Update on Fermion Mass Models with an Anomalous Horizontal U(1) Symmetry

0.65 MB·English

by  P. H. Chankowski

IFT-2004/31
CERN-PH-TH/2004-263
February 2, 2008
Update on Fermion Mass Models with an
U(1)
Anomalous Horizontal Symmetry
5
0
0
2

n
a a a b,
J Piotr H. Chankowski , Kamila Kowalska , Stéphane Lavigna
1 and
a,b
0
Stefan Pokorski
1

1 a
v Institute of Theoreti
al Physi
s, Warsaw University, Ho»a 69, 00-681, Warsaw, Poland
b
1
Theory Division, Physi
s Department, CERN, CH-1211 Geneva 23, Switzerland
7
0
1
0
5
0
/
h Abstra
t
p
-
p
e We re
onsider models of fermion masses and mixings based on a gauge anomalous horizontal
U(1)
h
symmetry. In the simplest model with a single (cid:29)avon (cid:28)eld and horizontal
harges of the
:
v
same sign for all Standard Model (cid:28)elds, only very few
harge assignements are allowed when
i
X
all experimental data, in
luding neutrino os
illation data, is taken into a

ount. We show that
r
a a pre
ise des
ription of the observed fermion masses and mixing angles
aUn(1ea)sily be obtained
by generating sets of the order one parameters left un
onstrained by the symmetry. The

orresponding Yukawa matri
es show several interesting features whi
h may be important for
(cid:29)avour
hanging neutral
urrents and CP violation e(cid:27)e
ts in supersymmetri
models.
1
Permanent address: Servi
e de Physique Théorique, CEA-Sa
lay, F-91191Gif-sur-Yvette Cedex, Fran
e.
1 Introdu
tion
The origin of the fermion mass and mixing textures is a
hallenge for physi
s beyond the
Standard Model. One promising approa
h to that problem is based on hypotheti
al horizontal
symmetries whi
h are spontaneously broken by va
uum expe
tation values of some (cid:16)(cid:29)avon(cid:17)
Φ
(cid:28)elds . Hierar
hi
al patterns in the fermion mass matri
es(
Φan/Mth)enn be expMlained by the
Froggatt-Nielsen me
hanism [1℄, as due to suppression fa
tors h i , where is the s
ale
n
of integrated out physi
s and the power depends on the horizontal group
harges of the
fermion, Higgs and (cid:29)avon (cid:28)elds.
Fermion mass models based on abelian [2℄(cid:21)[10℄ and non-abelian [11℄ horizontal symmetries
have been widely dis
usssed in the literature. Ea
h approa
h has its own virtues and short
om-
U(2)
ings, parti
ularly when realized in supersymmetri
models, whi
h we
onsider here. and
SU(3)
symmetries are quite predi
tive for fermion mass matri
es on
e the pattern of symmetry
breaking is spe
i(cid:28)ed. However, the quantitative des
ription of fermion masses and mixings
U(1)
requires a rather
ompli
ated stru
ture of symmetry breaking. The main short
oming of
horizontal symmetries as models of fermion masses is the dependen
e of the quantitative pre-
di
tions on arbitrary order one
oe(cid:30)
ients, the remnant of the integrated out unknown physi
s.
In prin
iple also the arbitrariness in the
hoi
e of the abelian
harges for fermions looks less
U(2) SU(3)
appealing than the rigid stru
tures of the essentially unique
hoi
es of and as
on-
tinuous horizontal non-abelian symmetry groups. But that last point is balan
ed by the fa
t
U(1)
that the breaking of the symmetry is mu
h simpler.
U(1)
gaugegroupfa
torsaregeneri
instringmodels. Oneparti
ularlysimpleandattra
tive
U(1)
horizontal symmetry is the anomalous often found in heteroti
string
ompa
ti(cid:28)
ations.
Anomaly
an
ellation by the Green-S
hwarz me
hanism [12℄ is possible only under
ertain
U(1)

onditionswhi
h strongly
onstrain the possible
hoi
es of horizontal
harges for fermions.
It has already been noti
ed long ago that those
onstraints are at a qualitative level amazingly

onsistent with the quark masses and mixings [4℄(cid:21)[9℄. A parti
ularly appealing feature of the
U(1) Φ /M
anomalous is that the symmetry breaking parameter h i
an be
omputed in terms
of the horizontal
harges, and that its value in expli
it models turns out to be very
lose to the
Cabibbo angle. Furthermore, with positive
harges for matter (cid:28)elds and negative for the (cid:29)avon
(cid:28)eld (or vi
e-versa), this va
uum is unique.
The purpose of this note is to update the predi
tions for fermion masses of the simplest,
U(1) Φ
string-inspired gauge anomalous models with a single (cid:29)avon (cid:28)eld . We are motivated by
several fa
tors. One is the re
ent experimental progress in neutrino physi
s. Neutrino masses
U(1)
and mixings
an be well des
ribed by models based on horizontal symmetries, and it is
therefore interesting to
he
k what phenomenologi
al
onstraints, in addition to the Green-
S
hwarz anomaly
an
ellation
onditions, are put on the horizontal
harges by the requirement
U(1)
that the same gauged symmetry explains both the quark and lepton mass hierar
hies. In
fa
t, assuming that all Standard Model (cid:28)elds have horizontal
harges of the same sign, we (cid:28)nd
a very limited number of possible
harge assignments.
Furthermore, it is interesting to
he
k the su

ess of these models beyond the qualitative
1
level. In this paper, we perform a
omplete numeri
al (cid:28)t of a few representative models to
fermion masses and mixings, with the in
lusion of
omplex order one
oe(cid:30)
ients and with
tanβ
attention paid to the dependen
e.
Finally, in supersymmetri
models there are new sour
es of (cid:29)avour and CP violation in-
du
ed by virtual sfermion ex
hanges. Sfermion mass matri
es depend both on the pattern of
supersymmetry breaking and on the rotations to the super-CKM basis whi
h are determined
by the fermion mass matri
es. Thus, it is useful to have an expli
it set of su

essful fermion
mass models, de(cid:28)ned by a set of horizontal
harges and
omplex order one
oe(cid:30)
ients. The
fa
t that only a small number of
harge assignments are allowed makes it possible to test the
phenomenologi
al impli
ations of an anomalous gauge horizontal symmetry. In a forth
oming
paper, we shall use this set of models to study (cid:29)avour
hanging neutral
urrents (FCNCs) in
D
supersymmetry breaking s
enarios with dominant -term breaking.
The paper is organized as follows. In the next se
tion, we review the basi
properties of
U(1)
fermion mass models based on an anomalous and list the phenomenologi
ally allowed

harge assignements. In se
tion 3 we generate sets of
omplex order one
oe(cid:30)
ients giving a
pre
ise des
ription of fermion masses and mixings, and study the features of the asso
iated
Yukawa matri
es relevant for FCNC and CP violating pro
esses. Finally, we present our
on-

lusions in se
tion 4.
U(1)
X
2 Constraints on
harges
R
The models we
onsider in this paper are extensions of the ( -paritˆy
onserving) MSSM with a
U(1) Φ X
X
horizontal gauge abelian symmetry and a
hiral super(cid:28)eld with -
harge normalized
1 X
to − , whose va
uum eQˆxpe
UtˆactioDnˆcvalLˆue bErˆecaksAth=e h1o,r2i,z3ontaHˆl symmeHtˆry. The -
hargesqof
the M¯ SSM super(cid:28)elds A, A, A, A, A ( ), u and d are denoted by A,
u¯ d l e¯ h h
A A A A u d
, , , Nˆc, and , resn¯pe
tively. The model also
ontains three right-handed neutrino
super(cid:28)elds A with
harges A, whi
h are needed to generate the neutrino masses via the
seesaw me
hanism [13℄. No additional matter
harged under the SM gauge group is assumed,
although the forth
oming analysis would not be altered by the presen
e of ve
tor-like matter
U(1)
X
under both the SM gauge group and . The Yukawa
ouplings of quarks and leptons
are generated via the Froggatt-Nielsen me
hanism [1℄, from nonrenormalizable superpotential
terms of the form: Φˆ u¯A+qB+hu
CABUˆcQˆ Hˆ .
u A B u M (1)
!
These terms arise upon integrating out heavy ve
tor-like (super)(cid:28)elds, the so-
alled Froggatt-
M
Nielsen (cid:28)elds, whose
hara
teristi
mass s
ale is . In string
ompa
ti(cid:28)
ations, the rle of the
M
Froggatt-Nielsen (cid:28)elds is played by massive string modes, and is identi(cid:28)ed with the Plan
k
s
ale (although ve
tor-like (cid:28)elds may also be present among the massless string modes). Aftˆer
φ Φ
breaking of the horizontal symmetry by the VEV of the s
alar
omponent of the super(cid:28)eld ,
2
ǫ φ /M
oneobtainse(cid:27)e
tive Yukawa
ouplingssuppressed bypowers ofthesmallparameter ≡ h i :
YAB = CAB ǫu¯A+qB+hu .
u u
(2)
CAB
The fa
tors u are not
onstrained by the horizontal symmetry and are assumed to be of
order one. Then the hierar
hy of Yukawa
ouplings is determined, up to these unknown fa
-
tors, by the
harges of the MSSM (cid:28)elds. Sin
e holomorphi
ity of Φˆthe superpoYtenAtBia=l fo0rbids
nu¯on+reqnor+mhaliz<ab0le terms witYh AaBn=eg0ativue¯ p+owqer+ofhthe super(cid:28)eld , one has u if 2
A B u u A B u
. Similarly, if is not an integer.
Within the above assumptions, one
an show that the horizontal abelian symmetry has to
be anomalous [9℄ (see also Refs. [4, 5℄). Indeed, one has the following relations:
detM detM v3v3 ǫC3+3(hu+hd) ,
u d ∼ u d (3)
detMd/detMe ǫ−21(C1+C2−83C3)+hu+hd ,
∼ (4)
C (2q +u¯ +d¯ ) C (3q +l )+h +h C 1 (q +8u¯ +2d¯ +
where 3 ≡ A A A A , 2 ≡ A A A u d and 1 ≡ 3 A A A A
3l +6e¯ )+h +h SU(3) SU(3) U(1) SU(2) SU(2)
A A u d C c X L L
P are the
oe(cid:30)
ientPs of the mixed - - P , - -
U(1) U(1) U(1) U(1) v
X Y Y X u,d
and - - anomalies, respe
tively, and are the va
uum expe
tation
valuesofthetwoHiggsdoublets. Ifallanomaly
oe(cid:30)
ientswere vanishing,thehierar
hyamong
h +h h +h = 6 8 ǫ = λ λ
u d u d
quark masses would require a large, positive value of ( − for , where
tanβ
is the sine of the Cabibbo angle, depending on the value of ), while the relation (4) would
h +h = 1 2 ǫ = λ
u d
require a mu
h smaller value ( or for ). Therefore, the horizontal symmetry
has to be anomalous if it is to explain the observed fermion mass hierar
hy.
This fa
t provides the main motivation for
onsidering a gauged horizontal symmetry. It
is well-known that abelian gauge anomalies
an be
ompensated for by the Green-S
hwarz
3
me
hanism [12℄, as is
ommon in four-dimensional heteroti
string
ompa
ti(cid:28)
ations . This
requires that the following relations between anomaly
oe(cid:30)
ients be satis(cid:28)ed:
C C C C X
1 2 3 X
= = = = Tr , C = 0 ,
XXY
k k k k 12 (5)
1 2 3 X
C U(1) X
X X
where is the
oe(cid:30)
ient of the
ubi
anomaly, Tr is the
oe(cid:30)
ient of the mixed
U(1) k G
X a a
-gravitational anomaly, and is the Ka
-Moody level of the gauge group , whi
h
2
This
on
lusion
an be evaded if the Froggatt-Nielsen (cid:28)elds, instead of being ve
tor-like under both the
U(1) U(1)
X X
Standard Model gauge group and the group as usually assumed, are
hiralunder . In su
h a
ase
X
e(cid:27)e
tive operators
arrying a negative -
harge
an beUi(n1d)u
ed in the low-energyLˆe(cid:27)Lˆe
tHiˆveHˆtheory. Moreover,
X A B u u
even if the Froggatt-Nielsen(cid:28)elds areve
tor-likeunder , e(cid:27)e
tive operators with a negative
X
-
harge may be indu
ed by the seesaw me
hanism, sin
e right-handed neutrino super(cid:28)elds are
hiral under
U(1)
X
(see e.g. the model of Ref. [14℄).
U(1)
3
Anomalous 's are also
ommon in four-dimensional open string
ompa
ti(cid:28)
ations. However, while
heteroti
string
ompa
ti(cid:28)
ations
ontain at most one anomalousabelian gauge group fa
tor, open string
om-
U(1)
pa
ti(cid:28)
ations may
ontain several anomalous 's whose anomalies are
ompensated for by a generalized
Green-S
hwarzme
hanism [15℄. In this
ase, the Green-S
hwarzanomaly
onditions are less
onstraining than
U(1)
in the heteroti

ase, and the s
aleof breakingof the anomalous 'sdepends on twisted moduli vevs,whi
h
are(cid:28)xedbyunknownnonperturbativephysi
s. Forthesereasons,wepreferto
onsiderthe
aseofananomalous
U(1)
of the heteroti
type, with its anomalies
an
eled by the universal Green-S
hwarzme
hanism.
3
depends on the
ompa
ti(cid:28)
ation. The Ka
-Moody level of a non-abelian gauge group is an in-
teger, while the Ka
-Moody level of an abelian gauge group
an be fra
tional. In the following,
k = k
2 3
we shall assume , as is the
ase in most of (if not all of) the heteroti
string
ompa
ti(cid:28)-
U(1) U(1) U(1)
X X Y

ations
onstru
ted so far. Sin
e the - - anomaly
annot be
ompensated for
C
XXY
by the Green-S
hwarz me
hanism, its
oe(cid:30)
ient has to vanish by itself.
A ni
e feature of the Green-S
hwarz me
hanism is that the value of the Weinberg angle
C C
1 2
at the string s
ale is determined by the ratio of the anomaly
oe(cid:30)
ients and [16℄.
Indeed, upon
ombining the anomaly
onditions with the gauge
oupling uni(cid:28)
ation relation
k g2 = k g2 = = g2
1 1 2 2 ··· string valid at the string s
ale, one obtains:
C
sinθ2 (M ) = 2 .
W string C +C (6)
1 2
sin2θ = 3/8
W
The
anoni
al value of the Weinberg angle at the GUT s
ale ( ) is obtained for
C /C = 5/3 k /k = 5/3
1 2 1 2
, or equivalently for the normalization , whi
h is well known from
k = k
2 3
string model builders. As observed in Refs. [4, 5, 8℄, this normalization (together with )
mdmsmb ǫhu+hd
leads through Eq. (4) to memµmτ ∼ , whi
h given the un
ertainty due to the order one
h +h = 0, 1 2 h +h
4 u d u d

oe(cid:30)
ients is
onsistent with data for ± or even ± . Su
h values of are
µ
suitableforele
troweak symmetry breakingifthe -termisgenerated fromthe Giudi
e-Masiero
me
hanism [17℄. In the following, we shall (cid:28)x the Weinberg angle at its
anoni
al GUT value
C /C = 5/3
1 2
and impose .
U(1)
X
Another ni
e feature of an anomalous symmetry is that its breaking s
ale, hen
e
ǫ
the small expansion parameter , is determined by the value of the gauge
oupling at the
C X = 0
2
uni(cid:28)
ation s
ale and by the anomaly
oe(cid:30)
ient [14℄. Indeed, due to the fa
t that Tr 6 ,
a Fayet-Iliopoulos term is generated at the one-loop level [18, 19℄:
g2
ξ2 = string (XM2) ,
192π2 Tr (7)
g M X X < 0
string Φ
where is the string
oupling and the redu
ed Plan
k mass. Provided that Tr ,
φ Φˆ
this triggers a nonzero VEV of the s
alar
omponent of the super(cid:28)eld , whi
h sets the value
ǫ
of the parameter to [14℄:
g2 α
string 2
ǫ = X = C ,
s192π2 Tr 4π 2 (8)
r
k g2 = g2 C /k = X/12 X = 1
where we have used the relations 2 2 string and 2 2 Tr (re
all that Φ − ). As
ǫ λ 0.22
we shall see later, in realisti
models the value of is very
lose to the Cabibbo angle, ≃ .
We now look forsolutionsto the anomaly
onstraints. In addition,we require the absen
e of
U(1) (XY) = 0
kineti
mixing between the anomalous and the hyper
harge, i.e. Tr . As noti
ed
in Ref. [20℄, this
ondition also prevents the generation of a large one-loop Fayet-Iliopoulos
term for the hyper
harge [21℄ in the s
enario in whi
h the dominant
ontribution to soft s
alar
4
Assumingapproximatebottom-tauYukawa
ouplinguni(cid:28)
ationattheGUTs
ale,thismassratioisbetween
tanβ 5 60 h +h =0
u d
0.75 and 0.9 for the
entral values of the fermion masses in the range − , whi
h favours .
4
D
masses
omes from the anomalous -term. Su
h a Fayet-Iliopoulos term
ould have indu
ed
X

harge and
olour breaking minima. We lookfor models with the -
harges of allquark, lepton
andHiggssuper(cid:28)elds positive orzero, whi
h issu(cid:30)
ient toensure theuniqueness of theva
uum
U(1)
X
that breaks the anomalous (see Ref. [22℄ for a detailed dis
ussion of (cid:29)at dire
tions).
The relevant anomaly
oe(cid:30)
ients read:
1
¯
C = (q +8u¯ +2d +3l +6e¯ )+h +h ,
1 A A A A A u d
3 (9)
A
X
C = (3q +l )+h +h ,
2 A A u d
(10)
A
X
¯
C = (2q +u¯ +d ) ,
3 A A A
(11)
A
X
C = (q2 2u¯2 +d¯2 l2 +e¯2)+h2 h2 ,
XXY A − A A − A A u − d (12)
A
X
2
C = (6q3 +3u¯3 +3d¯3 +2l3 +e¯3 +n3)+h3 +h3 1+C′ ,
X 3 A A A A A A u d − X (13)
A
X
X = (6q +3u¯ +3d¯ +2l +e¯ +n¯ )+2(h +h ) 1+C′ ,
Tr A A A A A A u d − g (14)
A
X
C′ C′
where X and g stand for the
ontributions of additional SM singlets or ve
tor-like represen-
U(1)
X
tations
harged under (whi
h may be
harged under some hidden gauge group) to the
U(1) (XY)
X

ubi
anomaly and to the mixed gravitational anomaly, respe
tively. Finally, Tr
reads: ¯
(XY) = 2 (q 2u¯ +d l +e¯ )+2(h h ) .
A A A A A u d
Tr − − − (15)
A
X
5
The
onstraints :
3
C = C , C = C , C = 0 (XY) = 0 ,
2 3 2 1 XXY
5 and Tr (16)

an be rewritten as:
1
(u¯ q ) h = 0 ,
A A u
− − 2 (17)
A
X
1
¯
(d l ) h h = 0 ,
A A u d
− − 2 − (18)
A
X
(u¯ e¯ ) h = 0 ,
A A u
− − (19)
A
X
(q2 2u¯2 +d¯2 l2 +e¯2)+h2 h2 = 0 .
A − A A − A A u − d (20)
A
X
C′ C′ k
5We assume that X, g and the Ka
-Moody level X are su
h that the Green-S
hwarz
onditions are
U(1)
X
satis(cid:28)ed for the
ubi
anomaly and for the mixed gravitational anomaly.
5
We look forsolutionsto Eqs. (17)-(20) whi
h su

essfully reprodu
e the observed quark and
lepton mass and mixing hierar
hies with all matter
harges non-negative. Sin
e the Yukawa
U(1)
X

ouplings are generated at the s
ale of breaking of the anomalous (whi
h for de(cid:28)niteness
M 2 1016
GUT
we identify with the s
ale at whi
h gauge
ouplings unify in the MSSM, ≃ ×
GeV), we must
onsider the masses and mixings renormalized also at this s
ale. In the quark
se
tor, the renormalization of the CKM matrix a(cid:27)e
ts, to a very good approximation, only the
A λ ρ η
parameter of the Wolfenstein parametrization, while , and almost do not evolve with
m /m m /m
d s u c
energy [23, 24, 25, 26℄. The s
ale dependen
e of the quark mass ratios and is
tanβ 3.5 50
also very weak. Numeri
ally for values in the range − we (cid:28)nd:
m (µ) m (M ) m (µ) m (M )
s = χ(µ) s Z , c = χ3(µ) c Z ,
m (µ) m (M ) m (µ) m (M ) (21)
b b Z t t Z
χ(M ) (0.75 0.9)
GUT
with ≃ − . This leads to the following approximate hierar
hies of quark
mixingangles and mass ratios
lose to the GUT s
ale, expressed inpowers of the Cabibbo angle
λ 0.22
≃ :
V V λ , V λ4 , V λ2 , V λ4 λ3 , V λ2 ,
us cd ub cb td ts
| | ≃ | | ≃ | | ∼ | | ∼ | | ∼ − | | ∼ (22)
m m 1 m m
d λ2 , s λ2 , u λ4 , c λ4 .
m ∼ m ∼ 2 m ∼ m ∼ (23)
s b c t
Forlightquarkswehaveusedherethe
entralvaluesofthePDGestimates[27℄. Thedependen
e
tanβ
on of the GUT s
ale values of these mass ratios and CKM elements is weak in the
tanβ

onsidered range of values.
In the lepton se
tor, the
harged lepton mass hierar
hy is:
m 1 m
e λ3 , µ λ2 .
m ∼ 2 m ∼ (24)
µ τ
m ∆m2 m ∆m2
If one assumes a hierar
hi
alneutrino mass spe
trum, i.e. ν3 ≈ atm and ν2 ≈ sol,
the s
ale dependen
e of lepton mixing angles and neutrino maspses is very weak [28,p26℄, and
one
an take, at the s
ale where right-handed neutrinos de
ouple:
m m
U 1 , (α,A) = (e,3) , U < λ , ν2 λ , ν1 < 1 .
αA e3
| | ∼ 6 | | m ∼ m (25)
ν3 ν2
X
Eqs. (17) to (20), together with the assumption of positive -
harges for the MSSM su-
per(cid:28)elds, are very restri
tive and only a limited number of
harge assignments
ompatible with
these
onstraints give a satisfa
tory a

ount of the observed mass and mixing hierar
hies. Let
us (cid:28)rst
onsider the quark se
tor. The large top quark Yukawa
oupling is a

ounted for by
q = u¯ = h = 0
3 3 u

hoosing . Furthermore, the quark mass ratios and mixing angles are given
by the simple expressions:
muA ǫ(qA−qB)+(u¯A−u¯B) , mdA ǫ(qA−qB)+(d¯A−d¯B) , mt tanβ ǫ−(d¯3+hd) ,
muB ∼ mdB ∼ mb ∼
V ǫ|qA−qB| .
AB
∼ (26)
6
The symbol ∼ reminds us that the above relations
ontain unknown fa
tors of order one, so
that the a
tual values of the mass ratios and mixing angles may slightly depart from the naive
ǫ λ
(cid:16)power
ounting(cid:17). With this remark in mind, and assuming ≃ , we (cid:28)nd that the CKM
q = (3,2,0) q = (4,3,0) q = (4,2,0)
A A A
matrix is
orre
tly reprodu
ed by , or by . In ea
h
ase,
λ
the naive (cid:16)power
ounting(cid:17) disagrees by at most one power of with the measured value of
V q = (3,2,0) V V q = (4,3,0) V V
ub A cb ts A us cd
one or two CKM angles: | | for , | | and | | for , | | and | |
q = (4,2,0)
A
for . In the lepton se
tor, the experimental data on neutrino os
illations strongly
l
A

onstrain the (for a re
ent review, see e.g. Ref. [29℄). In the absen
e of holomorphi
zeroes in
the Dira
and Majorana mass matri
es, the seesaw me
hanism yields an e(cid:27)e
tive light neutrino
mass matrix of the form:
ǫ2(l1−l3) ǫ(l1−l3)+(l2−l3) ǫ(l1−l3)
v2ǫ2(l3+hu)
M u ǫ(l1−l3)+(l2−l3) ǫ2(l2−l3) ǫ(l2−l3) ,
ν
∼ M   (27)
R ǫ(l1−l3) ǫ(l2−l3) 1
 
M
R
where isthes
aleofright-handedneutrinomasses. Su
h amassmatrix
aneasilyreprodu
e
the hierar
hi
al neutrino mass spe
trum. In order to reprodu
e both the large atmospheri
mixing angle and the hierar
hy between the atmospheri
and solar mass s
ales, one must
l = l 2 2
2 3

hoose Mand allow for a m∆imld2t/u∆nimng2betwe0e.n2 the order one entries in the lower right ×
submatrix of ν, of order sol atm ≈ . Due to this tuning, the solar mixing angle
θ l l = 1 2
12 1 3

omes out (cid:16)large(cid:17) providped that − or (see Appendix B for details). This in turn
U ǫl1−l3
e3
implies that | | ∼ should be rather large, and even
lose to its present experimental
l l = 1
1 3
limit in the
ase − .
q l
A A
Foragiven
hoi
eofthe andthe di
tatedbytheCKMandPMNS mixingmatri
es,the
X
remaining -
hargesof themodelare
onstrained bothby the quark and
harged leptonmasses
X
and by Eqs. (17) to (20). We list below the solutions for the -
harge assignments satisfying
the
onstraints (17) to (20) for whi
h the predi
tions obtained from naive power
ounting are
in reasonably good agreement with experimental data on fermion masses and mixings, when
extrapolated to the GUT s
ale:
¯
q = u¯ = e¯ = (3,2,0), d = l = (l ,l ,l ), h = h = 0,
A A A A A 1 3 3 u d
1:
[l l = 1 2; l = 0,1,2 3]
1 3 3
− or or (28)
¯
q = u¯ = e¯ = (3,2,0), d = (2,1,0), l = (1,0,0), h = 0, h = 2,
A A A A A u d
2: (29)
¯
q = u¯ = e¯ = (3,2,0), d = (1,1,0), l = (1,0,0), h = 0, h = 1,
A A A A A u d
3: (30)
¯
q = u¯ = e¯ = (3,2,0), d = (2,1,0), l = (2,0,0), h = 0, h = 1,
A A A A A u d
4: (31)
¯
q = u¯ = e = (4,2,0), d = l = (l +1,l ,l ), h = h = 0,
A A A A A 3 3 3 u d
5:
[l = 0,1,2 3]
3
or (32)
¯
q = u¯ = e¯ = (4,2,0), d = (1,1,0), l = (1,0,0), h = 0, h = 1.
A A A A A u d
6: (33)
Several of these solutions
an be found in the literature. The
onstraints (17) to (20) are
h = h = 0 SU(5)
u d
automati
ally satis(cid:28)ed if and the horizontal symmetry
ommutes with , as
already noti
ed in Ref. [20℄. This is the
ase for solutions 1 and 5. We were not able to (cid:28)nd
h = h = 0 SU(5)
u d
solutions with that are not
ompatible with the symmetry. In all solutions,
7
the naive predi
tions slightly depart from the observed values for some quantities, and one has
to rely on the e(cid:27)e
t of the un
onstrained order one parameters to
orre
t them. In parti
ular,
m /m λ2 V λ3
u c ub
solutions 1 to 4 naively predi
t ∼ and | | ∼ , while solutions 5 and 6 predi
t
V V λ2 m /m l l = 1
us cd u c 1 3
| | ∼ | | ∼ , but give the
orre
t ratio. In addition, solution 1 with − and
m /m λ2 l l = 2 m /m λ3
e µ 1 3 d s
solutions 2, 3 yield ∼ ; solution 1 with − and solution 5 yield ∼ ,
m /m λ l l = 1 l l = 2
d s 1 3 1 3
and solution 3 yields ∼ . Finally, − is preferred over − by the solar
mixing angle [30℄.
l = 2 tanβ tanβ . 15
3
Solutions 1 and 5 with and solution 2 predi
t a low value of , ;
l = 1 tanβ (15 50)
3
solutions 1 and 5 with and solutions 3, 4 and 6 require ∼ − ; and solutions
l = 0 tanβ & 50 tanβ
3
1 and 5 with require . The range of values
ompatible with a given
solution is rather broad due to the e(cid:27)e
t of the order one
oe(cid:30)
ients.
ǫ
UsingEq. (8), one
an
omputethepredi
tedvalueoftheexpansionparameter . Assuming
α (M ) = α = 1 ( αU )1/2
2 string U 24, one obtains (the quoted values s
ale as 1/24 ):
ǫ = 0.23, 0.25, 0.27 l l = 1 l = 0,1,2,
1 3 3
for solution 1 with − and (34)
ǫ = 0.24, 0.26, 0.28 l l = 2 l = 0,1,2,
1 3 3
for solution 1 with − and (35)
ǫ = 0.24 ,
for solution 2, 3 and 4 (36)
ǫ = 0.25, 0.27, 0.29 l = 0,1,2,
3
for solution 5 with (37)
ǫ = 0.26 .
for solution 6 (38)
As already stressed in Ref. [14℄, it is a remarkable su

ess of (cid:29)avour models based on an
U(1) ǫ
anomalous that the predi
ted value of
omes out so
los¯e to the Cabibbo angle. Note
q = u¯ = e¯ = (2,1,0) l = d = (1,0,0) + l
A A A A A 3
that the
harge assignment , , sometimes
ǫ = (0.18 0.23)

onsidered in the literature, predi
ts − , whi
h is too large for this
harge
assignment to be
ompatible with the observed fermion masses and mixings.
3 Pre
ision des
ription of fermion masses and mixings
The fa
t that only a small number of horizontal
harge assignments are phenomenologi
ally
a

eptable, together with the interesting theoreti
al aspe
ts dis
ussed in the previous se
tion,
U(1)
makes the simplestanomalous modelsforfermionmasses worth further, morequantitative
study. Su
h a study may also be useful for studying various aspe
ts of the supersymmetri
(cid:29)avour problem and CP violation with hierar
hi
al fermion mass matri
es.
The
harge assignments 1-6 in Eqs. (28)-(33) predi
t the hierar
hi
al stru
ture of the quark
CAB
and
harged lepton Yukawa
ouplings up to order one fa
tors u,d,e, whi
h should be viewed as
(cid:28)xed by some unknown physi
s whi
h has been integrated out. The freedom in these fa
tors
6

an be used to obtaina pre
ise des
riptionof the fermionmasses and mixings ; itis the purpose
CAB
of this se
tion to (cid:28)nd su
h sets of
oe(cid:30)
ients u,d,e and to dis
uss their properties.
6
Foraquantitativestudyof the predi
tions in the neutrinose
tor,werefer to Ref. [30℄, whi
h howeveronly
l1 l3 = 1

onsidered the
hoi
e − ((cid:16)semi-anar
hi
al(cid:17)
ase). We argued, on the basis of the analyti
al formulae
l1 l3 =2
given in appendix B, that the
hoi
e − also leads to a satisfa
tory des
ription of neutrino masses and
8
U(1)
X
Before doing so, let us noti
e that, among the a

eptable
harge assignments, two
SU(5) U(1)
X
(namely
harge assignments 1 and 5) are
onsistent with × symmetry, and
an
therefore be re
on
iled with Grand Uni(cid:28)
ation of elementary for
es. Moreover, those are the
h = h = 0
u d
only a

eptable
harge assignments for , for whi
h, as mentioned earlier, there
is some (although not strong) phenomenologi
al preferen
e. Charge assignments 3 and 6,
SU(5)
although not
ompatible with symmetry, are interesting in the
ontext of the supersym-
metri
(cid:29)avour problem be
ause the right-handed down and strange quark super(cid:28)elds have the
same horizontal
harge, whi
h in some supersymmetry breaking s
enarios may suppress the
squark
ontribution to kaon mixing.
SU(5)
If one interprets the fa
t that the
harge assignments 1 and 5 are
ompatible with ×
U(1)
X
symmetry as the manifestation of an underlying Grand Uni(cid:28)ed Theory, one should
CAB = CBA
u u
impose the following (GUT-s
ale)
onstraints on the order one
oe(cid:30)
ients: and
CAB = CBA SU(5) m /m = m /m
d e . The se
ond
onstraint leads to the well-known relations e µ d s
m /m = m /m
µ τ s b
and , whi
h are in gross disagreement with the measured fermion masses
and must be
orre
ted [31℄. This
an be done through the
ontribution of renormalizable [31℄
or non-renormalizable [3U2℄(1o)perators to the YukΣaˆwa matri
es. Follow7i5ng RSeUf.([53)3℄, we shall
X
introdu
e an additional singlet super(cid:28)eld tra5¯ns1f0orHm¯iΣˆn/gMas a of , whi
h has
non-renormalizable
ouplings to fermions of the form . The Yukawa
ouplings of
SU(5) U(1)
X
the down-type quarks and
harged leptons then arise from the two × invariant
superpotential terms:
Σˆ Φˆ d¯A+qB+hd
W = 5¯ACAB10BH¯ + 5¯ACAB10BH¯ ,
1 M 2 ! M! (39)
Φˆ Σˆ
whi
h, after the s
alar
omponents of and a
quire VEVs, lead to:
YAB = CAB +κCAB ǫd¯A+qB+hd,
d 1 2
YeAB = (cid:0)C1BA −3κC2BA(cid:1) ǫd¯B+qA+hd, (40)
(cid:0) (cid:1)
κ Σ /M κ = 0.3
where ≡ h i . In our numeri
al (cid:28)ts, we take , whi
h makes it easy to a

ount for
the di(cid:27)eren
e between down-type quark and
harged letpon masses.
U(1)
X
In order to test in a quantitative way the ability of the symmetry to des
ribe the
24 CAB = CBA CAB
fermion masses and mixings, we look for sets of
omplex
oe(cid:30)
ients u u , 1 and
CAB 9 4
2
that reprodu
e the quark and
harged lepton masses and the parameters of the CKM
matrix. It is also interesting to
he
k how well the various
harge assignments reprodu
e the
observed values of the fermion masses and mixings for randomly generated
oe(cid:30)
ients. Thi¯s is
l = d
A A
illustrated in Figs. 1 and 2 for
harge assignments 1 (with two di(cid:27)erent
hoi
es for ),
CAB
5 and 6. For randomly generated sets of
oe(cid:30)
ients { u,1,2} with arbitrary phases and moduli
0.3 3
in the range − , ea
h histogram shows the relative number of sets of
oe(cid:30)
ients leading
R(Q) Q(M ) /Q(M )
GUT predicted GUT evolved
to a given value of the ratio ≡ for some parti
ular
mixings. In the following, we shallnot dis
ussthe neutrino se
toragainand shall fo
us on the
hargedfermion
se
tor.
9