331 Models and Grand Unification: From Minimal SU(5) to Minimal SU(6)

We consider the possibility of grand unification of the $\mathrm{ SU(3)_c \otimes SU(3)_L \otimes U(1)_X}$ model in an SU(6) gauge unification group. Two possibilities arise. Unlike other conventional grand unified theories, in SU(6) one can embed the $\mathrm{ SU(3)_c \otimes SU(3)_L \otimes U(1)_X}$ model as a subgroup such that different multiplets appear with different multiplicities. Such a scenario may emerge from the flux breaking of the unified group in an E(6) F-theory GUT. This provides new ways of achieving gauge coupling unification in $\mathrm{ SU(3)_c \otimes SU(3)_L \otimes U(1)_X}$ models while providing the radiative origin of neutrino masses. Alternatively, a sequential variant of the $\mathrm{ SU(3)_c \otimes SU(3)_L \otimes U(1)_X}$ model can fit within a minimal SU(6) grand unification, which in turn can be a natural E(6) subgroup. This minimal SU(6) embedding does not require any bulk exotics to account for the chiral families while allowing for a TeV scale $\mathrm{ SU(3)_c \otimes SU(3)_L \otimes U(1)_X}$ model with seesaw-type neutrino masses.


I. INTRODUCTION
The discovery of the Higgs boson established the existence of spin-0 particles in nature and this opened up the new era in looking for extensions of the Standard Model (SM) at accelerators. It is now expected that at higher energies, the SM may be embedded in larger gauge structures, whose gauge symmetries would have been broken by the new Higgs scalars.
So, we can expect signals of the new gauge bosons, additional Higgs scalars as well as the extra fermions required to realize the higher symmetries. One of the extensions of the SM with the gauge group SU(3) c ⊗ SU(3) L ⊗ U(1) X provides strong promise of new physics that can be observed at the LHC or the next generation accelerators [1,2]. Recently there has been a renewed interest in this model as it can provide novel ways to understand neutrino masses [3,4].
The SU(3) c ⊗ SU(3) L ⊗ U(1) X model proposed by Singer, Valle and Schechter (SVS) [1] has the special feature that it is not anomaly free in each generation of fermions, but only when all the three generations of fermions are included the theory becomes anomaly free.
As a result, different multiplets of the SU(3) c ⊗ SU(3) L ⊗ U(1) X group appear with different multiplicity and as a result it becomes difficult to unify the model within usual grand unified theories. For this reason string completions have been suggested [5]. In this article we study how such a theory can be unified in a larger SU (6) gauge theory that can emerge from a E (6) Grand Unified Theory (GUT) [6]. We find that the anomaly free representations of the SVS 331 model can all be embedded in a combination of anomaly free representations of SU (6), which in turn can be potentially embedded in the fundamental and adjoint representations of the group E(6) motivated by F-theory GUTs with matter and bulk exotics obtained from the flux breaking mechanism [7][8][9][10].
Interestingly, the SVS 331 model can also be refurbished in an anomaly free multiplet structure which can be right away embedded in a minimal anomaly free combination of representations of SU (6) as an E(6) subgroup. We refer to this new 331 model as the sequential 331 model. This scheme is particularly interesting since its embedding in SU (6) does not require any bulk exotics to account for the chiral families; and in that sense it provides a truly minimal unification scenario in the same spirit akin to the minimal SU (5) construction [11].
The article is organized as follows. In Section II we discuss the basic structure of the SVS SU(3) c ⊗ SU(3) L ⊗ U(1) X model whereas Section III describes the sequential SU(3) c ⊗ SU(3) L ⊗ U(1) X model. In Section IV we then analyze the resulting renormalization group running of the gauge couplings in the SVS model with and without additional octet states, and discuss necessary conditions for gauge unification. In Section V we then embed the different variants of the SU(3) c ⊗ SU(3) L ⊗ U(1) X model in an SU(6) unification group and demonstrate successful unification scenarios. Section VI concerns the experimental constraints from achieving the correct electroweak mixing angle and satisfying proton decay limits. We conclude in Section VII.
The SU(3) c ⊗ SU(3) L ⊗ U(1) X extension of the SM was originally proposed to justify the existence of three generations of fermions, as the model is anomaly free only when three generations are present. Such a non-sequential model, which is generically referred to as the 331 model, breaks down to the SM at some higher energies, usually expected to be in the TeV range, making the model testable in the near future. The symmetry breaking: we can readily identify the SM hypercharge and the electric charge as This allows us to write down the fermions and the representations in which they belong as The generation index i = 1, 2 corresponds to the first two generations with the quarks u L,R , d L,R , D L,R and c L,R , s L,R , S L,R . For the leptons, the generation index is a = 1, 2, 3.
There are several variants of the model that allow slightly different choices of fermions as well as their baryon and lepton number assignments. Here we shall restrict ourselves to the one which contains only the quarks with electric charge 2/3 and 1/3 and no lepton number (L). In this scenario all quarks (usual ones and the exotic ones) carry baryon number (B = 1/3) and no lepton number (L = 0), while all leptons carry lepton number (L = 1) and no baryon number (B = 0). Notice that in Ref. [4] the lepton number is defined as Here we assume k 0,1,2 ∼ m W to be of the order of the electroweak symmetry breaking scale and n 1,2 ∼ M 331 to be the SU(3) c ⊗ SU(3) L ⊗ U(1) X symmetry breaking scale. We shall not discuss here the details of fermion masses and mixing, which can be found in Refs. [3,4].
In this model the fields are assigned in a way such that the anomalies are cancelled for each generation separately. The multiplet structure is given by It is straightforward to check that each family is anomaly free. In order to drive symmetry breaking and generate the charged fermion masses, we assume a Higgs sector and vevs similar to the SVS 331 model 1 . The Yukawa Lagrangian for the quark sector can be written as with i = 1, 2 and where we neglect any flavour mixing. After the chain of spontaneous symmetry breaking the up-type quarks obtain a mass term m ua = y ua k 0 , while the down-type and vectorlike down-type quarks form a mass matrix in the (d, D) basis given by Note that in the cases y 1 da = y 2 da = y 1 Da = y 2 Da ≡ y d ; or k 1 = k 2 = k and n 1 = n 2 = n the determinant of the above Yukawa matrix vanishes giving m da = 0 and m Da = y 1 da k 1 +y 2 da k 2 + y 1 Da n 1 + y 2 Da n 2 . However, in the absence of any symmetries forcing the above conditions, the down quarks obtain mass as a result of the mixing with the vector-like quarks. One can determine it perturbatively by expanding the Yukawa contributions in terms of k i /n i 1 so as to obtain m da = y 1 da k 1 + y 2 da k 2 − y 1 Da k 1 + y 2 Da k 2 y 1 da n 1 + y 2 da n 2 y 1 Da n 1 + y 2 Da n 2 + · · · , 1 A model with similar fermion content and with k 1 = n 2 = 0 in the scalar sector was discussed in Ref. [12] using the trinification group SU where |M a dD | = y 1 da k 1 + y 2 da k 2 y 1 Da n 1 + y 2 Da n 2 − y 1 Da k 1 + y 2 Da k 2 y 1 da n 1 + y 2 da n 2 .
This structure can be used to account for the SM quark masses and CKM mixing, as well as the heavier vector-like quark mass limits from the LHC.
Turning now to the lepton sector, the relevant Yukawa terms are given by where α, β, γ are the SU(3) L tensor indices ensuring antisymmetric Dirac mass terms, C is the charge conjugation matrix, and i = 1, 2. After the symmetry breaking, these Yukawa terms give rise to the mass matrices for charged and neutral leptons. In the basis (e, E) the mass matrix is given by with the eigenvalues given by m e = − y 1 2 k 1 + y 2 2 k 2 + y 1 3 k 1 + y 2 3 k 2 y 1 2 n 1 + y 2 2 n 2 y 1 3 n 1 + y 2 3 n 2 where |M a eE | = y 1 2 n 1 + y 2 2 n 2 y 1 3 k 1 + y 2 3 k 2 − y 1 2 k 1 + y 2 2 k 2 y 1 3 n 1 + y 2 3 n 2 .
For the case of neutral leptons the mass matrix can be written as: in the basis (ν, N 1 , N 3 , N 2 , N 4 ), where N 1 , N 3 are SU(2) L isosinglets and ν, N 2 , N 4 are components of doublets. Next, we rotate the above mass matrix by an orthogonal transformation This yields the rotated mass matrix m νN given by where u = y 1 k 0 , X = y 1 2 n 1 + y 2 2 n 2 , x = y 1 2 k 1 + y 2 2 k 2 , z = y 1 3 k 1 + y 2 3 k 2 , Z = y 1 3 n 1 + y 2 3 n 2 .
Now we recall that k 0,1,2 ∼ m W is of the order of the electroweak symmetry breaking scale, while and n 1,2 ∼ M 331 is of the order of the SU(3) c ⊗ SU(3) L ⊗ U(1) X symmetry breaking scale, and hence one expects that X, Z u, x, z. If we further assume X +Z X −Z, then we can identify the 44 and 55 entries as the heaviest in the mass matrix given in Eq. (12) and these rotated isodoublet states form a pair of heavy quasi Dirac neutrinos with mass of the order of the SU(3) c ⊗ SU(3) L ⊗ U(1) X symmetry breaking scale. We can now readily use perturbation theory to obtain the masses for the three remaining lighter states. Up to second order in perturbation theory we obtain two Dirac states with mass of the order of the electroweak symmetry breaking scale ±u = ±y 1 k 0 and a light seesaw Majorana neutrino with mass 2u(z − x)/(X + Z). With this we see that the model has enough flexibility to account for the observed pattern of fermion masses. It is not our purpose here to present a detailed study of the structure of the fermion mass spectrum, but only to check its consistency in broad terms.

IV. RENORMALIZATION GROUP EQUATIONS AND GAUGE COUPLING UNIFICATION
In this section we study the SVS model RGEs to explore if unification of the three gauge couplings [13] can be obtained in the SU without any presumptions about the nature of the underlying group of grand unification [4].
Using the RGEs we express the hypercharge (and X) normalization and the unification scale Next we study the allowed range of driving an interesting radiative model for neutrino mass generation [4].
The evolution for running coupling constants at one loop level is governed by the RGEs which can be written in the form where α i = g 2 i /4π is the fine structure constant for i-th gauge group, µ 1 , µ 2 are the energy scales with µ 2 > µ 1 . The beta-coefficients b i determining the evolution of gauge couplings at one-loop order are given by Here, C 2 (G) is the quadratic Casimir operator for the gauge bosons in their adjoint representation, On the other hand, T (R f ) and T (R s ) are the Dynkin indices of the irreducible representation R f,s for a given fermion and scalar, respectively, and d(R f,s ) is the dimension of a given representation R f,s under all gauge groups except the i-th gauge group under consideration. An additional factor of 1/2 is multiplied in the case of a real Higgs representation.
The electromagnetic charge operator is given by where the generators (Gell-Mann matrices) are normalized as Tr(T i T j ) = 1 2 δ ij . We define the normalized hypercharge operator Y N and X N as such that we have and the normalized couplings are related by where Now using Eqs. (13,21,22) we obtain Here, the SM running is described by the the SU Similarly, n 2 Y can be expressed as a function of M X , The above two relations are valid provided , which are satisfied in the cases that we shall discuss below. Furthermore, we take M X ≤ M U ≤ In Fig. 3 (left) we plot the allowed range for M X for which unification is guaran-  There will be some extra fermions and the multiplicity of the different representations are now different. It is to be noted that these states can be naturally embedded in an E (6) theory. We start with the maximal SU(2) × SU (6)  geometry to the flux breaking, we identify the different states with the different algebraic varieties, and then the intersection numbers would give us the multiplicities of the different representations. A detailed study of such E(6) F-theory GUTs [7][8][9][10] is beyond the scope of this article and we shall rather take a phenomenological approach to the problem. We consider the required representations to match the low energy phenomenological requirements.
The first step is to keep the known fermions light and also to have SU(3) c ⊗ SU(3) L ⊗ U(1) X symmetry breaking scale as low as TeV, while at the same time requiring for gauge coupling unification.
Considering the6 representation of SU (6), which contains the down antiquarks d c L with hypercharge Y = 1/3, isospin lepton doublet containing e L and ν L with Y = −1/2, and N L with Y = 0; we can get the normalization for the hypercharge from T r(Y 2 ) = 5/6n −2 Y in the notation of Eq. (19). The U(1) Y normalization defined in Eq. (19) is given by n Y = 5/3 and using Eq. (20) we obtain the U(1) X normalization given by n X = 2/ √ 3, which is below the normalizations required for a guaranteed unification in Fig. 1 and Fig. 3. However, it is still possible to obtain gauge coupling unification following the prescription in Ref. [4], and n X = 2/ √ 3, the relation between normalized couplings at the scales M Z and M X are given by Using Eq. (30) it is straightforward to obtain Finally, using Eqs. (14), (31) the above equation can be written in the form we obtain sin 2 θ w (M Z ) 0.231, which is consistent with the electroweak precision data [14].
Turning to the prediction for proton decay, we note that being a non-supersymmetric scenario the gauge d = 6 contributions for proton decay are most important here. An analysis of all SU(3) c ⊗ SU(2) L ⊗ U(1) Y invariant operators [16][17][18][19] that can induce proton decay in SU (6) is beyond the scope of this article and will be addressed in a separate communication. Here we will consider the decay mode p → e + π 0 , which is constrained by experimental searches to have a life time τ expt p ≥ 1 × 10 34 [14]. The relevant effective operators in the physical basis are given by [20,21] O(e c α , where c(e c α , d β ) = k 2 1 V 11 Here i, j, k = 1, 2, 3 are the color indices and α, β = 1, 2; (37) Now noting that M U = 10 15.5 GeV and α −1 GUT ∼ 35 in the SVS and sequential 331 models, the lifetime of the proton decay mode p → e + π 0 comes out to be 2 ∼ 10 34 yrs, which is consistent with the current experimental limit [14]. In both cases the gauge coupling unification is associated to the presence of sequential a leptonic octet at some intermediate scale between the 331 scale, which lies in the TeV range, and the unification scale. It is important to stress that the presence of the octet plays a key role in the mechanism of neutrino mass generation. In other words, the same physics that drives unification is responsible for the radiative origin of neutrino masses [4].