###### Abstract

The supersymmetric SU() Yang-Mills theory coupled to matter fields in the fundamental representation has meta-stable vacua with broken supersymmetry when . By gauging the flavor symmetry, this model can be coupled directly to the standard model. We show that it is possible to make a slight deformation to the model so that gaugino masses are generated and the Landau pole problem can be avoided. The deformed model has simple realizations on intersecting branes in string theory, where various features of the meta-stable vacua are encoded geometrically as brane configurations.

SLAC-PUB-12252

CALT-68-2621

hep-ph/0612139

Direct Mediation of Meta-Stable Supersymmetry Breaking

Ryuichiro Kitano, Hirosi Ooguri and Yutaka Ookouchi

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309

Physics Department, Stanford University, Stanford, CA 94305

California Institute of Technology, Pasadena, CA 91125

## 1 Introduction

Although there is no clear evidence yet, it is plausible that softly broken supersymmetry is realized in nature. Not only because it is a symmetry possessed by string theory, there are many phenomenologically attractive features in supersymmetric models, such as cancellation of quadratic divergences and unification of the gauge coupling constants [1, 2, 3].

It is then a question how supersymmetry is broken and how we feel it. There have been many studies on this subject, but, as is often the case, one of the earliest proposals [4, 5] among them seems to be the most elegant and simple idea. The idea is that there is a QCD-like strong interaction which breaks supersymmetry dynamically, and the standard model gauge group is identified with a subgroup of flavor symmetry in this sector. The standard model gauge sector can, therefore, feel the supersymmetry breaking directly via one-loop diagrams.

This idea has been discarded for a long time because of its difficulty in realistic model building. First, Witten has shown that there is a supersymmetric vacuum in supersymmetric QCD by using an index argument [6]. Therefore, we are forced to think of the possibility of chiral gauge theories for supersymmetry breaking, which is already a bit complicated. (See [7, 8, 9] for dynamical supersymmetry breaking in chiral gauge theories, and [10, 11] for models of direct gauge mediation in that context.) There is also a problem of Landau poles of the standard model gauge interactions. Once we embed the gauge group of the standard model into a flavor group of the dynamical sector (this itself is not a trivial task), there appear many particles which transform under the standard model gauge group. These fields contribute to beta functions of the gauge coupling constants and drive them to a Landau pole below the unification scale. Finally, even though the gauge sector of the standard model directly couples to the supersymmetry breaking dynamics, it is non-trivial whether we can obtain the gaugino masses. It is often the case that the leading contribution to gaugino masses cancels out.

Very recently, there was a break-through on this subject. Intriligator,
Seiberg and Shih (ISS) have shown that there is a meta-stable
supersymmetry breaking vacuum in some of supersymmetric QCD
theories [12]. The model is simply SU() gauge
theory with massive (but light) quarks. Within a range , supersymmetry is broken in the meta-stable vacuum.
The possibility of direct gauge mediation in this model is also
discussed in Ref. [12].^{*}^{*}*See also
[13] for a related work. Because of its simplicity of
the model, it is straightforward to embed the standard model gauge group
into the SU() flavor symmetry. However, it is concluded that there
are still problems regarding the Landau pole and the gaugino masses.
In the ISS model, there is an unbroken approximate U(1) symmetry
which prevents us from obtaining the gaugino masses.

The U(1) problem is a common feature in models of gauge mediation. As is discussed recently in Ref. [14], if the low energy effective theory of the dynamical supersymmetry breaking model is of the O’Raifeartaigh type, there is an unbroken -symmetry at the minimum of the potential (the origin of the field space). It has been proposed that the inverted hierarchy mechanism [15] can shift the minimum away from the origin by the effect of gauge interactions [16, 17, 18, 19, 14]. An alternative possibility that the shift is induced by an -symmetry breaking term in supergravity Lagrangian (the constant term in the superpotential) is recently discussed in Ref. [20]. It is, however, still non-trivial whether we obtain the gaugino masses even with the -symmetry breaking vacuum expectation values in direct gauge mediation models. For example, a model in Ref. [21] generates gaugino masses only at the order even though the -symmetry is broken by assuming the presence of the local minimum away from the origin. Since the scalar masses squared are obtained at the order as usual, gaugino masses are much smaller than the scalar masses unless the messenger scale is , that is difficult in models of direct gauge mediation because of the Landau pole problem. In fact, as we will see later, the structure of the messenger particles in the ISS model is the same as that in this model. (The same structure can be found in many models, for example, in Ref. [22] and also in very early proposals of gauge mediation models in Ref. [23, 24, 25].) Therefore, it is not sufficient to destabilize the origin of the field space for generating both gaugino and scalar masses.

In this paper we propose a slight deformation to the ISS model with which we can obtain gaugino masses by identifying a flavor subgroup with the standard model gauge group. We add a superpotential term which breaks -symmetry explicitly so that non-vanishing gaugino masses are induced. The vacuum structure becomes richer by the presence of the new term. In addition to the vacuum that is obtained by a slight perturbation to the ISS meta-stable vacuum, which we will call the ISS vacuum, there appear new (but phenomenologically unacceptable) meta-stable vacua. We find that decays of the ISS vacuum into the other vacua are sufficiently slow so that it is phenomenologically viable.

We also show that the Landau pole problem can be avoided by keeping the dynamical scale of the ISS sector sufficiently high in a way that is compatible with phenomenological requirements. In addition, if meta-stable vacua exist in a model with the same number of colors and flavors, as suggested by ISS, we can also consider the case where the ISS sector is in the conformal window, . In this case, we can take the scales of the ISS sector as low as (100–1000 TeV).

The deformed ISS model can be realized on intersecting branes in string theory, where a rich vacuum structure and the meta-stability of vacua can be understood geometrically.

## 2 The ISS model

We first review the ISS model. The model is simply a supersymmetric QCD with light flavors. Perturbative corrections to a scalar potential are calculable in the magnetic dual picture, and they have been found to stabilize a supersymmetry breaking vacuum. The model has an unbroken -symmetry, which prevents it from generating gaugino masses. An explicit one-loop computation of the masses suggests a natural solution to this problem, which we will discuss in the next section.

### 2.1 Supersymmetry breaking

The model is an SU() gauge theory with flavors. The quarks have mass terms:

(1) |

The index runs for . The masses are assumed to be much smaller than the dynamical scale . There is a meta-stable supersymmetry breaking vacuum when , where there is a weakly coupled description of the theory below the dynamical scale . The gauge group of the theory is SU() and degrees of freedom at low energy are meson fields and dual quarks and . There are superpotential terms:

(2) |

A dimensionful parameter is introduced so that the dimensionality of the superpotential is correct. A natural scale of is .

With this superpotential, the condition for all components of cannot be satisfied. The rank of the matrix is at most whereas the mass matrix has the maximum rank, . The lowest energy vacuum is at

(3) |

where and runs from 1 to , and is sorted in descending order. The -components of with have non-vanishing value . At this vacuum, the gauge symmetry SU() is completely broken.

We parametrize fluctuations around this vacuum to be:

(4) |

We put dimensionless parameter of so that components have
canonically normalized kinetic term.
Again, and .
Among these fields and the trace part of , ,
remains massless at tree level. The other fields obtain masses of
.
One-loop correction to a potential for the pseudo-moduli and
Re is shown to give positive masses squared,
which ensures the stability of the vacuum.^{†}^{†}†Imaginary part of
is a Goldstone boson associated with a broken
U(1) symmetry.
Once we take into account the non-perturbative effect, the true
supersymmetric vacuum appears far away from the origin of the meson
field . The life-time of the false vacuum can be arbitrarily long if
. Also, interestingly, the supersymmetry breaking
vacuum is preferred in the thermal history of the universe
[26, 27, 28].

### 2.2 Gaugino masses

It is possible to embed the standard model gauge group into a flavor symmetry group of this model. When we take and , there is a global symmetry; SU() SU() U(1). With or , we can embed SU(3) SU(2) U(1) into SU() or SU(), respectively. In the case where we embed SU(3) SU(2) U(1) into the SU() flavor symmetry, the standard model gauge group at low energy is a diagonal subgroup of SU(3) SU(2) U(1) in SU() dual gauge interaction (under which and transform and is neutral) and that in the SU() flavor group.

As discussed in Ref. [12], there is an unbroken
-symmetry under which carries charge two and and are
neutral.
Since the -symmetry forbids the gaugino masses, there is no
contribution to the gaugino masses of the standard model gauge group
even though it is directly coupled to a supersymmetry breaking sector.
It is instructive to see how the gaugino masses vanish at one-loop. The
fields and carry quantum numbers of both SU() and SU() and couple to which has
non-vanishing vacuum expectation value in the -component.
Therefore and play a role of messenger fields in
gauge mediation.^{‡}^{‡}‡The standard model gauge group at low energy
partly comes from SU() when we embed the SU(3) SU(2)
U(1) into SU(). One-loop diagrams with the
and fields, therefore, contribute to the gaugino
masses also in this case, although they are not charged under SU().
The relevant superpotential for this discussion is

(5) |

where we suppressed indices and defined . The and fields have mixing terms. In a matrix notation,

(6) |

where is a mass matrix for the messenger fields

(7) |

The formula for the gaugino masses can be generalized for this multi-messenger case as follows:

(8) |

where is or depending on whether we embed the standard model gauge group into the SU() or the SU() flavor symmetry. This formula is valid when . Since there is no dependence in , we obtain .

We can now clearly see that the gaugino mass would vanish at the leading order in even if we could obtain a non-vanishing vacuum expectation value for which breaks the -symmetry [21]. In the following section, we consider a model with explicit -symmetry breaking which generates the gaugino masses at the leading order in .

## 3 Deformed ISS model

Motivated by discussion in the previous section, we consider a modification of the ISS model which contains a mass term for the meson fields and so that has dependence. In the electric description, this corresponds to adding the following superpotential term

(9) |

where the color SU() indices are contracted in . Though this is a non-renormalizable interaction, it can be generated by integrating out extra massive fields coupled to in a renormalizable theory. In section 4, we will show that such a theory can be realized on intersecting branes in string theory. This interaction preserves the global symmetry SU() SU() U(1). We assume the same structure for mass terms of and as that in the model in the previous section, i.e.,

(10) |

so that the global symmetry is preserved. In the magnetic description, the mass terms correspond

(11) |

where and . In terms of component fields, the full superpotential is given by

(12) |

We could have added other terms compatible with the global symmetry. Although the theorem of [29] implies that a generic deformation to the superpotential generates a supersymmetry preserving vacuum at tree-level, it may not cause a problem with our scenario as far as the new vacuum is far from the one we are interested in and the transition rate between the vacua is small. However, since there are tree-level flat directions in , a deformation by destabilizes the ISS vacuum. Whether such a deformation is prohibited is a question of ultra-violet completions of the theory, but there is an interesting observation we can make from the point of view of the low energy effective theory. As we will see later, we need a certain level of hierarchy between and () to suppress a tunneling rate into unwanted vacua and also to avoid a Landau pole of the gauge coupling of the standard model gauge interaction. With this hierarchy this model possesses an approximate (anomalous) -symmetry which is softly broken by the small mass term . The charge assignment is and . This symmetry justifies the absence or suppression of other higher dimensional operators such as which destabilize the supersymmetry breaking vacua. (The supersymmetry breaking vacua remain stable as far as the coefficient for is smaller than .)

### 3.1 Vacuum structure

The introduction of the mass term for and makes the vacuum structure of this model quite rich. In addition to the supersymmetric and supersymmetry breaking vacua in the ISS model, there are also several stable supersymmetry breaking vacua. The stability and decay probability between these vacua are controlled by parameters in superpotential.

Meta-stable supersymmetry breaking vacua

As long as is smaller than , we can think of as a small perturbation to the ISS model, and thus there exists a similar meta-stable supersymmetry breaking vacuum. The effect of a finite value of is a small shift of of . The pseudo-moduli and Re obtain masses of as in the ISS model. We show in Figure 1 the one-loop effective potential for the pseudo-moduli . We see a small shift of the minimum. For , this vacuum is destabilized. Therefore, we assume in the following that is smaller than .

The small , in fact, modifies the vacuum structure drastically at far away from the origin of the field space. We can find other supersymmetry breaking vacua with

(13) |

(14) |

where the number of in the first equation, denoted , runs from to . Since these vacua have energy that are lower than that of the ISS vacua, , it has non-zero transition probability to these vacua. Below, we show that the decay rate can be made parametrically small by a mass hierarchy, . Although the vacuum with is the global minimum of the classical potential, they are not phenomenologically viable since gauginos cannot get masses at the leading order in for the same reason as in the original ISS model when we embed the standard model into some of unbroken global symmetry.

As we have seen, our vacuum is not the global minimum of the potential. It can decay into lower energy vacua specified by (13) and (14). We estimate the decay rate by evaluating the Euclidean action from our vacuum to others. The barrier by the one-loop potential is not high, of order . Thus, the most efficient path is to climb up the potential of and then slide down to more stable supersymmetry breaking vacua. The distance between and is of order and is wide compared to the height of the potential. Thus, we can estimate bounce action with triangle approximation [30],

Even if we choose , which will be required below, the Euclidean action can be made parametrically large by taking . Thus, the decay rate is parametrically small.

One might think that we can find more efficient path through tree level potential barrier. However at least it has to climb up that is very high, compared to the difference between two supersymmetry breaking vacua, of order . In this case, we can use the thin wall approximation [31] to estimate the bounce action and obtain . Again, we can make it parametrically large when .

Supersymmetry preserving vacua

So far we studied supersymmetry breaking vacua. In addition to these, the model also has supersymmetric vacua. Here, we will show that these supersymmetry preserving vacua can also be identified in the free magnetic dual description. Following [12], we look for a supersymmetric vacuum where meson fields get large expectation values. By the vacuum expectation value of and , dual quarks and become massive and can be integrated out. Also in the energy scale , and should be integrated out. Thus, we are left with the superpotential,

where the last term is generated by non-perturbative dynamics of a pure SU gauge theory. The low energy scale after decoupling of dual quarks, is given by the matching conditions at the two mass scales and ,

Note that and are singlets for the gauge group and do not contribute to running of gauge coupling. With the non-perturbative superpotential, -term conditions for light field and have solutions of the form,

Since and the difference of the vacuum expectation value between supersymmetric vacua and supersymmetry breaking vacua is very large, compared to the height of supersymmetry breaking vacua, we can estimate the Euclidean action for the decay process by triangle approximation [30],

(15) |

The factor is always positive. Therefore, with the mass hierarchy and , we can make the action arbitrarily large, and thus make the meta-stable vacua arbitrarily long-lived. These conditions also allow us to ignore higher order correction to the Kähler potential.

### 3.2 Gaugino and scalar masses

With the explicit -symmetry breaking by , direct mediation of supersymmetry breaking happens. The standard model gauge group can be embedded into either the SU() or the SU() flavor symmetry which is remained unbroken at low energy.

The gaugino masses are, in this case, given by the same formula in Eq. (8) with mass matrix :

(16) |

Therefore

(17) |

with the gauge coupling constant of the standard model gauge interaction. The factor is again when we embed the standard model gauge group into SU() (SU()).

Scalar masses are also obtained by two-loop diagrams. It is calculated to be

(18) |

is a quadratic Casimir factor for a field labeled . For having a similar size of gaugino and scalar masses, is required. It is possible to have this relation as long as without destabilizing the meta-stable vacuum.

### 3.3 Mass spectrum and the Landau pole problem

We summarize the mass spectrum at the ISS vacuum here. The massless modes are the Goldstone boson, Im, and the fermionic component of . The pseudo-moduli and Re have masses which are similar size to the gauginos, i.e., . Other component fields in the chiral multiplets , , , , , and have masses of or eaten by the gauge/gaugino fields.

Discussion of the Landau pole depends on a way of embedding of the standard model gauge group into flavor symmetries. We separately discuss two cases. We find that it is possible to avoid a Landau pole if we embed the standard model gauge group into the SU() flavor symmetry and take the dynamical scale and the mass parameter to be large enough. We also comment on an alternative possibility that the SU() gauge theory above the scale is a conformal field theory (CFT). This possibility allows us to take the mass parameter and the dynamical scale to be much lower than the unification scale without the Landau pole problem.

#### 3.3.1 Embedding SU(3) Su(2) U(1) into SU()

In this case, the pseudo-moduli is a singlet under the standard model gauge group, and thus it does not contribute to the beta function.

The beta function coefficients of the SU(3) gauge coupling is

(19) |

where is a renormalization scale. Above the mass scale , which is defined in Eq. (9), the theory should be replaced by a renormalizable theory, where it neccesarily contains additional fields. Therefore, there are contributions from those fields above the scale . The size of the contributions depends on a specific ultra-violet completion of the theory.

In order for the embedding to be possible, , and from the condition , we obtain

(20) |

There is a quite large contribution to the beta function. To avoid a Landau pole below the unification scale, GeV, the mass scales and should be high enough. For example, and GeV can avoid the Landau pole.

Model in the conformal window

Although it is not conclusive, the authors of Ref. [12] suggested that there is a meta-stable supersymmetry breaking vacuum also when the numbers of colors and flavors are the same. If it is the case, there is an interesting possibility that we can go into the conformal window, . If is in the conformal window, the gauge coupling of SU() flows into the conformal fixed point at some scale . The theory stays as a CFT until the mass term becomes important, and eventually at a lower scale , the theory exits from the CFT and becomes strongly coupled. The effective theory below the scale is described by an SU() gauge theory with flavors with a mass term . This is exactly the ISS model with flavors. Once we assume the existence of the meta-stable supersymmetry breaking vacuum, direct gauge mediation should happen as we discussed in the previous section although we have lost the control of the perturbative calculation. (See [32] for a similar model.) The beta function coefficient is in this case,

(21) |

where we have included a contribution from anomalous dimensions of ’s in CFTs [33, 34, 35]. The factors and are unspecified contributions from the fields which generate the term.

With and , we find

(22) |

Therefore, the dynamical scale can be much lower than the unification scale in this case. For example, if we take the ultra-violet completion to be simply adding a pair of massive fields and which couple to and , respectively, the additional contributions are . In this case, we can take the dynamical scale to be as low as without a Landau pole problem for and .

We implicitly took the scale , where the term is generated, to be in Eq. (21) because of the requirement for the sizes of the gaugino and scalar masses to be similar. With (see Eq. (9)) and , we need to take . However, the actual scale at which new fields appear can be much higher than or even when the anomalous dimensions of and are large in the CFT. For example, when , is a marginal or a relevant operator. In this case, it is not required to have an ultra-violet completion of the theory up to or higher, i.e., , while satisfying . This can be understood by the running of the parameter in the CFT:

(23) |

The unspecified contribution is not important if is high enough.

If , there are flavors with mass which are not charged under the standard model gauge group. If we reduce the masses of those fields to be slightly smaller than , the low energy effective theory below has more flavors and we can perform a reliable perturbative calculation of the potential for pseudo-moduli.

#### 3.3.2 Embedding SU(3) Su(2) U(1) into SU()

In this case, is given by

(24) |

The condition for the embedding to be possible is . Therefore

(25) |

With this constraint, there is always a Landau pole below the unification scale. The situation does not improve even if we consider the possibility of the CFT above the mass scale .

To summarize, by embedding the standard model gauge group in the SU() subgroup of the flavor symmetry, we can couple the ISS model to the standard model. The gaugino masses are generated at one-loop, and the Landau pole problem can be avoided if the gauge coupling scale of the ISS sector is sufficiently high or if the theory above the mass scale is a CFT.

## 4 Ultra-violet completions

The perturbation to the ISS model we considered in the previous section is non-renormalizable in the electric description. In this section we will show that the model can be regarded as a low energy effective theory of a renormalizable gauge theory at high energy. Moreover, this renormalizable theory itself can be realized as a low energy effective theory on intersecting branes and on branes on a local Calabi-Yau manifold in string theory. In order to decouple Kaluza-Klein and string excitations from the gauge theory, the length scale of these brane configurations as well as the string length must be smaller than that of the gauge theory. These brane configurations are so simple that it may be possible to incorporate them in the on-going effort to construct the minimal supersymmetric standard model from string theory compactifications.

One way to generate the non-renormalizable interaction (9) is as follows. Consider an quiver gauge theory with the gauge group U() U() U() with

(26) |

and identify U() with the gauge group U() of the ISS
model.^{§}^{§}§In the previous sections, we consider the case when the
gauge group is SU(). When the gauge group is U(), the
“baryon” symmetry is gauged and one of the pseudo-moduli becomes massive at tree-level due to the additional
D-term condition. Otherwise, there is no major difference in properties
of meta-stable vacua. We assume that the scales
for the other gauge group factors are so low that we can treat U()
U() as a flavor group. We then deform the theory by
turning on the superpotential for the
adjoint fields in the vector multiplets
given by

This breaks supersymmetry into , and the total tree level superpotential of the deformed theory is

After integrating out massive fields , the superpotential can be written as

This reproduces the interaction (9) and the mass terms for if we set

(27) |

Since we suppose , all the equations (27) can be satisfied by appropriately choosing parameters and .

### 4.1 Embedding in string theory

In the perturbative string theory, the collective coordinates of D-branes are open strings ending on them [40]. Since the lightest degrees of freedom of open strings include gauge fields, variety of gauge theories arise on intersecting branes in the low energy limit where the string length becomes small and the coupling of D-branes to the bulk gravitational degrees of freedom becomes negligible [41, 42, 43]. We will present an intersecting brane configuration where the deformed ISS model is realized as a low energy effective theory. One should not be confused that our use of the intersecting brane model implies that the theory above the dynamical scale is replaced by string theory or a higher dimensional theory. The string length and the compactification scale are much shorter than the gauge theory scale. It is one of the string miracles that quantum moduli spaces of low energy gauge theories are often realized as actual physical spaces such as brane configurations or Calabi-Yau geometry, allowing us to discuss deep infrared physics in the ultra-violet descriptions of the theories. This phenomenon has been well-established for moduli spaces of supersymmetric vacua, and it has just begun to be explored for supersymmetry breaking vacua [44, 45, 46, 47, 48, 49]. (For earlier works in this direction, see for example [50, 51, 52].) Here, we will find that meta-stable supersymmetry breaking vacua of the deformed ISS model are realized as geometric configurations of branes.

Consider Type IIA superstring theory in the flat 10-dimensional Minkowski spacetime with coordinates . Introduce four NS5 branes located at and at different points in the direction, and extended in the and directions. Let us call these NS5 branes from the left to right along the direction as NS5, NS5, NS5, and NS5. We then suspends D4 branes between NS5 and NS5, D4 branes between NS5 and NS5, and D4 branes between NS5 and NS5. The brane dynamics in the common directions is described by the supersymmetric quiver gauge theory with the gauge group U() U() U(). Note that the gauge coupling constant for the three gauge group factors are given at the string scale by

(28) |

where and are string coupling constant and string length, and are the lengths of the three types of D4 branes suspended between NS5 branes. The gauge couplings set the initial conditions for the renormalization group equation at ultra-violet. The quiver gauge theory is realized in the low energy limit where , keeping fixed. We choose so that the gauge coupling constants for U() U() are small.

We can turn on the superpotentials by rotating NS5 and NS5 into the directions. More precisely, we use the complex coordinates and and rotate the two NS5 branes on the plane so that they are extended in the direction of . The holomorphic rotation preserves supersymmetry. In the field theory, this corresponds to turning on with [53]. We can also turn on the quark masses and by moving NS5 and NS5 in the direction. The resulting configuration is shown in Figure 2.

We can also T-dualize the NS5 branes to turn the D4 branes suspended between the NS5 branes into D branes wrapping compact cycles in a local Calabi-Yau manifold [54]. Realizations of meta-stable vacua on branes partially wrapping cycles in Calabi-Yau manifolds have been discussed, for example, in [55, 48].

The brane configuration shown in Figure 2 is similar to the one appeared recently in [47]. However, there are some important differences. In the model of [47], the quark masses and in the electric description are set equal to zero. Moreover the strong coupling scales of the three gauge group factors are chosen as in the model of [47], whereas in our model. These differences have led to different ways of supersymmetry breaking in these models. Despite the differences, some of the results in [47] may be useful for further studies of our model.

### 4.2 Meta-stable supersymmetry breaking vacua on the brane configuration

In [44, 45], the ISS model and its magnetic dual were studied by
realizing them on intersecting branes, and brane configurations for the
supersymmetry breaking vacua were identified. The brane
configurations provide a geometric way to understand the vacuum structure of
the model.^{¶}^{¶}¶See [46] on
issues that arise when one turns on finite string coupling in
these brane configurations. These issues are not relevant to our
discussion below since we mostly deal with tree-level
properties of Type IIA superstring theory.
Recently it was used, for example, to study solitonic states
on the meta-stable vacuum in the ISS model [56]. Here, we
will present brane configurations that correspond to the meta-stable
vacua in the deformed ISS model.

To identify the meta-stable vacua, we need to go to the magnetic description, which is realized on branes by exchanging NS5 and NS5. Since we assume , it is reasonable to expect that the first duality transformation involves only these two NS5 branes. To avoid confusion, let us call the resulting NS5 branes as NS5, NS5, NS5, and NS5 from the left to right in the direction. Note that NS5 and NS5 are parallel to each other, and so are NS5 and NS5. There are D4 branes between NS5 and NS5, anti-D4 branes between NS5 and NS5, and D4 branes between NS5 and NS5.

The ISS vacuum is obtained by bending the D4 branes between NS5 and NS5 toward NS5, disconnect each of them at NS5, and annihilate their segments between NS5 and NS5 with the anti-D4 branes by the tachyon condensation. The resulting brane configuration is shown in Figure 3. Note that this configuration breaks supersymmetry since the D4 branes between NS5 and NS5 and the D4 branes between NS5 and NS5 are in angles. Since their end-point separation is of the order of whereas the supersymmetry breaking is of the order of their relative angles , an open string stretched between these D4 branes does not contain a tachyon mode provided . Since NS5 and NS5 are parallel to each other, the D4 branes between them can move along them. This freedom corresponds to pseudo-moduli . These D4 branes are stabilized by a potential induced by closed string exchange between them and the D4 branes between NS5 and NS5, which is the closed string dual of the Coleman-Weinberg potential.

We can also identify the other meta-stable vacua of the deformed ISS model. Let us take of the D4 branes between NS5 and NS5 and move them toward the D4 branes between NS5 and NS5. Doing this costs energy since these D4 branes have to climb up the Coleman-Weinberg potential. Eventually, as they approach the D4 branes between NS5 and NS5, open strings between the two kinds of D4 branes start developing tachyonic modes. The tachyon condensation then reconnects pairs of D4 branes, leading to the brane configuration as shown in Figure 4. This process lowers the vacuum energy since the length of the single D4 brane between NS5 and NS5 is shorter than the sum of the two D4 branes before the reconnection.

One can show that these brane configurations reproduce various features of the corresponding meta-stable vacua, such as their vacuum energies, expectation values of various fields (such as , , and ), and their decay processes. This can be done by a straightforward application of the brane configuration analysis in [44, 45, 46], and we leave it as an exercise for the readers.

## 5 Meta-stability at finite temperature?

It has been shown that the meta-stable supersymmetry breaking vacuum in the ISS model is favored in the thermal history of the universe [26, 27, 28]. The essential observation is that there are more light degrees of freedom in the supersymmetry breaking vacuum compared to the supersymmetric one. Finite temperature effects make the meta-stable vacuum more attractive in this circumstance.

In the deformed ISS model we discussed in this paper, there are many other meta-stable vacua. However, interestingly, the desired vacuum (the ISS vacuum) possesses the largest symmetry group among those vacua. In other vacua, number of degrees of freedom of the pseudo-moduli is reduced because some components of have masses at tree level. Therefore, the desired vacuum is the most attractive in the thermal history of the universe.

## Acknowledgments

We would like to thank K. Intriligator, H. Murayama, E. Silverstein, T. Watari, and T. Yanagida for discussions. RK thanks the hospitality of the high energy theory group at Rutgers University. The work of RK was supported by the U.S. Department of Energy under contract number DE-AC02-76SF00515. The work of HO and YO is supported in part by the U.S. Department of Energy under contract number DE-FG03-92-ER40701. HO is also supported in part by the U.S. National Science Foundation under contract number OISE-0403366. YO is also supported in part by the JSPS Fellowship for Research Abroad.

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