Summary of the PhD thesis: Construction and investigation of a neutrino mass model with A4 flavour symmetry by pertubation method

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The thesis is devoted to constructing and evaluating versions of the SM extended with an A4 symmetry to explain some of the problems of neutrino physics. Within these extended models, the results on neutrino masses and mixing, derived by the perturbation method, are very closed to the global fit. In two models, a relation between the Dirac CPV phase and the mixing angles is established. In particular, the models predict Dirac CPV phase δCP and effective mass of neutrinoless double beta decay jhmeeij in good agreement with the current experimental data.

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Nội dung Text: Summary of the PhD thesis: Construction and investigation of a neutrino mass model with A4 flavour symmetry by pertubation method

MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY ……..….***………… PHI QUANG VAN CONSTRUCTION AND INVESTIGATION OF A NEUTRINO MASS MODEL WITH A4 FLAVOUR SYMMETRY BY PERTUBATION METHOD Speciality: Theoretical and mathematical physics Code: 62 44 01 03 SUMMARY OF THE PHD THESIS Hanoi – 2017
This thesis was compled at the Graduate University Science and Technology, Viet Nam Academy of Science and Technology. Supervisors: Assoc. Prof. Dr. Nguyen Anh Ky Institute of Physics, Viet Nam Academy of S cience and Technology. Referee 1: Prof. Dr. Dang Van Soa Referee 2: Assoc. Prof. Dr. Nguyen Ai Viet Referee 3: Dr. Tran Minh Hieu This dissertation will be defended in front of the evaluating assembly at academy level, Place of defending: meeting room, Graduate University Science and Technology, Viet Nam Academy of Science and Technology. This thesis can be studied at: - The Vietnam National Library - Library of the National Academy of Public Administration
Introduction Motivation of thesis topic Neutrino masses and oscillations are always a challenge in elementary particle physics. We have seen in the standard model (SM) that neutrinos do not have mass, but the experiment has shown that neutrinos have mass. The problem of neutrino masses and mixings is among the problems beyond the SM. This problem is important for not only particle physics but also nuclear physics, astrophysics and cosmology, therefore, it has attracted much interest. At present, there are many standard model extensions to studying neutrino masses and oscillations: the supersymmetry model, the grand unified theory, the left-right symmetry model, the 3-3-1 model, the mirror symmetry model, Zee model, Zee-Babu model, the flavour symmetry model, etc. One of the standard model extentions to explain neutrino mass is to add a flavor symmetry to the SM symmetry, such as SU (3)C × SU (2)L × U (1)Y × GF , in which GF is a flavour symmetry group, for example S3 , S4 , A4 , A5 , T 7, ∆(27) [1],... A popular flavour symmetry intensively investigated in the literature is that described by the group A4 (see, for instance, [2, 3]) allowing obtaining a tribi-maximal (TBM) neutrino mixing corresponding to the mixing angles θ12 ≈ 35.26◦ (sin2 θ12 = 1/3), θ13 = 0◦ and θ23 = 45◦ . The recent experimental data that showing a non-zero mixing angle θ13 and a possible non-zero Dirac CP-violation (CPV) phase δCP , rejects, however, the TBM scheme [4, 5]. There have been many attempts to explain these experimental phenomena. In particular, for this purpose, various models with a discrete flavour symmetry [1], including an A4 flavor symmetry, have been suggested [1–3]. The objectives of the thesis The thesis is devoted to constructing and evaluating versions of the SM extended with an A4 symmetry to explain some of the problems of neutrino physics. Within these extended models, the results on neutrino masses and mixing, derived by the perturbation method, are very closed to the global fit [4, 5]. In two models, a relation between the Dirac CPV phase and the mixing angles is established. In particular, the models predict Dirac CPV phase δCP and effective mass of neutrinoless double beta decay |hmee i| in good agreement with the current experimental data. 1
Introduction The main contents of the thesis In general, the models, based on A4 flavour symmetry, have extended lepton and scalar sectors containing new fields in additions to the SM ones which now may have an A4 symmetry structure. Therefore, base on an A4 flavour symmetry, these fields may also transform under A4 . At the beginning, the A4 based models were build to describe a TBM neutrino mixing (see, for example, [2]) but later many attempts, such as those in [1, 3, 6], to find a model fitting the non-TBM phenomenology, have been made. On these models, however, are often imposed some assumptions, for example, the vacuum expectation values (VEV’s) of some of the fields, especially those generating neutrino masses, have a particular alignment. These assumptions may lead to a simpler diag- onalization of a mass matrix but restrict the generality of the model. Since, according to the current experimental data, the discrepancy of a phenomenological model from a TBM model is quite small, we can think about a perturbation approach to building a new, realistic, model [7, 8]. The perturbative approach has been used by several authors (see for example, [9]) but their methods mostly are model-independent, that is, no model realizing the ex- perimentally established neutrino mixing has been shown. On the other hand, most of the A4 -based models are analyzed in a non-perturbative way. There are a few cases such as [10] where the perturbative method is applied but their approach is different from ours and their analysis, sometimes, is not precise (for example, the conditions imposed in section IV of [10] are not always possible). Besides that, in many works done so far, the neutrino mixing has been investigated with a less general vacuum structure of scalar fields. In this thesis we will introduce two versions of A4 flavor symmetric standard model, which can generate a neutrino mixing, deviating from the TBM scheme slightly, as requested and explained above. Since the deviation is small we can use a perturbation method in elaborating such a non-TBM neutrino mixing model. The corresponding neutrino mass matrix can be developed perturbatively around a neutrino mass matrix diagonalizable by a TBM mixing matrix. As a consequence, a relation between the Dirac CPV phase δCP and the mixing angles θij , i, j = 1, 2, 3 (for a three-neutrino mixing model) are established. Based on the experimental data of the mixing angles, this relation allows us to determine δCP numerically in both normal odering (NO) and inverse ordering (IO). It is very important as the existence of a Dirac CPV phase indicates a difference between the probabilities P (νl → νl0 ) and P (¯ νl → ν¯l0 ), l 6= l0 , of the neutrino- and antineutrino transitions (oscillations) in vacuum νl → νl0 and ν¯l → ν¯l0 , respectively, thus, a CP violation in the neutrino subsector of the lepton sector. We should note that for a three-neutrino mixing model, as considered in this paper, the mixing matrix in general has one Dirac- and two Majorana CPV phases. Since the Majorana CPV phases do not effect these transition probabilities they are not a subject of a detailed analysis here. In the framework of the suggested models and the perturbation method our ap- proach allows us to obtain δCP , within the 1σ region of the best fit value [5]. Further, 2
Introduction knowing δCP we can determine the Jarlskog parameter (JCP ) measuring a CP viola- tion. The determination of δCP and JCP represents an application of the present model and, in this way, verifies the latter (of course, it is not a complete verification). Structure of thesis Chapter 1 presents the basis of the standard model and the problem of neutrino mass. (1) Chapters 2 and 3 are designed for constructing and evaluating the two models A4 (10) and A4 for neutrino masses and mixing. Both models are constructed perturbatively around a TBM model but objects of perturbation are different: vacuums in A14 and Yukawa coupling coefficients in A104 . In each model, physical quantities such as neu- trino mass, mixing angles θij , δCP , JCP , and the relation between δCP with angle θij are investigated and calculated. Conclusions and discussion of the thesis’s results are presented in the final chapter. 3
Chapter 1 Standard model and neutrino masses problem 1.1 Standard model 1.1.1 Local gauge invariance in Standard model First, we can consider the free Lagrangian of the field ψ(x) L0 = ψ(x) iγ λ ∂λ − m ψ(x), (1.1) 0 To the invariant theory with the SU (2) local gauge transformation ψ (x) = U (x)ψ(x), suppose ψ(x) interacts with the vector field and has covariance derivative 1 ~ Dλ ψ(x) = ∂λ + ig ~τ Aλ (x) ψ(x), (1.2) 2 here, g is a dimensionless constant and Aiλ (x) is the vector field. Then the free La- grangian becomes LI = ψ(x) iγ λ Dλ − m ψ(x), (1.3) and it will invariant to local gauge transformation. In the electroweak interaction model (GWS) with local gauge group SU (2)L ×U (1)Y , the derivative ∂λ ψ(x) must be replaced by the covariant derivative Dλ ψ(x), where Dλ ψ(x) is 1 ~ 01 Dλ (x) = ∂λ + ig ~τ Aλ (x) + ig Y Bλ (x) ψ(x), (1.4) 2 2 where Aλ (x) and Bλ (x) are the vector gauge fields of symmetry SU (2)L and U (1)Y , g 0 and g are the corresponding coupling constants. 4
Standard model Standard model and neutrino masses problem 1.1.2 Spontaneous Symmetry Breaking. Higgs Mechanism After spontaneous symmetry breaking, the mass Lagrangian terms of W, Z and H have the form 1 1 Lm = m2W Wλ† W λ + m2Z Zλ Z λ − m2H H 2 , (1.5) 2 2 here 1 1 0 m2W = g 2 v 2 , m2Z = (g 2 + g 2 )v 2 , m2H = 2λv 2 = 2µ2 . (1.6) 4 4 In summary, in the model after the spontaneous symmetry break, the vector bosons ± W , Z 0 become a mass field, the field Aλ has no mass. 1.1.3 Yukawa interaction and fermion masses Lagrangian of standard model LSM = LF + LG + LS + LY , (1.7) where LF is the kinetic Lagrangian of quark and lepton section, LG is the free La- grangian of vector fields Bλ and Aiλ , LS is the Lagrangian of Higgs field and LY is Lagrangian Yukawa interaction of quarks and leptons. From LY can be obtained 0 0 0 0 0 0 LQ mass = −U m (U ) U − D m(D) D − L m(lep) L . (1.8) We see that after spontaneous symmetry breaks, quarks and leptons become masses. 1.1.4 The electroweak interaction current From the Lagrangian interactive model can be written as interactive current g CC µ g LI = − √ Jµ W + h.c. − JµN C Z µ − eJµEM Aµ , (1.9) 2 2 2 cos θW where JµN C = 2Jµ3 − 2 sin2 θW JµEM , (1.10) X 2 X 0 0 1 0 0 X JµEM = U i γµ Ui + − Di γµ Di + (−1) lγµ l. (1.11) 3 i=u,c,t 3 i=d,s,b l=e,µ,τ Standard model have achieved great success in elementary particle physics, but in the model, neutrinos are considered as massless, this is a suggestion for physicists to extend the standard model to solve neutrino mass problems. 5
Neutrino mass and osillation Standard model and neutrino masses problem 1.2 Neutrino mass and osillation 1.2.1 Dirac-Majorana mass term We consider a neutrino mass term in the simplest case of two neutrino fields, Dirac and Majorana mass term in this case have the form 1 1 − Ldm = mL ν L νLc + mD ν L νR + mR νRc νR + h.c.. (1.12) 2 2 We can rewrite the expression as a matrix 1 − Ldm = η L Mdm (ηL )c + h.c., (1.13) 2 here ! ! νL mL mD ηL = , and Mdm = . (1.14) νRc mD mR The matrix Mdm can be diagonalized by the matrix U and obtained ! m1 0 M≡ = U T Mdm U, (1.15) 0 m2 where ! 1 1 cos θ sin θ q m1,2 =| (mR + mL ) ± (mR − mL )2 + 4m2D |, and U = , (1.16) 2 2 − sin θ cos θ 2mD mR − mL with tan 2θ = , cos 2θ = p . (1.17) mR − mL (mR − mL )2 + 4m2D From (1.13) and (1.15) we have 1 1X − Ldm = νmν = mi ν i νi , (1.18) 2 2 i=1,2 ! ν1 here ν M = U † nL + (U † nL )c = , so νic = νi . From here we have the mixing ν2 expression νL = cos θν1L + sin θν2L , (1.19) νRc = − sin θν1L + cos θν2L . (1.20) We can see that the fields of the flavour neutrinos state ν are mixtures of the left- handed components of the fields of neutrinos with definite masses. 6
Neutrino mass and osillation Standard model and neutrino masses problem 1.2.2 Seesaw mechanism In the case of two neutrino fields, section 2.1, from (1.16) and condition mD MR , mL = 0, we obtained the neutrino mass m2D m1 ' mD , m2 ' MR mD . (1.21) MR From (1.16) we find θ ' mD /MR 1. Thus, we obtain the mixing expression between the flavour neutrino and the neutrino mass  ν = ν + m D ν L 1L MR 2L (1.22) ν c = − mD ν1L + ν2L . R MR The factor mD /MR is characterized by the ratio of the electroweak scale and the scale of the violation of thelepton number. If we estimate mD ' mt ' 170GeV and m1 ' 5.10−2 eV , then MR ' m2D /m1 ' 1015 GeV . From the above calculations, we can derive conditions for constructing the mech- anism of neutrino mass generation, seesaw mechanism, [14]: The left-handed Majo- rana mass term equal to zero, mL = 0. The Dirac mass term mD is generated by the standard Higgs mechanism, i.e. that mD is of the order of a mass of quark or lepton. The right-handed Majorana mass term breaks conservation of the lepton number, the lepton number is violated at a scale which is much larger than the electroweak scale, mR ≡ MR mD . 1.2.3 Neutrino oscillations From quantum field theory, states depend on time and satisfy Schrodinger’s equations [14], ∂|να (t)i i = H|να (t)i, (1.23) ∂t where H is total Hamiltonian, α = e, µ, τ . Here, we will consider state transformations in a vacuum, in which case H is a free Hamiltonian. The equation (1.23) has a general solution |να (t)i = e−iHt |να (0)i, (1.24) where, |να (0)i is the state at the initial time t = 0. From here, the neutrino and antineutrino left-handed states at t ≥ 0 are of the form 3 X 3 X |να (t)i = e−iHt |να i = e−iEi t Uαi ∗ |νi i, |ν α (t)i = e−iHt |ν α i = e−iEi t Uαi |ν i i. (1.25) i=1 i=1 With (1.25), we can obtain the amplitude of the transition να → να0 and ν α → ν α0 at time t 3 X 3 X −iEi t ∗ Aνα →να0 (t) = Uα0 i e Uαi , Aν α →ν α0 (t) = Uα∗0 i e−iEi t Uαi . (1.26) i=1 i=1 7
Neutrino mass and osillation Standard model and neutrino masses problem In the quantum mechanics, probabilities of the transitions is equal to the squared amplitude of the transition, thus probabilities of the transitions να → να0 and ν α → ν α0 has the form 1 1 Pνα →να0 (E, L) = δα0 α + Bα0 α + ACP 0 , Pν α →ν α0 (E, L) = δα0 α + Bα0 α − ACP 0 , (1.27) 2 αα 2 αα where ! X ∆m2ji Bα0 α = −2 < Uα0 i Uα∗0 j Uαi ∗ Uαj 1 − cos L , (1.28) i>j 2E X ∆m2ji ACP α0 α =4 = Uα0 i Uα∗0 j Uαi ∗ Uαj sin L. (1.29) i>j 2E From the ACP 0 αα in the (1.29), we can calculate ∆m212 ∆m223 (∆m212 + ∆m223 ) ACP α0 α = 16J sin L sin L sin L. (1.30) 2E 2E 2E where ∆m2ij = m2j −m2i and J = −c12 c23 c213 s12 s23 s13 sin δ is called the Jarlskog parameter. 8
Chapter 2 Neutrino masses and mixing in the (1) A4 model (1) 2.1 Extended standard model A4 Compared with the SM, the model studied here contains an extended lepton- and scalar sector (the quark sector is not considered here yet). The transformation rules under SU (2)L , A4 , Z3 and Z4 of the leptons and the scalars in this model are summa- rized in Table 2.1. Let us look at a closer distance the scalar- and the lepton sector. 0 00 `L e˜R µ ˜R τ˜R φh N ϕE ϕN ξ ξ ξ SU(2)L 2 1 1 1 2 1 1 1 1 1 1 0 00 0 00 A4 3 1 1 1 1 3 3 3 1 1 1 Z3 ω2 1 1 1 ω2 ω 1 ω ω ω ω2 Z4 i 1 1 1 1 i i -1 -1 i i Table 2.1: Lepton- and scalar sectors of the model and their group transformations A4 , Z3 , Z4 , where ω k = e2kπ/3 , k = 0, 1, 2. 2.2 Scalar sector The scalar potential has the form 0 00 0 00 0 00 V(φh , ϕE , ϕN , ξ, ξ , ξ ) = V1 (φh ) + V2 (ξ, φh ) + V3 (ϕE , ξ , ξ ) + V4 (ϕN , φh , ξ, ξ , ξ ). (2.1) 0 00 Let us denote the VEV’s of these scalar fields ξ, ξ , ξ , ϕE := (φ1 , φ2 , φ3 ) and ϕN := (ϕ1 , ϕ2 , ϕ3 ) as follows 0 00 hξi = σa , hξ i = σb , hξ i = σc , hφh i = vh , hϕE i = (v1 , v2 , v3 ) , hϕN i = (u1 , u2 , u3 ) . (2.2) To get a VEV of ϕE = (φ1 , φ2 , φ3 ) imposes the extremum condition on the potential V . This extremum potential equation system have the solutions the equality −α6r σb σc v12 = v 2 = 0 , v2 = v3 = 0, (2.3) 2(α1 + α3 ) 9
(1) Lepton sector Neutrino masses and mixing in the A4 Next, we consider extremum potential for the VEV of ϕN = (ϕ1 , ϕ2 , ϕ3 ), they have the solutions λ0 Type 1: (0, 0, 0) , u1 = u2 = u3 = 0; Type 2: (u, 0, 0) , u2 = 0 ; (2.4) 2(λ1 + λ3 ) λ0 + β2 + β3 Type 3: (u, u, u) , u2 = − ; Type 4: (u1 , u2 , u3 ) , u1 6= u2 6= u3 6= u1 , ui 6= 0. 6(λ1 + 2λ2 ) 2.3 Lepton sector Basing on the A4 × Z3 × Z4 symmetry, we can construct the following Yukawa terms of the effective Lagrangian for the lepton sector of the present model ϕE 00 ϕE 0 ϕE −Lnew Y = λe (lL φh )˜ eR + λµ l L φ h µ ˜R + λτ lL φh τ˜R + λD `L φ˜h N Λ Λ Λ + gN N c N ϕN + gξ N c N 1 ξ + H.c., (2.5) where λe , λµ , λτ , λD , gN and gξ are interaction coefficients of Lagrangian. This La- grangian, the corresponding charged lepton mass matrix automatically has a diagonal form   y e vh 0 0 λe v λµ v λτ v Ml =  0 yµ vh 0 , where ye = , yµ = , yτ = . (2.6)   Λ Λ Λ 0 0 yτ vh From the seesaw mechanism presented in Chapter 2, we have a neutrino mass matrix Mν = −MDT MN−1 MD , (2.7) where MD and MN are Dirac and Majorana neutrino mass matrices in Lagrangian (2.5). From the above relation we obtain the neutrino mass matrix   −b21 + 2b1 d − d2 + 4b2 b3 2b22 + b3 (b1 − d) 2b23 + b1 b2 − b2 d 1  Mν =  2b22 + b3 (b1 − d) −b23 + 4b1 b2 + 2b2 d 2b21 − b1 d − d2 + b2 b3  , (2.8)  D 2b23 + b1 b2 − b2 d 2b21 − b1 d − d2 + b2 b3 2b3 (2b1 + d) − b22 D = det(MN ) = −2b31 + 3b21 d + 6b1 b2 b3 − 2b32 + 6b2 b3 d − 2b33 − d3 . We see that, if b1 = b2 = b3 = b or u1 = u2 = u3 = u then the diagonalization of Mν ≡ Mν Mν† (equivalent the diagonalization matrix Mν ) matrix will obtain UT BM  q q  2 1 0  q3 q3 q  diag(Mν0 ) = UTTBM Mν0 UT BM , where UT BM =  − 16 1 − 1 . (2.9)    q q3 q 2  − 16 1 3 1 2 For the VEV alignment u1 6= u2 6= u3 6= u1 in (2.4) the neutrino mass has a general form (2.8). We have to diagonalize the Mν matrix or Mν ≡ Mν Mν† † diag(Mν ) = Upmns Mν Upmns , (2.10) here, Upmns is a mixing matrix 3 × 3. It is a difficult task to find a realistic (phe- 10
(1) Neutrino masses and mixing Neutrino masses and mixing in the A4 nomenological) model to realize Upmns . To solve this problem, different methods and tricks have been used. Since Upmns slightly differs from the TBM form (2.9) we will follow a perturbation approach. This approach allows us to find a theoretical mixing matrix, say U , which must be compared with the empirical PMNS matrix. 2.4 Neutrino masses and mixing The current experimental data (θ13 ≈ 9◦ , θ23 ≈ 42◦ , θ12 ≈ 33◦ ) [5] shows that the matrix Upmns can be obtained from Utbm by a small correction as seen from their difference. Therefore, we can consider Upmns as a perturbative development around Utbm . Thus, Mν in (2.8) can be written as Mν = M0 + V, (2.11) where, V matrix has extremely small elements. Since a homogeneous VEV align- ment hϕN i = (u1 , u1 , u1 ) such as that in Type 3 of (2.4) leads to a TBM mixing but the experiment tells us a mixing slightly deviating from the TBM one, we must con- sider an inhomogeneous VEV alignment Type 4 of (2.4) to deviate from a homoge- neous alignment just slightly, that is (u1 , u2 , u3 ) = (u1 , u1 + 2 , u1 + 3 ) with 2 , 3 1. 0 0 This condition is satisfied if λ0 λ1 ≈ λ2 ≈ λ3 ≈ λ ≡ λ, and β2 ≈ β3 λ. So (b1 , b2 , b3 ) = (b1 , b1 + e2 , b1 + e3 ); e2 , e3 1. Now, we will using the perturbation method akn |k 0 i + ..., with akn = (|m0n |2 − |m0k |2 )−1 Vkn , Vkn = hk 0 |M0† V + V † M0 |n0 i, X |ni = |n0 i + (2.12) k6=n to diagonalize the matrix Mν , and we obtain  q q q q q q  2 1 ∗ 1 2 2 1 + X − X − Y − Z  q3 q q3 q3 q3 q q q3 q3  U =  − 16 + 13 X ∗ − 12 Y ∗ 1 + 1 X − 1 ∗ Z − 12 + 16 Y − 13 Z , (2.13)    q q q q3 q6 q2 q q q  − 16 + 13 X ∗ + 12 Y ∗ 1 3 + 1 6X + 1 ∗ 2Z 1 2 + 1 6Y − 1 3Z where X = −a12 , Y = −a13 , Z = −a23 . (2.14) To check the model how it works, let us make a numerical analysis. We can calculate X = 0, 326 + 0, 034i, Y = −0, 007 + 0, 003i, Z = −0, 082 + 0, 251i. (2.15) From (2.15), we obtain s13 ≈ 0.0156 (or θ13 ≈ 8.97◦ ) and δ ≈ 1.39π. The latter value of s13 is very close to the experimental data in [4, 5]. Interestingly, the Dirac CPV phase, δCP ≡ δ, obtained here, accidentally coincides with its global fit given in [5]. A more detailed analysis on δCP will be make in the next section. 2.5 Dirac CP violation phase and Jarlskog parame- ter In order to determine all variables in the matrix (2.13), or, at least, their relations, we must compare this matrix with the experimental one. Denoting the elements of 11
(1) δCP and JCP Neutrino masses and mixing in the A4 the matrix (2.13) by Uij , i, j = 1, 2, 3, we get the equation (up to the first perturbation order) √ 2 |U21 |2 − |U31 |2 − |U22 |2 − |U32 |2 = −2 2Re(U13 ). (2.16) we obtain the following relation between the Dirac CPV phase δCP ≡ δ and the neutrino mixing angles θij (s2 − c2 )(2s212 − c212 ) cos δ = √ 23√ 23 . (2.17) 2 2(3 2s23 c23 s12 c12 + 1)s13 Based on the relation (2.17) and experimental inputs, δCP can be calculated numer- ically. With using the experimental data of the mixing angles within 1σ around the BFV [4, 5], the distributions of δCP are plotted in Fig. 2.1 for a normal ordering (NO) and in Fig. 2.2 for an inverse ordering (IO). Distribution of δCP (for an NO) Mean 2.265 Mean 4.018 δCP versus sinθ213 (for an NO) Mean x 0.02347 Mean y 3.141 RMS 0.38 RMS 0.3792 RMS x 0.001961 Entries δCP 1200 7 RMS y 0.9551 6 1000 5 800 4 600 3 400 2 200 1 0 0 0 1 2 3 4 5 6 7 0.018 0.02 0.022 0.024 0.026 0.028 0.03 δCP sinθ213 Figure 2.1: Distribution of δCP and δCP versus sin2 θ13 in an NO . Distribution of δCP (for an IO) Mean 1.769 Mean 4.514 δCP versus sinθ213 (for an IO) Mean x 0.02405 Mean y 3.142 RMS 0.6554 RMS 0.655 RMS x 0.002062 Entries δCP 7 600 RMS y 1.521 6 500 5 400 4 300 3 200 2 100 1 0 0 0 1 2 3 4 5 6 7 0.018 0.02 0.022 0.024 0.026 0.028 0.03 δCP sinθ213 Figure 2.2: Distribution of δCP and δCP versus sin2 θ13 in an IO . 12
(1) δCP and JCP Neutrino masses and mixing in the A4 Distribution of JCP Mean 0.0244 Mean 0.0272 RMS 0.007251 RMS 0.007421 Entries 800 700 NO 600 IO 500 400 300 200 100 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 JCP Figure 2.3: Distribution of JCP in an NO and an IO. Having all mixing angles and Dirac CPV phase it is not difficult to determine the Jarlskog parameter JCP ≡ J. Indeed, using the expression [14] |JCP | = |c12 c23 c213 s12 s23 s13 sin δ|, (2.18) we obtain |JCP | ≤ 0.038 and |JCP | ≤ 0.039 (rough bounds) for an NO and an IO, respec- tively (see the distribution of JCP in Fig. 2.3). These mean values of δCP and JCP are closer to the global fits than their corre- sponding values obtained at the BFV’s of the mixing angles (by inserting the latter in the analytical expressions (2.17) and (2.3) for δCP and |JCP |, respectively). To have a better view in comparing the two cases, the NO and the IO, the BFV’s of δCP and JCP for both cases are summarized in Table 2.2. Normal ordering Inverse ordering δCP /π 1.28 1.44 |JCP | 0.024 0.027 Table 2.2: The mean values of δCP and |JCP | in an NO and an IO. In the case of an NO, δCP has a mean value of 2.265 ≈ 0.72π for one of the solutions, and a mean value of 4.018 ≈ 1.28π for the other solution and its distribution gets maximums at 2.35 ≈ 0.75π and 3.95 ≈ 1.26π, respectively. We see that the second solution (for both its mean value and the value at its maximal distribution) lies in the 1σ region from the best fit value (BFV) 1.39π given in [4, 5]. In the case of an IO, δCP gets a mean value around 1.769 ≈ 0.56π (for the first solution), and around 4.514 ≈ 1.44π (for the second solution). Its distribution reaches maximums at about 2.15 ≈ 0.68π and 4.17 ≈ 1.33π. Again, the second solution lies within the 1σ region of the BFV 1.31π given in [4, 5]. To avoid any confusion, let us stress that the mean values of δCP and |JCP | do not coincide, in fact and in principle, with their values obtained at the BFV’s of the mixing angles. It means that a value of δCP or |JCP | obtained at a BFV of the mixing angles 13
(1) δCP and JCP Neutrino masses and mixing in the A4 should not in any way be identified with the mean value of the quantity concerned, although in some case they may be close to each other. It is also important to note that the equation (2.17) is ill-defined in the 3σ region of the mixing angles. It means that this equation of determination of δCP restricts the dissipation of the mixing angles (that is, the values scattered too far, in the 3σ region of distribution, are automatically excluded). 14
Chapter 3 Neutrino masses and mixing in the (10) A4 model (10) 3.1 Extended standard model A4 We present in this section an A4 -flavour symmetric extension of a type-I see-saw model.There could be different versions of this extended model but the one given in Tab. 3.1 requires a minimal and “natural” extension, thus, this model can be referred to as the minimally extended model of the type-I see-saw model, or, just, the minimal model, for short. `L `Ri Φh ΦS ΦS 0 ΦS 00 NT NS NS 0 NS 00 Spin 1/2 1/2 0 0 0 0 1/2 1/2 1/2 1/2 SU (2)L 2 1, 1, 1 2 1 1 1 1 1 1 1 0 00 0 00 0 00 A4 3 1, 1 , 1 3 1 1 1 3 1 1 1 Table 3.1: An A4 -flavour symmetric extended standard model The group transformation nature of the fields in the considered model is shown on the table, where `L = (`L1 , `L2 , `L3 ) and `Ri , i = 1, 2, 3, are the three generations of the left-handed- and the right-handed charged leptons, respectively. The three iso- doublets `Li form together an A4 -triplet, while the iso-singlets `Ri are also A4 -singlets, 1, 10 and 100 . The new fermions NT , NS , NS 0 and NS 00 , being neutral fields and iso- singlets, are an A4 -triplet and three A4 -singlets 1, 10 and 1”, respectively. Below, we shall choose to consider the type I see-saw model given in Tab. 3.1 and illustrated in Fig. 3.1. 15
(10) Scalar sector Neutrino masses and mixing in the A4 model Figure 3.1: An A4 -flavour-symmetric type-I see-saw model. 3.2 Scalar sector This sector consists of four scalars which are an iso-doublet A4 -triplet Φh , and three 0 00 iso-singlets ΦS , ΦS 0 and ΦS 00 transforming as A4 -singlets 1, 1 and 1 , respectively. Here, we briefly discuss the Higgs potential in the model. It has the general form V = V (Φh ) + V (Φh , ΦS , ΦS 0 , ΦS 00 ) + V (ΦS , ΦS 0 , ΦS 00 ), (3.1) The VEV’s of ΦS , ΦS 0 and ΦS 00 are hΦS i = σ1 , hΦS 0 i = σ2 , hΦS 00 i = σ3 . To get a VEV of φh imposes the extremum condition on the potential V . This extremum potential equation system have the solutions the equality −µ20 v12 = v22 = v32 := v 2 = · (3.2) 2(3λ1 + λ3 + λ4 + λ5 ) 3.3 Lepton sector The lepton sector contains charged leptons and neutrinos. In the present model, the charged lepton masses can be generated by the Yukawa terms of the Lagrangian 00 0 −LYcl = y1 (`L Φh )`R1 + y2 (`L Φh ) `R2 + y3 (`L Φh ) `R3 + h.c. (3.3) This Lagrangian has charged lepton mass matrix     me 0 0 1 1 1 1 1  Mlept = √ UL  0 mµ 0  , with UL = √  1 ω ω 2  , (3.4)    2 3 0 0 mτ 1 ω2 ω here me = y1 v, mµ = y2 v, mτ = y3 v are charge lepton mass e, µ, τ . For the neutrinos, their masses can be generated by Yukawa terms of Dirac and Majorana type. First, Lagrangian Dirac Yukawa −LD ν ¯e · NT + yTν b `¯L Φ · NT (3.5) Yν = yT a `L Φh eh 3 32 1 + ySν `¯L Φ e h · NS + y ν 0 `¯L Φ S eh 00 · N ν ¯e S 0 + yS 00 `L Φh 0 · NS 00 + h.c.. 1 1 1 16
(10) Neutrino masses and mixing Neutrino masses and mixing in the A4 model The Majorana Yukawa terms have the form 00 0 −LM M S + yTM2 N T NT 10 S + yTM2 N T NT 100 S Yν =yT1 N T NT 1 0 + y1M N S NS 1 S + y2M N S 0 NS 00 1 S + y3M N S 0 NS 0 100 S 0 00 00 + y4M N S NS 00 100 S + y5M N S 00 NS 00 10 S + y6M N S NS 0 10 S + h.c.. (3.6) Taking into account (3.5) and (3.6), we write the Yukawa terms for neutrinos 1 L Yν = L D M c Yν + LYν = nL Mseesaw (nL ) + h.c, (3.7) 2 here, nL = (νL , (NT , NS , NS 0 , NS 00 )c )T with NT = (NT 1 , NT 2 , NT 3 )T , and ! 0 MD Mseesaw = , (3.8) MDT MR with, MD and MR are Dirac and Majorana neutrino mass matrix (3.5) and (3.6). From here, we get the type-I see-saw mass matrix Mν = −MDT (MR )−1 MD , which in the charged-lepton diagonalizing basis becomes Mν = UL† Mν UL∗ . 3.4 Neutrino masses and mixing It is observed from (3.6) that if ySν = ySν 0 = ySν 00 , yT a , yT b ySν , M55 = M66 , (3.9) the neutrino mass matrix Mν can be diagonalized by the matrix UT BM . That means, the model with the condition (3.9) is a TBM model. Therefore, the (actual) neutrino mass matrix diagonalisable by the (experimental) PMNS mixing matrix, can be devel- oped pertubatively around the TBM one. In other words, to obtain a realistic model we could replace (3.9) by another condition which at the first order of approximation reads ySν 0 = ySν + 1 , ySν 00 = ySν + 2 , M55 = M66 + σ, (3.10) with i , i = 1, 2, and σ are small number (i.e., |i |, |σ| 1). From here we can develope perturbation of the neutrino mass matrix Mν . Mν = Mν0 + W, (3.11) Now we can diagonalize the neutrino mass matrix square M†ν Mν by the matrix  q q q q q q  2 1 ∗ 1 2 2 1 3 + 3x − x − y − z  q q q q3 q3 q q q3 q3  ˜ = U  − 1 + 1 ∗ + 12 y ∗ 1 1 1 ∗ 1 1 − 1 , (3.12) 3x 3 + 6x + 2z + y z   q 6 q q q q q q2 q6 q3  1 1 ∗ 1 ∗ 1 1 1 ∗ 1 1 1 6 − 3 x + 2y − 3 − 6x + 2z 2 − 6y + 3z with x = λ12 , y = λ13 , z = λ23 . (3.13) 17
(10) Neutrino masses and mixing Neutrino masses and mixing in the A4 model Below, to check how our model works we will consider the case of a real param- eter x. Deriving x, y and z by matching the matrix elements (UP M N S )11 , (UP M N S )13 and (UP M N S )23 in Eq. (3.12) with the corresponding elements of the experimentally measured PMNS matrix at the current best fit value of neutrino oscillation data [4,5]. In the following, to check the present model we are going to analyze and discuss in the model scenario several physics quantities, which can be verified experimentally, such as the neutrinoless double beta decay effective mass |hmee i|, CPV phase δCP ≡ δ and Jarlskog parameter JCP , using the PMNS matrix (3.12) matched with the current neutrino oscillation data [4, 5]. In the scenario of the present model, the (0νββ) decay effective mass has the form