Gauge mediated supersymmetry breaking without exotics in orbifold compactification

We suggest SU(5)$'$ in the hidden sector toward a possible gauge mediated supersymmetry breaking scenario for removing the SUSY flavor problem, with an example constructed in $\Z_{12-I}$ with three families. The example we present has the Pati-Salam type classification of particles in the observable sector and has no exotics at low energy. We point out that six or seven very light pairs of ${\bf 5}'$ and $\bar{\bf 5}'$ out of ten vectorlike $\five'$ and $\fiveb'$ pairs of SU(5)$'$ is achievable, leading to a possibility of an unstable supersymmetry breaking vacuum. The possibility of different compactification radii of three two tori toward achieving the needed coupling strength is also suggested.


I. INTRODUCTION
The gauge mediated supersymmetry breaking (GMSB) has been proposed toward removing the SUSY flavor problem [1]. However, there has not appeared yet any satisfactory GMSB model from superstring compactification, satisfying all phenomenological constraints.
The GMSB relies on dynamical supersymmetry breaking [2]. The well-known GMSB models are an SO(10) ′ model with 16 ′ or 16 ′ + 10 ′ [3], and an SU(5) ′ model with 10 ′ + 5 ′ [4]. If we consider a metastable vacuum also, a SUSY QCD type is possible in SU(5) ′ with six or seven flavors, satisfying N c + 1 ≤ N f < 3 2 N c [5]. Three family standard models (SMs) with this kind of hidden sector are rare. In this regard, we note that the flipped SU (5) model of Ref. [6] has one 16 ′ and one 10 ′ of SO(10) ′ , which therefore can lead to a GMSB model. But as it stands, the confining scale of SO(10) ′ is near the GUT scale and one has to break the group SO(10) ′ by vacuum expectation values of 10 ′ and/or 16 ′ . Then, we do not obtain the spectrum needed for a GMSB scenario and go back to the gaugino condensation idea. If the hidden sector gauge group is smaller than SU(5) ′ , then it is not known which representation necessarily leads to SUSY breaking. The main problem in realizing a GMSB model is the difficulty of obtaining the supersymmetry (SUSY) breaking confining group with appropriate representations in the hidden sector while obtaining a supersymmetric standard model (SSM) with at least three families of the SM in the observable sector.
In this paper, we would like to address the GMSB in the orbifold compactification of the E 8 ×E ′ 8 heterotic string with three families at low energy. A typical recent example for the GMSB is where Q is a hidden sector quark and f is a messenger. Before Intriligator, Seiberg and Shih (ISS) [5], the GMSB problem has been studied in string models [7]. After [5] due to opening of new possibilities, the GMSB study has exploded considerably and it is known that the above idea is easily implementable in the ISS type models [8]. Here, we will pay attention to the SUSY breaking sector, not discussing the messenger sector explicitly. The messenger sector {f, · · · } can be usually incorporated, using some recent ideas of [8], since there appear many heavy charged particles at the GUT scale from string compactifications.
The three family condition works as a strong constraint in the search of the hidden sector representations. In addition, the GUT scale problem that the GUT scale is somewhat lower than the string scale is analyzed in connection with the GMSB. Toward the GUT scale problem, we attempt to introduce two scales of compactification in the orbifold geometry. In this setup, we discuss physics related to the hidden sector, in particular the hidden sector confining scale related to the GMSB. If the GMSB scale is of order 10 13 GeV, then the SUSY breaking contributions from the gravity mediation and gauge mediation are of the same order and the SUSY flavor problem remains unsolved. To solve the SUSY flavor problem by the GMSB, we require two conditions: one is the relatively low hidden sector confining scale (< 10 12 GeV) and the other is the matter spectrum allowing SUSY breaking.
Toward this kind of GMSB, at the GUT scale we naively expect a smaller coupling constant for a relatively big hidden sector nonabelian gauge group (such as SU(5) ′ or SO(10) ′ ) than the coupling constant of the observable sector. But this may not be needed always.
The radii of three two tori can be different in principle as depicted in Fig. 1. For simplicity, we assume the same radius r for (12)-and (56)-tori. A much larger radius R is assumed for the second (34)-torus. For the scale much larger than R, we have a 4D theory. In this case, we have four distance scales, R, r, α ′ = M −2 s , and κ = M −1 P , where α ′ is the string tension and M P is the reduced Planck mass. The Planck mass is related to the compactification scales by M 2 P ∝ M 8 s r 4 R 2 . Assuming that strings are placed in the compactified volume, we have a hierarchy 1 R < 1 r < M s < M P . The customary definition of the GUT scale, M GUT , is the unification scale of the QCD and electroweak couplings.
For the 4D calculation of the unification of gauge couplings to make sense, we assume that the GUT scale is below the compactification scale 1 R , leading to the following hierarchy where we have not specified the hierarchy between M s and M P .
In Sec. II, we discuss phenomenological requirements in the GMSB scenario toward the SUSY flavor problem. In Sec. III, we present a Z 12−I example. In Sec. IV, we discuss the hidden sector gauge group SU(5) ′ where a GMSB spectrum is possible.

II. SUSY FCNC CONDITIONS AND GAUGE MEDIATION
The MSSM spectrum between the SUSY breaking and GUT scales fixes the unification coupling constant α GUT of the observable sector at around 1 25 . If a complete SU(5) multiplet in the observable sector is added, the unification is still achieved but the unification coupling constant will become larger. Here, we choose the unification coupling constant in the range α GUT ∼ 1 30 − 1 20 . The GMSB scenario has been adopted to hide the gravity mediation below the GMSB effects so that SUSY breaking need not introduce large flavor changing neutral currents (FCNC) [1]: where M P is the reduced Planck mass 2.44 × 10 18 GeV, M X is the effective messenger scale Now the expression (4) is used to give constraint on α h GUT . Defining the inverse of unification coupling constants as we express A ′ in terms of the scale Λ h as 1 If M GUT ≃ 2 × 10 16 GeV and Λ h ≃ 2 × 10 10 GeV, we obtain A ′ in terms of −b h j as shown in Eq. (7).
If we consider a metastable vacuum, a SUSY QCD type is possible in SU(5) ′ with six or 1 One can determine Λ h where α h = ∞ for which near Λ h the one loop estimation is not valid. So we seven flavors, 6(5 ′ + 5 ′ ) or 7(5 ′ + 5 ′ ) [5]. The reason that we have this narrow band of N f is that the theory must be infrared free in a controllable way in the magnetic phase. Three family models with α ′ < 1 25 are very rare, and we may allow at most up to 20% deviation from α GUT value, i.e. α ′ > 1 30 . Then, from Fig. 2 we note that it is almost impossible to have an SO(10) ′ model from superstring toward the GMSB. The reason is that SO (10)  The ISS type models are possible for SO(N c ) and Sp(N c ) groups also [5]. In this paper, however we restrict our study to the SU(5) ′ hidden sector only. We just point out that SO(N c ) groups, with the infrared free condition in the magnetic phase for N f < 3 2 (N c − 2), are also very interesting toward the unstable vacua, but the study of the phase structure here is more involved. On the other hand, we do not obtain Sp(N c ) groups from orbifold compactification of the hidden sector E ′ 8 .

III. A Z 12−I MODEL
We illustrate an SSM from Z 12−I . The twist vector in the six dimensional (6d) internal space is The compactification radius of (12)-and (56)-tori is r and the compactification radius of (34)-torus is R, with a hierarchy of radii r ≪ R.
We obtain the 4D gauge group by considering massless conditions satisfying P · V = 0 and P · a 3 = 0 in the untwisted sector [9]. This gauge group is also obtained by considering the common intersection of gauge groups obtained at each fixed point.

SU(4)
: The SU(2) V is like SU(2) R in the Pati-Salam(PS) model [11]. The gauge group SU(4) will be broken by the vacuum expectation value (VEV) of the neutral singlet in the PS model.
As shown in Table I Certainly, these conditions can be satisfied. At this point, we are content merely with having three SSM families without exotics, and let us proceed to discuss SUSY breaking via the GMSB scenario, using the hidden sector SU(5) ′ .
Out of ten SU(5) ′ quarks, there may result any number of very light ones according to the choice of the vacuum. A complete study is very complicated and here we just mention that it is possible to have six or seven light SU(5) ′ quarks out of ten. The point is that we have enough SU(5) ′ quarks. For example, one may choose the T 3 T 9 coupling such that one pair of SU(2) W doublets (two SU(5) ′ quarks) becomes heavy with a mass scale of m 1 .
For the sake of a concrete discussion, presumably by fine-tuning at the moment, one may 5 Details of the rules for Z 12−I are given in [6,10].  consider the T 6 T 6 coupling such that the following 5 ′ · 5 ′ mass matrix form is not broken by hidden sector squark condensates because their values are vanishing [5]. 6 For m 1,2 ≪ Λ h , an unstable minimum is not obtained [5]. Note that the unification of α c and α W is not automatically achieved as in GUTs because light (1, 2, 1; 1; 5 ′ , 1) 1/10 quarks do not form a complete representation of a GUT group such as SU (5). Unification condition must be achieved by mass parameters of the fields surviving below the GUT scale, and the condition depicted in Fig. 2 Fig. 2, we have α h ≃ 1 9 for Λ h = 10 12 GeV. These values are large. 7 To introduce this kind of a large value for the hidden sector coupling constant, we can introduce different radii for the three tori. In this way, a relatively small scale, M GUT ∼ 2 × 10 16 GeV compared to the string scale, can be introduced also via geometry through the ratio r/R. Let the first and third tori are small compared to the second tori as depicted in Fig. 3.
If the radius R of the second torus becomes infinite, we treat the second torus as if it is a fixed torus. Then, one might expect a 6D spacetime, expanding our 4D spacetime by including the large (34)-torus. One may guess that the spectrum in T 1 , T 2 , T 4 , and T 7 sectors would be three times what we would obtain in T i 0 (i = 1, 2, 4, 7). For T 3 and T 6 , the spectrum would be the same since they are not affected by the Wilson line from the beginning. But this naive consideration does not work, which can be checked from the spectrum we presented. If the size of the second torus becomes infinite, we are effectively dealing with 4d internal space, and hence we must consider an appropriate 4d internal space compactification toward a full 6D Minkowski spacetime spectrum. This needs another set of twisted sector vacuum energies and the spectrum is not what we commented above. A more careful study is necessary to fit the hidden sector coupling constant to the needed value. 7 A naive expectation of the hidden sector coupling, toward lowering the hidden sector confining scale, is a smaller α h GUT compared to 1 25 . Because of many flavors, α h GUT turns out to be large.
Here we just comment that in our example SU(5) ′ is not enhanced further by neglecting the Wilson line. Even though SU(5) ′ is not enhanced between the scales 1/r and 1/R, the SU(5) ′ gauge coupling can run to become bigger than the observable sector coupling at the GUT scale since in our case the bigger group SU(5) ′ , compared to our observable sector SU(4) group even without the Wilson line, results between the scales 1/r and 1/R.
The example presented in this paper suggest a possibility that the GMSB with an appropriate hidden sector scale toward a solution of the SUSY flavor problem is realizable in heterotic strings with three families.

V. CONCLUSION
Toward the SUSY flavor solution, the GMSB from string compactification is looked for.
We pointed out that the GMSB is possible within a bounded region of the hidden sector gauge coupling. We find that the hidden sector SU(5) ′ is the handiest group toward this direction, by studying the gauge coupling running. We have presented an example in Z 12−I orbifold construction where there exist enough number of SU(5) ′ flavors satisfying the most needed SM conditions: three observable sector families without exotics. Toward achieving the needed coupling strength of the hidden sector at the GUT scale, we have suggested different compactification radii for the three tori.