Direct Gauge Mediation of Uplifted Metastable Supersymmetry Breaking in Supergravity

Abstract

We propose a direct gauge mediation model based on an uplifted metastable SUSY breaking coupled to supergravity. A constant superpotential plays an essential role to fix the moduli as well as breaking SUSY and R-symmetry and the cancellation of the cosmological constant. Gaugino masses are generated at leading order of SUSY breaking scale, and comparable to the sfermion masses as in the ordinary gauge mediation. Landau pole problem for QCD coupling can be easily solved since more than half of messengers become superheavy, which are heavier than the GUT scale. Comment: 14 pages, 2 figures, references added

Full text

Page 1

arXiv:1008.1440v2 [hep-ph] 11 Aug 2010
Direct Gauge Mediation of
Uplifted Metastable Supersymmetry Breaking
in Supergravity
Nobuhito Maru
Department of Physics, Chuo University, Tokyo 112-8551, Japan
Abstract
We propose a direct gauge mediation model based on an uplifted metastable SUSY
breaking coupled to supergravity. A constant superpotential plays an essential role to
fix the moduli as well as breaking SUSY and R-symmetry and the cancellation of the
cosmological constant. Gaugino masses are generated at leading order of SUSY breaking
scale, and comparable to the sfermion masses as in the ordinary gauge mediation. Landau
pole problem for QCD coupling can be easily solved since more than half of messengers
become superheavy, which are heavier than the GUT scale.

Page 2

1 Introduction
Supersymmetry (SUSY) is one of the fascinating scenarios solving the hierarchy problem.
However, it has to be broken at low energy to be relevant to nature. According to Witten
index argument [1], SUSY can be broken nonperturbatively in chiral gauge theories except
for the vector-like model in the special case with the color and flavor numbers [2]. Model
building by use of chiral gauge theories is not so easy although the nonperturbative
dynamics of SUSY theories has been clarified by Seiberg [3]. Several years ago, Intriligator,
Seiberg and Shih (ISS) have discovered a metastable SUSY breaking vacuum in (light)
massive SUSY QCD in a free magnetic phase [4]. A remarkable fact is that the models
they proposed are vector-like models in a wide range of the number of flavors larger than
the number of the color. For SUSY breaking model builders, this fact immediately leads
to the idea that the Landau pole problem of QCD coupling, which was very hard to solve
for a long time, can be easily solved. However, the ISS model has a basic problem that an
R-symmetry is not broken in their vacuum, so the (Majorana) gaugino masses cannot be
generated after the mediation of SUSY breaking. In order to overcome this problem, there
have been many works [5] from various viewpoints by the way to break an R-symmetry
explicitly or spontaneously.
Once SUSY and R-symmetry breaking are realized, the next task is to transmit its
breaking to our world. Of various mediation mechanisms, the direct gauge mediation [6]
is phenomenologically attractive since there is no FCNC problem and it is economical
from the model building viewpoint. However, only breaking R-symmetry is not enough
and the anomalously small gaugino mass problem has been arisen for the proposed direct
gauge mediation models [5].1Gaugino masses are not generated at linear order of SUSY
breaking scale, but at least generated at the third order. This immediately leads to the
fine-tuning in the Higgs mass since the sfermion masses become heavy comparing to the
gaugino masses as in the split SUSY scenario [8]. This problem has been recently studied
in [9] and the authors have shown that the smallness of gaugino masses is related to the
global structure of vacua in renormalizable theories. In particular, it has been shown that
models which breaks SUSY in the lowest energy state necessarily lead to anomalously
small gaugino masses. To avoid this situation, namely to obtain the gaugino masses at
leading order of SUSY breaking, SUSY must be broken in uplifted vacua. Along this
direction, several interesting models has been known [10, 11, 12, 13, 14, 15].
In this paper, we propose a direct gauge mediation model based on an uplifted SUSY
breaking [11] coupled to supergravity (SUGRA). The model of [11] deforms not only a
moduli space such that the rank of the dual quark bilinears is not maximal, but also the
mass terms of quarks in an ISS model. Then, SUSY breaking vacuum is uplifted by the
deformation and the gaugino masses comparable to the sfermion masses are generated
1This problem has already been recognized before in [7].
1

Page 3

as in the ordinary gauge mediation [16]. The model is quite interesting but has some
unappealing points as discussed in [11]. The first point is that an explicit R-symmetry
breaking term of the moduli has to be introduced since the moduli is undetermined as it
stands, and what is worse the overall coefficient is required to be tiny so that the potential
have the minimum. The second point is that the perturbative coupling unification is
difficult to achieve since the number of messengers is large. This is a similar situation as
the direct gauge mediation using the chiral gauge theories These drawbacks are solved
in our model by only coupling an uplifted SUSY breaking model [11] to supergravity as
follows. R-symmetry breaking is realized by introducing a nonzero constant superpotential
with R-charge 2. As a result, the moduli is lifted by the constant superpotential. The
magnitude of the constant superpotential is determined by canceling the cosmological
constant. Moreover, the Landau pole problem is easily avoided because many of the
messengers becomes superheavy, which are larger than the GUT scale.
This paper is organized as follows. In the next section, we discuss our model. We
will see that a constant superpotential plays an essential role to lift the moduli as well
as canceling the cosmological constant and the breaking of SUSY and R-symmetry. The
messenger mass spectrum is calculated in section 3, which is needed in the Landau pole
analysis. In section 4, the direct gauge mediation is discussed. We will show that Landau
pole problem can be avoided in a wide range of parameters in section 5. The last section
6 summarizes the results of our paper.
2 Model
In this paper, we consider an uplifted metastable SUSY breaking model [11] of Intriligator-
Seiberg-Shih (ISS) [4] coupled to supergravity (SUGRA). The uplifted ISS model is a
deformed model of an N = 1 SUSY SU(N)(N ≡ Nf− Nc) gauge theory with Nc+ 1 <
Nf<3
The matter content and symmetries are summarized as follows.
2Ncflavor quarks and the gauge singlet couples to the quarks in the superpotential.
SU(N)
N
¯N
1
SU(Nf)
¯Nf
1
Nf
SU(Nf)
1
Nf
¯Nf
U(1)B
1
−1
0
U(1)
1
1
−2
U(1)R
0
0
2
q
˜ q
Φ
(2.1)
where SU(N) is a gauge group and others are global symmetries. The superpotential and
K¨ ahler potential are given by
W= hqΦ˜ q − hµ2
1
k
?
i=1
Φi
i− hµ2
2
Nf
?
i=k+1
Φi
i+ c, (2.2)
K = Tr[Φ†Φ + q†q + ˜ q†˜ q + ···]. (2.3)
2

Page 4

where the trace is taken in flavor space. c is a constant superpotential necessary for
the cancellation of the cosmological constant. The ellipsis in the K¨ ahler potential means
higher dimensional terms. Note that this model is dual to N = 1 SU(Nc) SUSY gauge
theory with Nfflavors of quarks Q in the range Nc+1 < Nf<3
2Ncand the superpotential
W = m1
k
?
i=1
Qi¯Qi+ m2
Nf
?
i=k+1
Qi¯Qi.(2.4)
The mass scales are related as µ2
gauge theory.
1,2= −m1,2Λ where Λ is a dynamical scale of the SU(Nc)
Let us first study the vacuum structure of the theory. The classical flat directions are
parameterized as follows.
?
?
˜ q =
(˜ ρ2)(Nf−k)×k
(˜ ρ2)(Nf−k)×(N−k)
Φ =
Vk×k
Yk×(Nf−k)
Z(Nf−k)×(Nf−k)
(ρ1)k×(Nf−k)
(ρ2)(N−k)×(Nf−k)
(˜ χ2)k×(N−k)
˜Y(Nf−k)×k
(χ1)k×k
(χ2)(N−k)×k
(˜ χ1)k×k
?
?
?
→
?
?
?
0k×k
0k×(Nf−k)
Z(Nf−k)×(Nf−k)
0(Nf−k)×k
µ11k×k
0(N−k)×k
µ11k×k
0(Nf−k)×k
?
, (2.5)
q =
→
0k×(Nf−k)
0(N−k)×(Nf−k)
0k×(N−k)
0(Nf−k)×(N−k)
?
?
, (2.6)
?
→
(2.7)
where k ≤ N [11]. For k = N, the ISS model is recovered. Using the degrees of freedom
of the global symmetries SU(k) × SU(Nf− k), we find the classical solution as in the
right-hand side. The vacuum expectation values (VEVs) of χ1, ˜ χ1will be determined later
by the F-flatness condition.
The VEV of the above fields are determined by the minimization of the scalar potential.
In the classical flat directions, the superpotential and the K¨ ahler potential are decomposed
as
K = Tr(|V |2+ |˜Y |2+ |Z|2+ |Y |2+ |χ1|2+ |χ2|2+ |˜ χ1|2+ |˜ χ2|2)
+Tr(|ρ1|2+ |ρ2|2+ |˜ ρ1|2+ |˜ ρ2|2+ ···),
N−k
?
It is useful to combine the K¨ ahler potential and the superpotential into a single form
(2.8)
W= −hµ2
2TrZ + h
i
Zρi
2(˜ ρ2)i+ h
k
?
i
?
Zρi
1(˜ ρ1)i+ µ1ρi
1˜Yi+ µ1Yi(˜ ρ1)i
?
+ c.(2.9)
G = K + ln|W|2
(2.10)
where the Planck scale is set to be the unity. Throughout this paper, this convention is
understood.
Using this function G, the scalar potential in supergravity is given by
?
+Gχ1χ†
+Gρ1ρ†
?
V= eGTrGV V†|GV|2+ GZZ†|GZ|2+ GY Y†|GY|2+ G˜Y˜Y†|G˜Y|2
1|Gχ1|2+ Gχ2χ†
1|Gρ1|2+ Gρ2ρ†
|h(χ1˜ χ1− µ2
2|Gχ2|2+ G˜ χ1˜ χ†
2|Gρ2|2+ G˜ ρ1˜ ρ†
1)|2+ |Z†W − hµ2
1|G˜ χ1|2+ G˜ χ2˜ χ†
1|G˜ ρ1|2+ G˜ ρ2˜ ρ†
2|2+ |χ†
2|G˜ χ2|2
2|G˜ ρ2|2− 3
1W|2+ |˜ χ†
?
= eKTr
1W|2− 3|W|2?
(2.11)
3

Page 5

where Gφiφ†
SUSY vacuum conditions are given by
j≡
∂2G
∂φi∂φ∗
j.
0 = χ1˜ χ1− µ2
0 = Z†W − hµ2
0 = W = −hµ2
1, (2.12)
2= Z†(−hµ2
2TrZ + c,
2TrZ + c) − hµ2
2, (2.13)
(2.14)
which are obviously incompatible, namely SUSY is broken. The vacuum energy is
V0= Tr|FZ|2= (Nf− k)|hµ2
2|2
(2.15)
where V0means the leading terms of the potential independent of the Planck scale.2At
this order TrZ is undetermined. On the other hand, the next leading terms of order
O(1/M2
V1 ≈ −hµ2
= −2hµ2
P) are given by
2Tr(Z†W + ZW†) − 3|W|2+ (Nf− k)(2kµ2
2TrZ(−hµ2
1+ Tr|Z|2)|hµ2
2|2
1+ TrZ2)(hµ2
2TrZ + c) − 3(−hµ2
2TrZ + c)2+ (Nf− k)(2µ2
2)2
(2.16)
where ?Z? = ?Z†? is assumed for simplicity.
Minimization condition
0 =
∂V1
∂TrZ= −2hµ2
2
?(−2hµ2
2TrZ + c) − 3(−hµ2
2TrZ + c) + (Nf− k)TrZ(hµ2
2)?
determines its VEV as
?TrZ? ≈
2c
(Nf− k + 1)hµ2
2
. (2.17)
Note that the moduli is lifted by the constant superpotential c and U(1)R symmetry
is spontaneously broken, which is natural because the constant superpotential has an
R-charge 2. This should be contrasted with the work [11] in which the U(1)R explicit
breaking term δW = Z2is added by hand and its overall coefficient must be fine-tuned
to be very small to have a minimum. In our case, the fine tuning is reduced to the
cancellation of the cosmological constant and it is a generic fine-tuning in the models
based on supergravity.
To cancel or obtain a tiny positive cosmological constant, we need the following fine
tuning.
0 ≈ V0+ V1≈ (Nf− k)(hµ2
?
2)2− 2
?2
2c2
Nf− k + 1
c2+ (Nf− k)(hµ2
?
−
2
Nf− k + 1+ 1
?
?
−3
−
2
Nf− k + 1+ 1
2)2
2kµ2
2+
4c2
(hµ2
2)2(Nf− k + 1)2
?
(2.18)
2Throughout this paper eK≃ e|?Z?|2≃ 1 is taken by (2.17).
4