Soft Scalar Masses in String Models with Anomalous $U(1)$ symmetry

17 Pages·0.16 MB·English

by  Yoshiharu Kawamura

DPSU-96-2
LMU-TPW 96-3
January, 1996
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9
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1 Soft Scalar Masses in String Models
n with Anomalous U(1) symmetry
a
J

0
3 Yoshiharu Kawamura a ∗ and Tatsuo Kobayashi b †

1
v
5 a Department of Physics, Shinshu University
6
3 Matsumoto, 390 Japan
1
and
0
6 b Sektion Physik, Universit¨at Mu¨nchen
9 Theresienstr. 37, D-80333 Mu¨nchen, Germany
/
h
p
-
p
e
h
Abstract
:
v
i
X We obtain the low-energy effective theory from string models with
r anomalous U(1) symmetry. Thefeature of soft supersymmetrybreak-
a
ing scalar masses and some phenomenological implications are dis-
cussed. We show that it is, in general, difficult to keep the degeneracy
and the positivity of squared soft scalar masses at the Planck scale.
∗e-mail: [email protected]
†Alexander von Humboldt Fellow
e-mail: [email protected]
1 Introduction
Superstring theories are powerful candidates for the unification theory of
all forces including gravity. There are various approaches to explore 4-
dimensional (4D)string models, for example, thecompactification onCalabi-
Yau manifolds[1], the construction of orbifold models[2, 3] and so on. The ef-
fectivesupergravitytheories(SUGRAs)havebeenderivedbasedontheabove
approaches[4]. The structure ofSUGRA[5]is constrained by considering field
theoretical non-perturbative effects such as a gaugino condensation[6] and
stringy symmetries such as duality[7] besides of perturbative results.
Though the origin of supersymmetry (SUSY) breaking is unknown, soft
SUSYbreakingtermshavebeenderivedundertheassumptionthattheSUSY
is broken by F-term condensations of the dilaton field S and/or moduli fields
T[8]. Some phenomenologically interesting features are predicted from the
structure of soft SUSY breaking terms which are parameterized by a few
number of parameters, for example, only two parameters such as a goldstino
angle θ and the gravitino mass m in the case with the overall moduli and
3/2
the vanishing vacuum energy[9]. Further cases with multimoduli fields are
discussed in Refs. [10].
Most of 4D string models have anomalous U(1) symmetries. Some inter-
esting features are pointed out in those models. Fayet-Iliopoulos D-term[11]
is induced at one-loop level for anomalous U(1) symmetry[12].∗ As a result,
some scalar fields necessarily develop vacuum expectation values(VEVs) and
some gauge symmetries can break down[14].
Such a symmetry breaking generates an intermediate scale M , which is
I
defined as the magnitude of VEVs of scalar fields, below the Planck scale
M . Using the ration M /M , higher dimensional couplings could explain
Pl I Pl
hierarchical structures in particle physics like the fermion masses and their
mixing angles. Recently much attention has been paid to such a study on
the fermion mass matrices [15, 16]. In Refs. [15], U(1) symmetries are used
to generate realistic fermion mass matrices and some of them are anomalous,
while stringy selection rules on nonrenormalizable couplings are used in Refs.
[16]. Hence it is interesting to examine what features 4D string models
with anomalous U(1) symmetry can show at low energy or whether we can
construct a realistic model.
∗ Some conditions for absence of anomalous U(1) are discussed in ref.[13].
1
In this paper, we derive the low-energy theory from 4D string models
with the anomalous U(1) symmetry. We discuss the feature of soft super-
symmetry breaking scalar masses and some phenomenological implications.
In particular, we study the degeneracy and the positivity of squared scalar
masses. This subject has not been completely examined in the literatures
[17, 18].
2 Structure of soft scalar mass
Let us explain our starting point and assumptions first. We assume that
4D string models are described as the effective SUGRA at the Planck scale
M . The gauge group is G = G′ U(1) where G′ is a group which
Pl SM × A SM
contains the gauge group of the standard model, SU(3) SU(2) U(1)
C L Y
× ×
as a subgroup and U(1) is anomalous. This anomaly is canceled by the
A
Green-Schwarz mechanism[19]. The chiral multiplets are classified into two
categories. One is a set of G′ singlet fields with large VEVs denoted as Φi.
SM
It is assumed that the SUSY is broken by those F-term condensations. Some
of them have non-zero U(1) charges and induce to the U(1) breaking. S
A A
and T belong to Φi . Here we treat only the overall moduli field T, but
{ }
not several moduli fields. The second one is a set of G′ non-singlet fields
SM
Φκ. For simplicity, we treat all Φκ’s as light fields whose masses are small
compared with M . We denote the above two types of multiplet as ΦI .
I
We study only a simple case with the following assumptions to avoid a
complication.
1. The U(1) breaking scale is much higher than that of G′ . We intro-
A SM
duce one chiral matter multiplet X with a large VEV of order M to
I
break U(1) .
A
2. The VEV of X is much smaller than those of S and T, i.e.
X S , T = O(M), (1)
h i ≪ h i h i
whereM isthegravitationalscaledefined asM M /√8π. Hereafter
Pl
≡
we take M = 1.
3. Effects of threshold corrections and a S-T mixing are small and ne-
glected.
2
It is straightforward to apply our method to more complicated situations.
We will comment on some of them later.
Our starting SUGRA is determined by the following three gradients, that
is, the K¨ahler potential K, the superpotential W and the gauge kinetic func-
tion f . Orbifold models lead to the following K¨ahler potential K: [4, 20]
α
K = ln(S +S∗ +δA V ) 3ln(T +T∗)
− GS A −
+(T +T∗)nX X 2 +(T +T∗)nκ Φκ 2 + , (2)
| | | | ···
where δA is a coefficient of the Green-Schwarz mechanism to cancel U(1)
GS A
anomaly and V is a vector superfield of U(1) . The dilaton field S trans-
A A
forms nontrivially as S S iδA θ(x) under U(1) with the transformation
→ − GS A
parameter θ(x). The coefficient δA is given as
GS
1
δA = TrQ . (3)
GS 96π2 A
We estimate as δA /q = O(10−1) O(10−3) by using explicit models.†
| GS X| ∼
Here q is a U(1) charge of X. And n ’s are modular weights of matter
X A I
multiplets Φ . The same K¨ahler potential is derived from Calabi-Yau models
I
with the large T limit. If the VEV of X is comparable with one of T, we
should replace the second and third terms in Eq. (2) as
3ln(T +T∗ X 2), (4)
− −| |
for the untwisted sector and
ln[(T +T∗)3 (T +T∗)nX+3 X 2], (5)
− − | |
for the twisted sector. The superpotential W has U(1) invariance. We
A
examine its consequence at low energy without specifying the form of W
in this paper. Note that the term dependent on only X is forbidden by the
U(1) invariance. ThetotalK¨ahlerpotentialGisdefinedasG K+ln W 2.
A
≡ | |
The gauge kinetic function f is given as f = S. For simplicity, here we
α α
assume the Kac-Moody levels satisfy k = 1 , because our results on soft
α
terms are independent of a value of k . The scalar potential is given as
α
V = V(F) +V(D), (6)
† For example, see Refs.[14, 21].
3
V(F) eG(GI(G−1)JG 3), (7)
≡ I J −
1 1
V(D) (DA)2 + (Da)2
≡ S +S∗ S +S∗
1 δA
= ( GS +q K X +q K Φκ)2
S +S∗ S +S∗ X X κ κ
1
+ (K (TaΦ)κ)2, (8)
S +S∗ κ
where the indexes I, J,... run all scalar species, the index a runs the gen-
erators of G′ gauge group and the U(1) charge of matter multiplet ΦI
SM A
is denoted as q . Note that the Fayet-Iliopoulos D-term[11] appears in V(D)
I
for U(1) if we replace S by its VEV. The U(1) is broken by the conden-
A A
sations of S and X. The U(1) breaking scale is of order N where N is a
A
h i
Nambu-Goldstone multiplet. The gravitino mass is obtained as m2 = eG.
3/2
We assume that V(F) O(m2 M2).
≤ 3/2
Next we explain the procedure to obtain the low-energy theory.
1. We write down the scalar potential V by using the variations ∆Φi =
Φi Φi . We treat Φi’s as dynamical fields.
−h i
2. We identify the Nambu-Goldstone multiplet N related to U(1) break-
A
ing whose mass is the same order of that of U(1) gauge boson by
A
calculating the scalar masses.
3. Then we solve the stationary conditions of the potential for Φi while
keeping the light fields arbitrary and integrate out the heavy field N
by inserting the solutions into the scalar potential. Simultaneously we
take the flat limit, while fixing m finite.
3/2
We can obtain the scalar potential Veff of the effective theory by the
straightforward calculation[22]. Here we write down the result in a model-
independent manner as follows,
Veff = V +Veff +Veff, (9)
0 SUSY Soft
V = eG(Gi(G−1)jG 3) , (10)
0 h i j − i
∂Wˆ 1
Veff = eff 2 + g2(K (TaΦ)κ)2, (11)
SUSY | ∂Φκ | 2 a κ
4
∂Wˆ
Veff = AWˆ +Bκ(Φ) eff +H.c.
Soft eff ∂Φκ
+(m2)λΦκΦ∗ +C ΦκΦλ +H.c., (12)
κ λ κλ
where g ’s are the gauge coupling constants of G′ . Here we use the relation
a SM
S = 1/g2 and omit the terms whose magnitudes are less than O(m4 ).
h i α 3/2
Note that there is no D-term contribution on the cosmological constant V .
0
Wedonotwritedowntheexplicit formsfortheeffective superpotentialWˆ ,
eff
parameter A, field-dependent functions Bκ or C since it is irrelevant to the
κλ
later discussions.‡ We are interested in only chirality-conserving scalar mass
(m2)λ in this paper. The formula is given as
κ
(m2)λ = (m2 +V ) Kλ + Fi (Kµ(K−1)νKjλ Kjλ) F∗
κ 3/2 0 h κi h ih iκ µ ν − iκ ih ji
+q g2 DA Kλ (13)
κ Ah ih κi
DA = 2M−2 Fi F∗ (DA)j , (14)
h i A h ih jih ii
where M is the mass of U(1) gauge boson and g is a gauge coupling
A A A
constant of U(1) . The last term in Eq. (13) is so-called D-term contribution
A
to the scalar masses[23].
We can apply the above result (9) – (14) to the effective SUGRA de-
fined by (2) – (8). For the analysis of soft SUSY breaking parameters, it is
convenient to introduce the following parameterization
eG/2(KS)−1/2GS = √3Cm eiαS sinθ, (15)
h S i 3/2
eG/2(KT)−1/2(GT +(KT)(KT)−1GX) = √3Cm eiαT cosθ, (16)
h T T X i 3/2
where (Ki) is a reciprocal of (Ki)−1. The vacuum energy V is written as
j j 0
V = 3(C2 1)m2 +V (X), (17)
0 − 3/2 0
V (X) eG([(KX)−1 (KT) KT −2] GX 2) . (18)
0 ≡ h X − T | X| | | i
Since C2 should be positive or zero, we have a constraint V (X) 3m2 +V
0 ≤ 3/2 0
from Eq. (17). In the case with V = 0, it becomes as
0
[(KX)−1 (KT) KT −2] GX 2 3. (19)
X − T | X| | | ≤
‡ Consult the reference [22] if necessary.
5
Itgivesaconstraint ontheVEVsofX andT. FurtheralargervalueofV (X)
0
in the above region means C 1. Such a limit as C 0 corresponds to the
≪ →
“moduli-dominated” breaking, that is, FS 1 and FT and FX contribute
≪
to the SUSY-breaking. Note that this situation does not agree with the
case ofthemoduli-dominated breaking without theanomalousU(1) breaking
sinθ 0.
→
The stationary conditions lead to D = O(m2 ), i.e. δA /(S + S∗) +
h i 3/2 h GS
q K X = 0 up to such an order. When we expand the scalar potential by
X X
i
using the variations forΦi, we canidentify the variationof Nambu-Goldstone
multiplet N as
1 ∆S +∆S∗ K X 2 ∆X +∆X∗
ξξ
∆N = [ +(1+ h ih i )
a 2 S K X
ξ
h i h i h i
K
+h ξTi(∆T +∆T∗)], (20)
K
ξ
h i
where ξ = X 2 and
| |
1 1 K X 2 K
a2 = + (1+ h ξξih i )2 +(h ξTi)2. (21)
4 S 2 X 2 K K
ξ ξ
h i h i h i h i
This field ∆N has a heavy mass of order O( q δA 1/2M), and the other
| X GS|
linear combinations of S, T and X have light masses.§ The mass of U(1)
A
gauge boson M is given as
A
M2 = 2g2(δA 2 KS +q2 KX X 2). (22)
A A GS h Si Xh Xi|h i|
We can find that δA /q 1 corresponds to X S , T . In this limit,
| GS X| ≪ h i ≪ h i h i
∆N and M are as follows,
A
∆N = ∆X +∆X∗, M2 = 2g2q2 KX X 2. (23)
A A Xh Xi|h i|
Further V (X) is negligible in this limit. Thus the scalar mass is rewritten
0
as
m2 = m2 +V +(m2) +(m2 ) , (24)
κ 3/2 0 F κ D κ
V 3(C2 1)m2 , (25)
0 ≡ − 3/2
(m2) m2 C2n cos2θ, (26)
F κ ≡ 3/2 κ
q
(m2 ) m2 κ (1 C2n cos2θ+6C2sin2θ), (27)
D κ ≡ 3/2q − X
X
§ We can estimate the order of their masses as O(m3/2)[22].
6
where (m2)λ = m2 Kλ . Note that our result is not reduced to that obtained
κ κh κi
fromthetheorywithFayet-IliopoulosD-term,whichisderivedfromtheeffec-
tiveSUGRAbytakingtheflatlimitfirst, eveninthelimitthat δA /q 1.
| GS X| ≪
This disagreement originatesfromthe factthat we regardS andT asdynam-
ical fields, that is, we use the stationary conditions ∂V/∂Φi = 0 to calculate
D-term condensation. This mass formula plays a crucial role in the following
discussion.
3 Phenomenological implications of soft scalar
mass
Now we shall discuss phenomenological implications of our results, especially
thedegeneracy andthepositivity ofthesquared soft scalarmasses. Hereafter
we take V = 0, i.e. C2 = 1. First we give a general argument by using the
0
mass formula
q
m2 = m2 [1+n cos2θ+ κ (7 n cos2θ 6cos2θ)]. (28)
κ 3/2 κ q − X −
X
Note that the coefficient of q /q in Eq. (28) is sizable. That could lead to
κ X
a strong non-universality of soft scalar masses.¶
3.1 Degeneracy of soft scalar masses
We obtain the difference of the soft masses as
∆m2 ∆q
= ∆ncos2θ + (7 n cos2θ 6cos2θ), (29)
m2 q − X −
3/2 X
by using Eq. (28). The experiments for the process of flavor changing neu-
tral current (FCNC) require that ∆m2/m2 < 10−2 for the first and the
3/2
second families in the case with m2 O(1)TeV∼[25]. Hence we should derive
q˜ ∼
∆m2/m2 0 within the level of O(10−2). Hereafter a 0 denotes such a
3/2 ≈ ≈
meaning.
¶ Recently much work is devoted to phenomenological implications of the non-
universality[24].
7
If ∆q/q 0, we have ∆m2/m2 = ∆ncos2θ. In this case the limit
X ≈ 3/2
cos2θ 0 leads to ∆m 0. It corresponds to the dilaton-dominated
→ →
breaking, where soft masses are universal [26, 9]. Unless ∆q/q 0, we
X
≈
needs “fine-tuning” on the value of cosθ as
7
cos2θ . (30)
≈ 6+n q ∆n/∆q
X X
−
This “fine-tuning” is possible only in the case wherek
∆n
1+ q n . (31)
X X
∆q ≤
Let us study the implication of Eq. (30). In the case with ∆n = 0, Eq. (30) is
reduced cos2θ 7/(6+n ). Such a value of cosθ is possible if n 1. Since
X X
≈ ≥
such modular weights require at least two oscillators, they are not obtained
naturally[28]. If we take more natural value, e.g. n = 1, Eq. (31) is
X
−
reduced to ∆nq /∆q 2.
X
≤ −
If Eq. (30) is satisfied, the soft scalar mass is written as
7(n ∆nq /∆q)
m2 = m2 [1+ κ − κ ]. (32)
κ 3/2 6+n ∆nq /∆q
X X
−
3.2 Positivity of squared soft scalar masses
The condition for the positivity of m2 is written as
κ
q q
n + κ (n +6) (1+7 κ )cos−2θ. (33)
κ X
− q ≤ q
X X
If 1+7q /q is positive, we can find a solution cosθ of the above constraint
κ X
for any n ,n ,q and q . On the other hand, if 1 + 7q /q is negative, it
κ X κ X κ X
leads to the following constraint;
q
κ
1+n (n 1), (34)
κ X
≥ q −
X
because cos−2θ 1.
≥
k In Ref. [27], the relation between modular weights and U(1) charges is discussed as
q ∆n/∆q =n . This relation does not satisfy Eq. (31).
X X
8
Let us consider two extreme examples for the SUSY-breaking. One is the
case of dilaton-dominated breaking (cosθ = 0). In this case we have
q
m2 = m2 (1+7 κ ). (35)
κ 3/2 q
X
The positivity of m2 requires that 1 + 7q /q 0. The other is that of
κ κ X ≥
moduli-dominated breaking (cos2θ = 1). In this case we have
q
m2 = m2 [1+n + κ (1 n )]. (36)
κ 3/2 κ q − X
X
For example in the case with n = n = 1, the positivity is realized only
κ X
−
if q /q is positive.
κ X
In both cases of Eqs. (35) and (36), the fields with q /q < 0 can easily
κ X
have negative squared scalar mass of O(m2 ) at the Planck scale. That im-
3/2
plies that several fields develop VEV’s and they trigger symmetry breakings.
We can show that there exist fields with q /q < 0 for each gauge group
κ X
other than U(1) . The reason is as follows. Let us assume the gauge group
A
is U(1) Q G . The Green-Schwarz anomaly cancellation mechanism re-
A × ℓ ℓ
quires that C = δA k for any ℓ, where C is a coefficient of U(1) G2
Gℓ GS ℓ Gℓ A × ℓ
anomaly and k is a Kac-Moody level of G . Through the U(1) breaking
ℓ ℓ A
due to the Fayet-Iliopoulos D-term, the field X develops its VEV. Here its
charge should satisfy q TrQ < 0 and q C < 0. Each gauge group G
X A X Gℓ ℓ
always has fields Φκ which correspond nontrivial representation on its group
and whose U(1) charges satisfy q q < 0 because of q C < 0. The D-
A X κ X Gℓ
term contribution on soft terms is very sizable. That could naturally lead to
m2 < 0 except a narrow region and cause G breaking.
κ ℓ
Next we show that the scalar mass can be a source to break G′ by using
SM
the explicit model[21]. The model we study is the Z orbifold model with a
3
shift vector V and Wilson lines a and a such as
1 3
1
V = (1,1,1,1,2,0,0,0)(2,0,0,0,0,0,0,0),
3
1
a = (0,0,0,0,0,0,0,2)(0,0,1,1,0,0,0,0),
1
3
1
a = (1,1,1,2,1,1,1,0)(1,1,0,0,0,0,0,0).
3
3
9