Table Of ContentIFT-02/02
Neutrino bimaximal mixing
∗
and unitary deformation of fermion universality
2 Woj
ie
h Królikowski
0
0
Institute of Theoreti
al Physi
s, Warsaw University
2
n Ho»a 69, PL(cid:21)00(cid:21)681 Warszawa, Poland
a
J
1
Abstra
t
2
1
An e(cid:27)e
tive texture is presented for six Majorana neutrinos, three a
tive and three
v
6 6 3 3
6
(
onventional) sterile, based on a × mass matrix whose × Dira
omponent (i.e.,
8
1
a
tive(cid:21)sterile
omponent) is
onje
tured to get a hierar
hi
al fermion universal form,
1
3 3
0
similar to the previously
onstru
ted × mass matri
es for
harged leptons as well as
2
0
for up and down quarks. However, for neutrinos this form be
omes unitarily deformed
/
h 3 3
p by the a
tion of bimaximal mixing, spe
i(cid:28)
in their
ase. The × lefthanded and
- 6 6
p
righthanded
omponents (i.e., a
tive-a
tive and sterile-sterile
omponents) of the ×
e
h
mass matrix are diagonal with degenerate entries of opposite sign. They dominate over
:
v 3 3 m = m
1 4
i the × Dira
omponent. In su
h a texture the neutrino masses are − ≃
X
m = m m = m ∆m2 ∆m2 ∆m2
r 2 − 5 ≃ 3 − 6 with the mass-squared di(cid:27)eren
es 21 ≪ 32 ≃ 31
a
∆m2 = ∆m2 = ∆m2 = 0
41 52 63
and . The last equality implies in our texture the absen
e
of os
illations for three (
onventional) sterile neutrinos. Thus, these neutrinos are here
e(cid:27)e
tively de
oupled, what is realized evidently in another way than through the popular
seesaw me
hanism. There remain the os
illations of three a
tive neutrinos, getting the
form as for bimaximal mixing, but with the mass spe
trum following from our texture.
PACS numbers: 12.15.Ff , 14.60.Pq , 12.15.Hh .
January 2002
∗
Supported in part by the Polish State Committee for S
ienti(cid:28)
Resear
h (KBN), grant 5 P03B 119
20 (2001(cid:21)2002).
ν(D) = ν +ν (α =
α αL αR
1. Introdu
tion. As is well known, three Dira
neutrinos are
e, µ, τ)
, while three Majorana a
tive neutrinos and three Majorana (
onventional) sterile
ν(a) = ν +(ν )c ν(s) = ν +(ν )c (α = e, µ, τ)
α αL αL α αR αR
neutrinos be
ome and , respe
tively.
The neutrino mass term in the Lagrangian gets generi
ally the form
1 M(L) M(D) ν(a)
= (ν(a), ν(s)) αβ αβ β .
−Lmass 2 α α M(D)∗ M(R) ν(s) (1)
αβ βα αβ β
X
(L) (R)
M M
αβ αβ
If and are not all zero, then in nature there are realized six Majorana neutrino
ν ν (i = 1,2,3,4,5,6)
i i
mass(cid:28)elds orstates | i
onne
tedwithsixMajorananeutrino(cid:29)avor
ν ν (α = e, µ, τ , e , µ , τ )
α α s s s
(cid:28)elds or states | i through the unitary transformation
ν = U ν or ν = U ν ,
α αi i | αi α∗i| ii (2)
i i
X X
ν ν(a) ν ν(s) α = e, µ, τ
where we passed to the notation α ≡ α and αs ≡ α for . Of
ourse,
ν(a) = ν ν(a) = (ν )c ν(s) = ν ν(s) = (ν )c α = e, µ, τ
αL αL αR αL αR αR αL αR
, and , for . Thus, the neutrino
6 6 M = (M ) (α,β = e, µ, τ , e , µ , τ )
αβ s s s
× mass matrix is of the form
M(L) M(D)
M = .
M(D)† M(R) ! (3)
6 6 U = (U ) (i = 1,2,3,4,5,6)
αi
The neutrino × mixing matrix appearing in Eqs. (2) is,
6 6
at the same time, the unitary × diagonalizing matrix,
U MU = M diag(m , m , m , m , m , m ) ,
† d 1 2 3 4 5 6
≡ (4)
3 3
if the representation is used, where the
harged-lepton × mass matrixis diagonal. This
will be assumed hen
eforth.
2. Model of neutrino texture. In the present paper we study the model of neutrino
3 3
texture, where the × submatri
es in Eq. (3) are
1 0 0
M(L) = m0 0 1 0 = M(R),
−
0 0 1
1 1 0 tan2θ 0 0
√2 √2 14
M(D) = m0 −21 12 √12 0 tan2θ25 0 (5)
1 1 1 0 0 tan2θ
2 −2 √2 36
1
0
m > 0 tan2θ (ij = 14, 25, 36)
ij
with beingamasss
aleand denotingthreedimensionless
M(D) M
parameters. Noti
e that in the Dira
omponent of the neutrino mass matrix
there appears the familiar matrix of bimaximal mixing that deforms unitarily a diagonal
hierar
hi
al stru
ture (
f. Eq. (10) later on).
U M
One
an show that the unitary diagonalizingmatrix for the mass matrix de(cid:28)ned
in Eqs. (3) and (5) gets the form
1 0 1 U(3) 0(3) 0 C(3) S(3)
U = UU , U = , U = − ,
0(3) 1(3) ! S(3) C(3) ! (6)
where
1 1 0 1 0 0
√2 √2
U(3) = −21 12 √12 , 1(3) = 0 1 0 ,
1 1 1 0 0 1
2 −2 √2
c 0 0 s 0 0
14 14
C(3) = 0 c 0 , S(3) = 0 s 0
25 25
(7)
0 0 c 0 0 s
36 36
s = sinθ c = cosθ
ij ij ij ij
with and , while the neutrino mass spe
trum is
0
m = m 1+tan22θ
i,j ij
± (8)
q
and, equivalently,
an be des
ribed by the equalities
c2 s2 m = m0
ij − ij i,j ± (9)
(cid:16) (cid:17)
(ij = 14, 25, 36)
. The easiest way to prove this theorem isto start withthe diagonalizing
U M
matrix given in Eqs. (6) and (7), and then to
onstru
t the mass matrix de(cid:28)ned
M = U m U
αβ i αi i β∗i
in Eqs. (3) and (5) by making use of the formula , where the mass
P
spe
trum (8) or (9) is to be taken into a
ount.
c =
12
From Eqs. (6) and (7) we
an see that in our texture the mixing angles give
1/√2 = s c = 1/√2 = s c = 1 s = 0 c , s (ij = 14, 25, 36)
12 23 23 13 13 ij ij
, and , , whilst
s = 0
13
are to be determined from the experiment. Evidently, these values, in parti
ular ,
may be a subje
t of
orre
tion.
3 3 M
Note that the Dira
×
omponent of the neutrino mass matrix transformed
1
U U
unitarily by means of the fa
tor of the diaginalizing matrix be
omes diagonal and
may get a hierar
hi
al stru
ture,
2
tan2θ 0 0
U1 †MU1 (D) = U(3)†M(D) = m0 0 14 tan2θ25 0 ,
(10)
(cid:18) (cid:19) 0 0 tan2θ
36
similar to the Dira
-type mass matri
es for
harged leptons as well as for up and down
1
U
quarks, all dominated by their diagonal parts. The transforming fa
tor given in Eq.
3 3 U(3)
(6) works e(cid:27)e
tively thanks to its × submatrix that is just the familiarbimaximal
mixing matrix [1℄, spe
i(cid:28)
for neutrinos, des
ribing satisfa
torily the observed os
illations
ν ν
e µ
of solar 's and atmospheri
's.
3 3
Spe
i(cid:28)
ally, the Dira
×
omponent of the neutrino mass matrix, when the bimax-
imal mixing
hara
teristi
for neutrinos is transformed out unitarily, may be
onje
tured
in a fermion universal form that was shown to work very well for the mass matrix of
harged leptons [2℄ and neatly for mass matri
es of up and down quarks [3℄ (obviously,
in those three
ases of
harged fundamental fermions there exist only Dira
-type mass
matri
es). Then, for neutrinos we get [4℄
µε 2α 0
1 1 (D) 1
U†MU = 2α 4µ(80+ε)/9 8√3α ,
29 (11)
(cid:18) (cid:19) 0 8√3α 24µ(624+ε)/25
µ > 0 α > 0 ε > 0
where , and are some neutrino parameters. Sin
e already for
ε(e) = 0.172329 ε 0
harged leptons is small [2℄, we will put for neutrinos → . We will
α/µ
also
onje
ture that for neutrinos is negligible, as for
harged leptons the small
2
α(e)/µ(e) = 0.023+0.029 m = mexp = 1777.03+0.30
0.025 τ τ 0.26
− [2℄ gives the predi
tion − MeV [5℄,
(cid:16) m (cid:17)= mexp m = mexp α(e)/µ(e) 2 = 0
e e µ µ
when and are used as inputs, while with the
(cid:16) (cid:17)
m = 1776.80
τ
predi
tion be
omes MeV. In su
h a
ase, from Eqs. (10) and (11) we
an
on
lude that
µ µ 4 80 µ 24 624
0 0 0
mtan2θ = ε 0, mtan2θ = · = 1.23µ, mtan2θ = · = 20.7µ
14 25 36
29 → 29 9 29 25
(12)
in Eqs. (5), (8) and (10). Hen
e, from Eqs. (8)
0 0 0
m = m, m = m2 +1.50µ2, m = m2 +427µ2 .
1,4 2,5 3,6
± ±r ±r (13)
3
3. Neutrino os
illations. A
epting the formulae (12) and making tentatively the
0
µ m 0 tan2θ 1
ij
onje
ture that ≪ , we
an operate with the approximation, where ≤ ≪
(ij = 14, 25, 36) m m m
1 2 3
. Then, we get the
ase of nearly degenerate spe
trum ≃ ≃
m = m m = m m = m
1 4 2 5 3 6
(and − , − , − ), but withhierar
hi
almass-squareddi(cid:27)eren
es
∆m2 ∆m2 ∆m2
21 ≪ 32 ≃ 31, where
∆m2 = 1.50µ2 , ∆m2 = 425µ2 , ∆m2 = 427µ2
21 32 31 (14)
∆m2 = ∆m2 = ∆m2 = 0
(and 41 52 63 ).
The familiar neutrino-os
illation probabilities on the energy shell,
P(ν ν ) = ν eiPL ν 2 = δ 4 U U U U sin2x
α → β |h β| | αi| βα − β∗j βi αj α∗i ji (15)
j>i
X
with
∆m2L m2
x = 1.27 ji , ∆m2 = m2 m2 , p E i ,
ji E ji j − i i ≃ − 2E (16)
U = U
α∗i αi
valid when CP violation
an be ignored (then ), lead to the formulae
P(ν ν ) = 1 sin2x = 1 P(ν ν ) P(ν ν ) ,
e e 21 e µ e τ
→ − − → − →
1 1
P(ν ν ) = 1 sin2x (sin2x +sin2x ) = P(ν ν ) = 1 P(ν ν )
µ µ 21 31 32 τ τ µ τ
→ − 4 − 2 → − →
1
P(ν ν ) = sin2x = P(ν ν ),
µ e 21 τ e
→ 2 → (17)
U
αi
when the matrix elements are
al
ulated from Eqs. (6) and (7) [see Eq. (A.2)℄. Here,
m = m m = m m = m x (i,j = 1,2,3,4,5,6)
1 4 2 5 3 6 ji
the relations − , − , − work in .
3 3
Note thattheos
illationformulae(17)holdalsointhe × textureofa
tiveneutrinos,
3 3 U = (U ) (α = e, µ, τ , i = 1,2,3)
αi
if the × mixing matrix is put equal to the
U(3) = U(3)
αi
bimaximal mixing matrix given in Eq. (7), while the (
onventional) sterile
(cid:16) (cid:17)
neutrinos are to be de
oupled through the familiar seesaw me
hanism. In this
ase, for
m , m , m 3 3 M = (M )
1 2 3 αβ
any mass spe
trum the × mass matrix be
omes
m1 0 0 m1+2m2 −m21√+2m2 m21√−m22
M = U(3) 0 m2 0 U(3)† = −m21√+2m2 m1+m42+2m3 −m1−m42+2m3 . (18)
0 0 m m1 m2 m1 m2+2m3 m1+m2+2m3
3 2√−2 − −4 4
4
0 0
m = m m m M m1(3)
1 2 3
In parti
ular, for ≃ ≃ one obtains ≃ . The reason, why
ν , ν , ν
e µ τ
in both
ases the os
illation formulae for a
tive neutrinos get the same forms
(17), is(cid:21)inourtexture(cid:21) theabsen
e ofos
illationsforthe(
onventional) sterileneutrinos
ν , ν , ν U
es µs τs αi
. In fa
t, one
an show from Eqs. (15) that the matrixelements evaluated
P(ν ν ) = 0 P(ν
with the use of Eqs. (6) and (7) [see Eq. (A.2)℄ lead to α → βs and αs →
ν ) = δ (α, β = e, µ, τ) ∆m2 = ∆m2 = ∆m2 = 0
βs βsαs 41 52 63
in
onsequen
e of , where
m = m , m = m , m = m
1 4 2 5 3 6
− − − .
The os
illation formulae (17) provide the relations
P(ν ν ) = 1 sin2(x ) ,
e e sol 21 sol
→ −
1 1
P(ν ν ) = 1 sin2(x ) sin2(x ) +sin2(x )
µ µ atm 21 atm 31 atm 32 atm
→ − 4 − 2
h i
1 sin2(x ) ,
32 atm
≃ −
P(ν¯ ν¯ ) = 1 sin2(x ) 1 ,
e e Chooz 21 Chooz
→ − ≃
1
P(ν ν ) = sin2(x ) 0 ,
µ e LSND 21 LSND
→ 2 ≃ (19)
ν ν
e µ
implying bimaximal mixing for solar 's and atmospheri
's, negative result for Chooz
ν¯ ν ν¯
e µ µ
rea
tor 's [6℄ and no LSND e(cid:27)e
t for a
elerator 's (and 's) [7℄.
ν
µ
From the se
ond formula (19) de
ribing atmospheri
's we infer due to the Su-
perKamiokande results [8℄ that
1 = sin22θ 1 , ∆m2 = ∆m2 3 10 3 eV ,
atm ∼ 32 atm ∼ × − (20)
what gives
µ2 7.1 10 6 eV2 or µ 2.7 10 3 eV ,
− −
∼ × ∼ × (21)
when Eq. (14) is used.
ν
e
Then, the (cid:28)rst formula (19) referring to solar 's predi
ts
sin22θ = 1 , ∆m2 = ∆m2 1.1 10 5 eV2 ,
sol sol 21 ∼ × − (22)
ν
e
where Eqs. (14) and (21) are applied. Su
h a predi
tion for solar 's is not in
onsistent
with the Large Mixing Angle (LMA) solar solution [9℄.
5
4. Con
lusions. We presented in this note an e(cid:27)e
tive texture for six Majorana
6 6
neutrinos, three a
tive and three (
onventional) sterile, based on the × mass matrix
de(cid:28)ned in Eqs. (3) and (5), and leading to the mixingmatrix given in Eqs. (6) and (7), as
3 3
well as to the mass spe
trum (8) or (9). We
onje
tured that the Dira
×
omponent
of su
h a neutrino mass matrix, when the bimaximal mixing spe
i(cid:28)
for neutrinos is
3 3
transformed out unitarily, gets a fermion universal form (11) similar to the × mass
3 3
matrix for
harged leptons and × mass matri
es for up and down quarks,
onstru
ted
previously with a
onsiderable su
ess [2,3℄.
ν
e
This texture predi
ts reasonably os
illations of solar 's in a form not in
onsistent
with LMA solar solution, if the SuperKamiokande value of the mass(cid:21)squared s
ale for
ν
µ
atmospheri
's is taken as an input. In both
ases, neutrino os
illations are maximal.
The proposed texture predi
ts also the negative result of Chooz experiment for rea
tor
ν¯ ν ν¯
e µ µ
's and the absen
e of LSND e(cid:27)e
t for a
elerator 's (and 's). The new miniBooNE
experiment may
on(cid:28)rm or revise the original LSND results.
As far as the neutrino mass spe
trum is
on
erned, our model of neutrino texture is
< <
m = m m = m m = m
1 4 2 5 3 6
of 2+2+2 type (with | | ∼ | | ∼ | |), in
ontrast to the models of
ν , ν , ν
e µ τ
3+1 or2+2 types [10℄ dis
ussed inthe
ase when, beside three a
tive neutrinos ,
ν
s
there is one extra sterile neutrino . In those models, three Majorana
onventionalsterile
ν , ν , ν
es µs τs
neutrinos are de
oupled through the familiar seesaw me
hanism, as being
ν , ν , ν
4 5 6
pra
ti
ally identi
al with three very heavy mass neutrinos (of the GUT mass
ν , ν , ν
4 5 6
s
ale). In our model, on the
ontrary, the mass neutrinos are
onstru
ted to
ν , ν , ν
1 2 3
be degenerate in mass magnitude with the light mass neutrinos , respe
tively.
ν , ν , ν
es µs τs
In
onsequen
e of this degenera
y, the
onventional sterile neutrinos do not
os
illate in our neutrino texture.
6
Appendix
In our neutrino texture, the expli
it forms of the neutrino mass matrix and its diago-
nalizing matrix are
1 0 0 tan2θ14 tan2θ25 0
√2 √2
0 1 0 tan2θ14 tan2θ25 tan2θ36
− 2 2 √2
0 0 1 tan2θ14 tan2θ25 tan2θ36
2 − 2 √2
0
M = m (A.1)
tan2θ14 tan2θ14 tan2θ14 1 0 0
√2 − 2 2 −
tan2θ25 tan2θ25 tan2θ25 0 1 0
√2 2 − 2 −
0 tan2θ36 tan2θ36 0 0 1
√2 √2 −
[from Eqs(3) and (5)℄ and
c14 c25 0 s14 s25 0
√2 √2 −√2 −√2
c14 c25 c36 s14 s25 s36
− 2 2 √2 2 − 2 −√2
c14 c25 c36 s14 s25 s36
2 − 2 √2 − 2 2 −√2
U = (A.2)
s 0 0 c 0 0
14 14
0 s 0 0 c 0
25 25
0 0 s36 0 0 c36
[from Eqs. (6) and (7)℄, respe
tively, where the neutrino mass spe
trum is given as
0 0
m = m 1+tan22θ (c2 s2)m = m c = cosθ s = sinθ (ij =
i,j ± ij or ij − ij i,j ± with ij ij and ij ij
q
14,25,36) M = U m U U M U = m δ (α,β =
αβ i αi i β∗i αβ α∗i αβ βj i ij
. Of
ourse, and
0
e, µ, τ , e , µ , τ , i,j = 1,2,P3,4,5,6) tPan2θ 0 tan2θ = 1.23µ/m
s s s 14 25
. Here, → ,
0
tan2θ = 20.7µ/m µ 2.7 10 3
36 −
and [from Eqs. (12)℄ with ∼ × eV [from Eq. (21)℄.
M(D) M
TheDira
omponent ofthemassmatrix ,breakingtheele
troweaksymmetry,
may be of the Higgs origin. On the
ontrary, the lefthanded and righthanded
omponents
7
M(L) M(R) M
and of , the former breaking the ele
troweak symmetry, may be given
expli
itly. The bimaximal mixing matrix
1 1 0
√2 √2
U(3) = 1 1 1 , (A.3)
−2 2 √2
1 1 1
2 −2 √2
M(D) = m0 U(3)diag(tan2θ , tan2θ , tan2θ )
14 25 36
operating within the Dira
omponent of
M
asafa
torandso, breakingtheele
troweak symmetry by itself,maybe introdu
edalso
U(3) 3 3 θ = π/4 , θ = π/4
12 23
expli
itly. Note that is the generi
× mixing matrix with
0
θ = 0 mdiag(tan2θ , tan2θ , tan2θ ) U(3)
13 14 25 36
and . Deforming unitarily , the matrix
introdu
es the signi(cid:28)
ant di(cid:27)eren
e between the hierar
hi
al stru
tures of neutrinos and
harged leptons.
ν
α
From Eqs. (2) and (A.2) one
an see that the (cid:29)avor neutrinos are built up as
ν c ν s ν ν
ij ij i ij j i
the simplest orthogonal
ombinations of the superpositions ≡ − ≃ and
ν s ν +c ν ν (ij = 14,25,36) ν
i′j ≡ ij i ij j ≃ j of the mass neutrinos i:
1 1
ν = (ν +ν ) (ν +ν ) ,
e 14 25 1 2
√2 ≃ √2
1 1 1 1
ν = (ν ν )+ν (ν ν )+ν ,
µ 14 25 36 1 2 3
√2 "−√2 − # ≃ √2 "−√2 − #
1 1 1 1
ν = (ν ν )+ν (ν ν )+ν ,
τ √2 "√2 14 − 25 36# ≃ √2 "√2 1 − 2 3#
ν = ν ν ,
es 1′4 ≃ 4
ν = ν ν ,
µs 2′5 ≃ 5
ν = ν ν ,
τs 3′6 ≃ 6 (A.4)
0
s c µ m
ij ij
where the se
ond step is taken for ≪
orresponding to ≪ .
0
s 0 (ij = 14,25,36) µ/m 0
ij
In the mathemati
al limit of →
orresponding to →
0 1 0
M mdiag(1(3), 1(3)) U U = diag(U(3), 1(3)) m m
i,j
one obtains → − and → , where → ± .
M(D)
Then, together with , the neutrino hierar
hi
al stru
ture vanishes.
8
Referen
es
[1℄ For a theoreti
al summary
f. J. Ellis, Nu
l. Phys. Pro
. Suppl. 91, 503 (2001); and
referen
es therein.
[2℄ W. Królikowski, in Spinors, Twistors, Cli(cid:27)ord Algebras and Quantum Deformations
(Pro
. 2nd Max Born Symposium 1992), eds. Z. Oziewi
z et al., Kluwer A
ad. Press,
1993; A
ta Phys. Pol. B 27, 2121 (1996); and referen
es therein.
[3℄ W. Królikowski, A
ta Phys. Pol. B 30, 2631 (1999); hep(cid:21)ph/0108157; and referen
es
therein.
[4℄ W. Królikowski, hep(cid:21)ph/0109212; hep(cid:21)ph/0201004; in both papers the neutrino spe
-
m = m = m = 0
4 5 6
trum is of 3+3 type, in the se
ond with .
[5℄ The Parti
le Data Group, Eur. Phys. J. C 15, 1 (2000).
[6℄ M. Appolonio et al., Phys. Lett. B 420, 397 (1998); B 466, 415 (1999).
[7℄ G. Mills, Nu
l. Phys. Pro
. Suppl. 91, 198 (2001); R.L. Imlay, Talk at ICHEP 2000
at Osaka; and referen
es therein.
[8℄T.KajitaandY.Totsuka,Rev. Mod. Phys.73,85(2001);T.Toshito,hep(cid:21)ex/0105023.
[9℄Forare
entanalysis
f.M.V.GarzelliandC.Giunti,hep(cid:21)ph/010819;hep(cid:21)ph/0111254;
f. also V. Barger, D. Marfatia and K. Whisnant, hep(cid:21)ph/0106207.
[10℄ Cf. e.g. V. Barger, B. Kayser, J. Learned, T. Weiler and K. Whisnant, Phys. Lett.
B 489, 345 (2000); M.C. Gonzalez(cid:21)Gar
ia, M. Maltoni and C. Peña(cid:21)Garay, hep(cid:21)
ph/0108073;andreferen
estherein;
f.alsoW.Królikowski,hep(cid:21)ph/0106350;O.Yasuda,
hep(cid:21)ph/0109067.
9
