Single-sector supersymmetry breaking, chirality and unification

Single-sector supersymmetry breaking, chirality and unification

Siavosh R. Behbahani, Nathaniel Craig, Gonzalo Torroba
SLAC National Accelerator Laboratory, Stanford, CA 94309
Department of Physics, Stanford University, Stanford, CA 94305
Institute for Advanced Study, Princeton, NJ 08540
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854

Calculable single-sector models provide an elegant framework for generating the flavor textures via compositeness, breaking supersymmetry, and explaining the electroweak scale. Such models may be realized naturally in supersymmetric QCD with additional gauge singlets (SSQCD), though it remains challenging to construct models without a surfeit of light exotic states where the Standard Model index emerges naturally. We classify possible single-sector models based on Sp confining SSQCD according to their Standard Model index and number of composite messengers. This leads to simple, calculable models that spontaneously break supersymmetry, reproduce the fermion flavor hierarchy, and explain the Standard Model index dynamically with little or no additional matter. At low energies these theories realize a “more minimal” soft spectrum with direct mediation and a gravitino LSP.


1 Introduction

A very appealing idea is that some of the Standard Model (SM) quarks and leptons are secretly composites of more fundamental “preons” that explain their attributes in a simple way. Seemingly unrelated puzzles like the origin of the Yukawa couplings and the gauge hierarchy problem could then be solved by the same underlying dynamical mechanism. This more fundamental gauge theory has to produce approximately massless fermionic bound states, protected by a chiral symmetry.111These ideas have been explored over the years in different ways, mainly motivated by the work of ’t Hooft [1]. Early examples of nonsupersymmetric models were given in [2]; for a review and references on supersymmetric constructions see [3]. A very interesting model combining technicolor with composite SM fermions was given in [4]. Supersymmetric gauge theories provide a natural framework for these ideas. Indeed, supersymmetric QCD (SQCD) can produce exactly massless bound states. It may also be used to generate dynamically the Fermi scale and to break supersymmetry. Furthermore, Seiberg duality [5] gives, in many cases, weakly coupled dual theories where the massless mesons and baryons are described as elementary excitations.

Recent developments in SUSY gauge theories have motivated a renewed interest in constructing realistic composite models. The works [6, 7] proposed that the strong dynamics responsible for supersymmetry breaking may also produce composite SM fermions. These “single-sector models” can in principle give a simultaneous explanation for the observed flavor textures and the stabilization of the electroweak hierarchy. They are also quite economical in that they do not have a modular structure with messengers put in by hand –supersymmetry is communicated directly to the composites. These beautiful constructions have the drawback of not being calculable, and a detailed understanding of the spectrum was not possible.

The insight of Franco and Kachru [8] was to combine these ideas with the ISS mechanism (Intriligator, Seiberg and Shih [9]), building calculable single-sector models in SQCD. Next [10] constructed models with a fully realistic texture using a dimensional hierarchy mechanism. The flavor hierarchies are generated by coupling the SQCD mesons to elementary Higgs fields via higher dimensional operators, produced at a certain scale larger than the dynamical scale . After confinement, these irrelevant operators give rise to marginal Yukawa interactions, naturally suppressed by powers of


Moreover, following the same single sector philosophy it was shown in [11] that the strong gauge dynamics can also yield a composite Higgs, break and solve the problem.

The main conclusion from these works is that SQCD in the free magnetic range, supplemented by adequate superpotential deformations, can provide a unified explanation for the flavor hierarchies, supersymmetry breaking and Higgs physics, simultaneously generating some of the SM fields as composites. The requirement of being in the free magnetic range is essential for calculability: the asymptotically free “electric theory” becomes strong in the IR but admits a weakly coupled “magnetic dual”. The magnetic theory gives a description of the SM composites as elementary excitations, allowing for a direct perturbative analysis of their interactions and spectrum.

In this context, there are two important aspects that have to be addressed. The first one is how the SM chirality emerges from the confining gauge theory. Is it possible to generate dynamically a nonzero index222The index for a complex representation is defined as . for the representations of ? Is there a simple explanation for the family structure of the SM? The other point regards the construction of single sector models with perturbative unification. This has been quite difficult to achieve in constructions so far. The goal of this work is to address these points.

1.1 Overview

Let us summarize the main points of the present work. In §2 a general analysis of calculable single-sector models is presented, with emphasis on SM chirality and possible flavor textures. In models constructed to date, the confining interactions generate, for each desired composite generation, unwanted matter in the conjugate . The problem of conjugate representations was avoided in [8, 10] by introducing additional spectator fields in multiplets, and coupling them to the unwanted matter. The procedure is somewhat artificial, because the index is put in by hand directly in the UV. This is overcome by allowing the electric quarks to transform under chiral representations of the SM gauge group.

The problems of generating the SM chirality and achieving unification are related: models with chiral SM representations have in general less amount of unwanted composites than their vector-like counterparts. This helps to keep the gauge couplings perturbative. Motivated by these two points, we perform in §3 a general classification of confining SQCD theories that produce composites in chiral representations of the SM gauge group. This group theory analysis will be used to construct more efficient single sector models with less amount of extra matter and spectators.

This reveals a rich set of possible spectators (denoted by ), both in chiral and vector-like representations of . The supersymmetry breaking structure of single-sector models is based on the ISS mechanism [9], but it includes an important novelty: some of the mesons are coupled to the magnetic singlets . In the electric theory, these couplings are generated from marginal interactions between the electric quarks and spectators,


These become relevant in the IR. In §4 we present a detailed analysis of metastable supersymmetry breaking in the presence of the deformations (1.2) for in real or complex representations of the flavor group.

In §§5 and 6 we apply our results to construct examples where a nonzero index is dynamically generated, and which perturbatively unify. The first two composite generations can either arise from a single electric meson (in which case there is an approximate flavor symmetry) or from mesons of different classical dimension (“dimensional hierarchy” models). The third generation and are necessarily elementary. An interesting outcome of our analysis is that the more economical examples are in fact models with flavor symmetry. However, the flavor textures generated via Eq. (1.1) are not in full agreement with experimental values. We will present a new mechanism for generating fermion masses and mixings that circumvents this problem.

Insofar as the calculability of our models relies on preserving perturbativity of SM gauge couplings up to the GUT scale, we briefly review perturbativity constraints on additional SM-charged matter in Appendix A. Finally, we end in Appendix B with a detailed discussion of FCNC constraints on the pattern of soft supersymmetry-breaking masses in the various models under consideration. While -symmetric models are essentially unconstrained by FCNCs, mild constraints arise for models with a dimensional hierarchy.

2 Chirality and flavor hierarchies

We begin by describing our approach in general terms; some of the results can be applied to mechanisms different than the ones presented in [8, 10, 11]. In order to clarify our motivations and the classification given in §3, some necessary results from these papers will also be summarized in this section.

In searching for microscopic gauge theories that can produce a composite SM it is useful to review the way in which the SM itself explains the observed hadrons. In the IR, becomes strong and confines, while the gauge fields are ‘spectators’ of the strong color dynamics. The SM also contains spectator fermions that only couple to . These are of course the leptons and, in particular, they are crucial for anomaly cancellation. The theory of preons that could underlie the SM will be built upon a similar pattern. This approach was advocated for instance in [1].

2.1 Basic SQCD setup

For concreteness we will consider a SQCD theory with gauge group. The analysis for and theories is quite similar, and it turns out that theories lead to more economical models. Recall that an gauge theory333The notation is and, more generally, . with fundamental quarks , ( flavors) admits, for , a dual description with gauge group

containing magnetic quarks together with a meson singlet in the antisymmetric of the flavor group. The dynamical scale of the theory is denoted by . Following the discussion in §1 we require that the magnetic dual is IR free, .

In some cases we will also be interested in adding a field in the “traceless” antisymmetric () of the gauge group, with a superpotential which restricts the mesons to


Both are in the antisymmetric of the flavor group . Color indices are contracted with . The motivation for adding this field is to have mesons of different classical dimension that can lead to realistic fermion masses [10]. The duality for this case was studied in [16].

The SM quantum numbers are explained by weakly gauging and taking the electric quarks to transform under (possibly) chiral representations of . Finally, the theory contains elementary spectator fields which, by definition, are singlets under and transform under . In summary, the matter content is given by


(By a slight abuse of notation, the same indices , are used to denote the flavors and the possible representations ). According to our definition, all the elementary SM fields (third generation, Higgs, etc.) are contained in the spectator fields . Anomaly cancellation may also require extra elementary spectators not present in the SM.

The structure of the magnetic dual is


where in the absence of , while including this field with a cubic superpotential leads to . The SM quantum numbers of the mesons are obtained from the antisymmetric part of . Also, the magnetic superpotential includes cubic couplings that will be important for supersymmetry breaking.

2.2 Standard Model chirality

The mesons and defined in (2.1) have SM quantum numbers given by the antisymmetric part of . We wish to identify the composite SM generations


with appropriate meson components,


In principle it is also possible to use SQCD baryons as composite SM fields, but in the simple examples below this does not lead to realistic models.

At this stage, composite models can be classified into

  • models where the two composite generations , , arise from the same meson. In the absence of superpotential interactions, there is then an unbroken flavor symmetry;

  • dimensional hierarchy models, where the lightest first generation arises from the dimension 3 meson , while the second generation is identified with .

We also distinguish between ‘democratic’ models where both and are composites, and ‘ten-centered’ models where only the ’s are composites. As we discuss below, such models tend to be very minimal and lead to quite realistic flavor textures.

Next, we wish to determine whether it is possible to generate dynamically the SM chirality. This entails calculating the index for the and representations


where is the number of fields in the magnetic theory transforming under the representation of . Even though the SM particles are identified from components of and , the magnetic quarks are also included in the calculation of the index. The reason for this is that in the single-sector models that we explore, the magnetic gauge group is completely higgsed and the ’s couple to some of the components of the mesons, producing vector-like composite messengers. This is explained in more detail in §4.

Three different levels of chirality may be distinguished:

a) Models where the Standard Model index is not explained, . It is generated in the UV using spectators. In the absence of spectators the composites are vector-like.

b) Models where the index is correct, but there are also fermions in other chiral representations. These are better in that, although spectators are still needed, they are in general “predicted” by the cancellation of SM anomalies directly in the electric theory.

c) Models where the total index is explained, and it agrees with the index. The only light chiral fields are then in and representations, and no spectators are needed besides the usual SM elementary fields ( and the third generation).

These qualifications apply for ten-centered models with simple modifications; the goal in this case is to produce and among . §3 is devoted to the general group theoretic analysis of the SM index. In single-sector models with 2-index representations, we will find a few models of type b) and, surprisingly, just one model of type c) is allowed.

2.3 Generating the flavor textures

Let us now explain how the flavor textures are generated [8, 10]. At a scale before the electric theory confines, there is some new dynamics that generates interactions between the elementary Higgs and the mesons, . These operators, being irrelevant in the UV, are suppressed by powers of . After confinement they become marginal and give rise to the SM Yukawa couplings, with hierarchies controlled by powers of


In models where the first two composite generations () come from a single meson , these new interactions are of the form


where denotes the elementary third generation. In the IR, after canonically normalizing the meson by , Eq. (2.6) gives rise to Yukawa couplings


where order one coefficients are being omitted. For , these simple Yukawa textures are a good starting point for generating the hierarchies in fermion masses, but more structure is required to obtain fully realistic masses and mixings. In §5, we present an alternative mechanism for obtaining realistic Yukawa matrices in models with flavor symmetry. This will open up new model-building possibilities.

Next, consider dimensional hierarchy models; here the lightest first generation is identified with the dimension 3 meson , while the second generation arises from . The superpotential at the scale now reads


After confinement, gives rise to Yukawa couplings


Realistic flavor textures are obtained for  [10].

Finally, let us discuss ten-centered models, where the flavor hierarchies come entirely from the ’s: the corresponding to the first two generations are composites, while are elementary. As before, in these models the composites and can either be produced by the same dimension two meson, or we can have , . In the former case, the analog of Eq. (2.6) gives


Similarly, for dimensional hierarchy models,


Predictions for masses and mixings are discussed below, after analyzing concrete models.

So far we have assumed that both and are elementary. However, as pointed out in [11], it is also possible to have a composite . This leads to various phenomenologically desirable consequences: the electroweak scale is generated dynamically (and to a smaller extent radiatively), the problem is solved, and the hierarchy between top and bottom/tau masses is naturally explained. While it is not the purpose of this work to discuss in detail the Higgs physics of single-sector models, we point out that a composite may also lead to attractive ten-centered models, providing an additional suppression for in (2.10), (2.11).

3 Chirality and unification: analysis of the SM index

Having explained the gauge dynamics that produces composite generations and the various types of flavor textures that can be obtained, we are now ready to perform a group-theoretic classification of single sector models according to the SM representations that are obtained in the IR and their messenger content. We will compute the index of the complex representations and use these results to construct models with massless and/or composites. This procedure will allow us to find all the examples where perturbative unification may be achieved, and exhibit an interesting connection between SM chirality and unification.

3.1 Calculation of the index

Recall that the microscopic theory is SQCD with gauge group and flavors , in the free magnetic range. Absent superpotential interactions, the flavor symmetry group is , with the electric quarks transforming in the fundamental representation. The SM quantum numbers are explained by weakly gauging a subgroup

Then the quarks decompose as


where the color index is not shown. For our purposes it is not necessary to consider higher dimensional representations. In general the embedding will be chiral, namely ; SM anomalies are canceled by adding spectators that are singlets under . See discussion in §2.1.

In the dual magnetic description, the magnetic quarks decompose into


The SM index contributed by the magnetic quarks is then


where is the color multiplicity. Note also that we have defined the index for 5-dimensional representations such that positive index indicates an excess of ’s.

The mesons transform in the antisymmetric of . Taking the antisymmetric part of in Eq. (3.1) gives:


The final expression for the total index associated to the composites is thus


As we anticipated above, we are computing the total composite index by adding the contributions of and , although the latter have a magnetic gauge group index while the former are singlets under . We will explain in §4 that in our models the gauge group is completely higgsed at low energies; the elements of with nontrivial SM quantum numbers pair with corresponding conjugate fields from to give vector-like messengers. Accordingly, we introduce the messenger index (see e.g. [19])


In what follows, we specialize to the smallest possible magnetic gauge group, namely .

3.2 Solutions

Having determined the index for arbitrary combinations of simple matter, let us now classify the most economical models. For simplicity, we will restrict ourselves to models with , so that the flavor symmetry is at most . In order that these models produce successful theories of Standard Model flavor, we will require . From the embedding Eq. (3.1),


thus the sum must be even. Finally, in the models of §4 the supersymmetry breaking vacuum requires or , in order not to prematurely break the .

The models with matter satisfying the above criteria, ordered by the messenger index of Eq. (3.6) are:

0 0 0 0 0 0

1 0 0 0 0 0 0 0 6

0 0 2 1 0 0 0 1 2 6

1 0 0 1 0 0 0 0 8
Sp-5 1 0 1 0 0 1 0 8
Sp-6 0 0 3 1 0 0 0 3 5 8

1 0 1 1 0 0 1 0 10
Sp-8 0 1 2 0 0 0 1 10
Sp-9 0 0 3 2 0 0 0 2 3 10
Sp-10 1 0 2 0 0 2 1 10

1 0 1 2 0 1 1 12
Sp-12 1 0 2 1 0 0 2 1 12
Sp-13 2 0 0 0 0 0 1 12

The indices in this table are calculated using Eq. (3.5), which count net number of chiral composites from and ; this classification then applies to single-sector models with flavor symmetry. Dimensional hierarchy models have additional composites from , so that (3.1) should be added again. Hoping that it does not lead to confusion, the notation from this classification will be used for both and dimensional hierarchy models. Notice that Eq. (3.7) restricts the multiplicities . For instance in the class , and have to be both even or both odd. In class , must be even, while in and is odd.

One of our goals is to find models where perturbative unification is possible. In general, a messenger index of is the largest that allows perturbative gauge coupling unification if the messengers are between GeV, as is the case for the single-sector models considered here. Therefore we will focus on models . Nevertheless, we will also consider model in detail, as it is the unique model of category (c) (see discussion in §2.2). It is useful to explain the salient features of models with small messenger index. After analyzing the supersymmetry breaking mechanism in §4, we will study these examples in detail in §§5 and 6.

Let us begin by considering class , the unique class of strictly ten-centered models with no extra chiral matter in small representations (). The minimal model of this class that we may construct is a -symmetric model with

(the model with has only ’s in messenger fields, but not in the components of the meson in which Standard Model fermions are embedded). Conveniently, the messenger index is compatible with perturbative gauge coupling unification. Using Eq.(3.5), we find for this model

This gives a net number of two massless composite in the , to be identified with the first two SM generations. These meson components have a flavor symmetry in the absence of an electric superpotential. Notice that and also give two pairs of in the minimal case . These become messengers in the supersymmetry breaking model. Once elementary Standard Model fields have been included, anomaly cancellation requires the addition of spectators to cancel the anomaly contribution of the .

We may also use this class to build models with a dimensional hierarchy. In this case, the minimal embedding is identical to that of the model. This will lead to extra ’s in the pseudomoduli of both meson fields, which must be removed with spectators. However, we will find that this doubling is a desirable feature for models where the doubling of matter fields is required to obtain the correct more-minimal spectrum of MSSM soft masses.

These are all the examples for ten-centered models with . But perhaps this is too restrictive and we may have some number of ’s lifted by spectators. This is a natural situation in our framework, because the SM generations already give three spectators. We may then consider ten-centered models in classes and , both of which have chiral ’s that must be removed by spectators.

Consider first class , which is particularly attractive. The minimal model of this class is

For this model the chiral indices are

From the matter content of , two of the ’s become messengers in this model, leaving only one that needs to be lifted by spectators, as can be seen from the index. Of course, since this model has only one , it must necessarily be a dimensional hierarchy model in order to produce the correct number of composite generations. This model is quite compact, with a small messenger index guaranteeing perturbativity at all scales.

What else can we build in class ? There is no model with just two ’s. We may build a model with three ’s (and ) via

for which the chiral indices are

The SM anomalies are canceled by adding the following SQCD singlets: one , one and 6 fields in the –two of which are the elementary first and second generation SM fields. Generic superpotential deformations lift the extra unwanted matter at long distance, leaving a confined theory with composites in the .

We close out our discussion of ten-centered models with those of class . The simplest such model is

The chiral indices from composites in this case are

Here four ’s and two ’s become messengers. Two ’s may be given mass terms, and likewise with a and pair. Anomalies are canceled by adding and spectators, leaving two chiral ’s to become Standard Model states. Such a model can give rise either to a theory or a dimensional hierarchy with doubled matter, much as in the models of the class. This concludes our study of all the simple ten-centered models with minimal spectator content and sufficiently low messenger index.

Now let us analyze models of type c). Here we require all except for the and/or . If this can be accomplished, then no spectators are needed and the SM chirality is generated by the gauge dynamics. Interestingly, there is a unique example (up to addition of singlets),


This corresponds to class above, with nonvanishing index


There are no ten-centered models of type c). On the other hand, models in class a) (where ) were presented in [8, 10] so we will not describe them again here.

Let us pause to summarize what we have achieved so far. We have found the gauge theories that give rise to massless meson composites in and SM representations, with smallest messenger number. The masslessness is guaranteed, in the absence of EWSB, by a nonvanishing index computed in the table above. In some of the simpler examples, the number of composite SM families is determined by fields that are neutral under . For instance, in classes and the number of mesons in the is proportional to the number of electric quarks neutral under the SM gauge group. Similarly, in dimensional hierarchy models the families are associated to mesons and containing the SM-neutral field (an antisymmetric of the electric gauge group).444Class is an interesting exception, where a fixed index is obtained from the product of two preons in the .

Models with SM-chiral electric quarks give a small messenger index and make perturbative unification possible. This reveals an interesting connection between SM chirality and unification, the basic reason being that in these models of direct mediation the dynamics that produces composite generations also gives messenger fields. Another consequence of this approach is the existence of spectator fields that are singlets under the electric gauge group. These are required to make the theory anomaly-free. Once we allow for generic superpotential deformations, these fields will have the desirable effect of lifting extra SM exotics produced by the dynamics. So, let us now study these and other aspects of the low energy theory in detail.

4 Metastable SUSY breaking in SQCD plus singlets

Having explained the flavor hierarchies via compositeness, the next step is to use the same gauge dynamics to break supersymmetry dynamically and calculably. While the original proposals in [6, 7] were incalculable, [8] found that metastable supersymmetry breaking can occur quite naturally in these models. Indeed, ISS showed [9] that adding small masses to the electric quarks,


leads, in the free magnetic range, to metastable vacua at the origin of field space. (These formulas refer to an gauge group; the case will be studied shortly). The macroscopic theory becomes


(where is set by the dynamical scale) and supersymmetry is broken by the rank condition.

Single-sector models have one new ingredient, namely additional singlets under the electric gauge group, that can potentially modify the supersymmetry breaking vacua.555We thank D. Green, A. Katz and Z. Komargodski for pointing out to us that superpotential couplings between electric quarks and singlets in the adjoint of the flavor group produce new metastable vacua with lower energy than the ISS configuration. For further discussion or other applications see [17]. We saw in §3 that anomaly cancellation in general requires adding spectators that are neutral under the confining dynamics but have nontrivial quantum numbers. In fact, this is phenomenologically attractive because, allowing for generic (gauge invariant) superpotential interactions,


can lift unwanted exotics and leave us with just the SM matter fields in the IR.

This naturally leads us to consider SQCD with electric quark masses and couplings to singlets,


in the free-magnetic phase.666The global symmetry limit was already discussed in [20] as a way to generate the quark masses dynamically. We consider singlets (and masses) that transform nontrivially under the flavor symmetry group, such that the weakly gauged subgroup of (3.1) is left unbroken. To set some notation, we begin by reviewing the ISS construction (i.e. no extra singlets) for theories with unequal electric masses. Then we study in detail the case of interest Eq. (4.4); the analysis for gauge theories is similar.

4.1 Metastable vacua in theories with different masses

As in §2.1 we consider an gauge theory with fundamentals , in the range . We also turn on masses for the flavors,


Color indices are contracted with ; subsequent formulas will be sometimes simplified by omitting this contraction. Quark masses are ordered according to and are chosen so that of Eq. (3.1) is left unbroken; they also have to be parametrically smaller than the dynamical scale, .

The matter content and symmetries of the magnetic theory in the limit are

1 antisym 2

where and and have canonical kinetic terms. The theory has a superpotential


where the matrix in the linear term is given by


The diagonal entries are chosen to be real and positive, and ordered so that if . For simplicity, the cubic coupling is also taken to be real.

The F-terms


cannot all vanish because the second term has a smaller rank than the first. The metastable vacuum is obtained by turning on the maximum number of expectation values to cancel the largest F-terms,


Fluctuations around the vacuum are parametrized by


The tree-level nonzero F-terms and vacuum energy are


Importantly for what follows, the expectation values are set by the largest and the F-terms are controlled by the smaller ones. The nonzero expectation value higgses completely the magnetic gauge group , and the SM gauge group is a weakly gauged subgroup from


(this group is left unbroken in the limit ). Notice that give of and is an antisymmetric. This justifies our previous procedure for computing the SM index.

The tree-level spectrum is as follows. The field is a pseudo-modulus; it is flat at tree-level but generically receives quantum corrections and will be lifted. are supersymmetric at tree level. On the other hand, couples directly to the pseudo-modulus which has a nonzero F-term. Also, and have a supersymmetric mass and have nontrivial SM quantum numbers. Therefore, in the macroscopic theory are composite messengers with supersymmetric mass and splittings given by . More precisely, some of these fields are Nambu-Goldstone modes in the limit ; see [9] for more details.

While this derivation has been for general , for our purposes it suffices to have two different values , with the first entries equal to and the last ones equal to . Then the superpotential becomes


with the expectation values simplifying to


We assume (4.13) for the rest of the calculations –the more general case can be approached along similar lines. Here the expectation value of has been fixed by the D-terms.

Integrating out the heavy messengers gives an effective Coleman-Weinberg potential that lifts the pseudo-moduli [9]


where, up to order one numerical factors, the CW mass is


This implies that the MSSM composite sfermions acquire masses of order (4.16).

4.2 Metastable vacua in SQCD plus singlets

Now let us discuss the case relevant for our constructions: an gauge theory with extra singlets and superpotential


Here run over an arbitrary subset of the flavor indices and to avoid a proliferation of indices we have identified all the cubic couplings into a single . The masses and cubic couplings are singlets under the weakly gauged . It is worth exploring first the global limit , which is of more general interest beyond single-sector models. 777SQCD plus singlets has a very rich dynamics in the conformal window; see for instance [18].

The cubic couplings are marginally relevant and their effects are easiest to understand in the magnetic dual, where we have


The fields have been grouped into a -rowed antisymmetric matrix, with the understanding that only the elements corresponding to the set have nonzero fields. The F-term for now becomes


The rank condition of (4.8) is modified in an important way because the extra singlets make it possible to cancel more than F-terms from .

Due to the form of the linear terms, it is convenient to split into ‘diagonal’ and ‘off-diagonal’ pieces,


and both contributions can be treated separately. Let us look first at the diagonal terms assuming that there are nonzero fields and setting . If these elements are all independent, then there are vacua with that have lower energy than Eq. (4.14). For these vacua are supersymmetric –here we focus on the supersymmetry breaking case .

If has elements in the upper block (the analog of in Eq. (4.10)) and elements in the lower block, with and