Gauge Mediation in String Theory
arXiv:0704.1085v2  [hep-th]  11 Apr 2007
CALT-68-2642
UT-07-12
Gauge Mediation in String Theory
Teruhiko Kawano,1 Hirosi Ooguri,1,2 and Yutaka Ookouchi2
1Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
2California Institute of Technology, Pasadena, CA 91125, USA
Abstract
We show that a large class of phenomenologically viable models for gauge mediation
of supersymmetry breaking based on meta-stable vacua can be realized in local Calabi-Yau
compactifications of string theory.
April, 2007
http://arxiv.org/abs/0704.1085v2
1. Introduction
The use of meta-stable vacua in supersymmetric model building has attracted much
attention lately, especially after the discovery [1] that generic supersymmetric field theories
in four dimensions such as the supersymmetric QCD with massive flavors have meta-
stable vacua with broken supersymmetry. In [2], realistic models of direct mediation were
constructed using superpotentials without U(1)R symmetry. Though explicit breaking of
the U(1)R symmetry generates meta-stable vacua, there is a range of parameters where one
can make them sufficiently long lived, while satisfying the phenomenological constraints
on the masses of the gauginos, the gravitino, and the scalars without artificially elaborate
constructions. The models can also avoid producing Landau poles in standard model gauge
interactions below the unification scale. Recently beautiful realizations of these models in
string theory, including a natural mechanism to generate small parameters of these models,
were found in [3].
Gauge mediation models were also constructed using meta-stable vacua with similar
phenomenological benefits [4,5]. Related ideas have been explored in [6-10]. Accepting
the possibility that our universe may be in a meta-stable state allows us to circumvent
the theoretical constraints due to the Nelson-Seiberg theorem on R-symmetry [11] and the
Witten index [12] and gives us greater flexibility in model building, as emphasized in [13]
among other recent papers.
Among the models constructed recently based on meta-stable vacua, the ones dis-
cussed in [4] are particularly simple. In this paper, we will show that they have ultra-violet
completions in supersymmetric quiver gauge theories which can be realized in string com-
pactifications. Moreover, our construction can be naturally generalized to a large class of
quiver gauge theories, providing a basis for the speculation in [4] that “gauge mediation
may be a rather generic phenomenon in the landscape of possible supersymmetric theo-
ries.” In this paper, we will demonstrate the idea by explicitly working out one example:
a model based on type IIB superstrings compactified on the A4-fibered geometry [14]. We
will also give an outline of generalizations of this construction to a large class of quiver
gauge theories. Detailed analysis of meta-stable vacua in these models will be given in a
separate paper [15].
2. The Model
The model we will consider in this paper is realized in string theory compactified on
the local Calabi-Yau manifold described by the equation,
x2 + y2 +
(z + ti(w)) = 0,
ti(w) = 0,
ti(w)− ti+1(w) = µi(w − xi), (x, y, z, w) ∈C
(2.1)
Since ti’s are functions of w, this gives the A4 singularity fibered over w ∈C. In particular,
there exist four two-cycles S2 on which D branes can be wrapped. The low energy limit
of D5 branes wrapping the two-cycles S2 and extending along the four uncompactified
dimensions is the A4 quiver gauge theory with the gauge group U(N1)×U(N2)×U(N3)×
U(N4) with the adjoint chiral multiplets Xi=1,2,3,4 for the four gauge group factors and
the bi-fundamental chiral multiplets (Q12, Q21), (Q23, Q32), and (Q34, Q43). This quiver
gauge theory can also be realized on intersecting brane configuration with NS5 and D4
branes, as expected from the T-duality between the An singularity and NS5 branes [16].
N 1 N 2 N 3 N 4
Q 12,21 Q 23,32 Q 34,43
X1 X2 X3 X 4
Fig. 1: A4 quiver diagram
From the Calabi-Yau singularity (2.1), one can read off the superpotential of the quiver
theory as [17,18]
WA4 =
tr (Qi+1,iXiQi,i+1 −Qi,i+1Xi+1Qi+1,i) +
(Xi − xi)
2. (2.2)
Note that the dimensionful parameters µi and xi are the moduli of the Calabi-Yau manifold
given by (2.1), namely they are closed string moduli. The dynamical scales Λi=1,···,4 of
the four gauge group factors are also closed string moduli, related to the sizes of the S2’s.
These closed string moduli are frozen and can be regarded as parameters of the low energy
theory. Let us suppose that µi are sufficiently larger than Λi so that we can integrate out
all the adjoints Xi to obtain the effective superpotential
Weff =
mitrQi i+1Qi+1 i −
tr (Qi i+1Qi+1 i)
trQ21Q12Q23Q32 +
trQ32Q23Q34Q43,
(2.3)
where
mi = ci − ci+1, µ̃i =
2µiµi+1
µi + µi+1
(i = 1, 2, 3).
This quiver gauge theory can be used as a gauge mediation model as follows. We
identify the bi-fundamentals (Q34, Q43) as messenger fields. One way to incorporate the
standard model sector would be to identify a subgroup of the U(N4) gauge group with the
standard model gauge group or a GUT gauge group. Alternatively, we can replace the 4th
node of the quiver diagram of Fig. 1 carrying the U(N4) gauge group with a string theory
construction of the standard model. For example, if the standard model is realized on
intersecting branes, messengers can be open strings connecting the 3rd node carrying the
U(N3) gauge group to the standard model branes.
1 In the following, we will denote the
bi-fundamental fields (Q34, Q43) as (f, f̃) to distinguish them from the rest of the quiver
gauge theory and to emphasize their role as the messengers.
The rest of the quiver gauge theory is treated as a hidden sector, where supersymmetry
is broken dynamically. To use the result of [1], let us assume that the ranks of the gauge
group factors satisfy
N2 + 1 ≤ N1 +N3 <
N2 (2.4)
and that
Λ1,Λ3,Λ4 ≪ Λ2 ≪ µi.
In this case, one can identify the gauge group SU(Nc) of the model of [1] with SU(N2) ⊂
U(N2) of the quiver theory. Since the metastable vacuum can be found near the origin of
the meson fields M11 ∼ Q12Q21, M33 ∼ Q32Q23, the terms tr (Q12Q21)
, tr (Q32Q23)
tr (Q12Q21Q32Q23) in the superpotential (2.3) are irrelevant in our discussion below, if the
masses µi of the adjoints satisfy the following bounds [4,5],
µ̃1,2
≤ min
m1,2Λ2,
. (2.5)
In this range of the parameters, the hidden sector and its interaction with the mes-
senger sector is described by the superpotential,
W = mtrQ12Q21 +mtrQ32Q23 +
trQ32Q23f f̃ +m3tr f f̃ −
. (2.6)
1 In this case, we can still use the effective potential (2.3) to describe the interaction of the
messengers and the hidden sector, but we should set 1/µ̃4 = 1/(2µ3) since we do not have the
adjoint field X4.
Here, we set the mass parameters m1 = m2 = m, for simplicity. Consider the case when
N1 = N2 = 3 and N3 = 1 so that the Landau pole problem can easily be avoided. The
resulting model is a variant of the models proposed in [4]. The model [4] has the global
symmetry U(4) × U(1)mess, where U(4) is the flavor symmetry of the ISS model and
U(1)mess acts on the messengers (f, f̃). The meta-stable vacuum spontaneously breaks the
U(4) symmetry, giving rise to Nambu-Goldstone bosons, when m1 ≃ m2. In our model,
the would-be Nambu-Goldstone bosons are eaten by the gauge symmetry. This difference
is not important in the low energy analysis of supersymmetry breaking effects.
Let us discuss phenomenological constraints on the parameters in (2.6). We will focus
on the following part of the superpotential (2.6),
Wmess =
M33f f̃ +m3 f f̃ , (2.7)
where M33 = Q32Q23/Λ2 is neutral under the U(N3) = U(1) gauge group. We have
dropped the irrelevant quartic term (f f̃)2 because the messengers (f, f̃) are weakly inter-
acting at energies above the electroweak scale, if the mass parameter µ̃3 is large enough.
The F -component of the meson superfield M33 develops the vacuum expectation value
and breaks supersymmetry [1]. The supersymmetric mass and the soft supersymmetry
breaking mass of the messenger fields (f, f̃) are then given by
Wmess ≃
m3 + θ
f f̃ . (2.8)
Following the analysis in [4,5], we find that all the phenomenological requirements for the
messenger sector can be satisfied, for example, in the following range of parameters,
Λ2 ≃ 10
11GeV, m ≃ 108GeV, m3 ≃ 10
7GeV,
µ1 ≥ µ2 ≥ 10
13GeV, µ3 ≃ 10
18GeV.
(2.9)
3. Generalization
We found that both the messenger sector and the hidden sector of the models proposed
in [4] can be realized in the A4 quiver gauge theory. This construction naturally suggests
the following generalization. Consider a quiver diagram which can be separated into two
disjoint diagrams Γ1 and Γ2 by cutting at one node, which we denote by a. If the scale
Λa associated to the gauge group on the a-node is sufficiently low, and if superpotential
interactions between them are small, we have effectively two separate quiver gauge theories
for phenomena much above the scale Λa, one associated to Γ1 and another associated to
Γ2, which are weakly interacting with each other through the a-node. If supersymmetry is
broken in the sector Γ1, it can be communicated to the sector Γ2 by the gauge mediation
mechanism. The beauty of the quiver gauge theory construction is that, because of the
presence of bi-fundamental and adjoint fields on links and nodes, an effective superpotential
of the form (2.8) is naturally generated when supersymmetry is broken in a part of the
diagram connected to the a-node.
It follows trivially that any quiver theory that is vector like with adjustable mass terms
has meta-stable supersymmetry breaking vacua in some range of its parameter. All one has
to do is to identify a part of the diagram where supersymmetry can be broken using a known
mechanism, for example as in [1] or its variant [19], and to have its effect communicated to
the rest of the diagram by messengers. One can also consider the scenario where the quiver
theory associated to a sub-diagram Γ2 has a supersymmetric vacuum with dynamically
generated small scales, which can be used to set parameters of the theory associated to
another sub-diagram Γ1, where supersymmetry is broken. The supersymmetry breaking
effect can then be communicated back to the sub-diagram Γ2. This would give a string
theory realization of the idea of [20]. These and other mechanisms of supersymmetry
breaking will be explored further in [15].
These supersymmetry breaking quiver gauge theories can be coupled to the messenger
sector. In fact, as in the case of the A4 model discussed in the previous section, the mes-
senger sector itself can be included in quiver theories. If the messenger sector is attached
at the end of the quiver diagram, the effective low energy superpotential always takes the
form (2.3). Thus, one can see that the models in [4] and their generalizations are robust
and naturally appear in this large class of string compactifications.
Acknowledgments
We thank D. Berenstein, M. Dine, R. Kitano, J. Marsano, C. S. Park, N. Seiberg,
M. Shigemori, and T. Watari for discussions. H.O. thanks the hospitality of the high
energy theory group at the University of Tokyo at Hongo.
H.O. and Y.O. are supported in part by the DOE grant DE-FG03-92-ER40701. The
research of H.O. is also supported in part by the NSF grant OISE-0403366 and by the 21st
Century COE Program at the University of Tokyo. Y.O. is also supported in part by the
JSPS Fellowship for Research Abroad. The research of T.K. was supported in part by the
Grants-in-Aid (#16740133) and (#16081206) from the Ministry of Education, Culture,
Sports, Science, and Technology of Japan.
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