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TEXTURES, FAMILY SYMMETRIES, AND NEUTRINO MASSES
provided by CERN Document Server
ELENA PAPAGEORGIU
Laboratoire de Physique Theorique et Hautes Energies, Universite de Paris XI, B^atiment 211, 91405 Orsay, France
We review recentwork on the interplaybetween broken family symmetries which give rise to approximate
texture zeros in the mass matrices of the quarks and leptons and the presence of certain hierarchy patterns
in the sp ectrum of massive neutrinos.
1 Intro duction 2 Generating the texture of the quark and lep-
~
ton mass matrices from an extra U (1)
In an attempt to understand the hierarchy of the fermion
masses and the mixing of the quarks in terms of a few fun-
~
damental parameters the idea of extending the Standard
Let us assume the existence of a family-dep endent U (1)
~
Mo del (SM) byaU(1) horizontal symmetry of stringy
symmetry at the Planck scale, with resp ect to which the
1;2;3;4;5;6
origin has b een reinvestigated in the prosp ect
quarks and leptons carry charges and a resp ectively,
i i
of anomaly cancellation a la Green Schwarz, whereby,
where i =1;2;3 is the generation index. Due to the
~
the mixed anomalies of the U (1) with the SU (3), the
electroweak symmetry the lefthanded up (quark/lepton)
SU (2) and the U (1) gauge groups are cancelled bya
Y
states and the corresp onding down (quark/lepton) states
shift in the dilaton eld in 4D string theories, a sucient
carry the same charges. The generation of symmetric
condition for obtaining also the canonical uni cation of
textures implies on the other hand equal charges for left-
16
the three gauge couplings at M =10 GeV, as well as
G
handed and righthanded states.
the phenomenologically successful
We generate rst the texture of the quark mass ma-
trices M and M . Given the role played by the third
u d
m ' m m m ' m m (1)
b e d s
generation we start with rank-one matrices and makea
choice for the charges such that only the (3,3) renormal-
c c
izable couplings t th and b bh are allowed, the other
1 2
relations without the need of a grand uni ed group.
entries b eing zero as long as the symmetry is exact. This
Besides, since with the thirteen observables of the
choice xes the charges of the light Higges h to 2
1;2
Yukawa sector one cannot uniquely determine the entries
( ). On the other hand the presence of scalar elds,
3
of the quark and lepton mass matrices, even by assuming
some of which acquire a vev at the uni cation scale, and
that the latter are hermitian, it has b ecome increasingly
of higher-dimension op erators, as this is common in most
p opular to assume that some of the entries are suciently
string compacti cation schemes, can lead to the sp onta-
suppressed with resp ect to others so that they can b e
neous breaking of the symmetry and the generation of
replaced by so-called \texture zeros" which re ect the
2;3
small nonzero entries. The most economical scenario
nature of the underlying horizontal symmetry and give
requires only one singlet eld or a pair of elds ~ devel-
relations b etween the fermion masses and the mixings,
p
opping equal (vev's) along a \D- at" direction and car-
like the successful jV j' m =m prediction. We shall
us d s
rying opp osite charges 1. These in turn can give rise
therefore start with a discussion of how to generate such
<>~
c j2 j
i j
to higher-order couplings q h ( ) q where
1 j
i
M
phenomenologically successful symmetric textures in the
M is a scale typical of these higher-dimension op erators,
context of a broken horizontal symmetry with a minimum
18
e.g. the string uni cation scale M ' 10 GeV or M .
S P
numb er of scalar elds and higher-dimension op erators.
<>~
~
The p ower with which the scale E = will ll in
M
The extension of this approach to the lepton sec- the (i; j )entry is such as to comp ensate the charge of
c
tor has given di erent neutrino mass and oscillation pat- q hq . Notice that when the exp onent is p ositive (nega-
j
i
6;7;8;9;10;11
terns which are able to explain some of the tive) only the eld ~ (~ ) can contribute. Most likely
+
three neutrino-mass related puzzles, the solar neutrino such a theory will contain also heavy Higgs multiplets
de cit (SN), the atmospheric neutrino de cit (AN), and H , needed for breaking the GUT symmetry and giving
i
<>~
j2 j c
i j
the need of hot dark matter (HDM) for galaxy formation. ) q , where by 2 we denote rise to q H (
j i
i
M
This will b e discussed in the second part of the talk. the charge of H . As a consequence one can generate i
Let us discuss next the generation of Dirac neu-
Table 1: Possible sets of M , M textures corresp onding to dif-
D c u
d
trino mass terms: M N where N are the three
j i=1;2;3
i
~
ferentchoices of U (1) charges and classi ed according to eq.(3).
righthanded neutrino states which are present in most
The ones and the zeros corresp ond to an order of magnitude esti-
GUT's. Since these are of the same typ e as the mass mate, since in our approach it is p ossible to generate the p owers of
lamb da but not the co ecients in front.
terms in the quark and charged lepton sector it is nat-
ural to adopt again the same approach. Then, b ecause
the charges a have b een xed through the charged lepton
i
D
~
Ansatz, M = M .Furthermore for the choice a =
Set ~ ~
e 1 1
and a = one obtains the other well known GUT re-
2 2
lation:
A 1 1 0 0 1 1 1
D D
M = M or M = M : (5)
u d
B 1 0 1 0 1 1 1
On the other hand, the heavy Ma jorana mass terms
c
M N N , which are needed in the (6 6) neutrino mass
R j
i
C 1 1 1 0 1 1 0
matrix in order to suppress the otherwise unacceptably
large masses for the ; ; , need not b e generated the
e
D 0 1 1 1 1 1 0
same way. In compacti ed string mo dels, due to the
absence of large Higgs representations, righthanded neu-
E 1 1 1 0 1 1 1
trinos donot get tree-level masses, so all entries in M
R
12 c
are due to nonrenormalizable op erators N H H N
k l j
i
2
(k; l =1;2; :::) whose scale is of O (M =M )multiplied
P
G
mass matrices of the following typ e:
4
for some orbifold suppression factor C'1 10 :
1 0
2jz j jz +z j jz +z j
1 1 2 1
~ ~ ~
E E E
9 13
jz +z j 2jz j jz +z j
R = C ' 10 10 GeV : (6)
1 2 2 2
~ ~ ~
A @
; (2) M
E E E
u;d
M
P
jz +z j 2jzj jz +z j
2 1 ~ ~ ~
E 1+ E E
Nothing is a priori known concerning the particular tex-
whith jz j = j j, and jz j = j j. Assuming two
ture of M or the existence of hierarchy in this sector.
i i
R
2
~ ~
scales E''0:2 and E'E, where 'jV j is
One can therefore follow either of the two paths: As-
us
c
the Wolfenstein parameter, and the right combination of
sume no hierarchy and use the op erators N H H N to
k l j
i
a
charges one can generate four sets of M M textures:
generate a nonsingular Ma jorana mass texture
u d
A, B, C and D (Table 1 and eq.(3)) which contain ve
0 1
6
R R R
1 4 5
zeros in total , without counting zeros of symmetric en-
@ A
M = R R R ; (7)
R 4 2 6
tries twice, and lead to the most predictive Ansatz for
R R R
5 6 3
quark masses and mixings in agreement with exp eriment.
A fth set, E, with four zeros emerges from the partic-
as in refs.[3,6,8], or generate the entries along the same
2;3
ularly economical scenario with the two singlets . All
principle whichwas used to generate the hierarchyinthe
ve sets can b e obtained from
quark and charged lepton sector, refs.[9-11]. In any case
1 0 1 0
3
6 4
0 ~ 0
the sp ectrum of the light neutrino states ; ; de- 0
e
M M
u d
6 4 2
2 3
~ A @ A @
p ends very delicately on the (2,3) and/or the (1,3) entry ' ; '
~ ~
m m
4 2
6;7;8 t b
1
of M and the symmetries of M , as will b e shown
0 ~ 1
u;d R
next.
(3)
and Table 1.
Let us now turn to the charged lepton mass matrix
3 The neutrino mass sp ectrum
M . Assuming simply the gauge symmetries of the SM
e
D
the U (1) charges of the leptons are not related to those
Starting from the Ansatz: M ' 1 R or M M
X
R R
of the quarks but adopting the philosophy that all the one is led, up on diagonalisation of the reduced (3 3)
entries except the (3,3) entry are zero b efore symmetry
light neutrino matrix
breaking leads to a = . Another constraint comes
3
1
D y D ef f
' M M M ; (8) M
R
from the second mass relation of eq.(1) which implies
a + a = + . This relation can b e satis ed when
1 2 1 2
to the quadratic seesaw sp ectrum:
a = and a = and leads to the GUT relation:
1 1 2 2
8 4
m : m : m ' (z : z :1)m ; (9)
e
M M (4)
e d
a
otherwise some of the standard neutrinos will b e to o heavy
with a slight mo di cation of the (2,2) entries as in the
and will have to decay as required byvarious astrophysical and
13
Georgi-Jarlskog mo del. , cosmological considerations
where r 6= 0. This gives rise to sp ectra which are only slightly
3
distorted with resp ect to the quadratic seesaw sp ectrum
2
m
t
D 2
when due to the texture zeros and/or the symmetries of
if M = M ; (10) z = m =
u
R
M some of the other minors are zero. In contrast when
R
or,
r = 0, the hierarchy in the neutrino sp ectrum can b e
3
2
m
reversed as this is the case for the quark texture solu-
D
b
z = m if M = M ; (11) =
d
R
tion A where a texture zero app ears at the (3,3) entry of
ef f
M . Then, dep ending up on the p osition of the zeros in
and the lepton mixings:
M , the muon neutrino is heavier than the tau neutrino,
R
2 8
2 3 4 5
or, they are mass degenerate and sin 2 O(1) .
jV j ; jV j ; jV j ; (12)
e e
3
Large mixing is on the other hand typical of
e
8
the quark texture solutions B and C . As a matter of
giving rise to neutrino oscillations with
fact, mass-degenerate neutrinos and large mixing angles
2 2 3 2 5
emerge more naturally from the ve-zero texture solu-
sin 2 0:02; sin 2 10 ; sin 2 10 :
e e
6;7;8
tions A-D than from the four-zero solution E which
(13)
requires very particular conditions of strong hierarchy
The eqs.(10-13) lead to an elegant resolution of the SN
9;10;11
in M to give mass degenerate neutrinos . This
problem through matter enhanced ! oscillations R
e
14
may b e so b ecause the more texture zeros there are in
while the mass falls whithin the range needed for it
the quark mass matrices the more the symmetries of the
to b e the HDM candidate. The quadratic seesaw sp ec-
righthanded neutrino sector can in uence the light neu-
trum is typical of a much larger class of mo dels when
trino sp ectrum.
M do es not contain any texture zeros nor any symme-
R
7 ef f
try . This b ecomes transparent when M is expressed
in terms of the minors of the matrix M , r , which
R i=1;:::6
are obtained by omitting the row and column containing
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2
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