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Gaugino Mediated Supersymmetry Breaking

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We consider supersymmetric theories where the standard-model quark and lepton fields are localized on a "3-brane" in extra dimensions, while the gauge and Higgs fields propagate in the bulk. If supersymmetry is broken on another 3-brane, supersymmetry breaking is communicated to gauge and Higgs fields by direct higher-dimension interactions, and to quark and lepton fields via standard-model loops. We show that this gives rise to a realistic and predictive model for supersymmetry breaking. The size of the extra dimensions is required to be of order 10-100 times larger than fundamental scale (e.g. the string scale). The spectrum is similar to (but distinguishable from) the predictions of "no-scale" models. Flavor-changing neutral currents are naturally suppressed. The \mu term can be generated by the Giudice-Masiero mechanism. The supersymmetric CP problem is naturally solved if CP violation occurs only on the observable sector 3-brane. These are the simplest models in the literature that solve all supersymmetric naturalness problems. Comment: Refs. added. 12 pages, 1 figure
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arXiv:hep-ph/9911323v3 30 Nov 1999
UMD-PP-00-040, UW-PT/99-26
Gaugino Mediated Supersymmetry Breaking
Z. Chacko,
Markus A. Luty,
Ann E. Nelson,
Eduardo Ponon
Department of Physics, Box 351560
University of Washington
Seattle, Washington, 98195, USA
Department of Physics
University of Maryland
College Park, Maryland 20742, USA
Abstract
We consider supersymmetric theories where the standard-model quark and
lepton ﬁelds are localized on a ‘3-brane’ in extra dimensions, while the
gauge and Higgs ﬁelds propagate in the bulk. If supersymmetry is broken
on another 3-brane, supersymmetry breaking is communicated to gauge
and Higgs ﬁelds by direct higher-dimension interactions, and to quark and
lepton elds via standard-model loo ps. We show that this gives rise to
a realistic and predictive model for supersymmetry breaking. The size of
the extra dimensions is required to be of order 10–100 times larger than
fundamental scale (e.g. the string scale). The spectrum is similar to (but
distinguishable from) the predictions of ‘no-scale’ models. Flavor-changing
neutral currents a re naturally suppressed. The µ term can be generated
by the Giudice-Masiero mechanism. The supersymmetric CP problem is
naturally solved if CP violation occurs only on the observable sector 3-brane.
These are t he simplest models in the literature that solve all supersymmetric
naturalness problems.
November 12, 1999
1 Introduction
Supersymmetry (SUSY) provides an attractive framework for solving the hierarchy
problem, but it introduces naturalness puzzles of its own. Perhaps the most serious
is the ‘SUSY ﬂavor problem:’ why do the squark masses conserve ﬂavor? A natural
solution is given by models of gauge-mediated SUSY breaking [1] or by ‘anomalous
U(1)’ models [2]. In Ref. [3], Randall and Sundrum suggested another solution in
theories where the visible sector ﬁelds are localized on a ‘3-brane’ in extra dimensions
and the hidden sector ﬁelds are localized on a spatially separated ‘3-brane’. (Models
of this type were introduced in the context of string theory by Hoˇrava and Witten
[4].) Ref. [3] pointed out that in such theories contact terms between the visible and
hidden ﬁelds are suppressed if the separation r between the visible and hidden branes
is suﬃciently large. The reason is simply that contact terms arising from int egra t ing
out states with mass M are suppressed by a Yukawa factor e
Mr
if M
>
r. Because
the suppression is exponentia l, the separation need only be an order of magnitude
larger than the fundamental scale (e.g. the string scale) to strongly suppress contact
int eractions.
If contact interactions between the hidden and visible sector ﬁelds can be ne-
glected, other eﬀects become important for communicating SUSY breaking. One
possibility is the recently-discovered mechanism of anomaly-mediation [3, 5], a model-
independent supergravity eﬀect that is always present. (For a careful discussion
of anomaly mediation in a speciﬁc higher-dimensional model, see Ref. [6].) Unfor-
tunately, if anomaly-mediation dominates, and if the visible sector is the minimal
sup ersymmetric standard model (MSSM), then slepton mass-squared terms are neg-
ative. This problem can be avoided in extensions of the MSSM [7]. In this paper,
we will explore the a lternate po ssibility that standard-model gauge and Higg s ﬁelds
propagate in the bulk and communicate SUSY breaking between the hidden sector
and visible-sector matter ﬁelds. Models with the all the MSSM superﬁelds except
the gauge and Higgs ﬁelds localized on a 3 -brane were also considered in R ef. [8]. In
those models supersymmetry was directly broken by the compactiﬁcation b oundary
conditions, requiring a rather large extra dimension (radius of order TeV
1
) in or -
der to explain the gauge hierarchy. Models similar to the one considered here , i.e.
with a hidden supersymmetry breaking sector sequestered on a diﬀerent 3-brane and
standard-model gauge and Higg s ﬁelds in the bulk, were considered in Ref. [9]. These
models contained an additional U(1) gauge multiplet; the present paper shows that
this is not required to obta in a realistic theory of SUSY breaking.
In the higher-dimensional theory, the standard-model gauge and Higgs ﬁelds can
1
int eract only through non-renormalizable interactions. We therefore treat the higher-
dimensional theory as an eﬀective theory with a cutoﬀ M, which may be viewed as
the fundamental scale of the theory. Below the compactiﬁcation scale µ
c
1/r, the
theory matches onto a 4-dimensional eﬀective theory. In this theory, the couplings
of the gauge and Higgs ﬁelds are suppressed by 1/(Mr)
D 4
, where D is the number
of ‘large’ spacetime dimensions. Therefore, the size of the extra dimensions cannot
be too large in units of the fundamental scale. However, because the suppression of
cont act terms is exponential, there is a range of radii with suﬃcient suppression of
cont act terms to avoid ﬂavor-changing neutral currents without exceeding the strong-
coupling bounds on the couplings in the higher-dimension t heory [9].
In this scenario, SUSY breaking masses for gauginos and Higgs ﬁelds are gener-
ated by higher-dimension contact terms between the bulk ﬁelds a nd the hidden sector
ﬁelds, assumed to arise from a more fundamental theory such as string theory. In par-
ticular, the µ t erm can be generated by the Giudice-Masiero mechanism [10]. Other
direct contact interactions between the hidden and visible sectors are suppressed be-
cause o f their spatial separation. The leading contribution to SUSY breaking for
visible sector ﬁelds arises from loops of bulk gauge and Higgs ﬁelds, as illustrated in
Fig. 1. These diagrams are ultraviolet convergent (and hence calculable) because the
spatial separation of the hidden and visible branes acts as a physical point-splitting
regulator. In eﬀective ﬁeld theory language, the contribution from loop momenta
above the compactiﬁcation scale is a (ﬁnite) matching contribution, while the con-
tribution from loop momenta below the compactiﬁcation scale can be obtained from
the 4 -dimensional eﬀective theory. The higher-dimensional theory therefore gives ini-
tial conditions for the 4-dimensional renormalization group at t he compactiﬁcation
scale µ
c
: nonzero gaugino masses and Higgs mass parameters, and loop-suppressed
soft SUSY breaking parameters for the squarks and sleptons. This is similar to the
boundary conditions of ‘no-scale’ supergravity models [11], but in the present case
the boundary conditions are justiﬁed by the geometry of the higher-dimensional the-
ory. Since the SUSY breaking masses for a ll chiral matter ﬁelds other than the third
generation squarks are dominated by the gaugino loop, we call this scenario ‘ga ugino
mediated SUSY breaking’ (˜gMSB).
The renormalization group has a strong eﬀect on the SUSY breaking parameters,
and the soft masses at the weak scale are all of the same order. In fact, the Bino can
be the lightest superpartner (LSP) in this scenario. The spectrum is similar to that
of ‘no-scale’ supergravity models [11], with the important diﬀerence that the present
scenario allows a Fayet-Iliopoulos term for hypercharge that can have an important
eﬀect on the slepton spectrum. We obtain realistic spectra without excessive ﬁne-
2
Fig. 1. The leading diagram t hat contributes to SUSY-breaking scalar
masses in the models considered in this paper. The bulk line is a
gaugino propagator with two mass insertions on the hidden brane.
tuning for neutralino and slepton masses below approximately 200 GeV, suggesting
that these superpartners are relatively light in this scenario.
This paper is organized as follows. In Section 2, we discuss the higher-dimensional
theory. We show that the size of the extra dimensions can be large enough to suppress
FCNC’s while still having gauge and Higgs couplings of order 1 at low energies. We
also show how the SUSY CP problem can be naturally solved in this class of models.
In Section 3, we discuss the phenomenology of this class of models. Section 4 contains
our conclusions.
2 Bulk Gauge and Higgs Fields
In this Section, we discuss some general features of higher-dimensional theories with
gauge and Higgs ﬁelds in the bulk and other ﬁelds localized on ‘3-branes.’ We use
the term ‘3 -branes’ to mean either dynamical surfaces (e.g. topological defects or
string-theory D-branes) or non-dynamical features of t he higher-dimensional space-
time (e.g. o r bifo ld ﬁxed point s). All of these ingredients occur in string theory, but we
will not concern ourselves with the derivation of the model from a more fundamental
theory. We simply write an eﬀective ﬁeld theory valid below some scale M, which
may be the string scale, the compactiﬁcation scale a ssociated with additional small
3
dimensions, or some other new physics.
We t herefore consider an eﬀective theory with D spacetime dimensions, with 3 +1
non-compact spacetime dimensions and D 4 compact spatial dimensions with linear
size of order r. The D-dimensional eﬀective lagrangian takes the form
L
D
= L
bulk
(Φ(x, y)) +
X
j
δ
D 4
(y y
j
)L
j
(Φ(x, y
j
), φ
j
(x)), (2.1)
where j runs over the various branes, x are coordinates for the 4 non-compact space-
time dimensions, y are coordinates for the D 4 compact spatial dimensions, Φ is
a bulk ﬁeld, and φ
j
is a ﬁeld localized on the j
th
brane. This eﬀective theory can
be treated using the usual techniques of eﬀective ﬁeld theory, and parameterizes the
most general interactions of the assumed degrees o f freedom below the scale M.
1
We assume that the D 4 extra spatial dimensions are compactiﬁed on a distance
of order r 1/M. We also assume that the distance between diﬀerent branes is also
of order r. This ensures that contact interactions between ﬁelds on diﬀerent branes
arising from states above the cutoﬀ are suppressed by the Yukawa factor e
Mr
.
We assume that the standard-model gauge and Higgs ﬁelds propagate in the bulk.
Bulk gauge ﬁelds have a gauge coupling with mass dimension 4 D, which is an
irrelevant interaction fo r all D > 4. When we match onto the 4-dimensional theory
at the compactiﬁcation scale, the eﬀective 4-dimensional gauge coupling is
2
g
2
4
=
g
2
D
V
D 4
, (2.2)
where g
D
is the gauge coupling in the D-dimensional theory a nd V
D 4
r
D 4
is the
vo lume of the compact dimensions. If g
D
1/M
D 4
, we have g
4
1/(Mr)
D 4
1,
which is unacceptable. In order to have g
4
1 (a s observed), we require the gauge
coupling to be larger than unity in units of M. However, it presumably does not make
sense to take g
D
larger than its strong-coupling value, deﬁned to be the value where
loop corrections are order 1 at the scale M. This follows from ‘na¨ıve dimensional
analysis’ (NDA) [14, 15], which is known to work extremely well in supersymmetric
theories [16]. If we assume that the loop corrections are suppressed by ǫ at the scale
M, the lagrangian is [9]
L
D
M
D
ǫℓ
D
L
bulk
(
ˆ
Φ/M, /M) +
X
j
δ
D 4
(y y
j
)
M
4
ǫℓ
4
L
j
(
ˆ
Φ/M,
ˆ
φ
j
/M, /M). (2.3)
1
For a n explicit supersymmetric example and calculations, see Ref. [12].
2
We neglect the eﬀects of gravitational curvature.
4
Torus: Sphere:
D
ML
max
e
ML
max
/2
Mr
max
e
Mr
max
5 740 3 × 10
162
118 4 × 10
52
6
63 2 × 10
14
18 2 × 10
8
7
29 6 × 10
7
11 3 × 10
5
8
20 5 × 10
5
8.7 2 × 10
4
9
16 3 × 10
4
8.0 3 × 10
4
10
14 9 × 10
4
7.8 4 × 10
4
11
13 1 × 10
3
7.8 4 × 10
4
Table 1. Estimates of the maximum size and exponential suppression
factor for propagation between two branes of maximal separation. L
max
is the maximum length of a cycle of a symmetric torus, and r
max
is the
maximum radius of the sphere.
where
D
= 2
D
π
D /2
Γ(D/2) is the geometrical loop factor for D dimensions, and all
couplings in L
bulk
and L
j
are order 1. Note that the ﬁelds
ˆ
Φ and
ˆ
φ in Eq. (2.3) do not
have canonical kinetic terms. The idea behind Eq. (2.3) is that the fa ctors multiplying
L
bulk
and L
j
act as loop-counting parameters (like ¯h in the semiclassical expansion)
that cancel the loo p factors and ensure that loop corrections are suppressed by ǫ.
Strong coupling corresponds to ǫ 1.
We can use Eq. (2.3) to read oﬀ the value of the D-dimensional gauge coupling
g
2
D
ǫℓ
D
M
D 4
. (2.4)
We can obtain the maximum value for the size of the extra dimension consistent with
the fact that g
4
1 by setting ǫ 1 and using Eq. (2.2) . The results are shown
in Table 1. We see that the exponential suppression factor due to the large size of
the extra dimensions can be substantial even for many extra dimensions [9]. Similar
conclusions hold for the Higgs interactions.
To see how much suppression is required, note that the dangerous contact terms
have the form (using Eq. (2.3))
L
brane
e
Mr
M
2
Z
d
4
θ (
ˆ
φ
hid
ˆ
φ
hid
)(φ
obs
φ
obs
), (2.5)
where the observable ﬁelds (but not the hidden ﬁelds) have been canonically normal-
ized. This must be compared with the operators that give rise to the gaugino and
5
Higgs SUSY breaking par ameters. From Eq. (2.3) we obtain
L
brane
D
4
Z
d
2
θ
1
M
D 3
ˆ
φ
hid
W
α
W
α
+ h.c.
+
D
4
Z
d
4
θ
(
1
M
D 3
ˆ
φ
hid
H
u
H
d
+ h.c.
+
1
M
D 4
ˆ
φ
hid
ˆ
φ
hid
h
H
u
H
u
+ H
d
H
d
+ (H
u
H
d
+ h.c.)
i
)
,
(2.6)
where W
α
is the gauge ﬁeld strength and H
u,d
are the Higgs ﬁelds, normalized to
have canonical kinetic terms in D dimensions. (More precisely, these are N = 1
sup erﬁelds obtained by projecting the bulk supermultiplets onto the branes. For a
speciﬁc example, see Ref. [12].) Matching to the D-dimensional theory, we ﬁnd
m
1/2
, µ
ˆ
F
hid
M
D
/ℓ
4
M
D 4
V
D 4
, Bµ, m
2
H
u
, m
2
H
d
ˆ
F
2
hid
M
2
D
/ℓ
4
M
D 4
V
D 4
. (2.7)
Note that the Bµ term and the Higg s mass-squared terms are enhanced by a volume
factor.
3
For example,
Bµ
m
2
1/2
4
D
M
D 4
V
D 4
ǫℓ
4
, (2.8)
where we have imposed g
4
1 to obtain the last estimate. We see that if the theory is
strongly coupled at the fundamental scale, we require a ﬁne tuning of order 1/ℓ
4
1%
to obtain all SUSY breaking para meters of the same size [9]. However, for a small
number of extra dimensions, the fundamental theory need not be strongly coupled at
the fundamental scale, and we can naturally obtain all SUSY breaking terms close
to the same size. For example for D = 5 compactiﬁed on a circle with circumference
L 20/M, the exponential suppression is e
10
5 × 10
5
and Bµ/m
2
1/2
4. As
the number of extra dimensions increases, the strong coupling estimate is approached
rapidly. See Table 2.
The contribution to visible sector scalar masses from contact terms is
m
2
vis
e
Mr
ˆ
F
hid
M
!
2
. (2.9)
3
This point was missed in an earlier version of this paper . It was pointed out in Ref. [13], which
appeared while this paper was being completed. See also Ref. [9].
6
Torus: Sphere:
D
Bµ/m
2
1/2
Bµ/m
2
1/2
5 3.4 3.4
6
10 16
7
28 43
8
69 85
9
160 130
Table 2. Estimates of Bµ/m
2
1/2
for the symmetric torus and the sphere.
The size of the extra dimension is chosen so that the exponential sup-
pression factor is of order e
8
3 × 10
4
(approximately the maximum
suppression for large D) . This means that the tor us has cycle length
L = 16/M, and the sphere has radius r = 8/M.
The values Eq. (2.7) are the values renormalized at the compactiﬁcation scale; we will
later see that we require
ˆ
F
vis
/M 200 GeV. Using the experimental constraints
4
m
2
˜
d˜s
m
2
˜s
<
(6 × 10
3
)
m
˜s
1 TeV
, Im
m
2
˜
d˜s
m
2
˜s
!
<
(4 × 10
4
)
m
˜s
1 TeV
, (2.10)
so e
Mr
10
3
to 10
4
is plausibly suﬃcient to suppress FCNC’s.
We now discuss the loop eﬀects that communicate SUSY breaking to the visible
sector ﬁelds, such as those illustrated in Fig. 1. These are ultraviolet convergent be-
cause the separation of the hidden and visible branes acts as a physical point-splitting
regulator for these diagrams. Another way to see this is that there is no local coun-
terterm in the D-dimensional theory that can cancel a possible overall divergence.
5
Given a speciﬁc D-dimensional theory, this diagram is therefore calculable. From
the point of view of 4-dimensional eﬀective ﬁeld theory, the extra dimensions act as a
cutoﬀ of order µ
c
1/r. The eﬀects of this cutoﬀ can be absorbed into a counterterm
for the visible sector scalar masses and A terms of order
m
2
vis
g
2
4
16π
2
m
2
1/2
, A
vis
g
2
4
16π
2
m
1/2
, (2.11)
where m
1/2
is the gaugino mass. The precise value of the counterterms is calculable if
we fully specify the D-dimensional theory. However, we will see that the RG running
of the soft masses f r om µ
c
to the weak scale gives large additive contributions to the
4
For a complete discussion, s ee e.g. Ref. [17].
5
Multiloop diagrams may have subdivergences, but these can always be cancelled by counterterms
localized on one of the branes.
7
visible soft masses, and the ﬁnal results are rather insensitive to the precise value of
the counterterm. We will therefore be content with the simple estimate above.
Note that the soft terms arising from cont act terms are larger than the anomaly-
mediated contributions, which give
m
λ
g
2
4
16π
2
F
hid
M
4
, m
2
H
u
, m
2
H
d
g
2
4
16π
2
F
hid
M
4
!
2
, (2.12)
where M
4
>
M is the 4-dimensional Planck scale [3, 5]. The other soft terms also get
cont ributions larger than their anomaly-mediated values from the RG, as discussed
above. Therefore, we can neglect the a no maly-mediated contribution in this class of
models.
The higher-dimensional origin of these theories can a lso solve the ‘SUSY CP prob-
lem’ [18]. This problem arises from the fact that the phases in the SUSY breaking
terms must be much less than 1, otherwise t hey give rise t o electron and neutron
electric dipole moments in conﬂict with experimental bounds. This is a natural-
ness problem because CP is (apparently) maximally violated in the CKM matrix,
and it must be explained why it is not violated in all terms. In the present model,
CP-violating phases can appear in µ, B, and m
1/2
, generated from higher-dimension
operators in Eq. (2.6). The phases in µ a nd B can be rota t ed away using a combi-
nation of U(1)
PQ
and U(1)
R
transformations, leaving a single phase in m
1/2
. This
phase can vanish naturally in the present model if CP is violated only by terms in the
lagrangian localized on the visible brane. This is a natural assumption because loop
eﬀects do not g enerate local CP-violating terms in the bulk or the hidden brane. This
situation can arise (for example) if CP is broken spontaneously by ﬁelds localized on
the visible brane. (In order to avoid a lar ge neutron electric dip ole moment, we must
also assume that the eﬀects o f the QCD vacuum angle are suppressed [19].)
There are many ot her aspects of the higher-dimensional theory that we could
discuss, but the basic features of the scenario depend only on the qualitative feature
that the visible and hidden sectors are spatially separated. A complete speciﬁcation
of the higher-dimensional model would have to take into a ccount the fact that there
are more supersymmetries in higher dimensions. This may be broken spontaneously
or explicitly (e.g. by an orbifold), and couplings between bulk and boundary ﬁelds
must be consistent with SUSY. An explicit example with 5 dimensions compactiﬁed
on a S
1
/Z
2
orbifold is easily constructed [12, 9]. Another important feature o f the
higher-dimensional theory is the stabilization of the extra dimensions. Stabilization
mechanisms that are appropriate for the scenario we are considering ar e discussed in
Refs. [20, 6]. We conclude that there is no obstacle to constructing realistic eﬀective
ﬁeld theory models of the type outlined here. The question of whether a model of
8
this type can be derived from a more fundamental theory such as string theory is left
for future work.
3 Phenomenology
We now turn to the phenomenology of these models. We have seen that the SUSY
breaking parameters in the eﬀective 4-dimensional theory are determined at the com-
pactiﬁcation scale µ
c
1/r. We have also seen that µ
c
is one to two orders of
magnitude below the fundamental scale M, which is most natura lly taken to be
close to the string scale. Therefore, we expect µ
c
to be close to the uniﬁcation scale
M
GUT
2 × 10
16
GeV. We therefore identif y µ
c
and M
GUT
in making our estimates.
We will further assume that the theory is embedded in a grand-uniﬁed theory
(GUT) at the scale M
GUT
, as suggested by the success of gauge coupling uniﬁca-
tion in the MSSM. We therefore consider the following SUSY breaking parameters
renormalized at M
GUT
:
Gaugino masses: M
1
= M
2
= M
3
= m
1/2
,
Higgs masses: m
2
H
u
, m
2
H
d
m
2
1/2
, µ, B m
1/2
,
Squark and slepton masses: m
2
m
2
1/2
16π
2
,
A terms: A
m
1/2
16π
2
.
(3.1)
We have arg ued above that these conditions can emerge naturally in this scenario for
D = 5 or 6. If we neglect the small loop-suppressed parameters, the model is deﬁned
by the 6 para meters m
1/2
, m
2
H
u
, m
2
H
d
, µ, B, and y
t
renormalized at M
GUT
. (We do
not consider large tan β solutions, so we neglect all other Yukawa couplings.) The
value of y
t
at the weak scale ﬁxes tan β fro m the observed value of the top quark.
The requirement that electroweak symmetry breaks with the correct value of M
Z
and
tan β then ﬁxes two more parameters. We see that we are left with essentia lly 4
parameters.
An important issue when analyzing the spectrum at the weak scale is the radiative
corrections t o the lightest neutral Higgs mass [21]. The largest eﬀect can be viewed
as a top loop contribution to a n eﬀective quartic term in the eﬀective potential below
the stop mass [22]. We include an estimate of this eﬀect by adding the term
V
H
=
3y
4
t
8π
2
ln
m
˜
t
m
t
!
(H
u
H
u
)
2
(3.2)
9
to the Higgs p otentia l.
We evolve the 1-loop RG equations from the scale M
GUT
= 2 × 10
16
GeV down
to the weak scale µ
W
= 5 00 GeV, using α
GUT
= 1/(24.3). We use input values of
m
1/2
, m
2
H
u
, m
2
H
d
, and y
t
at M
GUT
and determine µ and B by imp osing electroweak
symmetry breaking. The value of the top quark mass is used to ﬁx tan β; we use
m
t
(µ
W
) = 1 65 GeV, which includes 1-loop QCD corrections. We minimize the Higgs
potential including the term Eq. (3.2) with m
˜
t
taken to be the heaviest of the stop
mass eigenstates, and y
t
renormalized at µ
W
. These approximations could be reﬁned,
but they will suﬃce to illustrate the main features of the spectrum of this class of
models.
Some parameter choices that give rise to realistic spectra are given in Table 3.
We ﬁnd that the dependence on the overall scale of the initial SUSY breaking masses
is what would be expected: the superpartners become heavier, a nd the amount of
ﬁne-tuning required to achieve electroweak symmetry breaking increases (see below).
The right-handed sleptons get an important positive contribution from a hypercharge
Fayet-Iliopoulos term if m
2
H
d
> m
2
H
u
at the GUT scale. This distinguishes this model
from ‘no-scale’ models. This is illustrated in the second and third parameter points
in Table 3. For m
2
H
d
> m
2
H
u
, we easily obtain spectra where the LSP is a neutralino.
The value of y
t
mainly inﬂuences the value of tan β, which is important because the
lightest Higgs boson is light for small tan β. We also ﬁnd that tan β
>
2.5 is preferred
in order to have a suﬃciently large mass for the lightest neutral Higgs.
An important feature of these results is the amount of ﬁne-tuning required to
achieve electroweak symmetry breaking. We deﬁne the fractional sensitivity to a
parameter c (a coupling renormalized a t M
GUT
) to be [23]
sensitivity =
c
v
v
c
, (3.3)
where v is the Higgs VEV and the derivative is ta ken with all other couplings at the
GUT scale held ﬁxed. The largest sensitivity is to m
1/2
and µ, and the values of
the sensitivity parameter are given in Table 3. We see that the sensitivity increases
strongly as the superpartner masses are increased. Note that even for parameters
where the superpartner masses are close to the experimental limits, the sensitivity
parameter is large (
>
20). However, it is argued by Anderson and Casta˜no in Ref. [24]
that sensitivity does not capture the idea of ﬁne-tuning: the theory is ﬁne-tuned only
if the physical quantities signiﬁcantly more sensitive than a priori allowed choices of
parameters. From this point of view, the ﬁne-tuning of points with low superpartner
masses is much less severe, and naturalness clearly favors regions of parameters with
light superpartner masses [2 4]. In particular, requiring that the naturalness parameter
10
Point 1 Point 2 Point 3
inputs: m
1/2
200 400 400
m
2
H
u
(200)
2
(400)
2
(400)
2
m
2
H
d
(300)
2
(600)
2
(400)
2
µ 370 755 725
B 315 635 510
y
t
0.8 0.8 0.8
neutralinos: m
χ
0
1
78 165 165
m
χ
0
2
140 315 315
m
χ
0
3
320 650 630
m
χ
0
4
360 670 650
char ginos: m
χ
±
1
140 315 315
m
χ
±
2
350 670 645
Higgs: tan β 2.5 2.5 2.5
m
h
0
90 100 100
m
H
0
490 995 860
m
A
490 1000 860
m
H
±
495 1000 860
sleptons: m
˜e
R
105 200 160
m
˜e
L
140 275 285
m
˜ν
L
125 265 280
stops: m
˜
t
1
350 685 690
m
˜
t
2
470 875 875
other squarks: m
˜u
L
470 945 945
m
˜u
R
450 905 910
m
˜
d
L
475 950 945
m
˜
d
R
455 910 905
gluino: M
3
520 1000 1050
sensitivity: m
1/2
16 50 50
µ 19 78 78
Table 3. Sample points in parameter space. All masses are in GeV. In
the ﬁrst two points, the LSP is mostly Bino, while in the third it is a
right-handed slepton. The sensitivity parameter is deﬁned in the main
text.
11
deﬁned in Ref. [24] be less than 10 implies that the parameter m
1/2
should b e less
than 400 GeV.
4 Conclusions
This model is the simplest supersymmetric theory in the literature that generates
an acceptable spectrum for the sup erpartners while explaining the absence of non-
standard ﬂavor-changing processes a nd electric dipole moments. It is highly pre-
dictive, with squark and slepton masses qualitatively similar to those of ‘no-scale’
sup ergravity models. The nonuniversality o f the up- and down-type Higg s masses at
the GUT scale can distinguish this theory from ‘no-scale’ supergravity—the expected
diﬀerence between the up and down type Higgs masses generates a hypercharge Fayet-
Iliopoulos term which aﬀects the slepton mass spectrum. The right-handed sleptons
and the lightest neutralino are signiﬁcantly lighter than the other superpartners, and
obtaining natural electroweak symmetry breaking requires that these be lighter than
roughly 200 GeV.
While this work was being completed, we received Ref. [13 ], which considers very
similar ideas.
Acknowledgments
M.A.L. and E.P. are supported by the NSF under grant PHY-98-02551. Z.C. and
A.E.N. are supported by the DOE under contra ct DE-FGO3-96-ER40956
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14
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