A Model of Neutrino Mass Matrix With δ=−π/2 and θ 23 =π/4
aa r X i v : . [ h e p - ph ] N ov Model of Neutrino Mass Matrix With δ = − π/ and θ = π/ Xiao-Gang He , , ∗ INPAC, SKLPPC and Department of Physics,Shanghai Jiao Tong University, Shanghai, China CTS, CASTS and Department of Physics,National Taiwan University, Taipei, Taiwanand National Center for Theoretical Sciences and PhysicsDepartment of National Tsing Hua University, Hsinchu, Taiwan (Dated: May 19, 2018)
Abstract
Experimental data have provided stringent constraints on neutrino mixing parameters. In the standardparameterization the mixing angle θ is close to π/
4. There are also evidences show that the CP violatingphase is close to − π/
2. We study neutrino mass matrix reconstructed using this information and findseveral interesting properties. We show that a theoretical model based on the A symmetry naturallypredicts δ = − π/ θ = π/ µ − τ exchange and CP conjugate symmetry limit. In this case CP violation solely comesfrom the complex group theoretical Clebsh-Gordan coefficients. The model also predicts | V e | = 1 / √ δ and θ can be significantly deviateaway from the symmetry values − π/ π/
4, respectively. But | V e | = 1 / √ ∗ [email protected] V P MNS matrix[1] are not always small[2–4]. In the standard parameterization[2, 5] for three neutrino mix-ing commonly used[3, 4], the mixing angle θ is close to π/ θ is large, θ is relatively small butaway from zero, and also s c is close to 1 / √
3. Since the mixing angle θ is non-zero, the famoustri-bimaximal mixing[6] is ruled out. There are now evidences show that the CP violating phase δ is close to − π/
2. This also implies that the tri-bimaximal mixing is in trouble since it predicts δ = 0. The phase δ is sometimes referred as Dirac phase which shows up in neutrino oscillations.If neutrinos are Majorana particles, there are also new CP violating Majorana phases α i . Thereare many discussions about implications for data available emphasizing the particular values for | δ | = π/ θ = π/ | δ | is π/
2. This is, strictly speaking, an incorrectstatement because that the value of the Dirac phase is parametrization dependent. For example,even the absolute value of the Dirac phase is π/ π/ δ = − π/ θ = π/
4. In the basis where the charged lepton mass matrix isalready diaganolized, the neutrino mass matrix defined by the term giving neutrino masses in theLagrangian (1 / ν L m ν ν cL , has the following form m ν = V P MNS ˆ m ν V TP MNS , (1)where ˆ m ν = diag ( m , m , m ) with m i = | m i | exp ( iα i ). Here we have put Majorana phase informa-tion in the neutrino masses. The standard form for V P MNS is given by V P MNS = 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 , (2)where c ij and s ij are cos θ ij and sin θ ij , respectively. With δ = − π/ θ = π/ m ν has thefollowing form[8, 10] m ν = a c + iβ − ( c − iβ ) c + iβ d + iγ b − ( c − iβ ) b d − iγ , (3)where a = m c c + m s c − m s , b = − (cid:0) m ( s + c s ) + m ( c + s s ) − m c (cid:1) ,c = − √ m − m ) s c c , d = 12 (cid:0) m ( s − c s ) + m ( c − s s ) + m c (cid:1) ,β = 1 √ s c (cid:0) m c + m s + m (cid:1) , γ = − ( m − m ) s c s . (4)Note that in the most general case, because non-zero Majorana phases, the parameters a , b , c , d , β and γ are all complex. 2he above matrix has a high level regularity pattern implying some underlying symmetry may beat work to produce it. Searching an underlying theory guided by symmetry principle may achievethis. Before doing this, however, it is worthwhile to understand more about the mass matrix ineq.(3). An immediate question one may ask is that if, in general, the neutrino mass matrix ineq.(3) always predicts δ = − π/ θ = π/
4. The answer is negative. If δ = π/ θ = π/ β and γ need to bemultiplied by a “-” sign. Therefore without further information given, a general mass matrix inthe form given by eq.(3) can give δ = ± π/ θ = π/
4. Whether they predict + π/ − π/ δ and θ must take ± π/ π/
4, respectively, neither. Thiscan be understood by studying the following quantity m ν m † ν = V P MNS ˆ m ν ˆ m † ν V † P MNS . (5)The general form for neutrino mass in eq.(3) will give the “12” and “13” entries A , of m ν m † ν as A + A = − i aβ ∗ + cγ ∗ − βd ∗ − βb ∗ )= − ( | m | − | m | ) s c c ( c − s ) − ( | m | c + | m | s − | m | ) s c ( c + s ) e − iδ ,A − A = 2( ac ∗ + cd ∗ − cb ∗ + βγ ∗ )= − ( | m | − | m | ) s c c ( c + s )+( | m | c + | m | s − | m | ) s c ( c − s ) e − iδ . (6)If the parameters in the set P : { a , b , c , β , γ } , are complex, the above equations can find solutionsfor other values of θ and δ . Therefore the general neutrino mass matrix form does not imply that δ and θ must be ± π/ π/
4. If, however, the parameters in the set P are all real, as long assin δ = 0, one must have s = c and δ = ± π/ m i are real (the Majorana phases arezero or π ). In this case the neutrino mass matrix can be rewritten as m ν = A C − C ∗ C D ∗ B − C ∗ B D , (7)with A = a , B = b , C = c + iβ , and D = d − iγ . The most general m ν can be written as[8] m ν = e ip e ip
00 0 e ip A C − C ∗ C D ∗ B − C ∗ B D e ip e ip
00 0 e ip , (8)where the phases p i are arbitrary.All neutrino mass matrices which can be written in the above form, will predict δ = ± π/ θ = π/ p i to obtainforms of m ν for convenience of analysis. For example the “-” sign for the “13” and “31” entries canbe removed by choosing p = p = 0 and p = π , the resultant matrix can be written in a morefamiliar forms m ν = A C C ∗ C D ∗ ˜ BC ∗ ˜ B D , (9)3here ˜ B = − B .The simplicity of the above mass matrix may serve as a good starting point to understand thepossible underlying theory. If this has something to do with reality, one should not stay at the purephenomenological level for analysis, but go further to study whether there are theoretical modelswhich can obtained such a neutrino mass matrix in some consistently way. Several attempts formodel buildings have been made[8, 9]. It has been shown in ref. [7] by Grimus and Lavoura thatthe above form of mass matrix is symmetric under a transformation of e → e , µ − τ exchange witha CP conjugation. We will refer this as the Grimus-Lavoura symmetry (GLS). In this work westart with a simple model proposed earlier based on A symmetry[13] to realize the tri-bimaximalneutrino mixing, and then modify it to allow a non-zero θ to find the conditions for having theGLS limit for neutrino mass matrix with δ = − π/ θ = π/ s c = V e = 1 / √ A is serving as a family symmetry[13]. The Higgs sector is enlarged to have threeHiggs fields, Φ = (Φ , Φ , Φ ) (SM doublet), φ (SM doublet) and χ = ( χ , χ , χ ) (SM singlet).Under the A , Φ and χ both transform as 3, and φ as 1. Three right-handed SM singlet neutrinos ν R = ( ν R , ν R , ν R ) are introduced allowing seesaw mechanism to be in effective. The standardleft-handed leptons l L = ( l L , l L , l L ), and standard right-handed charged leptons ( l R , l R , l R ), and ν R transform as a 3 , (1 , ′′ , ′ ) and 3, respectively. We refer the readers for more details on A group properties to Refs.[8, 13, 14]. The Lagrangian responsible for the lepton mass matrix is L = λ e (¯ l L ˜Φ) l R + λ µ (¯ l L ˜Φ) ′ l R + λ τ (¯ l L ˜Φ) ′′ l R + H.C. + λ ν (¯ l L ν R ) φ + m (¯ ν R ν CR ) + λ χ (¯ ν R ν CR ) χ, (10)where (¯ l L ˜Φ) l R = (¯ l L ˜Φ + ¯ l L Φ + ¯ l L ˜Φ ) l R , (¯ l L ˜Φ) ′ l R = (¯ l L ˜Φ + ω ¯ l L Φ + ω ¯ l L ˜Φ ) l R , (11)(¯ l L ˜Φ) ′′ l R = (¯ l L ˜Φ + ω ¯ l L Φ + ω ¯ l L ˜Φ ) l R . Here ω = exp ( i π/
3) and ω = exp ( i π/
3) are the C-G coefficients of the A group products.If the vev structure is of the form < Φ , , > = v Φ , < χ , > = 0, < χ > = v χ , and < φ > = v φ ,one would obtain the charged lepton mass term as (cid:0) ¯ l L ¯ l L ¯ l L (cid:1) U l √ λ e v Φ √ λ µ v Φ
00 0 √ λ τ v Φ l R l R l R , U l = 1 √ ω ω ω ω . (12)From the above, we can identify the charged lepton mass to be m i = √ λ i v Φ . The neutrino massmatrix has the seesaw form with M = (cid:18) M D M TD M R (cid:19) , M R = m m χ m m χ m , (13)4here M D = Diag (1 , , λ ν v φ , and m χ = λ χ v χ . From this one obtains the light neutrino massmatrix M ν of the form given by M ν = − M D M − R M D = w x y x z , (14)where w = z = − ( λ ν v φ ) m/ ( m − m χ ), x = ( λ ν v φ ) m χ / ( m − m χ ) and y = − ( λ ν v φ ) /m .The above model leads to the tri-bimaximal mixing which predicts θ = 0. It had been thefocus of A symmetry studies for a few years[13, 15, 16]. But it is now ruled out because a non-zero θ has been measured. In this scheme, in order to obtain the tri-bimaximal mixing, the neutrinomass matrix with “11” and “33” entries to be equal is crucial. It has been pointed out[13] that amore natural form of vev structure will lead to the “33” entry in the neutrino mass matrix to bedeviate from the “11” entry which leads to a non-zero θ . To achieve this, for our purpose here,we will introduce two scalars S ′ and S ′′ which are SM singlet but transform as 1 ′ and 1 ′′ under A . This results in two new terms for M R in the Lagrangian Y S ′ (¯ ν R ν CR ) ′′ S ′ + Y S ′′ (¯ ν R ν CR ) ′ S ′′ + H.C. (15)After S ′ , ′′ develops a non-zero vev, v S ′ ,S ′′ , we have M R = m m χ m m χ m , (16)where m = m + Y S ′ v S ′ + Y S ′′ v S ′′ , m = m + ω Y S ′ v S ′ + ωY S ′′ v S ′′ and m = m + ωY S ′ v S ′ + ω Y S ′′ v S ′′ .The resulting light neutrino mass matrix M ν no longer has w = z , but has w = − λ ν v φ m / ( m m − m χ ) , z = − λ ν m / ( m m − m χ ) , (17)and x and y are changed to x = λ ν v φ m χ / ( m m − m χ ) , y = − λ ν v φ /m . (18)In the basis where the charged lepton mass matrix is diagonalized, the neutrino mass matrixbecomes m ν = U † l M ν U ∗ l = 13 w + 2 x + y + z w − ω x + ω y + ωz w − ωx + ωy + ω zw − ω x + ω y + ωz w + 2 ωx + ωy + ω z w − x + y + zw − ωx + ωy + ω z w − x + y + z w + 2 ω x + ω y + ωz . (19)Inserting ω = exp ( i π/
3) in the above, m ν can be transformed into the form in eq.(3) by redefineright-handed charged leptons. The parameters in the set P A : { w, x, y, z } are in general complexwhich will not always have δ = − π/ θ = π/
4. One needs to work in the GLS limit whichcan be realized if the parameters in the set P A are all real. In this case the complexity of the massmatrix is purely due to the A group theoretical C-G coefficients ω and ω . This is a case whereCP violation is caused by C-G coefficients providing a concrete example of intrinsic CP violation.Before we analysis the general features of the neutrino mass matrix with complex parametersin the set P A , we would like to analysis the constraints on the model parameters to have the GLS5imit, that is, to have w, x, y, z to be real. The complexity of the parameters can appear in theYukawa couplings, in the vevs, and also in places where ω i appear in m i . To make the Yukawacouplings and scalar vevs real, one can require the model Lagrangian to satisfy a generalized CPsymmetry under which( l L , l L , l L ) → (( l L ) CP , ( l L ) CP , ( l L ) CP ) , ( ν R , ν R , ν R ) → (( ν R ) CP , ( ν R ) CP , ( ν R ) CP ) , (Φ , Φ , Φ ) → (Φ † , Φ † , Φ † ) , ( χ , χ , χ ) → ( χ † , χ † , χ † ) , ( S ′ , S ′′ ) → ( S † ′ , S † ′′ ) , (20)and all other fields transform the same as those under the usual CP symmetry. Here the superscript CP in the above indicates that the fields are the usual CP transformed fields.The above transformation properties will transform relevant terms into their complex conjugateones. Requiring the Lagrangian to be invariant under the above transformation dictates the Yukawacouplings to be real. The same requirement will dictates the scalar potential to forbid spontaneousCP violation and vevs to be real. One, however, notices that the parameters m , are in generalcomplex even if the Yukawa couplings and the vevs of the scalar fields are made real because of theappearance of ω i . To make them real to reach GLS limit, it is therefore required that Im ( ω Y S ′ v S ′ + ωY S ′′ v S ′′ ) = Im ( ωY S ′ v S ′ + ω Y S ′′ v S ′′ ) = 0 . (21)The above can be achieved by the absent of the scalar fields S ′ , ′′ in the theory or set Y S ′ v S ′ = Y S ′′ v S ′′ . If the vev structure of χ is fixed as given previously, absence of S ′ , ′′ will not have aphenomenologically acceptable mass matrix. Therefore, we will take the later possibility as exampleof GLS limit case to show some detailed features. In this case M ν can be diagonalized by V ν as thefollowing M ν = V ν ˆ m ν V Tν , V ν = c − s s c , (22)where s = sin θ and c = cos θ . One obtains the mixing matrix to be V P MNS = U † l V ν = 1 √ c + s c − sc + ωs ω ωc − sc + ω s ω ω c − s , (23)Normalizing the above mixing matrix to the standard parametrization in eq.(2), one obtains s = 1 p cs ) , s = 1 √ , s = (1 − cs ) / √ . (24)Here we have normalized c ij and s ij to be all positive. The neutrino eigen-masses are all real, butin general they can take positive or negative values depending on the values of w , x , y and z . Notethat the absolute values of elements in the second column of V P MNS are all 1 / √ δ = − π/ δ = + π/
2. An easy way of doing thisis to study the Jarlskog invariant quantity[18] J = Im ( V e V ∗ e V ∗ µ V µ ). Eqs.(2) and (23) give J = c s c s c s sinδ = − √ c − s ) , (25)6hich leads to δ = π × (cid:26) − , if c > s , +1 , if s > c . (26)Note that J is not zero implying CP violation which is caused by the complexity of C-G coeffi-cients. Eq. (23) can be transformed into the standard parameterization by multiplying the V P MNS on the right and left by diagonal matrices P r = diag (1 , , i ) and P l = diag (1 , ( ω c − s ) / | ω c − s | , ( ωc − s ) / | ωc − s | ), respectively. P l does not have physical effect because it can be absorbed byredefinition of right-handed charged leptons. The physical effects of P r is to change the sign of m .Let us now compare experimental data with the model predictions for the mixing angles and CPviolating phase. There are several global fits of neutrino data[3, 4]. The latest fit gives the centralvalues, 1 σ errors and the 2 σ ranges as the following[3] δ/π s s / − s N H . +0 . − . . ± .
016 2 . ± .
12 0 . +0 . − . σ region 0 . ∼ . . ∼ .
357 2 . ∼ .
50 0 . ∼ . IH . ± .
31 0 . ± .
016 2 . ± .
12 0 . +0 . − . σ region 0 . ∼ . . ∼ . . ∼ .
357 2 . ∼ .
52 0 . ∼ .
621 (27)Here
N H and IH indicate neutrino mass hierarchy patterns of normal hierarchy and invertedhierarchy, respectively. In the model above, adjusting the values, w , x , y and z , both NH and IHmass patterns can be obtained. There is strong hint that the Dirac phase should be close to 3 π/ − π/ c > s . The value − π/ σ range. Although for NH case δ isoutside of 1 σ range, there no problem with 2 σ range. For s , the model predicts s = 0 .
5. Thisvalue is outside of 1 σ range for both the NH and IH cases. However, they are, again, in agreementwith data within 2 σ .In the model s = (1 − cs ) / / √ s to fix cs = 0 . ± .
018 to predict s = 0 . ± .
004 for both NH and IH cases. This is inagreement with data within 1 σ . Note that V e = ( s c ) = 1 /
3. It agrees with data within 1 σ .It is remarkable that neutrino mixing matrix in this model with just one free parameter can be inreasonable agreement with data. This may be a hint that it is the form for mixing matrix, at leastas the lowest order approximation, that a underlying theory is producing.If w , x , y and z are allowed to be complex, the GLS is explicitly broken, there are modificationsto the mixing angles. There is additional source for CP violation other than the intrinsic onefrom complexity of C-G coefficient, and also the mixing angles will be modified. The eigen-masseswill contain Majorana phases. Detailed analysis of how to diagonalize the mass matrix has beendiscussed in Ref.[12]. In general this model does not always predicts δ = ± π/ θ = π/
4. Themixing matrix can be, in general, written as V P MNS = U † l V ρ V ν = 1 √ c + se iρ ce iρ − sc + ωse iρ ω ωce iρ − sc + ω se iρ ω ω ce iρ − s , (28)where V ρ is a diagonal matrix diag (1 , , e iρ ) with tan ρ = Im ( xw ∗ + x ∗ z ) /Re ( xw ∗ + x ∗ z ). It isinteresting that the phase ρ does not show up in J which is still − ( c − s ) / √
3. This implies that7P violation related to neutrino oscillation is still purely due to intrinsic CP violation. The mixingangles and the Dirac phase δ are all modified with s = 1 √ cs cos ρ ) / , s = (1 + cs cos ρ + √ cs sin ρ ) / √ cs cos ρ ) , s = (1 − cs cos ρ ) / √ , (29)and sin δ = (1 + 4 c s sin ρ ( c − s ) ) ) − / (1 − c s sin ρ (1 + cs cos ρ ) ) − / × (cid:26) − , if c > s , +1 , if s > c . (30)In this case, the new parameter ρ can be used to improve agreement of the model with data. Inboth NH and IH cases, δ and s can be brought into agreement with data at 1 σ level. To see howthis can be done, as an example, we take the largest value of cs so that s takes its lower 1 σ allowedvalue, and then varying cos ρ to obtain the upper 1 σ allowed value. This fixes cs and cos ρ to be0.468 and 0.992, respectively. With these values, s and δ are determined to: 0 .
534 and 1 . π ,respectively. These values are in agreement with data at 1 σ level. When more precise experimentaldata become available, the model with complex model parameters can be distinguished from thatwith the parameters are all real and other models.In summary we have shown that neutrino mass matrix reconstructed with δ = − π/ θ = π/ A symmetry naturally realize the GLS limit and predicts such a neutrino mixing pattern togetherwith the prediction | V e | = 1 / √
3. In this model, CP violation can be solely come from the complexgroup theoretical C-G coefficients if the neutrino Majorana phases are zero or π . This model fitsexperimental data very well and can be taken as the lowest order neutrino mass matrix for futuretheoretical model buildings. If there are additional source of CP violation other than those intrinsi-cally existed in the C-G coefficients, the CP violating phase δ and the mixing angle θ can be awayfrom − π/ π/
4. The models discussed can fit data within 1 σ . Future improved experimentaldata will be able to further test the model and provide more hints for the underlying theory ofneutrino mixing. Acknowledgments
The work was supported in part by MOE Academic Excellent Program (Grant No: 102R891505)and MOST of ROC, and in part by NSFC(Grant No:11175115) and Shanghai Science and Tech-nology Commission (Grant No: 11DZ2260700) of PRC. [1] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. , 870(1962); B. Pontecorvo, Sov. Phys.JTEP , 984(1968).[2] K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014).[3] D. V. Forero, M. Tortola and J. W. F. Valle, Phys. Rev. D , no. 9, 093006 (2014)[4] F. Capozzi, G. L. Fogli, E. Lisi, A. Marrone, D. Montanino and A. Palazzo, Phys. Rev. D , 093018(2014); M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, JHEP , 052 (2014).[5] L.L. Chau and W.Y. Keung, Phys. Rev. Lett. , 1802(1984).
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