Theory of Massive neutrinos

In the SM, due to the absence of SU(2)_L singlet right handed singlet neutrinos, a Dirac mass term is absent and hence neutrinos are massless. However above mentioned evi- dences from wide range of physical phenomena hints towards massive neutrinos and it is important to introduce new physics scenarios in order to explain neutrino mass. Note that SM is a renormalizable theory and lepton number conservation turns out to be an accidental symmetry. Once we demand that the accidental lepton number conservation is only an approximate symmetry of SM, we can incorporate Majorana masses for the neutrinos through higher dimensional operators. Such a simple and economic way to incorporate neutrino mass via non-renormalizable dimension-5 operator (also known as Wienberg Operator [24]), constructed from lepton and Higgs doublets, is given by

δL^d=5 = 1 2cij

L^c_iH˜^∗ H˜^†Lj

Λ . (1.22)

Here Λ is the new physics scale involved and cij is the dimensionless complex coefficient.

Interestingly this is the only allowed dimension-5 operator invariant under SM gauge group, constructed from SM fields and also violates lepton number by two units. After electroweak symmetry breaking, neutrino mass obtained from Eq. (1.22) can be written as

m_νij = cij

2Λv², (1.23)

wherevis the SM Higgs vev. Clearly the smallness of the neutrino massm_ν is mainly dic- tated by largeness of the new physics scale Λ appearing in the dimension-5 operator. Such a dimension-5 operator always appears (as in Eq. (1.22)) in theories which incorporates neutrino mass as discussed below.

From the above discussion it is now obvious that, to explain neutrino mass we must go beyond the SM. Since the true nature of neutrinos is still elusive to us, we are left with two different possibilities: neutrinos are Dirac particles or they are Majorana particle. In the SM (Section 1.1) we have already talked about the Dirac fermions and we know that a mass term for such a particle can be written as

mDψ_LψR+h.c., (1.24)

when there exists fields of both chirality. In the simplest extension of the SM, if we incorporate right-handed (RH) neutrinos in the theory, we can write a Dirac mass term for the neutrinos (similar to Eqs. (1.10, 1.24)). However to explain neutrino mass in the eV range, the associated Yukawa coupling becomes∼10⁻¹¹. Since there is no appealing reason for this extremely small coupling and hence the theory is considered to be ‘un- natural’. Note that it is possible to introduce Majorana mass for RH neutrinos. We know the action of a particle-antiparticle conjugation operator (C) is defined as ψ^c = Cψ¯^T stands for charged conjugate of ψ with C = iγ₂γ₀. It can flip the chirality of a field, via (ψL)^c = (ψ^c)R and (ψR)^c = (ψ^c)L. Now in case of Majorana fermions, the left and right-handed components are related by a C conjugation as ψ=ψ_L+ψ_R=ψ_L+η(ψ_L)^c where η is a arbitrary phase factor (η =e^iζ) following ψ^c =η^∗ψ. Now for these neutral fermions, the mass term can be written as

mL(ψ^c)_RψL+MR(ψ^c)_LψR, (1.25) In a nutshell, we must look for a mechanism which can provide tiny neutrino mass in a more natural way. Few such possibilities of neutrinos mass generation are: (i) Seesaw mechanisms (both high and low scale seesaw, some of these scenarios will be reviewed in this section). (ii) Loop mechanisms (also known as radiative mechanism) [25–28], (iii) R-parity violating supersymmetry [29, 30]. (iv) Extra Dimensions [31, 32] and String Theories [33]. In the following subsections we discuss mostly different seesaw mechanism of neutrino mass generation relevant for the landscape of the studies in this thesis.

1.4.1 Type-I Seesaw Mechanism

Perhaps the most elegant way to explain the small neutrino mass is the type-I seesaw mechanism [34–37]. In the minimal extension of the SM, gauge singlet RH Majorana (N_Ri,i= 1,2,3 for three generations) fermions, are included in the theory. Therefore in presence of both Dirac and Majorana fermions the Lagrangian responsible for neutrino mass can be written as

−L^Type−I=YD`¯HNe R+ 1

2MRN_R^cNR+h.c., (1.26) and after electoweak symmetry breaking generates

−L^Type−I=mDνLNR+1

2MRN_R^cNR+h.c.. (1.27) In Eq. (1.27), mD = YDv is the Dirac mass matrix for the neutrinos where YD is the coupling matrix. MR is the symmetric mass matrix for the Majorana neutrinos. In the basis (νL, N_R^c), the effective neutrino mass matrix can be written as

Mν = 0 m^T_D mD MR

!

. (1.28)

For three generations of neutrinos, each entries in Eq. (1.28) are 3×3 matrices. Con-

H

ℓ

NR

×

NR

H

ℓ

(a) Type-I

H H

∆

ℓ ℓ

(b) Type-II H

ℓ

Σ

×

_Σ

H

ℓ

(c) Type-III

Figure 1.2: Feynman diagrams for Type-I, Type-II and Type-III seesaw.

sideringmD to be much lighter thanMR (mD<< MR), after block diagonalization, mass matrices for light and heavy neutrinos can be written as [38]

m^I_ν ' −m^T_DM_R⁻¹mD and M_heavy'MR. (1.29)

From Eq. (1.29) it is now clear that observed smallness of the light neutrinos can be explained form the heaviness ofMR. This is known as type-I seesaw mechanism of neutrino mass generation. A typical Feynman diagram for such process is given in the left panel of Fig. 1.2. Eq. (1.29) indicates that with YD ∼ O(1), to generate neutrino mass of the order of 0.1 eV, makes MR ∼10¹⁴ GeV¹. Presence of these heavy neutrinos not only explains tiny neutrino mass, it can also play an important role in cosmology to explain the matter-antimatter asymmetry of the universe. Such possibility will be discussed in Section 1.7 and Chapter 2 is devoted in realizing a framework to establish a connection between neutrino masses, mixing and the matter-antimatter asymmetry of the universe in a type-I seesaw scenario.

1.4.2 Type-II Seesaw Mechanism

In type-II seesaw mechanism [40–43], SM model is extended by introducingSU(2)Lscalar triplet field (∆) having hypercharge 2. Combining with SM lepton doublets it can give rise to a Majorana mass term at renormalisable label provided ∆ gets a vev. In its 2×2 matrix representation the scalar triplet can be written as

∆ = ∆⁺/√

2 ∆⁺⁺

∆⁰ −∆⁺√ 2

!

, (1.30)

where ∆⁰, ∆⁺ and ∆⁺⁺ are neutral, singly and doubly charged components of it. The Lagrangian relevant for neutrino mass can be written as

−L^Type−II = Y∆`<sup>T</sup>C∆`+h.c.

+

µHe^T∆He +h.c.

+M_∆²Tr(∆^†∆), (1.31) hereY∆is the complex coupling matrix responsible for providing Majorana masses to the neutrinos,µis a dimensionful coupling constant associated with the lepton number violat- ing term andM∆ represents the mass of the scalar triplet. When the neutral component of the scalar gets vev (h∆⁰i=u∆), the neutrino mass matrix obtained via type-II seesaw can be written as

m^II_ν = 2Y∆u∆ with u∆= µv²

M_∆², (1.32)

1Such a huge mass scale indicates that the origin of neutrino mass can also be related to some Grand Unified Theory (GUT) symmetry group likeSO(10) [39].

and hence neutrino mass gets the required seesaw suppression once we make the scalar triplets very heavy. In Eq. (1.32), (v/M∆)² provides this suppression. Feynman diagram for type-II seesaw is illustrated in the middle panel of Fig. 1.2.

In presence of both RH neutrinos and scalar triplets, neutrino mass will get contribu- tion from both type-I and type-II seesaw. In such a type-I+II scenario the effective mass matrix for neutrinos can be written as

m_ν =m^II_ν +m^I_ν =m^II_ν −m^T_DM_R⁻¹mD. (1.33) In Chapter 3 of this thesis, we study interplay of both of these terms in explaining neutrino mass, mixing and its consequence in leptogenesis.

1.4.3 Type-III Seesaw Mechanism

In an alternate scenario, instead of fermionic singlet neutrinos as included in pure type-I seesaw, if we introduce SU(2)_L fermionic triplet of zero (Σ) hypercharge :

Σ = Σ⁰/√

2 Σ⁺

Σ⁻ −Σ⁰√ 2

!

, (1.34)

the Lagrangian relevant for neutrino mass can be written as [44]

−L^Type−III =Y_Σ`Σ ˜H+1

2M_ΣTr(Σ^cΣ). (1.35)

Here YΣ is the Yukawa coupling matrix whereas MΣ represents the mass matrix for the fermionic triplet. The relevant Feynman diagram for this case is illustrated in the right panel of Fig. 1.2. Hence the neutrino mass matrix for type-III seesaw mechanism can be written as

m^III_ν =−Y_Σ^T v²

M_ΣYΣ. (1.36)

Therefore, in the limit MΣ >> v, i.e. when the fermionic triplets are heavy enough one can reproduce small neutrino mass using type-III seesaw mechanism also.

1.4.4 Inverse Seesaw Mechanism

Introducing heavy RH neutrinos N_Ri (i= 1,2,3 for three generations) in original type-I seesaw scenario one can explain tiny neutrino mass very easily. Here the energy of the lepton number violation responsible for generating neutrino is dictated by its mass, M_R. With the Yukawa couplings of order unity or so, we can obtain the light neutrino mass of observed magnitude, provided the new physics scale M_R is sufficiently high ∼10¹⁴ GeV or so. Though it suggests an interesting and natural explanation of why neutrinos are so light, such a high new physics scale is beyond the reach of present and future neutrino experiments. On the other hand, inverse seesaw [33, 45] turns out to be a potential alternate scenario where the new physics scale responsible for neutrino mass generation can be brought down near TeV scale at the expense of involving additional SM gauge singlet fermions Si (i= 1,2,3 i.e. one per each generation of RH neutrinos). Therefore in presence of these two set of SM gauge singlets (having opposite lepton number), the inverse seesaw Lagrangian can be written as

− LISS=YDL¯HNe _R+M N_R^cS+1

2µS^cS+h.c.. (1.37) Here YD is the Dirac Yukawa coupling,M is the complex mass matrix forN_R−S inter- action. µ is a complex symmetric matrix originated from S^cS interaction and turns out to be the only source of lepton number violation in the theory with small µ. The lepton number conservation is an approximate symmetry of this model hence theN_R−N_Rterm is not allowed as it provides a large lepton number violation. After electroweak symme- try breaking, the Lagrangian in Eq. (1.37) yields a 9×9 mass matrix M_ν in the basis (ν_L^c, NR, S)

M_ν =









0 mD 0

m^T_D 0 M

0 M^T µ









. (1.38)

Now, with the assumption µ << mD << M, the effective mass matrix for the light neutrinos can be written as [46]

m_ν =mDM⁻¹µ(M^T)⁻¹m^T_D=F µF^T, (1.39)

where mD =YDv and F = mDM⁻¹ which represents mixing between active and sterile neutrinos. Here we observe that, in Eq. (1.39), the neutrino mass is double suppressed by M. Hence to get the light neutrino mass in the eV range with Yukawa coupling of the order of unity, heavy neutrino mass M can be brought down to TeV scale once µ is at keV scale. In the limit µ → 0 the lepton number conservation of the theory is restored and we find the massless neutrinos just like SM. Therefore a small non-vanishing value of µ is more natural in ’t Hooft sense [47]. Such a scenario have many important phenomenological consequences. It can lead to enhanced active-sterile neutrino mixing as well as make heavy neutrinos accessible to the direct search experiments at LHC. In Chapter 4 we explore one such scenario to study all the related phenomenological issues.