Renormalizable Adjoint SU(5)

We investigate the possibility to find the simplest renormalizable grand unified theory based on the SU(5) gauge symmetry. We find that it is possible to generate all fermion masses with only two Higgs bosons, 5_H and 45_H. In this context the neutrino masses are generated through the type III and type I seesaw mechanisms. The predictions coming from the unification of gauge couplings and the stability of the proton are discussed in detail. In this theory the leptogenesis mechanism can be realized through the out of equilibrium decays of the fermions in the adjoint representation.


I. INTRODUCTION
The possibility to unify all fundamental interactions in nature is one of the main motivations for physics beyond the Standard Model (SM). The so-called grand unified theories (GUTs) are considered as one of the most natural extensions of the Standard Model where this dream is partially realized. Two generic predictions of those theories are the unification of gauge interactions at the high scale, M GUT ≈ 10 14−16 GeV, and the decay of the lightest baryon [1], the proton, which unfortunately still has not been observed in the experiments.
The first grand unified theory was proposed by Georgi and Glashow in reference [2]. As is well known this model, based on SU (5) gauge symmetry, has been considered as the simplest grand unified theory. It offers partial matter unification of one Standard Model family in the anti-fundamental 5 and antisymmetric 10 representations. The Higgs sector is composed of 24 H and 5 H . The GUT symmetry is broken down to the Standard Model by the vacuum expectation value (VEV) of the Higgs singlet field in 24 H , while the SM Higgs resides in 5 H . The beauty of the model is undeniable, but the model itself is not realistic. This model is ruled out for three reasons: the unification of the gauge couplings is in disagreement with the values of α em , sin 2 θ W and α s at the electroweak scale, the neutrinos are massless and the unification of the Yukawa couplings of charged leptons and down quarks at the high scale in the renormalizable model is in disagreement with the experiments.
Recently, several efforts has been made in order to define the simplest realistic extension of the Georgi-Glashow model. The simplest realistic grand unified theory with the Standard Model matter content was pointed out in reference [3] where the 15 H has been used to generate neutrino masses and achieve unification. For different phenomenological and cosmological aspects of this proposal see references [4], [5], [6], and [7]. This theory predicts for the first time the existence of light scalar leptoquarks and that the upper bound on the proton lifetime is τ p 2×10 36 years. Therefore, this realistic grand unified theory could be tested at future collider experiments, particularly at LHC, through the production of scalar leptoquarks and at next generation of proton decay experiments. Now, if we extend the Georgi-Glashow model adding extra matter, there is a realistic grand unified model where the extra matter is in the 24 representation. This possibility has been proposed recently by Bajc and Senjanović in reference [8]. In this scenario using higher-dimensional operators the neutrino masses are generated through the type I [9] and type III [10] seesaw mechanisms. In this case the theory predicts a light fermionic SU (2) L triplet [8] which is responsible for type III seesaw. See references [8], and [11] for more details.
The type III seesaw mechanism has been proposed for the first time in reference [10]. In this case adding at least two fermionic SU (2) L triplets with zero U (1) Y hypercharge the effective dimension five operator relevant for neutrino masses are generated once the neutral components of the fermionic triplets are integrated out [10]. In the context of grand unification Ma studied for the first time the implementation of this mechanism in SUSY SU (5) [12]. In this case a fermionic chiral matter superfield in the 24 representation has to be introduced [12] and the neutrino masses are generated through type I and type III seesaw mechanisms since in the 24 representation one has the fermionic triplet responsible for type III seesaw and a singlet responsible for type I seesaw. Therefore, if we want to realize the type III seesaw mechanism one must introduce extra matter in the adjoint representation. The implementation of this mechanism in non-SUSY SU (5) has been understood in reference [8]. In this case they have introduced extra matter in the 24 representation and use higher-dimensional operators in order to generate at least two massive neutrinos and a consistent relation between the masses of charged leptons and down quarks [8].
The models mentioned above include the whole set of higher dimensional operators in order to have a consistent relation between the Yukawa couplings at the unification scale. In this work we want to stick to the renormalizability principle and focus our attention on renormalizable extensions of the Georgi-Gashow model. Following the results presented in references [12] and [8] we investigate the possibility to write down the simplest renormalizable grand unified theory based on the SU (5) gauge symmetry. We find that it is possible to generate all fermion masses at the renormalizable level , including the neutrino masses, with the minimal number of Higgs bosons: 5 H and 45 H . The implementation of the leptogenesis mechanism [13] is possible. In this model the leptogenesis mechanism can be realized through the out of equilibrium decays of the fermions in the adjoint representation. Notice that in the model proposed in reference [8] only resonant leptogenesis could be possible since the fermionic triplet is very light. As we will show in the next section there is no problem to satisfy the experimental lower bounds on the proton decay lifetime. We propose a new renormalizable grand unified theory based on the SU (5) gauge symmetry with extra matter in the adjoint representation. We refer to this theory as "Renormalizable Adjoint SU (5)". The model proposed in this letter can be considered as the renormalizable version of the model given in reference [8] and is one of the most appealing candidates for the unification of the Standard Model interactions at the renormalizable level. In the next sections we discuss some of the most relevant phenomenological aspects of this proposal.

II. RENORMALIZABLE ADJOINT SU (5)
In order to write down a realistic grand unified theory we have to be sure that all constraints coming from the unification of gauge couplings, fermion masses and proton decay can be satisfied. In this letter we stick to the simplest unified gauge group, SU (5), and to the renormalizability principle. Now, if we want to have a consistent relation between the masses of charged leptons and down quarks at the renormalizable level we have to introduce the 45 H representation [14]. Therefore, our Higgs sector must be composed of 24 and the masses for charged leptons and down quarks are given by: where 5 H = v 5 . Y 1 and Y 2 are arbitrary 3 × 3 matrices. Notice that there are clearly enough parameters in the Yukawa sector to fit all charged fermions masses. See reference [15] for the study of the scalar potential and [5] for the relation between the fermion masses at the high scale which is in agreement with the experiment. There are three different possibilities to generate the neutrino masses [12] at tree level in this context. The model can be extended in three different ways: i) we can add at least two fermionic SU (5) singlets and generate neutrino masses through the type I seesaw mechanism [9], ii) we can add a 15 of Higgs and use the type II seesaw [16] mechanism, or iii) we can generate neutrino masses through the type III [10] and type I seesaw mechanisms adding at least two extra matter fields in the 24 representation [12]. In reference [8] it has been realized the possibility to generated the neutrino masses through type III and type I seesaw adding just one extra matter field in 24 and using higher-dimensional operators. Notice that the third possibility mentioned above is very appealing since we do not have to introduce SU (5) singlets or an extra Higgs. If we add an extra Higgs, 15 H , for type II seesaw mechanism the Higgs sector is even more complicated. In this letter we focus on the possibility to generate the neutrino masses at the renormalizable level through type III and type I seesaw mechanisms.
The predictions coming from the unification of the gauge couplings in a renormalizable SU (5) model where one uses type I or type II seesaw mechanism for neutrino masses were investigated in reference [17]. However, a renormalizable grand unified theory based on SU (5) where the neutrino masses are generated through the type III seesaw mechanism has not been proposed and this is our main task. The SM decomposition of the needed extra multiplet for type III seesaw is given by: 24 = (ρ 8 , ρ 3 , ρ (3,2) , ρ (3,2) , ρ 0 ) = (8, 1, 0) (1, 3, 0) (3, 2, −5/6) (3, 2, 5/6) (1, 1, 0). In our notation ρ 3 and ρ 0 are the SU (2) L triplet responsible for type III seesaw and the singlet responsible for type I seesaw, respectively. Since we have introduced an extra Higgs 45 H and an extra matter multiplet 24, the Higgs sector of our model is composed of 5 H , 24 H and 45 H , and the matter is unified in the5, 10 and 24 representations.
The new relevant interactions for neutrino masses in this context are given by: Notice from Eq. (1) and Eq. (4) the possibility to generate all fermion masses, including the neutrino masses, with only two Higgses : 5 H and 45 H . The first term in the above equation has been used in reference [12] in the context of SUSY SU (5) and in reference [8] in the context of non-SUSY SU (5). Notice that the main difference at this level of our model with the model presented in reference [8] is that we do not need to use higher-dimensional operators and with only two Higgses we can generate all fermion masses. Notice that in SU (5) models usually that is not possible.
The masses of the fields responsible for the seesaw mechanisms are computed using the new interactions between 24 and 24 H in this model: Once 24 H gets the expectation value, 24 H = v diag(2, 2, 2, −3, −3)/ √ 30, the masses of the fields living in 24 are given by: where we have used the relations M V = v 5πα GUT /3, λ = λ/ √ 50π and chose M V as the unification scale. Notice that when the fermionic triplet ρ 3 , responsible for type III seesaw mechanism, is very light the rest of the fields living in 24 have to be heavy if we do not assume a very small value for the λ parameter.
Before study the unification constraints and discuss the different contributions to proton decay let us summarize our results. We have found that it is possible to write down a renormalizable non-supersymmetric grand unified theory based on the SU (5) gauge symmetry where the neutrino masses are generated through type I and type III seesaw mechanisms using just two Higgses 5 H and 45 H . In this context, as in the model proposed in reference [8], the implementation of leptogenesis is possible. However, in their case one could have only resonant leptogenesis. Those issues will be discussed in detail in a future publication.

III. UNIFICATION CONSTRAINTS AND NUCLEON DECAY
In order to understand the constraints coming from the unification of gauge couplings we can use the B-test relations: B 23 /B 12 = 0.716 ± 0.005 and ln M GUT /M Z = (184.9 ± 0.2)/B 12 , where the coefficients B ij = B i − B j and B i = b i + I b iI r I are the so-called effective coefficients. Here b iI are the appropriate one-loop coefficients of the particle I and r I = (ln M GUT /M I )/(ln M GUT /M Z ) (0 ≤ r I ≤ 1) is its "running weight" [19]. To obtain the above expressions we have used the following experimental values at M Z in the M S scheme [18]: sin 2 θ W (M Z ) = 0.23120 ± 0.00015, α −1 em (M Z ) = 127.906 ± 0.019 and α s (M Z ) = 0.1176 ± 0.002. In the rest of the paper we will use the central values for input parameters in order to understand the possible predictions coming from the unification of gauge interactions.
As is well known the B-test fails badly in the Standard Model case since B SM 23 /B SM 12 = 0.53, and hence the need for extra light particles with suitable B ij coefficients to bring the value of the B 23 /B 12 ratio in agreement with its experimental value. In order to understand this issue we compute and list the B ij coefficients of the different fields in our model in Tables I, II, and III. Notice that we have chosen the mass of the superheavy gauge bosons as the unification scale. From the tables we see clearly that Σ 3 , Φ 3 and ρ 3 fields improve unification with respect to the Standard Model case since those fields have a negative and positive contribution to the coefficients B 12 and B 23 , respectively.    Before we study the different scenarios in agreement with the unification of gauge interactions let us discuss the different contributions to proton decay. For a review on proton decay see [20]. In this model there are five multiplets that mediate proton decay. These are the superheavy gauge bosons V = (3, 2, −5/6) (3, 2, 5/6), the SU (3) triplet T , Φ 3 , Φ 5 and Φ 6 . The least model dependent and usually the dominant proton decay contribution in non-supersymmetric scenarios comes from gauge boson mediation. Its strength is set by M V and α GUT . Notice that we have identified M V with the GUT scale, i.e., we set M V ≡ M GUT . We are clearly interested in the regime where M V is above the experimentally established bounds set by proton decay, M V (2 × 10 15 ) 5 × 10 13 GeV if we do (not) neglect the fermion mixings [21].
In this theory the value of M GUT depends primarily on the masses of Σ 3 , ρ 3 , Φ 1 and Φ 3 through their negative contributions to the B 12 coefficient. The Φ 3 field cannot be very light due to proton decay constraints. The Φ 3 contributions to proton decay are coming from interactions Y 4 Q T ıσ 2 Φ 3 Q and Y 2 Q T ıσ 2 Φ * 3 L. The field Φ 3 should be heavier than 10 11 GeV in order to not conflict experimental data. Of course, this rather naive estimate holds if one assumes most natural values for Yukawa couplings. If for some reasons one of the two couplings is absent or suppressed the bound on Φ 3 would cease to exist. For example, if we choose Y 4 to be an anti-symmetric matrix, the coupling Y 4 Q T ıσ 2 Φ 3 Q vanishes. Therefore, Φ 3 could be very light. In general the field Σ 3 could be between the electroweak and the GUT scales, while ρ 3 has to be always below the seesaw scale, M ρ3 10 14 GeV. Let us study several scenarios where the unification constraints are quite different: • The first scenario corresponds to the case when Σ 3 is at GUT scale, while Φ 3 and/or ρ 3 could be below the unification scale. The rest of the fields are at the unification scale. Using the B ij coefficients listed in Tables. I-III we find that if Φ 3 is at GUT scale it is possible to achieve unification at It is important to know which is the minimal value for the mass of the fermionic triplet, responsible for type III seesaw, consistent with unification. The minimal value of M ρ3 corresponds to the case when Σ 3 and Φ 3 are at the GUT scale, while Φ 1 and Σ 8 are close to the electroweak scale. In this case M ρ3 ≈ 1.5 TeV and the unification scale is 3 × 10 16 GeV. Therefore, we can conclude that in this case the seesaw mechanism could be tested at Notice that in all the scenarios studied in this section we can satisfy the constraints coming from proton decay and neutrino masses. The theory proposed in this letter can be considered as the renormalizable version of the theory given in reference [8]. As we have discussed in this letter, the predictions coming from proton decay, the unification constraints and leptogenesis are quite different in this case. Our theory can be considered as the simplest renormalizable grand unified theory based on the SU(5) gauge symmetry.

IV. SUMMARY
We have investigated the possibility to find the simplest renormalizable grand unified theory based on the SU(5) gauge symmetry. We find that it is possible to generate all fermion masses with only two Higgs bosons, 5 H and 45 H . In this context the neutrino masses are generated through the type III and type I seesaw mechanisms. The predictions coming from the unification of gauge couplings and the stability of the proton have been discussed in detail. In this theory the leptogenesis mechanism can be realized through the out of equilibrium decays of the fermions ρ 3 and ρ 0 in the adjoint representation. We refer to this theory as "Renormalizable Adjoint SU(5)".