Cascade supersymmetry breaking and low-scale gauge mediation

We propose a new class of models with gauge mediated supersymmetry breaking, the cascade supersymmetry breaking. This class of models is consistent with the gravitino mass as low as O(1)eV without having suppressed gaugino masses, nor the Landau pole problems of the gauge coupling constants of the Standard Model below the scale of the grand unification. In particular, there is no supersymmetric vacuum in the vicinity of the supersymmetry breaking vacuum even for such a low gravitino mass. Thus, the model does not have a vacuum stability problem decaying into supersymmetric vacua.


Introduction
The gauge mediated supersymmetry breaking (GMSB) models [1][2][3][4][5][6][7] provide one of the most attractive realizations of the phenomenologically acceptable minimal supersymmetric standard model (MSSM). The gravitino is the lightest supersymmetric particle whose mass ranges from the eV to the GeV. This feature of gauge mediation motivated a lot of theoretical works on phenomenological and cosmological aspects of the models [8].
Light gravitino models are constrained by cosmological and astrophysical problems.
In particular, the analysis of CMB data and the Lyman-α forest data put an upper bound of 16 eV on the gravitino mass [9]. Furthermore, future cosmic microwave background observations will probe gravitino mass down to the eV range [10]. Thus, models with very light gravitino are particularly interesting. Such models require a very low fundamental scale of supersymmetry (SUSY) breaking and as a result may have very rich collider phenomenology.
Models with such a light gravitino (see for example Ref. [11][12][13][14][15]) generically suffer from a range of theoretical problems: the Standard Model couplings hit the Landau pole below the scale of the Grand Unification Theory (GUT); gaugino masses are suppressed compared to the sfermion masses; there are supersymmetric ground states in the vicinity of desired SUSY breaking vacua. Much of the parameter space in this class of models has been excluded by the neutralino/chargino mass limit [16,18,24] at the Tevatron experiments [19] or by the vacuum stability problem [20] for the gravitino lighter than 16 eV.
In this paper, we propose a new class of the GMSB models, the cascade supersymmetry breaking, which may avoid these problems.
The organization of the paper is as follows. In section 2, we briefly review problems arising in attempts to construct low scale GMSB models. In section 3, we introduce a generic idea of the cascade supersymmetry breaking. An example based on dynamical supersymmetry breaking (DSB) is introduced in section 4. In sections 5, we study phenomenological features of the model, messenger and superpartner spectra as well as possible generalizations. In section 6, we comment on a relation of our model to the conformal gauge mediation mechanism developed in Ref. [21]. The final section is devoted to the conclusions and some discussions.

A brief review of low scale gauge mediation 2.1 Scales in light gravitino scenario
Since we are interested in models with the gravitino mass in the eV range, the fundamental SUSY breaking scale must be low, Such a low SUSY breaking scale can be achieved in GMSB models. Indeed, in the simplest GMSB models, the superpartner masses are given bỹ where m is the messenger scale and F S is a mass splitting within messenger multiplets.
One must also require F S < m 2 to avoid tachyonic messengers and charge-color breaking.
Thus, a requirement that the superpartners have mass at the electroweak scale leads to the lower bound on the mass parameters in the messenger sector, We see that light gravitino scenarios can only be realized when the SUSY breaking effects in the messenger sector are comparable to the fundamental scale of SUSY breaking, F S ∼ F . To avoid the separation of scales, one must look at models of direct or semidirect gauge mediation. Furthermore, one expects that successful models will necessarily be strongly coupled.

R-symmetry and the messenger sector
Let us briefly review difficulties encountered in search for models of low scale gauge mediation. We begin by considering a simple example of the messenger sector 1 : Where S is a supersymmetry breaking field and ψ,ψ are messenger fields. This model possesses an R-symmetry (under which S has charge 2) if the charges of the messenger 1 Low energy description of direct GMSB models is often given by superpotentials of this type [11,12].
fields can be chosen such that [22,23]: We will restrict our attention to models with det m = 0. Indeed, models with det m = 0 are problematic from phenomenological perspective because the true vacuum in this case is a charge-color breaking one (for detailed analysis of the vacuum structure of this class of models see Ref. [23]). When parameters are chosen so that the gravitino mass is in the O(10 eV) range, the charge-color breaking ground states are found in the vicinity of the charge-color preserving minima of the potential. This leads to two problems with the desired minimum: first, the lifetime of this minimum is expected to be too short; second, initial conditions must be fine-tuned for it to be selected in the course of cosmological evolution. 2 Furthermore, the lifetime of the vacuum in the models with light gravitino and supersymmetric runaway directions is too short. We will, therefore, require that λm −1 λ = 0 which guarantees the absence of the runaway directions [24]. With these assumptions, the model in Eq.(4) possesses the following important features • The supersymmetry is broken; • The effective mass matrix for messenger fields is independent of the vacuum expectation value (vev) of the pseudo-modulus: A phenomenologically viable model requires that the R-symmetry is broken (otherwise the Standard Model gauginos remain massless). This can be achieved through introduction of additional fields and interactions that lead to either spontaneous or explicit R-symmetry breaking 3 . However, this is often insufficient. Indeed, as long as the effective messenger mass matrix satisfies Eq.(6), the gaugino masses vanish at the leading order in 2 A very simple model can be constructed [25] if the requirement of a very light gravitino is relaxed. See also Ref. [26] for models with det m = 0.
3 Such interactions generically lead to appearance of supersymmetric vacua elsewhere on the moduli space. We will discuss problems associated with the existence of such vacua shortly.
The first contribution to gaugino masses appears only at the order F 3 S [11], The sfermion masses are still generated at the leading order in F S and the requirement that gaugino and sfermion masses are comparable can only be satisfied in models of low scale SUSY breaking. However, it turns out that the coefficient c is small and sufficiently large gaugino masses can not be achieved without fine-tuning. Furthermore, detailed numerical analysis, has shown that, even with c = O(1), the predicted gaugino masses have been almost excluded by Tevatron constraints on the neutralino/chargino masses for m 3/2 16 eV [16,18,24].
Thus, we must turn to models where Eq. (5) is not satisfied in the messenger sector. In such models, gaugino masses are unsuppressed, however, the SUSY breaking vacuum is only a local one. To see this, note that at least at one point along the pseudo-flat direction the matrix m ij + λ ij S has a zero eigenvalue and at least one pair of messengers becomes massless. They can now acquire vevs and restore supersymmetry. To ensure sufficiently long vacuum lifetime, one must increase the messenger mass m. This suppresses all superpartner masses if the gravitino mass (and, therefore, the fundamental scale of SUSY breaking) is kept fixed. In particular, for m 3/2 16 eV, the detailed numerical analysis [20] has lead to an upper bound on superpartner masses of about 1 TeV.

Direct and semi-direct gauge mediation
A toy model discussed above implied direct or semi-direct gauge mediation. Additional problems often arise when UV complete realizations of direct gauge mediation is considered (a precise definition of direct gauge mediation is given in [28,29]  gravitino. This latter difficulty may be avoided in semi-direct gauge mediation [21,31] (see also Refs. [32,33] for earlier attempts), where messengers are charged under the DSB group but do not play a direct role in SUSY breaking. This allows to construct models with small number of messengers and avoid Landau pole problems. However, the leading contribution to the gaugino mass is again vanishing due to the R-symmetry (see also Ref. [34]).

Cascade supersymmetry breaking
In order to solve problems discussed in the previous section we will employ some of the tools proposed in the original GMSB models [5][6][7] -namely we will introduce a secondary SUSY breaking sector. Despite the existence of several sectors, our model will have all the desirable features of direct (and semi-direct) gauge mediation. Finally, in a strong coupling limit, we will be able to obtain low SUSY breaking scale. We will refer to this class of models as cascade SUSY breaking 5 .
To illustrate the idea of cascade gauge mediation, we consider a model with the following superpotential Here, S represents a field of the primary SUSY breaking sector, while X, ψ, andψ are fields in the secondary sector (with latter two serving as messengers). We have assumed that the superpotential couplings between the two sectors are suppressed. In section 4 we will introduce a dynamical model with absence of the superpotential interactions between the two sectors will be ensured by symmetries. The two sectors will interact only through the Kähler potential 6 (see Fig. 1): Cascade SUSY breaking Figure 1: A schematic picture of the cascade supersymmetry breaking. Supersymmetry breaking in the secondary sector is induced by the primary supersymmetry breaking via the connections in the Kähler potential.
In the decoupling limit, c SP = 0, SUSY is broken in the primary sector, while the secondary sector possesses a supersymmetric minimum with spontaneously broken Rsymmetry 7 : Once the coupling between two sectors is turned on, c SP = 0, supersymmetry breaking in the primary sector induces supersymmetry breaking in the secondary sector through higher dimensional operators. Indeed X potential becomes where We see that X obtains a non-vanishing F -term vev which, in the limit |m soft | Λ 2 , is given by Now SUSY breaking effects are mediated to the MSSM sector due to the coupling kXψψ.
One finds a standard expression for gaugino masses The very light gravitino is realized when all the dimensioneless coefficients are of order one while the mass scales in the Lagrangian are comparable, i.e., In this limit, the supersymmetry breaking and the messenger mass scales are of the same order of magnitude, as required to obtain light gravitino.
Before closing this section, we comment on the sign of c SP . The SUSY breaking effects discussed above shift the X vev independently of the sign of c SP . The model may be further simplified when c SP is positive. In this case, the following superpotential is sufficient to induce SUSY breaking effects in the secondary sector: There exist a supersymmetric minimum at X = 0 where messengers are massless. However, if c SP > 0 the supersymmetric vacuum is "destabilized", and both the scalar and the F -term components obtain non-vanishing expectation values.

A model of cascade supersymmetry breaking
The superpotential Eq.
In this section, we introduce UV complete description of the cascade SUSY breaking based on the SP (N c ) × SU (4) gauge theory with the matter contents given in Table 1.
The Standard Model gauge groups will be embedded into the global SU (5) SM symmetry.
We choose tree level superpotential to be where λ's is a coupling constant, and m R,F denote mass parameters. Let us first choose parameters that will simplify the analysis of SUSY breaking: where Λ 1 and Λ 2 are dynamical scales of SP (N c ) and SU (4) groups respectively. In this regime, we can integrate out massiveR and R fields and the dynamics of two gauge groups, SP (N ) and SU (4), decouples to the leading order . Dynamical scales of the two low energy gauge groups are given by Notice that the SU (4) gauge group is asymptotically free above m R for N c < 4, while SP (N c ) group is asymptotically free for N c > 1.

Primary supersymmetry breaking sector
In the limit m R → ∞, the SP (N c ) dynamics breaks SUSY through IYIT mechanism [38] and we will refer to this sector as a primary SUSY breaking sector. Let us briefly review the dynamics of this sector. Below m R the physics is described by an SP (N c ) gauge group and N c + 1 flavors and a set of gauge singlet fields. The full superpotential is given by the sum of tree level terms and the quantum constraint: where X is a Lagrange multiplier. This superpotential is inconsistent with the supersymmetric ground state 8 . A convenient description of the dynamics can be obtained in terms of the meson fields V ij = Q i Q j . These fields marry singlets and become massive. Thus, the low energy theory contains only a single gauge singlet field with the superpotential where S = Pf(S ij ) 1/(Nc+1) . The Kähler potential has the form For small λ, the coefficient η S is calculable and negative [41] ensuring that the vacuum is at the origin:

Secondary supersymmetry breaking sector
Below m R the secondary sector is described by an s-confining SU (4) gauge theory [42], where and α's are the SU (4) indices while a and b are SU (5) SM indices. We further define rescaled fields, X,M ,B andB by The supersymmetric minimum of the secondary sector is located at In this vacuum, all the fields charged under the global SU (5) SM symmetry are massive. 9 For sufficiently small m F , the secondary sector also possesses a metastable SUSY breaking vacuum [43]. Typically, models of direct gauge mediation make use of this metastable SUSY breaking vacuum. However, the maximal global symmetry of the secondary sector is spontaneously broken in non-supersymmetric minimum which, in turn, requires a gauge group larger than SU (4) and generically leads to Landau poles for the Standard Model couplings below the GUT scale. We will instead use a supersymmetric vacuum of the secondary sector. The presence of R,R fields induces Kähler potential interactions between the primary and secondary sectors and generates non-vanishing F X .

Interaction between the two sectors
After R,R fields are integrated out, the superpotential of the low energy theory is while the interactions between the two sectors are given by corrections to the Kähler potential: It is useful to note that since coupling constants in Eq.(30) are radiatively generated, they are small in the large m R limit. On the other hand, these couplings are of order one in a The vacuum used in the cascade supersymmetry breaking is denoted by "×". We have shown a local supersymmetry breaking minimum at X = 0 discussed in Ref. [43].
Let us analyze the spectrum of the model at weak coupling. For |c SP | 1, the X vev is slightly shifted from its supersymmetric value in Eq. (28), and the size of the secondary supersymmetry breaking is expected to be much smaller than F S and Λ 2 2 . Furthermore, the effect of the interactions on the vevs of the fields in the primary sector is negligible.
Thus, the effective scalar potential of X can be approximated by 10 10 The scalar potential of X possesses a discrete Z 4 symmetry which is a subgroup of the Z 8 R-symmetry. The discrete symmetry is spontaneously broken by X = 0, which leads to the domain wall production at the phase transition. The domain wall is, however, unstable since the Z 4 symmetry is explicitly broken by the constant term in the superpotential through the supergravity effects, and hence, it does not cause the cosmological domain wall problem [44].
As a result, the supersymmetric vacuum in Eq.(28) is shifted to Here, we have assumed that |m soft | is much smaller than m F , Λ 2 in the weak coupling regime.
As we have noted before the secondary supersymmetry breaking is achieved regardless of the sign of c SP . A a schematic cartoon of the supersymmetry breaking shift in the scalar potential and its dependence on the sign of c SP is shown in Fig.2. The implications of the possibility of positive c SP are discussed in the appendix.

Superparnter masses
We These corrections to the Kähler potential generate D-type scalar masses for messengers: Since c M ∼ c B ∼ c SP , these terms are of order m 2 soft .Moreover, in the strong coupling regime, self-interactions in the secondary sector result in order 1 corrections to these masses.
In addition messengers receive holomorphic soft masses that can be obtained from Eq. (26): Finally, supersymmetric terms in the messenger mass matrix are given by Note that one must choose parameters of the model in such a way that all eigenvalues of the mass squared matrix for scalar messengers are positive. This must be achieved while avoiding too large positive values of D-type soft masses squareds (otherwise gauge mediated masses for the sfermions could become negative).
With the knowledge of the messenger spectrum, we can obtain superpartner masses.
To the leading order in SUSY breaking parameters, gaugino masses only depend on the holomorphic soft masses of the messenger fields [45,46] and are given by A general formula describing soft sfermion masses is presented in Ref. [45]. The contribution due to non-vanishing messenger supertrace dominates the result in the large m R limit leading to log divergent terms where C a (r) denotes the quadratic Casimir invariant of the MSSM gauge symmetries for each sfermion of a representation r and we made the use of the fact that the D-type masses are of the order m soft . Effect of the holomorphic soft messenger masses is small in the large m R limit, To guarantee that the sfermion and gaugino masses are comparable we must take strong coupling 11 limit Eq. (31). In this limit we can only give an estimate of superpartner masses, Nevertheless this is a desirable region of the parameter space since it results in low scale GMSB. We also note that strong coupling effects could have significant consequences for the superpartner spectrum and further suppress the hierarchy between the superpartner masses and fundamental SUSY breaking scale [47]. Since such modifications would only improve the plausibility of the light gravitino scenario, we will use the estimate (42) in the rest of the paper.
In table 2, we show a summary of scales in model that allow comparable gaugino and sfermion masses (see the appendix for the detailed analysis). In the table, we listed the appropriate mass scales for c SP < 0 and c SP > 0 separately. Notice that the sign of c SP is not the parameter of the model but is determined by model by model. Our case study approach just reflects our inability to calculate the sign of c SP due to the strong interactions 12 .
As we have explained earlier, the model possesses a local charge-color and SUSY breaking minimum at X = 0. When c SP < 0, it is possible that this minimum has lower energy and in the strongly coupled regime the lifetime of the phenomenologically viable vacuum is too short (see Fig.3 for a schematic picture). Let us study this question in the calculable regime. If |c SP | 1, the phase transition does not occur as long as the mass parameters satisfy, Table 2: Summary of the model of cascade supersymmetry breaking based on SP (N c ) × SU (4) gauge symmetries. In the parameter regions listed below, the gaugino masses and the sfermion masses are comparable. The discussions for c SP > 0 are given in the appendix. The very light gravitino is realized by taking m R close to Λ 2 in the cases with low scale gauge mediation (see discussion around Eq. (48)).
mass parameters mediation scale or In the final expression, we have used Eqs. (34) and (37). Thus, in the calculable regime, we can guarantee the stability of the vacuum by assuming Eq. (45). The situation is more complicated in a strongly coupled regime. In particular, when m R ∼ Λ 2 composites involving R andR fields can not be integrated out and must be included into effective low energy description. While the vacuum structure may be more complicated in this regime, it appears plausible that charge-color preserving vacuum will remain the lowest energy minimum of the potential near the origin of the field space. In the following, we assume that this is indeed the case.

Gravitino Mass
Let us now obtain a lower bound on the gravitino mass in this model. The fundamental scale of the SUSY breaking is bounded from below by the experimental limits on the sfermion masses, in particular the slepton masses [48],  In our model we can convert this constraint into the lower bound on soft masses both in the primary and secondary SUSY sectors: which in turn leads to a lower bound on the gravitino mass 13 We see that in the strong coupling limit ( Thus, the effects from the composite messengers are very small for m F ∼ Λ 2 ∼ |m soft |. Above the the renormalization scale Λ 2 , on the other hand, the model has only four fundamental representations of the SU (5) SM gauge groups, and hence, the model allows perturbative coupling unification even for low SUSY breaking scale.

Conformal Gauge Mediation
The light gravitino mass can be achieved in the model of section 4 only if several a priori unrelated mass parameters are comparable. Here, we show that this apparent coincidence of scales may be a consequence of conformal symmetry of the underlying model (see Ref. [21] for the detailed discussion of conformal gauge mediation). Note that in the case where both gauge groups are asymptotically free, N c = 2, 3, both SP (N c ) and SU (4) are in conformal window. Thus, the existence of the IR conformal fixed point is plausible.
At the fixed point, beta functions for gauge and Yukawa couplings must vanish leading to the following conditions These equations do not uniquely determine anomalous dimensions of the matter fields.
To do so we use an a-maximization method [50]. The a-function of the model is given by, where R i 's are related to the anomalous dimensions by R i = 2(1 + γ i /2)/3. The values of anomalous dimensions obtained both by using a-maximization method and perturbative one-loop calculation are presented in the table 3. We see a good agreement between the two calculations in the case N c = 2, strongly suggesting that a (relatively weakly coupled) fixed point exists. In the case N c = 3, the perturbative calculation breaks down but the existence of the IR fixed point is still plausible.
Let us now assume that, in the UV, the coupling constants are chosen close to the fixed point values while mass parameters m R and m F are small but non-vanishing. The model then quickly flows to the fixed point. and remains conformal down to scales of order m R (for m R > m F ). Below m R conformal symmetry is broken and the low energy Notice that there is no arbitrariness in the relations between the mass scale m R and the dynamical scales since the values of the gauge coupling constants at around the energy scale m R are fixed by the conformal symmetry.
We still need one coincidence, namely IR values of m R and m F must be of the same order. It is reasonable to assume that these explicit mass scales are comparable in the UV, m R ∼ m F ∼ m 0 . However, the effects of RG evolution are significant where M CFT denotes the scale at which the model approaches the fixed point. Since γ R < γ F we find that m F is naturally much smaller than m R . As discussed in the appendix, the light gravitino scenario is still viable for small m F if c SP > 0.

Conclusions and discussion
In this paper, we proposed a new class of models with gauge mediated supersymmetry breaking, the cascade supersymmetry breaking, which admits a low scale gauge mediation and very light gravitino. This class of models is reminiscent of early GMSB models with several scales in that there is a primary and a secondary SUSY breaking sectors 14 .
However, in our model supersymmetry breaking in the secondary sector is itself achieved through gauge interactions. As we have demonstrated, it is possible to implement low scale gauge mediation in this class of models while avoiding light gaugino, Landau pole, and vacuum stability problems generic in direct GMSB models. Furthermore, a specific model presented in this paper may allow gravitino mass as low as to be as low as O(1) eV, in the range that will be probed by the future cosmic microwave background observations [10].
Several comments are in order. As discussed in Refs. [51][52][53][54], the models where messengers are charged under the strong gauge dynamics of the DSB sector generically suffer from the existence of the unwanted stable composite fields with MSSM quantum numbers.  [55,56]. Such a low relic density, however, cannot be achieved even when the annihilation cross section saturates the unitarity limit [57].
To avoid these constains, at least one global U (1) must be broken explicitly. For example, if B is the lightest composite with the Standard Model quantum numbers, the simplest operator that allows one to avoid direct detection constraints is [54], If, on the other hand,F RQ is the lightest composite with the Standard Model quantum numbers, the U (1) symmetry may be broken by a lower dimension operator Since this operator is suppressed by only M * , the resultant lifetime of the composite particle is much shorter than 1 second.
A second global U (1) symmetry allows the possibility that a neutral composite, R 4 or R 4 , is also stable. Such a composite provides a plausible dark matter candidate. Despite their large mass the the relic density of these particles can be consistent with observations since their annihilation cross section is expected to be close to the unitarity limit [57]. 15 15 The stable gravitino with mass in the eV range is a sub-dominant component of dark matter.
We would also like to note tha in the model of section 4 the continuous R-symmetry is broken down to the discrete Z 8 R-symmetry by the explicit mass terms in the secondary sector. Thus, the model does not have an R-axion even after the X obtains non-vanishing vev. One exception is the choice of c SP > 0 which, as discussed in the appendix allows m F to be small compared to other mass scales in the theory (see table 2). In this case, the spontaneous breaking of an approximate continuous R-symmetry leads to a very light R-axion with mass Here, the second contribution comes from the explicit R-symmetry breaking by the constant term in the superpotential through the supergravity effects [59]. Therefore, it is possible that there is a very light R-axion with mass in the hundreds MeV range for m F = O(100) keV. Decay of such an axion inside the detector could lead to displaced vertex with a low-mass muon pair and be detactable at the LHC [60].
Finally, we would like to point out the primary SUSY breaking sector in the model of section 4 may give rise to pseudo Nambu-Goldstone bosons with mass in a TeV range. As disscussed in Refs. [61,62] these particles may provide another plausible dark matter candidate. In this case, one can explain the observed excesses of cosmic ray electron/positron fluxes at PAMELA [63], ATIC [64], PPB-BETS [65], and Fermi [66] experiments, while evading the limit on anti-proton flux from PAMELA experiment [67]. 16 in a strongly coupled regime of our model, a large region of the parameter space leads to viable superpartner spectrum.
This is a consequence of the fact that for positive c SP the phase transition to the vacuum at X = 0 does not happen (see Fig.2) even for (m 3 F Λ 2 ) 1/2 < m 2 soft . The vevs in the ground state are then largely shifted from the supersymmetric vevs in Eq. (28) and to leading order are independent of m F which is now a free parameter and, in particular, can be taken small. One must also verify that the messengers are non-tachyonic for positive c SP . A detailed analysis of the messenger mass matrix shows that this is the case when