Towards Minimal SU(5)

We propose a simple SU(5) model that connects the neutrino mass generation mechanism to the observed disparity between the masses of charged leptons and downtype quarks. The model is built out of 5-, 10-, 15-, 24-, and 35-dimensional representations of SU(5) and comprises two (three) 3 × 3 (3 × 1) Yukawa coupling matrices to accommodate all experimentally measured fermion masses and mixing parameters. The gauge coupling unification considerations, coupled with phenomenological constraints inferred from experiments that probe neutrino masses and mixing parameters and/or look for proton decay, fix all relevant mass scales of the model. The proposed scenario places several multiplets at the scales potentially accessible at the LHC and future colliders and correlates this feature with the gauge boson mediated proton decay signatures. It also predicts that one neutrino is massless. ∗ E-mail: dorsner@fesb.hr † E-mail: shaikh.saad@okstate.edu ar X iv :1 91 0. 09 00 8v 1 [ he pph ] 2 0 O ct 2 01 9


Introduction
The promising proposal of unification of the Standard Model (SM) matter fields and their interactions using SU (5) group as the supporting gauge structure has been around for more than four decades [1]. The initial efforts, naturally, have been devoted to understanding of what turned out to be a finite number of possible ways to generate masses for the SM charged fermions [2,3]. The subsequent need to accommodate the neutrino mass generation mechanism, on the other hand, has revealed that there are many potentially viable paths that could be taken and the majority of the studies within the SU (5) theory framework, over the last two decades, has been focused on various ways of implementing it [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
We combine, in this manuscript, several known mechanisms of mass generation for both the charged and neutral fermions of the SM within the SU (5) framework to produce unexpectedly economical and predictive model that might take us a step closer towards minimal SU (5). The scalar sector of our proposal contains only three representations, whereas the fermion sector, besides the SM fields, contains only one vector-like representation. In addition to that the proposal allows for only two (three) 3 × 3 (1 × 3) Yukawa coupling matrices that need to accommodate all experimentally measured fermion masses and mixing parameters, where one of the 3 × 3 matrices can be taken, without the loss of generality, to be diagonal and real whereas the other 3 × 3 matrix is symmetric. Finally, the gauge coupling unification analysis, coupled with the need to accommodate experimentally observed parameters in the neutrino sector, constrains all relevant mass scales of the model to reside within very narrow ranges of viability.
The manuscript is organized as follows. We specify particle content of our proposal in Sec. 2. There we also discuss the particularities of the gauge coupling unification, spell out the connection between different mass generation mechanisms, and present numerical study that showcases the phenomenological viability of the proposal. We conclude in Sec. 3.

Particle content
We propose an extension of the original Georgi-Glashow (GG) model that connects the neutrino mass generation to the observed disparity between the masses of charged leptons and down-type quarks. The scalar sector of the GG model comprises one 5-dimensional and one 24-dimensional representation. We add to it a single 35-dimensional representation. The decomposition of these representations under the SM gauge group SU (3) × SU (2) × U (1) is

Gauge coupling unification
We want to demonstrate that the model generates gauge coupling unification at the one-loop level. The introduction of 35 H and 15 F +15 F is crucial in that regard since the SM multiplets in 24 H and 5 H cannot produce viable unification on their own. There are, however, two mass relations between the multiplets in 35 H and in 15 F + 15 F that one needs to take into account in order to perform a proper unification analysis as we discuss next.
It is well-known that the masses of φ 1 ∈ 24 H and φ 8 ∈ 24 H can be treated as arbitrary parameters, as far as the gauge coupling unification is concerned, since the potential contains enough non-trivial contractions of 24 H with itself to allow for that to happen. The model can also accommodate the splitting between the masses of Λ 1 ∈ 5 H and Λ 3 ∈ 5 H , where we take that M Λ 3 ≥ 3 × 10 11 GeV in order to satisfy experimental bounds on the partial proton decay lifetimes [22] while the mass of Λ 1 is the mass of the SM Higgs field, i.e., M Λ 1 ≡ M H .
The interaction of 15 F + 15 F with 24 H can induce mass splitting between fermions in 15-dimensional representation, which is a rank 2 symmetric tensor. Namely, the two where α, β, γ(= 1, 2, 3, 4, 5) are the SU (5) indices, yield 3) The masses of multiplets in 35 H , which is a rank 3 completely symmetric tensor, are determined by the following interactions and the squares of the masses read The last equality is the other mass relation that we need to use in our unification analysis.
To summarise, relevant degrees of freedom for the gauge coupling unification considera- The gauge coupling unification analysis reveals that the field Φ 1 (1, 4, −3/2) (Σ 1 (1, 3, 1)) prefers to be very heavy (light) if M GUT is to be sufficiently large. But, for the reasons that will become clear when we discuss the neutrino mass generation mechanisms, the masses of Φ 1 and Σ 1 should be of the same order of magnitude if one is to have potentially viable scenario. Also, Φ 1 cannot be too heavy since the neutrino masses would not come out right. Due to these conflicting needs we are left with rather limited parameter space where one can simultaneously address experimental results on proton decay lifetimes and neutrino masses in a proper manner. We accordingly present, in Fig. 1, the gauge coupling unification scenario that corresponds to the case when M Σ 1 = 10 11 GeV, M Φ 1 = 10 12 GeV, and M GUT = 7.7 × 10 15 GeV. We use that particular unification scenario in our numerical study of the fermion masses and mixing parameters to showcase viability of our proposal.
The vertical lines in Fig. 1 correspond to the mass scales of the relevant multiplets that we explicitly specify for clarity of exposition.

Neutrino mass generation
The SU (5) contractions that generate contributions towards Majorana neutrino masses read where Y a and Y b are arbitrary 1 × 3 Yukawa coupling matrices and λ is a dimensionless parameter. The neutrino mass contributions are generated both at the tree level via d = 7 operator and at the one-loop level via d = 5 operator [24]. The Feynman diagram for the former (latter) contribution is shown in the left (right) panel of Fig. 2.
This, in turn, is the right neutrino mass scale if the entries in Y a and Y b are O(1) parameters.
The tree level contribution for the unification scenario under consideration is very suppressed compared to the one-loop level one. Namely, we find that the prefactor of the term GeV. Note that one neutrino is always massless in our proposal since det(M ν ) = 0.

Charged fermion masses
The contractions that are relevant for the charged fermion mass generation read (2.14) Here, we use the freedom to freely rotate 10 F i and 5 F j in an independent manner to go into the basis where Y d is a real and diagonal matrix. Y u is a symmetric 3 × 3 complex matrix whereas Y c is a 3 × 1 complex matrix. The contraction proportional to the matrix Y a also appears in Eq.  (1, 2)), where the second number in the parentheses represents electric charge in units of absolute value of the electron charge. The mixing between the fermions in 10 F i and 15 F +15 F that appears as a result of the SU (5) symmetry breaking reads L ⊃ − 1 4 whereas the electroweak symmetry breaking induces the following mixings among the fermions

17)
where we define dimensionless parameter δ ≡ 5/3v 24 /(4M Σ 3 ) and take v H to be real. The gauge coupling unification scenario we present in Fig. 1 yields δ = 2.49 × 10 4 . Note that without the mixing one retrieves the GG prediction M e = M T d , at the unification scale, that is in conflict with experimental observations after the measured values of the down-type quarks and charged leptons are evolved to the GUT scale.
The neutrino mass matrix elements, again, are To summarise, our proposal uses one vector-like 15-dimensional representation that simultaneously generates neutrino masses with the aid of one 35-dimensional scalar representation and creates viable mismatch between the masses of the down-type quarks and charged leptons. There are only two (three) 3 × 3 (3 × 1) Yukawa coupling matrices Y u and Y d (Y a , Y b , and Y c ) to accommodate experimentally measured fermion masses and mixing parameters. Moreover, one can go, without the loss of generality, into a basis where Y d is diagonal and real matrix while Y u is a symmetric matrix. Note that the dimensionless parameter y is, strictly speaking, Yukawa coupling but the unification considerations imply that it can be neglected for all practical purposes.

Numerical analysis
We perform, in this section, a numerical fit of the SM fermion masses and mixing parameters that corresponds to the unification scenario of Fig. 1 to demonstrate viability of our proposal.
The fermion mass matrices are given in Eqs.  Table I.
Since the down-type quark and neutrino mass matrices share the same Yukawa couplings Y a i , i = 1, 2, 3, we perform a combined fit of these two sectors to reproduce the correct down-type quark masses and neutrino observables. We obtain the following fit parameters where we normalise Y a 3 and Y b 3 to 1. To be consistent with our unification scenario and this normalisation the overall scale for the neutrino mass matrix is fixed to be m 0 = 9.28 × 10 −12 GeV that, in turn, requires λ = 0.239. We summarize the best fit values in Tables I   and II. Clearly, all the observables can be fitted to their experimentally measured central values given in Tables I and II while one neutrino is predicted to be massless.
The Yukawa couplings in Y a,b,c are, in general, complex numbers but we have, for simplicity, taken them to be real. Note that the up-type quark mass matrix is proportional to the complex symmetric matrix Y u . This provides enough freedom for one to simulta-  8.56 Table II: The fit input for the neutrino observables [27] and the corresponding fit output.

Conclusion
We propose a simple SU (5) model that relates the neutrino mass generation mechanism to the observed disparity between the masses of charged leptons and down-type quarks. The entire structure of the model is based on the first five non-trivial SU (5)  The gauge coupling unification considerations, coupled with phenomenological constraints inferred from experiments that probe neutrino masses and mixing parameters and/or look for proton decay, fix all relevant mass scales of the model. The proposed scenario places several multiplets with non-trivial assignments under SU (3) and/or SU (2) at the scales potentially accessible at the LHC and future colliders and correlates this feature with the gauge boson mediated proton decay signatures. Two particular decay channels, i.e., p → K + ν and p → π + ν, depend only on the scale of unification and the SM parameters. The model contains only one scalar leptoquark whose contribution towards proton decay is negligible.
It also predicts the existence of one massless neutrino.