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These studies are based on the traditional assumption that the renormalization group (RG) flow of the visible world masses does not depend on details of the hidden sector dynamics, which, as is realized recently [12, 13], turned out not necessarily to be the case. Based on this observation, effects of hidden sector on the renormaliza- tion group equations (RGE) are studied in [14]. They use a toy model of the hidden sector with a self-interaction coupling in the constrained MSSM, and show that the soft mass spectrum at low energy is altered by hidden sector effects, suggesting that such spectra can be used to determine the hidden scale and the strength of the self-coupling. In the present paper we shall carry out a similar analysis on the GMSB scenario that gives rise to gravitino cold dark matter, by taking into account the contributions from the hidden sector and reexamining the RG analysis of the minimal GMSB [3, 4]. We find that the effect of the renormalized self-interaction coupling of the hidden sector field renders the soft scalar masses smaller than the conventional GMSB results, and con- sequently the low energy mass spectrum of the MSSM particles can be drastically altered. A phenomenologi- cally interesting consequence of our analysis is that stau can be the NLSP even in the minimal GMSB scenario. Our model thus provides a simple stau NLSP model with gravitino cold dark matter. 

The plan of the paper is as follows. In the next sec- tion we present the setup of our model, focusing on how the hidden sector dynamics contributes to the RG of the visible sector. In Section 3 we present our numerical results, and in Section 4 we discuss constraints of the parameter space from the collider and cosmological ex- periments. We conclude in Section 5 with comments, and explicit forms of the hidden sector contributions to the RGE are summarized in the Appendix. 

## II. THE HIDDEN SECTOR IN THE GAUGE-MEDIATED SUPERSYMMETRY BREAKING 

In the GMSB scenario, the messengers couple directly to the hidden sector fields (which is responsible for the supersymmetry breaking), and indirectly to the MSSM matter fields through the standard model gauge interac- tions. Once the supersymmetry is broken in the hidden sector, the gauginos and the scalars in the MSSM become massive via one-loop and two-loop corrections, respec- tively. In the minimal GMSB based on the N$_{5}$ = 1 5 + ¯ 5 messenger belonging to an SU (5), the gaugino masses at the messenger scale M are given by 

where 

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t = ln( µ/M ) and Λ = F/M with F the supersymmetry breaking scale. The index a = 1 , 2 , 3 is for the MSSM gauge groups U (1), SU (2), SU (3), and g$_{a}$ are the corre- sponding gauge couplings. The soft scalar masses at the messenger scale are given by 

where the index i denotes the MSSM scalars (˜ q, ˜ u, ˜ d, ˜ l, ˜ e, H$_{u}$, H$_{d}$ ) in this order, and C a $_{2}$( R$_{i}$ ) are the quadratic Casimir for the matter fields in represen- tation R$_{i}$ of the a -th MSSM gauge group. As indicated, these masses are generated at the messenger scale M . Physical masses at the electroweak scale are evaluated through RG evolution. Below we follow [13, 14] and derive the RGE, taking the hidden sector effects into account. 

We model the hidden sector by a single chiral superfield X with Lagrangian, 

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where the superpotential is 

This does not itself break the supersymmetry but we as- sume that the F -term component of X acquires a non- zero vacuum expectation value by some mechanism. We shall be interested in the effects of the hidden sector renormalization. The hidden sector field X couples only to the messenger fields, and the messenger fields couple to the MSSM fields through the gauge interactions. Be- low the messenger scale M , the messenger fields are in- tegrated out, yielding the following effective interactions between X and the MSSM fields, 

with real and complex coefficients k$_{i}$ and w$_{a}$ . Here Φ$_{i}$ are the MSSM matter superfields (their scalar components are φ$_{i}$ = ˜ q, ˜ u, ˜ d, ˜ l, ˜ e, H$_{u}$, H$_{d}$ ) and W a α are the MSSM field- strengths. As the F -term component of X obtains a non- zero vacuum expectation value at the messenger scale M , the first and the second terms of (6) respectively give the scalar masses (3) and the gaugino masses (1). 

Let us consider quantum corrections to (6). There are no 1 particle irreducible diagrams that renormalize operators linear in X , and hence the second line of (6) receives only the wave function renormalization coming from the hidden sector field. By the renormalization X → Z − 1 / 2 X X , the second line of (6) becomes 

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