Three-loop neutrino masses via new massive gauge bosons from E_6 GUT

We propose an SU(3)_C\times SU(2)_L \times SU(2)_N \times U(1)_Y model arising from E_6 grand unified theory. We show that the tiny neutrino masses in this model can be generated at the three-loop involving the SU(2)_N gauge bosons. With Yukawa couplings around 0.01 or larger and TeV-scale SU(2)_N gauge bosons, we show that the neutrino oscillation data can be explained naturally by presenting a concrete benchmark set of input parameters. All new particles are around the TeV scale. Thus our model can be tested at the ongoing/future collider experiments.


I. INTRODUCTION
One of the great achievements in particle physics during the last few decades is the discovery of the neutrino oscillations [1,2], which can be explained by assuming nonzero masses of neutrinos. However, neutrinos are massless in the Standard Model (SM). Therefore, the neutrino oscillations provide a solid evidence for new physics beyond the SM.
The lightest charged particle in the SM is the electron, and its mass is at least six orders of magnitude larger than the predicted neutrino mass [3]. Thus, any new physics theory beyond the SM should explain why the neutrino masses are so tiny. Several attempts have been made in last a few decades. In the minimal SM extension, there is a unique Weinberg's dimension five operator [4] where φ (m) is scalar field and can be one or more; l is the lepton doublet; α, β, γ and δ are the SU (2) L indices; i and j are the generation indices. The f and f ′ are roughly of the order 1/M , where M is the mass scale of new physics. At the tree level, there exist only three different mechanisms to realize this operator [5]: Type-I [6][7][8][9], Type-II [10][11][12][13][14][15], and Type-III [16] see-saw mechanisms involving singlet fermion, scalar triplet, and Majorana triplet fermion, respectively, as heavy intermediate particles with the mass of the order of M . We can obtain a tiny neutrino mass by integrating out the heavy fields, which is roughly given by v is the Vacuum Expectation Value (VEV) of the scalar φ (m) . The neutrino mass is suppressed by the heavy mass scale M , which is generally close to the unification scale in Grand Unified Theory (GUT) for standard high energy seesaw models where not all the Yukawa couplings are very small. Such a high energy scale is inaccessible at experiments like LHC.
In order to get a testable new physics scale, we need a suppression mechanism different from the usual see-saw mechanisms. One such mechanism could be the radiatively generated neutrino masses [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. The suppression arises from the loop integrals and the new physics scale M is usually the TeV scale. At the n-loop order, a dimension d diagram estimates the neutrino mass as where c is a dimensionless quantity contains all the coupling constants and other mass ratios. The existing works on 3 loop masses, for example, the KNT model [25], the AKS model [29] and the Cocktail model [34], involve new particles assuming SM gauge symmetry extended by an additional discrete symmetry. In this work, we present a model with an additional SU (2) N gauge symmetry [35,36] where the gauge symmetry group SU (2) N can arise as a subgroup in the decomposition of the E 6 GUT model [37][38][39][40][41][42][43]. The particle content of the model restricts the Majorana neutrino masses to be generated below the three-loop level. The SU (2) N gauge bosons play an important role in the determination of the neutrino masses at three loops. Due to a large suppression factor, 1 16π 2 3 ∼10 −7 , arising from the loop integrals, the TeV mass scale can be the new physics scale of our model. The new gauge symmetry SU (2) N can be broken around the TeV scale, so our model can be tested at the ongoing LHC and/or HE-LHC, FCC, and SppC, etc.
This paper is organized as follows: in Section II, we present the model in details and discuss the possible Yukawa coupling terms. We study the Higgs potential and its minimization in Section III. In section IV, we calculate the masses for different scalar particles and obtain the physical states. Section V includes the details about the gauge sector of the model, such as gauge boson masses and their couplings with scalars and fermions. We obtain an analytical expression for the neutrino mass matrix in Section VI. A numerical analysis, to show the consistency of the analytical expression with the experimental data, is given in section VII and we conclude in Section VIII.

II. MODEL BUILDING
Our model can arise from the E 6 GUT. One possible maximal subgroup of E 6 is SU (6) × SU (2) N . The SU (6) group has maximal subgroup SU (5) × U (1) ′ . We assume that the U (1) ′ gauge symmetry is broken around the GUT scale. Because SU (5) group contains the SM gauge symmetry, the low energy gauge symmetry of our model is The SU (2) N has no component to the electric charge operator in our model, so the charge operator is defined as Q = T 3L + Y . We assume that the SU (2) L doublet assignments are vertical while the SU (2) N doublets are horizontal.
Under the gauge symmetry SU

the quantum numbers for the fermions are
The H d , H u , and S 0 come from 27 representation, while the bidoublet scalar T arises from the 650 representation of E 6 . The SU (2) N gauge symmetry is broken when S 0 acquires a VEV, and the electroweak gauge symmetry is broken by the VEVs of H d and H u . The Lagrangian for the Yukawa sector and vector-like mass terms are where α, β, γ and δ are SU (2) indices; i and j are generation indices; and ǫ αβ is the totally antisymmetric SU (2) tensor with ǫ 12 = +1. For simplicity, we assume M ij = M i δ ij , and µ ij = 0. N c i s are needed (and the related terms in the above Lagrangian) only if the SU (3) C × SU (2) L × SU (2) N × U (1) Y symmetry of our model has an E 6 origin. However, if we choose to work with the E 6 GUT model then we introduce a discrete Z 2 symmetry to forbid the Type I seesaw mechanism in this model. Under this Z 2 symmetry, only N c i is odd, while all the other particles are even. In such a situation, the y 7ij and y 9ij terms in Eq. (4) are forbidden and the lightest fermion of N c i can be a dark matter candidate.
Using the explicit components of the fields, we get We consider three nonzero VEVs φ 0 gives the down-type quark masses and charged lepton masses, v2 √ 2 gives the up-type quark masses, and vs √ 2 gives masses to the vector-like particle ( However, there is no neutrino mass term at tree level. a dark matter candidate.

III. THE HIGGS POTENTIAL
We need the complete Higgs potential to get the physical scalar states and their masses. The most general renormalizable scalar potential for the Higgs scalars of our model is where all the parameters are real. Here α, β, γ, δ, ρ, σ, µ and ν are the SU (2) indices and ǫ αβ is the totally antisymmetric SU (2) tensor with ǫ 12 = +1.
The minimum of the potential is given by The minimisation conditions are After H d , H u and S 0 acquire VEVs, we can write them as

IV. SCALAR MASSES
With the scalars in Eqs. (10) and (11), we can now obtain the terms in the Lagrangian density which gives masses to the different scalars from Eq. (5). The mass terms for the single charged scalars are (12) First, we get a mixing between φ ± 1 and φ ± 2 . That mixing gives four scalars h ± 1 and h ± 2 with mass squared zero and respectively. The states are where the mixing angle is given by tan β = v2 v1 . The two massless states h ± 1 are corresponding to two charged Goldstone modes, and the other two states h ± 2 are two single charged physical scalars. The scalars φ ± 3 and T ± 2 will mix and give the following four mass eigenstates with mass squared and respectively. The mixing angle is given by tan 2θ = and The four states H ± 1 and H ± 2 are identified as four single charged physical scalar. From Eq. (12) we get two more single charged physical scalar T ± 1 with mass squared The following term of the Lagrangian density gives the masses of the double charged scalars The mass matrix is already diagonalized and gives the mass squared of the four doubly charged physical scalar and respectively. Next we consider the mass terms for the five neutral scalars Here, ρ 3 and ρ 1s are the states to mix and give one neutral scalar Goldstone mode and one neutral physical scalar with mass squared equal to We get three more neutral physical scalars from the mixing of ρ 1 , ρ 2 and ρ 2s . The term below gives the masses of pseudoscalars where η 1 , η 2 and η 2s will mix and give two neutral pseudoscalars Goldstone mode and one physical neutral pseudoscalar with mass squared λ ′

v1v2vs
. η 3 and η 1s will mix and give another neutral pseudoscalar Goldstone mode and another physical neutral pseudoscalar with mass squared equal to 2 . We start with 24 scalar degrees of freedom and end up with 18 physical scalars. The other six degrees of freedom correspond to the six Goldstone mode are eaten by the massless gauge bosons. The Goldstone modes will become the longitudinal modes of gauge bosons, which will become massive. So there will be six massive gauge bosons and one massless gauge boson.

V. GAUGE BOSONS
In this Section, we discuss the gauge boson masses and physical gauge boson states, as well as their interactions with the physical scalars and fermions. The Lagrangian density, which gives the gauge boson masses and their interactions with the scalars, is where α and β are the SU (2) indices. The covariant derivative is defined as where g, g ′ 2 and g ′ are the coupling constant corresponding to the SU (2) L , SU (2) N , and U (1) Y groups respectively. W µ , W ′ µ , and B µ are the gauge bosons of the SU (2) L , SU (2) N , and U (1) Y groups respectively.

V.I. Gauge Boson Masses
We define √ 2W ± µ = W 1µ ∓ iW 2µ and √ 2X 1,2µ = W ′ 1µ ∓ iW ′ 2µ . After the spontaneous symmetry breaking of the gauge groups the massless gauge boson will become massive. We write the part of Eq. (27) that gives the masses of the gauge bosons (29) −L mass gauge = After the spontaneous symmetry breaking, B µ , W 3µ and W ′ 3µ will mix and give three physical gauge bosons, which can be written as where the mixing angles are given by, tan θ W = g ′ g and tan 2θ N = b a− . The definitions of b and a ± are There are four other physical gauge bosons, which are W ± µ and X 1,2µ . The mass squared of all the physical gauge bosons are then given by, Also, there are exactly six massive gauge bosons corresponding to six Goldstone modes.

V.II. Gauge Bosons Interactions
Next, we study the interactions between the physical gauge bosons and scalars. A few important terms in Eq.
We rewrite Eq. (40) in terms of the physical scalars using Eqs. (13)- (16), and then derive the necessary Feynman rules for the interactions in the following (41) Next, we consider the kinetic energy terms of the L i leptons. These terms give us the interactions of the leptons with the gauge bosons. Let us first write down the kinetic term where i is the generation index; α and β are SU (2) index; and a = 1,2,3. The Eq. (42) will give us the important interaction term between the neutral leptons and gauge bosons X µ 1,2 as below We need one more interaction term which will play an important in the next section. We write the Yukawa sector, given by Eq. (4), in terms of the physical scalar and fermions. We write all the relevant terms here:

VI. NEUTRINO MASSES
We obtain an analytical expression for the neutrino mass matrix elements in this Section. As mentioned before, the particle content of our model does not allow us to generate the neutrino masses below three loop, thus, the leading contributions to neutrino masses arise from the three loop diagrams shown in Fig. 1. The new gauge bosons, X 1 and X 2 , are responsible for these three loop diagrams 1 . We have all the necessary physical particle masses and the interaction terms to calculate the three loop diagram in Fig. 1. In unitary gauge, the Majorana mass matrix elements are given by (45) (M ν ) ji = 1 4 g ′ 2 4 y 1jl y 2li sin 2θ sin 2 β × I 3loop , 1 We have used the package TikZ-Feynman [50] to draw the diagram.
where i, j , l = 1,2,3. And I 3loop is the three-loop integral given by 2 The definitions of the integral functions appeared in Eq. (46) are The mass matrix elements get large suppression from g ′ 2 4 (16π 2 ) 3 ∼ 10 −11 , which pushes the new scale to TeV. The numerical analysis depends on the choice of various parameters in the model, particularly the contribution from the gauge bosons X 1,2 will be very crucial.

VII. NUMERIAL ANALYSIS
In this Section, we show that the neutrino mass matrix given by the Eq. (45) can fit the neutrino oscillation data. We only consider the normal hierarchy of the neutrino masses. The discussion of the inverted hierarchy case will be similar. The best fit of the neutrino oscillation data for normal hierarchy at 3σ range [46] We define the matrix M dν = diag(m 1 , m 2 , m 3 ) as the diagonalised neutrino mass matrix. In the normal hierarchy scenario, the oscillation data correspond to m 1 < m 2 < m 3 . In the simplest scenario, the lightest neutrino can be assumed to be massless. We take the neutrino mass eigenvalues as follows, We then obtain the Majorana mass matrix from the M dν matrix as where U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [47,48] where c ab ≡ cos θ ab and s ab ≡ sin θ ab . P is the unit matrix for Dirac neutrinos or a diagonal matrix with two phase angles for Majorana neutrinos. We take both the Majorana phase angles to be zero, and the central values of the parameters from Eq. (53). Without loss of generality, we choose δ CP to be zero as well. We can now obtain the mass matrix from Eq.
We have the constraint v 2 1 + v 2 2 ≃ 246 2 GeV 2 and tan β = v2 v1 . The v 1 and v 2 are also constrained by the top and bottom quark masses. The small values of tan β will give large value of the top quark Yukawa coupling (y 4 ) 33 , which will make the model non-perturbative. To avoid that, we take tan β = 2, which gives us the mixing angle β to be equal to 63 • . We then get v1 at the TeV scale, to be 17 TeV. Now choosing g ′ 2 to be equal to 0.35, we get the mass of the gauge boson X to be 5 TeV. By choosing appropriate values for different λ parameters in Eqs. (19) and (20), we can take m 2 2 and m 2 3 to be 2.5 ×10 7 GeV 2 and 2.5 × 10 5 GeV 2 respectively. Now choosing λ ≃ .03, we obtain m H1 and m H2 to be 5 TeV and 500 GeV respectively from Eqs. (17) and (18) using the mixing angle, θ= 0.005 • . Similarly, Eq. (21) gives m T1 to be 500 GeV. The other mass parameters needed are the vector-like particle masses. The lower bound on the vector-like lepton mass is 101 GeV, which comes from the LEP experiment [49]. We take m 0 to be 115 GeV, 125 GeV, and 135 GeV respectively for the first, second, and third generations. We use M, the mass for the XE particle to be 110 GeV, 120 GeV, and 130 Gev respectively for the three generations. All parameters are as follows, We can use these parameters in Eq. (45) to fit the neutrino mass matrix given in Eq. (57) for Yukawa couplings that satisfy y 1 × y 2 to be of the order of 0.001 to 0.0001. We have presented one set of viable input parameters. There exist many other possible sets as well. The Yukawa coupling constants can be made even larger by taking larger values of vs √ 2 . The mass m X controls the value of the numerical integration. The other mass parameters do not play an important role in the calculations. The mass gap between m H1 and m H2 can be small, which does not affect the numerical result in any significant way. Another important factor, which affects the value of the numerical result, is the loop suppression factor. The value of tan β can change the numerical results as well.

VIII. CONCLUSION
To construct a natural radiative neutrino mass model which can be tested at the future collider experiments, we have extended the SM gauge symmetry by the SU (2) N gauge group, which comes from the decomposition of the E 6 GUT. We have presented the particle content and all the possible Yukawa interactions and studied the scalar and gauge sectors in details. Interestingly, the tiny neutrino masses are found to be only generated at three loops where the SU (2) N gauge bosons play an important role. The new gauge bosons X 1,2 and vector-like fermions enter into the three loop diagrams. Because of the large suppression from the loop integral, the new physics scale can be around TeV, which is testable, unlike the high-scale tree-level see-saw mechanism, as well as the one-loop and two-loop neutrino mass models.
We have obtained an analytical expression for the Majorana neutrino masses. This mass expression depends on the spontaneous symmetry breaking scale of the SU (2) N gauge group. From the three loop calculation, we have shown that the analytical expression, in our radiative neutrino mass model, is consistent with the neutrino oscillation data. For example, for vs √ 2 to be 17 TeV and the new gauge boson mass to be 5 TeV. The other mass parameters are chosen to be between the electroweak and TeV scale, which is consistent with our goal of obtaining neutrino mass at experimentally testable scale. Using these input parameters along with the neutrino mass matrix obtained from the oscillation data, we found the Yukawa couplings to be 0.01 or larger. For larger values of vs √ 2 the Yukawa couplings will be larger.