Retrofitting O'Raifeartaigh Models with Dynamical Scales

We provide a method for obtaining simple models of supersymmetry breaking, with all small mass scales generated dynamically, and illustrate it with explicit examples. We start from models of perturbative supersymmetry breaking, such as O'Raifeartaigh and Fayet models, that would respect an $R$ symmetry if their small input parameters transformed as the superpotential does. By coupling the system to a pure supersymmetric Yang-Mills theory (or a more general supersymmetric gauge theory with dynamically small vacuum expectation values), these parameters are replaced by powers of its dynamical scale in a way that is naturally enforced by the symmetry. We show that supersymmetry breaking in these models may be straightforwardly mediated to the supersymmetric Standard Model, obtain complete models of direct gauge mediation, and comment on related model building strategies that arise in this simple framework.


Introduction and General Idea
Dynamical supersymmetry (SUSY) breaking [1] and the mediation of SUSY breaking to the supersymmetric Standard Model (SSM) have been studied extensively. A particularly important objective is to identify simple models of dynamical SUSY breaking that may be straightforwardly mediated to the SSM, yielding predictive and phenomenologically attractive superpartner spectra [2,3,4]. In early examples with gauge-mediated SUSY breaking [5], the problems of SUSY breaking and its mediation were addressed by postulating separate SUSY breaking and messenger sectors. These models motivated many advances [6,7,8], culminating in a few genuinely simple and viable models of direct gauge mediation [9,10,11,12,13,14,15], in which fields of the SUSY breaking sector also play the role of messengers, transmitting SUSY breaking to the SSM.
In this note, we develop a straightforward method for obtaining simple models of SUSY breaking in which all small scales are generated dynamically. We show further that SUSY breaking in these models may be rather simply communicated to the SSM, providing new avenues for direct gauge mediation and gravity mediation. To illustrate the method, we work through two complete gauge mediation examples that are representative of large classes of models, and we discuss the method's application to more general model building problems.
The basic strategy in its simplest realization can be summarized as follows: (1) Start with a model of perturbative SUSY breaking, such as an O'Raifeartaigh or Fayet model, whose small input parameters m i break an R symmetry that would be restored if the m i transformed as the superpotential does.
(2) Couple the system to a SUSY preserving sector with a dynamically small operator vacuum expectation value (VEV). Our prototypical example will be pure SU (2) Yang-Mills theory, with gauge field strength superfield W α and dynamical scale Λ. Replace dimensional parameters m i in the superpotential by W α W α suppressed by appropriate powers of a high scale M * . At low energies, W α W α ∼ Λ 3 . This renders the m i dynamically small in a way naturally enforced by the symmetries and preserves a local SUSY breaking minimum.
We will refer to this procedure (1)- (2) as retrofitting the old-fashioned perturbative SUSY breaking models. Elementary ingredients suffice to bring such models up to modern model building standards of naturalness, while preserving some of the simplicity of early constructions [16]. In effect, we consider a supersymmetric hidden sector to obtain dynamically small scales, which allows the SUSY breaking sector to be more directly coupled to the SSM.
If desired, the couplings to W α W α can arise from purely renormalizable interactions by integrating out massive flavors in the SU (2) SUSY gauge theory [17]. In any case, the coupling to the SU (2) sector does not destroy the local SUSY breaking minimum of the perturbative model (1), though it often introduces SUSY vacua far away in field space. As discussed, for example, in [8,11,12,18,19,20], we need not impose that the SUSY breaking configuration be the global minimum of the potential.
Indeed, one element of many successful models is metastability. In field theory models of dynamical SUSY breaking, the requirement that SUSY be broken in the global minimum is very restrictive, and allowing for metastable vacua greatly simplifies the problems of model building, especially for gauge mediation. This point was emphasized clearly, for example, in [8,7,11,12]; more recently it has found application in the problem of moduli stabilization [18] and dynamics [21,22,23], in the vacuum structure of large N gauge theories arising in generalizations of AdS/CFT [19], and in supersymmetric QCD [20].
As we will see, simple constructions lead almost trivially to a large class of dynamical SUSY breaking models and suggest an array of further model building possibilities. It is worth remarking that the models need not be chiral and can have non-vanishing Witten index, like the models of [19,20]. They can possess interesting (discrete) symmetries, which naturally protect the structures required for model building goals. This simple method allows construction of theories with direct gauge mediation as well as gravity-mediated models with appropriately large (non-loop-suppressed) gaugino masses.
As discussed recently in [20], some basic classes of supersymmetric gauge theories reduce at low energies to infrared-free O'Raifeartaigh models with metastable SUSY breaking. In some circumstances, the direct mediation models of the type we consider here may be UV completed by asymptotically-free quantum field theory. In other circumstances, the models may be completed by string theory, where metastable SUSY breaking [18] has played a crucial role.
From the perspective of weakly coupled string theory, one might worry that there are additional approximate moduli that affect the value of the gauge coupling. It is worth noting in this connection that the current state of the art in string moduli stabilization -via a combination of a tree level potential, orientifolds, and Ramond-Ramond fluxes -can fix the dilaton and other moduli at a high scale. In the context of low energy supersymmetric models, this allows for a gaugino condensate which does not vary with extra moduli beyond those evident in the low energy field theory of interest here, whose couplings are fixed by discrete symmetries.
In the next section, we consider retrofitting a class of O'Raifeartaigh models and work through a simple example in detail. We next simplify the model further to extract some lessons about the role of chirality and symmetry. We follow this in §3 with another general class of models including a Fayet-Iliopoulos parameter. In the final section, we summarize and discuss further model building applications.

Retrofitting O'Raifeartaigh Models
In this section we will implement the procedure outlined above in concrete examples and comment on model building lessons that arise in this framework. We begin with a brief review of O'Raifeartaigh models and their challenges. Next we consider a simple explicit example which we retrofit to render its scales dynamical in a way consistent with symmetries. This model is complete in that it readily incorporates messengers appropriate for gauge mediation, generating Standard Model superpartner masses. In the final subsection we extract lessons illustrated by even simpler systems, emphasizing the role of metastability in avoiding the unnecessary constraints of chirality and vanishing Witten index.
Consider O'Raifeartaigh models, with n fields Z 1 , . . . , Z n , n ′ fields φ 1 , . . . , φ n ′ , n ′ < n, and superpotential We will address each of these, illustrating the technique with perhaps the simplest version of (2.1). Let us first summarize the method. With regard to point (i), the Coleman-Weinberg potential expanded about an appropriate point in field space generically lifts the flat direction; one can explicitly check for self consistent metastable solutions as in [16,12].
Point (ii) can be addressed by coupling in messengers and including their contribution to the Coleman-Weinberg potential self-consistently. Finally point (iii) can be addressed by coupling in an otherwise supersymmetric SU (2) gaugino condensate, or any other more general SUSY sector with a dynamically small operator VEV.

A Complete, Simple Example
As a very simple illustrative example, consider a model with messengers η andη in, say, the 5 and5 of SU (5), and three fields Z 1 , Z 2 , and φ. A natural superpotential based where M * is a high scale corresponding to new(er) physics, such as a grand unified or Kaluza-Klein scale. We will obtain the parameter µ 2 ∼ Λ 3 /M * dynamically from a coupling d 2 θW α W α λZ 2 M * between the SU (2) sector and the O'Raifeartaigh model. This theory is invariant under the following two symmetries: a discrete Z Z 2N R symmetry, with N > 2, under which the superpotential transforms with charge 2 and the fields φ, Z 1 , Z 2 , W α , and ηη have charges 1, −1, 0, 1, and 1; and a continuous R symmetry, under which φ is neutral and Z 1 , Z 2 , and ηη transform, which governs the renormalizable terms, but is broken by the M * -suppressed operators. The superpotential of (2.2) is the most general one respecting these symmetries, up to terms higher order in M −1 * . In the absence of the messengers, the model has a massless combination of Z 1 and Z 2 , and a φ VEV Plugging in this solution yields F Z 1,2 terms of order is forced on the model by the discrete symmetries it respects.) An F term for Z combined with a VEV for φ will produce naturally small superpartner masses as we will discuss further below.
The Coleman-Weinberg potential obtained by integrating out φ, η,η yields a metastable minimum for Z at the origin in a self-consistent expansion about φ = φ 0 , η =η = 0. Let us begin by integrating out the fluctuations of φ; we will show that these dominate over messenger loops in this model. 1 Writing φ = φ 0 + δφ, the mass terms for the fluctuations δφ ≡ δφ 1 + iδφ 2 are of the form plus contributions subleading in the regime µ/M * ≪ 1. The fermion loops cancel the δφδφ contribution here, and so the leading contribution to the potential from the φ multiplet is plus subleading contributions. The messenger loops are subleading relative to the φ loops; the η,η mass terms are of the form (|η| 2 + |η| 2 )µ 2 |λ + Z 2 /M * | 2 plus a SUSY breaking term proportional to ηηµ 5 /(M 3 * λ), which will be much smaller than that in (2.5). Although subleading in the Coleman-Weinberg potential, the messengers provide the dominant transmission of SUSY breaking to the SSM. As we just noted, the leading contribution to the messenger masses is from the supersymmetric λφηη coupling, giving m η,η ∼ λµ, while the leading SUSY breaking contribution to their masses is . In application to gauge mediation, this yields gaugino and squark masses of the order ofm where g represents SSM gauge couplings.
For F Z i ≤ 10 20 GeV 2 , the gravity mediated contribution to superpartner masses is suppressed relative to the gauge mediated contribution. Imposing this, we find that, for example, m η,η ∼ λµ ∼ 10 11 GeV and M * ∼ 10 15 GeV produces a viable model, with λ ∼ 0.1. Of course, if M * were much lower than the GUT scale, then the messenger scale could be lower as well. As we will discuss further in the next subsection, another application of our method is to models where gravity mediation dominates. 1 In the model of §3, the messengers themselves will play a leading role in stabilizing the scalar fields, providing a particularly direct mediation mechanism.
To retrofit the model, as discussed above, we couple in a pure SUSY Yang-Mills sector with gauge superfield W α , replacing the superpotential (2.2) with (2.8) Integrating out the gauge interactions yields Expanding in Z 2 yields at leading order a model of the form (2.2), with It is self-consistent to integrate out the gauge degrees of freedom because they have M *suppressed couplings to the rest of the system, too weak to compete against the forces in the Yang-Mills sector proper, which appear at the scale Λ.
Including the Z 2 dependence in solving for φ 0 yields generalizing (2.4). This Z 2 -dependence can lead to the presence of supersymmetric minima far away for appropriate ranges of parameters, but it does not destabilize our local minimum, as we can see easily as follows. Expanding in Z 2 , the term |F Z 2 | 2 in the effective potential produces a tadpole of order Z 2 Λ 12 /(M 9 * λ 2 ). The Coleman-Weinberg potential (2.6) produces a mass term of order λ 2 |Z 2 | 2 Λ 9 /M 7 * , sufficient to stabilize Z 2 close to its original minimum at the origin.

Remarks on Retrofitting O'Raifeartaigh Models
In the previous subsection, we implemented the retrofitting procedure in a complete model, which was natural, given the specified symmetries, and incorporated messengers generating sparticle masses. The method has wider applicability, and it is interesting to extract and separate some of the essential elements of the procedure and consider independently the role of symmetry, chirality (or lack thereof), and metastability. Before rendering the mass parameters dynamical, note that a small deformation of the model makes the SUSY breaking minimum merely metastable. If we write We are assuming g is fixed. If there are other moduli-like fields contributing to the gauge coupling, we assume that they are fixed at a higher scale, e.g., by fluxes or other dynamics.
The parameter a is naturally of order one, if χ is neutral under the discrete R symmetry.
In contrast to our complete models in §2.1 and §3, this structure is not enforced by symmetries, but it is meant only to be illustrative. The addition of small, symmetry-preserving couplings does not alter its basic features.
All of this illustrates that it is easy to construct metastable models of dynamical SUSY On the other hand, it is also a very simple to consider these theories as hidden sectors for gravity mediation. These models are promising from this viewpoint since no symmetry forbids a coupling of Z to the SSM gauginos. In this case, the scalar and gaugino masses are of the same order, rather than being suppressed by a loop factor, as in anomaly mediation.

Retrofitting Fayet Models
Another class of illustrative examples includes Fayet models, another of the classic models of perturbative SUSY breaking. We will start by describing a version with two input parameters, at least one of which needs to be small for natural SUSY breaking.
We then upgrade the model to obtain the necessary small scale dynamically. This class of examples has the feature that the fields generating the leading contributions to the Coleman-Weinberg potential also can play the role of messengers of gauge-mediated SUSY breaking.

The Perturbative SUSY Breaking Model
Begin with gauge group U (1) and chiral fields X, φ, andφ with charges 0, 1, and −1, respectively. The model has superpotential and D-term So far the model has two parameters input by hand: r and M .
Taking φ,φ to be messengers, a SUSY breaking configuration with X , F X = 0 would transmit SUSY breaking to the Standard Model a la gauge mediation. This model has such a minimum, as follows. The potential energy of the model is where ∆V is the Coleman-Weinberg potential expanded about the point of interest in field space.
To obtain the structure described above, let us expand the theory about φ =φ = 0 and X ≈ M/ √ λ. We assume X 2 ≫ eD, and also take eD ≫ F X ≡ M 2 − λX 2 ; we will verify that the latter assumption is self-consistent at the end. With these hierarchies, the φ,φ origin is stable, with m 2 φ = |X| 2 + eD and m 2 φ = |X| 2 − eD. Setting φ =φ = 0, we find the following potential for X: 4) where, at the present level, D = r is an input constant. In the dynamical version to follow, we will render D dynamically small in the vacuum.
The Coleman-Weinberg potential ∆V is straightforward to calculate here, particularly Here the first term comes from the φ,φ loops, and the second comes from the fermion loops which must cancel the first term up to the subdominant F -breaking effects. Performing the integration over momentum gives the result This potential (3.4) has extrema at of which X + ≡ X 0 is a metastable minimum. In the regime defined above, this yields As a self consistency check, for M ≫ eD, we have F X ≪ eD, as assumed above in the calculation of the Coleman-Weinberg potential.
It is worth noting that the result (3.8) for F X follows from a simple scaling argument, which could be useful in more complicated examples. Before including the D-term breaking effect and resulting Coleman-Weinberg potential, the theory had a supersymmetric vacuum at X = M/ √ λ, with X mass m X = M . The perturbative correction to the potential produces a tadpole ∂∆V /∂X evaluated at X 0 ∼ M , which shifts the field by an amount ∆X ∼ (∂∆V /∂X)/m 2 X . The resulting F -term is then of order

Dynamical D
To render eD dynamically small, we first trade it for a superpotential term using the original Fayet model. Add two chiral fields a,ã of charge ±1 under the U (1) symmetry, and a superpotential W a0 = m a0 aã . (3.10) As above we will be interested in large X, where φ =φ = 0. In this regime, for er ≥ m 2 a0 , the minimization in a,ã yields a vacuum Thus the input Fayet-Iliopoulos parameter r itself can be of order the large scale M * , and the problem of obtaining a naturally small eD reduces to that of obtaining m a0 dynamically.
This can be done as follows. First, note that the model would respect a Z Z 2 R symmetry under which a,ã are neutral (and under which X is neutral, with φφ transforming nontrivially), if m a0 , M 2 , λ were replaced with a dynamical operator which transforms nontrivially under the symmetry. Introduce a pure SU (2) sector, with kinetic term d 2 θW α W α .
Here W α W α transforms nontrivially under the Z Z 2 R symmetry so that this kinetic term is invariant under the symmetry. Imposing this symmetry, we cannot write down a bare m a0 aã term, but we can write which weakly couples the SU (2) degrees of freedom to the O'Raifeartaigh/Fayet SUSY breaking sector. In the last step in (3.12), we replaced W α W α with its holomorphic VEV Λ 3 . As in the O'Raifeartaigh case discussed above, it is consistent to integrate out the Yang-Mills degrees of freedom, since they couple weakly via M * -suppressed couplings to the rest of the theory.
By the same token, the above symmetry prevents the pure superpotential M 2 X − λX 3 /3 from appearing, but this times W α W α /M 3 * can appear, along with an M X 2 term. (Adding an additional symmetry-respecting term proportional to φφ has no effect as it can be absorbed by a shift in X.) This modification leaves fixed the scaling of the X VEV found above, X 0 ∼ M , and the scaling (3.9) of the resulting F term. Altogether this produces a theory in which the small parameter eD has been effectively replaced with m 2 a ∼ Λ 6 /M 4 * . This leads to F X ∼ Λ 9 /M 7 * . 2 This much is sufficient to obtain very high scale gauge mediation naturally, with weak scale SUSY breaking obtained via the above method for rendering eD dynamically small, and with M an order of magnitude or two below M * ≡ M GUT as the only input parameter.
If we take M ∼ 10 −1 M GUT , then we obtain a high scale gauge mediation model with a naturally small SSM gaugino mass arising from the dynamically small eD we obtained via the retrofitting procedure.

Discussion and Future Directions
In this paper we combined simple ingredients in a straightforward way to obtain SUSY breaking models with all hierarchically small scales naturally explained dynamically. This procedure of retrofitting simple models can, of course, also be applied to more intricate examples; for example one can similarly retrofit the model of [20] to render the input quark mass scale dynamically small, as was done recently in a footnote in [24]. In retrospect, however, perhaps the simplest possibility for model building is to obtain the small scale as a supersymmetric but dynamically small VEV, while obtaining the breaking of SUSY the old fashioned way.
There are several future directions to pursue. Here we focused on perhaps the very simplest models of perturbative SUSY breaking, but there are more general classes containing gauge fields for which one can systematically analyze the vacuum structure and retrofitting. It will also be interesting to investigate the realization of these models in string compactifications.
It would also be interesting to investigate retrofitting models to yield low scale messenger masses. Gauge mediation models with messenger masses below ∼ 10 7 GeV have the desirable feature that they do not require non-standard cosmology to avoid overclosing the universe with gravitinos, and they predict the spectacular prompt photon and multi-lepton high scale messenger masses. However, as noted in §2.2, low scale models may be possible, especially given the simplicity of the class of models discussed here.
Realistic application of these models requires an assessment of their cosmological stability. The metastable vacua themselves are very long-lived, but whether the universe finds its way into them cosmologically is an a priori separate question. This is very plausible given the symmetries governing our system [25]. It is under investigation in a similar class of models along the lines of [20] in [26], and may be affected by the process described in [27].