Conformal Gauge Mediation and Light Gravitino of Mass m_{3/2}

We discuss a class of gauge mediated supersymmetry breaking models with conformal invariance above the messenger mass scale (conformal gauge mediation). The spectrum of the supersymmetric particles including the gravitino is uniquely determined by the messenger mass. When the conformal fixed point is strongly interacting, it predicts a light gravitino of mass m_{3/2}<O(10) eV, which is attractive since such a light gravitino causes no problem in cosmology.


Introduction
"Conformal gauge mediation" proposed in [1] is a novel class of gauge mediation of supersymmetry (SUSY) breaking with strong predictive power. In the conventional gauge mediation models [2]- [8], although the gaugino and sfermion masses are related, the gravitino mass is essentially a free-parameter that depends on the detail of the messenger couplings. In contrast, an advantage of the conformal gauge mediation is that the spectrum only depends on the conformal breaking scale, and, in particular, the gravitino mass is completely fixed by the SUSY breaking dynamics, enhancing our low-energy predictability.
The fundamental reason why we obtain this strong predictability in the conformal gauge mediation is due to the conformal invariance near the cut-off scale where the theory is defined. The assumption of the conformal invariance fixes the coupling constants of the SUSY breaking sector at their fixed point values, and they do not take arbitrary values in the low energy prediction. The only relevant deformation -mass of the messengers in our construction, will yield the scale of the theory, determining the messenger scale, the conformal breaking scale and eventually the SUSY breaking scale as well.
This "uniqueness" of the theory leads us to the analogy [1] between QCD and the conformal gauge mediation. QCD, in the massless quark limit, is a marvelous unification of the Hadron physics in that the low energy predictions only depend on the QCD scale.
Similarly, the conformal gauge mediation unifies the dynamics of the SUSY breaking and the messenger physics so that the low energy predictions only depend on the conformal breaking scale.
In this paper, as announced in [1], we further examine strongly interacting examples of the conformal gauge mediation. We show that the requirement to avoid the splitting SUSY spectrum naturally gives rise to the strongly interacting conformal gauge mediation.
Surprisingly, the strongly interacting conformal gauge mediation reveals an attractive feature from the cosmological viewpoint. In this model, the gravitino mass is as small as O(1) eV, in which case there is no astrophysical nor cosmological problems associated with gravitino.
The organization of the paper is as follows. In section 2, we briefly review the conformal gauge mediation scenario and discuss the spectrum of the SUSY standard model (SSM) sector in the case of strongly interacting conformal gauge mediation. In section 3, we show explicit examples of the strongly interacting conformal gauge mediation. The last section is devoted to our conclusions with a further discussion.

Conformal Gauge-Mediation Scenario
The conformal gauge-mediation scenario [1] is based on an extension of a dynamical SUSY breaking model, which is also a variant of the conformal SUSY breaking [11], where the SUSY breaking model is extended by introducing vector-like representations (P,P ) as new flavors with the superpotential mass term We choose the number of the new flavors so that the extended dynamical SUSY-breaking sector has a non-trivial infrared (IR)-fixed point in the massless limit of the new flavors (m → 0).
where γ P denotes the anomalous dimension of P andP at the IR-fixed point, and M UV is the scale of the UV cut-off [1]. Notice that since all the coupling constants of the SUSY-breaking sector are fixed on the IR-fixed point, the relation between the mass term of the new flavors and the dynamical SUSY-breaking scale is uniquely determined, that is, the SUSY-breaking scale is related to the mass of the new flavors by Λ susy ≃ c susy m phys , with a coefficient c susy . Notice that the ratio c susy is not a free parameter of the model but with dimensionless coefficients c gaugino ∝ n mess c 9 susy and c scalar ∝ n 1/2 mess c 3 susy , where n mess is the number of the messengers. 1 We emphasize that the coefficients c gaugino and c scalar include no free parameters but have definite values depending on the model [1]. 2

Mass estimation in strongly interacting models
As we discussed in Ref. [1], for c susy ≪ 1, the gaugino mass is suppressed by about a factor of n 1/2 mess c 6 susy than the sfermion masses. Thus, if the model is weakly interacting at the IR-fixed point, the gaugino is much lighter than the sfermions. 1 The mediation mechanism we are discussing here is similar to so-called mediator model in Ref. [9] for m phys ≫ Λ SUSY . As pointed out in Ref. [10], the O(F ) contribution to the gaugino mass is suppressed by higher-loops of the SSM gauge interactions compared to the O(F ) contribution to the scalar masses in the mediator model. Here, F denotes the F -term supersymmetry breaking in the SUSY breaking sector. On the other hand, the O(F 3 ) contributions starts at the one-loop diagram of the SSM (with higher loop diagrams of the SUSY breaking sector interactions). Since we are interested in the model with √ F ∼ m phys , the dominant contribution to the gaugino mass is not O(F ) but O(F 3 ). 2 Here, we also assume that the R-symmetry is also broken spontaneously at the scale of the order of Λ susy .
When the model is strongly interacting at the IR-fixed point, however, the ratio c susy can be O(1). In that case, we expect that the hierarchy between the gaugino and sfermion masses dissolves. The weak scale SUSY breaking without fine-tuning (i.e. without splitting SUSY spectrum) forces us to investigate the strongly interacting conformal gauge mediation. In section 3, we will show explicit models of the conformal gauge mediation where the model is strongly interacting at the IR-fixed point. Unfortunately, the precise prediction of soft masses is difficult in such cases since the messenger particles also take part in the strong interaction when they decouples. 3 Here, instead, we estimate the gaugino and scalar masses as 4 in the spirit of the naive dimensional analysis by assuming c susy = O(1). 5 Notice that as the SUSY breaking scale is uniquely determined by the messenger mass (or equivalently by the SUSY breaking scale), the same holds for the gravitino mass, Here, M PL ≃ 2.4 × 10 18 GeV denotes the reduced Planck scale. Thus, in the conformal gauge mediation, there is a strict relation between the soft masses in Eq. (6) and the gravitino mass. Interestingly, the relation predicts a very light gravitino in the case of the strongly interacting models, That is, by requiring that the gaugino and the scalar masses are of the order of 1 TeV, we obtain which corresponds to the gravitino mass 3 Recently, generic properties of the gauge mediation associated with the strongly interacting SUSYbreaking sector have been discussed in Ref. [12], although it is still difficult to obtain soft masses numerically. 4 For c susy = O(1), the above approximation of the ratio, m gaugino /m scalar ≃ n 1/2 mess c 6 susy , breaks down. 5 The sign of the sfermion squared mass cannot be determined by perturbative analysis. In this paper, we simply assume that the sfermions obtain positive mass squared.
Therefore, we find that the conformal gauge mediation with no large hierarchy between the gaugino and scalar masses predicts the gravitino mass m 3/2 < ∼ O(1) eV. Notice that such a small gravitino mass may be determined at the future collider experiments, e.g. by measuring the branching ratio of the decay rate of the next to lightest superparticle [13].
From the cosmological point of view, the light gravitino of mass m 3/2 < O(10) eV is very attractive since it shows no conflict with astrophysical and cosmological observations [14]. Moreover, as we will see in section 3, we can construct models with a stable SUSY breaking vacuum in our framework. In such cases, the conformal gauge mediation model is quite successful in cosmology regardless of the detail of the thermal history of the universe.
So far, there have been some attempts to obtain models of gauge mediation with m 3/2 < O(10) eV, where the SUSY breaking vacuum is stable (see Refs. [15]- [19] for example). In those models, however, the motivation to choose the parameter to do so would still need to be explained. In the conformal gauge mediation, however, the prediction of the light gravitino is rather compulsory because there is no parameter to tune.
Before closing this section, we comment on another attractive feature of the conformal gauge mediation. As briefly discussed in Ref. [1], the messenger quarks are expected to be heavier than the messenger leptons by the QCD wave function renormalization effects to the messenger quarks. 6 Thus, the colored superparticles obtain relatively lighter masses compared with the usual gauge mediated SUSY breaking models, which makes superparticles more accessible at the Large Hadron Collider experiments than the usual gauge mediation models.

Examples of Conformal Gauge Mediation
In this section, we present two examples of the conformal gauge mediation where the ratio between the messenger scale and the SUSY breaking scale is expected to be O(1), i.e. c susy = O(1). Although there are many choices for the dynamical SUSY-breaking which would be extended to the conformal gauge mediation model, we concentrate on the scenario in which the SUSY-breaking vacuum is stable with the consistent cosmology in mind.
The first example is a model based on the dynamical SUSY breaking of SO(10) h gauge theory with a spinor representation [20,21]. According to a general procedure to realize conformal gauge mediation, we add N f vector-like representation 10 with a mass term in Eq. (1). For 7 < N f < 21, this model is known to have an non-trivial IRfixed point [22,23]. As analyzed in Ref. [24,25], the anomalous dimensions of the chiral superfields at the conformal fixed point can be computed by using the a-maximization technique [26,27]: For N f = 10, we have γ P ≃ −0.97. Since the anomalous dimension of the messengers is close to the unitarity bound: γ P ≃ −1, the model is expected to be strongly interacting at the IR-fixed point, and hence, the ratio c susy is expected to be O(1). By identifying subgroups of the flavor symmetry SU(5) ⊂ SU(10) with the gauge groups of the SSM, we obtain an example of the strongly interacting conformal gauge mediation. 7 Another example is a model based on the dynamical SUSY breaking of SU(5) h with 10 +5 [28,21]. Again, we add N f vector-like quarks 5 +5 (for 5 < N f < 13) to make the model have an non-trivial IR-fixed point [1] (see also [29]). We identify five out of N f flavors are messenger fields which are charged under the SSM gauge group. The anomalous dimensions of the chiral superfields at the conformal fixed point can be computed by For N f = 6, we have γ P ≃ −0.82, and hence, this model is also expected to have c susy = O(1). Thus, another example of the strongly interacting conformal gauge mediation model is obtained by identifying the flavor symmetry SU(5) ⊂ SU(6) with the gauge groups of the SSM.

Perturbative GUT?
One unavoidable property of the strongly interacting conformal gauge mediation is the large beta function contribution to the SSM gauge coupling constant. This is due to the fact that the anomalous dimensions of the messengers will increase the number of messengers charged under the SSM gauge group. The perturbativity of the standardmodel gauge interactions demands that the number of the messengers n mess should satisfy where we have included the higher loop effects of the SUSY-breaking sector through the anomalous dimension γ P of P andP . Here, we have used the NSVZ exact formula [30]- [32] of the beta functions of the SSM gauge interactions. For γ P ≃ −1 and m phys = O(10 5 ) GeV this condition is reduced to In the above two examples, the numbers of the messengers are n mess = 10 for the SO(10) h model and n mess = 5 for the SU(5) h model, respectively. 8 Therefore, the standard model coupling constants blow up below the GUT scale as long as the perturbative formula for the beta function (13) is valid. However, this does not necessary mean that the theory is ill-defined above that scale: it is just a breakdown of the low-energy effective field theory description. It rather suggests the presence of a dual description of the standard model at the high-energy scale, where the standard model itself can be realized as a weakly interacting dual gauge group (we refer e.g. to [33] for an attempt).
Leaving the above interesting possibility aside, there are several possible ways to avoid the problem if we wish. One way to recover the perturbative unification is to separate the messenger gauge group and the SSM subgroup of SU (5)

Conclusion and Discussion
In this note, we have shown that the conformal gauge mediation admits the non-hierarchical SSM spectrum by considering a strongly interacting theory. An interesting prediction of the strongly interacting conformal gauge mediation is the very light gravitino (m 3/2 < O(10) eV), which is very attractive from a cosmological point of view. As another attractive feature, we can construct models with the stable SUSY breaking vacuum. In such models, there is no constraint on the thermal history of the universe which is severely constrained if the vacuum is meta-stable.
Several comments are in order. As we have discussed, the gravitino mass is predicted to be O(1) eV for strongly interacting conformal gauge mediation models. In this case, the gravitino abundance cannot provide the mass density of the observed dark matter. Thus, there must be other candidates for the dark matter. The most interesting candidate 9 The perturbative unification of the SSM gauge coupling is realized when the following three conditions are satisfied. 1) M 5 is required to be close to m phys , so that the SSM gauge couplings do not receive large renormalization effects between m phys and M 5 . 2) The SU (5) F gauge theory is perturbative enough so that the perturbativity condition of the SSM gauge couplings similar to Eq. (13) admits the newly introduced bi-fundamntal field. 3) The gauge coupling constant of SU (5) F is rather large at M 5 , so that the gauge coupling constants of the SSM do not change so much at the threshold scale M 5 .
for the dark matter is the QCD axion [34,35] which is involved in a solution to the strong CP-problem by the spontaneously breaking of the anomalous Peccei-Quinn (PQ) symmetry [36] (with the breaking scale f P Q ≃ 10 11 GeV [37]). By assuming that the strong CP-problem is solved by the axion mechanism, we can picture the SSM with m 3/2 < O(10) eV, fully consistent with cosmology.
The introduction of the PQ-symmetry also provides us with an interesting perspective on the origin of the µ-term. With appropriate charge assignments for the PQ-breaking field (with a breaking scale f P Q ) and the Higgs doublets under the PQ-symmetry, we can write down a higher dimensional term in the superpotential Thus, for f P Q ≃ 10 11 GeV, we obtain an appropriate size of for µ-term, µ = O(1) TeV, without causing another CP-problem.
We also comment on the dynamical tuning of the cosmological constant [11]. As discussed in Ref. [11], the dynamical tuning of the cosmological constant is realized in strongly interacting conformal SUSY breaking models for γ

A Cousin model of the conformal gauge mediation
In this appendix, we consider a cousin model of the conformal gauge mediation where the SSM spectrum has a strict relation with the gravitino mass in which we again make use of the conformal SUSY breaking.
The model is based on the conformal SUSY breaking model of SO(10) gauge group with a spinor representation. We introduce N f = 10 numbers of vector representation P to make it conformal. In addition, for messengers, we add SO(10) singlet superfield X andX which are charged under SU(5) GUT as 5 and5 respectively. The superpotential is given by We regard the mass term for P as a small perturbation as before, but we assume that the quartic coupling λ is in the vicinity the strongly interacting fixed point value (i.e. λ * ∼ 1). 12 Before turning on the mass deformation, the model is supposed to be in the conformal regime. The anomalous dimension of P can be re-computed as γ 10 ≃ −0.97 by using the a-maximization.
The conformal symmetry is broken by the mass term. As a consequence, the SUSY is dynamically broken at m phys in Eq. (2) near the origin of singlet fields X,X. The effective dynamics of the messengers X andX can be represented by the superpotential due to the anomalous dimension of X,X and P .
At this stage, the effective dynamics of the model has been reduced to the conventional gauge mediation scenario. where we have m gaugino ∼ m sfermion ∼ αΛ susy /4π which are independent of the parameter λ. Therefore, the scale of the SSM spectrum is determined by only Λ susy as in the conformal gauge mediation model. Notice that, by the same argument we made in section 2, this model also predicts the light gravitino (m 3/2 < O(1) eV).
An important feature of the cousin model is that the perturbativity of the SSM gauge couplings is intact up to the GUT scale. Thus, we can easily justify the assumption that the UV cut-off scale of the conformal SUSY breaking sector to be the Planck scale, i.e.
M U V ≃ M PL . Therefore, in this model, we can also realize the dynamical tuning of the