GMSB at a stable vacuum and MSSM without exotics from heterotic string

We show that it is possible to introduce the confining hidden sector gauge group SU(5)' with the chiral matter 10 plus 5-bar, which are neutral under the standard model gauge group, toward a gauge mediated supersymmetry breaking (GMSB) in a Z_{12-I} orbifold compactification of E_8xE_8 heterotic string. Three families of MSSM result without exotics. We also find a desirable matter parity P (or R-parity) assignment. We note that this model contains the spectrum of the Lee-Weinberg model which has a nice solution of the mu problem.


I. INTRODUCTION
The supersymmetric (SUSY) extension of the standard model (SM) encounters a few naturalness problems, the SUSY flavor problem [1], the little hierarchy problem [2], the µ problem [3], etc. The hierarchichal magnitude is worst in the µ problem but here there are nice solutions [4]. The little hierarchy problem has weakened the nice feature of the SUSY solution of the gauge hierarchy problem and we hope that it will be understood somehow in the future. On the other hand, the SUSY flavor problem seems to require family independence of the interactions at the GUT scale. The attractive gravity mediation scenario for transmitting SUSY breaking down to the observable sector probably violate the flavor independence of interactions violently. This observation has led to the gauge mediated supersymmetry breaking (GMSB) [5]. However, the superstring attempt toward a GMSB model has not been successful phenomenologically, even though the possibility of SUSY breaking spectra was pointed out [6].
Recently, dynamical SUSY breaking (DSB) at an unstable minimum at the origin of the field space got quite an interest following Intrilligator, Seiberg and Shih (ISS) [7,8,9], partly because it has not been successful in deriving a phenomenologically attractive model in the stable vacuum. Among the results on SU(N), SO(N) and Sp(2n) groups, the result is especially simple for SU(N c ) with N f flavors, showing an unstable minimum for N c + 1 ≤ N f < 3 2 N c . This mechanism is easily applicable to SU(5) ′ models with 6 or 7 flavors, which can be realized in string compactifications [6]. Nevertheless, it is better to realize a phenomenologically successful SUSY breaking stable minimum, not to worry about our stability in a remote future. In this paper, therefore, we look for a GMSB spectrum in the orbifold compactification of the E 8 ×E ′ 8 heterotic string with three families, trying to satisfy all obvious phenomenological requirements.
The well-known DSB models are an SO(10) ′ model with 16 ′ or 16 ′ + 10 ′ [10], and an SU(5) ′ model with 10 ′ + 5 ′ [11]. It is known that GMSB with 16 ′ + 10 ′ can be obtained from heterotic string [12], but the beta function magnitude is too large (in the negative) so that SO(10) ′ confines somewhat above 10 13 GeV against a meaningful GMSB. If the hidden sector gauge group is large, the content of matter representation is usually small and the beta function magnitude (in the negative) turns out to be too large to implement the GMSB scenario. If the confining group is SU(4) ′ or smaller, it is not known that one can obtain a SUSY breaking stable minimum. Thus, SU(5) ′ is an attractive choice for the GMSB [6].
To solve the SUSY flavor problem along this line of the GMSB, we require two conditions: relatively low hidden sector confining scale ( 10 12 GeV) and appearance of matter spectrum allowing SUSY breaking.
A nice feature of the ISS type model at an unstable vacuum toward model building is that the SUSY breaking can be mediated through dimension-4 superpotential given in 1 where Q is a hidden sector quark and f is a messenger. It is possible because the vectorlike representations, for example six or seven (Q + Q), are present and the QQff interaction is suppressed by one power of mass parameter. So this mass parameter can be raised up to the GUT scale.
On the other hand, the uncalculable model with 10 ′ + 5 ′ of SU(5) ′ does not have such a simple singlet direction in terms of chiral fields. For example, the term ǫ ijklm 10 ij 10 kl 10 mn5 n = 0 since taking n = 1 without generality it is proportional to ǫ 1jklm 10 1j 10 kl 10 m15 1 which can be shown to be vanishing using the antisymmetric symbol ǫ. The singlet combination is possible in terms of the chiral gauge field strength, W ′ α W ′ α . It is pointed out that the F -term of this singlet combination can trigger the SUSY breaking to low energy [13], where the effective parameters of M and M f can be lower than the GUT scale.
The GMSB problem in string models is very interesting. For example, quite recently but before ISS, it has been reviewed [14], but the phenomenological requirements toward the minimal supersymmetric standard model (MSSM) have made it difficult to be found in string models. The three family condition works as a strong constraint in the search of the hidden sector representations. If we require the exotics free condition, the possibility reduces dramatically.
In a Z 12−I orbifold compactification, we find a model achieving the GMSB at a stable vacuum together with three families of quarks and leptons without any exotics. Since there is no exotics, it is hoped that the singlet VEVs toward successful Yukawa couplings have much more freedom, most of which are set at the string scale. We find a successful embedding of matter parity P and a nice solution of the µ problem. One unsatisfactory feature is that sin 2 θ W is not 3 8 . Thus, to fit the weak mixing angle to the observed value, we must assume intermediate state vectorlike particles. Anyway, another kind of intermediate state particles is needed also for a successful messenger mass scale. (1) We obtain the 4D gauge group by considering massless conditions satisfying P · V = 0 and P · a 3 = 0 in the untwisted sector [15]. We embed the discrete action Z 12−I in the E 8 ×E ′ (a) Gauge group: The 4D gauge groups are obtained by P 2 = 2 vectors satisfying P ·V = 0 and P · a 3 = 0 mod integer, The gauge group SU(3) W will be broken down to SU(2) W by the vacuum expectation value The hypercharge direction is the combination of U(1)s of Eq. (4) and some generators of nonabelian groups and W 8 , F 3 , F 8 are nonabelian generators of SU(3) W and SU(3) ′ . We defineỸ = Y Abel + 1 We included the SU(3) ′ generators in Y of (8) so that there does not appear exotics.
The five U(1) generators of (4) are defined as (b) Matter representations: Now there is a standard method to obtain the massless spectrum in Z 12−I orbifold models. The spectra in the untwisted sectors U 1 , U 2 , and U 3 , and twisted sectors, T 1 0,+,− , T 2 0,+,− , T 3, T 4 0,+,− , T 5 0,+,− , and T 6, are easily obtained [16]. The representations are denoted as where we already use the broken SU(3) W andỸ = Y Abel + 1 √ 3 W 8 given in Eq. (10). For obvious cases, we will use the abbreviated notation But when SU(3) ′ triplets or antitriplets are involved, the hypercharge isỸ . We list all matter fields below, where 1 = (1, 1, 1; 1, 1). Breaking SU(3) ′ , we assign Then 3 ′ has extra entries of 2 Eq. (14) with (16) gives the SM quantum numbers. From these, we note that there is no exotics. Other exotics free orbifold compactifications [6,16] have E ′ 8 sector contribution to Y as in the present case. But, we do not know whether this is a necessary condition for exotics free models or not.

A. Three families with no exotics
Removing vectorlike representations and neutral singlets, we obtain the following chiral representations, where 10 ′ 0 = (1; 10 ′ , 1) 0 and 5 ′ 0 = (1; 5 ′ , 1) 0 . In Table I, we list three families except the charged lepton singlets. Note that SU(3) c triplets with underlined entries mean, for example, We choose its generator Γ such that the light quarks carry odd U(1) Γ charges while Higgs doublets carry even U(1) Γ charges. This is necessary to remove the baryon number violating u c d c d c term. For the lepton number violation, the condition is not so strong and furthermore in our model there are so many possibilities in choosing the charged singlets e c , and here we do not discuss them. Then, one successful choice of Γ is whereW The Γ quantum numbers are also listed in Table I. Breaking U(1) Γ by VEVs of even integer SM singlets, a discrete symmetry Z 2 , which is called matter parity P , survives, Thus, looking at the light quarks only the dangerous term u c d c d c is not allowed. However, we have to consider mixing of light quarks with heavy quarks which can be dangerous in principle [16]. In our model, there are ten quark flavors: six SM quarks and four extra Therefore, if P is not broken, light d c and heavy Ds and D ′ can never mix and we achieve an exact matter parity P . But a successful matter parity assignment should not be in conflict with other phenomenological requirements. The most severe constraint comes from making exotic particles massive [16]. In passing, we point out that the other vectorlike particles, such as D − D, D ′ − D ′ , doublet pairs, and unit charge lepton pairs E − − E + , are not so dangerous as exotics. Since our model does not include any exotics, we do not need VEVs of any odd Γ singlets for obvious phenomenological reasons. A detailed study of singlet VEVs is outside of the scope of the present discussion, and will be presented elsewhere.

E. The µ term
A possible large µ term arises from the coupling between three pairs of3 W,0 −3 W,+ as Because of ǫ IJ , the same Higgs family does not have the coupling and the 3 × 3 H u − H d mass matrix is an antisymmetric one whose determinant is zero. Therefore, we obtain two massive Higgs doublet pairs and one massless Higgs doublet pair. Thus, there remains only one massless Higgs doublet pair, achieving the MSSM spectrum at low energy. In this scheme also, there are methods to generate an electroweak scale µ term [3,4].

III. HIDDEN SECTOR SU(5) ′ , GAUGE MEDIATION AND MESSENGERS
As shown in Table III, there are SU(5) ′ fields. But some of these obtain masses by Yukawa couplings at the string scale. Below the string scale vectorlike pairs become massive by VEVs of singlets, and hence we consider only the chiral representations. We need the mass scale of the vectorlike pairs are much above the SU(5) ′ confining scale so that the SUSY breaking by 10 ′ and 5 ′ is intact.   In Table III, we list all the SU(5) ′ non-singlet fields. From these, one can easily check that SU(5) ′ gauge anomaly is absent. One conspicuous feature is that we obtained one Note that the singlet combination 10 ′ 10 ′ 10 ′ 5 ′ is not possible with one 10 ′ . The SU(5) ′ singlet combination in this uncalculable model can be parameterized by the gauge field strength field W ′ α W ′ α as discussed in [13]. The interaction between the messenger f and the hidden sector gauge fields can appear from string compactification as where we have in general the holomorphic functions ξ and η of singlet chiral fields, S 1 , S 2 , · · · .
The quantum number of ξ(S 1 , S 2 , · · · )ff is the same as that of dilaton, where the H-With this GMSB scenario, firstly the observable sector gaugino obtains mass of order while the gravitino mass is around m 3/2 ≈ Λ 3 h /M 2 P l . To obtain 1 TeV gluino mass (but much smaller gravitino mass of order 0.2 GeV) with α = 1 25 and Λ h = 10 12 GeV, for example, we need (M 2 M mess ) 1/3 ≈ 1.5 × 10 14 GeV. The Higgsino mass matrix and the B matrix take the following form, Finally, we comment on possible higher order terms in the Kähler potential. Even though all the important hidden sector matter 10 ′ does not appear in the superpotential, it can appear in the Kähler potential. Possible terms of the form 10 ′ 10 ′ * f f * /M 2 K might appear. The higher order Kähler terms was calculated for the compactification T 6 = (T 2 ) 3 (with the volume moduli T s and the complex structure moduli Cs) in Ref. [20] for two matter fields Q α , where n i α and h 2,1 = 1 (for our Z 12−I ) are the modular weight and a Hodge number, respectively. Also, l α is an integer. The term 10 ′ 10 ′ * f f * /M 2 K is not appearing in the above expression, and at present there does not exist a K matter calculation for four matter fields of our interest. Even if it appears, the mass suppression scale M K is expected to be of order the string scale and hence is much larger than M appearing in Eq. (28) toward the GMSB scenario. However, if it appears with the same order of the suppression factor as in Eq. (28), the idea of our GMSB is not successful phenomenologically. We may need M 2 /M 2 K < 0.03 [21].

IV. CONCLUSION
We have shown that there exists a possibility of the hidden sector SU(5) ′ with 10 ′ 0 plus 5 ′ 0 matter below the GUT scale so that a GMSB at the stable vacuum is successful. Toward achieving the needed coupling constant α ′ 5 of the hidden sector at the GUT scale, we may need different compactification radii for the three tori [6]. The model is very interesting in that it contains three MSSM families without any exotics. We find a desirable U(1) Γ gauge symmetry whose Z 2 discrete group can be a matter parity P or R-parity. Due to our Lee-Weinberg type model, there remains only one light pair of Higgs doublets, achieving the MSSM spectrum. On the other hand, the weak mixing angle at the unification scale is not 3 8 . Various mass scales in addition to the different compactification radii may enable us to fit the mixing angle to the observed one at the electroweak scale. A detail analysis of the model for the R-parity problem, weak mixing angle, compactification radii, D and F flat directions, and Yukawa couplings will be discussed elsewhere.