F-term Induced Flavor Mass Spectrum

0.16 MB·

by  Naoyuki Haba

OU-HET 596/2008
F
-term Induced Flavor Mass Spectrum
8
0
0
Naoyuki Haba and Yoshio Koide†
2

n
a
J
Department of Physics, Osaka University, 1-1 Machikaneyama, Toyonaka, 560-0043, Japan
2
2 [E-mail address: [email protected]]

] †IHERP, Osaka University, 1-16 Machikaneyama, Toyonaka, 560-0043, Japan
h
p [E-mail address: [email protected]]
-
p
e
h
Abstract
[

1
v
New mechanism of generating flavor mass spectrum is proposed by us-
1
0 ing an O’Raifeartaigh-type supersymmetry breaking model. A desired
3
bilinear form of fermion mass spectrum is naturally realized through
3
. F-components of gauge-singlet (nonet of SU(3) flavor symmetry) su-
1
0 perfields, and the suitable charged-lepton mass relation is reproduced.
8 The charged-slepton mass spectrum is non-degenerate in general, and
0
: can be even hierarchical (proportional to the charged-lepton masses in
v
the specific case). Flavor changing neutral processes are suppressed since
i
X
the charged-lepton and slepton (except for right-handed sneutrino) mass
r
a matrices are diagonalized simultaneously in the flavor space. The right-
handed sneutrinos are light with the similar ratio to the lepton sector
(m˜ /m˜ m /m ).
νR e ∼ ν e
1 Introduction
Investigating an origin of a flavor mass spectrum will provide an important clue to the un-
derlying theory of quarks and leptons. In the standard model, the flavor mass spectrum is
originated in the structure of Yukawa coupling constants Yf (f = u,d,ν,e; i,j = 1,2,3),
ij
where fermion mass matrices Mf are given by Mf = Yf H (H is the Higgs doublet). When
ij ijh i
there is a flavor symmetry in the mass matrices Mf, we have predictions for the masses and
mixings.
On the other hand, there is another idea for the origin of the flavor mass spectrum, which
is originated in a structure of vacuum expectation values (VEVs) of scalar fields (all couplings
are of (1)). This paper would like to take this approach, and one of the authors (YK) has
O
proposed a prototype of such a model[1], where there are three Higgs doublets with VEVs of
v = H0 satisfying
i h ii
2
v2 +v2 +v2 = (v +v +v )2. (1.1)
1 2 3 3 1 2 3
It can lead to the charged-lepton mass relation[2],
2
m +m +m = (√m +√m +√m )2, (1.2)
e µ τ e µ τ
3
where Me = diag.(m ,m ,m ) v2, which gave a remarkable prediction of m = 1776.97
e µ τ ∝ i τ
MeV from the observed values of m and m .1 However, a model with multi-Higgs doublets
e µ
causes the large flavor changing neutral current (FCNC) in general[4]. Therefore, we must
change the scenario in which the origin of the mass spectrum is separated from the electro-
weak symmetry breaking. A typical example is the Froggatt-Nielsen (FN) model[5], which
has new scalars φ and their VEVs (v ) induce mass spectrum which has nothing to do with
i
the electro-weak symmetry breaking. For example, in Ref.[6], a U(3) -flavor symmetry has
F
been introduced, where φ is regarded as a U(3) -nonet field. Then Yukawa interactions of the
F
charged-lepton sector are induced from a higher dimensional operator [y (φ φ /Λ2)L H E ]
0 ik kj j d i F
in the supersymmetry (SUSY) theory, where Λ is the cutoff scale of this effective Lagrangian.2
L and E are SU(2) -doublet and singlet lepton superfields, respectively, and H is the Higgs
L d
doublet which couples to the charged-lepton superfields. This model can avoid the FCNC
problem. However, we cannot explain why only the bilinear term of FN field (φ φ ) is
ik kj
introduced in the superpotential.3
In this paper, we propose another mechanism which naturally induces bilinear form of
mass spectrum. For example, the charged-lepton masses are induced through F-components
1The observed value is mobs =1776.99+0.29 MeV[3].
τ −0.26
2Reference [7] proposed a similar model but without higher dimensional operators.
3AcontinuousflavorsymmetryinducesmasslessNGbosons,whileadiscreteflavorsymmetrycannotforbid
higher order terms, such as φ4 in Z -symmetry.
2
1
of the superfield φ as
φ†
ij
y L H E . (1.3)
0Λ2 j d i
" #
D
Since a F-component has mass dimension two, so that Eq.(1.3) can be a good candidate to
reproduce the mass relation of Eq.(1.2).4 The vacuum of this model induces the mass relation
Eq.(1.2) as well as SUSY-breaking, so we can expect fruitful byproducts, such as sfermion
mass spectrum.
2 A Model
We mainly focus on the lepton sector. An application to the quark sector might be possible.
Let us start showing our model.
2.1 Lagrangian
We adopt the following O’Raifeartaigh-type superpotential[9],
W(Φ,A,B) = W (Φ )+λfTr[A Φ Φ ]+λf Tr[B Φ Φ ] µ2Tr[ξ A ] . (2.1)
Φ f A f f f B f f f − f f f
fX=u,d(cid:16) (cid:17)
The K¨ahler potential is set as
= + + , (2.2)
0 1 Y
K K K K
= Tr[A†A ]+Tr[B†B ]+Tr[Φ†Φ ]+H†H +L†L+E†E +N†N, (2.3)
K0 f f f f f f f f
fX=u,d(cid:16) (cid:17)
1
= Tr[A†A ]2 +Tr[B†B ]2 , (2.4)
K1 −Λ2 f f f f
fX=u,d(cid:16) (cid:17)
1 1
= Tr ydA† +ydB† LH E + Tr yuA† +yuB† LH N +h.c., (2.5)
KY Λ2 A d B d d Λ2 A u B u u
h(cid:16) (cid:17) i
(cid:2)(cid:0) (cid:1) (cid:3)
where Φ , A and B are U(3) -nonet superfields. ξ is a 3 3 numerical matrix while all
f f f F f
×
other couplings (y’s, λ’s) are complex numbers (not matrices). W is a superpotential which
Φ
contains only Φ . is introduced to realize A = B = 0 and also avoids massless scalers
f 1 f f
K h i h i
as will be shown later.5 Tr[A†LH E] and Tr[B†LH E] in play crucial roles of generating
d d d d KY
effective Yukawa interactions for the charged-leptons and neutrinos.
We take R-charge assignments as R(A ) = R(B ) = 2 and R(Φ ) = 0 in order to
f f f
forbid phenomenologically unwanted interactions, Tr[A2], Tr[A3], Tr[A2Φ ], Tr[A LH E],
f f f f d d
4A similar type of Eq.(1.3), [y′φ†ijL H N ] , was used for tiny Dirac neutrino masses in Ref.[8], where N
0Λ2 j u i D
is the right-handed neutrino and H is the Higgs doublet which constructs up-type Yukawa interactions.
u
5The minus sign of is assumed to be derived by the underlying theory.
1
K
2
Tr[B LH E], and so on. We also introduce an additional Z -symmetry, under which only
d d 2
Φ transforms as odd-parity fields. It forbids the terms, Tr[A Φ ], Tr[B Φ ], Tr[Φ LH E],
f f f f f d d
and so on. As for a tadpole term, µ′2Tr[ξ B ] can be eliminated by the field redefinition of
f f
A and B .6 We will omit the indices “u” and “d” when they are obvious.
f f
2.2 Vacuum of the model
The scalar potential (V = ) is given by
scalar
−L
∂W ∂W ∂W
V = Tr F +Tr F +Tr F . (2.6)
A B Φ
− ∂A ∂B ∂Φ
(cid:18) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:19)
F’s are calculated from the equations of motions (∂ /∂F = 0) as
L
∂W
†
F + = 0, (2.7)
Φ ∂Φ
∂W 2 1
F† + Tr[A†A]F† +Tr[F†A]A† + y∗ † = 0, (2.8)
A ∂A − Λ2 A A Λ2 AX
∂W 2 (cid:16) (cid:17) 1
F† + Tr[B†B]F† +Tr[F†B]B† + y∗ † = 0, (2.9)
B ∂B − Λ2 B B Λ2 BX
(cid:16) (cid:17)
where = F HE + LF E + LHF . Notice that V in Eq.(2.6) contains fields L, E and H
L H E
X
through the K¨ahler potential, . The equation of motion of F† is given by
KY L
∂W 1
F† + + HE(y F† +y F†) = 0, (2.10)
L ∂L Λ A A B B
and equations of motions of F† and F† are similar to it.7
H E
It is hard to obtain exact solutions of Eqs.(2.8) and (2.9), thus we use a perturbation
method of expanding 1/Λn. Then we obtain
∂W 2 ∂W ∂W 1
F† = + Tr[A†A] +Tr A A† + , (2.11)
A − ∂A Λ2 ∂A ∂A O Λ4
(cid:18) (cid:20) (cid:21) (cid:19) (cid:18) (cid:19)
and the similar equation of B (which is calculated by a replacement of A B). As Eq.(2.1)
→
derives
∂W ∂W
Φ
= +λ (AΦ+ΦA)+λ (BΦ+ΦB), (2.12)
A B
∂Φ ∂Φ
∂W
= λ ΦΦ µ2ξ, (2.13)
A
∂A −
∂W
= λ ΦΦ, (2.14)
B
∂B
6Precisely speaking, we should determine the K¨ahler potential Eq.(2.2) after this redefinition.
7Of cause Eqs.(2.8) (2.10) are also obtained directly from the inverse of the K¨ahler potential, such as
∼
F† = −1 dW/dA.
A −KAA†
3
the scalar potential becomes
V = V +V , (2.15)
0 1
2
∂W
V = Tr Φ +λ(CΦ+ΦC) +Tr λ ΦΦ µ2ξ 2 + λ ΦΦ 2 , (2.16)
0 A B
∂Φ | − | | |
" #
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:2) (cid:3)
2 (cid:12) (cid:12)
V = (cid:12) λ 2Tr[A†A]+ λ 2(cid:12)Tr[B†B] Tr[ΦΦΦ†Φ†]
1 Λ2 | A| | B|
(cid:8)λ(cid:0)∗µ2Tr[A†A]Tr[ξΦ†Φ†]+h.c. +(cid:1) µ2 2Tr[A†A]Tr[ξ†ξ]
− A | |
+(cid:0)λ Tr[AΦΦ] µ2Tr[Aξ] 2 + λ(cid:1) 2 Tr[BΦΦ] 2 , (2.17)
A B
− | | | |
where C and C′ are d(cid:12)(cid:12)efined by (cid:12)(cid:12) o
λC λ A+λ B, λC′ = λ A+λ B (2.18)
A B B A
≡ −
with λ = λ2 +λ2. Notice that V (V ) is the potential of (Λ0) ( (Λ−2)).
A B 0 1 O O
Nine scalars of C′ are massless in the tree level potential V , because Eq.(2.16) does not
p 0
contain C′. Thus, in the direction of C′, A and B are not determined at (Λ0). They are
h i h i O
determined byincluding (Λ−2)corrections ofV . StationaryconditionsdV/dA = dV/dB = 0
1
O
decide the vacuum at
A = B = 0 (2.19)
h i h i
with the condition
∂W
Φ
= 0. (2.20)
∂Φ
Under the Eqs.(2.19) and (2.20), the condition of dV/dΦ = 0 determines Φ as
h i
λ
Φ Φ = A µ2ξ. (2.21)
h ih i λ2 +λ2
A B
Then the height of the scalar potential at this vacuum is given by
λ2
V = 1 A µ4 Tr[ξ†ξ]. (2.22)
min − λ2 +λ2
(cid:18) A B(cid:19)
We will show that the minimum exists at L = E = 0 in Section 5, and then F†, F†
h i h i Φ A
and F† are given by
B
∂W ∂W λ ∂W
F† = = 0, F† = = λ BΦΦ, F† = = λ ΦΦ, (2.23)
Φ − ∂Φ A − ∂A Bλ B −∂B − B
A
from Eqs.(2.7) (2.9), respectively. Thus, the effective Yukawa interaction of the charged-
∼
lepton sector is given by
λd Φ Φ
(ydλd ydλd) B h dikih dkjiL H E , (2.24)
A B − B A λd Λ2 j d i
A
4
which is the desirable bilinear form and might produce the mass relation m v2 (v =
ei ∝ di di
Φ ) in the diagonal basis of Φ .
dii d
h i h i
Notice that Eq.(2.21) seems to imply the flavor indices of Φ 2 are completely fixed by
h i
the parameter ξ. However, it is not correct because Eq.(2.20) must be also satisfied at the
vacuum. We take a standpoint that the parameter ξ is basically free, and it is constrained by
W (Φ) through the relation of Eq.(3.6) which will be shown in Section 3.
Φ
U
2.3 Mass spectra of (3) -nonet fields
F
Now let us calculate fermion masses of U(3) -nonet particles of Φ, A and B which are denote
F
as ψ , ψ and ψ , respectively. The fermion masses are induced from the 2nd and 3rd terms
Φ A B
in Eq.(2.1) as
= λ (ψ ) (v +v )(ψ ) +2λ (ψ ) vˆ (ψ ) , (2.25)
mass C ij i j Φ ji d Φ ii j Φ jj
L
i,j i j
X X X
where vˆ = v (v + v + v )/3 and v = Φ . The superfields A and B couple to Φ only
i i 1 2 3 i ii
− h i
through the term Tr[(λAA+λBB)ΦΦ] in the superpotential W so that ψC′ are massless while
ψ have Dirac masses of ( Φ ) with ψ . This situation is not affected by the existence of
C Φ
O h i
the K¨ahler potential due to A = B = 0.
1
K h i h i
By taking account of R-parity conservation, A and B should be regarded as R-parity
even. Thus the massless ψC′ are the lightest superparticle (LSP). We should remind that
one degree of freedom in ψC′ is absorbed into the longitudinal mode of the gravitino, and its
mass becomes m F/M (M : four-dimensional Planck scale). Other eight components
3/2 P P
≃
of ψC′ are remaining as massless fermions, which seem problematic from the view point of
phenomenology. But, is it true? ψC′ interacts with the leptons through the 1st and 2nd terms
of Eq.(2.5) which contains the interactions
1
(Tr[∂ γµψ† ψ H E]+Tr[∂ γµψ† Lψ E]+Tr[∂ γµψ† LH ψ ]). (2.26)
L ∋ Λ2 µ Cd′ L d µ Cd′ Hd µ Cd′ d E
These interactions in the diagrams with ψC′ in the external lines vanish by using the equation
of motion, ∂µγµψC′ = 0.8 Thus, although ψC′ are massless LSP, the decay processes to ψC′
d
are strongly suppressed. The FCNC processes, such as µ eγ, mediated by ψC′ in the loop
→
diagrams are also suppressed due to the small coupling of (lepton)-(slepton)-(ψC′) and no
chiral-flip of ψC′.
Next let us estimate the scalar masses of nonet fields. The scalar masses of Φ are of order
8Theinteractions 1 (Tr[ψ† ∂ γµψ H E]+Tr[ψ† L∂ γµψ E]+Tr[ψ† LH ∂ γµψ ])alsovanishbyusing
Λ2 C′ µ L d C′ µ H C′ d µ E
d d d
partial integral.
5
Φ which are induced from
h i
2
4λ2 v2 Φ +2λ2 (v v +δ )Φ Φ , (2.27)
d i jj i j ij ij ji
! !
i j i,j
X X X
in the diagonal basis of Φ . On the other hand, the masses of C and C′ are of orders Φ and
h i h i
Φ 2/Λ, respectively, which are induced from
h i
λ2 Tr[ Φ 4]
2λ2 Tr[ Φ Φ CC + Φ C Φ C]+ B h i Tr[AA†]+(Tr[BB†] . (2.28)
h ih i h i h i λ2 Λ2
(cid:26) A (cid:27)
(cid:0) (cid:1)
It should be emphasized that C′ scalars become massive through V . Although C and C′ in-
1
teract with sleptons through the K¨ahler interactions 1 Tr[∂2C†LH E] and 1 Tr[∂2C′†LH E],
Λ2 d Λ2 d
respectively, their contributions to the FCNC processes are negligibly small. It is because
these interactions only exit in higher loops as well as C and C′ are heavy enough.
3 Lepton mass spectra
The mass relations of quarks and leptons (and also their superpartners) are generated in
our model. A main purpose of this paper is to propose a new mechanism of generating the
bilinear form of the mass spectrum. So we briefly show the derivation of the charged-lepton
mass relation in Eq.(1.2).
For our goal, we take the simplest superpotential W as9
Φ
2
W (Φ ) = λ Tr[Φ ] Tr[Φ Φ ] (Tr[Φ ])2 +m Tr[Φ Φ ]. (3.1)
Φ d d d d d d d d d
− 9
(cid:18) (cid:19)
The form of the cubic term (λ -term) in Eq.(3.1) is equivalent to
d
(8) (8) (8)
λ Tr[Φ Φ Φ ] Tr[Φ Φ Φ ] , (3.2)
d d d d − d d d
(cid:16) (cid:17)
(8) (8) (8) (8)
where Φ = Φ Tr[Φ ]/3. We assume to drop Φ Φ Φ -term from the U(3) -invariant
d d − d d d d F
cubic term Tr[Φ Φ Φ ]. It might be possible by imposing an additional symmetry as shown
d d d
in Ref.[6].10
From Eq.(3.1), we obtain
∂W
Φ = 0 = cdΦ +cd1, (3.3)
∂Φ 1 d 0
d
9We assume R- and Z -symmetries are (spontaneously) broken only in W (Φ ) sector, in which R(λ ) =
2 Φ f d
R(m ) = 2 and Z -parity odd λ are induced from VEVs of some unknown fields possessing R-charges and
d 2 d
Z -parity odd.
2
10Another Z -parity with +1 for Φ(1) =Tr[Φ ]/3 and 1 for Φ(8) worked well.
2 d d − d
6
where
cd = 2(m +λ Tr[Φ ]), (3.4)
1 d d d
2
cd = λ Tr[Φ Φ ] (Tr[Φ ])2 . (3.5)
0 d d d − 3 d
(cid:18) (cid:19)
The coefficients cd and cd in Eq.(3.3) must be zero[10] in order to obtain non-zero and non-
1 0
degenerate eigenvalues of Φ . Then, the condition cd = 0 just gives the VEV-relation of
h di 0
Eq.(1.1) in the diagonal basis of Φ .
d
h i
WeshouldalsonoticethatEqs.(2.20) and(2.21)requireonerelationamongsixparameters,
λ ,λ ,µ ,ξ , and λ ,m in W (Φ ) as
A B d d d d Φ d
λ µ2 2m2
A d Tr[ξ ] = d. (3.6)
λ2 +λ2 d 3 λ2
A B d
Taking the same order of µ and m might be natural for the setup of the present model.
d d
4 Slepton mass spectra
Now let us investigate the slepton mass spectra in our model. The K¨ahler potential of (Λ−2)
O
derives the SUSY breaking slepton masses
1
= Tr (ydA +ydB )L 2 +Tr ydA +ydB 2 Tr[ L 2]
KL Λ2 1 d 2 d 3 d 4 d | |
+Tr(cid:16) (yhu(cid:12)A +yuB )L 2 (cid:12)+iTr yuhA(cid:12) +yuB 2 (cid:12)Tri[ L 2]
| 1(cid:12) u 2 u | (cid:12) | 3 (cid:12) u 4 u| (cid:12) | |
+Tr(cid:2) (ydA +ydB )E 2(cid:3) +Tr(cid:2) ydA +ydB 2(cid:3) Tr[ E 2]
5 d 6 d 7 d 8 d | |
+ Trh(cid:12) (yuA +ydB )N(cid:12) i2 +Trh(cid:12) yuA +yuB(cid:12) i2 Tr[ N 2] (4.1)
(cid:12) 5 u 6 u (cid:12) (cid:12)| 7 u 8 (cid:12)u| | |
h i (cid:17)
(cid:12) (cid:12) (cid:2) (cid:3)
as well as makes L and E(cid:12) zero in the scal(cid:12)ar potential.
h i h i
Reminding that the charged-lepton masses Me have been given by
i
λd v2
Me = (ydλd ydλd) B di H (4.2)
i A B − B A λd Λ2h di
A
from Eq.(2.24), the 1st and 2nd terms of Eq.(4.1) induce the left-handed slepton masses as
ydλd ydλd 2 Λ2 ydλd ydλd 2 Λ2
1 B − 2 A e˜ (Me)2 e˜ + 3 B − 4 A e˜ e˜ (Me)2 . (4.3)
ydλd ydλd H 2 Li i Li ydλd ydλd H 2 Li Li k
(cid:18) A B − B A(cid:19) h di (cid:18) A B − B A(cid:19) h di k
X
The1stterminducesgeneration-dependentmasses(proportionaltothecharged-leptonsquared
masses) while the 2nd term gives the universal soft mass. The neutrino masses Mν are given
i
7
by the similar equation to Eq.(4.2), and the matrix Φ is not diagonal in the diagonal basis
u
h i
of Φ because the neutrino mixing matrix U is not U = 1. So the slepton mass matrix
d ν ν
h i
from the 3st and 4th terms in Eq.(4.1) is not diagonal in general. Here we take a standpoint
that neutrinos are Dirac particles with tiny Dirac masses. It is because an introduction of Ma-
jorana masses of right-handed neutrinos heavier than Λ might be unnatural11 (See Eq.(4.8)).
In this case a small value of µ2 is required which induces the small values of Φ 2 as shown in
u h ui
Eq.(2.21), and then non-degenerate effects from the neutrino sector are negligibly small due
to the lepton mass relation m2 /m2 < 10−14 (even for the inverted-hierarchical neutrino mass
νi ei
spectrum). Then the contributions from A and B in the 3rd and 4th terms in Eq.(4.1) can
u u
be neglected.12
Therefore the left-handed slepton masses (m˜ )2 have the form13
LL i
(m˜ )2 = k [(Me)2 +m2 ] (4.4)
LL i L i L0
where m2 is the universal soft mass for all three generations. It should be noticed that non-
L0
degenerate masses can dominate the universal masses, in which charged-slepton masses are
almost proportional to the charged-lepton masses. The right-handed charged-slepton masses
are also calculated from Eq.(4.1) as
(m˜ )2 = k [(Me)2 +m2 ]. (4.5)
RR i R i R0
Notice that the left-right mixing masses m˜2 are zero due to the (approximate) R-symmetry.
LR
As for sneutrino sector, right-handed sneutrinos have masses of (MνΛ/ H ), for example,
u
O h i
right-handed τ-sneutrino has a mass of (100) eV in case of Λ 105 GeV. Due to the tiny
O ∼
neutrino Yukawa couplings, the FCNC processes induced by the (s)neutrinos are suppressed.
Anyhow the light sneutrinos might be interesting for the cosmology.
We emphasize again that the charged-lepton and charged-slepton mass matrices are diag-
onalized simultaneously in the flavor space. Therefore, the FCNC processes in the lepton sec-
tor, for example µ eγ, are suppressed, although the charged-sleptons have non-degenerate
→
masses in general.
As for the gaugino masses, it is difficult to generate the suitable scale of them. It is because
the R-symmetry is broken only in W sector. (Reminding that gaugino masses require both
Φ
SUSY and R-symmetry breakings, only higher order operators can induce gaugino masses in
the present model.) So here we assume that the gaugino masses are induced another source,
such as moduli F-terms. The µ-term is also assumed to be induced from another mechanism.
11We might also take another standpoint that the origin of Majorana masses is beyond our model and can
be heavier than Λ. In this case the following results are changed.
12We do not consider the case of (yν/ye)2 > 1014 (i = 1,2,3) and accidental cancellations among y’s, λ’s,
i i
and Φ ’s.
h i
13Here k =[(ydλd ydλd)/(ydλd ydλd)]2(Λ/ H )2, and so on.
L 1 B − 2 A A B − B A h di
8
Here let us fix the scale of Λ in our model. We take the soft SUSY breaking masses as
F/Λ (1) TeV (4.6)
∼ O
(m˜ 1 TeV), while the τ-Yukawa should be
τ
≃
F/Λ2 (10−2). (4.7)
∼ O
Combining Eqs.(4.6) and (4.7), the scale of Λ should be
Λ (105)GeV. (4.8)
∼ O
One example of derivation Λ is considering the large extra dimensional theory[11], in which
the Λ is regarded as the D-dimensional Planck scale M with a relation of
∗
M2 = M2+d(2πR)d , (4.9)
P ∗
where d is a number of the extra dimension (D = 4 + d). M 105 GeV means R
∗
∼ ∼
108 GeV−1( 10−7m) in the case of D = 6. (The D = 5 case is experimentally excluded.)
∼
Thepresent modelrequires asetup inwhich allfields except forgravity multiplets arelocalized
on the 4-dimensional brane.
In a short summary we present the mass spectra of the model. The input parameters are
m 1 GeV, m 103 GeV and m 10−10 GeV. Then the outputs are Φ 104 GeV,
τ τ˜ ν3 d
∼ ∼ ∼ h i ∼
Φ 10−1 GeV and Λ 105 GeV. The following table shows the order of mass spectra of
u
h i ∼ ∼
U(3) -nonet fields.
F
fields VEV fermion masses scalar masses
Φ Φ Φ Φ
f f f f
h i h i h i
C 0 Φ Φ
f f f
h i h i
C′ 0 0 Φ 2/Λ
f h fi
We should notice that the gravitino is the next-LSP (NLSP), m 0.1 eV, and the mass
3/2
∼
of right-handed τ-sneutrino is of order 100 eV. The lightest right-handed sneutrino is the
next-to-next-LSP (NNLSP) in this model.14
Finally we comment on the case of introducing a soft term
V m µ2 Tr[ξ A ]+h.c., (4.10)
soft ∼ 3/2 f f f
in the supergravity (SUGRA) setup. Notice that this term breaks the R-symmetry explicitly,
and then the vacuum is shift to non-vanishing A and B Λ2/M . Then massless eight
P
h i h i ∼
14The cosmologicalstudies might be interesting.
9