Embedding the neutrino four-group Z_2 x Z_2 into the alternating group of four objects A_4

0.14 MB·English

by  Wojciech Krolikowski

IFT-04/32
Z2 Z2
Embedding the neutrino four-group ×
A4∗
into the alternating group of four obje
ts
Woj
ie
h Królikowski
5
0 Institute of Theoreti
al Physi
s, Warsaw University
0
2 Ho»a 69, PL(cid:21)00(cid:21)681 Warszawa, Poland

n
a
J

3
Abstra
t

1 s
v If and only if the small neutrino mixing sine 13 is put zero, the e(cid:27)e
tive neutrino
8
M s s Z
0 12 23 2
mass matrix with any mixing sines and is invariant under a
y
li
group
0
m m 0 Z Z
1 2 1 2 2
of the order two and (cid:22) in the limit of − → (cid:22) also under the group ×
0
5
often
alled the four-group. However, the elements of this group do not build up fully the
0
/ M m m
h term of proportional to 2− 1. When the four-group is embedded into the group of
p
A M
- 4
p even permutations of four obje
ts (whi
h is of the order twelve), then the term of
e m m A
2 1 4
h proportional to −
an be fully
onstru
ted from elements of . In
ontrast, when
: D
v 4
the four-group is embedded into the dihedral group of the order eight, then the term
i
X M m m D
2 1 4
r of proportional to −
annot be fully built up from elements of .
a
PACS numbers: 12.15.Ff , 14.60.Pq , 12.15.Hh .
De
ember 2004
∗
Work supported in partby the PolishState CommitteeforS
ienti(cid:28)
Resear
h(KBN), grant2P03B
129 24 (2003(cid:21)2004).
1. Introdu
tion
U = (U ) (α =
αi
As is well known, the generi
form of e(cid:27)e
tive neutrino mixing matrix
e,µ,τ , i = 1,2,3)
reads,
c s 0
12 12
U = c s c c s ,
23 12 23 12 23
 −  (1)
s s s c c
23 12 23 12 23
−
 
s
13
if is put zero
onsistently with the nonobservation of the neutrino os
illations in the
ν¯ s2 < 0.04
short-baseline experiments for rea
tor e's (the estimated upper bound is 13 ) [1℄.
c = cosθ s = sinθ
ij ij ij ij
Here, and . Two possible CP-violatingMajorana phases irrelevant
fortheos
illationpro
esses areomittedinEq. (1). One CP-violatingDira
phase relevant
s = 0
13
for these pro
esses disappears together with .
The formula(1) is
onsistent with allwell established neutrino-os
illationexperiments
θ 32 θ 45 ∆m2 =
that provide the globalestimates: 12 ∼ ◦ and 23 ∼ ◦ for mixingangles, and 21
m2 m2 8 10 5 eV2 ∆m2 = m2 m2 2.5 10 3 eV2
2 − 1 ∼ × − and | 32| | 3 − 2| ∼ × − for mass-squared
di(cid:27)eren
es [2℄.
U ν = U ν ν = ν ,ν ,ν
The unitary matrix gives the neutrino mixing α i αi i, where α e µ τ
νi = ν1,ν2,ν3 P
and are the (cid:29)avor and mass a
tive neutrinos, respe
tively. In the (cid:29)avor
U
representation, where the
harged-lepton mass matrix is diagonal, the matrix gives
U M U = m δ
also the diagonalization α,β α∗i αβ βj i ij of the e(cid:27)e
tive neutrino mass matrix
M = (Mαβ) (α,β = e,µ,Pτ). Then, the inverse formula Mαβ = iUαimiUβ∗i enables one
Mαβ P mi
to
al
ulate the mass-matrix elements in terms of neutrino masses and mixing-
U U
αi
matrix elements . In this way, using the form (1) of , one obtains the following
s = 0
13
generi
form of e(cid:27)e
tive neutrino mass-matrix valid in the
ase of :
1 0 0 0 0 0
m +m
M = 1 2 0 c2 c s +m 0 s2 c s
2  23 − 23 23  3 23 23 23 
0 c s s2 0 c s c2
− 23 23 23 23 23 23
   
1 0 0 0 c s
+m2 −m1 c −0 c2 c s +s c 203 − 203 ,
2  sol 23 − 23 23  sol 23  (2)
0 c s s2 s 0 0
− 23 23 23 − 23
    
c = c2 s2 = cos2θ s = 2c s = sin2θ
where sol 12 − 12 12 and sol 12 12 12.
1
3 3 1(m +m ) m 1(m m )
In the formula(2), allthree × matri
es standing at 2 1 2 , 3 and 2 2− 1
3 3

ommute, while two
omponents of the third × matrix anti
ommute. Diagonalizing
both sides of Eq. (2), one gets
onsistently
m 0 0 1 0 0 0 0 0
1 m +m
1 2
0 m 0 = 0 1 0 + m 0 0 0
2 3
  2    
0 0 m 0 0 0 0 0 1
3
   1 0 0  
m2 m1 −
+ − 0 1 0 ,
2   (3)
0 0 0
 
U MU = (m ,m ,m )
† 1 2 3
where diag .
Z Z
2 2
2. The four-group ×
s = 0
13
The e(cid:27)e
tive neutrino mass matrix (2) valid in the
ase of
an be rewritten as
follows:
m +m 1 1
1 2
M = (ϕ ϕ )+m (ϕ +ϕ )
1 4 3 1 4
2 2 − 2
m m 1
2 1
+ − [c (ϕ ϕ )+s (c λ s λ )] ,
sol 2 3 sol 23 1 23 4
2 2 − − (4)
3 3
if the × matri
es
1 0 0 1 0 0
−
ϕ = 0 1 0 , ϕ = 0 c s ,
1 2 atm atm
   − 
0 0 1 0 s c
atm atm
− −
 1 0 0   1 0 0 
−
ϕ = 0 1 0 , ϕ = 0 c s
3 4 atm atm
 −   −  (5)
0 0 1 0 s c
atm atm
−
   
and
0 1 0 0 0 1
λ = 1 0 0 , λ = 0 0 0
1 4
    (6)
0 0 0 1 0 0
   
c = c2 s2 = cos2θ s = 2c s = sin2θ
are introdu
ed, where atm 23 − 23 23 and atm 23 23 23. Here, the
c = 1/√2 = s c = 0 s = 1
23 23 atm atm
values (i.e., and ) are
onsistent with experimental
estimates.
2
c s 3 3
atm atm
It is easy to
he
k for any and that four × matri
es (5) are mutually
multiplied a

ording to the Cayley table of the form
ϕ ϕ ϕ ϕ
1 2 3 4
ϕ ϕ ϕ ϕ ϕ
1 1 2 3 4
ϕ ϕ ϕ ϕ ϕ
2 2 1 4 3
ϕ ϕ ϕ ϕ ϕ
3 3 4 1 2
ϕ ϕ ϕ ϕ ϕ
4 4 3 2 1
Z Z
2 2
what
hara
terizes the Abelian (cid:28)nite group × of the order four, often
alled the
four-group [3℄. It is isomorphi
to the simplest dihedral group of the order four. Thus,
c s 3 3 3
atm atm
for any and , four × matri
es (5)
onstitute a redu
ible representation of
the Abelian four-group [4℄. In fa
t, they
an be simultaneously diagonalized, leading to a
3 = 1+1+1 = 1,1,1
redu
ed representation diag( ).
M m m
2 1
One
an see from Eq. (4) that the term of proportionalto − is not fullybuilt
3 3 c λ s λ
23 1 23 4
up from elements of four-group: the × matrix − lies outside the four-group.
ϕ ϕ ϕ ϕ
1 4 2 3
This matrix
ommutes with and , and anti
ommutes with and .
c s
atm atm
It is not di(cid:30)
ult to verify for any and that the formula (4) implies four
M
dis
rete invarian
es of the e(cid:27)e
tive neutrino mass matrix , two exa
t:
ϕ Mϕ = M
1,4 1,4
(7)
m m 0
2 1
and two valid in the limit of − → :
ϕ2,3Mϕ2,3 = M (m2 m1)ssol(c23λ1 s23λ4) m2−m1→0 M
− − − −→ (8)
ϕ c s
1 atm atm
(of
ourse, in the
ase of the invarian
e is trivial). Thus, for any and , the
M m m 0
2 1
four-group is the symmetry group of in the limit of − → .
c = 1/√2 = s
23 23
Further on, in
onsisten
y with experimental estimates, one will put
c = 0 s = 1
atm atm
(i.e., and ).
3
A
4
3. The alternating group of four obje
ts
A
4
The group of even permutations of four obje
ts was applied to the neutrino mass
matrix in Refs. [5℄. This non-Abelian (cid:28)nite group
onsists of twelve permutations
(1,2,3,4) , (2,1,4,3) , (3,4,1,2) , (4,3,2,1) ,
(2,3,1,4) , (1,4,2,3) , (4,1,3,2) , (3,2,4,1) ,
(3,1,2,4) , (4,2,1,3) , (1,3,4,2) , (2,4,3,1)
(9)
A
4
of whi
h the (cid:28)rst four
onstitute an Abelian subgroup of , isomorphi
to the four-
Z Z
2 2
group × . The twelve permutations (9)
an be represented, respe
tively, by the
4 4
following twelve × matri
es
onstru
ted from the formal Dira
matri
es in the Dira
representation:
g = 1(D) , g = σ(D) , g = γ , g = γ σ(D) ,
1 2 1 3 5 4 5 1
1(D) β 1(D) +β 1(D) σ(D) 1(D) +σ(D)
g = − + σ(D) − 3 +γ 3 ,
5 2 2 1 2 5 2
!
(cid:18) (cid:19)
(D)
g = γ g , g = g γ , g = σ g ,
6 5 5 7 5 5 8 1 5
1(D) β 1(D) σ(D) 1(D) +β 1(D) +σ(D)
g = − 1(D) +γ σ(D) − 3 + 1(D) +γ σ(D) 3 ,
9 2 5 1 2 2 5 1 2
(cid:16) (cid:17) (D) (cid:16) (cid:17)
g = g σ , g = g γ , g = γ g ,
10 9 1 11 9 5 12 5 9
(10)
1(D) = 1(2),1(2) γ = 1(2),1(2) β = 1(2), 1(2) σ(D) =
where diag( ) , 5 antidiag( ) , diag( − ) and 1,3
σ ,σ 1(2) = σ = σ =
1,3 1,3 1 3
diag( ) with diag(1,1) , antidiag(1,1) and diag(1,-1).
4 4 4
Twelve × matri
es (10) form a redu
ible representation of the non-Abelian group
A 4 = 1 + 3 = 1,3
4
. In fa
t, it
an be redu
ed to a representation diag( ) through the
unitary transformation realizing the transitions
γ diag(1(2), 1(2)) , β antidiag( 1(2), 1(2))
5
→ − → − − (11)
and
(D) (D)
σ diag(σ ,σ ) , σ diag( σ , σ )
1 → 3 3 3 → − 1 − 1 (12)
4
g g (a = 1,2,...,12) g = 1,ϕ ϕ (a = 1,2,...,12)
Then a → a′ with a′ diag( ′a), where ′a are the
3 3
following twelve × matri
es:
1 0 0 1 0 0 1 0 0 1 0 0
− −
ϕ = 0 1 0 , ϕ = 0 1 0 , ϕ = 0 1 0 , ϕ = 0 1 0 ,
′1   ′2   ′3  −  ′4  − 
0 0 1 0 0 1 0 0 1 0 0 1
− −
       
0 1 0 0 1 0 0 1 0 0 1 0
− −
ϕ = 0 0 1 , ϕ = 0 0 1 , ϕ = 0 0 1 , ϕ = 0 0 1 ,
′5   ′6  −  ′7  −  ′8  
1 0 0 1 0 0 1 0 0 1 0 0
− −
       
0 0 1 0 0 1 0 0 1 0 0 1
− −
ϕ = 1 0 0 , ϕ = 1 0 0 , ϕ = 1 0 0 , ϕ = 1 0 0 ,
′9   10 −  ′11   ′12 − 
0 1 0 0 1 0 0 1 0 0 1 0
− −
       (13)
3 A 1
4
forming an irredu
ible representation of the group . Its irredu
ible representation
g (a = 1,2,...,12)
a′
involved in
onsists of twelve numbers 1,1,...,1. There are also two
1 A g = 1,ϕ
4 a′ ′a
other irredu
ible representations of , not involved in the diag( )
onsidered
here.
Note the identities
T T T T
ϕ = ϕ , ϕ = ϕ , ϕ = ϕ , ϕ = ϕ
′5 ′9 ′6 ′11 ′7 ′12 ′8 ′10 (14)
and
ϕ +ϕ +ϕ +ϕ = 0 ,
′1 ′2 ′3 ′4
ϕ +ϕ +ϕ +ϕ = 0 ,
′5 ′6 ′7 ′8
ϕ +ϕ +ϕ +ϕ = 0 ,
′9 ′10 ′11 ′12 (15)
as well as
0 1 0
1
(ϕ +ϕ +ϕ +ϕ ) = 1 0 0 = λ .
2 ′5 ′6 ′9 ′11   1 (16)
0 0 0
 
λ ϕ ϕ ϕ ϕ
1 ′1 ′4 ′2 ′3
The matrix
ommutes with and , and anti
ommutes with and .
From Eqs. (15) one gets the "strong"
onstraint
5
1 0 0 0
0 0 0 0
ga′ = 12 0 0 0 0  . (17)
a
X  0 0 0 0 
 
 
Thus, the "weak"
onstraint
1
2
ga′  3  = 0, (18)
a
X  4 
 
 
(1,2,3,4)T (1,0,0,0)T
if imposed on the state of four obje
ts, impliesthat the state of the
(0,2,3,4)T
obje
t 1 vanishes, while the state of the obje
ts 2,3,4 isnot
onstrained. When
ν ν ,ν ,ν
s e µ τ
interpreting1 asa lightsterileneutrino and 2,3,4asthree a
tive neutrinos , the
ν ν ,ν ,ν
s e µ τ
"weak"
onstraint (18) may eliminate as an "unphysi
al" obje
t, leaving only
as "physi
al". However, in absen
e of the "weak"
onstraint (18) all four neutrinos may
be "physi
al" obje
ts.
3 A 3
4
Another irredu
ible representation of , isomorphi
to the previous
onsisting of
3 3
twelve × matri
es (13),
an be de(cid:28)ned through the unitary transformation
1 0 0 1 0 0
ϕ = 0 1 1 ϕ 0 1 1 (a = 1,2,...,12) .
a  √2 √2  ′a √2 −√2  (19)
0 1 1 0 1 1
−√2 √2 √2 √2
   
3 3 3
Then, one obtains twelve new × matri
es also forming an irredu
ible representation
A
4
of :
1 0 0 1 0 0 1 0 0 1 0 0
− −
ϕ = 0 1 0 , ϕ = 0 0 1 , ϕ = 0 1 0 , ϕ = 0 0 1 ,
1 2 3 4
   −   −   
0 0 1 0 1 0 0 0 1 0 1 0
− −
       
0 1 1 0 1 1
1 ± ∓ 1 ± ∓
ϕ = 1 1 1 , ϕ = 1 1 1 ,
5,8 √2 ± √2 √2  6,7 √2 ∓ −√2 −√2 
1 1 1 1 1 1
± −√2 −√2 ∓ √2 √2
   
0 1 1 0 1 1
1 ± ± 1 ∓ ∓
ϕ = 1 1 1 , ϕ = 1 1 1 .
9,10 √2 ± √2 −√2  11,12 √2 ± −√2 √2  (20)
1 1 1 1 1 1
∓ √2 −√2 ∓ −√2 √2
   
Noti
e the identities
6
ϕT = ϕ , ϕT = ϕ , ϕT = ϕ , ϕT = ϕ
5 9 6 11 7 12 8 10 (21)
and
ϕ +ϕ +ϕ +ϕ = 0 ,
1 2 3 4
ϕ +ϕ +ϕ +ϕ = 0 ,
5 6 7 8
ϕ +ϕ +ϕ +ϕ = 0 ,
9 10 11 12
(22)
as well as
1 0 0 0 0 0
1 1
(ϕ ϕ ) = 0 1 1 , (ϕ +ϕ ) = 0 1 1 ,
2 1 − 4  2 −2  2 1 4  2 2 
0 1 1 0 1 1
−2 2 2 2
   
1 0 0
1 −
(ϕ ϕ ) = 0 1 1
2 2 − 3  2 −2  (23)
0 1 1
−2 2
 
and
0 1 1
1 1 − 1
(ϕ +ϕ +ϕ +ϕ ) = 1 0 0 = (λ λ ).
5 6 9 11 1 4
2 √2   √2 − (24)
1 0 0
−
 
(λ λ )/√2 ϕ ϕ ϕ ϕ
1 4 1 4 2 3
The matrix −
ommutes with and , and anti
ommutes with and .
c = 1/√2 = s c = 0 s = 1
23 23 atm atm
In the
ase of (i.e., and ), the e(cid:27)e
tive neutrino
mass matrix (2) takes the form
1 0 0 0 0 0
m +m
M = 1 2 0 1 1 + m 0 1 1
2  2 2  3 2 2 
0 1 1 0 1 1
−2 2 2 2
 1 0 0   0 1 1
+ m2 −m1 c −0 1 1 +s 1 1 0 −0 .
2  sol 2 −2  sol√2   (25)
0 1 1 1 0 0
−2 2 −
    
Thus, making use of Eqs. (23) and (24), one
an write
m +m 1 1
1 2
M = (ϕ ϕ )+m (ϕ +ϕ )
1 4 3 1 4
2 2 − 2
m m 1
2 1
+ − [c (ϕ ϕ )+s (ϕ +ϕ +ϕ +ϕ )] .
sol 2 3 sol 5 6 9 11
2 2 − (26)
7
Here, the identities
ϕ +ϕ = (ϕ +ϕ ) , ϕ +ϕ +ϕ +ϕ = (ϕ +ϕ +ϕ +ϕ )
1 4 2 3 5 6 9 11 7 8 10 12
− − (27)
hold due to Eqs. (22).
1(m +m ) m 1(m m )
In the formula (26) there appear at 2 1 2 , 3 and 2 2− 1 three
ommuting
3 3 ϕ
a

ombinations of eight (from twelve) × matri
es given in Eqs. (20). In the third
ϕ ϕ ϕ +ϕ +ϕ +
2 3 5 6 9

ombination there are two anti
ommuting
omponents sin
e − and
ϕ = (λ λ )√2 M
11 1 4
− anti
ommute. For the e(cid:27)e
tive neutrino mass matrix all these

ombinations play the role of dynami
al variables.
M m m
2 1
In
on
lusion, one
an see from Eq. (26) that the term of proportional to −
A 3 3
4
is fully
onstru
ted from elements of the group embedding the four-group: the ×
λ λ A
1 4 4
matrix − lies inside the group . As
an be inferred from an analogi
al dis
ussion
D
4
for the dihedral group of the order eight, also embedding the four-group, the term of
M m m D 3 3
2 1 4
proportional to − is not fully built up from elements of the group : the ×
λ λ D D
1 4 4 4
matrix − lies outside the group (
f. Appendix in Ref. [4℄). The group was
applied to the neutrino mass matrix in Refs. [6℄.
Of
ourse, in the simple
ase of four-group as well as in the more involved
ases of
A D λ λ M
4 4 1 4
group and group , the matrix − appearing in the term of proportional to
m m M ϕ
2 1 2
− violates the invarian
e of under the transformations generated by matri
es
ϕ m m 0 c = 1/√2 = s
3 2 1 23 23
and , unless the limitof − → is
onsidered (
f. Eq. (8) with ).
As is well known, the neutrino mass matrix, being nonzero, breaks the ele
troweak
SU(2) U(1)
gaugesymmetry × asdothemassmatri
esof
hargedleptonsaswellasupand
down quarks. Thus, it is natural to expe
t that, in general, the e(cid:27)e
tive family dis
rete
symmetries of fermion mass matri
es (cid:22) appearing after the ele
troweak symmetry is
broken (cid:22) do not preserve the ele
troweak symmetry. In parti
ular, the e(cid:27)e
tive family
Z Z ϕ ϕ
2 2 2 3
symmetry × of neutrino mass matrixworks: under and transformationsinthe
m = m ϕ ϕ
2 1 1 4
limit of , and always under and transformations. But, at the same time,
in the (cid:29)avor representation, the
harged-lepton mass matrix is diagonal with the entries
m m m ϕ ϕ
e µ τ 2 4
≪ ≪ ex
luding its potential invarian
e under and transformations valid
m = m c s ϕ
τ µ atm atm 1
only at (for any and ), though it is always invariant under diagonal
ϕ
3
and transformations.
8
Referen
es
[1℄ M. Apollonio et al. (Chooz Collaboration), Eur. Phys. J. C 27, 331 (2003).
[2℄ For a re
ent review
f. M. Maltoni et al., hep(cid:21)ph/0405172; M.C. Gonzalez-Gar
ia,
hep(cid:21)ph/0410030; G. Altarelli, hep(cid:21)ph/0410101.
[3℄ E.P. Wigner, Group theory and its appli
ation to the quantum me
hani
s of atomi
spe
tra, A
ademi
Press, New York and London, 1959; B. Simon, Representations of
(cid:28)nite and
ompa
t groups, Ameri
an Mathemati
al So
iety,1996.
[4℄ W. Królikowski, hep(cid:21)ph/0410257 (to appear in A
ta Phys. Pol. B , where Appendix
is added); and referen
es therein.
[5℄ E. Ma, hep(cid:21)ph/0307016; hep(cid:21)ph/0409075; and referen
es therein.
[6℄ W. Grimus, L. Lavoura, A
ta Phys. Pol., B 34, 5393 (2003); W. Grimus, A.S. Yoshi-
pura,S.Kaneko,L.Lavoura,M.Tanimoto,hep(cid:21)ph/0407112;W.Grimus,A.S.Yoshipura,
S. Kaneko, L. Lavoura, H. Savanaka, M. Tanimoto, hep(cid:21)ph/0408123; and referen
es
therein.
9