Trinification with sin 2 θ W = 3 8 and seesaw neutrino mass

We realize a supersymmetric trinification model with three families of (27tri + 27tri + 27tri) by the Z3 orbifold compactification with two Wilson lines. It is possible to break the trinification group to the supersymmetric standard model. This model has several interesting features: the hypercharge quantization, sin θ0 W = 3 8 , naturally light neutrino masses, and introduction of R-parity. The hypercharge quantization is realized by the choice of the vacuum, naturally leading toward a supersymmetric standard model. [


I. INTRODUCTION
Unification of fundamental forces in the last three decades [1] has been a partially successful endeavor. Probably, the most attractive feature of the unification is the quantisation of the electromagnetic charge, Q em (proton) = −Q em (electron). However, the most serious problem in this old grand unification(GUT) is the existence of the fine-tuning problem the so-called gauge hierarchy problem. To understand the gauge hierarchy problem, supersymmetry has been considered, which is then extended to a consistent superstring theory with a big gauge group in ten dimensions(10D). One particularly interesting superstring model is the E 8 × E ′ 8 heterotic string [2], because E 8 contains a chain of symmetry breaking down to E 8 → E 6 [3] and then down to E 6 → SO(10) → SU (5). In this E 8 × E ′ 8 heterotic string, the intermediate step E 6 seems to play a crucial role in the classification of standard model (SM) particles. This is because the spinor representation of SO (10) is included in the fundamental representation 27 of E 6 .
For the unification of all fundamental forces, the old GUT idea has to be unified with gravity also, which seems to be possible in string theory [4]. Thus, the unification of all forces is better studied in the E 8 × E ′ 8 heterotic string. 1 In this theory, there must be a reasonable compactification down to four dimensions(4D) so that the SM results as an effective theory at low energy world. The most powerful compactification toward applications in obtaining 4D effective theories seems to be the orbifold compactification [5,6]. However, the adjoint representation needed for breaking the GUT group is not present at the Kac-Moody level k = 1. 2 This led to 4D string constructions toward standard-like models [8] and flipped SU(5) models [9].
Here, in obtaining an effective 4D model we include all the possibilities of assigning the However, the standard-like models and the flipped SU(5) models suffer from the sin 2 θ 0 W (≡ the value at the GUT scale) problem toward unification [10]. The SU(5), SO(10) or E 6 GUT's with the SM fermions in the spinor(or fundamental) representation gives sin 2 θ 0 W = 3 8 , which will be called the U(1) Y hypercharge quantization, or simply hypercharge quantization. The sin 2 θ 0 W problem is the hypercharge quantization problem. The hypercharge quantization problem can be understood in the orbifold constructions [10] if the 4D gauge group is the trinification type, SU(3) 3 gauge group with 27 chiral fields(let us define this as 27 tri ) in one family, suggested in the middle of eighties [11]. Nevertheless, supersymmetrization of the trinification model does not lead to naturally small neutrino masses. Therefore, it was suggested that in the supersymmetric trinification one must add another vectorlike 27 tri + 27 tri [12].
So far, the trinification with small neutrino masses from the orbifold compactification was possible with the bare value of sin 2 θ 0 W = 1 4 , where one obtains just vectorlike lepton humors in addition to 27 tri [13], which however does not satisfy the hypercharge quantization. If the U(1) Y hypercharge quantization is not satisfied, one must introduce an intermediate scale to fit with data. This is called the optical unification [14], which depends on details of the intermediate scale particles and the magnitudes of the intermediate scales. For the U(1) Y hypercharge quantization, one needs vectorlike (27 tri + 27 tri )'s, not just vectorlike lepton-humor(s) [13].
Therefore, for the U(1) Y hypercharge quantization it is of utmost importance to obtain vectorlike (27 tri + 27 tri )'s. In this paper, we fulfil such an objective with an orbifold compactification, and hence obtain the bare value of sin 2 θ 0 W = 3 8 naturally.

II. TRINIFICATION WITH THREE MORE (27 ⊕ 27)'S
Choosing the hypercharge generator as Y = − 1 2 (−2I 1 + Y 1 + Y 2 ), let us denote the trinification spectrum under SU(3) 3 as, where where the representations will be called carrying three different humors as denoted by subscripts: lepton-, quark-, and antiquark-humors. These names are convenient to remember since they contain the designated SM fields. Note that lepton-humor field contains also a pair of Higgs doublets which do not carry color charge. With three sets of trinification spectrum, there exist three pairs of Higgs doublets.
We take the following orbifold model with two Wilson lines [6],  Table I.
To obtain all the possible vacuum structure of the compactification, we consider all the possible VEV's also as commented in the Introduction. In this spirit, let there exist the following vacuum expectation values of the scalar components of the fields appearing in Table I, so that the last factors in SU (3) , are completely broken, and furhermore, the following link fields for identifications of the primed and unprimed SU(3)'s, Namely, we identify 3 and 3 of SU (3) Table II. Note that there results the needed three families, Since the full trinification spectrum is added with the vectorlike combination of 3 [27 tri ⊕ 27 tri ], the bare weak mixing angle is sin 2 θ W = 3 8 , fulfilling the hypercharge quantization [10].

III. PHENOMENOLOGY
There are many indices we deal with: the untwisted and the twisted sector number, the humor(gauge group), and the family indices. So, we use the following convention For example, Ψ [2] (T0) (a) represents the second (out of the three) antiquark humor (3, 1, 3), appearing in the twisted sector T0. This notation will be generalized to respective fields such as c c , after the symmetry breaking  Tables I and II comes in three copies: so it is more accurate to represent for example the T0 sectors as T0-1, T0-2, and T0-3. The family indices can be dropped off if unnecessary.

A. Neutrino mass
The trinification fields of (27 tri ⊕ 27 tri ) in Table II can be removed at a large mass scale of order M G by giving VEV's to all singlets in T0 The superpotential can be taken as where g ABCDE are the couplings and we multiplied three singlet fields to satisfy the point group selection rule [16].
For the case of (11), the three light fermions result from T0. On the other hand, if we change indices in Eq. (11) from 0 ↔ 2, then there result light fermions from T2. Also, a more complicated family structure can be obtained by assigning couplings and VEV's judiciously. One can see that (11) gives only the Dirac neutrino masses. For a see-saw mechanism, we need a huge Majorana neutrino mass at high energy scale. So, we consider the following nonrenormalizable couplings in the superpotential allowed by Z 3 orbifold, where M is of order the string scale. We will assign huge VEV's to N 5 's in T6 and singlets in T7. Inserting these VEV's to (12), if allowed. This is a choice of a specific string vacuum from a multitude of vacua.
The existence of the above discrete symmetry can be understood in the following way.
The N 5 in (3, 3, 1) and N 5 in (3, 3, 1) has the following SM quantum numbers in terms of (2): Thus, N 5 can couple to while there is no field which N 5 can couple to. We assign a huge VEV to N 5 , but forbid a VEV of N 5 . As stressed before, this is chosen by the string vacuum. Note, however, there are two sectors(T0 and T6) where N 5 appear. With our example (11)  Thus, the conditions for the D-flatness lead to where the subscripts of the generators represent the SU (3) Table II has more structures such as (1, 1, 3, 1) ′ → (3, 1, 1, 1), but SU(3) ′ 3 and SU(3) 1 are completely broken. Some string orbifold models contain a mechanism for the doublet-triplet splitting by not allowing extra vectorlike quarks but allowing Higgsinos [8]. This has been reconsidered in field theoretic orbifolds by assigning appropriate discrete quantum numbers to the bulk fields so that extra massless vectorlike quarks are forbidden [17]. In essence, the string theory interpretation of the doublet-triplet splitting must arise from the study of the selection rules, summarized in Ref. [16]. In our case, the doublet-triplet splitting must occur after the breaking of the trinification gauge group down to the standard model gauge group by VEV's of N 10 , N 10 , and N 5 . In principle, N 10 can remove all the D − D c fields(D − D c also) and H 1 − H 2 fields (H 1 − H 2 also). But phenomenologically, we need just one pair of light Higgsinos of the MSSM, surviving this removal process. It is the old µ-problem [18] or the MSSM problem [13]. At the perturbative level, we have not found such a mechanism yet. But, there may be strong dynamics at high energy so that the determinant of the Higgsino mass matrix vanishes [13], which we do not pursue here. In our case, there is no anomalous U(1) symmetry from the string compactification since rank 16 is saturated by SU(3) 8 . Thus, it is possible to consider the model-independent axion degree which can translate to a Peccei-Quinn symmetry at low energy [19]. This may help to allow a pair of light Higgs doublets [18].

IV. CONCLUSION
We constructed a supersymmetric trinification model with three families of (27 tri + 27 tri + 27 tri ) by the Z 3 orbifold compactification with two Wilson lines. It is shown that a correct symmetry breaking pattern to the supersymmetric standard model can be achieved. One of the most attractive features is that the hypercharge quantization, i.e. the bare value sin 2 θ 0 W = 3 8 , is realized by the choice of vacuum. It is an important observation since there is no Z 3 orbifold model with any number of Wilson lines which can directly lead to the needed spectrum (27 tri + 27 tri + 27 tri ). The model presented in this paper gives naturally light neutrino masses, and allows an introduction of R-parity. For one choice of the R-charge, the D=4 and D=5 baryon number violating operators are excluded, closing the window to the proton decay experiment. A natural solution of the MSSM problem, however, has to be implemented, which we hope to discuss in a future communication.