Neutrinos, Symmetries and the Origin of Matter

Joao Penedo

Under Supervision of Filipe Joaquim

CFTP, Departamento de Fısica, Instituto Superior Tecnico, Lisboa, Portugal(Dated: November 2013)

In this thesis, we explore seesaw extensions of the Standard Model, where heavy states mediateneutrino mass generation and the smallness of these masses is naturally accounted for. Nonvanishingneutrino masses allow for leptonic mixing, whose structure strongly differs from that of quark mixing.The closeness of the lepton mixing matrix to the tribimaximal pattern points to the presence of discretesymmetries in the underlying high-energy theory – such as invariance under transformations of the A4

group, considered in this work.The Standard Model also fails to provide a satisfactory mechanism for the generation of the baryon

asymmetry of the Universe. A remarkable feature of the seesaw extensions is the possibility that theout-of-equilibrium decays of the new heavy states are responsible for the dynamical generation of thisasymmetry. This corresponds to the leptogenesis mechanism, whose efficiency is here determined bynumerically solving a system of Boltzmann equations.

A particular model for spontaneous leptonic CP violation is analysed where neutrino masses andmixing are explained imposing an A4 discrete symmetry. The implementation of the leptogenesismechanism in this context is discussed in detail.

Keywords: Baryon asymmetry of the Universe; CP violation; Leptogenesis; Neutrino masses and mixing; Seesaw

mechanism; Symmetries.

I. INTRODUCTION

Symmetries have played a fundamental role in thedevelopment of modern physics. They are to be un-derstood as laws of invariance under specified opera-tions, the description of which relies on the languageof group theory, transcending a purely geometrical no-tion. In particular, gauge symmetries are at heartof our current understanding of the subatomic world,which relies on the Standard Model (SM) of particlephysics.

Despite its remarkable successes and repeated ex-perimental verification, the SM cannot provide a sat-isfatory answer to some questions which remain open.Among these is the fact that neutrino masses and mix-ing cannot be accounted for in the model, as requiredby neutrino oscillations. An additional puzzle arisesas the observed leptonic mixing deviates strongly fromthe structure of quark mixing. Exploiting the norma-tive aspect of symmetries, one may impose a familysymmetry at high-energy (now broken) – operatingin the leptonic sector – which produces the observedmixing pattern.

One might also wonder if the theory is sufficient toexplain the observed imbalance between matter andantimatter, which allows for the richness of phenom-ena we observe in Nature (namely ourselves), as op-posed to a dull, symmetric Universe filled by a seaof photons. As it turns out, the SM is insufficientto account for the observed baryon asymmetry of theUniverse (BAU), even though the basic ingredientsmay seem to be there at a first glance. An alter-nate mechanism for the generation of this asymmetry(baryogenesis) is required.

A natural solution to both the neutrino mass andthe BAU generation problems – to be addressed se-quentially in what follows – is found within the see-saw framework, where the exchange of heavy parti-cles induces low-energy neutrino masses. The out-of--equilibrium decays of these heavy states in the early

Universe might then be responsible for producing theobserved BAU with the aid of SM non-perturbativeprocesses, in a mechanism known as leptogenesis.

II. NEUTRINOS IN THE SM

The Standard Model is a relativistic quan-tum Yang-Mills theory based on an underlyingSU(3)c×SU(2)L×U(1)Y gauge symmetry group. Thedynamics of the fields are encoded in the Lagrangianof the theory, which can be split into the followingcontributions:

LSM = LHiggs + LGauge + LFermions + LYuk . (1)

Following the requirement of gauge invariance un-der SU(2)L×U(1)Y (generators Ii and Y , respec-tively), a change to the usual Dirac and Klein-Gordonkinetic terms of the fermions and scalar Higgs doubletfields is in order, namely the substitution of the ordi-nary derivative by a covariant one [1]:

∂µ → Dµ ≡ ∂µ − i g2Aiµ I

i − i gY Bµ Y, (2)

where g2 and gY are coupling constants associatedwith the gauge groups SU(2)L and U(1)Y , respec-tively. This leads to the introduction of four realboson fields Aiµ and Bµ in the theory, whose trans-formation properties are such that the kinetic termsare kept invariant under the action of the gauge group.

The term LHiggs in (1) contains the kinetic term forthe Higgs doublet φ as well as the Higgs potential,

V (φ) ≡ µ2φ†φ+ λ(φ†φ

)2. (3)

Fermion and gauge boson kinetic terms are includedin LFermions and LGauge, respectively. Finally, LYuk

contains the Yukawa interactions between fermion chi-ral components and the Higgs doublet:

LYuk =−Ylαβ `αL φ lβR −Yu

αβ qαL φ uβR

−Ydαβ qαL φdβR + H.c. ,

(4)

2

which produce fermion mass terms of the Dirac-typeafter electroweak (spontaneous) symmetry breaking(EWSB). In the above, `αL and qαL denote leptonand quark doublets, while the fields uα and dα cor-respond to up- and down-type quarks, and να and lαto neutrinos and charged leptons. Greek indices runover three generations of fermions.

Since a right-handed component for the neutrinofield is missing from the SM field content, one seesthat the appearance of a Dirac mass term in LSM uponEWSB, as in the case of other fermion fields, is notpossible. Hence, neutrinos are strictly massless in theSM. This implies that performing a rotation of theneutrino fields in flavour space is inconsequential asfar as LYuk is concerned, since there is no mass termto spoil: neutrinos have definite (zero) mass in allbases. Lepton mixing is thus absent from the SM.

Flavour neutrino fields νe,µ,τ are defined as the neu-trino field combinations which are coupled by thecharged current (CC) to each massive charged leptone, µ, τ . Since interactions do not mix lepton flavours,lepton flavour numbers Li – associated to the possi-bility of invariant rephasings of lepton fields – and, byextension, total lepton number L =

∑i Li, are con-

served in the SM.The SM depends on nineteen parameters, which are

to be taken as experimental input. Fermion massescomprise a large number of these parameters. Sincethe theory offers no prediction for their values, theirorigin remains a mystery.

The question of why the experimental values arewhat they are seems to be metaphysical and unpro-ductive. However, both the closeness of the VCKM tothe identity matrix as well as the presence of masshierarchies between generations can be seen as sug-gestive of hidden relations between parameters.

Undeniable evidence for physics beyond the Stan-dard Model arises when one considers the experimen-tal evidence for neutrino oscillations [2]: neutrinoshave small but nonzero masses. In the next section,simple extensions of the SM will be considered in or-der to generate neutrino masses which are naturallysmall.

III. NEUTRINOS BEYOND THE SM

A Dirac mass term for neutrinos can be defined con-sistently with the gauge symmetry in a straightfor-ward extension to the SM, obtained by adding threeright-handed sterile neutrino fields ναR to the particlecontent. These extra fields are two-component spinorsand weak isospin singlets with null hypercharge, andthe extra Lagrangian terms read, before and afterspontaneous symmetry breaking:

LνYuk = −Yναβ `αL φ νβR + H.c.

EWSB−−−−−→ −(v +

H√2

)Yναβ ναL νβR + H.c. .

(5)

Diagonalization of the neutrino mass matrix Mν ∝Yν is achieved by rotating the neutrino fields intoa basis of massive states ν1,2,3 and entails nonzeroleptonic mixing.

In this framework, and due to the large abyss be-tween the masses of neutrinos and those of otherfermions, one sees that the entries of Yν must be un-naturally smaller than those of Yl,u,d. One might ex-pect that by virtue of having the same origin, namelyelectroweak symmetry breaking, the twelve elemen-tary fermions would have comparable masses. Thefact that neutrinos are a strong outlier seems to signalthe presence of an alternative mass generation mech-anism at work.

A. The Majorana Mass Term

As neutrinos are the only neutral fermions, it is pos-sible to construct for them a Majorana mass term[3]. This is achieved by assuming that the chiralcomponents of the neutrino field are not indepen-dent, identifying νCL with νR (the superscript C de-notes the charge conjugation transformation). Theneutrino field then satisfies the Majorana condition,νC = ν = νL + νCL . A Majorana-type mass termreads:

Lν massMaj = −1

2mν ν = −1

2m(νCL νL + H.c.

). (6)

One notices that the above mass term is allowedsince νCL not only behaves as a right-handed field(PL ν

CL = 0, as expected), but also because it trans-

forms as νL under a Lorentz transformation and, thus,LνMaj preserves Lorentz invariance.

There is, nonetheless, a symmetry that is clearlybroken by the Majorana Lagrangian, namely leptonnumber L – which automatically implies the breakingof B−L and B+L (B is baryon number) – as one haslost the liberty to rephase the neutrino field.

A mass term of the type (6) is forbidden in the SMsince the νL are components of an SU(2)L doublet andthere is no field with the required electroweak assign-ments to aid in producing an invariant term of thattype which is also renormalizable. If one forgoes thislast requirement, then the lowest dimensional termwhich can be constructed using SM fields and inducesMajorana neutrino masses is the 5-dimensional Wein-berg operator [4], which can be written as [5]:

LWein = cαβ1

Λ

(`αCL φ∗)(φ† `βL

)+ H.c. (7)

where the cαβ are complex coefficients and Λ corre-sponds to an energy cutoff. Upon EWSB, one obtainsa Majorana neutrino mass term from (7):

−1

2Mν

αβ ναCL νβL + H.c. , (8)

where Mν is the neutrino (Majorana) mass matrix.The above considerations are independent of the

form of the high-energy theory which gives rise to theeffective Weinberg operator LWein at low-energy. Inthe next section, we consider seesaw extensions of theSM in which Majorana neutrino masses arise. Theirvalues turn out to be very small compared to the restof the fermions if a large enough scale Λ is considered,as suggested by (8).

3

ℓL ℓL

φ φ

NR,ΣR

YN,Σ YN,Σ

ℓL

ℓLφ

φ

∆Y∆ µ∗

FIG. 1. Exchange interactions which give rise to the Wein-berg operator of (7). Seesaw types I and III correspondto the exchange of fermion fields (left), while the type IIseesaw mechanism is implemented through the exchangeof scalar fields (right).

B. The Seesaw Mechanism

If one regards the SM as an effective descriptionof particle physics, one forcibly has to add new de-grees of freedom to the theory – taken to be heavy(with masses of order Λ) – which are typically avail-able at high-energies but decouple from the theory atlow-energy. In seesaw extensions of the SM, the 5--dimensional Weinberg operator arises after integrat-ing out massive states from interactions in which theyare exchanged. Such an interaction reduces, at lowerenergy, to a four-point interaction of the form φφ``,which produces neutrino mass terms following EWSB.Therefore, one must introduce fields ψ with interac-tions terms of the form ψφ`, corresponding to type Ior type III seesaw mechanisms, or both ψφφ and ψ``,as is the case with the type II seesaw mechanism (seeFig. 1).

1. Type I Seesaw

In the type I seesaw paradigm [6–10], n′ sterile neu-trino fields NiR ∼ (1, 0) are added to the SM fieldcontent. Aside from a Yukawa term of the form (5),one is free to add a Majorana mass term as that of(6) for the NiR. The relevant part of the extendedLagrangian reads [5, 11]:

−(YN †)

αi`αL φ NiR −

1

2MN

ij NiCRNjR + H.c., (9)

where YN is a general n′ × 3 complex matrix andMN is n′ × n′ and symmetric. Thanks to the steril-ity of the NiR, the covariant derivative in the kineticterm reduces to a regular one and the presence of anL-violating Majorana mass term is not forbidden bygauge symmetry.

Integrating out the heavy fields induces the Wein-berg operator and the neutrino mass matrix reads:

Mν = −YNT v2

MNYN . (10)

A type III seesaw implementation, where the addedfermions are arranged into SU(2)L triplets, results ina similar structure for the neutrino mass matrix.

2. Type II Seesaw

An alternative to the above scenario is found in typeII seesaw framework [12–16], which requires the addi-

tion of scalar triplets ~∆i = (∆++i ,∆+

i ,∆0i ) with hy-

percharge Y = 1 to the SM content.Defining the ∆i matrix as:

∆i ≡(

∆0i −∆+

i /√

2

−∆+i /√

2 ∆++i

), (11)

the relevant extended Lagrangian terms read:(Dµ ~∆i

)†(Dµ

~∆i

)−(M∆

ij

)2Tr[∆†i∆j

]−(Y∆i

αβ `αCL ∆i `βL + µi φ

T ∆i φ+ H.c.),

(12)

where the Y∆i are n×n symmetric matrices and M∆

is an n′ × n′ Hermitian matrix. The low-energy neu-trino mass matrix is in this context given by:

Mν = −∑i

2λ∗i v2

MiY∆i , (13)

where λi ≡ µi/Mi (Mi are triplet mass eigenvalues).One notices here an intersting feature, namely thatthe flavour structure at high-energies is muddled up,as in seesaw types I and III (see Eq. (10)), but pre-served when the low energy limit is taken.

An alternative way of obtaining the above contri-bution for the neutrino mass matrix is by consider-ing that, thanks to EWSB, both φ0 (the lower fieldcomponent of the Higgs double) and ∆0

i acquire non-vanishing VEVs. In the approximation Mi � v, theHiggs VEV v remains unchanged, while the triplet in-duced VEV is given by [17]:

ui ≡ 〈∆0i 〉 = −λ

∗i v

2

Mi, (14)

and one has |ui| � v for all i.Having bestowed neutrinos with nonzero masses

(and before addressing the production of the BAU),one turns to lepton mixing and the role of family sym-metries in the shaping of the mixing pattern.

IV. LEPTON MIXING AND FAMILYSYMMETRIES

As in the case of quarks, lepton mixing arises fromthe mismatch of lepton states which are massive andthose which participate in CC interactions. Assuminghenceforth that neutrinos possess Majorana masses,one sees that the rotation of the fields to the massbasis leads to

L`CC →g2√

2νiL γ

µ (U†PMNS)iα lαLWµ + H.c., (15)

where one has defined the lepton mixing matrixUPMNS, known as the Pontecorvo-Maki-Nakagawa--Sakata matrix [18–20]. The relation between states

4

of definite flavour (greek indices) and definite mass(roman indices) is given by:

ναL =(UPMNS

)αiνiL . (16)

The number of physical parameters contained inUPMNS can be determined as in the quark case withone important caveat: due to the Majorana natureof neutrinos one has lost the freedom to invariantly

rephase the three corresponding fields. Therefore, forn generations, one can only remove n phases throughthe rotation of charged leptons instead of 2n− 1 fromthe unitary mixing matrix (the global U(1)L is alsobroken). One ends up, for n = 3, with n(n− 1)/2 = 3mixing angles and n(n + 1)/2 − n = n(n − 1)/2 = 3physical phases. A parametrization of the UPMNS isgiven by [21]:

UPMNS =

c12 c13 s12 c13 s13 e−iδ

−s12 c23 − c12 s23 s13 eiδ c12 c23 − s12 s23 s13 e

iδ s23 c13

s12 s23 − c12 c23 s13 eiδ −c12 s23 − s12 c23 s13 e

iδ c23 c13

︸ ︷︷ ︸

≡VPMNS

1 0 00 eiα1 00 0 eiα2

︸ ︷︷ ︸

≡KPMNS

, (17)

where sij ≡ sin θij and cij ≡ cos θij refer to mixingangles θ12, θ13, θ23 ∈ [0, π/2]. The δ ∈ [0, 2π[ phase isa Dirac-type phase, while α1, α2 ∈ [0, 2π[ arise due tothe Majorana character of neutrinos. As mentionedin Section III A, lepton mixing breaks lepton flavournumbers (and the total L in the Majorana case). Thisleads to the possibility of observing charged leptonflavour violating processes such as µ → e γ, µ − econversion in nuclei, and µ→ 3 e [22]. So far, searcheshave yielded negative results, setting upper limits oforder 10−12 on branching ratios and conversion rates[23–25].

As a consequence of lepton mixing, neutrinos pro-duced with a definite flavour are allowed to oscillatebetween flavours. Neutrino oscillations are only sensi-tive to the mass-squared differences ∆m2

ij ≡ m2i −m2

j ,providing no information on the absolute neutrinomass scale. Since the sign of ∆m2

31 is indeterminate,whereas ∆m2

21 is positive, there is room for two possi-ble orderings of the mass spectrum: normal ordering,with m1 < m2 < m3, and inverted ordering, for whichm3 < m1 < m2.

Results from the latest global fit to neutrino oscil-lation data [26] are summarized in Table I, and canbe given a visual interpretation through the diagramof Fig. 2. From inspection of Fig. 2 it is apparentthat the experimental mixing data bears a close re-semblance to the tribimaximal (TBM) pattern. TheTBM hypothesis, put forward by Harrison, Perkins,and Scott [27], corresponds to taking |Ve3|2 = 0,|Vµ3|2 = 1/2, and |Ve2|2 = 1/3, where V here denotesVPMNS. By redefining Majorana phases and rephasingcharged leptons, one may write:

UTBM ≡ eiϕ

−√

23

1√3

0

1√6

1√3

1√2

1√6

1√3− 1√

2

︸ ︷︷ ︸

≡VTBM

KPMNS, (18)

where KPMNS = diag(1, eiα1 , eiα2) is given in terms ofthe new phases.

The possibility of a TBM mixing pattern motivatedthe study of discrete family symmetries which might

ParameterResult of Global Fit within 1σ {3σ}

Normal Order Inverted Order

∆m221 7.62 ± 0.19

{+0.58−0.50

}∆m2

31 2.53 +0.08−0.10

{+0.24−0.27

}−(

2.40 +0.10−0.07

{+0.28−0.25

})sin2 θ12 0.320 +0.015

−0.017

{±0.050

}sin2 θ23 0.49 +0.08

−0.05

{+0.15−0.10

}0.53 +0.05

−0.07

{+0.11−0.14

}sin2 θ13 0.026 +0.003

−0.004

{+0.010−0.011

}0.027 +0.003

−0.004

{+0.010−0.011

}δ

(0.83 +0.54

−0.64

)π{

any} (

0.07 +1.93−0.07

)π{

any}

TABLE I. Global fit results taken from [26] for the three--neutrino oscillation parameters and for both orderingpossibilities. ∆m2

21 and ∆m231 are presented in units of

10−5 eV2 and 10−3 eV2, respectively.

govern the leptonic sector. However, exact TBM isexcluded by the aforementioned results on a nonzeroreactor angle, given by the Daya-Bay, RENO andDouble Chooz collaborations [28–30] after hints fromthe T2K and MINOS collaborations [31, 32]. Nev-ertheless, one can still pursue the family symmetryapproach, without having to resort to mass anarchy[33]. Taking symmetry as a guiding principle, onemay accommodate the above results by regarding spe-cific mixing patterns – like the TBM – as a leading-order approximation, to be perturbed when higher-order corrections are taken into account.

V. AN A4 MODEL WITH SPONTANEOUSCP VIOLATION

In the present section, we turn to the analysis ofthe model of Ref. [34], which implements an A4 fam-ily symmetry (popular in the literature, [35–38]) in atype II seesaw framework with spontaneous CP vio-lation. Aside from two real flavon fields, Φ and Ψ,which are singlets of SU(2)L with null hypercharge,

5

νe ν ντ

ν1

ν2

ν3

FIG. 2. Depiction of lepton mixing for both a normallyordered and an inverted neutrino mass spectrum, wherethe global fit data of Table I has been considered (left), tobe compared with the tribimaximal ansatz (right).

one adds two SU(2)L triplets ∆1,2 with Y = 1 andone complex scalar field S, which is a singlet of the

gauge group, to the SM particle content. A Z4 shapingsymmetry is also considered. For details concerningthe scalar potential and symmetry assignments, thereader is referred to Section 3.3 of the present thesis.

An interesting property of the present model is theabsence of explicit CP violation at the Lagrangianlevel. The unique source all CP violation correspondsto the complex phase of the VEV 〈S〉 = vS e

iα. As forthe Yukawa Lagrangian, the usual SM charged leptonterm is forbidden by the Z4 symmetry, as well as thestandard type II seesaw coupling of triplets ∆1,2 tolepton doublets (a coupling of the Higgs type, ∆φφ,is only allowed for ∆2). Instead, terms of these formswill arise as effective operators. Up to O(1/Λ, 1/Λ′),where Λ and Λ′ are taken to correspond to an as-sumed unique flavon scale and to the scale of S decou-pling, respectively, one obtains the following YukawaLagrangian, compatible with the postulated symme-tries:

Ll = −yle

Λ

(`L Φ

)1φ eR −

ylµΛ

(`L Φ

)1′′φµR −

ylτΛ

(`L Φ

)1′φ τR + H.c.,

Lν =1

Λ′∆1

(`TL C

† `L)1

(y1 S + y′1 S

∗)+y2

Λ∆2

(`TL C

† `L Ψ)1

+ H.c.,

(19)

where the bold superscripts indicate which field com-bination is chosen from the decomposition of thetriplet tensor product.

A. Nonzero Reactor Neutrino Mixing Angle

In order to attain an acceptable agreement withexperimental data within the presented framework,one must account for the deviation θ13 6= 0 fromthe TBM pattern. This is achieved by consideringperturbations to the flavon alignment which directlyleads to TBM mixing and which might arise due tonon-renormalizable corrections to the flavon potential.Consider the perturbed alignment:

〈Φ〉TBM = (r, 0, 0), 〈Ψ〉 = s(1, 1 + ε1, 1 + ε2) . (20)

The (leptonic) Yukawa Lagrangian then becomes:

Ll =−Ylαβ `αL φ lβR + H.c., (21)

Lν = −Y∆1

αβ `αCL ∆1 `βL −Y∆2

αβ `αCL ∆2 `βL + H.c.,

where one recognizes the familiar charged leptonYukawa and type II seesaw terms, with:

Yl =1

Λ

ye 0 0

0 yµ 0

0 0 yτ

, Y∆1 = y∆1

1 0 0

0 0 1

0 1 0

, (22)

Y∆2 =y∆2

3

2 −1− ε2 −1− ε1

−1− ε2 2(1 + ε1

)−1

−1− ε1 −1 2(1 + ε2

) . (23)

In the above, ye,µ,τ ≡ r yle,µ,τ/Λ, y∆1≡ vS

Λ′

(y1 e

iα +

y′1 e−iα), and y∆2

≡ s y2/Λ. The useful definitions

2u1 y∆1 ≡ z1eiβ and 2u1 y∆1 ≡ z2 (with z1 and z2

real) are considered (recall Eq. (14)).The neutrino mass eigenvalues are given, as a func-

tion of perturbations, by m22 ' z2

1 and:

m21,3 ' z2

1 +1

3

(z2

2 (3 + 2ε1 + 2ε2)

± 2 z1 z2 (3 + ε1 + ε2) cosβ).

(24)

For small enough perturbations, |ε1,2| � 1, only nor-mal ordering is allowed. As for the mixing pattern,one obtains the following deviations to TBM:

sin2 θ12 '1

3

[1 +

2

3(ε1 + ε2)

],

sin2 θ23 '1

2

[1 +

1

3(ε1 − ε2)

],

sin2 θ13 '1

2

(ε1 − ε2

6 cosβ

)2

.

(25)

As far as mixing phases are concerned, one can mea-sure Dirac-type CP violation through the followinginvariant [11, 39] (U is short for UPMNS):

JCP =1

8

(∏θij

sin(2 θij))

cos θ13 sin δ . (26)

For ∆m231 � ∆m2

21, one obtains the result JCP 'tanβ (ε2 − ε1)/36⇒ sin δ ' ± sinβ.

One is now in a position to consider experimentalconstraints on the perturbed version of model. This

6

is done in Fig. 3: the left part illustrates the allowedregions of the (ε1, ε2) plane, while the right part showsthe corresponding possible values of JCP. One obtainsa limit 0.02 . |JCP| . 0.05 at 3σ. This perturbed

FIG. 3. Scatter plot of the experimentally allowed regionsin the

(ε1, ε2

)plane (up), where exact TBM is seen to be

excluded, and corresponding regions of the(JCP, β

)plane

(down).

version of the model will be reconsidered later on, aswe analyse the viability of producing the BAU withinits context.

VI. THE BARYON ASYMMETRY OF THEUNIVERSE

We live in an apparently biased Universe in whichthere is, as far as one can see [40], a clear preferencefor the presence of matter over antimatter. How couldthis imbalance come to be? One assumes that it is ei-ther a consequence of the initial conditions of the Uni-verse or it is dynamically generated as the Universecools down. The reason why the accidental asymme-try case is usually rejected is twofold. On the onehand, generating the observed asymmetry would re-quire one extra quark for every ∼ 107 antiquarks inthe early Universe [41] (an unnatural scenario). Onthe other hand, a constraint arises if the effects of in-flation – which is predicted to exponentially dilute anyinitial asymmetry – are considered.

In the dynamical case, the symmetry violating pro-cesses should be weak enough in order to produce theobserved values of the BAU, which can be quantifiedin the present epoch through the parameter η, definedas the ratio between the number density of baryonic

number, nB, and that of photons, nγ :

η ≡ nB

nγ≡ nb − nb

nγ' nbnγ, (27)

where nb and nb represent the densities of baryonsand antibaryons. The value of η also signals the im-balance between matter and radiation in the present-day Universe. The abundances of light elements andthe anisotropies of the cosmic microwave background(CMB) constrain it independently. One verifies thatboth conditions give compatible values for the BAUparameter, which when combined give [21]:

η = (6.19± 0.15)× 10−10. (28)

An equivalent description is given by YB ' η/7.04,corresponding to the ratio between nB and the currententropy density of the Universe s (Yi ≡ ni/s).

Equilibrium thermodynamics considerations lead tothe conclusion that the Universe must have possesseda baryon asymmetry already at early times. To putmatters into perspective, Fig. 4 depicts importantstages in the thermal history of the Universe.

The evolution of contents of the Universe has beendriven by departures from thermal equilibrium. Asa rule of thumb, one may consider an interaction tobe out of equilibrium if the corresponding rate Γ isnot fast enough to accompany the expansion of theUniverse, Γ � H(T ) (H represents the Hubble pa-rameter, and T the temperature of the Universe). Onthe contrary, if interactions are fast, Γ � H(T ), onetakes the corresponding particle species to be in equi-librium. The non-trivial case is located in between,with Γ ∼ H(T ), and demands a more careful andquantitative treatment, which relies on the Boltzmanntransport equation.

VII. THERMAL LEPTOGENESIS

Taking an initial state of zero baryon number, abaryon asymmetry can be generated if the followingsufficient conditions, presented by Andrei Sakharov in1967 [42], hold:

• B symmetry is violated.• C and CP symmetries are violated.• Departures from thermal equilibrium occur.

Although the Sakharov conditions are not necessaryconditions for the BAU generation [43], trying to getaround any one of the three is generally not easy andmight require troublesome assumptions.

The SM appears to be insufficient in explainingthe presently observed BAU in light of the Sakharovconditions: it offers neither a first-order electroweakphase transition (EWPT) nor enough CP violation.One interesting alternative scenario is that of lepto-genesis [44], in which the decays of heavy particles –such as the seesaw mediators of Section III B – violatelepton number L. Lepton-number asymmetries wouldthen be converted into baryon-number asymmetriesby electroweak sphaleron processes, whereas the re-quired CP violation would arise from complex cou-plings in the heavy particle interaction Lagrangian.

7

FIG. 4. Brief thermal history of the Universe. The family symmetry breaking scale is often taken to be of the order ofthe GUT scale.

We will henceforth focus on the case of thermal lepto-genesis, meaning that the heavy particles (here takento correspond to seesaw mediators) are produced ther-mally, following reheating, by scattering processes inthe plasma.

Having accounted for possible deviations from ther-mal equilibrium due to the extreme mass of the newstates and the expansion of the Universe, one turns tothe remaining Sakharov conditions.

If the decays of the new species violate L alone, thenB−L is also broken. This is a crucial aspect of baryo-genesis models operating at high scales T � TEW ,as electroweak sphalerons (which tend to wash outall asymmetries) cannot erase B−L. For a second-order EWPT, the following relation between leptonand baryon numbers at the electroweak scale is im-posed by fast sphaleron interactions [45, 46]:

YB =12

37YB−L ⇒ YB = −12

25YL = −0.48YL . (29)

The remaining Sakharov condition, CP violation,is satisfied through CP asymmetries in the quantumamplitudes for decay processes. These generally arise,

at leading order, from the interference between a tree-level and a one-loop diagram.

VIII. TYPE II LEPTOGENESIS

In the context of leptogenesis, Boltzmann equations(BEs) can be used to quantify the evolution of anasymmetry in lepton number. They are here appliedto the case where the decaying particles generatingthe lepton asymmetries are the heavy scalar triplets∆i of the type II seesaw mechanism. The flavouredparameters which quantify CP asymmetries in decaysand inverse decays of the ∆i are given by:

εαβi = 2Γ(∆∗i → `α `β)− Γ(∆i → `α `β)

Γ∆i + Γ∆∗i

(30)

where Γ∆iis the total triplet decay rate. These arise

from the interference bewteen the diagrams presentedin Fig. 5. For Mj � Mi, one obtains the CP asym-metries of Eq. (31).

εαβi = − 2

1 + δαβ

1

2π

∑j 6=i

Mi

Mj

Im(λ∗i λjY

∆i

αβY∆j∗αβ + (Mi/Mj) Tr

[Y∆i

†Y∆j

]Y∆i

αβY∆j∗αβ

)Tr[Y∆i

†Y∆i

]+ |λi|2

. (31)

The Boltzmann equation formalism has been de-veloped in Chapter 4 of the present thesis. The fullnetwork of Boltzmann equations, needed to computenumber densities of asymmeties is explicitly given, forthe type II seesaw leptogenesis framework, in Section5.2 of the thesis.

Consider, henceforth, the particular context of theperturbed A4 model developed in Section V. Someworking assumptions (see Section 5.4), including thatof hierarchical triplets, M2 � M1, have been made.

The maximum values which the CP asymmetries mayattain are, as a function of the parameters of themodel, given by:

ε01,max '

√∆m2

31

12√

6πv2

[1− 1

3(ε1 + ε2)

]M1 sinβ , (32)

ε02,max ' −

√∆m2

31

48πv2

[1− 1

9(ε1 + ε2)

]M2 tanβ . (33)

The magnitudes of these maximum values areshown as a contour plot in the (β,Mi)-plane in Fig. 6.

8

∆i

`β

`α

∆i

φ

φ

Tree-Level

∆i ∆j

φ

φ

`α

`β

∆i ∆j

`α

`β

`µ

`ν

One-Loop

FIG. 5. Tree-level diagrams for the decays of type II seesaw scalar triplets and one-loop diagrams contributing to thedecay process ∆i → `α `β .

FIG. 6. Contours of the (magnitude of the) maximum CPasymmetries in the decays of hierarchical scalar triplets∆1,2.

The plot shows that, for certain points in the (β,Mi)plane, relatively large CP asymmetries are obtained.To finally solve the system of BEs one needs to com-pute the reaction densites for the relevant processes,given in Fig. 7. These densities are obtained fromthe integration of reduced cross sections, whose ex-plicit form is presented (for the model at hand) inSection 5.3 of this thesis. The BE integration wasperformed numerically through a Fortran-based pro-gramme. Fig. 8 shows the results for the baryon asym-metry density YB for two choices of M1, after compat-ibility with neutrino oscillation data has been assured.

We now restrict ourselves to a particular point, cho-sen to lie in the region above the green band (detailsare given in Section 5.4). Reaction densities and theevolution of asymmetries for this particular perturbedversion of the model are given in Fig. 9. A baryonasymmetry YB ' 5.93 × 10−10 is obtained for thispoint, following the decoupling of the lightest triplet,by summing over all three YBLα and converting theresulting YB−L asymmetry into a purely baryonic onethrough (29).

We have seen that the model is sufficient to ac-count for the observed baryon asymmetry of the Uni-verse. One has assumed that triplet masses are hier-archical, taken to imply that asymmetries producedprior to the decay of the lightest triplet have been

washed out. The fact that CP asymmetries in theout-of-equilibrium decays of ∆2 can overshadow thoseof ∆1 by as much as an order of magnitude (seeFig. 6) challenges this assumption. Flavour consider-ations may nevertheless come to our rescue, as a suffi-ciently massive ∆2 may decay during the unflavouredregime, for which all different lepton flavours are out-of-equilibrium and indistinguishable. The structure ofthe model prevents, in such a case, the generation ofasymmetries.

IX. CONCLUDING REMARKS

The most popular SM extensions accommodatingmassive neutrinos in a natural way are those in whichneutrino masses are generated through the tree-levelexchange of new heavy particles. We have analysedthe phenomenological viability of a specific type IIseesaw model based on an A4 non-Abelian discretesymmetry. In this scenario, CP-violating effects stemfrom a single complex phase associated to the VEV ofa scalar singlet. The model is compatible with all dataprovided by neutrino oscillation experiments and pre-dicts large CP-violating effects, detectable in futureexperiments.

The connection between neutrino physics and cos-mology may be established within the leptogenesisscenario, where the excess of baryons in the Universerelies on a lepton asymmetry. For the specific case oftype II seesaw leptogenesis, the starting point to gen-erate that lepton asymmetry is the out-of-equilibriumdecay of the heavy-triplet scalars. The viability of theleptogenesis scenario in the framework of the afore-mentioned A4 model has also been explored. We haveconcluded that the model not only allows to reproducethe current neutrino mass and mixing pattern, butalso generates a sufficiently large baryon asymmetryof the Universe, in accordance with the experimentalresult.

Our conclusions justify the purpose of this thesis:to relate neutrino masses and mixing, symmetries andthe origin of matter. What we have presented is justan example (among many others) of how complemen-tary studies may shed some light on answering openquestions in fundamental physics, the future of whichhinges on the combined exploration of physical phe-nomena at the energy, intensity and cosmic frontiers.

9

`β

`α

Decays and Inverse Decays

∆i ∆i

φ

φ

`α

∆i

φ

φ`β

s-Channel Scatterings

Gauge Scatterings

Fermions and Higgs

∆i

∆i

f, φ

f, φ

Ai, B

`α

∆i

φ

φ`β

t-Channel Scatterings

t-channel

∆i

∆i

∆i

Aa, B

Ab, B

u-channel

∆i

∆i

∆i

Aa, B

Ab, B

cubic

Aa, B

∆i

∆i

quartic

Ab, B

Ac, B

∆i

∆i

Aa, B

Ab, B

FIG. 7. Scalar triplet interactions relevant to the BE out-of-equilibrium analysis, where one considers the diagramspresented for decays, inverse decays and s- and t-channel scatterings and their charge conjugates, as well as gaugescattering reactions.

90 95 100 105 110 115

10-8

10-9

10-10

10-11

10-12

10-13

10-14

245 250 255 260 265 270

10-8

10-9

10-10

10-11

10-12

10-13

10-14��

M1 = 1´1012

GeV

M1 = 5´1012

GeV

YB

Β HºL

FIG. 8. Scatter plot of the baryon asymmetry generated in randomly-chosen perturbed versions of the model, forM1 = 1012 GeV (black) and M1 = 5× 1012 GeV (cyan).

FIG. 9. Reaction densities normalized to the product H(T )nγ(T ) (left) and evolution of the various densities consideredin the BE network (right). Gauge scatterings keep triplets close to equilibrium. The relevant densities here are theflavoured YBLα ≡ YB/3− YLα (for a description of the remainder, see Chapter 5).

10

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