Parametrization of the Yukawa matrix in the scotogenic model and single-zero textures of the neutrino mass matrix
aa r X i v : . [ h e p - ph ] J un June 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2
International Journal of Modern Physics Ac (cid:13)
World Scientific Publishing Company
Parametrization of the Yukawa matrix in the scotogenic model andsingle-zero textures of the neutrino mass matrix
Teruyuki Kitabayashi ∗ Department of Physics, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292,Japan
Received Day Month YearRevised Day Month YearAs the first topic, we propose a new parametrization of the complex Yukawa matrix in thescotogenic model. The new parametrization is compatible with the particle data groupparametrization of the neutrino sector. Some analytical expressions for the neutrinomasses with the new parametrization are shown. As the second topic, we consider thephenomenology of the socotogenic model with the one-zero-textures of the neutrino flavormass matrix. One of the six patterns of the neutrino mass matrix is favorable for thereal Yukawa matrix. On the other hand, for the complex Yukawa matrix, five of the sixpatterns are compatible with observations of the neutrino oscillations, dark matter relicabundance and branching ratio of the µ → eγ process.PACS numbers:14.60.Pq, 95.35.+d, 98.80.Cq
1. Introduction
The nature of the dark matter and neutrinos cannot be explained within the stan-dard model of particle physics. The new physics beyond the standard model mayprovide the hints of these problems. The scotogenic model can simultaneously ac-count for dark matter candidates and the origin of tiny masses of neutrinos. In thismodel, neutrino masses are generated by one-loop interactions mediated by a darkmatter candidate. One-loop interactions related to dark matter and neutrino masshave been extensively studied in the literature.
On the other hand, there have been various discussions on flavor neutrino massmatrices with zero elements. The origin of such texture zeros was discussed inRefs.
In particular, texture of the flavor neutrino mass matrix with single-zero element is called one-zero-textures. There are six patterns for the one-zero-textures of the flavor neutrino mass matrix, which are usually denoted by G , G , · · · , G . The phenomenology of the one-zero-textures was studied, for examples, inRefs. Also, the experimental potential of probing the texture-zero models has ∗ [email protected] 1 une 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Teruyuki Kitabayashi been discussed. For example, see the Ref. for the possibility of probing differenttexture-zero flavor mass matrices at DUNE.Recently, the author has reported a relation of the scotogenic model with theone-zero-textures of the neutrino flavor mass matrix for the real Yukawa matrix. If the elements of the Yukawa matrix in the scotogenic model are real, one of sixpattern in the one-zero-textures, G , is favorable. Three patterns (G , G and G )are impossible in the scotogenic model with the real Yukawa matrix. Two patterns(G and G ) are not favorable within the scotogenic model with the real Yukawamatrix because the predicted neutrino oscillation data, the relic abundance of darkmatter, and the upper limit of the branching ratio of the µ → eγ process should beunlikely or be out of the range of the observed data.In this paper, we enlarge the previous argument on the relation of the scoto-genic model with the one-zero-textures of the neutrino flavor mass matrix to includethe complex elements in the Yukawa matrix. There are the following two main topicsin this paper: • We propose a new parametrization of the complex Yukawa matrix in thescotogenic model. The new parametrization is compatible with the par-ticle data group parametrization of the neutrino sector. For the phe-nomenology of the scotogenic model with the PDG parametrization, theparametrization of the Yukawa matrix which is proposed in this paper maybe useful. • In the previous study, all elements of the Yukawa matrix are real, so thatthree patterns (G , G and G ) in the one-zero-textures are impossible inthe scotogenic model. If we include the complex Yukawa matrix elements,or equivalently CP-violating phases, the results may have been different.Some analysis of this topic will be found in this paper.This paper is organized as follows. In Sec.2, a brief review of the scotogenicmodel is provided. In Sec.3, we propose a new parametrization of the complexYukawa matrix and show some analytical expressions for the neutrino masses in thescotogenic model with the new parametrization. In Sec.4, some phenomenologicalstudy of the scotogenic model with the complex Yukawa matrix in the one-zero-textures of the neutrino flavor neutrino mass matrix scheme will be shown. Finally,Sec.5 is devoted to summary.
2. Scotogenic model
We show a brief review of the scotogenic model. The scotogenic model has threeextra Majorana SU (2) L singlets N k ( k = 1 , ,
3) and one new scalar SU (2) L doublet η = ( η + , η ). N k and η are odd under Z symmetry while other fields are even under Z symmetry. The Lagrangian of the scotogenic model contains new terms for theune 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Parametrization of the Yukawa matrix in the scotogenic model and single-zero textures of the neutrino mass matrix new fields, L ⊃ Y αk (¯ ν αL η − ¯ ℓ αL η + ) N k + 12 M k ¯ N k N Ck + H.c., (1)and the scalar potential of the model contains the quartic scalar interaction V ⊃ λ (Φ † η ) + H.c., (2)where L α = ( ν α , ℓ α ) is the left-handed lepton doublet and Φ = ( φ + , φ ) is the Higgsdoublet in the standard model. The elements of the flavor neutrino mass matrix M ν = M ee M eµ M eτ − M µµ M µτ − − M ττ , (3)where the symbol “ − ” denotes a symmetric partner, are obtained as M αβ = X k =1 Y αk Y βk Λ k , (4)where Λ k = λv π M k m − M k (cid:18) − M k m − M k ln m M k (cid:19) , (5) m = 12 ( m R + m I ) , (6)and v , m R , and m I denote the vacuum expectation value of the Higgs field, and themasses of √ η ] and √ η ], respectively.The scotogenic model predicts the existence of particle dark matter. The lightest Z odd particle is stable in the particle spectrum. This lightest Z odd particlebecomes a dark matter candidate. We assume that the lightest Majorana singletfermion, N , becomes the dark matter and N is considered to be almost degeneratewith the next to lightest Majorana singlet fermion N , M . M < M . In this case,the (co)annihilation cross section times the relative velocity of annihilation particles v rel is given by
3, 5, 8, 11, 70 σ ij | v rel | = a ij + b ij v , (7)with a ij = 18 π M ( M + m ) X αβ (cid:12)(cid:12) Y αi Y ∗ βj − Y ∗ αj Y βi (cid:12)(cid:12) , (8) b ij = m − m M − M M + m ) a ij + 112 π M ( M + m )( M + m ) X αβ (cid:12)(cid:12) Y αi Y ∗ αj Y βi Y ∗ βj (cid:12)(cid:12) , where σ ij ( i, j = 1 ,
2) is annihilation cross section for N i N j → ¯ f f . The effectivecross section σ eff is obtained as σ eff = g g σ + 2 g g g σ (1 + ∆ M ) / e − ∆ M · x + g g σ (1 + ∆ M ) e − M · x , (9)une 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Teruyuki Kitabayashi where ∆ M = ( M − M ) /M depicts the mass splitting ratio of the degeneratesinglet fermions, x = M /T denotes the ratio of the mass of lightest singlet fermionto the temperature T and g and g are the number of degrees of freedom of N and N , respectively, and g eff = g + g (1 + ∆ M ) / e − ∆ M · x . (10)Since N is considered almost degenerate with N , we have ∆ M ≃ σ eff | v rel | = a eff + b eff v , (11)where a eff = a a a , b eff = b b b . (12)The thermally averaged cross section can be written as h σ eff | v rel |i = a eff +6 b eff /x and the relic abundance of cold dark matter is estimated to beΩ h = 1 . × x f g / ∗ m pl (GeV)( a eff + 3 b eff /x f ) , (13)where m pl = 1 . × GeV, g ∗ = 106 .
75 and x f = ln 0 . g eff m pl M h σ eff | v rel |i g / ∗ x / f . (14)In the scotogenic model, flavor-violating processes such as µ → eγ are inducedat the one-loop level. The branching ratio of µ → eγ is given by Br( µ → eγ ) = 3 α em π ( G F m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k =1 Y µk Y ∗ ek F (cid:18) M k m (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (15)where α em denotes the fine-structure constant, G F denotes the Fermi coupling con-stant and F ( x ) is defined by F ( x ) = 1 − x + 3 x + 2 x − x ln x − x ) . (16)
3. Parametrization of Yukawa matrix3.1.
Yukawa matrix
In order to obtain any phenomenological prediction in the scotogenic model, theelements of the Yukawa matrix Y = Y e Y e Y e Y µ Y µ Y µ Y τ Y τ Y τ , (17)should be determined. This matrix is closely connected with the neutrino sector.There are several ways for parametrization of the Yukawa matrix. For exam-ple, Suematsu, et.al. proposed a parametrization of the Yukawa matrix with theune 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Parametrization of the Yukawa matrix in the scotogenic model and single-zero textures of the neutrino mass matrix assumption of the tribimaximal mixing in the neutrino sector. The exact tribi-maximal pattern is approximately consistent with the observed solar and atmo-spheric neutrino mixings; however, it predicts a vanishing reactor neutrino mixingangle. The observed reactor neutrino mixing angle is small but moderately large.Although the exact tribimaximal pattern cannot be the correct description ofthe neutrino sector, the way to determine the Yukawa matrix elements for theexact tribimaximal pattern is still useful. For example, using the method in Ref., Singirala proposed the following parametrization Y = Y e Y e Y e − . Y e Y e . Y e . Y e − Y e . Y e , (18)for an modified tribimaximal mixing (see also ) U = U MTB = cos θ sin θ − sin θ √ θ √ √ θ √ − cos θ √ √ cos ϕ e − iζ sin ϕ − e iζ sin ϕ ϕ , (19)with θ = 35 ◦ , ϕ = 12 ◦ and ζ = 0. The corresponding three neutrino mixing angles θ , θ , θ , CP-violating Dirac phase δ and two Majorana phases α , α in theparticle data group (PDG) parametrization are θ = 35 . ◦ , θ = 38 . ◦ , θ =9 . ◦ and δ = α = α = 0.In the next subsection, we propose a new parametrization of the Yukawa matrixin the scotogenic model. We employ the similar strategy in as well as to determinethe elements of the Yukawa matrix; however, we do not take any assumption in themixing of the neutrino sector such as U = U MTB . The PDG parametrization ofthe neutrino mixing matrix: U = U PDG = U ( θ , θ , θ , δ ) P ( α , α ) is employedin the new parametrization of the Yukawa matrix. A Yukawa matrix with PDGparametrization have been already proposed in terms of Y τ , Y τ and Y τ by Ho andTandean with P = diag . ( e iα / , e iα / , In this paper, a new PDG compatibleYukawa matrix parametrization will be proposed in terms of Y e , Y e and Y e with P = diag . (1 , e iα / , e iα / ).It must be emphasized that the parametrization of the Yukawa matrix by Hoand Tandean is valuable parametrization. Moreover, the commonly used Casas-Ibarra parametrization
40, 76 is powerful to determine the numerical magnitude ofthe Yukawa matrix elements.
26, 29
Of course, the numerical determination of theYukawa matrix with some assumptions such as U = U MTB is worth way. Theaim of the next subsection is not denial of these excellent parametrizations of theYukawa matrix but proposing a new parametrization. For the phenomenology ofthe scotogenic model in terms of Y e , Y e and Y e with the PDG parametrizationwith P = diag . (1 , e iα / , e iα / ), the parametrization of the Yukawa matrix whichis proposed in this paper may be useful.une 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Teruyuki Kitabayashi
Parametrization
We propose a new parametrization of the Yukawa matrix in this subsection. ThePDG parametrization of the neutrino mixing matrix is given as U PDG = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c × e iα /
00 0 e iα / , (20)where c ij = cos θ ij , s ij = sin θ ij ( i, j =1,2,3). Because the relation U T PDG M ν U PDG = diag . ( m , m , m ) , (21)with Eqs.(3), (4) and (20) has to be satisfied, we obtain M diag11 = X k =1 { Λ k ( s ( − Y µk c + Y τk s )+ c ( Y ek c − e iδ s ( Y τk c + Y µk s ))) } = m , (22) M diag12 = X k =1 { e iα / Λ k ( − c ( Y µk c − Y τk s )( − Y ek c + e iδ s ( Y τk c + Y µk s )))+ s ( Y µk c − Y τk s )( − Y ek c + e iδ s ( Y τk c + Y µk s ))+ c s ( Y ek c − Y µk c − Y τk s − e iδ Y ek ( Y τk c + Y µk s ) sin 2 θ + e iδ s ( Y τk c + Y µk s ) + Y µk Y τk sin 2 θ ) } = 0 , (23) M diag13 = X k =1 { e i ( α − δ ) / Λ k ( Y ek s + e iδ c ( Y τk c + Y µk s )) × ( s ( − Y µk c + Y τk s )+ c ( Y ek c − e iδ s ( Y τk c + Y µk s ))) } = 0 , (24) M diag22 = X k =1 { e iα Λ k ( c ( Y µk c − Y τk s )+ s ( Y ek c − e iδ s ( Y τk c + Y µk s ))) } = m , (25) M diag23 = X k =1 { e i ( α + α − δ ) / Λ k ( Y ek s + e iδ c ( Y τk c + Y µk s )) (26) × ( c ( Y µk c − Y τk s ) + s ( Y ek c − e iδ s ( Y τk c + Y µk s ))) } = 0 , une 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Parametrization of the Yukawa matrix in the scotogenic model and single-zero textures of the neutrino mass matrix and M diag33 = X k =1 { e i ( α − δ ) Λ k ( Y ek s + e iδ c ( Y τk c + Y µk s )) } = m , (27)where m , m and m denote the neutrino mass eigenvalues.The Eqs.(23), (24) and (27) yield the following Yukawa matrix Y = Y e Y e Y e a Y e a Y e a Y e a Y e a Y e a Y e , (28)where a = − c t c − e − iδ s t , a = s t c − e − iδ c t ,a = c t c − e − iδ s t , a = − s t c − e − iδ c t ,a = e − iδ s t , a = e − iδ c t , (29)and t ij = tan θ ij ( i, j =1,2,3). The Eq.(28) with Eq.(29) is the new parametrizationof the Yukawa matrix which we proposed in this paper. This parametrization of theYukawa matrix elements is also relevant for extended scotogenic models if the flavorneutrino masses are expressed as M αβ = P k =1 Y αk Y βk Λ k .Now, we check the reproducibility of the Singirala’s result in Eq.(18). If we take θ = 35 . ◦ , θ = 38 . ◦ , θ = 9 . ◦ ,δ = α = α = 0 ◦ , (30)to compere the coefficients of the Yukawa matrix elements with Singirala’s numericalresult, we obtain a = − . , a = 1 . , a = 3 . ,a = 0 . , a = − . , a = 4 . , (31)and these values are consistent with Eq.(18).Some analytical expressions for the neutrino sector in the scotogenic model withthe new parametrization of the Yukawa matrix in Eqs.(28) and (29) are obtainedas follows.The Eqs.(22), (25) and (27) yield the following neutrino mass eigenvalues m i = b i Λ i Y ei , (32)where b = 1 c c , b = e iα s c , b = e i ( α − δ ) s . (33)une 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Teruyuki Kitabayashi
Although the neutrino mass ordering (either the normal mass ordering or theinverted mass ordering) is not determined, a global analysis shows that the prefer-ence for the normal mass ordering is mostly due to neutrino oscillation measure-ments.
77, 78
We assume the normal mass ordering (NO) for the neutrinos. In thiscase, the squared mass differences of the neutrinos are given by∆ m = m − m , ∆ m = m − m , (34)and we obtain the relations Y e = σ b Λ q ∆ m + b Λ Y e ,Y e = σ b Λ q ∆ m + b Λ Y e , (35)where σ , = ±
1. We take σ , = 1. Since of the relation Λ k = f ( λ, m , M k ), theelements of the Yukawa matrix can be determined as Y αk = f ( θ ij , δ, α i , ∆ m ij ; Y e ; λ, m , M k ) , (36)where θ ij , δ, α i , ∆ m ij are neutrino sector parameters, λ, m , M k are dark sectorparameter, and Y e bridges these two sectors.The ee -element of the flavor neutrino mass matrix M ν can be written as M ee = Y e Λ + Y e Λ + Y e Λ = Y e Λ + σ b q ∆ m + b Λ Y e + σ b q ∆ m + b Λ Y e . (37)Similarly, we obtain M eµ = a Y e Λ + σ a b q ∆ m + b Λ Y e + σ a b q ∆ m + b Λ Y e , (38) M eτ = a Y e Λ + σ a b q ∆ m + b Λ Y e + σ a b q ∆ m + b Λ Y e , (39) M µµ = a Y e Λ + σ a b q ∆ m + b Λ Y e + σ a b q ∆ m + b Λ Y e , (40) M µτ = a a Y e Λ + σ a a b q ∆ m + b Λ Y e + σ a a b q ∆ m + b Λ Y e , (41)and M ττ = a Y e Λ + σ a b q ∆ m + b Λ Y e + σ a b q ∆ m + b Λ Y e . (42)Thus, M αβ is independent of Λ and Λ as well as M and M : M αβ = f ( θ ij , δ, α i , ∆ m ij ; Y e ; λ, m , M ) . (43)une 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Parametrization of the Yukawa matrix in the scotogenic model and single-zero textures of the neutrino mass matrix
4. One-zero-textures4.1.
Real Yukawa matrix
As we addressed in introduction, we assume that the flavor neutrino mass matrix M ν has one zero element. There are the following six patterns for the flavor neutrinomass matrix M ν in the one-zero-textures:G : × ×− × ×− − × , G : × ×− × ×− − × , G : × × − × ×− − × , G : × × ×− ×− − × , G : × × ×− × − − × , G : × × ×− × ×− − . (44)First, we assume that all elements of the Yukawa matrix are real. Because thepresent data disfavor CP conservation, the assumption of a real Yukawa couplingmatrix is not realistic; however, we start our study without CP violation just as asimple case. In this case, the author has already reported that one of six patterns,G , is favorable for the observed neutrino oscillation data, the relic abundance ofdark matter, and the upper limit of the branching ratio of the µ → eγ process. In this previous study, a specific parametrization of the Yukawa matrix with anassumption of the neutrino mixing (modified tribimaximal mixing) was employed. In this subsection, we briefly reproduce the results in Ref. by using the newparametrization of the Yukawa matrix in Eq.(28) with Eq.(29).For the G pattern, the relation M ee = Y e Λ + Y e Λ + Y e Λ = 0 (45)is required by Eq.(4). Since Λ k > Y αk is real, Eq.(45) yields Y ek = 0. However,the vanishing Y ek yields M eµ = X k =1 Y ek Y µk Λ k = 0 ,M eτ = X k =1 Y ek Y τk Λ k = 0 , (46)as well as − × ×− − × , (47)and the one-zero-texture assumption should be violated. The G pattern is excludedin the scotogenic model. Similarly, the G and G patterns are also excluded. Thus,une 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Teruyuki Kitabayashi three patterns (G , G and G ) are impossible in the scotogenic model with the realYukawa matrix.To see that whether the G , G and G are consistent with observation or not,we performed numerical calculations with the following input values: • For neutrino sector, we fix the masses of the neutrinos with the best-fitvalues of the squared mass differences ∆ m ij ∆ m = 7 . × − eV , ∆ m = 2 . × − eV , (48)and vary the mixing angles in the following 3 σ region θ / ◦ = 31 . − . ,θ / ◦ = 38 . − . ,θ / ◦ = 7 . − . . (49) • For dark sector, we adopt the following standard criteria.
3, 40, 41
1) Thequartic coupling satisfies the relation | λ | ≪ N is darkmatter particle, we require M ≤ M , M , m . 3) The mass scale of newfields is a few TeV. We take1 × − ≤ λ ≤ × − , . ≤ r ≤ . , . ≤ r ≤ . , ≤ m ≤ , (50)where r k = M k m . (51)The G pattern is consistent with the observations. For example, the best-fitvalues of neutrino mixing angles θ / ◦ = 33 . , θ / ◦ = 41 . , θ / ◦ = 8 . , (52)with λ = 4 . × − , r = 0 . , r = 1 . , m = 3TeV , (53)yield Ω h = 0 . , Br( µ → eγ ) = 2 . × − , (54)for G , which are consistent with the observed energy density of the cold darkmatter component in the ΛCDM cosmological model by the Plank CollaborationΩ h = 0 . ± . and the measured upper limit of the branching ratio Br( µ → eγ ) ≤ . × − . Although the upper limits of the branching ratio of Br( τ → une 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Parametrization of the Yukawa matrix in the scotogenic model and single-zero textures of the neutrino mass matrix λ = 1 × -9 o ≤ α ≤ o Ω h α [deg] G4 0.010.101.0010.00 0 45 90 135 180 225 270 315 360 λ = 2.5 × -9 o ≤ α ≤ o Ω h α [deg] G40.010.101.0010.00 0 45 90 135 180 225 270 315 360 λ = 1 × -9 o ≤ α ≤ o Ω h α [deg] G4 0.010.101.0010.00 0 45 90 135 180 225 270 315 360 λ = 2.5 × -9 o ≤ α ≤ o Ω h α [deg] G4 Fig. 1. The dependence of the relic abundance of dark matter Ω h on the coupling λ and Ma-jorana CP phases α , α for Br( µ → eγ ) ≤ . × − in the G pattern. These panels showΩ h vs α (upper two panels) or α (lower two panels) for λ = 1 × − (left two panels) or λ = 2 . × − (right two panels). In all panels, other model parameters are fixed as r = 0 . r = 1 . m = 3 TeV. The horizontal line shows Ω h = 0 . µγ ) ≤ . × − and Br( τ → eγ ) ≤ . × − were also reported, we onlyaccount for Br( µ → eγ ) since it is the most stringent constraint.On the other hand, it turned out that the predicted values of Ω h and Br( µ → eγ ) for G seems to be unlikely with the observed data. Moreover, G patternis excluded from observation. The G and G patterns are not favorable for thescotogenic model with the real Yukawa matrix elements. For more detail, see Ref. Complex Yukawa matrix
In the previous subsection, all elements of the Yukawa matrix are real, there isno CP-violating source in the Yukawa sector, so that three patterns (G , G andG ) in the one-zero-textures are impossible in the scotogenic model. If we includeCP-violating phases, the results to be different. Some analysis of this topic will befound in this subsection.We performed numerical calculations with the following input values: • For neutrino sector, we fix the masses and mixing of the neutrinos withthe best-fit values in Eq.(48) and Eq.(52). The coefficients of the Yukawaune 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Teruyuki Kitabayashi = 0.5 0 o ≤ α ≤ o Ω h α [deg] G4 0.010.101.0010.00 0 45 90 135 180 225 270 315 360r = 0.9 0 o ≤ α ≤ o Ω h α [deg] G40.010.101.0010.00 0 45 90 135 180 225 270 315 360r = 0.5 0 o ≤ α ≤ o Ω h α [deg] G4 0.010.101.0010.00 0 45 90 135 180 225 270 315 360r = 0.9 0 o ≤ α ≤ o Ω h α [deg] G4 Fig. 2. The dependence of the relic abundance of dark matter Ω h on the mass ratio r andMajorana CP phases α , α for Br( µ → eγ ) ≤ . × − in the G pattern. These panels showΩ h vs α (upper two panels) or α (lower two panels) for r = 0 . r = 0 . λ = 1 × − , r = 1 . m = 3 TeV. The horizontal line shows Ω h = 0 . matrix elements and the mass eigenvalues are estimated to be a = − . − . e − iδ ,a = 0 . − . e − iδ ,a = 1 . − . e − iδ ,a = − . − . e − iδ ,a = 4 . e − iδ , a = 5 . e − iδ , (55)and b = 1 . , b = 3 . e iα ,b = 46 . e i ( α − δ ) . (56)The Dirac CP phase is fixed in the following best-fit values: δ = 261 ◦ (57)and Majorana CP phases are varied as0 ◦ ≤ α , α ≤ ◦ . (58)une 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Parametrization of the Yukawa matrix in the scotogenic model and single-zero textures of the neutrino mass matrix = 1.5 0 o ≤ α ≤ o Ω h α [deg] G4 0.010.101.0010.00 0 45 90 135 180 225 270 315 360r = 3.0 0 o ≤ α ≤ o Ω h α [deg] G40.010.101.0010.00 0 45 90 135 180 225 270 315 360r = 1.5 0 o ≤ α ≤ o Ω h α [deg] G4 0.010.101.0010.00 0 45 90 135 180 225 270 315 360r = 3.0 0 o ≤ α ≤ o Ω h α [deg] G4 Fig. 3. The dependence of the relic abundance of dark matter Ω h on the mass ratio r andMajorana CP phases α , α for Br( µ → eγ ) ≤ . × − in the G pattern. These panels showΩ h vs α (upper two panels) or α (lower two panels) for r = 1 . r = 3 . λ = 1 × − , r = 0 . m = 3 TeV. The horizontal line shows Ω h = 0 . • For dark sector, we adopt the same criteria in the previous subsection.For illustration, λ = 1 . × − , r = 0 . , r = 1 . , m = 3TeV ,δ = 261 ◦ , α = 34 . ◦ , α = 20 . ◦ , (59)yield Ω h = 0 . , Br( µ → eγ ) = 3 . × − , (60)for G , which are consistent with the observed energy density of the cold darkmatter component and the measured upper limit of the branching ratio of µ → eγ process.From more general parameter search, it turned out that G pattern is unfavor-able from observation. On the other hand, remaining five patterns of G , G , · · · ,G are consistent with observation. To see these results, first we show the resultsfrom a parameter search for G in FIG.1 - FIG.4. Figure 1 shows that the depen-dence of the relic abundance of dark matter Ω h on the coupling λ and MajoranaCP phases α , α for Br( µ → eγ ) ≤ . × − in the G pattern. These panelsune 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Teruyuki Kitabayashi = 2 TeV 0 o ≤ α ≤ o Ω h α [deg] G4 0.010.101.0010.00 0 45 90 135 180 225 270 315 360m = 4 TeV 0 o ≤ α ≤ o Ω h α [deg] G40.010.101.0010.00 0 45 90 135 180 225 270 315 360m = 2 TeV 0 o ≤ α ≤ o Ω h α [deg] G4 0.010.101.0010.00 0 45 90 135 180 225 270 315 360m = 4 TeV 0 o ≤ α ≤ o Ω h α [deg] G4 Fig. 4. The dependence of the relic abundance of dark matter Ω h on the scalar mass m andMajorana CP phases α , α for Br( µ → eγ ) ≤ . × − in the G pattern. These panelsshow Ω h vs α (upper two panels) or α (lower two panels) for m = 2TeV (left two panels) or m = 4TeV (right two panels). In all panels, other model parameters are fixed as λ = 1 × − , r = 0 . r = 1 .
5. The horizontal line shows Ω h = 0 . show Ω h vs α (upper two panels) or α (lower two panels) for λ = 1 × − (left two panels) or λ = 2 . × − (right two panels). In all panels, other modelparameters are fixed as r = 0 . r = 1 . m = 3 TeV. The horizontal lineshows Ω h = 0 .
12. FIGs.2, 3 and 4 are the same as FIG.1 but for the dependenceof Ω h on r , r and m . From FIG.1 - FIG.4, we see the existence of the allowedparameter set { α , α , λ, r , r , m } for the observed Ω h and Br( µ → eγ ) in G case. Similar results are obtained for G , G , G and G (the dependence of Ω h on α , α , λ , r , r or m is most clearly shown in the case of G , so that we chosethe case of G for illustration). Thus we conclude that the five patterns of G , G ,G , G and G are consistent with observations.On the contrary, G pattern is unfavorable with observations. Figure.5 showsthat the relic abundance of dark matter Ω h vs the Majorana phase α (upperpanel) or α (lower panel) in the G pattern for the parameter space in Eq.(50). Thehorizontal line shows Ω h = 0 .
12. The G pattern is excluded from observations ofthe neutrino oscillation (best-fit values), dark matter relic abundance and branchingratio of µ → eγ in the parameter space in Eq.(50). If we expand the model parameterspace and/or we allow 3 σ data of neutrino oscillation experiment instead of best-fitvalues, some points in G pattern might become consistent with observations. Evenune 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Parametrization of the Yukawa matrix in the scotogenic model and single-zero textures of the neutrino mass matrix λ = (0.1 - 5) × -9 r = 0.5 - 0.9r = 1.5 - 3.0m = 2 - 4 TeV 0 o ≤ α ≤ o Ω h α [deg] G1 0.101.00 0 45 90 135 180 225 270 315 360 λ = (0.1 - 5) × -9 r = 0.5 - 0.9r = 1.5 - 3.0m = 2 - 4 TeV 0 o ≤ α ≤ o Ω h α [deg] G1 Fig. 5. The relic abundance of dark matter Ω h vs the Majorana phase α (upper panel) or α (lower panel) for Br( µ → eγ ) ≤ . × − in the G pattern. The horizontal line showsΩ h = 0 . if it is correct, we can say that G pattern is unfavorable with observations.We also estimate the effective neutrino mass for the neutrinoless double betadecay. The neutrinoless double beta decay is allowed if neutrinos are massive Majo-rana particles. The half-life of the neutrinoless double beta decay is proportionalto the effective neutrino mass m ββ which depends on the Dirac and the Majo-rana CP phases. From numerical calculations, we obtain m ββ = O (0 . − .
1) eVfor Ω h = 0 . ± .
001 and Br( µ → eγ ) ≤ . × − in G , G , G , G , G ( m ββ = 0 for G ). The estimated magnitude of the effective Majorana neutrinomass from experiments is m ββ [eV] . . − . In future experiments, a desiredsensitivity of m ββ . .
01 eV will be reached and we may obtain some constraintson the parameters in the scotogenic model from the future neutrinoless double betadecay experiments.We would like to comment that the discrimination of realistic one-zero textureswith complex Yukawa matrix is much less powerful than the case of real couplings.This suggests that it may be useful to study two-zero textures.
44, 85–88
Indeed, forexample, one of the model parameters in Eq.(59), such as one of the Majoranaphases, will be predicted by using the relation of M eτ = M µµ = 0. We would like todiscuss about the interplay between the scotogenic model and the two-zero texturesof the flavor neutrino masses in the future work.
5. Summary
There are two main topics in this paper. As the first topic, we have proposed a newparametrization of the complex Yukawa matrix in the scotogenic model [Eq.(28)with Eq.(29)]. The new parametrization is compatible with the particle data groupparametrization of the neutrino sector. Some analytical expressions for the neutrinomasses in the scotogenic model with the new parametrization of the Yukawa matrixhave been obtained. Although there are many other ways to parametrize the Yukawamatrix, the way in this paper may be one of the useful method to consider theune 27, 2019 0:40 WSPC/INSTRUCTION FILE kitabayashi˙v2 Teruyuki Kitabayashi phenomenology of the scotogenic model with the standard PDG parametrization.As the second topic, we have considered some phenomenology of the socotogenicmodel with the one-zero-textures of neutrino flavor mass matrix. If the elements ofthe Yukawa matrix are real, one of the six patterns of the flavor neutrino massmatrix with single-zero element is favorable. On the other hand, if the Yukawamatrix is complex, the five patterns G , G , G , G and G are compatible withobservations of the neutrino oscillations, dark matter relic abundance and branchingratio of the µ → eγ process; however, G is unfavorable with observations.Finally, we would like to comment that the baryon asymmetry of the universe isclosely related to leptonic CP phases in the leptogenesis scenario. The scenariosof leptogenesis in the scotogenic model have been extensively studied in the litera-ture.
49, 51, 90–95
It seems that it is hard to realize ordinary thermal leptogenesis withflavor effects via the decay of the fermionic dark matter N i with a hierarchical massspectrum. In this paper, we assume that the lightest Majorana singlet fermion isthe dark matter with mass spectrum of M ≤ M < M . Thus, the vanilla leptoge-nesis scenario might not work properly for the model in this paper. This situationmay be changed if we employ some alternative mechanism, such as resonant lep-togenesis.
96, 97
Moreover, we know that in the bosonic dark matter scenario, thelightest neutral component in the scalar η is a dark matter, a successful low-scaleleptogenesis can be achieved. We would like to discuss the topics of the baryonasymmetry of the universe in the scotogenic model with the one-zero-textures ofthe neutrino flavor mass matrix as a separate work in the future.
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