USTC-ICTS-16-11
Alternative Schemes of Predicting Lepton Mixing Parameters from Discrete Flavor and CP Symmetry Jun-Nan Lu∗, Gui-Jun Ding†
arXiv:1610.05682v1 [hep-ph] 18 Oct 2016
Interdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
Abstract We suggest two alternative schemes to predict lepton mixing angles as well as CP violating phases from a discrete flavor symmetry group combined with CP symmetry. In the first scenario, the flavor and CP symmetry is broken to the residual groups of the structure Z2 × CP in the neutrino and charged lepton sectors. The resulting lepton mixing matrix depends on two free parameters θν and θl . This type of breaking pattern is extended to the quark sector. In the second scheme, an abelian subgroup contained in the flavor group is preserved by the charged lepton mass matrix and the neutrino mass matrix is invariant under a single remnant CP transformation, all lepton mixing parameter are determined in terms of three free parameters θ1,2,3 . We derive the most general criterion to determine whether two distinct residual symmetries lead to the same mixing pattern if the redefinition of the free parameters θν,l and θ1,2,3 is taken into account. We have studied the lepton mixing patterns arising from the flavor group S4 and CP symmetry which are subsequently broken to all of the possible residual symmetries discussed in this work.
∗ †
Email:
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1
Introduction
The neutrino oscillation experiments have made great progress in the last twenty years [1– 3]. It has been firmly established that neutrinos must be massive particles and different flavor eigenstates are mixed. The three lepton mixing angles θ12 , θ13 and θ23 as well as two mass squared difference ∆m221 and ∆m231 have been precisely measured [4–7]. However, we still don’t know the neutrino mass ordering (∆m231 > 0 or ∆m231 < 0) and the signal of CP violation in the lepton sector has not been observed. The preliminary T2K data favor a maximal Dirac CP violation phase δCP ' −π/2 [8], and the latest global fits of neutrino mixing parameters show a weak evidence for a negative Dirac phase −π < δCP < 0 [5–7]. The primary objectives of near future neutrino experiments are to determine the ordering of the neutrino masses and to measure the value of δCP . On the theoretical side, the origin of neutrino mass and lepton flavor mixing is still unknown although there has been lots of theoretical studies. Motivated by the observation that the simple tri-bimaximal mixing possibly originates from a A4 flavor group, non-abelian discrete flavor symmetry has been extensively exploited to explain the observed lepton mixing angles. Many other symmetries such as S4 , A5 , ∆(3n2 ) and ∆(6n2 ) etc have been considered over the years. Please see Refs. [9–13] for review on discrete flavor symmetry and its application in model building. A significant progress in recent years is the precise measurement of the reactor mixing angle θ13 [14–18]. The discovery of a somewhat large value of θ13 rules out the tri-bimaximal mixing patterns and many flavor models which predicted small or zero θ13 . Many approaches have been pursued to explain such a largish θ13 . Within the paradigm of the discrete flavor symmetry, model-independent scan of the lepton sector reveals that only large flavor symmetry groups (e.g. (Z18 × Z6 ) o S3 with the group id [648, 259]) can produce mixing patterns compatible with experimental data and the Dirac CP phase is generally trivial if the lepton mixing matrix is fully fixed by the symmetry alone [19–23]. In order to accommodate a non-zero θ13 and a nontrivial Dirac CP phase simultaneously, it is interesting to combine flavor symmetry with CP symmetry. This approach can generate a rich structure of mixing patterns which are in good agreement with the experimental data, and it allows us to predict all the mixing angles and CP phases in terms of a small number of input parameters [24–26]. From the bottom-up point of view, the generic neutrino and charged lepton mass matrices have both residual CP symmetry and residual flavor symmetry, and the residual flavor symmetry can be generated from the residual CP transformations [27– 29]. Hence it is natural to assume that the residual flavor and CP symmetry arise from a large flavor and CP symmetry group at high energy scale. In this approach, the CP symmetry nontrivially acts on the flavor space such that the so called consistency condition has to be fulfilled in order for the theory to be consistent [24,30–32]. There has been intense theoretical activity on flavor symmetry in combination with CP symmetry. Many flavor symmetry groups and their predictions for lepton mixing parameters have been studied such as A4 [33–37], S4 [24, 38–43], A5 [44–47], ∆(27) [48, 49], ∆(48) [50, 51] and ∆(96) [52] as well (1) as ∆(3n2 ) [53, 54], ∆(6n2 ) [53, 55, 56] and D9n,3n [57] group series for a generic integer n. Recently a comprehensive scan of leptonic mixing parameters which can be obtained from finite discrete groups of order less than 2000 and CP symmetry has been performed [58]. Moreover, the phenomenological implications of flavor and CP symmetry in neutrinoless double decay [37, 39, 43, 44, 56–59] and leptogenesis [58–60] have been investigated. It is remarkable that the residual CP symmetry provides a bridge between flavored leptogenesis and low energy leptonic CP violation. It is usually assumed that the residual flavor symmetry in the charged lepton is an abelian subgroup which can distinguish among the three generations, and the residual symmetry in the neutrino sector is a direct product of Z2 and CP . As a consequence, the lepton mix1
ing matrix turns out to depend on a single real parameter θ and all mixing parameters are strongly correlated with each other. In the present work, we shall discuss the other possible approaches to predict lepton mixing parameters from flavor and CP symmetry, and two scenarios would be considered. In the first one, the neutrino and charged lepton mass matrices are invariant under two distinct Z2 × CP subgroups. Consequently all mixing parameters including mixing angles and CP phases are predicted in terms of two real parameters θl and θν . In the second scenario, the postulated flavor symmetry is broken to a residual abelian subgroup with three or more elements in the charged lepton sector while a single residual CP transformation is preserved by the neutrino mass matrix, the PMNS mixing matrix would depend on three real parameters θ1,2,3 . As an example, we present a detailed analysis for the S4 flavor symmetry group and CP symmetry. All possible independent combinations of remnant symmetries and the predictions for lepton mixing parameters are studied analytically and numerically. The paper is organized as follows. In section 2 we study the symmetry breaking pattern in which a flavor symmetry combined with a CP symmetry is broken to Z2 × CP in both the neutrino and charged lepton sectors. The resulting consequence for the prediction of the lepton mixing matrix is discussed, and the technical steps in the derivation are explained. We derive the conditions under which two distinct residual symmetries give rise to the same mixing pattern. Moreover we analyze the independent mixing patterns which can be obtained from the popular flavor group S4 and CP in this scheme. In section 3 our approach is extended to the quark sector. In section 4 we explore another proposal in which the charged lepton and neutrino mass matrices are invariant under the action of a residual abelian subgroup and a single CP transformation respectively. Finally section 5 concludes this paper. Moreover, Appendix A contains the necessary group theory of S4 as well as its abelian subgroups. Appendix B gives the conditions under which two distinct residual symmetries of the structure Z2 × CP in both the up and down quark sectors lead to the same CKM mixing matrix in the case that the fixed element is neither 0 nor 1.
2
Lepton flavor mixing from residual symmetry Z2 ×CP in both charged lepton and neutrino sectors
In the widely studied direct and semidirect approaches [11–13], it is assumed that the neutrino mass matrix mν possesses residual symmetry Z2 ×Z2 and Z2 ×CP respectively, and the charged lepton mass matrix is invariant under an abelian subgroup contained in the flavor group. In this section, we shall be concerned with the scenario that the remnant symmetry preserved by both the neutrino and charged lepton mass matrices is of the structure Z2 ×CP . The three generations of left-handed leptons are assigned to a faithful irreducible triplet 3 of the flavor symmetry group.
2.1
General form of the PMNS matrix
We shall denote the residual Z2 flavor symmetry of the charged lepton sector as Z2gl ≡ {1, gl } with gl2 = 1, and the remnant CP transformation is Xl . In order for the theory to be consistent, the following consistency condition has to be fulfilled Xl ρ∗3 (gl )Xl−1 = ρ3 (gl ) ,
(2.1)
where ρ3 (gl ) denote the representation matrix of the element gl in the three dimensional representation 3. The charged lepton mass matrix m†l ml is invariant under the action of the 2
residual symmetry Z2gl × Xl , and it fulfills Xl† m†l ml Xl = (m†l ml )∗ , ρ†3 (gl )m†l ml ρ3 (gl )
=
m†l ml
.
(2.2a) (2.2b)
The unitary transformation Ul which diagonalizes the hermitian matrix m†l ml with Ul† m†l ml Ul = diag(m2e , m2µ , m2τ ) are strongly constrained by the postulated residual symmetry. In the following, we shall show how to determine Ul from ρ3 (gl ) and Xl . As the order of gl is 2, the eigenvalues of ρ3 (gl ) are (1, −1, −1) or (−1, 1, 1), we take the first case as an example without loss of generality. Assuming Σl1 is a diagonalization matrix of ρ(gl ) and it satisfies Σ†l1 ρ3 (gl )Σl1 = diag(1, −1, −1) ≡ ρˆ3 (gl ),
(2.3)
ρ3 (gl ) = Σl1 ρˆ3 (gl )Σ†l1
(2.4)
then we have The residual CP has to be consistent with the residual flavor symmetry, therefore the following consistency condition should be fulfilled [43, 57] Xl ρ∗3 (gl )Xl† = ρ3 (gl−1 ).
(2.5)
Inserting Eq. (2.4) into the above equation and considering gl2 = 1, we get Xl Σ∗l1 ρˆ3 (gl )ΣTl1 Xl† = Σl1 ρˆ3 (gl )Σ†l1 ,
(2.6)
(Σ†l1 Xl Σ∗l1 )ˆ ρ3 (gl )(ΣTl1 Xl† Σl1 ) = ρˆ3 (gl ) ,
(2.7)
(Σ†l1 Xl Σ∗l1 )ˆ ρ3 (gl ) = ρˆ3 (gl )(Σ†l1 Xl Σ∗l1 ) .
(2.8)
which leads to which means Therefore Σ†l1 Xl Σ∗l1 is a block diagonal and symmetric matrix and its most general form is given by iξ e 1 0 † ∗ Σl1 Xl Σl1 = (2.9) 0 ul2×2 where ξ1 is an arbitrary real number and ul2×2 is a two-dimensional symmetric unitary matrix. l l lT l We denote the Takagi factorization of ul2×2 as σ2×2 fulfilling ul2×2 = σ2×2 σ2×2 , where σ2×2 is † ∗ a two-dimensional unitary matrix. As a result, the matrix Σl1 Xl Σl1 can be written into iξ /2 iξ /2 e 1 0 e 1 0 † ∗ Σl1 Xl Σl1 = . (2.10) l lT 0 σ2×2 0 σ2×2 Then we can obtain the Takagi factorization of Xl as iξ /2 iξ /2 e 1 0 e 1 0 Xl = [Σl1 ][Σl1 ]T ≡ Σl ΣTl l l 0 σ2×2 0 σ2×2 with
Σl = Σl1
eiξ1 /2 0 l 0 σ2×2
(2.11)
.
(2.12)
It is straightforward to check that the remnant flavor transformation ρ3 (gl ) is diagonalized by Σl , Σ†l ρ3 (gl )Σl = diag(1, −1, −1). (2.13) 3
From Eq. (2.2a) we can obtain that the constraint on the unitary transformation Ul from the residual CP transformation Xl is Ul† Xl Ul∗ = diag(eiβe , eiβµ , eiβτ ) ≡ Q2l ,
(2.14)
where βe,µ,τ are arbitrary real parameters. Thus we have Ul† Σl ΣTl Ul∗ = Q2l ,
(2.15)
T T ∗ −1 (ΣTl Ul∗ Q−1 l ) (Σl Ul Ql ) = 1 .
(2.16)
which leads to is an orthogonal matrix, and it is also a unitary matrix. Hence the combination ΣTl Ul∗ Q−1 l is a real orthogonal matrix denoted by O3×3 . Then the unitary transTherefore ΣTl Ul∗ Q−1 l formation Ul takes the following form Ul = Σl O3×3 Q−1 l .
(2.17)
Furthermore, Eq. (2.2b) implies that Ul is also subject to the constraint of the residual flavor symmetry as follows, Ul† ρ3 (gl )Ul = Pl diag(1, −1, −1)PlT (2.18) where Pl is a generic permutation matrix, and it can take six possible forms 1, P12 , P13 , P23 , P23 P12 and P23 P13 with 010 001 100 P12 = 1 0 0 , P13 = 0 1 0 , P23 = 0 0 1 . (2.19) 001 100 010 Plugging the expression of Ul in Eq. (2.17) into Eq. (2.18), we obtain † PlT Ql O3×3 Σ†l ρ3 (gl )Σl O3×3 Q−1 l Pl = diag(1, −1, −1) .
Using Eq.(2.13) we have † −1 O3×3 Q−1 l Pl diag(1, −1, −1) O3×3 Ql Pl = diag(1, −1, −1)
(2.20)
(2.21)
Therefore the combination O3×3 Q−1 l Pl is a block diagonal unitary matrix, and it can be parameterized as iξ e 2 0 −1 O3×3 Ql Pl = , (2.22) l 0 v2×2 l where ξ2 is a real number and v2×2 is a two-dimensional unitary matrix. Thus we have 2iξ e 2 0 −1 −1 T T −2 (O3×3 Ql Pl ) (O3×3 Ql Pl ) = Pl Ql Pl = , (2.23) lT l v2×2 0 v2×2
which implies Hence
eiξ2 0 l 0 v2×2
eiξ2 0 l 0 v2×2
PlT Ql Pl
T
eiξ2 0 l 0 v2×2
PlT Ql Pl
= 1.
(2.24)
PlT Ql Pl is a block diagonal real orthogonal matrix, and it takes the form
eiξ2 0 l 0 v2×2
T PlT Ql Pl = S23 (θl ) ,
4
(2.25)
where S23 (θl ) is a rotation matrix with
1 0 0 S23 (θl ) ≡ 0 cos θl sin θl . 0 − sin θl cos θl
(2.26)
As a consequence, the unitary transformation Ul is fixed by the residual symmetry Z2 × CP to be T Ul = Σl S23 (θl )PlT Q−1 (2.27) l . Similarly the residual flavor symmetry of the neutrino mass matrix is denoted as Z2gν ≡ {1, gν } with gν2 = 1, the residual CP transformation is Xν , and CP should commute with Z2gν as well Xν ρ∗3 (gν )Xν−1 = ρ3 (gν ) . (2.28) The invariance of the neutrino mass matrix under the residual symmetry Z2gν × Xν requires ρT3 (gν )mν ρ3 (gν ) = mν ,
XνT mν Xν = m∗ν
(2.29)
Plugging UνT mν Uν = diag(m1 , m2 , m3 ) into this equation, we can derive the following constraints on the unitary transformation Uν , Uν† ρ3 (gν )Uν = diag(±1, ±1, ±1) , Uν† Xν Uν∗ = diag(±1, ±1, ±1) ≡ Q2ν ,
(2.30a) (2.30b)
where Qν is a diagonal and unitary matrix with non-vanishing entries equal to ±1 and ±i. Without loss of generality Qν can be parameterized as 1 0 0 Qν = 0 ik1 0 , (2.31) k2 0 0 i with k1,2 = 0, 1, 2, 3. Firstly we can diagonalize the residual flavor symmetry transformation ρ3 (gν ) by a unitary transformation Σν1 as Σ†ν1 ρ3 (gν )Σν1 = ρˆ3 (gν ) = diag(1, −1, −1)
(2.32)
The consistency condition of remnant symmetry is Xν ρ∗3 (gν )Xν† = ρ3 (gν−1 ) = ρ3 (gν ) ,
(2.33)
Xν Σ∗ν1 ρˆ∗3 (gν )ΣTν1 Xν† = Σν1 ρˆ3 (gν )Σ†ν1 .
(2.34)
(Σ†ν1 Xν Σ∗ν1 )ˆ ρ∗3 (gν )(Σ†ν1 Xν Σ∗ν1 )† = ρˆ3 (gν ) .
(2.35)
which leads to Thus we have Hence Σ†ν1 Xν Σ∗ν1 is a block diagonal matrix, and it is of the following form iζ e 1 0 † ∗ Σν1 Xν Σν1 = , 0 uν2×2
(2.36)
where ζ1 is an arbitrary real number and uν2×2 is a two-dimensional symmetric unitary matrix. ν† ν νT ν σ2×2 with σ2×2 σ2×2 = 1. Then we obtain uν2×2 can be factorized into the form uν2×2 = σ2×2 Σ†ν1 Xν Σ∗ν1 = Σν2 ΣTν2 , 5
(2.37)
where
Σν2 =
eiζ1 /2 0 ν 0 σ2×2
.
(2.38)
As a consequence, the Takagi factorization of the residual CP transformation Xν is given by Xν = Σν ΣTν .
(2.39)
with Σν = Σν1 Σν2 . It is easy to check that the residual flavor transformation ρ3 (gν ) is diagonalized by Σν as well, Σ†ν ρ3 (gν )Σν = Σ†ν2 Σ†ν1 ρ3 (gν )Σν1 Σν2 = Σ†ν2 diag(1, −1, −1)Σν2 = diag(1, −1, −1) .
(2.40)
Now we proceed to discuss the constraint on Uν from the remnant CP transformation. Substituting the relation Xν = Σν ΣTν into Eq. (2.30b), we get T Qν Uν† Σν Qν Uν† Σν = 1 . (2.41) This implies that Qν Uν† Σν is a real orthogonal matrix denoted as O3×3 . Therefore the unitary transformation Uν is of the form T Uν = Σν O3×3 Qν . (2.42) Subsequently we consider the constraint from the residual flavor symmetry given in Eq. (2.30a), Uν† ρ3 (gν )Uν = PνT diag(1, −1, −1)Pν ,
(2.43)
where Pν is a permutation matrix, since the neutrino masses are unconstrained in the present framework and the neutrino mass spectrum can be either normal hierarchy (NH) or inverted hierarchy (IH). Inserting Eq. (2.42) into Eq. (2.43), one finds † T −1 T T Q−1 ν O3×3 Σν ρ3 (gν )Σν O3×3 Qν = Qν O3×3 diag(1, −1, −1)O3×3 Qν = Pν diag(1, −1, −1)Pν . (2.44) which gives rise to T T diag(1, −1, −1) O3×3 Qν PνT = O3×3 Qν PνT diag(1, −1, −1) . (2.45) T Qν PνT is a block-diagonal unitary matrix, and we can parameterize it as Therefore O3×3 iζ e 2 0 T T O3×3 Qν Pν = , (2.46) ν 0 v2×2 ν where ζ2 is real and v2×2 is a two-dimensional unitary matrix. Both sides of this equation multiply with their transpose, we obtain 2iζ e 2 0 T T T T T 2 T O3×3 Qν Pν O3×3 Qν Pν = Pν Qν Pν = , (2.47) νT ν 0 v2×2 v2×2
which implies T iζ eiζ2 0 e 2 0 −1 T −1 T Pν Qν Pν Pν Qν Pν = 1 . (2.48) ν ν 0 v2×2 0 v2×2 0 T Pν Q−1 ν ν Pν is a block diagonal real orthogonal matrix, and it is of the
eiζ2 0 v2×2 following form Therefore
eiζ2 0 ν 0 v2×2
T Pν Q−1 ν Pν = S23 (θν ) ,
6
(2.49)
where θν is real. Consequently, the unitary transformation Uν is fixed to be Uν = Σν S23 (θν )Pν Qν .
(2.50)
The lepton mixing matrix UP M N S is a result of the mismatch between Ul and Uν . Hence we find UP M N S is of the form UP M N S = Ul† Uν = Ql Pl S23 (θl )Σ†l Σν S23 (θν )Pν Qν ,
(2.51)
where the phase matrix Ql can be absorbed by redefinition of the charged lepton fields. We see that the lepton mixing matrix depends on two free continuous parameters θl and θν , and one entry of the PMNS matrix is fixed to be some constant value by the postulated residual symmetry. Notice that S23 (θ + π) = S23 (θ)diag(1, −1, −1) = diag(1, −1, −1)S23 (θ) where the diagonal matrix can be absorbed into the matrices Ql and Qν , consequently the fundamental interval of the parameters θl and θν are [0, π). g0 g0 If two pairs of residual subgroups {Z2 l × Xl0 , Z2 ν × Xν0 } and {Z2gl × Xl , Z2gν × Xν } are related by a similarity transformation hgl h−1 = gl0 , hgν h−1 = gν0 ,
ρ3 (h)Xl ρ3 (h)T = Xl0 , ρ3 (h)Xν ρ3 (h)T = Xν0
(2.52)
with h ∈ S4 , then the unitary transformations of the changed lepton and neutrino fields are related by Ul0 = ρ3 (h)Ul , Uν0 = ρ3 (h)Uν . (2.53) Therefore the same result for the PMNS matrix would be obtained.
2.2
The criterion for the equivalence of two mixing patterns
In some cases, two distinct residual symmetries lead to the same mixing pattern, if a possible shift in the continuous free parameters θl and θν is taken into account. Then we shall call these two mixing patterns are equivalent. In this section, we shall derive the criterion to determine whether two resulting mixing patterns are equivalent or not. In our approach, the lepton mixing matrices derived from two generic residual symmetries take the form UP M N S = Ql Pl S23 (θl )Σ†l Σν S23 (θν )Pν Qν , UP0 M N S
=
0 0 0 0 Q0l Pl0 S23 (θl0 )Σ0† l Σν S23 (θν )Pν Qν
(2.54) .
(2.55)
Obviously the fixed element has to be equal if the two mixing patterns are equivalent, and without loss of generality we assume it is the (11) entry of the PMNS matrix. As a result, the permutation matrices Pl , Pν , Pl0 and Pν0 can only be 1 and P23 . Because the following identities P23 S23 (θl ) = diag(1, −1, 1)S23 (θl − π/2),
S23 (θν )P23 = S23 (θν + π/2)diag(1, −1, 1) (2.56)
are satisfied, and the diagonal matrix can be absorbed into the matrices Ql and Qν , we could choose Pl = Pν = Pl0 = Pν0 = 1. For any given values of θl , θν and the matrices Ql , Pl , Qν , Pν , if the corresponding solutions of θl0 , θν0 as well as Q0l , Pl0 , Q0ν , Pν0 can be found such that the equality UP M N S = UP0 M N S is fulfilled, these two mixing patterns would be equivalent, i.e., Ql S23 (θl )U S23 (θν )Qν = Q0l S23 (θl0 )U 0 S23 (θν0 )Q0ν , (2.57) 7
0 where U ≡ Σ†l Σν and U 0 ≡ Σ0† l Σν . Then we have
QL S23 (θl )U S23 (θν )QN = S23 (θl0 )U 0 S23 (θν0 ) ,
(2.58)
0† where QL = Q0† l Ql is a generic diagonal phase matrix, and QN = Qν Qν is also diagonal with entries ±1 and ±i. The matrices on both sides of Eq. (2.58) multiplying with their transpose leads to T T T QL S23 (θl )U S23 (θν )Q2N S23 (θν )U T S23 (θl )QL = S23 (θl0 )U 0 U 0T S23 (θl0 ) . (2.59)
Subsequently taking trace, we obtain T T Tr S23 (θl )Q2L S23 (θl )U S23 (θν )Q2N S23 (θν )U T = Tr U 0 U 0T .
(2.60)
Since the right-handed side of this equality is a constant and it doesn’t depend on θl and θν , the phase matrices QL and QN should be of the form iδ e 1 0 0 η1 0 0 QL = 0 eiδ2 0 , QN = 0 η2 0 , (2.61) iδ2 0 0 k1 e 0 0 k2 η2 where k1,2 = ±1, δ1,2 are real parameters, and η1,2 are ±1 and ±i with eiδ1 η1 = 1. Thus from Eq. (2.58) we can derive QL U QN = S23 (θl00 )U 0 S23 (θν00 ) , (2.62) with θl00 = θl0 − k1 θl , θν00 = θν0 − k2 θν .
(2.63)
Once the residual symmetries are specified, the unitary matrices U and U 0 can be determined by following the procedures listed in section 2.1. Generically U and U 0 can be written as a1 a2 a3 b1 b 2 b3 U = a4 a5 a6 , U 0 = b 4 b 5 b 6 . (2.64) a7 a8 a9 b7 b 8 b9 A necessary condition for the equivalence of UP M N S and UP0 M N S is a1 = b1 which can not be 0 or 1 in order to be compatible with experimental data. Firstly let’s consider a special case with 1 0 0 1 0 0 (2.65) QL = 0 eiδ 0 , QN = 0 1 0 . 0 0 1 0 0 eiδ Solving the equation Eq. (2.62) for the variables θl00 , θν00 and δ, we can obtain the condition for the existence of solution. • b22 + b23 6= 0, b24 + b27 6= 0 In this case, the solutions for θl00 , θν00 and δ are given by a4 b7 − a7 b4 iδ a24 + a27 a4 b4 + a7 b7 iδ 00 −2iδ e , sin θ = e , e = , l b24 + b27 b24 + b27 b24 + b27 a2 b2 + a3 b3 a3 b 2 − a2 b 3 cos θν00 = , sin θν00 = . 2 2 b2 + b3 b22 + b23 cos θl00 =
(2.66)
Since θl00 , θν00 and δ are real parameters, ai and bi should be subject to the following constraints (a4 b4 + a7 b7 )(a∗4 b∗7 − a∗7 b∗4 ) ∈ R, a24 + a27 = b24 + b27 , 8
(a2 b2 + a3 b3 )(a∗2 b∗3 − a∗3 b∗2 ) ∈ R,
a2 + a23 = b22 + b23 .
(2.67)
Inserting Eq. (2.66) into Eq. (2.62), we find that the equivalence of these two mixing patterns requires (xb5 + yb6 )z + (xb8 + yb9 )w (xb6 − yb5 )z + (xb9 − yb8 )w , a = , 6 (b22 + b23 )(b24 + b27 ) (b22 + b23 )(b24 + b27 ) (xb8 + yb9 )z − (xb5 + yb6 )w (xb9 − yb8 )z − (xb6 − yb5 )w a8 = , a9 = , 2 2 2 2 (b2 + a3 )(b4 + b7 ) (b22 + b23 )(b24 + b27 )
a5 =
(2.68)
with x = a2 b 2 + a3 b 3 , y = a2 b 3 − a3 b 2 , z = a4 b 4 + a7 b 7 , w = a4 b 7 − a7 b 4 .
(2.69)
• b22 + b23 = 0, b24 + b27 6= 0 This case requires b3 = is1 b2 ,
a3 = is1 a2 ,
with s1 = ±1 .
(2.70)
The parameters θl00 , θν00 and δ are determined to be a4 b4 + a7 b7 iδ a4 b7 − a7 b4 iδ a24 + a27 00 −2iδ e , sin θ = e , e = , l b24 + b27 b24 + b27 b24 + b27 cos θν00 =
