Zee Mass Matrix and Bi-Maximal Neutrino Mixing

15 Pages·0.17 MB·

by  C. Jarlskog

Lund-Mph-98/10
Zee Mass Matrix and Bi-Maximal Neutrino Mixing
Cecilia Jarlskog a, Masahisa Matsuda a
9 ∗ †
9
9
Solveig Skadhauge a and Morimitsu Tanimoto b
1 ‡ §

n
a a Department of Mathematical Physics, LTH, Lund University, S-22100, Lund, Sweden
J b Science Education Laboratory, Ehime University, 790-8577 Matsuyama, JAPAN

4
1

2 ABSTRACT
v
2
8 We investigate neutrino masses and mixings within the framework of the Zee mass
2
matrix, with three lepton flavors. It is shown that the bi-maximal solution is the only
2
1 possibility toreconcile atmospheric andsolarneutrino data, withinthisansatz. Weobtain
8
two almost degenerate neutrinos, which are mixtures of all three neutrino flavors, with
9
/ heavy masses ∆m2 . The predicted mass of the lightest neutrino, which should
h atm
≃
p consist mostly of qν and ν , is ∆m2/(2 ∆m2 ).
- µ τ atm
p ≃ ⊙
q
e
h PACS: 14.60.Pq, 14.60.Lm, 12.60.Fr
:
v
i
X
r
a
E-mail address: [email protected]
∗
Permanent address:Department of Physics and Astronomy, Aichi University of Education,
†
Kariya, Aichi 448, Japan. E-mail address:[email protected]
E-mail address: [email protected]
‡
E-mail address: [email protected]
§
In the past few years stronger experimental signals, than ever before, have been seen
for neutrino oscillations. Recent atmospheric neutrino experiments [1,2] indicate oscilla-
tions among neutrino flavors with large mixing angles [3]. The simplest solution to the
solar neutrino deficit problem, observed by the Super-Kamiokande experiment [4] as well
as other experiments [5], is again neutrino oscillations. Indeed, the field of neutrino oscil-
lations is expected to enter into a new era, with the start of long baseline (LBL) neutrino
experiments [6–8]. These experiments will hopefully solve the present neutrino anomalies.
Since the CHOOZ experiment [9] excludes oscillation of ν ν with a large mixing
µ e
→
anglefor∆m2 9 10 4eV2,largemixingbetweenν andν isthesimplest interpretation
− µ τ
≥ ×
of the atmospheric ν deficit. The difference between the quark-mixing and lepton-mixing
µ
matrices is striking. The Cabibbo-Kobayashi-Maskawa mixing matrix V [10] in the
CKM
quarksector isdescribed bysmallmixing anglesamongdifferent flavorsbut intheleptonic
sector [11] at least one large mixing angle seems to be needed. It will be important to
understand why these patterns are so different.
Another issue, to be understood, is why the neutrino masses are so small? The most
popular answer to the latter question is given by the see-saw mechanism [12] which intro-
duces heavy right-handed Majorana neutrinos with masses of the order 1010 1016GeV.
−
This attractive model has been extensively studied in the literature. However, it is im-
portant to consider also other possible scenarios with small neutrino masses, specially
extensions of the standard model (SM) at a low energy scale. The Zee model [13] is such
an alternative and has been studied in the literatures for almost twenty years [14–18]. In
this paper, we will discuss the present status of the Zee mass matrix, in the light of recent
experimental results.
In the Zee model [13] neutrino masses are generated by radiative corrections, and
hence the model may provide an explanation of the smallness of neutrino masses. In this
model, the following Lagrangian is added to the SM;
L = fll′ΨlLiσ2(Ψl′L)ch− +µΦT1iσ2Φ2h− +h.c.
l,l′=e,µ,τ
X
= 2f [ν (µ )c e (ν )c]h +2f [ν (τ )c e (ν )c]h
eµ eL L L µL − eτ eL L L τL −
− −
+2f [ν (τ )c µ (ν )c]h +µ(Φ+Φ0 Φ0Φ+)h +h.c. , (1)
µτ µL L − L τL − 1 2 − 1 2 −
where Ψ = (ν ,l)T, Φ = (Φ+,Φ0)T, i = 1,2. The Higgs potential is omitted here. The
lL l L i i i
charged Zee boson, h , is a singlet under SU(2) . We need at least two Higgs doublets
± L
in order to make the Zee mechanism viable, since the antisymmetric coupling to the Zee
boson is the cause of B L violation, and hence of Majorana masses. Note that only Φ
1
−
couples to leptons, as in the SM. The mass matrix, generated by radiative correction at
one loop level [13–18], is given by
0 m m
eµ eτ
m 0 m , (2)
 eµ µτ 
m m 0
eτ µτ
 
 
where
µv
m = f (m2 m2) 2F(M2,M2)
eµ eµ µ − e v 1 2
1
1
µv
m = f (m2 m2) 2F(M2,M2) (3)
eτ eτ τ − e v 1 2
1
µv
m = f (m2 m2) 2F(M2,M2)
µτ µτ τ − µ v 1 2
1
and
1 1 M2
F(M2,M2) = ln 1 . (4)
1 2 16π2M2 M2 M2
1 − 2 2
The parameter v is the vacuum expectation value of the neutral component of the
1(2)
Higgs doublet Φ . M and M are the masses of the physical particles defined by the
1(2) 1 2
fields
H+ = h+cosφ Φ+sinφ , H+ = h+sinφ+Φ+cosφ , (5)
1 − 2
where Φ+ is the charged Higgs boson that would have been a physical particle in the
absence of the h+. Finally, the mixing angle φ is defined by
4√2µM
W
tan2φ = . (6)
g (M2 M2)2 (4√2g 1µM )2
1 − 2 − − W
q
Due to the antisymmetry of the coupling matrix, fll′ = fl′l, the Zee model requires all
−
diagonal elements in Eq. (2) to vanish at one loop level. Small corrections will however be
obtained at higher orders in perturbation theory. Hereafter, we refer to the above matrix,
with vanishing diagonal elements, as the Zee mass matrix.
The parameters m ,m ,m in the Zee mass matrix are not described by the eigen-
eµ eτ µτ
values m (i = 1,2,3) due to the traceless property of the matrix, leaving only two
i
independent observable parameters. In the same way it is impossible to represent the
mixing matrix U by using m . In literature [14–18], it has been assumed that there is
i
a hierarchy such as m m ,m and the neutrino masses and mixings are discussed
eµ eτ µτ
≪
under such assumptions. This hierarchy is natural if the coupling constants fll′ are of the
same order of magnitude. However, we would like to explore all possibilities of masses and
mixings in the Zee model by relaxing this assumption. Instead we use recent atmospheric
neutrino data [1,2] as input to determine patterns in the Zee mass matrix that are viable.
The neutrino mass matrix M is generally, for Majorana particles, constructed by
ν
M = UMdiag.UT (7)
ν
due to its symmetric property M = MT. The mixing matrix U, called the MNS mixing
ν ν
matrix [11], is defined in the basis where the mass matrix of charged leptons is diagonal,
with masses m . Furthermore,
e,µ,τ
m 0 0
1
Mdiag. = 0 m 0 . (8)
 2 
0 0 m
3
 
 
2
For Majorana neutrinos there are three phases in the matrix U and this is generally given
by
c c s c eiβ s e i(δ α)
1 3 1 3 3 − −
U = s c c s s eiδ (c c s s s eiδ)eiβ s c eiα , (9)
 1 2 1 2 3 1 2 1 2 3 2 3 
− − −
s s c c s eiδ ( c s s c s eiδ)eiβ c c eiα
1 2 1 2 3 1 2 1 2 3 2 3
 − − − 
 
where c cosθ and s sinθ . The Zee mass matrix exhibits no CP violation [19] and
i i i i
≡ ≡
we will therefore neglect the phases in our investigation. The diagonal elements in M
ν
are given by
(1,1) m c2c2 +m s2c2 +m s2 ,
1 1 3 2 1 3 3 3
(2,2) m (s c +c s s )2 +m (c c s s s )2 +m s2c2 , (10)
1 1 2 1 2 3 2 1 2 − 1 2 3 3 2 3
(3,3) m (s s c c s )2 +m (c s +s c s )2 +m c2c2 .
1 1 2 − 1 2 3 2 1 2 1 2 3 3 2 3
These should be zero (in general small) in the Zee model and we arrive at the following
relations:
cos2θ tan2θ
1 3
m = − m , m = m m . (11)
2 −sin2θ tan2θ 1 3 − 1 − 2
1 3
−
The second equality is obvious by the traceless property of the Zee mass matrix. Two
of three masses have the same sign and the remaining one has opposite sign, which
implies that one of the fields has opposite CP parity as compared to the other two. The
dependence of mass eigenvalues on θ and θ is shown in Fig.1. Inserting (11) into (10)
1 3
gives the relation
1
cos2θ cos2θ cos2θ = sin2θ sin2θ (3cos2θ 2)sinθ . (12)
1 2 3 1 2 3 3
2 −
This equation means that the three mixing angles are not independent. For typical values
of θ ,θ ,θ the structure of the mixing matrix is discussed later, using this equation.
1 2 3
Before entering into the analysis of the Zee mass matrix, we give a short survey of
recent neutrino experiments. Our approach is to assume that oscillations account for the
solar and atmospheric neutrino data, thus pinning down two mass squared differences,
which is the maximal number of mass differences in the model we are investigating. If
the results of LSND [20] would be confirmed by KARMEN [21] or any other experiment,
the model examined in this paper would no longer be relevant. The deficit of ν in recent
µ
Super Kamiokande data [1], is interpreted as oscillation of ν ν with nearly maximal
µ τ
→
mixing angle in a two flavor analysis [3,22]. These results yield
∆m2 (0.5 6) 10 3eV2 , sin22θ > 0.82 (90%C.L) . (13)
atm ≃ − × − atm
Further, the deficit in the solar neutrino experiments suggests the following best-fit solu-
tions [23] as
(1) MSW small angle solution; ∆m2 5.4 10 6eV2 ,sin22θ 6 10 3 ,
− −
⊙ ≃ × ⊙ ≃ ×
3
π/2
m = m
1 2
m =−m
1 2
(a) m = m
1 3
3π/8 m =−m
1 3
m = m
2 3
m =−m
2 3
(b)
3 π/4
θ
(c)
(d)
π/8
(e) (f)
0
0 π/8 π/4 3π/8 π/2
θ
1
FIG. 1. The dependence of mass eigenvalues on the mixing angles θ and θ . The lines
1 3
represents special relations among the three masses as denoted in the figure. In the right hand
side the domains are running oppositely, starting with (a) in the bottom and ending with (f) in
the top. In region (a) we have m > m > m , (b) m > m > m , (c) m > m > m ,
1 2 3 1 3 2 3 1 2
− − − − −
(d) m > m > m , (e) m > m > m , and (f) m > m > m . Here the sign of m is
3 2 1 2 3 1 2 1 3 1
− − −
taken to be positive.
(2) MSW large angle solution; ∆m2 1.8 10 5eV2 ,sin22θ 0.76 ,
−
(3) ”just-so” vacuum solution; ∆m⊙2 ≃ 6.5×10 11eV2 ,sin22θ⊙ ≃ 0.75 .
−
⊙ ≃ × ⊙ ≃
It is noted that the large angle MSW solution seems to be excluded by the simulta-
neous fits to all the available data [23]. As there are still theoretical uncertainties about
this [24], we will keep this possibility in our considerations. The combined results of at-
mospheric and solar neutrino experiments suggest that there exist two hierarchical mass
squared differences ∆m2 ∆m2. Further, the component U in Eq.(9) should be
atm ≫ e3
small, as suggested by the CHOOZ⊙experiment [9]. The value sin22θ < 0.18 im-
CHOOZ
plies U < 0.22 for ∆m2 9 10 4eV2 [25]. For the MSW small angle solution, case
e3 −
| | ≥ ×
(1), together with the atmospheric results, a possible mixing matrix has the form
1 ǫ ǫ
1 2
U ǫ c s , (14)
1  3 
≃
ǫ s c
4
 − 
 
where c s 1/√2. This mixing matrix might be realized in the case;
≃ ≃
∆m2 ∆m2 ∆m2 , ∆m2 ∆m2 , (15)
| 32| ≃ | 31| ≃ atm | 21| ≃ ⊙
which demands m m m or m m m (i,j = 1,2 or 2,1) or
3 i j i j 3
≫ ≥ ≥ ≫
4
∆m2 ∆m2 ∆m2 , ∆m2 ∆m2 , (16)
| 32| ≃ | 21| ≃ atm | 31| ≃ ⊙
implying m m m or m m m (i,j = 1,3 or 3,1). Another solution for
2 i j i j 2
≫ ≥ ≥ ≫
MSW small angle mixing is
ǫ 1 ǫ
1 2
′
U c ǫ s . (17)
1 ≃  3 
s ǫ c
4
− 
 
This type of mixing suggests
∆m2 ∆m2 ∆m2 , ∆m2 ∆m2 (18)
| 32| ≃ | 31| ≃ atm | 21| ≃
⊙
or
∆m2 ∆m2 ∆m2 , ∆m2 ∆m2 (19)
| 31| ≃ | 21| ≃ atm | 32| ≃
⊙
and we obtain the solution m m m or m m m (i,j = 2,3 or 3,2) for the
1 i j i j 1
≫ ≥ ≥ ≫
latter case.
For the “just-so” and MSW large angle solutions a typical mixing matrix is
c s ǫ c s ǫ
1 1
s c ǫ or ǫ ǫ 1 . (20)
 2  2 3 
−
ǫ ǫ 1 s c ǫ
3 4 4
  − 
   
Neither of these is compatible with the maximal mixing pattern of the atmospheric ν
µ
→
ν oscillation, and must be discarded for this reason. We are left with one possibility;
τ
c s ǫ
′ ′
U cs cc s (21)
3  ′ ′ 
≃ −
ss sc c
′ ′
 − 
 
for interpreting large angle solar neutrino solutions, where c s c s 1/√2. In
′ ′
≃ ≃ ≃ ≃
the limit ǫ = 0 this is known as the bi-maximal mixing matrix [26]. Nearly bi-maximal
mixing is discussed in Ref. [27]. Taking
∆m2 ∆m2 ∆m2 , ∆m2 ∆m2 (22)
| 32| ≃ | 31| ≃ atm | 21| ≃
⊙
with m m m or m m m , (i,j = 1,2 or 2,1), it has been shown that
3 i j i j 3
≫ ≥ ≥ ≫
the mixing matrix U is consistent with vacuum solution for both solar and atmospheric
3
neutrino anomalies within the experimental uncertainties [28]. The MSW large angle
solution could also be accommodated here, but then the allowed parameter space is rather
small.
Furthermore thedegenerate case m m m is another possibility forallthe above
1 2 3
≃ ≃
cases. Using the definition;
δ δ +ǫ
m = m , m = m (1 ) , m = m (1 ), (23)
a 0 b 0 c 0
− 2 − 2
5
we can assign a,b,c to each of 1,2,3 according to the mass relation obtained above. Only
in this degenerate case is it possible to have a mass m = (1)eV for the neutrinos, as is
0
O
suggestedbythehotdarkmatterargument. Settingm = 1eVtheparametersδ andǫtake
0
the values (10 3 10 2) and (10 5) respectively to reproduce ∆m2 ∆m2 ∆m2
O − − − O − ab ≃ ac ≃ atm
and ∆m2 ∆m2.
bc ≃ ⊙
We now return to the analysis of the Zee mass matrix. In accordance with the data
of atmospheric neutrino experiments we require θ π/4. Now Eq.(12) implies that we
2
≃
have the following possibilities; θ 0 or θ 0 or θ arctan 1/2. The latter case will
3 1 3
≃ ≃ ≃
be commented on later. We will only consider the case of θ1 q0 together with θ3 0,
≃ ≃
due to the experimental constraints on U . To begin with we concentrate on the solution
e3
θ 0.
3
≃
If we take the extreme limit θ = π/4 and θ = 0, the mixing matrix becomes
2 3
c s 0
1 1
U s1 c1 1 . (24)
atm ≃ −√2 √2 √2 
s1 c1 1
 √2 −√2 √2 
 
It is noted that the extreme case does not change the qualitative structure of the model
and is consistent with combined SK and CHOOZ data in a three flavor analyses [3,22].
In this limit we obtain the constraints
m c2 +m s2 = 0, m s2 +m c2 +m = 0 (25)
1 1 2 1 1 1 2 1 3
from Eq.(10). Then the parameters satisfy the constraint
m
tan2θ = 1 > 0 , (26)
1
−m
2
and we arrive at
0 c1s1(m m ) c1s1(m m )
− √2 1 − 2 √2 1 − 2
Mν = −c√1s21(m1 −m2) 0 −12(m1s21 +m2c21 −m3)
c1s1(m m ) 1(m s2 +m c2 m ) 0
 √2 1 − 2 −2 1 1 2 1 − 3 
 0 m1m2 m1m2 
| | | |
± 2 ∓ 2
=  m1m2 q0 mq m , (27)
± | 2 | − 1 − 2
∓q|m12m2| −m1 −m2 0 
 q 
where the upper (lower) sign corresponds to the case m < 0 (m > 0). The mixing
1 1
matrix becomes
|m2| |m1| 0
m1+m2 m1+m2
 r| | | | r| | | | 
U = m1 m2 1 . (28)
| | | |
− 2(m1+m2) 2(m1+m2) √2 
 r | | | | r | | | | 
 
 m1 m2 1 
 2(m|1 +|m2) − 2(m|1+|m2) √2 
 r | | | | r | | | | 
 
6
As the two mass squared differences are hierarchical we have to stay close to the lines
in Fig.(1). Due to the symmetry we only have to survey the left part of the parameter
region, i.e θ [0,π/4]. Changing from the left to right side merely corresponds to an
1
∈
interchange of m m and has no physical effect. We now consider the three possible
1 2
↔
values for θ .
1
(I). Thecasewithθ 0. TheZeemassmatrixthenrequires thefollowing massrelations
1
≃
to hold;
m m m , ∆m2 ∆m2 = ∆m2 , ∆m2 = ∆m2 . (29)
| 2| ≃ | 3| ≫ | 1| 21 ≃ 31 atm 23
⊙
In this case we get the mixing matrix;
1 ǫ 0
1
UZee = ǫ1 1 1 , (30)
1 −√2 √2 √2 
ǫ1 1 1
 √2 −√2 √2 
 
with ǫ m /m . This corresponds to the small angle MSW solution U (Eq.(14)).
1 1 2 1
≃ | |
Neverthelesqs due to the mass relations in Eq.(29) the predicted probability of ν ν
µ τ
→
oscillation,
∆m2 L
P(ν ν ) = 4 U U U U sin2 ij
µ → τ − µi τ∗j µ∗j τi 4E
i>j
X
∆m2 L
4U2 U2 sin2 atm (31)
≃ µ1 τ1 4E
∆m2 L
= ǫ4sin2 atm ,
1 4E
is tiny in contradiction with the SK experiment. Here and in the following we neglect
terms with ∆m2 in P(ν ν ). Thus, the Zee mass matrix and large angle solution are
µ τ
→
not compatible,⊙for θ 0. Setting θ π/2 would correspond to m m compared
1 1 1 2
≃ ≃ ↔
to the current case. It also causes an interchange of the two first columns in UZee, which
1
yields the matrix in Eq.(17). However this must be discarded for the same reason.
(II). Here we take θ arctan1/√2 and obtain
1
≃
m m m , ∆m2 ∆m2 = ∆m2 , ∆m2 = ∆m2 . (32)
| 1| ≃ | 3| ≫ | 2| 12 ≃ 32 atm 13
⊙
This case requires the mixing matrix to be
2 1 0
3 3
UZee =  q 1 q1 1 . (33)
2 − 6 3 2
 q1 q 1 q1 
 
 6 − 3 2 
 q q q 
Giving the angles:
8 4
sin22θ = 4U2 (1 U2 ) , sin22θ = 4U2 U2 . (34)
CHOOZ e2 − e2 ≃ 9 atm µ2 τ2 ≃ 9
7
Hence this is also incompatible with experiments.
(III). In this third case θ π/4, whereby
1
≃
m m m , ∆m2 ∆m2 = ∆m2 , ∆m2 = ∆m2 . (35)
| 1| ≃ | 2| ≫ | 3| 13 ≃ 23 atm 12
⊙
The mixing matrix is
1 1 0
√2 √2
UZee =  1 1 1 . (36)
3 −2 2 √2
1 1 1
 
 2 −2 √2 
 
This corresponds to the solution given by U in Eq.(21). The oscillation probability reads
3
∆m2 L
P(ν ν ) sin2 atm , (37)
µ τ
→ ≃ 4E
in good agreement with experiments. Therefore we have a unique solution compatible
with large angle in ν ν within the ansatz of the Zee mass matrix. Due to the
µ τ
→
traceless property this implies
∆m2
m m ∆m2 , m , (38)
1 2 atm 3 ⊙
| | ≃ | | ≃ | | ≃ 2 ∆m2
q atm
q
This is the case of “pseudo-Dirac” since m m . The probabilities for other oscillation
1 2
≃ −
processes are as follows:
∆m2L
P(ν ν ) 1 sin2
e e ⊙
→ ≃ − 4E
1 ∆m2L
P(ν ν ) P(ν ν ) sin2 . (39)
e µ e τ ⊙
→ ≃ → ≃ 2 4E
Thus the solar neutrinos are converted into an equal amounts of ν and ν .
µ τ
The mixing patterns UZee corresponds to the small angle MSW solution, and the
1
matrix UZee corresponds to the large angle MSW or ”just-so” solution. It is interesting
3
to notice that the solution of three degenerate masses with (1)eV is not allowed due to
O
the relation m = m m and Eq.(26). This model requires naturally a hierarchical
3 1 2
− −
structure for mass matrix in the case of θ = π/4 and θ = 0. In conclusion only the
2 3
bi-maximal solution, given in Eq.(36), is feasible, within the framework of the Zee mass
matrix, whereas the solutions like UZee and UZee are not allowed.
1 2
Above, we have analyzed the Zee mass matrix at the limit θ = 0. It is also impor-
3
tant to discuss the case of nonzero θ in order to obtain three flavor angles and to have
3
predictions for future experiments. We parameterize this as
π π
θ = +δ , θ = +δ , θ = δ (40)
1 1 2 2 3 3
4 4
with δ (i = 1,2,3) 1 and we will neglect terms of order δ2. The traceless property
i ≪ i
of the Zee mass matrix requires the relation δ 8δ δ , by using Eq.(12), and yields the
3 1 2
≃
approximate mixing matrix
8
1 δ1c 1+δ1c δ
√−2 3 √2 3 3
UZee ≃ −1+δ1−2δ2+δ3 1−δ1−2δ2−δ3 1√+δ22c3 , (41)
1+δ1+δ2 δ3 1 δ1+δ2+δ3 1 δ2c
 2 − − − 2 √−2 3
 
where c = 1 δ2 and the mass matrix is described as
3 − 3
q
0 1 2δ1 δ2 1 2δ1+δ2
− −√2− −√2
Mν ≃ −1−2√δ12−δ2 0 −4δ1 m1 . (42)
1 2δ1+δ2 4δ 0
 −√2 − 1 
 
The ratio of lightest mass to heavy mass is approximately given by δ as shown in
1
Eqs.(27,42). Which restrictions do the present experiments give for the parameters
δ ,δ ,δ ? Taking Eq.(13) and the lower limit for large angle solution of solar neutrino as
1 2 3
sin22θ > 0.65 we obtain the following constraints;
⊙
(1 2δ2)(1 2δ2)
sin22θ 4 U 2 U 2 − 2 − 3 0.82 ,
atm µ3 τ3
≃ | | | | ≃ 2 ≥
(1 2δ2)(1 2δ2)
sin22θ 4 U 2 U 2 − 1 − 3 0.65 . (43)
e1 e2
⊙ ≃ | | | | ≃ 2 ≥
These inequalities with δ 8 δ δ leads to
3 1 2
| | ≃ | |
1 0.82(0.65)
δ2 1 . (44)
1(2) ≤ 128δ2  − 1 2δ2 
2(1) − 2(1)
 
We obtain the upper limit for δ to be
3
δ 0.28 . (45)
3
| | ≤
If the data will be improved as sin22θ > 0.95, sin22θ > 0.95 we get the upper limit
atm
⊙
δ < 0.1. A better estimate can nevertheless be deduced by noticing the following.
3
| |
The eigenvalues of M are m , (1 4δ )m ,4δ m . In order to adjust the mass squared
ν 1 1 1 1 1
− −
difference of ∆m2 using Eq(38), δ should be chosen as (10 3) for MSW or (10 8)
1 − −
O O
for “just-so”. Th⊙is demands also δ to be tiny due to the relation δ 8δ δ . The value
3 3 1 2
≃
δ (10 3) or (10 8) is well within the present experimental upper limit. In this case
3 − −
≃ O O
we expect no ν ν oscillations at the LBL experiments [6–8] since,
µ e
→
∆m2 L
P(ν ν ) 2δ2sin2 atm 0 . (46)
µ → e ≃ 3 4E ≃
We now briefly discuss the remaining solution: θ = arctan(1/√2)+δ , m m
3 3 1 2
≃ ≃
m /2, when requiring θ = π/4+δ in the Zee mass matrix. Here we take θ = π/4+δ
3 2 2 1 1
−
to obtain the “maximal” case [29]. The mixing matrix reads
1 δ1 1+δ1 1
−
√3 √3 √3
U  1+δ1−δ2 1−δ1+δ2 1−δ1−δ2 1+δ1+δ2 1+δ2  (47)
≃ − 2 − 2√3 2 − 2√3 √3
1+δ1+δ2 1 δ1 δ2 1 δ1+δ2 1+δ1 δ2 1 δ2
 2 − −2√−3 − − 2 − 2√−3 √−3 
 
9