac mass terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.4 Experimental tests on neutrino mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.5 Neutrinoless double-beta decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.6 Neutrino Mass Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.7 Seesaw Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Neutrino Oscillations 29
3.1 Neutrino oscillations in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Two-flavour case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.2 Three-flavour case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.3 Majorana mass and neutrino oscillation . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.4 Do charged leptons oscillate? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 The MSW effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 Constant density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Oscillations in non-uniform matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Solar Neutrino Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1
4 Neutrino Mass Models 48
4.1 Seesaw models in the Left-Right Symmetric model . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 The Gauge Sector and Symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.2 The seesaw mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Neutrino masses in GUT theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 SU(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.2 S0(10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Leptogenesis 63
5.1 Baryon asymetry of the universe (BAU) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.1 Evidence for BAU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.2 Basic Ingredients and direct baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Leptogenesis in the single flavour approximation . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.2 CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2.3 Out-of-equilibrium dynamics and wash-out . . . . . . . . . . . . . . . . . . . . . . 70
5.2.4 Lepton asymmetry and anomalous B + L violation . . . . . . . . . . . . . . . . . . . 78
5.2.5 Baryogenesis through leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2.6 Dependence on initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Conclusion 82
A Feynman rules for Majorana spinors 84
B Boltzmann equations for leptogenesis 85
B.0.7 Boltzmann equation for N1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B.0.8 Boltzmann equation for B−L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
C Calculation of the strength of the CP violation 88
C.1 Tree-level contribution to Nk → ` φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
C.2 Vertex contribution to the CP asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
C.3 Self-energy contribution to the CP asymmetry (1) . . . . . . . . . . . . . . . . . . . . . . . . 93
C.4 Self-energy contribution to the CP asymmetry (2) . . . . . . . . . . . . . . . . . . . . . . . . 95
2
Chapter 1
Introduction
The proposal of neutrinos goes back to the 1930’s when Wolfgang Pauli postulated their existence as a
mean to hold conservation laws. However, at that time he confessed that such doing was in fact obscene
from a point of view of testability, since such a particle would be undetectable. The establishment of the
Standard Model at CERN and Fermilab has helped us to understand and measure neutrino properties
as being a particle playing a role in a new interaction – the weak isopsin interaction – along with the
electron. Then several other neutrinos have been found, fuelling a new problem called the flavour
problem (which is yet unsuccesfully solved) which culminated with the discovery of neutrino masses
and neutrino oscillations. At first, the full importance of the discovery of neutrino masses didn’t strike
the scientific community, although it was nontheless seen as undeniable proof of physics beyond the
Standard Model. Nevertheless, Fukugita and Yanagida [27]-[28] would link in the late years of the
century this discovery to another puzzling mystery of the universe: the origin of matter. One knows that
the Standard Model is correct up to a incredible accuracy, but it nevertheless can’t be correct as a theory of
matter because it does not provide an answer to how the universe at some point chose matter over anti-
matter. Or, more truthfully, it does qualitatively, thanks to baryon and lepton number non-conserving
non-perturbative effects, but not quantitatively. The work of the two japanese physicists is based on the
premise that without imposing any new symmetries on the Lagrangian function of the Standard Model,
the simplest way (the same way as for the other fermions) to give mass to the neutrinos includes a term
which violates lepton number by two units. A class of models called leptogenesis grew from here and
were able to predict the baryon number asymmetry of the universe.
The first part of this thesis discusses neutrinos. We will go from their role in the Standard Model to
neutrino mass models, taking special interest on the phenomena of neutrino oscillations. The second
part concerns leptogenesisin the single flavour approximation . A selection of calculations performed to
study these topics is appended to this thesis.
3
Chapter 2
Massive Neutrinos
2.1 Fermions in the Standard Model
2.1.1 Gauge Interactions
Our knowledge nowadays goes down to the Fermi scale at a very high accuracy. The widely ap-
plauded Glashow-Weinberg-Salam model of electroweak interactions is based on a generalisation of a
principle from which electromagnetism can be derived: gauge invariance. Indeed by imposing that
L(φ(x), ∂φ(x)) = L(eiθ(x)φ(x))[29] in a covariant theory, Maxwell’s equations can be obtained. The same
premise applies to the so-called standard Model of particle physics, except that it makes use of the
SU(2)L×U(1)Y group where probability amplitudes of fermion fields are grouped in (iso)doublets which
bear a (hyper)charge Y. The three generators Tk of SU(2) in the doublet representation are the Pauli
matrices divided by 2. The covariant derivative has to be corrected in order to erase the terms which are
not gauge invariant (basically which depend on the amplitude and not the probability).
The relation between the third weak isospin generator, the hypercharge and the electric charge is
Q = T3 + Y (2.1)
If g is the coupling constant for SU(2)L and g′ is the coupling constant for U(1)Y then the covariant
derivative is
Dµ = ∂µ − igWµk Tk− ig′BµY = ∂µ − i
esw
(W+µT+ + W−µT−) + ie[QAµ−
1swcw
Zµ(T3 −Qs2w)]
=
∂µ + ieQA − i ecwsw
Z( 12 −Qs2
w) −i W+√
2−i W−√
2∂ + ieQA + i e
cwswZ( 1
2 + Qs2w)
(2.2)
where Wµ, Bµ and Aµ are gauge bosons, W± ≡W1 ∓W2, the neutral gauge bosons being redefined as
4
(B
W3
)=
(cw sw−sw cw
) (AZ
)(2.3)
where c2w + s2
w = 1 and define
g =e
sw(2.4)
g′ =e
cw(2.5)
The fermions are put in representations of SU(2)L ×U(1)Y. It has been found experimentaly that only
the left chirality of fermion fields have isospin. Therefore we may write the fermion spinors as
ψ =1 + γ5
2ψ +
1 − γ5
2ψ = ψL + ψR (2.6)
and only the first term is to be introduced in a isodoublet and the second one in an isosinglet. Hence,
QL =
(uLdL
)`L =
(eLν
)uR, dR, eR. (2.7)
No right-handed neutrino RHνexists in the GWS model. The covariant derivatives of the lepton fields
are immediately derived from (2.2)
DµeL =[∂µ + ieAµ
− ie
cwswZµ(
12− s2
w)]
eL − ie
sw√
2W+ν
Dµν =[∂µ + i
ecwsw
Zµ(12
+ s2w)
]ν − i
e
sw√
2W−eL
DµeR = (∂µ + ieAµ) eR
DµνR = ∂µνR
(2.8)
We included the last derivative for an hypothetical right handed neutrino to state the obvious that it
does not have any gauge interactions. The electroweak interaction is vectored by three different currents,
the electromagnetic, the charged and the neutral ones which are obtained from the conserved Noether
currents for each symmetry
−Lint = gWaµJaµ + g′BµJYµ
=g√
2(J+µW+
µ + J−µW−µ )(2.9)
with
5
Jaµ = ψLγµ
τa
2ψ,
JYµ = −ψLγµYψ
J±µ = J1µ ± iJ2
µ = νLγµeL + uLγµdL
(2.10)
From here, the gauge fields can be rewritten in this new basis upbringing two interactions, the charged
and neutral current interactions:
Lem = −eAµJµem
LCC =e
cw√
2(W+
µ uLγµdL + W−µ eLγ
µν)
LNC = eJemµ Aµ +
gcosθW
J0µZµ
(2.11)
where
Jemµ = −ψγµQψ
J0µ = J3
µ − sin2 θW Jemµ
=12
∑ψ
[gLψγµ(1 − γ5)ψ + gRψγµ(1 + γ5)ψ
] (2.12)
Unfortunately, the gauge symmetry we have shown above is not observed in nature at current
temperatures (T = 2.735 K). The only observed gauge symmetry are SU(3)C ×U(1)Q.
In fact, the absence of quadratic terms of any gauge boson fields shows that this theory is incomplete,
since LEP and Fermilab experiments have established that gauge bosons are massive.
the previous discussion tells us is that a pure gauge theory predicts no masses for either bosons
or fermions at tree-level, so all masses must be generated at a higher radiative order with a certain
mechanism which upon minimization yields solutions which are not gauge invariant even though its
Lagrangian respects the symmetry invariance of the theory.
2.1.2 Masses and Mixings
The solution to this problem was presented by several authors in parallel in 1964[33] with the introduction
of a scalar field based on a the same mechanism that gives mass to soft photons[32]. The simplest way to
break the SM into electromagnetism and QCD is to introduce a scalar field which is an weak isodoublet
φ =
(ϕ+
ϕ0
)(2.13)
6
The additional terms to the SM Lagrangian are
LH = (Dµφ)†(Dµφ) +LYuk + V(φ) (2.14)
with
Dµϕ+ =[∂µ + ieAµ
− ie
cwswZµ(
12− s2
w)]ϕ+− i
e
sw√
2W+ϕ0
Dµϕ0 =[∂µ − i
ecwsw
Zµ(12− s2
w)]ϕ0− i
e
sw√
2W−ϕ0 (2.15)
where V(φ) = −µ2φ†φ +µ2
v (φ†φ)2 (λ > 0 and µ2 < 0). Higher order terms in φ aren’t included to
preserve the theory’s renormalizability. When the Higgs field develops a vacuum expectation value
(VEV) along its uncharged component1
⟨φ0
⟩=
v√
2(2.16)
At spontaneous symmetry breaking (SSB) point, the covariant derivative of the Higgs field yields the
following masses for the gauge bosons
MW =gv2
MZ =MW
cw(2.17)
However, no fermion masses have been generated yet. These come fromtree level diagrams of the
Yukawa couplings which are the most general gauge invariant couplings between the fermion and Higgs
fields:
−LYuk = muQLφuR + mdQLφdR + me ¯LφeR (2.18)
But since masses are still undefined, we can add as many fields as we want with different coefficients
mu,md and me. Indeed, experiment has shown that fermions come in three xerox copies (generations):
there are three up-type quarks (up, charm and top), three down-type quarks (down, strange and bottom),
three charged leptons (electron, muon and tauon) and three neutrinos (νe, νµ and ντ) which have the
exact same gauge SM interactions but have different masses.
−L =∑a,b
[QL
aM(u)a,bφuR
b + QLaM(d)
a,bφdRb + ¯L
aM(`)a,bφ`R
b]
(2.19)
1The charged one can’t have a non-zero VEV because that would break electromagnetic gauge invariance giving mass to thephoton.
7
where the indices run over the three generations of fermions. When the Higgs develops a VEV, each
fermion gets a M( f )a,b = h( f )
a,bv/√
2 mass. The h(u) and h(`) matrices may be chosen freely to be diagonal.
However, because of the CC interaction term, the matrix h(d) cannot be rotated in order to give a diagonal
matrix in this basis. We have a unitary transformation
uL = ULu0L uR = VRu0
R dL = ULd0L dR = VRd0
R (2.20)
The M(d) is a complex non-diagonal matrix whose eigenvalues are the masses of the physical states,
by which we mean the masses of the linear combinations of different fermions fields in a basis where the
mass matrices are diagonal. These can be diagonalized by the following biunitary transformation
ULM(d)V†R = D(d) (2.21)
In the new basis where the mass matrices are diagonal, the fermion mass eigenstrates are
uct
L
→ UuL
u0
c0
t0
L
,
dsb
L
→ UdL
d0
s0
b0
L
,
eµτ
L
→ U`L
e0
µ0
τ0
L
,
νeνµντ
L
→ UνL
ν0
eν0µ
ν0τ
L
uct
R
→ VuR
u0
c0
t0
R
,
dsb
R
→ UdR
d0
s0
b0
R
,
eµτ
R
→ U`R
e0
µ0
τ0
R
(2.22)
A unitary transformations which takes a linear combination of the fermion fields and leaves the
gauge interactions and the kinetic terms unchanged is called a weak basis transformation (WBT). So let’s
picture a general WBT:
M′u = W†LMuWuR = Uu
LDuV†R
M′d = W†LMdWdR = Ud
LDdV†R (2.23)
By choosing WL = UuL, Wu
R = UuR and Wd
R = UdR, we get
Mu = Du
M′d = Uu†L Ud
LDd
(2.24)
So in this particular basis, one reduces the 36 real free parameters formely contained in the Mu and
Md matrices down to six masses and four mixing parameters from VCKM ≡ Uu†L Ud
L. The CKM matrix2
2owed to the names of Cabbibo-Kobayashi-Maskawa
8
has 18 real parameters for 3 generations of fermions. Its unitarity implies 9 constraints. Additionally,
5 phases unphysical can be cancelled through redefinitions of the fermion fields. Therefore, one has in
the end 4 parameteres, of which 3 are mixing angles and one is a phase (with physical meaning). One
common parametrization is[34]
VCKM =
c12c13 s12c13 s13e−iδ
−c23s12 − s23s13c12eiδ c23s12 − s23s13s12eiδ s23c13s23c12 − c23s13c12eiδ
−s23c12 − c23s13s12eiδ c23c13
(2.25)
where ci j ≡ cosθi j and si j ≡ sinθi j. Experimental values for the magnitudes of the 9 CKM elements
are[30]
|VCKM| =
0.97383+0.00024
0.00023 0.2272+0.001−0.001 (3.96+0.009
−0.009) × 10−3
0.2271+0.001−0.001 0.97296+0.00024
−0.00024 (42.21+0.1−0.8) × 10−3
(8.14+0.32−0.64) × 10−3 (41.61+0.12
−0.78) × 10−3 0.9991+0.000034−0.000004
(2.26)
The introduction of this subtilty of multiple generations changes the form of the interactions. The
weak charged current for quarks becomes
LqCC =
g√
2uiLγ
µ(VCKM)i jd jLW+µ + h.c. (2.27)
which violates flavour, meaning through this very same interaction there can be transitions between
different eigenstates of a particular weak basis, such as transitions between different mass eigenstates.
One interesting aspect of the multiple generation issue is that three generations imply CP violation. In
fact, one can show[36] that there is a quantity which is independent of the chosen weak basis and must
be zero for CP conservation[40]
Im(Vi jVklV∗ilV∗
kj) = JCKM
3∑m,n=1
εikmε jln, (i, j, k, l = 1, 2, 3) (2.28)
which in general is of course not zero. The value of this so-called Jarlskog invariant in the previsouly
introduced parametrization is
JCKM = c12s12c213s13c13s23 sin δ (2.29)
where δ is the only unknown parameter and is called the CP violating phase.
It is important to emphasize that flavour violation only occurs in the quark sector. The major difference
lies in the absence of neutrino masses. Indeed, the leptonic charged current is
9
LCC =g√
2¯eiL(VL)∗iαγ
µναLW−µ + h.c. =g√
2¯eiL(VL)∗iα(VL)α jγ
µν jLW−µ + h.c. =g√
2¯eiLγ
µνiLW−µ + h.c. (2.30)
where να = (UL)α jν j and we can chooseUνL = U`
L because there are no right-handed neutrinos. This
explains the absence of leptonic mixing processes.
2.2 Massive Neutrinos
2.2.1 Majorana Mass
A mass term is a second-order coupling of two fields which is Lorentz invariant. The following quantity
is invariant under Lorentz transformations [8]:
χTL (iσ2)ψL → χT
LSTL (iσ2)SLψL = χT
L (iσ2)ψL (2.31)
where ψL and χL are two left handed spinors. If one chooses χL = (iσ2)ψ∗R = ˆ(ψR) this term becomes
(iσ2ψ∗R)T(iσ2)ψL = ψ†RψL (2.32)
which is the so-called Dirac mass term which all matter bear (ψ now stands for a four-component
spinor):
m(ψ†RψL + ψ†LψR) = mψ†γ0ψ = mψψ = m(ψLψR + ψRψL) (2.33)
The free Dirac field operator ψ(x) is
ψ(x) =
∫d3p√
(2π3)2Ep
∑s=±1/2
[fs(p)us(p)e−ip·x + fs
†
(p)vs(p)e−ip·x]
(2.34)
where fs(p) and fs†
(p) are the annihilation and creation operators respectively of a Dirac particle with
momentum p and spin s along the momentum direction. The u and v functions are plane wave solutions
of positive and negative energy respectively, given by the Dirac equation
(/p −m)us(p) = 0 (2.35)
(/p + m)vs(p) = 0 (2.36)
(2.37)
10
In the Dirac representation, and when they when they are normalized according to
u†s (p)u†s′(p) = v†s (p)v†s′(p) = 2Epδss′ (2.38)
these functions are
us(p) =√
E + m(
χsσ·p
E+mχs
)(2.39)
vs(p) =√
E + m( σ·p
E+mχ′s
χ′s
)(2.40)
where
χ↑ = −χ′↓ =
(10
)χ↓ = −χ′↑ =
(01
)(2.41)
Now let’s imagine a Dirac field such an electron propagating along the x-direction. It has four degrees
of freedom which are the four spinors above used to describe it. For both electron and positron states
there are two helicity states: therefore the spinors are two with positive energy, with left-handed eL and
right-handed eR helicities, and two of negative energy eL and eR. Now let’s have an observer moving
slower than the electron along the same direction. Supposing it measures a positive spin along this
direction, the field it finds is eR, because the result of his measurement is that spin and momentum are
parallel. But another observer moving faster than the electron actually sees a left-handed field because
now the momentum will look reversed and spin is now anti-parallel to it. Still, there are two fields LH
fields, eL and eL. The right choice is eL of course, because it has the same charge as eR which is a Lorentz
invariant, so can’t be changed by a boost transformation.
Things are a little less clear in the case of a neutrino because it is neutral. There is no way to choose
– and no way to distinguish by measurement – between these two RH states. So instead of somewhat
unaturally postulating the existence of two extra states (νL and νR), we can wonder whether the neutrino
is not a particle with just two degrees of freedom, therefore imposing νL = νL and νR = νR; the neutrino is
its own anti-particle. Such a fermion field was first introduced by Majorana in 1937, and took his name.
To build such a field one would be tempted to write
ψ(x) = ψ∗(x) (2.42)
but these quantities do not transform the same way under a general Lorentz transformation. For
instance, under a transformation
11
x′µ = xµ + ωµνxν (2.43)
the spinor field changes
ψ′(x′) = exp(−
i4σµνω
µν)ψ(x) (2.44)
where
σµν =i2
[γµ, γν] (2.45)
and
ψ′∗(x′) = exp( i4σ∗µνω
µν)ψ∗(x) (2.46)
= exp( i4σ∗µνω
µν)ψ(x) (2.47)
so unless Re(σµν) = 0, equality (2.43) is not covariant. This is not true for the Dirac representation.
However the conjugate field
ψ(x) ≡ γ0Cψ∗(x) (2.48)
transforms the same way as ψ(x) if γ0Cσ∗µν = σµνγ0C. In the Dirac representation,
C = iγ2γ0 =
(0 iσ2
iσ2 0
)(2.49)
We can also define the a representation, called the Majorana represetation, where the γ-matrices are
purely imaginary and the relation ψ and ψ∗ transform the same way. In this representation obviously,
C = −γ0 =
(0 σ2σ2 0
)(2.50)
Thus, the general definition of a Majorana field is
ψ(x) = eiθψ(x) (2.51)
Its operator in the plane wave expansion is
12
ψ(x) =
∫d3p√
(2π3)2Ep
∑s=±1/2
[fs(p)us(p)e−ip·x + eiθ f †s (p)vs(p)e−ip·x
](2.52)
Relation (2.51) is easily verified.
2.2.2 Properties under discrete space-time symmetries
Charge conjugation Under the charge conjugation operation C, a free Majorana field transforms as
Cψ(x, t)C−1 = e−iφγ0Cψ∗(x, t) = e−iφφ(x, t) = e−i(θ+φ)φ(x, t) (2.53)
where φ is a phase. Using the last result and (2.52)
C fs(p)C−1 = e−i(θ+φ) fs(p) (2.54)
with φ = −θ. In a C-symmetric vacuum, (2.54) imples that
C
∣∣∣p, s⟩ = ei(θ+φ)∣∣∣p, s⟩ = ±
∣∣∣p, s⟩ (2.55)
showing that a free Majorana particle is an eigenstate of C3.
Charge-Parity conjugation A free Majorana field transforms under P as
Pψ(x, t)P−1 = eiϕγ0ψ(−x, t) (2.56)
Together with equation (2.53), we get
(CP)ψ(x, t)(CP)−1 = −ei(ϕ−φ)γ0ψ(−x, t) (2.57)
The plane wave expansion reads
γ0ψ(−x, t) =
∫d3p√
(2π3)2Ep
∑s=±1/2
[fs(−p)γ0us(−p)e−ip·x + eiθ f †s (−p)γ0vs(−p)e−ip·x
]=
∫d3p√
(2π3)2Ep
∑s=±1/2
[fs(−p)us(p)e−ip·x
− eiθ f †s (−p)vs(p)e−ip·x] (2.58)
Consequently,
3This is not true in general as an interacting Majorana particle might violate C.
13
(CP) fs(p(CP)−1 = −ei(φ−ϕ−θ) fs(−p) (2.59)
with φ − ϕ − θ + π = 0. Therefore, again,
CP
∣∣∣p, s⟩ = ±∣∣∣−p, s
⟩(2.60)
Thus, if interactions are CP conserving then a Majorana neutrino is an eigenstate of this transforma-
tion. This statement can also be used as a good approximation when CP is only slightly violated, such
as it is by the weak interaction in the quark sector and hopefully in the leptonic sector too.
Charge-Parity-Time reversal A Majorana field transforms under CPT as
(CPT )ψ(x)(CPT )−1 = −eiξγT5ψ∗(−x)
= −e−i(ξ+θ)γT5 C−1γ0ψ(−x)
(2.61)
The expansion in terms of creation and annihilation operators is
γT5 C−1γ0ψ(−x) =
∫d3p√
(2π3)2Ep
∑s=±1/2
(−1)s+1/2[
f−s(p)u∗s(p)eip·x− eiθ f †−s(p)v∗s(p)e−ip·x
](2.62)
where we have used
γT5 C−1γ0us(p) = (−1)s−1/2u∗−s(p) (2.63)
γT5 C−1γ0vs(p) = (−1)s+1/2v∗−s(p) (2.64)
from which we get
(CPT ) fs(p)(CPT )−1 = ei[(s−1/2)π−θ−ξ] f−s(p) (2.65)
(CPT ) f †s (p)(CPT )−1 = ei[(s+1/2)π+θ−ξ] f †−s(p) (2.66)
with ξ = ±π/2. Finaly, we prove once again that
CPT
∣∣∣p, s⟩ = ei[ξ+θ+π(s−1/2)] (2.67)
To say that a Majorana field is an eigenstate ofCPT is a more profound statement as it is believed that
interactions which can be described in field theory are CPT -invariant, so this statement (which means
that neutrino is its own anti-particle) is also valid for a physical particle in an interacting medium which
may not conserve C or CP.
14
2.2.3 Majorana and Dirac mass terms
As already stated above, let’s consider two spinors ψ and χ. Then one can build the following term
χTL (iσ2)ψL (2.68)
which is invariant under Lorentz transformations for which we can choose χL = (iσ2)ψ∗R = ˆ(ψR), hence
yielding the Dirac mass term
mψψ (2.69)
But by choosing χL = ψL one gets this other term which is also Lorentz invariant
m2
[ψTL (iσ2)ψL + ψ†L(iσ2)ψ∗L] (2.70)
The first term is called the Dirac mass term which the mass term all fermions bear. It is U(1) invariant
so it conserves all quantum numbers carried by the fields. However, the second term is called a Majorana
mass term and is not U(1) invariant. It therefore breaks any quantum number carried by the field. Thus,
no particles carrying charge or colour can have a Majorana mass term because of the forbidance imposed
by the SU(3)C × U(1)Q symmetry. This is the case of all fermions except for the neutrino which is both
neutral and black (colourless).
A Majorana field has only half the components of a Dirac field. Hence, it might be possible to describe
a Dirac field in terms of two Majorana spinors. Consider the field ψL and its CPT conjugate ψR which
can form a Majorana spinor. Similarly, another one can be formed out of χL and χR. The possible mass
terms are
12
mψψLψR +12
mχχLχR +12
mψLχR +12
m′χLψR + h.c. (2.71)
Now let’s suppose
Ψ =
(ψLχR
)and Ψ =
(χLψR
)(2.72)
The mass lagrangian can be rewriten as
−Lmass =12
(ψL χL
) ( mψ/2 mm′ mχ/2
) (ψRχR
)+ h.c. (2.73)
15
Let’s write it in a more compact way as
−Lmass =12
∑i, j
ΨLi Mi jΨ
Rj + h.c.
=12
∑i, j
ψTi C−1Mi jPRγ0CγT
0ψ∗
j + h.c.
=12
∑i, j
ψ†jγ0
(C−1PRC
)TMi jψ j + h.c.
=12
∑i, j
ΨLj M jiΨ
Ri + h.c.
(2.74)
from which under permutation of the dummy indices Mi j = M ji. So clearly, m = m′. Removing the
diagonal terms which break phse symmetries in the mass matrix, the final mass Lagrangian is
12
(ΨLΨR + ¯ΨLΨR
)+ h.c. (2.75)
which the mass term for a Dirac field, proving that it can be constructed with two Majorana
fermions[12].
2.2.4 Experimental tests on neutrino mass
Given current experimental results, at least two neutrinos are for sure massive. We present here a list of
experimental tests of the neutrino mass hypothesis.
Kinematic tests
• Nuclear β-decay: The energy distribution of electrons emitted in this decay can be calculated for
mνe = 0. Existence of electron neutrino mass will reduce the peak’s energy. Studies of tritium decay
have been conducted throughout the world to look for deviations of the Kurie plot. The most
stringent bound on neutrino mass from tritium experiments came from a russian group in Troisk
that obtained mνe < 2.5 eV[54].
• Pion decay: accurate measurements of the π+→ µ+νµ decay rate can determine the mass of the νµ.
One recent experimental bound sets mνµ < 190 keV[55].
• Tau decay: There are several decay modes which can be used to determine the mass of ντ, some of
them which are even semileptonic thanks to the tauon’s heavy mass. Experiment has set mντ < 18.2
MeV[56].
16
Exclusive tests The following processes only happen if neutrino masses exist. Therefore a valid sig-
nature of these processes is an undeniable proof of neutrino masses. These tests are based on neutrino
mixing which isn’t necessary for neutrino masses but is more than natural in gauge models.
• neutrino oscillation: this phenomenon which might explain the Solar Neutrino Problem according
to the Sudbury Neutrino Observatory data, changes the flavour of neutrinos as they propagate. This
is due tothe fact that the mass basis is different from the weak basis and different mass components
in a given neutrino flavour will evolve differently.
• neutrino decay: massive neutrinos with flavour mixing can decay in flavour violating processes
such as να → νβ + γ which arise as an effective interaction lagrangian of the form[24]
Lint =12νiσαβ(µi j + εi jγ5)ν jFαβ + h.c. (2.76)
where µi j and εi j are the magnetic and electric transition moments. If neutrino masses are in a
sub-eV scale, cosmic microwave background (CMB) simulations set the bounds on the neutrino
radiative lifetime at 1011− 1012yrs[21].
Two other tests are also important. Massless neutrinos only have one electromagnetic form factor,
its charge radius. Measurement of other properties such as a magnetic moment would imply a mass.
Neutrinoless double beta decay –where two neutrons in the nucleus decay into two protons and two
electrons – is an important signature of Majorana-like neutrinos since the process requires the Majorana
field’s invariance under CPT , making the process impossible with Dirac-type neutrinos.
2.2.5 Neutrinoless double-beta decay
Neutrinoless double-beta decay (ββ0ν) is a lepton number violating process which can only occur if
neutrinos are Majorana. For Dirac neutrinos or massless neutrinos in the Standard Model the only
similar process that occurs is
n + n→ p + p + 2e− + 2νe (2.77)
which has a lifetime of 1018−1021yr depending on what nucleus is involved, and the enegies released
in the beta rays are of about a few MeV. But, thanks to the Majorana neutrino’s CPT invariance, the
diagram in figure 2.2.5 is possible (c.f. Appendix A for the Feynmann rules of Majorana spinors).
17
p
pn
n
νL
νL
W
W
e−L
e−L
Figure 2.1: ββ0ν diagram possible thanks to the CPT invariance of the Majorana fermion propagator.
This process is the number one test to assess the type of fermion mass the neutrinos bear. We will do
a somewhat short kinematical estimation of the ratio between τββ0ν and τββ0ν[13]. Each interaction rate
can be written as
Γ2e2ν = 2π∫|M2e2ν|
2δ(Mi −M f − E1 − E2 − E3 − E4) dΠ1 dΠ2 dΠ3 dΠ4
∝
∫dE1p1E1
∫dE2p2E2
∫dE3p3E3
∫dE4p4E4|M2e2ν|
2δ(Mi −M f − E1 − E2 − E3 − E4)
∝
∫ ∆
0dy
∫ y
0dy2|M2e2ν|
2p1p2E1E2δ(∆ − y)
∼ Q5|M2e2ν|
2 (2.78)
where
∆ = Q/me = (Mi −M f − 2me)/me (2.79)
x = (E2 −me)/me (2.80)
y = (E1 + E2 − 2me)/me (2.81)
Similarly,
Γ2ν0β ∝
∫ ∆
0dy
∫ y
0dy2|M2e2ν|
2p1p2E1E2δ(∆ − y)
∼ Q5|M2e2ν|
2 (2.82)
18
We have assumed for this several crude approximations for which a short discussion is needed. We’ve
taken the amplitude matrix as being independent of the phase-space coordinates. For the case of a light
Majorana neutrino exchange as shown in figure 2.2.5 leads a amplitude matrix proporcional to
M = Mµνhadron
[ue(p1)γµPL]α[
/q + mν
q2 −m2ν + iε
C]αβ[ue(p2)γνPL]β − (1↔ 2)
= Mµνhadron
ue(p1)γµPL
/q + mν
q2 −m2ν
ue(p2)γνPRve(p2) − (1↔ 2)
=mν
q2 −m2ν
Mµνhadron
ue(p1)γµPLue(p2)γνve(p2) − (1↔ 2)
(2.83)
where we have written the full matrix as a product of a hadronic amplitude matrix times a leptonic
one (which we have explicited). This bit turns out to be more-or-less proportional to the energy of the
fermions. That’s why we have introduced a factor pEdE to mimic the dependence of the matrix on the
phase-space coordinates.
From our discussion so far, we’ve estimated that
Γ2e0ν
Γ2e2ν∼
1Q6|M2e0ν|
2
|M2e2ν|2(2.84)
A relation between these two amplitudes must be found. For that matter, picture the neutrino-
less diagram as the standard double-beta decay but where the two neutrinos have the same product
momentum k, plus some lepton violation. That way, we can empirically write
M2e0ν ∼ ε∆L
∫d3k
(2π)3M2e2ν (2.85)
and therefore
Γ2e0ν
2e2ν∼ ε2
∆ε(ωQ
)6 (2.86)
Q is usually of order ∼ 2 − 3MeV. On the other hand if we consider the nucleus to be a Fermi gas of
nucleons in a sphere of radius R = R0A1/3 = (1.2fm)A1/3, where A is the total number of neutrons then in
the simple Fermi-gas approximation, the state density is
n =2V
(2πh)3
∫ pF
0d3p =
Vp3F
3π2~3 (2.87)
which conversely gives
19
pF = ~(3π2 n
V
)=
pF,n = ~r0
(9πN4A
)1/3for the neutron
pF,n = ~r0
(9πZ4A
)1/3for the proton
(2.88)
where N and Z are the number of neutrons and protons respectively, and from which we can deduce
the average Fermi kinetic energy per nucleon (for N = Z = A/2)
〈E〉p =
∫ pF
0 Ed3p∫ pF
0 d3p=
35
p2F
2m=
310m~2
r20
(9π8
)2/3≈ 200MeV (2.89)
(where m is the mass of a nucleon) so that the energies carried by the neutrino lines will be of about
200MeV. At present, the experimental bounds on this process are Γ2e0ν/Γ2e2ν < 10−4, or more importantly
ε∆L . 10−8.
2.2.6 Neutrino Mass Terms
In regard to this new phenomenology one has to extend in some way the SM to include neutrino masses.
But the mass term for neutrinos does not have to be as constrained as the mass term for the other charged
fermions. Mass terms have to be both invariant under the Lorentz and gauge groups.
The Dirac mass is the one we have used for the fermion propagation so far. It is well behaved under
a gauge symmetry and therefore conserves any quantum number associated with the dirac field. On the
other hand, the majorana mass terms violate any quantum number the majorana field may carry for they
are not U(1) invariant. Since all other fermion fields are charged, they should only have dirac mass terms
because we know – up to a good experimental certainty and theoretical one4 – that the universe is neutral
and charge non-conservation in the theory is greatly undesired. Nevertheless, neutrinos lack this charge
which is casted upon every other known fermion and they can endure a Majorana mass term, but one
that one still violate lepton number (L) by two units. One such process that violates this number is the
neutrinoless double β-decay (i.e.next section for further analysis)whose detection stands in the front line
for majorana mass measurements. Now there are three ways to introduce neutrino masses with minimal
change to the current picture of the SM. All terms in the Lagrangian must be gauge invariant so we want
singlet operators. Possible extensions of the SM can be categorized crudely as:
• extend the fermion sector
• extend of the higgs sector
4It has been show by Landau and Lifschitz that for an spherically symmetric closed space net charge of the universe mustvanish[5]
20
• extend both
To take a better look at this matter we need to know what these new operators mean in terms of the
SM gauge group’s SU(3) × SU(2) ×U(1) representations [4]:field label SM representation
lepton doublet lL (1, 2,−1/2)charged lepton singlet eR (1, 1,−1)
Higgs doublet φ (1, 2,−1/2)Higgs doublet φ (1, 2, 1/2)
Extension of the fermion sectorProduct Operator SM rep. required field mass type
lL ⊗ lL (1, 1,−1) singlet scalar η++ LH MajoranalL ⊗ lL (1, 3,−1) triplet scalar ∆ LH MajoranaeR ⊗ eR (1, 1,−2) singlet scalar k++ RH MajoranaνR ⊗ νR (1, 1, 0) singlet scalar k0 RH Majorana
Extension of the Higgs sectorlL ⊗ φ (1, 1, 0) lepton singlet νR DiraclL ⊗ φ (1, 3, 0) lepton triplet Σ LH Majorana
Table 2.2.6 shows what products one can build and the fields one has to introduce in order to make
gauge invariant mass terms. The simplest extension of the SM is to include add RH neutrinos that couple
to the LH component via the usual Yukawa mass term MDlLφνR + h.c.. But in doing so one rejects the
possibility of lepton number violation through majorana mass terms. Plus, the coupling would have
to be extremely small in order to suppress the neutrino masses. One way to naturally suppress the
neutrino masses are the seesaw models which come in three different types and to which we will focus
our particular care because of their importance for leptogenesis.
One might argue that including RHνis not new physics because it is simply a matter of making neutrinos
just like the other fermions. However, if we include RHνthe Majorana mass term has to be included
as we have shown above, because it is allowed by both Lorentz and gauge symmetries. So, if RHνare
indeed included, no matter what experimental outcome might be proven, there is always new physics in
the neutrino sector, which will be either a new symmetry that forbids Majorana terms or, in the opposite
case, lepton number violation. And even if RHνaren’t found, new physics will appear not in the fermion
sector but in the scalar Higgs sector, with the introduction of scalar triplets. So the discovery of neutrino
masses is a very important sign of new physics.
21
2.2.7 Seesaw Models
Type I Seesaw
Motivation from the SO(10) group symmetry The simplest non-abelian group that can accomodate
the Standard Model is SU(5). Nevertheless, besides experimental bounds on the proton stability ruling
out the minimal SU(5) Grand Unification Model (GUT), nothing indicates that grand unification should
stop at SU(5) [8]. SU(5) ×U(1) is in its turn a subgroup of SO(10) and since SU(3) is a subgroup of SO(6)
and SU(2) is a subgroup of SO(4) this group can clearly accomodate the SM in this orthogonal group.
One way to assign fermions of the SM to representations of SO(10) is to first link a representation of
SO(10) to the 5 + 10 reps. of SU(5) (where one SM fermion family neatly fits).
It is striking that one fundamental representation of SO(10) alone can provide for all the Standard
Model fermions plus one SU(5) singlet which one can associate with the right-handed neutrino. The
simple Yukawa mass term that couples one 16 to another only hands a Dirac mass for the neutrino. Even
so, other couplings using the 120 and 126 reps. do allow Majorana masses. The fist type of seesaw deals
with the inclusion of a right-handed neutrino with both Dirac and RH Majorana mass terms.
Singlet fermions This model[38] introduces electroweak singlet right-handed neutrinos as one already
has for the other fermions. The only difference is that Yukawa couplings can also give RH majorana
masses whereas for the other fermions the only possible mass is the Dirac one. Because we want to link
them to some GUT scale at which B + L symmetry is broken it is only natural that these new particles
have fairly large masses. The extended neutrino mass Lagragian becomes then:
Lν = −Mi jDν
iLν
jR + 1
2 Mi jR
¯(νiR)cν
jR + h.c.
= − 12
¯(nL)c
(0 M∗D
M†D M∗R
)︸ ︷︷ ︸
CM∗
σ2nL + h.c. (2.90)
where nL = (νL, (νR)c)T we have used the identity:
νLνR =12
(νLνR + ¯(νR)c(νL)c
)(2.91)
The 3× 3 Dirac mass matrix MD is unitary mixing matrix times the Higgs VEV MD = λi jv/√
2, where
v = 〈φ〉 ≈ 174GeV. We now would like to find the mass eigenstates of M in the electroweak basis to write
the charged and neutral current interaction Lagrangians. Since M is symmetric, it is then diagonalized
by an orthogonal transformation such as [2]:
22
VTM∗V = D, V =
(K3×3 R3×3S3×3 T3×3
), D = Diag(d3×3,D3×3) (2.92)
The neutrino weak-eigenstates are twofold and related to the mass eigenstates by:
νi = Ki jνjL + Ri jνR
j
Ni = Si jνjL + Ti jνR
j (2.93)
Assuming MR MD, the first eigenstates will have a mass d and the second D. The first ones are the
light neutrinos mostly composed of the left-handed component as implied from low-energy observations.
Its mass is given by the usual seesaw relation; from (2.92)
S†MTDK∗ + K†MDS∗ + S†MRS∗ = d (2.94)
S†MTDR∗ + K†MDT∗ + S†MRT∗ = 0 (2.95)
T†MTDR∗ + R†MDT∗ + T†MRT∗ = D (2.96)
(2.95) leads to
S† ≈ −K†MDM−1R (2.97)
which holds up to a very high degree of accuracy. Together with (2.94) we get the usual seesaw
formula[39]
−K†MDM−1R MT
DK∗ = d (2.98)
Similarly,
D ≈ T†MRT∗ (2.99)
where the suppression mechanism becomes obvious become as MR gets bigger, d gets smaller, which in
view of a GUT scenario in which MD MR gives a natural explanation for the smallness of the observed
neutrino masses.
Let’s examplify what was written above for the single generation case. In this case, leptons only come in
one flavour and the mass matrix reads
M =
(0 mD
mD mR
)(2.100)
23
where the entries in this matrix are simply real positive numbers for the time being. An orthogonal
matrix can be paramatrized as
V =
(cosθ − sinθsinθ cosθ
)(2.101)
where tan 2θ = 2mD/mR and which then diagonalizes the mass matrix as
VTM∗V =
(−m1 0
0 m2
)(2.102)
where
m1,2 =12
(∓mR +
√m2
R + 4m2D
)(2.103)
Since both these values are either real or complex, they can’t be the physical value of the mass. So the
previous has to be rewritten as
VTM∗V =
(m1 00 m2
)·
(−1 00 1
)= dK2 (2.104)
and
M = VdK2VT (2.105)
Now, the mass eigenvectors are
(n1Ln2L
)= O
(νLNL
)=
(cosθ − sinθsinθ cosθ
) (νLNL
)(2.106)
and
(n1Rn2R
)= K2O
(νRNR
)=
(− cosθ sinθsinθ cosθ
) (νRNR
)(2.107)
In this basis the mass Lagrangian is diagonal. The one generation two eigenstates are
n1 = n1L + n1R = cosθ(νL − νR) − sinθ(NL −NR)
n2 = n2L + n2R = sinθ(νL + νR) + cosθ(NL + NR)(2.108)
with masses m1 and m2 respectively. Now let us see how this helps in explaining naturally the
smallness of neutrino masses. In a natural explanation, mD should be of the order of the fermion masses,
whereas mR being linked to a high-scale energy at which a GUT theory becomes evident then mR mD
and
24
m1 ∼m2
D
mRm2 ∼ mR (2.109)
and
| cosθ| ∼ 0.92
| sinθ| ∼ 0.38(2.110)
and it follows neatly that m1 mD, rendering the left handed neutrinos (n1 ∼ (νL − νR) - hence
much smaller than the usual fermion scale. The seesaw mechanism can hence explain the smallness of
the fermion scale. However, the cost of it is the same as in any GUT theory which is the insertion of a
high scale. Since cosmological observations now constrain the masses of the light neutrinos down to a
sub-eV scale, the mass of the RHνmust be of about 106TeV, which is far above the reach of any present
accelerator, and probably also of any future ones.
CC interaction The leptonic charged current interaction changes with the addition of these heavy mass
states. It bears now one extra term
LCC = −g
2√
2( ¯iKi jγ
µν j + ¯iRi jγµN j)W−µ + h.c. (2.111)
where `i are the charged leptons with flavour i = e, µ, τ and ν j and N j are the light and heavy mass
eigenstates respectively, with j = 1, 2, 3. In the total decoupling limit, R may be neglected and the
conventional CC lagrangian is recovered, with the light neutrino states interactions with the charged
leptons fields being described by the K matrix.
Parametrizations The light neutrino mass matrix is diagonalized by the K matrix[14]. The SM already
has nine low-energy parameters (three light masses, three mixing angles and three CP violating phases),
the insertion of three heavy RH neutrinos adds nine new parameters: three heavy masses, three mixing
angles and three other CP violating phases. Then there yet another three parameters which are the CP
parities of the RH neutrinos. Therefore there are in total 21 real degrees of freedom for the lepton sector.
There are several ways to parametrize these, namely the top-down approach in which one fixes the high-
energy sector to reconstruct the low-energy one or the other way round, the bottom-up parametrization
where the low-energy sector is fixed instead, or yet the Casas-Ibarra parametrizations[37] which is an
intermediate one and often very handy for calculations. The latter uses the fact the since K is a unitary
hermitian matrix it can be diagonalized by a 3 × 3 orthogonal matrix O which is parametrized in terms
25
of three real angles c(s)i j = (1− δi j)cos(sin)θi j and one Dirac-type CP violating phase δ and another two of
the Majorana-type α and β:
K =
c12c13 s12c13 s13e−iδ
−c23s12 − s23s13c12eiδ c23s12 − s23s13s12eiδ s23c13s23c12 − c23s13c12eiδ
−s23c12 − c23s13s12eiδ c23c13
·
1 0 00 e−iα 00 0 e−iβ
(2.112)
Out of the 18 parameters (3 of the 21 are CP parities choosable at will) three of them are fixed from
solar and atmospheric neutrino observations[1]:
rl∆m2 ≡ m2
2 −m21 =
(7.9+1.1−0.89
)× 10−5eV2 (2.113)
s212 = 0.31+0.07
−0.05 (2.114)
∆m2A ≡ |m
32 −m1
1| =(2.6+0.6−0.6
)× 10−3eV2 (2.115)
s223 = 0.47+0.17
−0.15 (2.116)
The following parameters remain free: neutrino mass scale mmin (smallest neutrino mass), the mass
ordering (sign of m2A), θ13, δ, α and β, and 3 RH masses.
Among the nine unconstrained parameters on the list, the first four are expected to be measured soon,
but the other eleven are beyond our hopes for a near future, which is a major predictive handicap for the
theory. Nonetheless, there are several ways to reduce this high number.
A set of invariants equivalent to the Jarlskog invariant, measuring CP violation in the leptonic sector can
be found by defining[41]:
sαi j = Im[UαiU∗α jR∗
i R j]
tαiβ j = Im[UαiUβ jU∗α jU∗
βi](2.117)
where U = UPMNS and R = diag(1, e−iα, e−iβ) are the two elements composing the K matrix. The
minimal set of independent rephasing invariant quantities is JCP = tαi13, J1 = s113, J2 = s123, which has the
advantage that JCP mimics the Jarlskog invariant for the CKM matrix and only enters processes which
violate CP, whereas J1 and J2 enter only lepton number violating interactions. After parametrization its
values are
JCP = −c12c23c213s12s23s13 sin δ
J1 = −c12c13s13 sin(α − δ) (2.118)
J2 = −s12c13 sin(α − β − δ)
26
which shows that there can still be lepton-number violating processes even if there is no CP violation
(δ = 0); on the other hand, if α = δ and β = 0 implies that there is no CP violation in any lepton number
conserving processes like neutrino oscillations.
One should still notice that current experiments only state that at least two neutrinos are massive.
Therefore, only two RH neutrinos are required. This aprticular hypothesis can be seen as the limit where
the third heavy neutrino decouples away because of its very high mass or relatively small couplings. In
such a case, the lightest neutrino can be aknowledged as effectively massless, thus removing another two
parameters from the theory (mmin and α).
Before moving ahead to the next seesaw mechanism, let’s study how WBTs happen in the leptonic sector.
These are
n′L = WLnL, (2.119)
`′L = WL`L, (2.120)
`′R = WR`R (2.121)
The mass matrices become
M′` = W†LM`WR, (2.122)
M′ν = WTL MνWL (2.123)
If we wish these to be the diagonal basis then
M` = U`LD`U`†
R , (2.124)
Mν = Uν∗L D`Uν†
L (2.125)
where Dν and D` are real and positive. The WBT used here is WL = U`L and W`
R = U`R
M′` = D`,M′ν = K∗DνK† (2.126)
It is important to notice that RHνrotations aren’t free. Hence, negative eigenvalues of the charged
lepton mass matrix are permitted since we are allowed to rotate it at will, but the same is not true for the
neutrinos, since any phases held by the eigenvalues will appear in the Majorana phases.
27
Type II Seesaw
In the previous section we kept ourselves from introducing left-handed Majorana masses because – i.e.??
– the product of two LH lepton fields breeds a singlet operator as well as an SU(2) triplet which must be
made gauge invariant thanks to some coupling to a new triplet Higgs field ∆ with Y=2:
σ · ∆ =
(∆+
√2∆++
√2∆0
−H+
)(2.127)
which has a non-zero VEV along its neutral component⟨H0
⟩= v∆/
√2. This triplet gives rise to the
additional Yukawa coupling and trilinear scalar coupling:
f ¯Lσ · ∆lL + µφσ · ∆φ + h.c. σ · ∆ ≡ σ2σ · ∆∗ (2.128)
which results in a v∆ f ˆLνL Majorana mass term for the neutrino as ∆ develops a VEV. It has been
shown [15] that for a general Higgs multiplet with weak isospin T and weak hypercharge Y:
ρ ≡M2
W
M2Zcos2θW
=ΣT,Y|vT,Y|
2[T(T + 1) − Y2/4]2ΣT,YY2/4
=v2φ + 2v2
∆
v2φ
+ 4v2∆
(2.129)
where the last result holds for our scalar triplet. Since current experiments have constrained the
value of rho to be ρ = 0.998 ± 0.005 which forces (v∆/vφ) < 0.17 (at 1σ) or v∆ < 30GeV which explains the
suppression for the neutrino masses. This model has eleven additional free parameters out of which nine
can be determined from the light neutrino parameters, while the remaining two (the triplet’s complex
coupling to the higgs doublets). The motivation for adding a different Higgs multiplet comes from GUT
theories where often one has to use higher representations of the Higgs to break the symmetry; e.g., only
SU(5) models with (no SUSY and) scalar representations of higher dimension than 5 allow the proton
decay predictions to behave decently in regard to observational constraints. Soft CP violation – where
the latter arises from SSB – also requires more complex multiplets to create the complex VEV.
Type III Seesaw
The third and last type of seesaw introduces the adjoint SU(2) representation triplet fermions Σ. The
relevant Lagrangian is:
λΣ lφΣ −12
MΣΣ (2.130)
and its contribution to the neutrino masses is ∝ v2λΣi jM−1j λΣkj, which is similar to the type I formula.
Again, this model has, like the first type, eighteen parameters beyond the SM’s.
28
Chapter 3
Neutrino Oscillations
3.1 Neutrino oscillations in vacuum
Neutrino oscllations mean that a beam of neutrinos of a specific flavour can change along their trajectory
into another flavour[16]. As stated earlier, if neutrinos are massive, their mass matrix linking weak and
mass eigenstates will be non-diagonal:
να =∑
i
Uαiνi (3.1)
where the greek index run the flavours and the latin one covers the mass eigenstates. U is a unitary
matrix which can be parametrized as the CKM matrix. Usually one assumes that all neutrinos have a
fixed momentum p1. Let’s suppose a neutrino in a beam which is born in a definite flavour α at time
t = 0, then the wave function will be
|να(x, t = 0)〉 =∑
i
eipxUαi |νi〉 (3.2)
and evolves with time as
|να(x, t)〉 =∑
i
ei(px−Eit)Uαi |νi〉 (3.3)
Since the masses are small p mi we may write x ≈ t and Ei =√
p2 + m2i ≈ p + (m2
i /2p):
|να(t)〉 ≈∑
i
e−im2
i2p tUαi |νi〉 (3.4)
From this expression the probability of finding a flavour β at distance x is readily obtained:
1This assessment is true as long as the number os oscillation lengths is not two big. Anyway, after too many lengths theneutrinos in the beam have become decoherent and oscillations have long since ceassed.
29
|φαβ(p, x)|2 =∑
i
U2βiU
2αi +
∑j,i
U∗βiUαiUβ jU∗α j cos 2πxli j
(3.5)
where we have introduced the oscillation length
li j =4πp
|m2i −m2
j |(3.6)
The interest of the phenomenon is its quantum mechanical nature and only exists thanks to mea-
surement theory. Let’s suppose the neutrino beam is created from a decaying beam of pions through
π+→ µ+νµ. The muon and pion’s momenta are measured so that the mass of the product neutrino be
determined and its flavour assessed. To do so, we must determine the mass with an error smaller than
|m2i −m2
j |. The error in the mass of the neutrino is given by
m2i = E2
i − p2⇒ δ(m2
i ) =√
4E2i (δEi)2 + 4p2(δp)2 (3.7)
the momentum error must then be
δ(m2i ) < |m2
i −m2j | ⇒ δp <
|m2i −m2
j |
2p(3.8)
and from Heisenberg’s uncertainty principle
δx &p
|m2i −m2
j |=
li j
4π(3.9)
The uncertainty in the interaction vertex becomes wider than the oscillation length and the oscillation
pattern is therefore lost.
We will first analyse in detail the two flavour case and three flavour one, and conclude with a brief review
of the effects of the majorana mass in neutrino oscillations. Neutrino oscillations are the best probe of
neutrino mass; solar and supernova experiments which aim at studying them can discover neutrino
mass differences down up 10−5 eV.
3.1.1 Two-flavour case
In this case the mixing matrix has only one mixing angle and can be written as
U =
(cosθ0 sinθ0− sinθ0 cosθ0
)(3.10)
and the weak and mass eigenstates are related by
30
|νe〉 = cosθ0 |ν1〉 + sinθ0 |ν2〉 (3.11)∣∣∣νµ⟩ = − sinθ0 |ν1〉 + cosθ0 |ν2〉 (3.12)
From (3.5) and because of CP (or T) symmetry, the transition (|φ(νe → νµ)|2 and |φ(νµ → νe)|2) and
survival probabilities (|φ(νe → νe)|2 and |φ(νµ → νµ)|2) are:
P(νe → νµ; t) = P(νµ → νe; t) ≡ Peµ(µe)(t) = sin2 2θ0 sin2(∆m2
4Et)
= sin2 2θ0 sin2(π
xl
)(3.13)
P(νe → νe; t) = P(νµ → νµ; t) ≡ Pee(µµ)(t) = 1 − sin2 2θ0 sin2(∆m2
4Et)
(3.14)
where ∆m2 = m22 −m2
1 and the oscillation length is
l =4πE∆m2 ≈ 2.48m
E(MeV)∆m2(eV2)
(3.15)
3.1.2 Three-flavour case
In this case the mixing matrix can be parametrized as the CKM matrix for quark mixing angles
νeνµντ
=
c1 s1c3 s1s3−s1c2 c1c2c3 − s2s3eiδ c1c2s3 + s2c3eiδ
−s1s2 c1s2s3 + c2s3eiδ c1s2s3 − c2c3eiδ
ν1ν2ν3
(3.16)
where ci ≡ cosθi and si ≡ sinθi. For a beam of neutrinos produced in the initial state νe the survival
and transition probabilities are
After many oscillation lengths it is natural to assume that the beam becomes decoherent and that the
oscillation pattern is lost. Indeed, when x li j the harmonics are smoothed off and only the average
intensity will be observable[15]:
〈Pee〉 = 1 − 2c21s2
1c23
[1 − cos
(2πxl12
)]− 2c2
1s21s2
3
[1 − cos
(2πxl13
)]− 2s4
1s23c2
3
[1 − cos
(2πxl23
)](3.17)⟨
Peµ⟩
= 2c21s2
1c22 + 2s2
1s23c2
3(s22 − c2
1c22) + 2s2
1s2s3c1c2c3 cos δ(s23 − c2
3) (3.18)
〈Peτ〉 = 2c21s2
1s22 + 2s2
1s23c2
3(c22 − c2
1s22) + 2s2
1s2s3c1c2c3 cos δ(s23 − c2
3) (3.19)
31
3.1.3 Majorana mass and neutrino oscillation
We have discussed neutrino oscillations in the case of Dirac neutrinos. If neutrinos are Majorana rather
than Dirac, the mass term in the lagrangian has to be replaced:
(mD)i jνiLν jR + h.c.→ (mM)i jνTiLCν jR + h.c. (3.20)
This term violates the total lepton number in addition to the individual lepton flavours. The Majorana
mass matrix is diagonalized by the transformation UTL mMUL = (mM)diag, so the problem becomes equiv-
alent to the Dirac one, just with a different matrix, but still with the same number of mass eigenstates as
weak ones. The oscillation pattern and probabilities have the same formulae. Therefore it is impossible
to test the Majorana hypothesis against the Dirac one with neutrino oscillation.
Things somewhat change if the most general case is analysed, where the Dirac and both Majorana terms
are considered. In this case the full mass matrix is a 2n × 2n matrix and a new kind of oscillations is
possible which convert weak-interacting neutrinos into sterile neutrinos να → νβ. These oscillations
violate the total lepton number L. Thus, one can distinguish the mixed Dirac-Majorana case from the
other two pure ones.
3.1.4 Do charged leptons oscillate?
The answer is obviously ’no’. The point here is that e, µ, τ refer to mass eigenstates. Neutrino mixing only
happens because we chose charged leptons to be in the mass basis, because neutrinos can’t be measured
directly but only by interactions such as CC or NC, so what we are really measuring are charged leptons
and associating the missing energy to an invisible particle called ’a neutrino’. And since mass eigenstates
don’t oscillate, charged leptons can’t oscillate. The proof of this goes as follows.[22]
Consider a muon being created at time t0 = 0 and position x0. After a time t and a distance x the state
evolves into
|Ψ(t, x)〉 = e−ipµxµ∣∣∣µ⟩ (3.21)
with a survival probability of
Pµµ = | < µ|Ψ(t, x) > |2 = 1 (3.22)
proving that no oscillation happens. Even if we consider a superposition of two states
32
|Ψ(0)〉 = cosθ∣∣∣µ⟩ + eiα sinθ |e〉 (3.23)
After propagation the state will be
|Ψ(t, x)〉 = e−ipµxµ cosθ∣∣∣µ⟩ + e−ipµxµeiα sinθ |e〉 (3.24)
and the survival probabilities will be again
Pµµ = cos2 θ (3.25)
Pee = sin2 θ (3.26)
and once again we see that charged leptons do not oscillate. But one may ask now: can we design an
experiment where a coherent superposition of e, µ, τ and then also detects their coherent superposition
rather than individual mass-eigenstate charged leptons? If this were possible, one would be able to
observe oscillations between such mixed charged lepton states.[23] In fact, the reason why neutrinos
oscillate and not charged leptons (since the CC interactions is symmetric in respect to neutrinos or
charged leptons) lies in the mass scale difference and in the decoherence properties of charged leptons.
3.2 The MSW effect
The neutrino oscillation problem doesn’t quite stop here. Matter may greatly enhance the oscillation
amplitude. The reason for this is that the interaction hamiltonian may be also flavour dependent. If such
is the case, then extra phases between the states might appear, inducing further levels of oscillation. For
example, ordinary matter is usually mainly made of electrons, protons and neutrons, whereas no taus ou
muons are present. Being so, only electron neutrinos will interact with the matter through charged current
showing an obvious assymetry in the Hamiltonian between the different flavours. Like electromagnetic
waves travelling through a medium and reducing their speed by acquiring an effective mass, neutrinos
also change their effective mass according to the interactions they suffer along their crossing of the
medium. Such processes are neutrino absorption and scattering by the matter constituents which change
both their energy and momentum. However, these processes are proporcional to the Fermi constant
G2F and are thus pretty small. Nevertheless, neutrinos can be forward scattered and their momentum
is conserved. This generates an effective potencial which varies between the electron neutrino and the
other flavoured neutrinos. Despite their being of order GF, their contribution to neutrino oscillation if
33
significant when they are similar or greater than ∆m2/2E. This enhancement can go as far as reaching
unity, unlike the vacuum phenomenon which is never higher than sin2 θ0. The study of neutrino
oscillations in matter starts with Wolfenstein[17] when he pointed out that in ordinary matter electron
neutrinos interact with electrons via charged and neutral current whereas tau and muon neutrinos only
interact through neutral current. The effect of oscillation enhancement in matter is owed to the work
of Mikheyev, Smirnov and Wolfenstein[18] (the MSW effect) and is one main part in solving the solar
neutrino problem.
In matter neutrinos of all three flavours interact with electrons, protons and neutrons through NC
interaction. Electron neutrinos react additionally with the medium’s electrons through CC interactions.
Check diagrams for these interactions in fig.3.2. The effective Hamiltonian for the CC interaction is the
W±e
νe
νe
e
Z0
p, n, e
p, n, e
νe,μ,τ
νe,μ,τ
Figure 3.1: Neutrino scattering diagrams via weak charged and neutral interactions.
usual V − A hamiltonian. The CC and NC effective hamiltonians for the electrons are:
HCC =GF√
2
[eγµ(1 − γ5)νe
] [νeγ
µ(1 − γ5)e]
(3.27)
HNC =GF√
2
∑f=p,n,e
[fγµ
( I3
2(1 − γ5) −Q f sin2θW
)f] [νeγ
µ (1 − γ5
)νe
](3.28)
For the other flavours only the NC interaction happens and the Hamiltonian is the same:
HNC =ρGF√
2
∑f=p,n,e
fγµ
I f3
2(1 − γ5) −Q f sin2θW
f
[ ¯νµ,τγµ(1 − γ5
)νµ,τ
](3.29)
In forward scattering the neutrino’s momentum is left unaffected so:
34
HCC =GF√
2
[¯e(p1)γµ(1 − γ5)νe(p2))
] [¯νe(p3)γµ(1 − γ5)e(p4)
](3.30)
=GF√
2
[¯e(p1)γµ(1 − γ5)νe(p))
] [¯νe(p)γµ(1 − γ5)e(p)
](3.31)
=GF√
2
[¯e(p)γµ(1 − γ5))e(p)
] [¯νe(p)γµ(1 − γ5)νe(p)
](3.32)
The coherent forward scattering contribution to the energy of νe in matter is equivalent to an effective
potencial which is found upon integrating over all the variables that correspond to the electron field:
HCCe f f (νe) = 〈HCC〉e (3.33)
=GF√
2
[¯νe(p)γµ(1 − γ5))νe(p)
]〈eγµ(1 − γ5)e〉 (3.34)
=GF√
2
[¯νe(p)γµ(1 − γ5))νe(p)
]〈eγµ(1 − γ5)e〉 (3.35)
=GF√
2
[¯νe(p)γµ(1 − γ5))νe(p)
] [〈e†e〉 + 〈e†−→α e〉 − 〈e†γ5e〉 − 〈e†−→αγ5e〉
](3.36)
=GF√
2
[¯νe(p)γµ(1 − γ5))νe(p)
] [Ne + 〈−→ve〉 − 〈
−→σe ·−→pe
Ee〉 − 〈−→σe〉
](3.37)
(3.38)
where Ne is the electron number density. In a neutral, unpolarized medium of zero total momentum
the only surviving term in the latter expression is Ne. Let us now calculate the NC contribution for the
effective potencial:
HNCe f f (νe) = 〈HNC〉p,n,e (3.39)
=GF√
2
[νeγ
µ (1 − γ5
)νe
] ∑f=p,n,e
〈 fγµ( I3
2(1 − γ5) −Q f sin2 θW
)f 〉 (3.40)
= GF√
2[νeγ
µ (1 − γ5
)νe
]〈pγµ
(12
(1 − γ5) − 2 sin2 θW
)p (3.41)
+ eγµ(−
12
(1 − γ5) + 2 sin2 θW
)e + nγµ
(−
12
(1 − γ5))
n〉 (3.42)
= GF√
2[νeγ
µ (1 − γ5
)νe
] [(12− 2 sin2 θW
)Np +
(−
12
+ 2 sin2 θW
)Np +
(−
12
)Nn
](3.43)
= GF√
2[νeγ
µ (1 − γ5
)νe
] (−
12
)Nn (3.44)
because Np = Ne in neutral matter. Hence, the effective potencial for the electron is:
35
He f f (νe) ≡ ¯nueVee = 〈HNC〉p,n,e + 〈HCC〉e ⇒ Ve =√
2GF
(Ne −
Nn
2
)(3.45)
The same works for tau and muon flavours except that Nτ = Nµ = 0 so:
Vµ = Vτ = −GF√
2Nn (3.46)
For anti-neutrinos one has to replace V → −V.
Let us consider the evolution of oscillating neutrinos. In vacuum the evolution is obviously easily tracked
in the mass basis whereas in matter – because the effective potencials are flavour-dependent – it is rather
favorable to follow the evolution of the system in the flavour basis. The flavour basis and the mass basis
are related by:
ν f l = Uνm (3.47)
and the Schrodinger equation is:
iddt|νm〉 = Hm |νm〉 , Hm = diag(E1,E2) (3.48)
iddt
∣∣∣ν f l⟩
= UHmU†∣∣∣ν f l
⟩(3.49)
iddt
(νeνµ
)≈ p + U
(m2
1/2E 00 m2
1/2E
)U†
(νeνµ
)(3.50)
= p +m2
1 + m22
4E+
∆m2
4E
(− cos 2θ0 sin 2θ0sin 2θ0 cos 2θ0
) (νeνµ
)(3.51)
For the modified Hamiltonian we must add the effective potencials for both flavours[1]:
iddt
(νeνµ
)= p +
m21 + m2
2
4E−
GF√
2Nn +
∆m2
4E
− cos 2θ0 + 4√
2EGF∆m2 Ne sin 2θ0
sin 2θ0 cos 2θ0
( νeνµ
)(3.52)
=∆m2
4E
− cos 2θ0 + 4√
2EGF∆m2 Ne sin 2θ0
sin 2θ0 cos 2θ0
( νeνµ
)(3.53)
The common diagonal terms to both flavours do not insert any phase difference between the neutrino
states and thus don’t contribute to the neutrino oscillations because they can only change the global
phase of the doublet and can be ommited for the purpose of this study. We shall now study the constant
matter density case for this oscillating system.
36
3.2.1 Constant density
In this case Ne = const. along the neutrino trajectory (so constant along both time and space). Upon
diagonalization, the effective Hamiltonian yields the following eigenstates:
νA = νe cosθ + νµ sinθ (3.54)
νB = −νe sinθ + νµ cosθ (3.55)
and the mixing angle is given by:
tan 2θ =2H12
H22 −H11=
sin 2θ0
cos 2θ0 − cos 2θ0 + 4√
2EGF∆m2 Ne
(3.56)
The difference between the two eigenenergies is:
∆E =∆m2
2E
√(cos 2θ0 −
2√
2GFE∆m2 Ne
)2
+ sin2 2θ0 (3.57)
It is now straight forward to find the transistion probability in matter:
P(νe → νµ; x) = sin2 2θ sin2(∆Ex
2
)= sin2 2θ sin2
(π
xlm
)(3.58)
where the oscillation length is now given by:
lm =2π
EA − EB=
4πE
∆m2
√(cos 2θ0 −
2√
2GFE∆m2 Ne
)2+ sin2 2θ0
(3.59)
The expressions are the same as for the vacuum case, except with a correction for the oscillation length
lm and the mixing angle θ which are given in terms of the vaccum ones losc, θ0 and Ne.
The oscillation amplitude is resonant when:
√
2GFNe =∆m2
2Ecos 2θ0 → sin2 2θ = 1 (3.60)
This is called the MSW resonance condition and when it is fulfilled the maximum value for θ = π/4
is achieved without regard to how small the vacuum mixing angle may be. Thus, one may have large
neutrino flavour transistions in matter even when they are very small in the vaccum.
Anyway, any matter enhancement requires:
37
∆m2
2Ecos 2θ0 =
12E
(m22 −m2
1)(cos2 θ0 − sin2 θ0) > 0 (3.61)
If ν2 is heavier than ν1 then θ ∈ [−π/4, π/4] ∪ [3π/4,−3π/4]. This means that the lighter mass-
eigenstate must have the largest νe component. If one chooses the convention cos 2θ0 > 0 then this
condition is merely a hierarquic one ∆m221 > 0. One the other hand anti-neutrinos have the opposite
condition ∆m221 < 0 which means that neutrinos and anti-neutrinos can’t experience both MSW enhanced
oscillations.
One may think that as long as the MSW condition is satisfied one can go as far as to have θ0 → 0
but if we do so lm → ∞ and the oscillation pattern is destroyed. Usually the neutrinos propagate in
polychromatic beams, so the the resonance condition A = 1 is almost certainly satisfied by some part of
the energy density as long as ∆m2 is of the right order of magnitude. Nevertheless, solar 7Be neutrinos
are monochromatic, so one may ask, can the resonance condition still be satisfied? The answer is yes,
provided that the corresponding resonance density is within the density range of the matter density
distribution in which neutrinos propagate.
3.2.2 Oscillations in non-uniform matter
For practical purposes let us define[12]:
A ≡ 2√
2GFENe = 1 −tan 2θ0
tan 2θ(3.62)
Now we wish to study the effect of matter density changes along the neutrino’s path in the oscillation
pattern. Let’s suppose they are emitted in a very dense medium like the core of the sun, where A is far
above the resonance value. In this region A ∆m2 cos 2θ0 θ→ π/2, the lighter state is almost purely
νµ whereas the heavier one is close to being plain νe. Now in vaccum the exact opposite happens. All this
is pretty clear when one looks at fig.3.2.2. The phenomenon that inverts the effective masses of νµ and νe
from νµ being lower than νe to νe being higher than νµ with the inscrease of A is called level-crossing.
At the core of the sun A 1⇒ θ ∼ πwhich means that the neutrino oscillation is strongly suppressed
by matter. Assuming that the Sun’s density decreases along its radius, A also decreases as the neutrino
propagates out of the star. The mixing then increases at first, reaching its maximum value θ = π/2
when A = 1 and then decreasing again towards vacuum values until θ = θ0. As said earlier, at first the
neutrinos mass eigenstates are almost pure, and if the system was to propagate only in the core of the
sun, any neutrino produced in one mass eigenstate will remain in that very same eigenstate. However,
the fact that it has to go through thinner matter changes the particle’s flavour composition, as long as
38
Figure 3.2: Neutrino energy levels versus electron density in the medium, where the dashed line standsfor the pure states and the solid line stands for the mixed states.
the density variation is made adiabatically so that the neutrino can adaptıtself to the changes. At the
final state νB’s transistion probability is P(νe → νµ) = cos2θ0 so for small θ0 the transition probability is
almost one; the adiabatic convertion is almost complete for νe to νµ which is the impressiveness of the
level-crossing mechanism. This is similar to the Landau-Zener mechanism in atomic physics that governs
the transition probability between two states separated by a time-dependent energy difference.
It is important to note once again that even though the smaller θ0 is the bigger the adiabatic conversion
works and therefore θ0 = 0 may seem – in a very unintuitive way – like a mean of full oscillation
improvement, when the vacuum mixing angle is null the conversion is no longer adiabatic and the
previous reasoning is no longer valid.
It is therefore important to set the validity of this adiabatic approximation. For that purpose we must
switch back to a quantitative description of the neutrino conversion in the adiabatic regime. The effective
Hamiltonian in the flavour basis can be diagonalized by a unitary transformation such as:
U(t)†H f l(t)U(t) = Hd(t) = diag (EA(t),EB(t)) (3.63)
and EA(t) and EB(t) are the instantaneous energies of νA(t) and νB(t). The evolution equation becomes:
iddt
(νAνB
)=
[Hd − iU†
(dUdt
)]=
(EA(t) −iθ(t)iθ(t) EB(t)
) (νAνB
)(3.64)
The adiabatic approximation is valid when the transitions between the instantaneous mass eigenstates
are suppressed and that happens when |θ| |EA − EB|. Only the energy difference counts because as
39
we’ve stated before, any term proporcional to the unity matrix is irrelevant for oscillations.
γ−1≡
2|θ||EA − EB|
=sin 2θ0
∆m2
2E
|EA − EB|3|
√
2GFNe| 1 (3.65)
This γ parameter is called the adiabacity parameter. In the adiabatic limit γ→ 0 the Hamiltonian in
(3.64) becomes diagonal and the time evolution of the matter eigenstates is then a simple matter of phase
factors. To set it more clearly, consider a neutrino born at time t = ti in the state:
ν(ti) = νe = cosθiνA + sinθiνB (3.66)
The adiabatic evolution leads at time t = t f to the state:
ν(t f ) = cosθi exp(−i
∫ t f
ti
EA(x)dx)νA + sinθi exp
(−i
∫ t f
ti
EB(x)dx)νB (3.67)
P(νe − νµ) =12−
12
cos 2θi cosθ f −12
sin 2θi sin 2θ f cos∫ t f
ti
(EA(x) − EB(x)) dx (3.68)
where x ≈ t for relativistic neutrinos. It becaomes evident now that only phase differences are relevant
for neutrino oscillations since:
Φ ≡
∫ t f
ti
(EA(x) − EB(x)) dx =
∫ t f
ti
m2A −m2
B
2E)
dx (3.69)
The only term oscillating with time is the last one. This term vanishes if the matter density at
production vertex is too far above the MSW resonance (sin 2θi ∼ 0), in which case the non-oscillatory
transition takes place with P(νe − νµ) = 12 (1 − cos 2θ f ) = cos2 θ f . If the final medium is just vacuum
θ f = θ0, the formula is exactly like the one we stated in our previous semi-quantitative discussion of
adiabatic conversion. Now let’s suppose some violation of adiabacity happen leading to transitions
between νA and νB. The probability of jumping from one state to another is the same as the Landau-
Zener-Stuckelberg probability denoted PLZS. In case no jump occurs, the transition probability is the
same as in the adiabatic case, and its conditional probability is then (1−PLZS)Pad(νe − νµ). One has to add
now the probability PLZSPad(νe − νe) that the νe survives but a jump occurs . The total probability is:
40
P(νe − νµ) = (1 − PLZS)Pad(νe − νµ) + PLZSPad(νe − νe) (3.70)
=12
(1 − PLZS)(1 − cos 2θi cosθ f − sin 2θi sin 2θ f cos Φ
)(3.71)
+12
PLZS
(1 + cos 2θi cosθ f + sin 2θi sin 2θ f cos Φ
)(3.72)
=12
[1 − (1 − 2PLZS)
(cos 2θi cosθ f + sin 2θi sin 2θ f cos Φ
)](3.73)
We can ommit the oscillating terms that average to zero:
12
[1 − (1 − 2PLZS)
(cos 2θi cosθ f
)](3.74)
If γ is not too small, the semi-classic approximation for the PLZS is valid and is given by:
PLZS ≈ e−2πΓ (3.75)
where
Γ = Φ∗ −Φ = −2Im∫
∆m2
2Edt = −Im
∫ Am
AMSW
∆m2
EdxA
(3.76)
where AMSW is the value of A at resonance and Am is the value of A when ∆m2 = 0. If A = const. as it
is approximately in the Sun, then
PLZS ≈ e−π2 γr (3.77)
where γr is the adiabacity parameter taken at the MSW resonance point. In the adiabatic limit,
γr 1, PLZS ∼ 0; conversely, in the opposite limit γr 1, PLZS ∼ 1 and the both survival and transition
probabilities swap values.
Let’s now discuss the adiabaticity condition. Since the energy differences are minimal and equal to
(∆m2/2E)sin2θ0 at resonance point, if adiabaticity is valid for AMSW then it is valid anywhere else. Let’s
write thus
γr =
(∆m2
2Esin 2θ0
)2 1√
2GFNe|MSW= tan 2θ0 sin 2θ0
∆m2
E
∣∣∣∣∣∣Ne
Ne
∣∣∣∣∣∣−1
MSW(3.78)
The resonance condition is approximately satisfied for ∆m2 sin 2θ ≥ 1/2 and within a range of electron
density
41
δNe ≈∆m2 cos 2θ
2√
2GFE(3.79)
and
δx ≈δNe
|Ne|r=
∆m2 sin 2θ0
|Ne|r2√
2GFE=
√2π
|Ne|rLrGF(3.80)
where Lr is the oscillation length at resonance point
Lr =4πE
∆m2 sin 2θ0(3.81)
and
γr = πδxLr
(3.82)
Therefore the adiabaticity condition sums up to δx Lr.
3.3 Solar Neutrino Problem
The Solar Standard models since the eighties have been widely successful in predicting and explaining
a vast range of experimental data. In these models the main idea is that thermonuclear reactions are at
the base of the solar combustion, and in our sun – being part of a family of stars called main sequence
stars – it comes mainly from a fusion of hydrogen and helium:
4p + 2e− → 4He + 2νe + 26.73MeV (3.83)
The main chain reaction happening in the sun is the pp cycle. The second major cycle is the CNO cycle
(carbon-oxygen-nitrogen) which is only responsible for 2% of the energy produced in the sun. Neutrinos
are emitted in both reaction schemes. Six of the eleven reactions in the pp chain produce neutrinos, which
are either nuclear beta decays or electron capture reactions. The pep and electron capture into 7Be reactions
emit beams of monochromatic electron neutrinos whereas the neutrinos bred in the other pp reactions
have non-discrete energy spectra. The fluxes of neutrinos are calculated based on these thermonuclear
reactions, on the assumption that the sun is in local hydrostatic equilibrium, that the energy is transfered
between different regions of the sun through radiation and convection. Additionally, the models are
calibrated in order to obtain the present values of solar radius, luminosity and He/H ratio. Some recent
42
models even incorporated diffusion processes of helium and heavy elements. The number of existent
models based on these premises in quite large (about 20), but all of them – except one – agree with each
other within a 3σ acceptance level when it comes to predicting fluxes of elements such as 8Be or 7Be[42]
and the most recent models also managed to reproduce succesfully with a very high accuracy the solar
sound velocities inferred from the heliosismological measurements[43]. The solar standard models are
hence widely accepted as accurate.
The neutrino fluxes predicted from these models are shown in table 3.3[44].
On the one hand, the pp flux is known to a good certainty ∼ 1% because it depends on the sun’s
luminosity which is relatively well known. On the other hand, higher energy fluxes are less well known:
∼ 20% for 8B and only the order of magnitude is known for the hep neutrinos. Nevertheless, these fluxes
are quite smaller than the pp ones, so the total flux is pretty much left untouched; however, it can affect
the high energy spectrum.
Several experiments have tried to measure the flux of solar neutrinos. The pioneering experiment is Ray
Davis’s 600 tonne chlorine tank in the Homestake mine[50], South Dakota. It’s based on the reaction:
νe + 37Cl→ 37Ar + e− (3.84)
The energy threshold of this reaction is 0.184MeV, so only the 8B (largest contributor), 7Be and pep
neutrinos can be detected. The Argon is extracted from the tank using chemical methods and counted
in proporcional counters. Davis’ radio-chemistry assay, begun in 1967, already finds evidence for only
one third of the expected number of neutrino events. In 1986, a light water Cherenkov experiment at
Kamioka, Japan, is altered to detect solar neutrinos through scattering with electrons:
νa + e− → νa + e− (3.85)
The only energy threshold E > 7.5MeV for this experiment comes from the background cuts. The later
43
upgrade of this experiment, Super-Kamiokande, has a better threshold of E > 5.5MeV. Because of these
cuts, both versions of the experiment are only sensitive to 8B neutrinos. This experiment also presents a
proof that the detected neutrinos come from the sun because when E me the angular distribution of
(3.85) is peaked around 180o degrees which proves finally that the flux is coming from the sun, unlike
in previous experiments where it was all a matter of common sense that no other source besides the sun
was intense enough to produce netrinos in such amounts of strength and energy. Like in Homestake, the
Kamioka detectors find one half of the expected events for the part of the neutrino spectrum for which
they are sensitive. There are yet another two important experiments contributing to the solar neutrino
problem (SNP), SAGE and GALLEX, both gallium detectors employing the reaction:
νe + 71Ga→ 71Ge + e− (3.86)
whose energy threshold is 0.234 MeV allowing these experiments to harvest neutrinos from the plen-
tiful pp flux. They also find about 60-70% of the expected rate[45]. Anyhow, the clear trend is that the
measured flux is found to be dramatically less than is possible for our present understanding of the
reaction processes in the sun and comes to odds with a widely accepted family of models which so far
appeared to be consistent with each other. The solutions to the SNP are multifold:
• astrophysical solution: insufficient knowledge of solar physics or an error in some input parameter;
• experimental solution: miscalculation of detection efficiencies or cross sections in the puzzling ex-
periments;
• particle physics solution: unknown neutrino physics.
If the first hypothesis is the solution, then only the total fluxes of each nuclear reaction may be differ-
ent from those predicted by the SSM because the energy spectra of the various components of the solar
neutrino flux are well known from standard nuclear physics. Unfortunately, the SNP has more than
just flux deficiencies: the results of different experiments seem to be inconsistent with each other. For
example, one can infer the flux of 8B directly from the Kamiokande and SKamiokande data, and find the
corresponding to the Homestake detection rate. If done so, the contribution is found to be larger than
the total detection rate, which means that the contribution appears to be negative! If we admit that the
solar spectra are undistorted for each reaction, then the SNP remains no matter what we do without new
neutrino physics. We remind again that the SSMs are well grounded, agree with each other and with
44
Solution Status Requirement Dependence Other features Original references
Resonant conver-sion to muon or tauneutrinos
a good fit, wellmotivated[49]
mixing of neutri-nos, mass O(10-3eV), see plot.
day night effect forlarge mixing
large angle solutionbest fit
Mikheyev andSmirnov[18] (1986),Wolfenstein (1978).
Resonant con-version to sterileneutrinos
disallowed by SNOdata[48]
mixing of neutri-nos, mass O(10-3eV)
day night effect forlarge mixing
vacuum oscillation not a good fitinconsistent withSNO[49]
mixing of neutri-nos, mass O(10-5eV)
annual variations Pontecorvo (1967)
helicity flip bad fit magnetic momentO(10-11 µB)
anticorrelation withsunspots
solar magnetic fieldunknown
Voloshin, Vysotskyand Okun[51](1986)
resonant spin-flavor conversion
still alive (hard tokill) unmotivated
magnetic momentO(10-11 µB), massO(10-3 eV)
anticorrelation withsunspots
taking the solarmagnetic field asfree parameter fitsall the results[53]
Akhmedov (88) Limand Marciano (88)
neutrino decay inconsistent lifetime < 8 min static constrained bySN1987A
Bahcall, Cabibboand Yahil (72)Pakvasa andTennakone (72)[52]
solar astrophysics desperate new physics insidethe sun
model dependent Conflicts helioseis-mology
Table 3.1: Different solutions to the SNP and their experimental counterparts.
heliosismological observations.
The second hypothesis is also very unlikely for most of the cited experiments. All of them but Homestake
have been calibrated and found to be in very good agreement with expectations. Even the Homestake
detector’s argon extraction efficiency was checked by doping it with a small number of radioactive argon
atoms, but no real calibration was caried out since no artificial sources of neutrinos exist with a suitable
energy spectrum. Also, to prove this hypothesis one would have to explain why experiments of different
unrelated types (chlorine, gallium or water Cherenkov) have been mislead, which is either very unlikely
or very depressing for a whole generation of experimentalists.
We are then left with the last hypothesis that there is some unknown neutrino physics dimming the
electron-flavoured flux. Several mechanisms have been proposed to explain the SNP with special re-
gards to neutrino oscillations. The proposed alternative solutions and their experimental validation
are summed up in table 3.1. The SNP can be considered almost as a solved problem by now. The
results from SNO clearly indicate that the total neutrino flux from the sun is in accordance with the
solar models of Bahcall and Pisconneault. The Kamland reactor neutrino experiment has measured the
electron antineutrino flux from nearby reactors and the results are consistent with the solar neutrino
oscillations. Neutrino oscilaltions can convert a fraction of solar νe into the other two flavours. Since
the energy of solar neutrinos is limited to a few MeV (which is already a very thin flux coming from
45
hep neutrinos) no CC reactions can happen in the τ and µ flavours because the lepton masses are two
high (mµ ∼ 1GeV and mτ ∼ 100MeV) and their corresponding neutrinos can’t therefore be detected in
chlorine- or gallium-based experiments. They can however still interact through NC interaction which
is visible in water Cherenkov detectors, but has a cross-section about six times smaller than CC, which
explains why K and SK experiments saw a deficit in neutrino flux.
The MSW effect greatly enhances oscillations inside the sun which allows the solar data to be fitted even
with a very small vacuum mixing angle, but no matter what the vacuum mixing angle is, in most cases it
will change the oscillation pattern. The most important of these experiments in validating the neutrino
flavour oscillation is most certainly SNO. SNO is an imaging Cherenkov detector that uses heavy water
(D2O) as both the interaction and detection medium. It has a spherical geometry covered with photo-
multipliers (PMTs). It can accomodate charged current (CC), neutral current (NC) and electron scattering
(ES) interactions. The comparison of each of these reactions (separating events through angular and
cross-section filtering) allows a flavour identification of the measured fluxes. The final results for the
fluxes are:
φCC = 1.76+0.06−.0.05(stat.)+0.09
−0.09(syst.) × 106cm−2s−1, (3.87)
φNC = 2.39+0.24−.0.23(stat.)+0.12
−0.12(syst.) × 106cm−2s−1, (3.88)
φES = 5.09+0.44−.0.43(stat.)+0.46
−0.43(syst.) × 106cm−2s−1, (3.89)
φe = 1.76+0.05−.0.05(stat.)+0.09
−0.09(syst.) × 106cm−2s−1, (3.90)
φµτ = 3.41+0.45−.0.45(stat.)+0.48
−0.45(syst.) × 106cm−2s−1, (3.91)
(3.92)
Adding the statistical and systematic errors in quadrature, we find that φµτ is 5.3σ away from its
null hypothesis value of zero. The total neutrino flux is on good agreement with the SSM’s predictions
for the 8B flux and the combined flux results are consistent with neutrino flavor transformation with no
distortion in the 8B neutrino energy spectrum[19] (cf. Fig.??). The comparison of day and night fluxes
in SNO provides a way to separate the pure-vacuum oscillation and MSW-enhanced oscillation theories
because in the second case one would see a difference between day and night fluxes since in the latter
the neutrinos must travel through the earth’s matter whereas if the MSW effect has no significance one
wouldn’t see any difference between fluxes measured in these two seperate day times. The day-night
assymetry is found to be:
46
Figure 3.3: The flux of νµ + ντ vs. the flux of νe. The fluxes as determined by the SNO CC, NC and ESreactions are shown, as well as the predictions of the Standard Solar Model (SSM). The errors representedare ±1σ, and the best fit values for ϕe and ϕµτ are shown.
Ae = 7.0 ± 4.9(stat.)−1.2+1.3%(syst.) (3.93)
which shows that the no-MSW effect hypothesis is inconsistent with SNO data. The results of all
solar neutrino data (SNO, SK, ...) and the Kamland experiment point to the values:
∆m212 ≈ 8 × 10−5eV2, sin22θ12 ≈ 0.8 (3.94)
47
Chapter 4
Neutrino Mass Models
4.1 Seesaw models in the Left-Right Symmetric model
All known interactions – gravitational, electromagnetic and strong – but the weak interaction respect
parity. If we expect to find some sort of common rule for all of Nature’s forces, it is only tempting to
suppose that at some high energy scale parity is conserved before it is somehow broken and from there
on maximally violated by the weak interaction, as observed nowadays in experiment. The left-right
symmetric model as we introduce it here is a minimal extension to the SM nothing is changed except for
the inclusion of an additional SU(2)R symmetry for the RH fermions which is the mirror of the known
SU(2)L making the model symmetric for left and right chiralities. As we will see in later on sections of
this review, the RHν – which we have already introduced to generate neutrino masses – is needed in
GUTs such as SO(10), and in this case is required in order to complete the left-right symmetry (LRS),
since all fermions must have an partner of opposite chirality. The LRSM is in fact a subgroup of SO(10).
Its attractiveness comes from the fact that parity is now a spontaneaously broken symmetry.
4.1.1 The Gauge Sector and Symmetry breaking
The simplest way to go from the SM to a LRSM is therefore SU(2)L × SU(2)R × U(1)Y where the lepton
and quark fields are assigned to the following irreducible representations1:
LL =
(νLlL
): (2, 1,−1), LR =
(NRlR
): (1, 2,−1);QL =
(uLdL
): (2, 1, 1/3), QR =
(uRdR
): (1, 2, 1/3).
(4.1)
We now require that the lagrangian of the model1We have safeguarded the possibility of the RH neutrino being of Majorana type and having a different mass from the LH
one by writing NR instead of νR
48
L = −14
WLµν ·W
µνL −
14
Bµν · Bµν + LLγµDµLL + QLγ
µDµQL
+ Tr∣∣∣Dµφ
∣∣∣2 + Tr∣∣∣Dµ (σ · ∆L)
∣∣∣2 + (L→ R) − V(φ,∆L,∆R, φ) − LY (4.2)
be invariant under P transformation. The covariant derivative is
∂ − i3∑
k=1
(gLWL
k TLk + gRWR
k TRk
)− ig′BY (4.3)
where WL,R and B are the gauge bosons for SU(2)L,R and U(1)Y respectively and TL,R are the SU(2)L,R
generators. The fields transform as
LL ↔ LR (4.4)
QL ↔ QR (4.5)
WLk ↔WR
k (4.6)
which needlessly to say implies gL = gR ≡ g. The electric charged is given analogously to the SM by
Q = TL3 + TR
3 +Y2
(4.7)
The quark doublets must have hypercharge Y = 1/3 and the lepton doublets Y = −1 since they
all bear a unit charge but with opposite signs. The formula then becomes physically meaningful with
Y = B − L. This formula has very important implications. At a scale where the LRSM has been broken
into SU(2)L ×U(1)B−L we have the relation[57]
∆TR3 = −
12
∆(B − L) (4.8)
If the interactions are baryon number conservating as they are in the SM, then |∆L| = 2∣∣∣∆TR
3
∣∣∣. If
symmetry breaking is chosen as to violate TR3 by one unit then L is violated by two units implying
Majorana neutrinos. Even if TR3 is only violated by half a unit Majorana neutrinos can still arise at higher
orders. The same happens in the hadronic sector if we choose leptons to be conserved: B is violated by
two units and neutron-antineutron oscillation happens.
The Higgs sector of this sector must be LRS so that the symmetry is maintained; the first candidate is the
bidoublet[20]
49
φ =
(ϕ0
1†
ϕ+2
−ϕ−1 ϕ02
), (2, 2, 0) (4.9)
as well as its conjugate field
φ ≡ σ2φ†T =
ϕ02†
ϕ+1−ϕ−2ϕ
01 (4.10)
This field can couple to the fermion bilinears QLQR and LLLR, which after symmetry breaking
⟨φ⟩
=
(k 00 k′
)(4.11)
can give masses to both quarks and leptons. Nevertheless, since the bidoublet is B − L neutral the
remaining symmetry is U(1)TL3 +TR
3×U(1)B−L and not U(1)Q. More scalar multiplets are therefore required.
One might choose simple SU(2) doublets for each chirality but they would only lead to Dirac neutrinos
and since our interest lies in neutrino masses and for the purpose of explaining their smallness we choose
two doubly-charged triplets[58]
∆L : (3, 1, 2), ∆R : (1, 3, 2) (4.12)
which lead to Majorana masses. To represent the triplet in a somewhat simpler and easier-to-compute
way, we note that the tensor product of two doublets is equal to 2⊗2 = 3 + 1, which means that the triplet
can be written down as a 2 × 2 traceless matrix (the product of two doublets minus a singlet). Hence,
∆L =
(∆+
L/√
2 −∆++L
∆0L −∆+
L/√
2
), ∆R =
(∆+
R/√
2 −∆++R
∆0R −∆+
R/√
2
)(4.13)
which develop a vacuum expectation value
⟨∆L,R
⟩=
(0 0
vL,R 0
). (4.14)
It is now useful to define
W± ≡W1 ∓ iW2√
2, T± ≡
T1 ∓ iT2√
2,
T+ =1√
2
(0 10 0
), T− =
1√
2
(0 01 0
). (4.15)
and a zero-mass eigenstate which is given by
50
A = −sw(W3
L + W3R
)+
√c2
w − s2wB (4.16)
where
cW = cosθw = −g′√
g2 + 2g′2
sW = sinθw =
√g2 + g′2
g2 + 2g′2
e = gsw.
(4.17)
We also define
Z = cwW3L −
s2w
cwW3
R +sw
cw
√c2
w − s2wB
Z′ =
√c2
w − s2w
cwW3
R −sw
cwB (4.18)
The covariant derivative becomes
g(WL3 + WR3TR3) + g′BY = −eAQ +g
cwZ(T3
L) −Qs2) +g√
c2w − s2
w
cw
(T3
R − Ys2
w
c2w − s2
w
)Z′ (4.19)
Hence the doublets yield at T=0
g2
2
∣∣∣W+L k1 −W+
Rk∗2∣∣∣2 ,
g2
2
∣∣∣W+L k2 −W+
Rk∗1∣∣∣2 ,
g2|k1|
2
4c2w
∣∣∣∣∣−Z +
√c2
w − s2wZ′
∣∣∣∣∣2 ,g2|k2|
2
4c2w
∣∣∣∣∣−Z +
√c2
w − s2wZ′
∣∣∣∣∣2 (4.20)
Likewise, the triplets yield
|gW+L vL|
2 , |gW+RvR|
2 , |g
cw
Z +s2
w√c2
w − s2wZ′
vL|2 , |
gcw√c2
w − s2w
Z′vR|2 (4.21)
The mass matrix for the neutral gauge bosons is
g2
c2w
(Z Z′
) |k1|
2+|k2|2
4 + |vL|2
−|k1|
2+|k2|2
4
√c2
w − s2w +
s2w√
c2w−s2
w
|vL|2
−|k1|
2+|k2|2
4
√c2
w − s2w +
s2w√
c2w−s2
w
|vL|2 |k1|
2+|k2|2
4 (c2w − s2
w) +s4
w|vL|2+c4
w|vR|2
c2w−s2
w
(
ZZ′
)(4.22)
51
The charged boson mass matrix is also obtained as
g2(
W−L W−R) |k1|
2+|k2|2
2 + |FL|2
−k∗1k∗2−k1k2
|k1|2+|k2|
2
2 + |FR|2
( W+L
W+R
)(4.23)
The charged gauge bosons mix. Since k1k2 has T3L = −1 and T3
R = 1 we may choose the mixing to be
real and the diagonalization of the matrix gives two physical charged bosons
(W1W2
)=
(cos ζ − sin ζsin ζ cos ζ
) (WLWR
)(4.24)
If the Higgs potential is chosen as to give |vL| , |vR|, parity is spontaneously broken. Morover, if
|vR| is assumed to be much smaller than |k1|, |k2| and |vL| (which strictly speanking is not required from
phenomenological considerations), the mixing angle and the charged boson masses are given by
ζ ≈k1k2
|vL|2,
m2W1≈ g2
(|vL|
2 +|k1|
2 + |k2|2
2
)(4.25)
m2W2≈ g2
(|vR|
2 +|k1|
2 + |k2|2
2
)Since the mixing angle is very small, W1 and W2 coincide to a good approximation with WL and WR:
M2WL≈ cos2 ζM2
W1+ sin2 ζM2
W2
M2WR≈ sin2 ζM2
W1+ cos2 ζM2
W2(4.26)
The charged current interaction takes the form in the LRSM
Lcc =g√
2
[(uLγµdL + νLγµeL
)Wµ
L(uRγµdR + NRγµeR
)Wµ
R
]+ h.c. (4.27)
For MWR MWR , the charged current interactions will violate almost maximally parity at low
energies. Therefore, any deviation from the V − A structure may constitute evidence for right handed
CC currents in a LRSM.
Similarly, Z and Z′ may be considered approximate eigenstates of mass
52
m2Z =
g2
c2w
(|k1|
2 + |k2|2
2+ 2|vL|
2)
m2Z′ =
g2
c2w
(|k1|
2 + |k2|2
2(c2
w − s2w) + 2
s4w|vL|
2 + c4w|vR|
2
c2w − s2
w
)(4.28)
The mixing is given by
(Z1Z2
)=
(cos ξ sin ξ− sin ξ cos ξ
) (ZZ′
)(4.29)
where for |vL| 1 and |k1|2 + |k2|
2 |vL|
2 the mixing angle is given by
tan 2ξ ≈(c2
w − s2w)3/4
2c4w
|k1|2 + |k2|
2
|vR|2≈ 2
√c2
w − s2wM2
Z/M2Z′ (4.30)
In the limit where |vR| 1, the mixing angle becomes zero. In this limit, the standard model relation
mW1 = MZ1 cosθw is changed into (assuming |k1|2 |k2|
2)
mW1 =
√|k1|
2 + 2|vL|2
|k1|2 + 4|vL|2
MZ1 cosθw (4.31)
Now, the squared value of the coeficient on the RH side of (4.31) has been constrained experimentally
down to 0.998 ± 0.050 which restricts |vL|/|k1| < 0.17. The VEV of a LH gauge interaction must indeed be
very small compared to the VEV the either one of the doublets.
The neutral current interaction is
Lnc =g
cW
KµLZµ +1√
c2w − s2
w
(sin2 θWKµL + cos2 θWKµR
)Z′µ
≈
gcW
LµL − ξ√c2
w − s2w
(sin2 θWKµL + cos2 θWKµR
)Z1µ +
1√c2
w − s2w
(sin2 θWKµL + cos2 θWKµR
)Z2µ
(4.32)
where
KµL,R =∑
f
fγµ(T3
L,RPL,R −Q sin2 θW)
f (4.33)
Phenomenology implies certain constraints on the masses of the bosons. Experimental bounds from
neutrino neutral current data set the Z2 mass above 389 GeV[60]. Further analysis of pp collider data
leads to a bound of MZ2 ≥ 445GeV[61]. Bounds for the Z2 mass are much easier to set than the ones for
MW2 and ζ because the NC interaction doesn’t depend on the RHν field (c.f. (4.32)), which would require
53
a better knowledge of the RHν mass. Unfortunately, such is needed to set bounds on MZ2 . If the RHν
is much lighter than the muon, then measurements of the ξ-parameter in µ-decay using fully stopped
polarized muons tell us that the mass of this charged boson must be higher than 432 GeV.[62] However,
the result is useless if the RHνis very heavy because these cease to have any detectable contribution to
muon decay. The best limits in this latter case are set by considering the mass difference of KL−KS which
gives[63]
M2WL
M2WR
<1
430⇒MWR ≥ 1.6TeV (4.34)
4.1.2 The seesaw mechanism
As stated before, two exclusive doublets (one for each SU(2) group) could have done the job of breaking
the LRSM into U(1)em but the RHνneutrinos would be of Dirac type and – as shown before in chap.3, the
seesaw mechanism which requires Majorana masses for the neutrinos provides a natural way to explain
the smallness of ν mass. Thus we wish to recreate it here, and such is obtained thanks to the two triplets
we have incorporated in our model. The most general LRSM-invariant Yukawa couplings for leptons are
−LY =∑
i, j
hi jLiLΦL jR + hi jLiLΦL jR (4.35)
+ fi j
(LT
iLC−1ε(τ · ∆L)L jL + LTiLC−1ε(τ · ∆L)L jL
)+ h.c. (4.36)
At a first stage, the right triplet acquires a VEV vR , 0 giving heavy Majorana masses to the RHνgiven
by the matrix fi jvR. At the second stage, the bidoublet Φ acquires a VEV but vL is set to zero for the
purpose of simplicity in our present discussion, so that we obtain a type-I seesaw formula with the
following 2ng × 2ng mass matrix
M =
(0 mD
mTD f vR
)(4.37)
where (mD)i j = hi jk1 + hi jk2. The matrix can be approximately (up to terms. O(ρ2)) block diagonalized
by an orthogonal matrix O[40]
O =
(1 − 1
2ρρT ρ
−ρT 1 − 12ρ
Tρ
)(4.38)
The mass matrix on the lower corner of OTMO can be associated with the heavy RHνbecause
54
mheavy = f vR +1
vR
(mT
DmD f−1 + f−1TmT
DmD
)= f vR + O
m2l
vR
≈ f vR (4.39)
where ml is the mass of the charged leptons. The other one is thus the light mass matrix given by
mlight = −1
vRmD f−1mT
D (4.40)
This formula reveals an additional interest for the LRSM as it connects V + A suppression with the
smallness of the neutrino mass since as vR →∞, mν → 0. Now, if we neglect the mixings and set mD ≈ ml,
we obtain
mνl ≈m2
l
MNl
(4.41)
If the heavy RHνmass is generation independent, this means
mνl −mν′l∝ m2
l −m2l′ (4.42)
This last result brings out the problems of our lack of knowledge about the Dirac mass term in (4.40).
If we wish to test this theory in soon-to-come colliders such as the LHC, the WR mass has to be in the
TeV scale. That will unfortunately imply by taking mD ≈ ml that for mνe ∼ 1eV, we get mνµ ∼ 40keV and
mντ ∼ 12MeV which is a spectrum absolutely ruled out by current data which sets . Therefore we may
conclude that a LRSM at the TeV scale where neutrino Dirac masses arise at Lagrangian level is ruled
out. Nevertheless, Dirac masses may still arise at higher loop levels because then mD ∼ αm f /4π, which
safeguards phenomenological possibilities for the model. This can occur in some SO(10) and E6 GUT
models which gives the model restored credibility.
4.2 Neutrino masses in GUT theories
The Standard Model is a wonderfully predictive model which so far works very well with present
experimental data. Even so, it has failed in blindly predicting any fermion masses. The truth is that any
fermion mass or mixing still is a mystery to the SM, which is related to our unsophisticated knowledge
of the Higgs mechanism. The solution to this and to other problems such as the strong CP problem yet
lies ahead of us. The discovery of neutrino mass and our doubts about how to recreate it within the SM’s
framework have only proven this ignorance of ours. Along with this obvious promise of new physics,
goes another hope for new physics which is more closely related to conceptual aesthetics rather than
actual phenomenological requirement, which is the unification of all forces of nature. The first successful
55
attempt was Maxwell’s unification of the electric force together with the magnetic one; the second one
is the actual SM which unites the weak and electromagnetic interactions. The idea of gauge theories
combined with a symmetry breaking mechanism is a very permisive theoretical groundbasis which has
allowed us to wonder – just like SU(3)C×U(1)em is only part of SU(3)C×SU(2)L×U(1)Y – whether the SM
is just a bit of something bigger such as SU(5), SO(10) or E6. Enlarging the group has many advantages
linked with reducing the model’s parameters but on the other hand also increases the SSB’s complexity
which is exactly what we lack knowledge about. The phenomenological interest of these models is that
they allow baryogenesis and predict charge quantization. We would therefore like to see them also
explain the smallness of neutrino masses or the greatnes of neutrino mixings. We will briefly present
how neutrino masses arise in SU(5) and SO(10).
4.2.1 SU(5)
SU(5) is the smallest group that can contain the SM. Its representations can be expressed in terms of
SU(3) × SU(2) ×U(1)’s representations as shown in Table4.2.1.
dimension (SU(3),SU(2))Y5 (3, 1)−2/3 ⊕ (1, 2)110 (3, 2)1/3 ⊕ (3∗, 1)−4/3 ⊕ (1, 1)215 (6, 1)−4/3 ⊕ (3, 2)1/3 ⊕ (1, 3)224=24∗ (8, 1)0 ⊕ (3, 2)−5/3 ⊕ (3∗, 2)5/3 ⊕ (1, 3)0 ⊕ (1, 1)045 (8, 2)1 ⊕ (6∗, 1)−2/3 ⊕ (3, 3)−2/3 ⊕ (3∗, 2)−7/3
⊕(3, 1)−2/3 ⊕ (3∗, 1)8/3 ⊕ (1, 2)1...
...
one sees quite immediately that all SM representations fit in a 5∗ + 10:
5∗ + 10 = (3∗, 1) + (3, 2) + (1, 1) + (1, 2∗) (4.43)
Let’s rewrite this with explicit reference to the SM’s one-generation family of fermions:
5 : (ψi)L = (d1, d2, d3, e−,−ν)L (4.44)
5∗ : (ψi)R = (d1, d2, d3, e+,−ν)R (4.45)
56
10 : (χi j)L =1√
2
0 uc
3 −uc2 −u1 −d1
−uc3 0 uc
1 −u2 −d2uc
2 −uc1 0 −u3 −d3
u1 u2 u3 0 −ec
d1 d2 d2 ec 0
L
(4.46)
Writing down the Yukawa interactions requires breaking down the group SU(5)→ SU(3)colour×U(1)em
with the simplest (fundamental) scalar representation, i.e., 5, along with 24. This is called the minimal
SU(5) model. Other symmetry breaking mechanisms which induce fermion masses may involve 10, 45
or 50. The mass structure of the fermion sector depends on that. The Higgs representation involved are
H = (h1, h2, h3, h+,−h0) = (3, 1) + (2, 1)under SU(3) × SU(2) (4.47)
Σ = 5 × 5∣∣∣traceless = φ5 ⊗ φ5 −
15φ5 · φ5 (4.48)
When Σ acquires a VEV it breaks SU(5) down to SU(3)C × SU(2)L ×U(1)Y as follows
〈Σ〉 = VDiag (1, 1, 1,−3/2,−3/2) (4.49)
giving masses to the gauge bosons[8]
M2X = M2
Y =258
g2V3 (4.50)
Later on, the 5 scalar acquires in its turn a VEV breaking the remaining symmetry into SU(3)C×U(1)Q
〈H〉 =(0, 0, 0, 0, v/
√
2)T
(4.51)
giving masses to the Z0 and the W± bosons
MW =gv2
MZ =gv
2 cosθW(4.52)
where θW is the usual Weinberg angle now expected to give at unification scale tanθW =√
3/5.
Unfortunately, as pretty as this model may seem, it has been ruled out by LEP and SLC data. The
unification scale V is found by running the coupling constants at low energies and having them unite
into only one. In the one-loop approximation we have
dgn
d lnµ= bn
g3n
16π2 (4.53)
57
where bn = 4/3ng − 11/3n for n = 2, 3 neglecting the Higgs contribution. For g =√
4πα,
dgn
d lnµ= bn
g3n
16π2 ⇔dαn
d lnµ= −
bn
2πα2
n (4.54)
which is solved by
α−1n =
1
α−1n (MU) + bn
2π ln(
MUMZ
)α−1
1 =1
α−11 (MU) − 7
2π ln(
MUMZ
)α−1
2 =1
α−12 (MU) − 19
12π ln(
MUMZ
) (4.55)
α−13 =
1
α−13 (MU) + 41
20π ln(
MUMZ
)For a successful unification one needs to have all αn (MU) equal. This condition is equivalent to:
α−11 − 3.08α−1
2 + 2.08α−13 = 0 (4.56)
which is satisfied by the LEP and SLC data for the gauge couplings:
α−11 (MZ) = 58.89 ± 0.11
α−12 (MZ) = 29.75 ± 0.11 (4.57)
α−13 (MZ) = 0.121 ± 0.004 ± 0.001
Nevertheless, even though the minimal non-supersymmetric model has been ruled out by accelerator
data, it is always interesting to analyse it because it is the simplest GUT and many techniques used
for higher–rank groups come out when studying SU(5). Fermion transform as 5 + 10. The product
representation of two fermion fields gives
(5 + 10) × (5 + 10) = 5 + 5 + 10 + 15 + 45 + 45 + 50 (4.58)
Clearly, there cannot be any bare mass terms or they would explicitly break the gauge symmetry.
Thus fermion masses can only arise via SSB through gauge invariant couplings to Higgs scalars. The
possible Higgs representations are shown in (4.58). Since in minimal SU(5) only the 5 and 24 Higgs
58
representations are taken into account, the most general Yukawa couplings are between the fermion fields
and the fundamental Higgs representation. Σ does not couple to fermions. The Yukawa Lagrangian is
hχTijC−1ψiH j + h′εi jklmχT
ijC−1χklHm + h.c. (4.59)
At SSB, the fermions acquire masses given by
mu = MU11v
mc = MU22v
mt = MU33v
md = me = MD11v (4.60)
ms = mµ = MD22v
mb = mτ = MD33v
The neutrinos remain massless because there are no RHν, so Dirac or RH Majorana terms cannot
form. Neither do LH Majorana mass terms arise at higher loop levels, because the Higgs bear no B − L
quantum number. To see this let’s define an operatorF to which we assignF (H) = −2/3,F (ψ) = 1/3 and
F (χ) = 1. Clearly, the Yukawa couplings given in (4.59) are invariant under U(1)F . This transformation
is related to B − L by the relation
B − L =35F −
2√
15λ24 (4.61)
which brings B − L values of h1,2,3 = −2/5 and h+,0 = 0. The adjoint representation does have a B − L
non-zero value, but it doesn’t couple to the fermion fields. Since the Higgs component which develops
a VEV has no B − L quantum number, it cannot generate any Majorana mass terms.
One might argue that the RHνwas left out of the picture purposefully and that nothing forbids us from
adding an SU(5) singlet to the fermion representations. A Dirac mass term can then arise by a coupling
of the 5 fermion representation with the 5 of Higgs and the fermion singlet νR. But this term would
naturally be expected to have mass at quark or lepton scale which is ruled out by experimental bounds.
It would not explain the smallness of neutrino masses. A seesaw mechanism would come in handy here.
The introduction of a symmetric 15-dimensional Higgs boson Πi j. The Yukawa couplings for this scalar
are
59
L15Y = fψT
i C−1ψ jΠi j + h.c. (4.62)
According to Table 4.2.1, the multiplet Π contains a doubly charged isotriplet ∆L which could induce
a type-II seesaw. We assign a non-zero VEV to
〈Π55〉 =u√
2(4.63)
If we assign F (Π) = 2 then the Lagrangian has exact B− L symmetry which is broken by the vaccum.
However, in the most general gauge-invariant potential B − L is explicitly broken by
AφTε (σ · ∆)φ + h.c. (4.64)
where φ is the isodoublet contained in H. This term has to be forbidden in order to have B−L broken
by SSB, which imposes the existence of a pseudo-scalar particle J produced by the VEV of ∆. Such a
model has already been ruled out by measurements of the Z-width. Yet if we allow B− L to be broken at
lagrangian level in the Higgs potential by
V = µΠHiH jΠi j + h.c. (4.65)
U(1)F is explicitly broken in this model and hence B − L as well. The VEV is
〈Π55〉 ≈ µΠv2
V2 (4.66)
Even if µΠ is of order V, we get u ∼ 10−12v ∼ 10−1eV. The smallness of neutrino mass (which is a
Majorana particle) comes naturally in this model.
4.2.2 S0(10)
The SO(10) GUT is very appealing to us interested in neutrino masses because the most fundamental
representation of this gauge group contains exactly all known fermions plus an overall singlet – the RHν.
Thus, unlike the SM or SU(5), neutrino masses arise naturally in this model. Fermion masses are tighted
up to the choice of the Higgs sector and the latter is tied with how we intend to break the symmetry.
Because SO(10) is of rank-5, there are many ways to break it down to SU(3)c × SU(2)L × U(1)Y. Its two
maxinmal continuous subgroups are [12]
• G224 = SU(2)L × SU(2)R × SU(4)C or G224D = SU(2)L × SU(2)R × SU(4)C ×D
60
• G5 = SU(5) ×U(1)
where D-parity is a discrete symmetry which interchanges (2, 1, 4) ↔ (1, 2, 4) in the SO(10) spinor;
its existence imposes gL = gR and ηB = ηB (ηB is the number density of baryons in the universe). SU(5)
has already been proved to be phenomenologically inviable unless we ”pollute” it with inumerable
Higgs representations or introduce supersymmetry, but it’s the simplest route and the philosophy of
it is basically the same for any route. Nevertheless, we still present other pathways to the SM, which
interestingly use the LRSM introduced in the previous above. Still, several roads lead down to the SM
from here. We state two important ones for neutrino masses2:
•
SO(10)MU→54 G224D
Mp→210 G224
MC=MW+R
→210 G2113
MZ′→126 G321 (4.67)
where MU ≈ 1016.6GeV, MWR = MC ≈ 105− 107GeV with MZ′ ≤ 1TeV, leading to a prediction
of sinθW ≈ 0.227[64]. This model is very seductive because it may be tested in soon-to-come
accelerators such as the LHC because of the TeV-scale of B − L breaking-
•
SO(10)MU=MC=Mp
→45 G3221
MWR ,MZ′→126 G321 (4.68)
where MU ≈ 1015.4GeV for sin2 θW ≈ 0.23 and MWR ≈MB−L ≈ 1012GeV.
Fermion masses are generated through SSB like in all previous gauge theories introduced. The
product of two fermion representations gives
16 ⊗ 16 = 10 ⊕ 120 ⊕ 126 (4.69)
Hence, the fermion masses arise through Yukawa couplings with the 10-, 120- and 126-dimensional
Higgs when these acquire VEV. The φ10 and φ126 couplings are symmetric in family indices while the
φ120 are antisymmetric. Under SU(5) these representations transform as
10 = 5 + 5
126 = 1 + 5 + 10 + 15 + 45 + 50 (4.70)
126 = 5 + 5 + 10 + 10 + 45 + 45
2We have indicated under the arrows the Higgs representation which is used for SSB and above it the masses which itgenerates, ie, the scale of SSB.
61
The components which can have non-zero VEV must be colour and charge singlets. The only Higgs
representations with such components are the 1, 5, 5, 45, 45. Their couplings to the fermions are
ψTLσ
2ψLφ10 = φ10(5) (uRuL + νRνL) + φ10(5)(dRdL + eReL
)ψT
Lσ2ψLφ126 = φ126(1)νT
Rσ2νT
R + φ126(15)νLσ2νL + φ126(5) (uRuL − 3νRνL) + φ10(5)
(dRdL − 3eReL
)(4.71)
ψTLσ
2ψLφ10 = φ120(5)νRνL + φ120(45)uRuL + φ120(5)(dRdL + eReL
)+ φ120(45)
(dRdL − 3eReL
)The mass matrix for the neutrinos is then
(νLνR
) (φ126(15) φ10(5) + φ120,5(−)3φ126(5)
φ10(5) + φ120(5) − 3φ126(5) φ126(1)
) (νL νR
)(4.72)
The φ126(1) breaks SO(10) into SU(5) × U(1) and therefore its VEV leaves the content of SU(5) intact
and its mass is expected to be higher than MX. Both Dirac and left and right Majorana mass terms
appear and the high scale of φ126(1) explains via the seesaw mechanism the small neutrino mass. But the
minimal breaking scheme involves no φ126 and is
SO(10)M′X→16 SU(5)
MX→45 SU(3) × SU(2) ×U(1)
MW→10 SU(3) ×U(1) (4.73)
However, B − L is not conserved and one can still generate RH Majorana masses with two-loop
diagrams at O(α2/π2). The RH-Moajorana mas is of order mu/MW × α2/π2MX′ which is heavy and can
still account for the LHν’s small mass. Indeed, the Dirac masses are given by[8]
mνl = mul
π2
εα2MW
MX′(4.74)
where ε is the associated with magnitude of the two-loop diagram couplings. For neutrino masses at
eV,sub-eV scale, ε & 10−2 which is reasonnable.
62
Chapter 5
Leptogenesis
5.1 Baryon asymetry of the universe (BAU)
Since Dirac’s equation which provided equally probable solutions of positive and negative energy (parti-
cle and anti-particle), the fact that the universe was mainly made of ordinary matter – and not anti-matter
– became puzzling, and because of particle-antiparticle anihilation the very reason why any bit of the
universe still exists is at least enigmatic. The next two sections focus on a new attempt to solve this
mistery.
5.1.1 Evidence for BAU
Within its reach, mankind has found that the world is mainly made of matter. On earth, the only evidence
of anti-matter’s existence comes from accelerators or nuclear decays which produce the greatest known
amounts, which still only add up to a trillionth of a gram. So all matter on earth is ordinary. The rest of
our solar system is also composed of the same kind of matter, because many probes have been sent to
many of the system’s orbiting objects and none of them desintegrated into light. Analysis of solar fluxes
allow us to probe the baryon-to-antibaryon ratio of the sun’s composition, and there again we find a
proton-to-antiproton ratio of and even this small amount of anti-protons appears to be consistent with
the assumption that it results as a byproduct of cosmic ray collision with the interstellar medium. The
same analysis is valid for the rest of the galaxy from which experiments such as Auger have detected
high energy rays coming from very far in the galaxy and the same results apply.
Studies of the large scale structure of the universe and of the cosmic microwave background’s (CMB)
anisotropy from WMAP have given us an estimate of the baryon asymetry. Because the universe went
through an e± anihilation era, it is important to compare the baryon number density to the photon density,
and the ratio of these two densities – since each one of them evolves with the inverse of a comoving
volume R−3 – is a conserved quantity as long as the B-violating reactions are occuring slowly.
63
ηCMBB ≡
ηB
ηγ= (6.1 ± 0.2) × 10−10 (5.1)
This result concurrs with the result from Big Bang Nucleosynthsis (BBN) where the abundances of3He, 4He,D,6Li and 7Li can be measured from astrophysical observations and depend crucially on the
value of the baryon-to-photon ratio[9]:
ηBBNB ≈ (5.6 ± 0.9) × 10−10 (5.2)
which is perfectly consistent, hence demonstrating the validity of BBN and standard cosmology.
For the rest of our universe things become a little more inferrative, because there is no way of telling
whether a galaxy is made of matter or antimatter just by looking at it. Nevertheless, X-Ray analysis of
the intergalactical medium has revealed the existence of hydrogen gas clouds bathing the galaxies in the
local cluster, and a non-neglegible presence of antimatter would certainly produce an accute emission
of gamma rays on the border between the antimatter patches and the gas, but no such flux has been
measured so far. Virgo, for instance, a 1013− 1014M⊙ nearby cluster of galaxies shows no abnormally
strong γ-ray flux. Therefore, if there is a significant amount of antimatter in the universe, it must be
parted from the rest of the universe in a scale of 1012M⊙ or even larger, which seems quite unreasonnable
and by no means indicates a baryon symmetric universe. More precisely, in a locally-baryon-symmetric
universe nucleons and antinucleons remain in chemical equilibrium at nb/s = nb/s ≈ 7 × 10−20 down to
temperatures of ∼22 MeV.[9] These estimated densities are several orders of magnitude smaller than the
observed values of baryon density nb/s ≈ 8× 10−11. In this symetric scenario, the universe would require
a segregation mechanism acting above 38 MeV, when the density was the observed one. Unfortunately,
the horizon at that time only contained 10−7M⊙, and causality obviously rules out by far separating
masses of 1012M⊙. We can without much further ado, say that antibaryons are nowhere to be found in
the universe compared to baryons.
Since B-violating interactions are unknown to us in the present state of our knowledge (not entirely
true at the non-perturbative level), the most reasonnable conclusion is that the universe already possessed
this asymmetry at early times. In the primordial soup (T & 1GeV, t . 10−6s) thermal quark-antiquark
pairs were highly abundant and the present maximal B-asymmetry was at the time a very small q − q
asymmetry:
nq − nq
nq≈ 3 × 10−8 (5.3)
64
In the past, the Universe was sought to have been created with this small yet crucial number right from
the start. With the advent of Grand Unification Theories (GUTs) which predict B-violating interactions,
a loophole was opened for a initially symmetric universe evolving later into this slightly asymmetric
one. But before we get into that, let’s have a short overview of what’s needed for the generation of this
number, a.k.a., baryogenesis.
5.1.2 Basic Ingredients and direct baryogenesis
Long before GUTs ever came to be in 1967, Sakharov had already laid down the threefold essential
features of a theory which aims at producing dynamically the BAU. These conditions are:
• B violation,
• C and CP violation,
• departure from equilibrium.
The technical and physical nature of each mechanism for baryogenesis may vary, but they must all
without exception contain these four crucial ingredients.
B Violation This requirement is fairly obvious; without it the present baryon asymmetry can only reflect
an initial one. GUTs provide an elegant framework for microphysics where B and L violating interactions
arise naturally thanks to putting together fermions and leptons in a reduced number of representations.
In such theories, gauge bosons mediate interactions which transform quarks into leptons and vice-
versa. The lifetime of the proton in these scenarios should be of order τp ∼ α−2GUTM4m−5
p & 1031 yrs
(current experimental limit[30]) which implies that these gauge bosons must have a mass over 1014GeV,
which is the reason why they are so invisible nowadays but could have been relevant in the past at
extreme temperatures. The present Standard Model’s Lagrangian has no such interactions, although the
symmetry is purely accidental and should neutrinos turn out to be Majorana-like, they would violate
lepton number. Also, as indicated by ’tHooft, the SM conserves B and L at the perturbative level but
there is an anomalous very small violation coming from non-perturbative transistions in the theory’s non-
trivial vacuum which is irrelevant at∼ 3K but then again could have been significant at early times thanks
to the high temperatures short after the Big Bang as noted by Kuzmin, Rubakov and Shaposhnikov[65].
C and CP violation Both are provided in the present state of our microscopic knowledge. C is maximally
violated in the electroweak sector. CP was first found to be violated in the neutral kaon system and since
65
it is linked to spontaneous symmetry breaking (SSB) which is still crudely understood one could expect
to see it in all sectors of the theory.
The reason why it is so necessary for baryogenesis seems pretty obvious at first, since we wish to have
a higher amount of baryons compared to antibaryons. Even so, let’s prove it is inevitable. Suppose, for
example, that the X baryons decay into two channels with branching ratios r and 1−r and baryon number
B1 and B2; X bosons decay with different branching ratios r, 1 − r into chanels with baryon number −B1
and −B2. The average baryon number produced by decays of X and X is
∆B = ∆B(X) + ∆B(X)
= B1|M(X→ B1)| + B2|M(X→ B2)|2 − B1|M(X→ B1)| − B2|M(X→ B2)|2 (5.4)
=12
(r − r) (B1 − B2)
One imediately concludes that baryon number (B1 , B2), C and CP (r , r) must be violated in order
to obtain a baryon number generation in the process.
Departure from thermal equilibrium Throughout the history of the universe many species have de-
parted from equilibrium and have left us with precious relics, namely, neutrino decoupling, decoupling of
the CMB, primordial nucleosynthesis and, hopefully, baryogenesis, inflation, decoupling of relic WIMPs
(weakly-interacting massive particles), etc. When a given species is in thermal equilibrium its state is
easy to follow. The densities of a particle species and its antiparticle in equilibrium are given by
nb ∝
∫∞
0
1
e(√
p2+m2b−µ)/T
+ 1dp
nb ∝
∫∞
0
1
e(√
p2+m2b−µ)/T
dp(5.5)
In chemical equilibrium the entropy is maximal when chemical potentials vanish for all nonconserved
quantum numbers. Furthermore, CPT invariance of gauge theories forces both particle and antiparticle
masses to be equal. Hence, in equilibrium, nb = nb.
5.2 Leptogenesis in the single flavour approximation
5.2.1 Overview
Because leptogenesis relies so heavily upon neutrino masses – on their various mass models and their
high parameter flexibility – as well as on many approximation routes, the number of approaches is quite
66
extensive. In this chapter we set ourselves to present a very simple model based on the decay of the lightest
RHνwhose mass is given by the type-I seesaw formula. The is called the one-flavour approximation
leptogenesis and often provides good estimates of the BAU generated, but most importantly illustrates
the main ideas behind the theory.
In this model, first introduced by Fukugita and Yanagida[66], the mass of the neutrinos is generated by
the type-I seesaw mechanism. In such a model, the RHνare very heavy particles which are SU(5)-sterile,
so their interactions are summed up to Yukawa couplings with the Higgs and leptons. Also, because
of their heaviness, the RHνdecouple from the primordial thermal bath and decay away into leptons,
anti-leptons and scalar bosons. The strength of CP-violation in the lepton sector will determine the
excess of leptons over anti-leptons at this stage. We also assume that the RHνhave a hierarchical mass
spectrum and therefore any lepton number created by the two heavier neutrinos N2,3 will be eventually
washed-out by the later decay of the lightest one N1. Once the decay reaction freezes-out, the lepton
asymmetry will evolve without being affected as long as all other processes present are CP-conserving.
At this stage, non-perturbative B + L violating processes that are in equilibrium partly convert this lepton
asymmetry into a baryon asymmetry. So let’s state it again[14]:
• During thermalization RHνare created thanks to inverse decays ` jφ→ Nk and Nk → ¯jφ. A lepton
asymmetry is generated
YL ≡nL − nL
s(5.6)
where YL is the comoving number density. At this stage it is ∼ −ε/CYN where
εkj ≡Γ(N→ ` jφ) − Γ(N→ ¯jφ)
Γ(N→ ` jφ) + Γ(N→ ¯jφ)(5.7)
and C is a wash-out factor greater than unity accounting for the partial depletion of the lepton
asymmetry during thermalization. The equilibrium density of RHνat the end of the thermal epoch
is roughly YN ∼ 10−3.
• When the temperature of the universe becomes . MRHν the heavy neutrinos decouple from the
thermal bath and decay giving way to a lepton asymmetry ∼ εYN. The total asymmetry is
YL ≈ ε(1 −
1C
)︸ ︷︷ ︸
η
YN (5.8)
In thermal leptogenesis, η ∼ 0.1.
67
• Finally, the baryon asymmetry is owed to the work of sphalerons which partly convert the lepton-
asymmetry into a baryon-asymmetry. This factor is of about 1/2. Hence, in thermal leptogenesis
YB ∼ 10−5ε which requires ε ∼ 10−6− 10−7 in order to mimic the observed BAU of YB ≈ 8.7 × 10−11
5.2.2 CP violation
The CP asymmetry is the first parameter which we wish to compute. In the mass eigenbasis the
Lagrangian for the heavy RHνis
LY = yαβ ¯αφeβ − h jk ¯jφNk −12
NkMkNk + h.c. (5.9)
where α, β, j are flavour indices and k = 1, 2, 3 is a hierarchical (from lightest to heaviest) index for the
RHνmass eigenstates. The relevant processes are
Nk →
` j + φ (with rate Γ)¯j + φ (with rate Γ) (5.10)
where L is violated by one unit. Because of CPT invariance, there is no difference between the rate
one process and its CP-opposite because Γ = |h jk|2I0 where I0 is a Lorentz phase space integral and is
therefore invariant under CP transformation. However, at higher radiative orders[4]
Γ =
∫[D]|M0 +M1 +M2 + . . . |2
=
∫[D]
(|M0|
2 +M†0M1 +M†1M0
)(5.11)
= |h jk|2I0 + h∗jkh jmhnmh∗nkI1 + h jkh∗jmh∗nmhnkI∗1 + O(h6)
where∫
[D] is a phase space integral and
Γ = |h jk|2I0 + h jkh∗jmh∗nmhnkI1 + h∗jkh jmhnmh∗nkI∗1 + O(h6) (5.12)
where I1 = I1 for the same reasons as I0 = I0. Consequently,
ε =Γ + Γ
Γ + Γ
= −4
Γ + ΓIm(h∗jkh jmhnmh∗nk)Im(I1)
(5.13)
From (5.13) we see that the couplings must be complex and that two heavy particles must exist in
the model. Furthermore, the mass of the RHνis required to be greater than the sum of the lepton and
68
Higgs mass together because the imaginary part of the integral corresponds to the interference with the
intermediate on-shell part of the one-loop integral. In this simplest model we assume that L evolves
regardless of flavour. Therefore our CP-violation parameter must be summed over the product leptons’
flavour, and since all asymmetry due to N2,3 is washed out by the later decay of N1 the relevant parameter
is
ε1 =∑
j
εkj =1
8π
∑m,1
Im[(h†h)21m]
(h†h)11]
f
M2m
M21
+ g
M2m
M21
(5.14)
where
f (x) =√
x[1 − (1 + x) ln
(1 + xx
)]and g(x) =
√x
1 − x(5.15)
The explicit calculation of ε1 is done in Appendix C. Since we’re dealing with a highly hierarchical
spectrum M1 M2,M3 we can approximate
f (x) + g(x) →x1 −3
2√
x⇒ |ε1| ≈
316π
M1
(h†h)11
∑j
Im[(h†h)1 j]M j
(5.16)
Further, this result, together the seesaw formula
mαβ = v2hαM−1k hβk (5.17)
where v is the isodoublet Higgs VEV, gives
|ε1| ≈3M1
16π(hTm∗h)11
v2(h†h)11≈
316π
(hTm∗h)11
m1(5.18)
where
m1 = 8πv2
M1ΓN1 = (h†h)11
v2
M1(5.19)
It was shown by Davidson and Ibarra [14] and further improved that
|ε1| .3M1
16πv2 (mmax−mmin) ×
1 − mmin
m1if mmin
mmax
1 −(
mmin
m1
)2 (5.20)
Because the baryon asymmetry produced by leptogenesis should at least reproduce the observed one
YB ≈ 1.38 × 10−3ηε1 & YobsB ≈ 8.7 × 10−11 (5.21)
69
it restricts M1 to
M1 & 6.5 × 108GeVη−1×
YB
YobsB
( matm
α(mmax,mmin, m1)
)(5.22)
where α(mmax,mmin, m1) ≈ 1 is anything in (5.20) apart for the 3M1/16πv2 factor. For now we may say
that η . 1. Such a boundary pushes the RHνmass scale far beyond the electroweak scale for leptogenesis
to work, which conversely makes the type-I seesaw mechanism a hardly (if not impossible) testable
model in laboratory experiments. Even so, our aim is to learn the basics of leptogenesis so we will
proceed with our study of this simply model.
5.2.3 Out-of-equilibrium dynamics and wash-out
The previous section shows that by adjusting a few parameters |ε1| can me made great enough to
generate the correct baryon-to-photon ratio ηB ≈ 10−10. However, such an analysis is incomplete; the
very solution to the problem must track the evolution of both N1 density and created B − L asymmetry.
There are various reasons for this, namely, the N1 density might be insufficient (or even completely
absent as could be implied by inflation scenarios) one has to find out whether processes involving the
SM particles can build up enough heavy neutrinos for a successful letogenesis; also, inverse decays
(`φ → N1) might be too strong and wash away any produced lepton asymmetry. It all depends upon
when processes couple or decouple from the thermal bath. Usually there is a good rule of thumb which
states that a given species is[9]
Γ & H (coupled) (5.23)
Γ . H (decoupled) (5.24)
where Γ is the interaction rate per particle for the species that keep the species in thermal equilibrium
and H is the Hubble constant. While this rule is usually very accurate, a proper study requires following
statiscally the microscopic processes involved which is done by solving the Boltzmann equations.
Dynamics of an expanding Universe Since the expansion of the universe is the key for decoupling
processes and leaving relic abundances, one must understand its mechanism. The evolution of the
Universe is described by the Einstein equation which relate the geometry of the Universe to its content:
Rµν −12
gµνR = 8πGTµν + Λgµν . (5.25)
70
In this equation R is the Ricci scalar, Rµν the Ricci tensor, Tµν the stress-energy tensor, gµν the space-
time metric, Λ is the cosmological constant and G is the Newton constant,
G =1
M2pl
,
Mpl ' 1.221 × 1019GeV . (5.26)
Assuming local homogeneity and isotropy (because energy and momentum are conserved locally),
the space-time metric is given by the maximally-symmetric Friedmann-Robertson-Walker (FRW) metric
ds2 = dt2− R2(t)
(dr2
1 − kr2 + r2dθ2 + r2 sin2 θdφ2)
(5.27)
where (t, r, θ, φ) are spherical comoving coordinates, R(t) is the cosmic scale factor and k = −1, 0,+1
gives the type of metric (hyperbolic, flat or spherical respectively).
Assuming further that the Universe content is a perfect fluid, we can write the stress-energy tensor as:
Tµν = −pgµν + (p + ρ)uµuν , (5.28)
where p is the pressure and ρ the energy density of the perfect fluid. The velocity vector of the fluid,
uµ is given in the rest frame of the plasma by u = (1, 0, 0, 0), so that Tµν = diag(ρ,−p,−p,−p).
With both metric and universe content informations we can now rewrite the 0-0 component of the Einstein
equations as
(RR
)2
+k
R2 =8πG
3ρ +
Λ
3(5.29)
known as the Friedmann equation. Setting Λ = 0 and defining H ≡ (R/R) and
Ω ≡ρ
ρC≡
3H2
8πG(5.30)
the Friedmann equation can be recast as
Ω =k
H2R2 + 1 (5.31)
WMAP has recently made measurements of the CMB and determined the density parameter at
present time to be[67]
Ω0 = 1.02 ± 0.02 (5.32)
71
which means that one can to a very good approximation assume that the universe’s curvature is flat
(k = 0). That being so, the Friedmann equation gives
H =
√8πρ
3m2Pl
≈1.66√
g∗T2
mPl(5.33)
where g∗ is the effective number of degrees of freedom contributing to the energy-density and is given
by
g∗ =78
∑f ermions
g f
(T f
Tγ
)4
+∑
bosons
gb
(Tb
Tγ
)4
(5.34)
Equilibrium thermodynamics The number density n is a particle species is given by
n =g
(2π)3
∫f (~p)d3p (5.35)
with
f (~p) =
[exp
(E − µ
T
)+ a
]−1
(5.36)
where a = −1, 0,+1 for Bose-Einstein (BE), Maxwell-Boltzmann (MB) or Fermi-Dirac (FD) distribu-
tions respectively. For MB statistics the equilibrium density is
neqMB =
gT3
2π2 z2K2(z) z = m
T , 0, µ = 0gT3
π2 m = 0.(5.37)
where g is the particle’s internal number of degrees of freedom and the modified Bessel function is
approximated by
z2K2(z) =
∫∞
zxe−x√
x2 − z2dx→
2 z 1(158 + z
) √πz2 e−z z 1
(5.38)
For FD statistics
neqFD =
gT3
2π2
(32ζ(3) +
µTζ(2) + . . .
)m T
neqMB µ T, m T
(5.39)
where ζ(2) = π2/6 and ζ(3) = 1.202 and ζ(4) = π4/90. For BE statistics
neqBE =
gT3
π2
(ζ(3) +
µTζ(2) + . . .
)m T
neqMB µ T, m T
(5.40)
72
The internal numbers of degrees of freedom g for the particles taking part in the relevant pprocesses
for leptogenesis are gN = 2 for the RHνbecause they are Majorana fermions, geR = 1 for the weak isosinglet
and g`,H = 2 for both fermion and scalar weak isodoublets.
Out-of-equilibrium thermodynamics Out-of-equilibrium dynamics require the study of the Boltz-
mann equations. The general Boltzmann equation can be written as
L[ f ] = C[ f ] (5.41)
where f (pµ, xµ) is the phase-space distribution of the species. The covariant Liouville operator is
given by
L[ f ] = pµ∂ f∂xµ− Γ
µνρpνpρ
∂ f∂pµ
(5.42)
where Γµνρ is the Christoffel symbol. Applying this to a Friedman-Robertson-Walker metric we get
L[ f (E, t)] = E∂ f∂t−
RR|−→p |2
∂ f∂E
(5.43)
Since
n(t) =g
(2π)3
∫d3p f (E, t) (5.44)
therefore with an integration by parts the Boltzmann equation becomes
dndt
+ 3Hn =12
∫C[ f ] dΠ (5.45)
where H is the Hubble constant and
dΠ ≡g
(2π)3
d3p2E
(5.46)
The collision term on the r.h.s of (5.45) for the process X + a + b + . . .↔ i + j + . . . is
12
∫C[ f ] dΠ = −
∫dΠX dΠa dΠb . . . dΠi dΠ j . . . (5.47)
× (2π)4δ4(pX + pa + pb + . . . − pi − p j − . . .) (5.48)
×
[|M|
2X+a+b+...→i+ j+... fa fb . . . fX(1 ± fi)(1 ± f j) . . . (5.49)
|M|2i+ j+...→X+a+b+... fi f j . . . fX(1 ± fa)(1 ± fb) . . . (1 ± fX)
](5.50)
73
We will assume an approximate Maxwell-Boltzmann distribution which allows us to simply
(1 + ± f ) ≈ 1 (5.51)
which is valid in absence of Bose condensation or Fermi degeneracy, so
nX + 3Hn = −
∫dΠX dΠa dΠb . . . dΠi dΠ j . . . (2π)4δ4(pX + pa + pb + . . . − pi − p j − . . .) (5.52)
×
[fX fa fb . . . |M(X + a + b + . . .→ i + j + . . .)|2 − fi f j . . . |M(i + j + . . .→ X + a + b + . . .)|2
](5.53)
Assuming that all particles are in kinetic equilibrium so that f = (n/neq) f eq with f eq = exp(−E/T) the
r.h.s. of (5.53) is
∫dΠX dΠa dΠb . . . dΠi dΠ j . . . (2π)4δ4(pX + pa + pb + . . . − pi − p j − . . .) (5.54)nX
neqX
f eqX
na
neqa
f eqa . . . |M(X + a + b + . . .→ i + j + . . .)|2
−ni
neqi
f eqi
n j
neqj
f eqj . . . |M(i + j + . . .→ X + a + b + . . .)|2
It is usually useful to write these equations in terms of the number density per comoving volume
which is done by normalizing it to the entropy s ∝ R−3 which is a conserved quantity. As so, let’s define
Y ≡ns
(5.55)
Since entropy is conserved in a comoving volume, the l.h.s of the Boltzmann equation can be rewritten
as
nX + 3HnX = sYX (5.56)
Let’s also write it in terms of z = mX/T. This can be done by cleverly noting that during a radiation-
dominated epoch, z and t are related by
t = 0.301g−1/2∗
mPl
T2 = 0.301g−1/2∗
mPl
m2X
z2≡ 2/H(z) = 2
mPl
1.66√
g∗m2 z−2 (5.57)
and that
74
s =2π2g∗s
45T3 (5.58)
where the internal number of degrees of freedom which contributes for the entropy is
g∗s =78
∑f ermions
g f
(T f
Tγ
)3
+∑
bosons
gb
(Tb
Tγ
)3
(5.59)
so that the Boltzmann equation becomes
dYdz
= −z
H(m)C (5.60)
where H(x) = H(m)/z2 = 1.66√
g∗m2/mPl. At leptogenesis temperatures, g∗s = g∗. At present state,
the main contribution to the entropy comes from photon, neutrino and anti-neutrino backgrounds at
Tν ≈ (4/11)1/3Tγ. Thus s0 ∝ g∗sT3γ with g∗s ∼ 3.9.
The collisional term in (??) also transforms as
dYX
dz=
YiY j . . .
(sYeqi )(sYeq
j ) . . .γXa...
i j... −YXYa . . .
(sYeqX )(sYeq
a ) . . .γ
i j...Xa... (5.61)
where the interaction density is defined as
γXa...i j... ≡ γ(X + a + . . .→ i + j + . . .)
=
∫dΠX dΠa . . . dΠi dΠ j . . . f eq
X f eqa . . . |M(X + a + . . .→ i + j + . . .)|2(2π)4δ4(pX + pa + . . . − pi + p j + . . .)
(5.62)
Now, the relevant processes are two-body decays and scatterings. In a decay, the four possible final
states, νβφ0,eβφ+,νβ and e−φ, the rates are (c.f. Appendix C)
|M(N1 → νβφ0)|2 = 2(h††)11(pNp`) = (h††)11M2
1 (5.63)
and with
∫(2π)4δ4(pX − pi − p j)dΠidΠ j =
∫|~pi|
16π2√
sdΩi =
|~pi − ~p j|
8π√
s=
√(pi · p j)2 −m2
i m2j
4πs(5.64)
we obtain for the decay rate density (check Appendix C for notation)
75
γ(N1 → `βφ, ¯βφ) = γNφ`β
+ γNφ ¯β
=
∫d3p
2E(2π)3 e−E/T4(h†h)211M2
1
∫(2π)4δ4(pX − pi − p j)dΠidΠ j
=1
8π3 T2M21(h†h)11
∫∞
ze−x√
x2 − z2 dx
= sYeqK1(z)K2(z)
Γ(N1 → `βφ, ¯βφ)
=gNT3
2π2 z2K1(z)Γ(N1 → `βφ) (5.65)
whereK1 is a modified Bessel function [25]:
zK1(z) =
∫∞
ze−x√
x2 − z2 dx→
1 z 1√π2 ze −z z 1
(5.66)
A more extensive discussion of the inclusive two-body decay rate density was evaluated in [26].
The equivalent result for two-body scatterings is given by
γXaij =
∫dΠX dΠa f eq
X dΠi dΠ j f eqa
∫|M(X + a→ i + j)|2(2π)4δ(pX + pa − pi − p j)
= 4gXga
∫dΠX dΠae−(EX+Ea)/T
√(pX · pa)2 −m2
Xm2a σ((pX + pa)2)
= gXga
∫dQ0 d3Q
(2π)4
e−Q0/T
πs[(pX · pa)2
−m2Xm2
a]σ(Q2)
=gXga
32π5
∫s ds dΩ
∫√
sdQ0e−Q0/T
√Q2
0 − s
1 −
m2X + m2
a
s2
2
− 4m2
Xm2a
s2
σ(s)
=T
64π4
∫∞
(mX+ma)2ds√
sK1
( √s
T
)σ(s) (5.67)
where s = p2X + p2
a and sσ(s) = 8[(pµXpaµ)2−m2
Xm2a]σ(s).
So we’re finally ready to approach leptogenesis with this new formalism. Let’s first parametrize the
decay rates to better reflect the CP violation:
γ(N1 → `φ) ≡ γ( ¯φ→ N1) = (1 + ε1)γDγ(N1 → ¯φ) ≡ γ(`φ→ N1) = (1 − ε1)γD (5.68)
where γD is the tree-level decay density
γD = neqXK1(z)K2(z)
ΓD =(h†h)11
16πM1neq
XK1(z)K2(z)
(5.69)
76
This parametrization garantees that
εN ≡γ(N1 → `φ) − γ( ¯φ→ N1)γ(N1 → `φ) + γ( ¯φ→ N1)
= ε1 (5.70)
Assuming that all particles except N1, ` and ¯ are in thermal equilibrium1,
nN1 + 3HnN1 = −2
nN1
neqN1
− 1
(γD + γφ,s + γφ,t) + O(ε1,
µ`T
)(5.71)
where µ` is the chemical potential for `. The other relevant variable is nB−L2:
nB−L + 3HnB−L = −2ε1
nN1
neqN1
− 1
γD −nB−L
neq`
γW + O(ε2
1,µ`T
)(5.72)
where
γW = γD +nN1
neqN1
γφ,s + 2γN,s + 2γN,t (5.73)
At first order – neglecting the neutrino-top scattering rates – the Boltzmann equations are written in
their final form as
dYN
dz= −D(z)K1
(YN(z) − Yeq
N (z))
(5.74)
dYL
dz= −ε1D(z)K1
(YN(z) − Yeq
N (z))−Wid(z)K1YL(z) (5.75)
where
z = M1/T, (5.76)
D(z) = zK1(z)K2(z)
, (5.77)
Wid(z) =12
YeqN
Yeq`
D(z) =14
z3K1(z), (5.78)
K1 =ΓN1
H(M1)(5.79)
The densities in equilibrium are
1c.f. Appendix C2c.f. Appendix C
77
YeqN (z) =
34ζ(3)
452π4g∗
z2K2(z) ' 3.9 × 10−3 for z 1 (5.80)
where the factor 3ζ(3)/4 is a corrective factor to the Maxwell-Boltzmann to make it more like the
Fermi-Dirac behaviour at high energies.
There are two possible regimes: either K1 1 meaning a very strong wash-out because decays are
in-equilibrium and lepton asymmetries are strongly washed out, whereas in the opposite case where
K1 1 the decays go quickly out of equilibrium and wash-out is much weaker. Two solutions are shown
in fig.5.2.3[4].
Figure 5.1: Evolution of comoving number densities as functions of z = M1/T . The grey dashed linerepresents Yeq
N1, the number density of N1 in thermal equilibrium, whereas the black line stands for YN,
the solution of the Boltzmann equation. In red (light grey) is depicted the lepton asymmetry YL. For allthese plots the CP asymmetry is taken equal to ε1 = 10−6, while the washout parameter is K1 = 0.01(100)on the left (right)..
It is important to note that all of Sakharov’s conditions for dynamical baryogenesis are respected in
this model. The production term for L – which is the first term on the r.h.s of the equation – depends on
ε1 , 0 and YN(z) , YeqN (z).
5.2.4 Lepton asymmetry and anomalous B + L violation
The plots in fig.5.2.3 show that there are basically three phases of leptogenesis:
• A thermalization phase where RHν’s are produced by scatterings and inverse decays of SM particles.
This process evolves until equilibrium is reached. In a strong washout regime (SWR), RHνare
thermalised fast and equilibrium is reached at high temperatures; conversely, in the weak washout
78
regime (WWR), equilibrium is only reached at lower energies. Anyhow, both cases produce a
lepton asymmetry which is ∝ YN.
• In the next phase the RHνdensity decreases. In the SWR YN tracks closely its equilibrium value due
to fast decays while in the WWR the decays come later, when YN has already left its equilibrium
value. Another asymmetry of opposite sign is created which threatens to cancel out the first one.
• Eventually, temperatures decrease and the RHν’s become too diluted freezing out the processes that
involve them. A small residual asymmetry survives. In the SWR, the resulting lepton asymmetry
is accurately given by[4]
YL ' ε1
0.4K1.16
1
YeqN (zin) (5.81)
ηs = 0.4/K1.161 (5.82)
in the WWR, the resulting lepton asymmetry is
YL ' 1.3ε1K21Yeq
N (zin)
ηw = 1.3K21
(5.83)
5.2.5 Baryogenesis through leptogenesis
In the SM there is no B or L violation at Lagrangian level. Still, it was pointed out by t’Hooft that non-
perturbative effects called instantons can lead to the violation of B+L – yet with B−L conservation. The is
due to the complex topology of the ground-state in a non-abelian gauge theory. When transitions between
degenerate vacua occur they violate B + L. Such tunneling involves going through field configurations
called sphalerons. These arise from a non-zero divergence of the leptonic and baryonic currents in the
ABJ triangle anomaly[15]:
∂µJBµ = ∂µJL
µ =ng
32π2
(−g2Wa
µνWµνa + g′2BµνWµν
)(5.84)
where g and g′ are the gauge couplings of SU(2)L and U(1)Y respectively, with Waµν and Bµν being the
corresponding field tensors and Fµν ≡ 12εµνλρFλρ . Obviously:
79
∆(B − L) =
∫d4x∂µ
(JBµ − JL
µ
)= 0
∆(B + L) =
∫d4x∂µ
(JBµ + JL
µ
)=
∫d4x
ng
16π2
(−g2Wa
µνWµνa
)+ g′2BµνWµν) (5.85)
= 2ngθ
where ∆Ncs is an integer called the Chern-Simons number. At T = 0 between two vacua with distinct
topological charges is exponentially suppressed, with
Γ ≈ exp(−
8π2
g2
)(5.86)
However at T & TEW Kuzmin, Rubakov and Shapovnikov showed that Higgs and gauge boson field
configurations can cause ”leap-overs” from one vacuum to another. The amplitude for this process is
roughly exp(−Esph/T) with
Esph ≈8πv
g(5.87)
being the height of the energy barrier between two vacua with different winding numbers. Hence, at
T Esph, the transition amplitude is non-neglegible and occurs rapidly. In the SMthere are two different
gauge groups with topologically non-trivial vacua, and therefore there will be effective interactions
mediated by either strong (SU(3)C) or weak sphalerons (SU(2)L):
vacuum ∑
generations qL + qL + uR + dR ⇒ ΓQCD/V ' 250α5s T4 at T . 1013 GeV∑
generations qL + qL + qL + `L ⇒ ΓQCD/V ' 250α5s T4 at T . 1012 GeV
(5.88)
where rates’ temperatures are calculated so that the sphaleron processes occur faster than Hubble
expansion. From here, one might be tempted – because of B − L conservation – to associate a B variation
to a B + L variation. Actually, things are a bit more complicated and require determination of which
interactions are in kinetic equilibrium when the lepton asymmetry is created. The actual formula is[59]
YB = −8ng + 4nH
14ng + 9nHYL (5.89)
where ng = 3 is the number of fermion generation and nH = 1 is the number of Higgs doublets in the
model. Going back to our leptogenesis scenario, as the universe cools down with t &M1/100, the lepton
asymmetry in the single flavour approximation is[4]
YB = −2851
YL ≈ −1.38 × 10−3ε1η (5.90)
80
Choosing ε1 ∼ 10−6 and a global parametrization η = (η−1w + η−1
s )−1' (0.8K−2
1 + 2.5K−1.161 )−1, the
observed value YobsB ≈ 8.7 × 10−11 implies K1 ∼ 3 × 0.3.
5.2.6 Dependence on initial conditions
We have assumed in the discussion above that RHνare inexistent before zin and are produced by inverse
decays and scatterings. This production is motivated by the wish for an initially symmetric universe.
Nevertheless, it is important to track an initial RHνdensity, because we have no idea what could have
happenned before leptogenesis, some GUT mechanism (or something even more exotic) could generate
RHνat a higher temperature scale. Considering three different scenarios, the earlier one YN(zin) = 0, the
equilibrium case YN(zin) = YeqN (zin) and the dominant YN(zin) > Yeq
N (zin), a strong washout scenario is the
one that depends the least on initial conditions and is thus the most robust in terms of predictions. Also,
the more RHνare present in an initial state the greater the efficiency, a difference which is maximal for
the dominant case in a weak washout regime.
81
Chapter 6
Conclusion
From the 1980’s till today, a considerable set of experiments of solar, atmospheric and reactor neutrinos
has established that the once massless neutrinos of the Standard Model have in fact mass (two of them at
least), though in a sub-eV scale. The simplest extension of the Standard Model as presented in the seesaw
mechanism provides a natural explanation for the smallness of their mass. Furthermore, it also provides
for the baryon asymmetry of the universe. The remarkable coincidence that this simple extension would
solve two apparently unrelated problems makes it a very attractive one. One might in fact say that the
discovery of neutrino masses is one of the most important experimental discoveries of recent times.
The main issue remains, whether such a theory of the BAU can be tested. Such a test would involve a
direct measurement of the CP asymmetry produced in heavy neutrino decays, which are far too heavy
to be produced in any even dreamed accelerator. And even if they’re lighter, their couplings become
smaller making it also impossible to measure anything. Indirect tests in cosmology, by measuring the
asymmetries produced by leptogenesis, are also ruled out there are two many high energy parameters
which we have have no idea yet on how to measure them. In the most optimistic case, these ar four:
• the BAU, known to a good accuracy;
• three neutrino cosmic flavour asymmetries, to which no experimental test has been proposed so
far.
Such information would yield information to us on either the flavour CP asymmetries (if leptogenesis
ocurred in the unflavoured regime) or else on a combination of the flavour CP asymmetries and on the
flavour depend washouts. Unfortunately, we haven’t even measured the cosmic neutrino background,
and the possibility of measuring O(10−10) asymmetries in the background is beyond our wildest hopes.
Nevertheless, even though no direct or indirect tests can prove leptogenesis is right, there are still tests that
can make it likely. The first one would obviously be the discovery of Majorana masses by measurement
82
of neutrinoless double-beta decay. A measurement of such a lepton number violating process will prove
for the first time the fulfillment of the first of the three Sakharov conditions. A failure in the discovery
of this process cannot however disprove leptogenesis alone, because if the neutrinos are found to be of
dirac type there is still a chance for leptogenesis with a normal hierarchic mass spectrum. So a complete
rejection of the theory requires data from neutrino oscillation experiments which can measure the mass
differences. In fact, if νe ↔ νµ oscillations are enhanced and νe ↔ νµ are suppressed then the mass state
involved in atmospheric oscillations is the heaviest one, which corresponds to a normal hierarchy. The
opposite goes for an inverted one. So no evidence of ββ0ν at |mee| . 10meV and an establishment of an
inverted hierarchy spectrum will disprove leptogenesis.
The second of Sakharov’s conditions requires the discovery of CP violation in the leptonic sector. Pro-
posed experiments such as SuperBeam and NOνA only probe the Dirac phases of the mixing matrix,
without any sensibility to the Majorana phases. So if ββ0ν is found and no CP violation is measured in
the Dirac sector, leptogenesis can’t be falsified by this test.
Another test which may falsify standard leptogenesis (type-I seesaw, unflavoured regime, heavy Majo-
rana decay) is the establishment of light neutrino masses ∼ 0.1− 0.2eV in which case leptogenesis would
fail to produce enough baryon asymmetry. The LHC at CERN may also help the plead of leptogenesis by
disproving electroweak baryogenesis making leptogenesis the most satisfying proposal for explaining
the BAU. Conversely, discovery of new physics such as leptoquarks or triplet Higgses which will estab-
lish that the neutrino masses don’t come from the seesaw mechanism will leave leptogenesis deprived
of its best motivational feature and probably discard it. So to conclude, the discovery of neutrino masses
is a very important fact which will be refined in the years to come by several experiments of various
types, closing the chapter on the electroweak standard model of interactions by giving mass to the once
awkwardly massless neutrinos and providing a non-supersymmetric chance for the SM to generate the
BAU. One also expects the LHC to give us a better understanding (which is still very crude) of the
symmetry breaking mechanism and of flavour physics, which if yielding a satisfactory and predictive
theory of flavour might help to establish leptogenesis as the standard model of matter generation, just
as in the past Big Bang Nucleosynthesis was for chemical elements.
83
Appendix A
Feynman rules for Majorana spinors
The self-conjugacy of the Majorana particles makes their treatment in Feynman integrals somewhat
delicate. Some rules remain unchanged when going from Dirac to Majorana spinors. For example, the
propagator of a Majorana field is the same:
〈0| T (ψA(x)ψB)(y) |0〉 =
∫d4p
(2π)4eip·(x−y)
[i
/p + mp2 −m2 + iε
]AB
(A.1)
where A and B are spinor indices andT is the time-ordering operator. This easily derived considering
the lagrangian for a Majorana field νR = νR + eiφ/2νR
Lν = iνRj/∂νRk −12
¯νRjM jkνRk −12
ˆνRjM∗jk ˆνRk
=12
[iNRk /∂NRk + i ¯NRk /∂NRk − e−iφk ¯NRkDNRk − eiφkNRkDNRk
]=
12[iNk /∂Nk −DkNkNk
]= −
12
e−iφkNTk C†[i/∂ −Mk]Nk
(A.2)
where NR are the mass eigenstates of the right handed neutrino and νRj = eiφk/2V†jkNRk. The propagator
is readily derived from the last result.
As for the vertices, they are easily defined by the Yukawa Lagrangian defines them
Nk → `Lj φ : N`
φ= −i h jk PR (A.3)
Nk → ¯Lj φ : N`
φ= i h∗jk C†PL (A.4)
For the Majorana fermions in external lines we used the following convention: u(p) for incoming
neutrinos and u(p) for outgoing neutrinos.
84
Appendix B
Boltzmann equations for leptogenesis
B.0.7 Boltzmann equation for N1
Starting with the Boltzmann equation for N1 and writing down all the relevant interactions on the RHS,
we obtain
dnN1
dt+ 3HnN1 = −
nN1
neqN1
γ(N1 → `φ) +n ¯
neq`
γ( ¯φ→ N1) −nN1
neqN1
γ(N1 → ¯φ)
+n`neq`
γ(`φ→ N1) −nN1n`neq
N1neq`
γ(`N1 → tRqL)s + γ(tRqL → `N1)s
−nN1n ¯
neqN1
neq`
γ( ¯N1 → tRqL)s + γ(tRqL → ¯N1)s −nN1
neqN1
γ(N1qL → `tR)t
−nN1
neqN1
γ(N1qL → tR ¯)t +n`neq`
γ(`tR → N1qL)t +n`neq`
γ(`qL → N1tR)t
−nN1
neqN1
γ(N1tR → qL ¯)t −nN1
neqN1
γ(N1tR → qL`)t +n ¯
neq`
γ( ¯tR → N1qL)t
+n ¯
neq`
γ( ¯qL → N1tR)t , (B.1)
where the subscripts s and t denote s- and t-channel processes respectively. In writing down (B.1), we
have assumed that neq`≡ neq
¯ and nφ,tR,qL ≡ neqφ,tR,qL
. Assuming that γ(A→ B) ≡ γ(B→ A) for all processes,
(B.1) becomes
dnN1
dt+ 3HnN1 = −
nN1
neqN1
(1 + ε1)γD +n ¯
neq`
(1 + ε1)γD −nN1
neqN1
(1 − ε1)γD +n`neq`
(1 − ε1)γD
+ 2γφ,s −nN1(n` + n ¯)
neqN1
neq`
γφ,s −4nN1
neqN1
γφ,t +2(n` + n ¯)
neq`
γφ,t , (B.2)
where
γφ,s = γ(`N1 → tRqL)s = γ(tRqL → `N1)s = γ(tRqL → ¯N1)s = γ( ¯N1 → tRqL)s , (B.3)
85
and
γφ,t = γ(N1qL → `tR)t = γ(N1qL → tR ¯)t = γ(`tR → N1qL)t = γ(`qL → N1tR)t ,
= γ(N1tR → qL ¯)t = γ(N1tR → qL`)t ,= γ( ¯tR → N1qL)t = γ( ¯qL → N1tR)t . (B.4)
Simplifying (B.2), we get
dnN1
dt+ 3HnN1 = −
2nN1
neqN1
γD +
n` + n ¯
neq`
γD +n ¯ − n`
neq`
ε1γD + 2γφ,s
−
n` + n ¯
neq`
nN1
neqN1
γφ,s −4nN1
neqN1
γφ,t + 2
n` + n ¯
neq`
γφ,t , (B.5)
= −2nN1
neqN1
γD + 2γD + 2γφ,s −2nN1
neqN1
γφ,s −4nN1
neqN1
γφ,t + 4γφ,t + O(ε1,
µ`T
), (B.6)
= − 2
nN1
neqN1
− 1
(γD + γφ,s + 2γφ,t) + O(ε1,
µ`T
), (B.7)
where in (B.6), we have used the definition of density
n` + n ¯
neq`
=
(g`
2π2
∫f eq`
(E) E2 dE)−1 (
g`2π2
∫ [f`(E) + f ¯(E)
]E2 dE
), m` T ,
=
(∫e−E/T E2 dE
)−1 (∫ [e−(E−µ`)/T + e−(E+µ`)/T
]E2 dE
),
=
(∫e−E/T E2 dE
)−1
2 cosh(µ`
T
) ∫e−E/T E2 dE ,
= 2 + O(µ`
T
). (B.8)
where we have used Maxwell-Boltzmann distribution for the phase space densities and imposed the
condition for kinetic equilibrium µ` ≡ −µ ¯.
86
B.0.8 Boltzmann equation for B−L
Writing down the evolution equation for particle density n ¯:
dn ¯
dt+ 3Hn ¯ =
nN1
neqN1
γ(N1 → ¯φ) −n ¯
neq`
γ( ¯φ→ N1) +n`neq`
γ(`φ→ ¯φ)s
−n ¯
neq`
γ( ¯φ→ `φ)s +n`neq`
γ(`φ→ ¯φ)t −n ¯
neq`
γ( ¯φ→ `φ)t
+ γ(tRqL → ¯N1)s −nN1n ¯
neqN1
neq`
γ( ¯N1 → tRqL)s +nN1
neqN1
γ(N1tR → qL ¯)t
−n ¯
neq`
γ( ¯qL → N1tR)t +nN1
neqN1
γ(N1qL → tR ¯)t −n ¯
neq`
γ( ¯tR → N1qL)t , (B.9)
=nN1
neqN1
(1 − ε1)γD −n ¯
neq`
(1 + ε1)γD +n`neq`
(γN,s + ε1γD) −n ¯
neq`
(γN,s − ε1γD)
+n`neq`
γN,t −n ¯
neq`
γN,t + γφ,s −nN1n ¯
neqN1
neq`
γφ,s +nN1
neqN1
γφ,t −n ¯
neq`
γφ,t +nN1
neqN1
γφ,t
−n ¯
neq`
γφ,t , (B.10)
dn ¯
dt+ 3Hn ¯ =
nN1
neqN1
(1 − ε1)γD −n ¯ − n` ε1
neq`
γD −n ¯ − n`
neq`
γN,s −n ¯ − n`
neq`
γN,t + γφ,s
−nN1n ¯
neqN1
neq`
γφ,s +2nN1
neqN1
γφ,t −2n ¯
neq`
γφ,t . (B.11)
Similarly, we can write down the equation for n` as
dn`dt
+ 3Hn` =nN1
neqN1
(1 + ε1)γD −n` + n ¯ ε1
neq`
γD −n` − n ¯
neq`
γN,s −n` − n ¯
neq`
γN,t + γφ,s
−nN1n`neq
N1neq`
γφ,s +2nN1
neqN1
γφ,t −2n`neq`
γφ,t . (B.12)
Subtracting (B.12) from (B.11), we have
dnB−L
dt+ 3HnB−L = − 2ε1γD
nN1
neqN1
+
n` + n ¯
neq`
ε1γD −nB−L
neq`
γD −2nB−L
neq`
γN,s
−2nB−L
neq`
γN,t −nN1nB−L
neqN1
neq`
γφ,s −2nB−L
neq`
γφ,t , (B.13)
where we have defined n ¯ − n` ≡ nB−L. Using (B.8) to simplify, we then get
dnB−L
dt+ 3HnB−L = −2ε1
nN1
neqN1
− 1
γD −nB−L
neq`
γD − 2γN,s − 2γN,t −nN1
neqN1
γφ,s − 2γφ,t
,= −2ε1
nN1
neqN1
− 1
γD −nB−L
neq`
γW + O(ε2
1,µ`T
), (B.14)
87
Appendix C
Calculation of the strength of the CPviolation
C.1 Tree-level contribution to Nk → ` φ
The Feynman diagram for this process is shown in Fig. C.1a. We can immediately write down the
amplitude for this decay as
M = iu j(−i h jkPR)uk ,
= u j(−i h jkPR)CuTk . (C.1)
|M|2 = u j(−i h jkPR)CuT
k
[u j(−i h jkPR)uc
k
]†,
= u j(−i h jkPR)CuTk (i h∗jk)(−uT
k C†PLu j) ,
= −(h∗jkh jk)u jPRCuTk uT
k C†PLu j , (C.2)
|M|2 = −(h∗jkh jk) PR C
12
∑s
ukuk
T
C† PL
∑s′
u ju j (C.3)
When the universe was hot enough, ` j and φ are strongly relativistic, so m` j ,mφ ≈ 0 and
|M|2 = −
(h∗jkh jk)
2Tr
[PR(−/p + Mk)PL( /p′)
], (C.4)
=12
(h∗jkh jk) Tr[PR/p /p′
], (C.5)
= (h∗jkh jk)(p · p′) . (C.6)
The four-momenta in the centre-of-mass frame are given by
p = (Mk , ~0) , p′ = (Mk/2 , −~q) , q = (Mk/2 , ~q) , (C.7)
and one can quickly deduce that |~q| = Mk/2 and p · p′ = p · q = p′ · q = M2k/2.
88
(b)
Nk
(V ∗h )jk
j
φ
pp′
q
Nk
(Vh)jk
j
φ
(a)
pp′
q
Figure C.1: (a) The Feynman graph for the process Nk → ` jφ. (b) The graph for Nk → ¯jφ. Here q = p−p′,and (Vh) jk ≡ −i h jk PR and (V∗h) jk ≡ i h∗jk C†PL are the vertex factors.
Therefore, we obtain
|M|2 = (h∗jkh jk)
M2k
2,
= (h†h)kkM2
k
2. (after summing over j) (C.8)
The decay rate for Nk → ` φ is then
Γ(Nk → ` φ) = 2 ×|~q|
8πE2cm|M|
2 ,
= 2(h†h)kkM2
k
21
8πMk
21
M2k
, (Ecm ≡Mk) ,
=(h†h)kk
16πMk , (C.9)
where the factor of 2 comes from the fact that there are two possible decay channels: Nk → νφ0 and
Nk → e−φ+.
C.2 Vertex contribution to the CP asymmetry
This contribution comes from the interference between the one-loop vertex graph in Fig. C.2 and its
tree-level counterpart in Fig. C.1a.
Im[(Mtree)†Mloop
]∝ εvertex . (C.10)
For the diagram in Fig. C.2, the orderings are:
[C]→ [A]→ [B] and [D]→ [E]→ [F]→ [B] . (C.11)
89
NmNk
n
j
φ
(V ∗h )nk
(Vh)jm
(Vh)nm
[B]
[A][E]
q3
p
p′
q1
q2
q
[C]
[D][F ]
Figure C.2: One-loop vertex correction graph for the process Nk → ` jφ. On the left: (Vh)ab ≡ −i hab PRand (V∗h)ab ≡ i h∗ab C†PL are the vertex factors; on the right: we have included the momentum flows andspinor indices [X], where q = p − p′, q2 = p − q1 and q3 = q1 − q.
From this, the amplitude of the interference term in index form is given by
I′vertex =
∫d4q1
(2π)4(−ih jm)(−ihnm)(ih∗nk)
[u j
]1C
[PR]CA[SNm(q3)
]AB [PR]FB
×[S`(−q1)
]EF [C†PL]DE
[uc
k
]D1
[D(q2)
]11
[−ih∗jkuT
k C†PLu j
]11︸ ︷︷ ︸
(Mtree)†
, (C.12)
where all symbols are as defined previously. So letting Ah = h∗jkh jmhnmh∗nk, this becomes (in matrix
notation)
I′vertex = Ah
∫d4q1
(2π)4
u j PR(−i)( /q3 + Mm)C PTR i(−M)TPT
LC∗uck(i)(−1)uT
k C†PLu j
(q23 −M2
m + iε)(q21 + iε)(q2
2 + iε),
= −iAh
∫d4q1
(2π)4
u j PR( /q3 + Mm)C C†PRC C†MCC†PLCC∗ CuTk uT
k C†PLu j
(q23 −M2
m + iε)(q21 + iε)(q2
2 + iε),
I′vertex =iAh
2
∫d4q1
(2π)4
PR( /q3 + Mm) PRMPL C (∑
s ukuk)T C†PL∑
s′ u ju j
(q23 −M2
m + iε)(q21 + iε)(q2
2 + iε),
=iAh
2
∫d4q1
(2π)4
Tr[PR( /q3 + Mm) PRMPL C
(/pT + Mk
)C†PL /p′
](q2
3 −M2m + iε)(q2
1 + iε)(q22 + iε)
,
...
= iAh Mk Mm
∫d4q1
(2π)4
q1 · p′
(q23 −M2
m + iε)(q21 + iε)(q2
2 + iε). (C.13)
Let us concentrate on the integral:
I′ = i Mk Mm
∫d4q1
(2π)4
q1 · p′
(q23 −M2
m + iε)(q21 + iε)(q2
2 + iε). (C.14)
To pick out the discontinuity, we apply Cutosky’s cutting rules[31]. Firstly, we note that of the three
possible ways to cut the diagram, only one of them can simultaneously put both cut propagators on-
90
shell, due to the heaviness of Nm: This only way (leftmost diagram) corresponds to cutting through the
propagators associated with momenta q1 and q2 (see Fig. C.2). Thus, we make the replacement
1q2
1 + iε→ −2πiδ(q2
1)Θ(E1) ,
1q2
2 + iε→ −2πiδ(q2
2)Θ(E2) = −2πiδ((p − q1)2)Θ(Mk − E1) , (C.15)
in (C.14), where q1 = (E1, ~q1) and q2 = (E2, ~q2). Using the definitions in (C.7) for p, p′ and q, we can evaluate
q1 · p′ = E1Mk
2− ~q1 · (−~q) = E1
Mk
2+ |~q1||~q| cosθ =
Mk
2(E1 + |~q1| cosθ) , (C.16)
where θ is the smaller angle between ~q1 and ~q. Putting all these together and substituting q3 = q1 − q, we
obtain (ε→ 0):
Disc(I′) =i M2
kMm
2
∫d4q1
(2π)4
(−2πi)2(E1 + |~q1| cosθ)δ(q21)δ((p − q1)2)Θ(E1)Θ(Mk − E1)
(q1 − q)2 −M2m
,
=−i M2
kMm
8π2
∫dE1d3q1 δ(E2
1 − |~q1|2) δ
[(Mk − E1)2
− |~q1|2]Θ(E1)Θ(Mk − E1)
×(E1 + |~q1| cosθ)
(E1 −Mk2 )2 − |~q1 − ~q|2 −M2
m
. (C.17)
Applying the identity: δ(x2− a2) = [δ(x − a) + δ(x + a)] /2|a|, we can rewrite δ(E2
1 − |~q1|2) as
δ(E21 − |~q1|
2) =1
2|~q1|
[δ(E1 − |~q1|) + δ(E1 + |~q1|)
]. (C.18)
Integrating over E1, the terms corresponding to the unphysical energy option of E1 = −|~q1|will drop out
automatically because of the step function Θ(−|~q1|) = 0 and one obtains
Disc(I′) =−i M2
kMm
16π2
∫|~q1|
2d|~q1|dΩ1
| − 2Mk|δ[|~q1| −
Mk
2
]Θ(Mk − |~q1|)
×1 + cosθ
(|~q1| −Mk2 )2 − |~q1|
2 −M2
k4 + |~q1|Mk cosθ −M2
m
, (C.19)
where we have used the identity δ(ax) = δ(x)/|a| and the fact that
−|~q1 − ~q|2 = −[|~q1|
2 + |~q|2 − 2|~q1||~q| cos(θ)]
= −|~q1|2−
M2k
4+ |~q1|Mk cosθ . (C.20)
Note that θ is again the smaller angle between ~q1 and ~q. We perform the d|~q1| integral in (C.19) to obtain
Disc(I′) =−i Mk Mm
32π2
∫dΩ
M2k
41 + cosθ
−M2
k4 −
M2k
4 +M2
k2 cosθ −M2
m
,
=−i Mk Mm
32π2
∫dΩ
1 + cosθ−1 − 1 + 2 cosθ − 4z
, (C.21)
91
where z ≡M2m/M2
k . We then evaluating∫
dΩ:
Disc(I′) =−i MkMm
32π2
∫dφ
∫d(cosθ)
1 + cosθ−2(1 − cosθ) − 4z
,
=i MkMm
32π2 (2π)∫ 1
−1dx
1 + x2(1 − x) + 4z
,
=i MkMm
16π[−z ln(−2z) − ln(−2z)
+z ln(−2(z + 1)) + ln(−2(z + 1)) − 1] ,
=−i M2
k
16π√
z(1 − (z + 1) ln
[z + 1z
]). (C.22)
Therefore, the imaginary part of I′ is given by
Im (I′) =12i
Disc (I′) ,
= −M2
k
32π√
z(1 − (z + 1) ln
[z + 1z
]). (C.23)
The two last pieces of information we require before evaluating the CP asymmetry is the total decay rate,
which we can get from (C.9)
Γtot = Γ + Γ = 2 ×(h†h)kk
16πMk =
(h†h)kk
8πMk , (C.24)
and the 2-body phase space factor which may be readily read off using (??) as
Vϕ = 2 ×︸︷︷︸two channels
|~q|8πE2
cm= 2 ×
18π
Mk
2 M2k
=1
8πMk. (C.25)
Putting all these together and summing over all heavy Majorana neutrino species m , k, as well as the
internal lepton species n, the expression for the CP asymmetry due to the vertex contribution is therefore
εvertex = −4
Γtot
∑m,k
∑n
Im(Ah) Im(I′Vϕ) , (C.26)
where we have used Im(I′Vϕ) ≡ Im(I′)Vϕ as Vϕ ∈ R, and so
εvertex = 4 ×8π
(h†h)kk Mk
∑m,k
∑n
Im(h∗jkh jmhnmh∗nk)
×M2
k
32π√
z(1 − (z + 1) ln
[z + 1z
]) 18πMk
,
=1
8π
∑m,k
Im[h∗jkh jm(h†h)km
](h†h)kk
√z(1 − (z + 1) ln
[z + 1z
]), (C.27)
with z ≡M2m/M2
k .
92
[D]
[C]
q
p′
p
[E] [F ][B]
[A]
p
q2
q1
j
φ
Nm
Nk
n
(Vh)jm
(Vh)nm
(V ∗h )nk
Figure C.3: One-loop self-energy correction graph (1) for the process Nk → ` jφ. On the left: (Vh)ab ≡
−i hab PR and (V∗h)ab ≡ i h∗ab C†PL are the vertex factors; on the right: we have included the momentumflows and spinor indices [X], where q = p − p′ and q2 = p − q1.
C.3 Self-energy contribution to the CP asymmetry (1)
The first self-energy contribution is given by interference between one-loop graph in Fig. C.3 and the
tree-level diagram. We have for the interference term:
Iself-(1) =
∫d4q1
(2π)4(ih∗jk)(−ih jm)(−ihnm)(ih∗nk)
[u j
]1C
[PR]CA[SNm(p)
]AB
× [PR]FB[S`(−q1)
]EF [C†PL]DE
[uc
k
]D1
[D(q2)
]11
[−uT
k C†PLu j
]11, (C.28)
where we have shown all spinor indices explicitly. Letting Ah = h∗jkh jmhnmh∗nk, this then becomes (mφ,m` j ≈
0)
Iself-(1) = Ah
∫d4q1
(2π)4
u j PR(−i)(/p + Mm)C PTR i(−M)TPT
LC∗uck(i)(−1)uT
k C†PLu j
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
,
= −iAh
∫d4q1
(2π)4
u j PR(/p + Mm)C C†PRC C†MCC†PLCC∗ CuTk uT
k C†PLu j
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
,
=iAh
2
∫d4q1
(2π)4
PR(/p + Mm) PRMPL C (∑
s ukuk)T C†PL∑
s′ u ju j
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
,
=iAh
2
∫d4q1
(2π)4
Tr[PR(/p + Mm) PRMPL C
(/pT + Mk
)C†PL /p′
](p2 −M2
m + iε)(q21 + iε)(q2
2 + iε),
...
= iAh Mk Mm
∫d4q1
(2π)4
q1 · p′
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
. (C.29)
To pick out the discontinuity of the integral
I(1) = i MkMm
∫d4q1
(2π)4
q1 · p′
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
, (C.30)
93
we note that there is only one sensible way to cut the diagram, namely, through the propagators associated
with q1 and q2. So, with the replacement:
1q2
1 + iε→ −2πiδ(q2
1)Θ(E1) , and1
q22 + iε
→ −2πiδ(q22)Θ(E2) , (C.31)
we have (using q2 = p − q1)
Disc(I(1)) = i MkMm
∫d4q1
(2π)4
(−2πi)2(q1 · p′)δ(q21)δ((p − q1)2)Θ(E1)Θ(Mk − E1)
p2 −M2m
. (C.32)
Simplifying this
Disc(I(1)) =−i MkMm
4π2(M2k −M2
m)
∫d4q1
[E1
Mk
2− (−~q · ~q1)
]δ(E2
1 − |~q1|2)
× δ[(Mk − E1)2
− |~q1|2]Θ(E1)Θ(Mk − E1) ,
=−i M2
kMm
8π2(M2k −M2
m)
∫dE1d3q1(E1 + |~q1| cosθ)
12|~q1|
δ(E1 − |~q1|)
× δ[(Mk − E1)2
− |~q1|2]Θ(E1)Θ(Mk − E1) ,
=−i M2
kMm
16π2(M2k −M2
m)
∫|~q1|
2d|~q1|dΩ |~q1|(1 + cosθ)1|~q1|
× δ[(Mk − |~q1|)2
− |~q1|2]Θ(Mk − |~q1|) ,
=−i M2
kMm
16π2(M2k −M2
m)
∫|~q1|
2d|~q1|dΩ (1 + cosθ) δ[M2
k − 2Mk|~q1|]Θ(Mk − |~q1|) ,
=−i M2
kMm
16π2(M2k −M2
m)
∫|~q1|
2d|~q1|dΩ(1 + cosθ)| − 2Mk|
δ[|~q1| −
Mk
2
]Θ(Mk − |~q1|) ,
=−i MkMm
32π2(M2k −M2
m)
∫dΩ
M2k
4(1 + cosθ) ,
=−i M3
kMm
32π2(M2k −M2
m)2π4
∫ 1
−1dx (1 + x)︸ ︷︷ ︸
=2
,
=−i M3
kMm
32π(M2k −M2
m). (C.33)
From this the imaginary part is given by
Im[I(1)
]=
12i
Disc[I(1)
]= −
M2k
64π
MkMm
M2k −M2
m
. (C.34)
The 2-body decay phase space for this case is given by
V′ϕ = 2 × 2 ×|~q|
8πE2cm
= 2 × 2 ×1
8πMk
2 M2k
=1
4πMk, (C.35)
94
where one of the factor of 2 is to account for the two channels of final decay products while the other
is to account for the two types of intermediate state (νφ0 or e−φ+) inside the self-energy loop. Putting
all these together and summing over all heavy Majorana neutrino species m , k, as well as the internal
lepton species n, we get a contribution to the asymmetry due to this interference as
εself-(1) = −4
Γtot
∑m,k
∑n
Im(Ah) Im(I(1)V′ϕ) , (C.36)
εself-(1) = 4 ×8π
(h†h)kk Mk
∑m,k
∑n
Im(h∗jkh jmhnmh∗nk)M2
k
64π
MkMm
M2k −M2
m
14πMk
,
=1
8π(h†h)kk
∑m,k
Im[h∗jkh jm(h†h)km
] MkMm
M2k −M2
m
,=
18π(h†h)kk
∑m,k
Im[h∗jkh jm(h†h)km
] √z
1 − z, z ≡
M2m
M2k
. (C.37)
C.4 Self-energy contribution to the CP asymmetry (2)
The interference term can be readily written down as
Iself-(2) =
∫d4q1
(2π)4(ih∗jk)(−ih jm)(ih∗nm)(−ihnk)
[u j
]1C
[PR]CA[SNm(p)
]AB
× [C†PL]BE[S`(q1)
]EF [PR]FD
[uc
k
]D1
[D(q2)
]11
[−uT
k C†PLu j
]11, (C.38)
where we have again shown all spinor indices explicitly. Letting Bh = h∗jkh jmh∗nmhnk, and putting into
matrix form, we get
Iself-(2) = Bh
∫d4q1
(2π)4
u j PR(−i)(/p + Mm)CC†PL i(M)PRuck(i)(−1)uT
k C†PLu j
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
,
= −iBh
∫d4q1
(2π)4
u j PR(/p + Mm)PL (M)PR CuTk uT
k C† PLu j
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
,
=−iBh
2
∫d4q1
(2π)4
PR(/p + Mm)PL (M)C[∑
s ukuk]T C† PL
∑s′ u ju j
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
, (index form)
=−iBh
2
∫d4q1
(2π)4
Tr[PR(/p + Mm)PL (M)(−/p + Mk) PL /p′
](p2 −M2
m + iε)(q21 + iε)(q2
2 + iε),
=iBh
2
∫d4q1
(2π)4
Tr[PR/pM/p /p′
](p2 −M2
m + iε)(q21 + iε)(q2
2 + iε),
= iBh
∫d4q1
(2π)4
2(p · p′)(p · q1) − p2(p′ · q1)
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
. (C.39)
95
We now concentrate on the integral
I(2) = i∫
d4q1
(2π)4
2(p · p′)(p · q1) − p2(p′ · q1)
(p2 −M2m + iε)(q2
1 + iε)(q22 + iε)
. (C.40)
Like before, there is only one way to cut the diagram (through q1 and q2), and the discontinuity is given
by (ε→ 0)
Disc(I(2)) = i∫
d4q1
(2π)4(−2πi)2δ(q2
1)δ((p − q1)2)Θ(E1)Θ(Mk − E1)
×2(p · p′)(p · q1) − p2(p′ · q1)
p2 −M2m + iε
,
=−i
4π2(M2k −M2
m)
∫dE1d3q1δ(q2
1)δ((p − q1)2)Θ(E1)Θ(Mk − E1)
×
2 × M2k
2MkE1 −M2
k
(Mk
2E1 + |~q1||~q| cosθ
) ,=
−i4π2(M2
k −M2m)
∫dE1d3q1δ(E2
1 − |~q1|2)δ
((Mk − E1)2
− |~q1|2)Θ(E1)
×Θ(Mk − E1)
M3kE1 −
M3k
2(E1 + |~q1| cosθ
) ,...
=−i M3
k
16π2(M2k −M2
m)
∫|~q1|
2d|~q1|dΩ (1 − cosθ)1
| − 2Mk|δ(|~q1| −
Mk
2
)×Θ(Mk − |~q1|) ,
=−i M4
k
32π2(M2k −M2
m)2π4
∫ 1
−1d(cosθ) (1 − cosθ)︸ ︷︷ ︸
=2
,
=−i M4
k
32π(M2k −M2
m). (C.41)
Hence, the imaginary part is given by
Im[I(2)
]=
12i
Disc[I(2)
]= −
M2k
64π
M2k
M2k −M2
m
. (C.42)
96
[D]
[C]
q
p′
p
[F ] [E][B]
[A]
p
q2
q1
j
φ
Nm
Nk
n
(Vh)jm
(V ∗h )nm
(Vh)nk
Figure C.4: One-loop self-energy correction graph (2) for the process Nk → ` jφ. On the left: (Vh)ab ≡
−i hab PR and (V∗h)ab ≡ i h∗ab C†PL are the vertex factors; on the right: we have included the momentumflows and spinor indices [X], where q = p − p′ and q2 = p − q1.
Therefore, asymmetry due to this interference term is given by
εself-(2) = −4
Γtot
∑m,k
∑n
Im(Bh) Im(I(2)V′ϕ) ,
= 4 ×8π
(h†h)kk Mk
∑m,k
∑n
Im(h∗jkh jmh∗nmhnk)M2
k
64π
M2k
M2k −M2
m
14πMk
,
=1
8π(h†h)kk
∑m,k
Im[h∗jkh jm(h†h)mk
] M2k
M2k −M2
m
.=
18π(h†h)kk
∑m,k
Im[h∗jkh jm(h†h)mk
] ( 11 − z
), z ≡
M2m
M2k
. (C.43)
It should be noted that upon summing over j, the above expression vanishes (because h∗jkh jm(h†h)mk is
real), thus in the one-flavor approximation, this term is absent. Nonetheless, combining this with result
(C.38), we get the full contribution due to the self-energy correction graphs:
εself =1
8π(h†h)kk
∑m,k
Im[h∗jkh jm
(h†h)km
√z
1 − z+ (h†h)mk
11 − z
], (C.44)
=1
8π(h†h)kk
∑m,k
Mk
M2k −M2
mIm
[h∗jkh jm
(h†h)km Mm + (h†h)mk Mk
]. (C.45)
97
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