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the gravitino dark matter is produced thermally. The BBN bound for the gravitino mass and the dark matter abundance (21) gives an upper bound of the reheating temperature. Using (18), we find T$_{R}$ ≲ 10 7 GeV. The model prediction (16) with M = 10 12 GeV and Λ = 10 5 GeV yields m$_{3}$$_{/}$$_{2}$ ≈ 11 MeV; if the right amount of grav- itino dark matter of this mass is to be produced thermally (Ω$_{CDM}$ = Ω TP ˜ G ), one needs T$_{R}$ ≈ 10 6 GeV. This is lower than the reheating temperature required for successful leptogenesis; the observed baryon asymmetry needs to be generated by some other mechanism, such as [44]. 

Stau NLSP is also of interest for collider phenomenol- ogy. Eq. (17) implies that the lifetime of stau τ$_{˜}$ τ may be long, depending on the gravitino and stau masses; for instance, τ$_{˜}$ τ ≈ 100 sec for m$_{˜}$ τ = 130 GeV and m$_{3}$$_{/}$$_{2}$ = 0 . 1 GeV. For a particle with such a long lifetime, the decay length well exceeds the size of detectors and the decay takes place outside detectors. There have been intrigu- ing proposals for trapping such long-lived charged NL- SPs outside detectors [6–10]. It has been pointed out that detailed studies of stau decay may provide precise measurements of the gravitino mass and the supersym- metry breaking scale. This would also be a crucial test of supergravity since the stau decay ˜ τ → τ ˜ G involves supergravity effects. 

In this paper we studied the effects of hidden sector renormalization in the minimal GMSB model, using a toy model of the hidden sector field X with self-coupling λ . We saw that the soft scalar and gravitino masses are affected by the hidden sector RG; they become smaller than the conventional GMSB RG results. Since the gaug- ino masses are unaltered, the resulting low energy mass spectrum of the superparticles can be quite different from the conventional GMSB. In particular, we find that stau, instead of neutralino, can be the NLSP in the minimal GMSB model. Based on this observation we also dis- cussed phenomenological implications of stau NLSP and gravitino cold dark matter arising from the model. With our typical choice of parameter values: the messenger scale M = 10 12 GeV, tan β = 10, Λ = F/M = 10 5 GeV and the hidden sector coupling λ = 3 . 8, the model yields stau NLSP with mass m$_{˜}$ τ ≈ 130 GeV. The gravitino mass arising from these parameter values is shown to be consistent with the BBN constraints. It was also shown that gravitino cold dark matter with observationally con- 

## V. DISCUSSION 

sistent abundance is attributed to thermal production in this scenario. 

We comment that the hidden sector dependence of the mass spectra, as discussed in this paper, does not nec- essarily mar the predictability of the GMSB scenario. Rather, the hidden sector only influences the soft scalar and the gravitino masses, and the beauty of the GMSB scenario is kept intact. In fact, as emphasized in [12–14], the effects of the hidden sector RG can be viewed as a new window to look into the hidden sector using the low energy mass spectrum that is expected to be revealed in experiments such as Large Hadron Collider. While the model discussed here represents a simple toy example of many possible hidden sector models, we believe essential features of our results to be generic. For example, only the soft scalar and the gravitino masses, and not the other masses neither the couplings, are subject to the hidden sector RG effects. This is a consequence of the powerful non-renormalization theorem and we do not expect this feature to be dependent on details of the hidden sector. Needless to say, however, there might be various other model-dependent issues that deserve further study. 

This work was supported in part by the Research Pro- gram MSM6840770029 and by the project of Interna- tional Cooperation ATLAS-CERN of the Ministry of Ed- ucation, Youth and Sports of the Czech Republic (M. A.), and the WCU grant R32-2008-000-10130-0 (S. K.). 

## Appendix A: The renormalization group equations 

In this appendix we list the RGE for the soft mass parameters that involve the hidden sector contributions. The new ingredient is the hidden-visible interaction (5), which give additional terms to the soft mass RGE. Within our approximation only the (3 , 3) family compo- nent of the three Yukawa matrices, y$_{t}$, y$_{b}$, y$_{τ}$ , are set to be non-zero. The corresponding non-zero components of the three trilinear A-term matrices are a$_{t}$, a$_{b}$, a$_{τ}$ . We use nor- malized variables A$_{t}$ = a$_{t}$/y$_{t}$ , A$_{b}$ = a$_{b}$/y$_{b}$ , A$_{τ}$ = a$_{τ}$/y$_{τ}$ . 

The RGE for the soft masses are 

## Acknowledgments 

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