Number of fermion generations from a novel Grand Unified model

Electroweak interactions based on a gauge group $\rm SU(3)_L \times U(1)_X$, coupled to the QCD gauge group $\rm SU(3)_c$, can predict the number of generations to be multiples of three. We first try to unify these models within SU(N) groups, using antisymmetric tensor representations only. After examining why these attempts fail, we continue to search for an SU(N) GUT that can explain the number of fermion generations. We show that such a model can be found for $N=9$, with fermions in antisymmetric rank-1 and rank-3 representations only, and examine the constraints on various masses in the model coming from the requirement of unification.


Introduction
In the standard model, the number of fermion generations appears as an arbitrary parameter, meaning that a mathematically consistent theory can be built up using any number of fermion generations. The same is true for many extensions of the standard model, including grand unification models based on the gauge groups SU (5) and SO (10). An interesting question related to extension of the standard model is whether the number of fermion generations can in any way be explained through the internal consistency of the model. In the literature, there is some discussion of grand unification models based on big orthogonal groups like SO(18), where one spinor multiplet contains all known fermion fields of all generations, and much more [1]. It was shown that [2,3], with suitable symmetry breaking scheme, only three generations can remain light, whereas others obtain masses at much above the electroweak scale.
In a quite different line of development, it was shown that if one extends the electroweak group to SU(3) × U(1) and tries to accommodate the standard fermions into multiplets of this gauge group which must include some new fermions, cancellation of gauge anomalies can restrict the number of generations and one can obtain consistent models with the number of generations equal to three or any multiple of it [4][5][6][7][8].
These models will be described briefly in § 2. Then, in § 3 and § 4, we try to see whether these models can be embedded into a simple SU(N) group. We conclude that, if one uses only completely antisymmetric tensor representations, such an embedding cannot be found. Then, in § 5, we start looking for general conditions that will specify the number of fermion generations for arbitrary SU(N) groups.
In § 6, we analyze one simple model, based on the group SU (9), that gives three generations. The renormalization group analysis of various scales in this models has been performed in § 7. We end with some concluding remarks in § 8.

The 3-3-1 models
The 3-3-1 models are based on the gauge group SU(3) c × SU(3) L × U(1) X . The first factor in the gauge group is the group of QCD, whereas the other two factors pertain to electroweak interactions. There are two versions of such models, and we discuss them one by one.
In one version, proposed by Pisano and Pleitez [5] and by Frampton [6] (to be denoted as the PPF model), the left-chiral fermions and antifermions belong to the following representations of the gauge group: Note that there are two kinds of generation indices: a goes from 1 to 3, whereas i takes only the values 2 and 3. An antifermion has been denoted by a hat. We emphasize that all representations given above pertain to the left-chiral components only. The representation of a right-chiral fermion would be the complex conjugate of that of the left-chiral antifermion, and vice versa. Note that there are extra quark fields, i.e., fields which are triplets of SU(3) c , but there is no neutrino field that is sterile under the standard model. Different generations of fermions are not copies of one another. Gauge anomalies cancel between the three generations [5,6], but not within a single generation. Thus, the consistency of the model requires three generations of fermions. Of course, this pattern of three generations can be repeated, obtained number of generations to be multiples of 3. The other version of 3-3-1 models was proposed by Singer, Valle and Schechter [4] (and hence will be referred to as the SVS model) and was examined by other authors later [7,8]. In this version, there are sterile neutrinos, the left-chiral component of which has been denoted byν in the list below: The notation for the generation indices is as before. The primed fields are extra quark fields which are not present in the standard model. Like the previous model, gauge anomalies cancel between various generations. The difference between the two models may be summarized in the following way. The standard model gauge group is not a maximal subgroup of the group SU(3) c × SU(3) L × U(1) X . In particular, the electroweak SU(3) L has two neutral generators, and some combination of these two, along with the generator of U(1) X , form Y and I 3L , the two neutral generators of the standard electroweak model. Using the standard normalization of the SU(3) generators, these combinations are given by in the first model, whereas in the second one, they are given by The SU(3) generators are given by where the λ's are the well-known Gell-Mann matrices.

Seeking embeddings into SU(N)
Since the group SU(3) c ×SU(3) L ×U(1) X is of rank 5, the smallest unitary group that contains it as a subgroup is SU (6). Therefore, in this section, we analyze whether the models discussed in § 2 can be embedded into an SU(6) grand unified model. For the sake of convenience, let us announce here the notation that will be used for denoting representations of various groups. For the grand unified group, the notations will be denoted by boldface. The SU(3) c ×SU(3) L ×U(1) X representations will be denoted by three numbers in parenthesis, e.g., a, b, c , as has already been done in Eqs. (1) and (2). And the representation of the standard model group, when required, will be denoted by square brackets, e.g., a, b, c . Thus, with this notation, the fundamental representation of SU(6) has the decomposition It is now easy to see that neither PPF nor SVS model can be embedded into SU (6). For the PPF model [5,6], note the representation of leptons given in Eq. (1a). Certainly, it is not contained in the fundamental representation of SU(6) or its complex conjugate, 6 * . Higher representations can be obtained by taking Kronecker products of the 6 and 6 * representations, and will be of the generic form (6) m (6 * ) n . Denoting the two different SU(3) c × SU(3) L × U(1) X representations that appear in Eq. (7) by A and B, we can write Take any term in the sum. Contributions to the U(1) X quantum number come from all four factors, and is given by Thus, for the lepton representation such as in Eq. (1a), we need in order to obtain X = 0. Only the B and B * contributes to non-trivial SU(3) L representations. In order to obtain a triplet, we need considering the triality of the representations. Moreover, the lepton must be a color singlet, which means that we should have These three conditions cannot be satisfied with integers, and hence it is impossible to obtain a (1, 3, 0) representation of SU(3) c × SU(3) L × U(1) X in any representation of SU (6). For the SVS model, the same kind of analysis can be performed keeping an eye towards the antilepton in Eq. (2b). In order to produce a singlet of both SU(3) factors, one needs Kronecker product of equal number of 6 and 6 * . However, such products will give X = 0.
We have thus proved that neither the PPF, nor the SVS model can be embedded into SU (6). This result should not be understood to imply that an SU(6) grand unified model is impossible. We can take a different embedding of the standard model generators I 3L and Y , viz., The SM reduction of the 6 * representation of SU(6) can be easily read from Eq. (7): Also, the antisymmetric rank-2 representation has the following decomposition under the SM gauge group: In both Eqs. (14) and (15), we have marked the known fermions which correspond to the SM representations. We observe that all known fermions of a single generation belong to these two representations. This is therefore like the minimal SU(5) grand unified model where the corresponding representations contain all known fermion fields of a single generation and nothing more. In this case, there are some extra fermions to complete the SU(6) representations. There is one big difference between the SU(5) and this SU(6) model. This can be seen from the anomaly coefficients of different representations. For the completely antisymmetric tensorial representations of SU(N), the anomaly coefficients are as follows [9]: Here, the completely antisymmetric representation of rank n has been denoted by n . The anomaly coefficient of any representation and its complex conjugate will be negative of each other. We see that for the SU(5) group, anomalies cancel between the 5 * and 10 representations. For the SU(6) gauge group, the 6 * and the 15 representations have anomaly coefficients −1 and 2 respectively. Thus, along with 3 copies of the 15 representation that is necessary in order to have three quark doublets, we need 6 copies of the 6 * representation in order to cancel anomalies. There will thus be 3 extra copies of the lepton doublet L. But these will form gauge invariant masses with the three copies of the 1, 2, 1 2 representation that appears in the three 15-plets, and these masses can be much heavier than the electroweak scale. Similarly, among the six copies of thed representation that appear in the 6 * multiplets, three will form gauge invariant masses with the 3, 1, − 1 3 representations present in the 15-plets, leaving only threed's for the electroweak scale. The sterile neutrino fields, i.e., the 1, 1, 0 representations shown in Eq. (14), can also have mass terms that are invariant under the SM gauge group. If all these masses are large, the only fields that are left over at the electroweak scale are no different from the fields that are obtained in three generations of 5 * and 10 multiplets of SU (5). Moreover, the model has no explanation for the number of generations. Any number of 15-plets, along with twice the number of 6 * -plets, would be anomaly-free. Hence this model is not interesting for our discussion.

Generalities about SU(N) models with N > 6
We now consider models based on the gauge groups SU(N), with N > 6. Henceforth we will use completely antisymmetric tensor representations only. Such a representation of rank n will be denoted by n , as was done in Eq. (16). The fundamental representation will be thus denoted by 1 in this notation, whereas its complex conjugate will be N − 1 . We will show that, with antisymmetric representations only, neither PPF nor SVS model can be embedded into an SU(N) grand unified group.
The crucial aspect of both PPF and SVS models is that, among the left-chiral fields, the quarks, i.e., color triplets, appear in triplet or antitriplet of SU(3) L , whereas the antiquarks, i.e., color antitriplets, are all singlets of SU(3) L . In particular, then, there is no multiplet that transforms like 3 * , 3, ⋆ or 3 * , 3 * , ⋆ . On the other hand, among the quark fields, some should be in 3, 3, ⋆ and some in 3, 3 * , ⋆ representations, thereby ensuring anomaly cancellation among different generations. These features are not available in the decomposition of any m representation of an SU(N) group, as we show now.
Since The situation is no different if we assume that the decomposition of the fundamental follows the rule In this case, following the same argument, we would conclude that the number of 3, 3, ⋆ and 3 * , 3 * , ⋆ are equal. Now, the 3 * , 3 * , ⋆ submultiplets can be got rid of by forming superheavy masses with 3, 3, ⋆ submultiplets. But then there will be no 3, 3, ⋆ left at the SU(3) c × SU(3) L × U(1) X level, which is unacceptable for both PPF and SVS models.

Other embeddings into SU(N)
At this point, we ignore the intermediate symmetry SU (3) where the number within parentheses indicate the number of copies of the SM gauge singlet in the last term. Similarly, we obtain Suppose now we consider a model with n 1 copies of 1 representation, n 2 copies of 2 , and n 3 copies of 3 . In counting the "copies", we will denote a complex conjugate representation by a negative number. We count the number of different SM multiplets in the antisymmetric representations of SU(N) and summarize the result in Table 1. We see that, in order to obtain n g generations of fermions, we need in order to ensure the correct number of quark doublets, as well asû andl. In addition, we need so that the correct number of lepton doublets andd are obtained. Anomaly cancellation between the representations will be ensured if we have making use of the anomaly co-efficients given in Eq. (16). However, only two of the three relations in Eq. (21) are independent. We find it simple to work with Eqs. (21a) and (21c). The general solution of these equations is given by If we take n 3 = 0, we obtain n g = n 2 , and hence no explanation of generations. This is what was done for the SU(6) grand unified model discussed in Sec. 3, a model which was found uninteresting precisely because it could not predict the number of generations. However, we can consider other kinds of solutions of Eq. (22). For example, if we take n 2 = 0, we obtain In this case, for the grand unified group SU(9), we find that the number of generations must be 3 or its multiple. It should be noted that it is just as easy to obtain solutions of Eq. (22) with n 1 , n 2 and n 3 all non-zero. One such model with an SU(8) gauge group was the subject matter of Ref. [10], where the authors took n 1 = −9, n 2 = 1, n 3 = 1 and obtained three generations of fermions. From our analysis, it seems that they could have obtained any other number of generations by adjusting the number of copies of various representations. For example, n 1 = −13, n 2 = 2, n 3 = 1 would give four generations. However, the merit of the n 2 = 0 solutions is that the number of generations cannot be arbitrary: it must be a multiple of 3.

Anatomy of an SU(9) model
As seen from our earlier analysis, an SU(9) model, in the absence of rank-2 antisymmetric multiplets, automatically gives three fermion generations, provided we take which means that there should be 9 copies of the anti-fundamental representation, and one copy of the rank-3 antisymmetric multiplet. To see in more detail how different fermion representations of the standard model are obtained from these representations of the grand unified group SU(9), we first discuss the decomposition of these multiplets under the group SU(3) c × SU(3) L × U(1) X . The decomposition of the fundamental of SU (9) can be taken to be as given in Eq. (17). We can choose the U(1) X quantum number in a way that it vanishes for all singlets of SU(3) c × SU(3) L . Then, choosing the normalization of the U(1) X arbitrarily, we can write 1 ≡9 → 3 * , 1, This gives, for the rank-3 representation of SU(9), the following decomposition into SU(3) c × SU(3) L × U(1) X multiplets: 3 ≡ 84 = 1, 1, 0 + 1, 1, 1 + 1, 1, −1 Looking at these decompositions, we find that the multiplets specified by Eq.  In addition, we find that the decomposition consists of several chiral multiplets of SU(3) c ×SU(3) L ×U(1) X . Under the group SU(3) L , these multiplets transform either like triplets or antitriplets, or like singlets. We want the triplets and antitriplets to contain the doublets of the standard electroweak gauge group. For that, we first have to discuss how the SM is embedded into SU(3) c × SU(3) L × U(1) X . Since the members of an SU(2) L doublet have the same value of hypercharge Y , we need to ensure that the diagonal Y generator has two equal entries in the 3×3 representation of SU(3) L . We notice that the PPF model was constructed in such a way that the second and the third elements of a triplet end up with the same value of Y through Eq. (4), whereas in the SVS model, the first two elements are equal. In order to obtain something different, we can now try a solution in which the first and the third elements of Y should have the same diagonal elements. This solution is given by With these assignments, we now present the chiral multiplets of SU(3) c × SU(3) L × U(1) X that are present in the choice of representations in Eq. (24), as well as their SM decompositions: We see that, at the SM level, there are three quark doublets, three multiplets that transform likeû L , and also three that transform like thel L : precisely the numbers necessary for obtaining three fermion generations in the SM. We find six SM multiplets that transform like thed L in Eq. (29b), but that is not a problem, because three of them can pair up with and equal number of 3, 1, − 1 3 multiplets that appear in Eq. (29a), and can obtain a bare mass term that is invariant under the SM gauge group. Similarly, the three 1, 2,

Gauge coupling unification and the intermediate scales
One addresses now the unification picture of the SU(9) model defined by the condition given in Eq. (24). For the sake of simplicity, we shall assume in our analysis that the SU(9) GUT model spontaneously breaks, through the scheme given by Eq.
(19), directly to the SM gauge group at a unique unification scale Λ. Thus, the unification condition for the SM gauge couplings α 1,2,3 at the scale Λ reads as The normalization constants k i are defined by where T and T i are the same unbroken generator properly normalized to the GUT group and its subgroup G i , respectively, for a given representation. Taking into account the decomposition of the fundamental representation given in Eq. (19) and the GUT normalization in Eq. (3), one easily derives that the normalization constants k i take the canonical values like in the SU(5) GUT models. The relation given in Eq. (31) is only valid at the unification scale and one has to relate these gauge coupling with measurable quantities at electroweak scale. The evolution of the gauge couplings in the one-loop approximation is ruled by the solutions of the Renormalization Group Equations (RGE's), which depend on the masses of the particles in the model. We take the SM particles and n H Higgs doublets at the electroweak scale. In addition, the vector-like combinations given in Eq. (30) can also have masses between M Z and the unification scale M U . We will assume, for the sake of simplicity, that all extra vector-like fermions with the same quantum number share the same mass scale, and denote these scales collectively as  We then find that Notice that the right-hand sides of Eq. (40) depend only on low-energy electroweak data and the group factors k i . Adopting the following experimental values at M Z [13] α −1 = 127.916 ± 0.015 , (41a) the above relations give in the canonical GUT models with k i = (5/3, 1, 1). The coefficients B ij that appear in the left-hand sides of Eq. (40) strongly depend on the particle content of the theory. For instance, considering the SM-like particles, with n g generations, together with n H light Higgs doublets, one has for the coefficients B 12 and B 23 :  [13] into e + π 0 . A million set of such random combinations were taken. A few of the allowed ones are presented in Fig. 1. From the entire set of combinations used in our numerics, we find that the allowed range for the unification scale obtained is 6 × 10 15 GeV ≤ Λ ≤ 2.2 × 10 16 GeV , which is also roughly what the limited data of Fig. 1 We see that the range of mass scales of the vector-like fermions are high, but they are indeed necessary for driving the evolution of the gauge couplings to a perfect unification. If one varies the number of Higss doublet, n H = 2, 3, the ranges of intermediate scales above do not change significantly.

Conclusions
To summarize, we have succeeded in building a grand unified model based on the group SU (9). The model uses fermions in antisymmetric representations only and the consistency of the model demands that the number of fermion generations is three. As mentioned in the introduction, earlier models based on the gauge group SU(3) c × SU(3) L × U(1) X and SO(18) also had this property of predicting the number of generations. It is interesting to note that our group SU (9) Our SU(9) model uses only the first two kinds of these submultiplets to predict the number of fermion generations.
We have also shown that our model can be consistent with unification requirements. In this part of the analysis, we have assumed a direct breaking of SU(9) into the SM gauge group. A more detailed analysis, including possibilities of intermediate symmetry breaking scales, will be taken up in a future work.