Scotogenic neutrino masses with gauged matter parity and gauge coupling unification

JHEP03(2022)034 Published for SISSA by Springer Received:September 18, 2021 Revised:January 26, 2022 Accepted:February 17, 2022 Published:March 7, 2022 Scotogenic neutrino masses with gauged matter parity and gauge coupling unification A.E. Cárcamo Hernández,a,b,c Chandan Hati,dSergey Kovalenko,e,b,c José W.F. Vallef and Carlos A. Vaquera-Araujog,h,i aUniversidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile bCentro Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile cMillennium Institute for Subatomic physics at high energy frontier — SAPHIR, Fernandez Concha 700, Santiago, Chile dPhysik Department T70, Technische Universität München, James-Franck-Straße 1, D-85748 Garching, Germany eDepartamento de Ciencias Físicas, Universidad Andres Bello, Sazié 2212, Piso 7, Santiago, Chile fAHEP Group, Institut de Física Corpuscular — CSIC/Universitat de València, Parc Científic de Paterna, C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia), Spain gDepartamento de Física, DCI, Campus León, Universidad de Guanajuato, Loma del Bosque 103, Lomas del Campestre C.P. 37150, León, Guanajuato, Mexico hConsejo Nacional de Ciencia y Tecnología, Avenida Insurgentes Sur 1582. Colonia Crédito Constructor, Alcaldía Benito Juárez, C.P. 03940, Ciudad de México, Mexico iDual CP Institute of High Energy Physics, C.P. 28045, Colima, Mexico E-mail: [email protected],[email protected], [email protected],[email protected],[email protected] Abstract: Building up on previous work we propose a Dark Matter (DM) model with gauged matter parity and dynamical gauge coupling unification, driven by the same physics responsible for scotogenic neutrino mass generation. Our construction is based on the extended gauge group SU(3)c⊗SU(3)L⊗U(1)X⊗U(1)N, whose spontaneous breaking leaves a residual conserved matter parity, MP, stabilizing the DM particle candidates of the model. The key role is played by Majorana SU(3)L-octet leptons, allowing the successful gauge coupling unification and a one-loop scotogenic neutrino mass generation. Theoretical consistency allows for a plethora of new particles at the .O(10) TeV scale, hence accessible to future collider and low-energy experiments. Keywords: Beyond Standard Model, Gauge Symmetry, Neutrino Physics ArXiv ePrint: 2109.05029 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP03(2022)034 JHEP03(2022)034 Contents 1 Introduction 1 2 The model setup 2 2.1 Field content 2 2.2 Majorana octet lepton fields 4 3 Symmetry breaking 6 4 Yukawa sector 9 4.1 Scotogenic neutrino masses 10 5 WIMP scotogenic dark matter 12 6 Gauge coupling unification 14 7 Summary and outlook 20 1 Introduction The supersymmetric approach to gauge coupling unification has so far not been vindicated experimentally, neither at colliders, nor through the observation of proton decay [1]. However, the historical discovery of neutrino oscillations and the growing evidence for a weakly interacting massive particle (WIMP) as, perhaps, the most viable candidate for cosmological Dark Matter motivate us to seek new roads to unification. We start from the phenomenologically safe foundations provided by the Standard Model (SM) with the gauge group SU(3)c⊗SU(2)L⊗U(1)Yextended to SU(3)c⊗SU(3)L⊗U(1)X⊗U(1)N. The use of SU(3)Las an extended electroweak symmetry has been advocated by its ability to explain the observed number of families through the anomaly cancellation requirement [2,3]. Moreover, a suggestion was made within the original SU(3)c⊗SU(3)L⊗U(1) framework proposed in [2] of how neutrino masses and gauge coupling unification could emerge together [4], so that the physics responsible for small neutrino masses could also drive the unification of the gauge couplings. This model provides a radiative seesaw mechanism for calculable neutrino masses, arising from quantum corrections mediated by new SU(3)c⊗SU(3)L⊗U(1) gauge bosons. Apart from its somewhat ad hoc nature, the model lacked an explanation for cosmological dark matter, another major drawback of particle physics. The key property of any dark matter candidate is its stability on cosmological time scales, suggesting the existence of a (nearly) preserved stabilizing symmetry. Recently it has been proposed that the latter could be a discrete residual matter parity symmetry, surviving the spontaneous breaking of the extended gauge symmetry [5–7]. This happens – 1 – JHEP03(2022)034 within a simple U(1)-extension of the gauge symmetry, allowing the implementation of a conserved matter parity MP= (−1)3(B−L)+2s.(1.1) analogous to the R-parity in supersymmetric theories. In this paper we put together all these attractive features, proposing a theory of calculable scotogenic Majorana neutrino masses [8] in the SU(3)c⊗SU(3)L⊗U(1)X⊗U(1)N framework, or 3-3-1-1, for short. Dark matter is a weakly interacting massive particle (WIMP) that mediates neutrino mass generation. Gauge couplings unify for a 3-3-1 scale just above the TeV range, making the model directly testable at the LHC. For that we introduce an SU(3)LMajorana octet, responsible for both neutrino mass generation and gauge coupling unification. The paper is organized as follows. In section 2we present the model setup, while in section 3we discuss the scalar sector and symmetry breaking. In section 4we analyse the neutrino mass matrix and the scotogenic mechanism. Dark matter and gauge coupling unification are discussed in sections 5and 6, respectively. A brief outlook is given in section 7. 2 The model setup 2.1 Field content We study a 3-3-1-1 model based on the SU(3)c⊗SU(3)L⊗U(1)X⊗U(1)Ngauge symmetry introduced in [7] and the implementation of a conserved matter parity according to eq. (1.1). This is very much analogous to the R-parity symmetry imposed on the supersymmetric theories. Except that the MPdiscrete symmetry is not imposed ad hoc as a global symmetry, but rather arises as a remnant of the spontaneously broken B−Lgauge group [5–7]. It leads to the stability of the lightest MP-odd particle, and hence to a potentially viable WIMP dark matter candidate. In our model, electric charge Qand the B−L generators are embedded into the gauge symmetry as Q=T3−T8 √3+X, (2.1) B−L=−2 √3T8+N, (2.2) with Ti(i= 1,2,3,...,8),Xand Nbeing the respective generators of SU(3)L,U(1)Xand U(1)N. The lepton sector of the model includes three leptonic triplets1 laL =     νa ea Na     L ,(2.3) 1Notice that in the original 3-3-1 formulation [2], left-handed leptons transform as anti-triplets of SU(3)L. This configuration can be recovered by exchanging all triplets and anti-triplets in the present work. – 2 – JHEP03(2022)034 Field SU(3)cSU(3)LU(1)XU(1)NQ MP= (−1)3(B−L)+2s qiL 330 0 (−1 3,2 3,−1 3)T(+ + −)T q3L3 3 1 3 2 3(2 3,−1 3,2 3)T(+ + −)T uaR 3 1 2 3 1 3 2 3+ daR 3 1 −1 3 1 3−1 3+ U3R3 1 2 3 4 3 2 3− DiR 3 1 −1 3−2 3−1 3− laL 1 3 −1 3−2 3(0,−1,0)T(+ + −)T eaR 1 1 −1−1−1 + νiR 1 1 0−4 0 − ν3R1 1 0 5 0 + ΩaL 1 8 0 0      010 −1 0 −1 010          − − + − − + + + −      η1 3 −1 3 1 3(0,−1,0)T(+ + −)T ρ1 3 2 3 1 3(1,0,1)T(+ + −)T χ1 3 −1 3−2 3(0,−1,0)T(−−+)T φ1 1 0 2 0 + σ1 1 0 1 0 − Table 1. 3311 model field content (a= 1,2,3and i= 1,2are family indices). Note the nonstandard charges of the νR. each with its new neutral lepton NaL,a= 1,2,3. The quark fields are arranged as qiL =     di −ui Di     L q3L=     u3 d3 U3     L .(2.4) Anomaly cancellation requires that two families of quarks qiL;i= 1,2transform as antitriplets and one q3Las a triplet, as in the original Singer-Valle-Schechter (SVS) scheme [2]. This way, the number of the fermion families is equal to the number of colors, thus allowing a natural explanation of the number of generations of the SM fermions. The full field content is shown in table 1. Notice that, in addition to the field content of the model of ref. [7], our present setup includes three Majorana octets ΩaL (a= 1,2,3). They are crucial for achieving a successful gauge coupling unification scenario, as well as for providing the tiny masses of the light active neutrinos via a one-loop scotogenic mechanism. The latter is possible thanks to the inclusion of a scalar singlet σ. This electrically neutral field has a nontrivial charge under the preserved remnant matter-parity symmetry, providing a viable scalar WIMP – 3 – JHEP03(2022)034 dark matter candidate. Our model also includes three right-handed neutrinos with nonstandard U(1)Ncharges, which have been introduced in [9] to ensure an anomaly free gauge symmetry. These fields do not take part in the neutrino mass generation mechanism, and they do not mix with the other neutral fermions of the model. The two νiR fermion fields can acquire a Majorana mass after spontaneous symmetry breaking, by the inclusion of a scalar field transforming as (1,1,0,8), while a mass term for ν3Rrequires a scalar with quantum numbers (1,1,0,−10). In order to keep the analysis of the scalar sector as simple as possible here we do not include those extra scalar fields. The gauged B−Lsymmetry is spontaneously broken by two units as the singlet scalar φdevelops a vacuum expectation value (VEV). As seen from the assignments in table 1 this leaves a discrete remnant symmetry MPspecified in eq. (1.1). The most general VEV alignment for the scalars consistent with a preserved MPsymmetry is [10] hηi=1 √2(v1,0,0)T,hρi=1 √2(0, v2,0)T,hχi=1 √2(0,0, w)T,hφi=1 √2Λ,hσi= 0. (2.5) Here we will assume the hierarchy w, Λ,v1, v2, leading to the following spontaneous symmetry breaking (SSB) pattern SU(3)C×SU(3)L×U(1)X×U(1)N ↓w, Λ SU(3)C×SU(2)L×U(1)Y×MP ↓v1, v2 SU(3)C×U(1)Q×MP.(2.6) 2.2 Majorana octet lepton fields The presence of the SU(3)Loctet fermions, Ω, is a key ingredient of the model in order to ensure gauge coupling unification and to generate radiative masses for the active neutrinos. In this subsection we will show the explicit derivation of the electric charge and matter parity assignments of the components of the Majorana leptonic octets. Our starting point is the SU(3)Lalgebra, which is described by [Ta, Tb] = ifabcTc, Ta=λa 2,(2.7) where λa,a= 1,...,8are the Gell-Mann matrices. The corresponding Cartan subalgebra is spanned by H1=T3and H2=T8. The SU(3)Lalgebra in the Cartan basis reads [Hi, Eαa]=(αa)iEαa, E† αa=E−αa, i = 1,2, a = 1,2,3,(2.8) – 4 – JHEP03(2022)034 where the ladder operators Eαaare given by Eα1=1 √2(T4+iT5) = 1 √2      0 0 1 0 0 0 0 0 0     ,(2.9) Eα2=1 √2(T6−iT7) = 1 √2      0 0 0 0 0 0 0 1 0     ,(2.10) Eα3=1 √2(T1+iT2) = 1 √2      0 1 0 0 0 0 0 0 0     ,(2.11) and the coefficients (αa)iare the components of the roots ~α1= 1 2,√3 2!, ~α2= 1 2,−√3 2!, ~α3= (1,0).(2.12) The SU(3)Lleptonic octet, Ω, can be decomposed in the Cartan basis of the SU(3)L generators as follows Ω=Ω(3)T3+ Ω(8)T8+ Ω(1) ±E±α1+ Ω(2) ±E±α2+ Ω(3) ±E±α3.(2.13) Using eqs. (1.1), (2.1), (2.2) and the charge assignments QX(Ω) = QN(Ω) = 0 given in table 1, we find the Qand B−Lcharge assignments as well as matter parity MPof the component fields in (2.13), namely Ω(T3,T8): Ω(3)(0,0),Ω(8)(0,0),Ω(1) ±±1 2,±√3 2,Ω(2) ±±1 2,∓√3 2,Ω(3) ±(±1,0) (2.14) Ω(Q,B−L;MP): Ω(3)(0,0;−1),Ω(8)(0,0;−1),Ω(1) ±(0,∓1;+1),Ω(2) ±(±1,±1;+1),Ω(3) ±(±1,0;−1). (2.15) Then, the leptonic Octet can be expressed in matrix form as ΩL=1 √2            1 √2Ω(3) +1 √6Ω(8) Ω(3) +Ω(1) + Ω(3) −−1 √2Ω(3) +1 √6Ω(8) Ω(2) − Ω(1) −Ω(2) +−2 √6Ω(8)           L ,(2.16) Note that (Ωc)ij = (Ωji)c. Therefore, Q(Ωij) = Q(Ωc ij)and MP(Ωij) = MP(Ωc ij), as should be for a Majorana field ΩM= ΩL⊕(ΩL)c. From eqs. (2.15) and (2.16) we find the charge Qand MPassignments of the leptonic octet, Ω, shown in table 1. – 5 – JHEP03(2022)034 In the following discussion, we will denote the components of Ωin the canonical normalization as Ω(3) =√2Ψ0,Ω(8) =√2˜ N,Ω(3) ±=√2E±,Ω(2) ±=√2˜ E±,Ω(1) +=√2∆, Ω(1) −=√2˜ ∆. In this notation, the octet takes the form ΩL=            1 √2Ψ + 1 √6˜ N E+∆ E−−1 √2Ψ + 1 √6˜ N˜ E− ˜ ∆˜ E+−2 √6˜ N           L ,(2.17) (ΩL)c=            1 √2(ΨL)c+1 √6(˜ NL)c(E− L)c(˜ ∆L)c (E+ L)c−1 √2(ΨL)c+1 √6(˜ NL)c(˜ E+ L)c (∆L)c(˜ E− L)c−2 √6(˜ NL)c            ,(2.18) ΩL=            1 √2ΨL+1 √6˜ NLE− L˜ ∆L E+ L−1 √2ΨL+1 √6˜ NL˜ E+ L ∆L˜ E− L−2 √6˜ NL            .(2.19) 3 Symmetry breaking The most general 3-3-1-1 gauge-invariant scalar potential of the model is given by V=µ2 1ρ†ρ+µ2 2χ†χ+µ2 3η†η+µ2 4φ†φ+µ2 5σ†σ +λ1(ρ†ρ)2+λ2(χ†χ)2+λ3(η†η)2 +λ4(ρ†ρ)(χ†χ) + λ5(ρ†ρ)(η†η) + λ6(χ†χ)(η†η) +λ7(ρ†χ)(χ†ρ) + λ8(ρ†η)(η†ρ) + λ9(χ†η)(η†χ) +λ10(φ†φ)(ρ†ρ) + λ11(φ†φ)(χ†χ) + λ12(φ†φ)(η†η) +λ13(σ†σ)(ρ†ρ) + λ14(σ†σ)(χ†χ) + λ15(σ†σ)(η†η) +λ16(φ†φ)2+λ17(σ†σ)2+λ18(φ†φ)(σ†σ) + λ19 h(σ†φ)(η†χ) + h.c.i +µt 2ρηχ +µs 2φ†σσ +µu 2(η†χ)σ+h.c., (3.1) where the λk(k= 1,2,··· ,19) are dimensionless parameters, whereas the µr(r= 1,2,··· ,5), µt,µs,µuhave dimension of mass. We emphasize that in our model we have no need of imposing any global symmetries, all the ingredients for the scotogenic neutrino mass generation are already contained in the gauge symmetry group. To ensure that the – 6 – JHEP03(2022)034 MPsymmetry remains conserved we require µ2 5>0, implying that the MP-odd scalar σ does not develop a nonzero VEV. The minimization conditions of the scalar potential yield the following relations: µ2 1=v1wµt−v2λ10Λ2+ 2λ1v2 2+λ5v2 1+λ4w2 2v2 , µ2 2=v1v2µt−wλ11Λ2+λ4v2 2+λ6v2 1+ 2λ2w2 2w, µ2 3=v2wµt−v1λ12Λ2+ 2λ3v2 1+λ5v2 2+λ6w2 2v1 , µ2 4=−1 22λ16Λ2+λ10v2 2+λ12v2 1+λ11w2.(3.2) The scalar potential of the model has been previoulsy analyzed in ref. [9]. Here we just quote the results relevant for the modified neutrino mass generation mechanism of the present model. Decomposing the scalar multiplets in components as η=     v1+s1+ia1 √2 η− 2 s0 3+ia0 3 √2     , ρ=     ρ+ 1 v2+s2+ia2 √2 ρ+ 3     , χ=     s0 1+ia0 1 √2 χ− 2 w+s3+ia3 √2     , φ=Λ+sφ+iaφ √2, σ=sσ+iaσ √2. (3.3) The CP-even neutral scalar sector of the model was studied in [9]. Besides the 125 GeV SM-like Higgs identified with h≈v1s1+v2s2 qv2 1+v2 2 ,(3.4) and having a mass m2 h≈Λ2v1v2wµt(λ4λ11−2λ2λ10)λ12−λ5λ2 11−4λ2λ16+λ6(λ10λ11−2λ4λ16)w2−λ16µ2 t m2 H1m2 H2m2 H3 . (3.5) There are three additional heavy Higgs bosons, given by the approximate expressions H1≈v2s1−v1s2 qv2 1+v2 2 , m2 H1≈v2 1+v2 2wµt 2v1v2 ,(3.6) H2≈cosξs3−sinξs4, m2 H2≈λ16Λ2+λ2w2−qλ2 16Λ4+λ2 2w4+λ2 11Λ2w2−2λ2λ16Λ2w2, H3≈sinξs3+cosξs4, m2 H3≈λ16Λ2+λ2w2+qλ2 16Λ4+λ2 2w4+λ2 11Λ2w2−2λ2λ16Λ2w2, valid under the assumption Λ, w, µtv1, v2. There are also two physical real scalars ϕ1, ϕ2and one Nambu-Goldstone boson G1, defined as      ϕ1 ϕ2 G1     =Us     s0 1 s0 3 sσ     =       v1cos θs √w2+v2 1 wcos θs √w2+v2 1 sin θs −v1sin θs √w2+v2 1−wsin θs √w2+v2 1 cos θs w √w2+v2 1−v1 √w2+v2 1 0             s0 1 s0 3 sσ     ,(3.7) – 7 – JHEP03(2022)034 where the mixing angle θssatisfies the relation tan2θs=2v1wqv2 1+w2(λ19Λ+µu) v1w−2µ2 5−Λ(λ18Λ+2µs)−λ13v2 2−λ15v2 1+λ9v2 1+w2−λ14w2+v2µtv2 1+w2. (3.8) The emergence of a Nambu-Goldstone boson in the CP-even scalar sector follows from of the existence of a non-hermitian gauge boson X0, whose real part must absorb G1after the SSB, so as to acquire a consistent mass, while its imaginary part absorbs an analogous CP-odd Goldstone boson, as we discuss below. The CP-odd neutral sector consists of four Nambu-Goldstone bosons, G2,3,4,5, and three massive states, denoted as A1,e ϕ1and e ϕ2. Three of these four Nambu-Goldstone bosons are given by G2=v1a1−v2a2 qv2 1+v2 2 , G3=v1a1−wa2 qv2 1+w2, G4=aφ,(3.9) and correspond to the longitudinal components of the physical gauge bosons, Z,Z0,Z00, respectively. On the other hand, the massive state A1is A1=v2wa1+v1wa2+v1v2a3 p(v2w)2+ (v1w)2+ (v1v2)2, m2 A1=µtv2 1w2+v2 2w2+v2 2v2 1 2v1v2w,(3.10) whereas the two physical states e ϕ1,e ϕ2and the fifth Goldstone G5are defined as     e ϕ1 e ϕ2 G5     =Ua     a0 1 a0 3 aσ     =       −v1cos θa √w2+v2 1 wcos θa √w2+v2 1 sin θa v1sin θa √w2+v2 1−wsin θa √w2+v2 1 cos θa w √w2+v2 1 v1 √w2+v2 1 0             a0 1 a0 3 aσ     ,(3.11) with mixing angle tan2θa=2v1wqv2 1+w2(µu−λ19Λ) v1w−λ18Λ2−2µ2 5+2Λµs−λ13v2 2−λ15v2 1+λ9v2 1+w2−λ14w2+v2µtv2 1+w2. (3.12) Notice that the Goldstone bosons G5and G1combine into a single complex neutral would-be Goldstone, absorbed by the longitudinal component of the non-Hermitian neutral gauge boson X0. The real scalars ϕ1,ϕ2,e ϕ1and e ϕ2acquire squared masses given by m2 ϕ1,2=1 4v1w(v1wλ18Λ2+2µ2 5+2Λµs+λ13v2 2+λ15v2 1+λ9v2 1+w2+λ14w2+v2µtv2 1+w2 ∓Fsnv1wλ18Λ2+2µ2 5+2Λµs+λ13v2 2+λ15v2 1+λ9v2 1+w2+λ14w2+v2µtv2 1+w22 −4v1wv2 1+w2v2µtλ18Λ2+2µ2 5+2Λµs+λ13v2 2+λ14w2+v1wλ9λ18Λ2+2µ2 5+2Λµs+λ14w2 −(λ19Λ+µu)2+λ9λ13v2 2+λ15v2v2 1µt+λ9λ15v3 1wo1/2), – 8 – JHEP03(2022)034 Here, C2(Gi)is the quadratic Casimir invariant corresponding to the adjoint representations, C2(G)≡   Nif SU(N), 0if U(1).(6.4) whereas T(Rf)and T(Rs)correspond to the Dynkin indices of the irreducible representation Rf,s for a given fermion and scalar, respectively. For the case of SU(N) they are T(Rf,s)≡         1/2if Rf,s is fundamental, Nif Rf,s is adjoint, 0if Rf,s is singlet. (6.5) The quantity d(Rf,s)in (6.3) is the dimension of a given representation Rf,s under all gauge groups except for the i-th gauge group under consideration. As shown above, in our 3-3-1-1 model the electric charge operator is defined as Q=T3−1 √3T8+X, (6.6) where the SU(3)Lgenerators are normalised as Tr (TiTj) = 1 2δij. Note that the U(1)X charge, X, enters in the definition of electric charge Qand hence can be related to the hypercharge after the breaking of SU(3)C⊗SU(3)L⊗U(1)X⊗U(1)Nto the SM gauge group SU(3)C⊗SU(2)L⊗U(1)Yas Y=−1 √3T8+X. (6.7) Therefore the initial value for αXat the 3-3-1-1 symmetry breaking scale MXcan be obtained using the hypercharge αYevolution from the Z-pole to MX. In addition, we can define the normalized charge operators XNand YN, which satisfy the relations X=nXXN, Y =nYYN,(6.8) with the normalizations of Xand the hypercharge Ybeing related by n2 Y=1 3+n2 X.(6.9) We recall that in an embedding of the SM gauge group into some unified simple group the hypercharge normalization is usually chosen so that it matches with the normalization for the SU(N) counterparts in the SM gauge group, Tr[TiTj] = 1 2δij .(6.10) For instance, in a SU(5) theory by fixing the normalisation of the fundamental representation one can fix the U(1)Ynormalisation to n2 Y= 5/3. Here, in the absence of any specific unification group we will treat nY,a priori, as a free parameter. – 15 – JHEP03(2022)034 Relevant Gauge group Scale of running Gauge group GiNotation for biValue of bi SU(3)Cb3C-7 SU(3)C⊗SU(2)L⊗U(1)YMZ< µ < MXSU(2)Lb2L−19 6 U(1)YbUN Y41 6 SU(3)CbX 3C−5 SU(3)C⊗SU(3)L⊗U(1)X⊗U(1)NMX< µ < M8SU(3)LbX 3L−13 2 U(1)XbUN X26 3 U(1)NbUN N163 3 SU(3)CbΩ 3C=bX 3C−5 SU(3)C⊗SU(3)L⊗U(1)X⊗U(1)NM8< µ < MUSU(3)LbΩ 3L−1 2 U(1)XbΩ;UN X=bUN X26 3 U(1)NbΩ;UN N=bUN N163 3 Table 2. Values of bifor different gauge groups (Gi) relevant for RG running of gauge couplings at different energy scales. We also notice that the U(1)Ncharge does not contribute to the electric charge, and therefore can be considered as an “electrically neutral new charge” (ENNC). As a result, the initial value for αNat the 3-3-1-1 symmetry breaking scale MXand the normalisation of U(1)N, N=nNNN,(6.11) remain free parameters which cannot be fixed by electroweak gauge coupling input values or hypercharge normalization. Furthermore, for the sake of generality, we take the 3-3-1-1 symmetry breaking scale (MX) and the fermionic octet mass scale M8> MXas independent scales. In table 2, we summarize the one-loop RGE beta coefficients governing the evolution of the relevant gauge couplings at different scales. Before addressing the evolution of the U(1)Ngauge coupling, it is straightforward to find the unification scale MUfor SU(3)c,SU(2)Land U(1)Xand the hypercharge normalization nYas a function of the intermediate symmetry breaking scales. First we note that the normalized couplings are related by n2 YαN Y−1=1 3α−1 3L+n2 Y−1 3αN X−1.(6.12) Taking the 3-3-1-1 symmetry breaking scale (MX) and the mass scale for the fermionic octets ΩaL,a= 1,2,3,M8> MXas independent parameters, and using eq. (6.2) we then obtain α−1 U=1 n2 Y−1 3α−1 em (MZ)cos2θw(MZ)−1 3α−1 2L(MZ)−bUN Y−1 3b2L 2πlnMX MZ−bUN X 2πlnMU MX,(6.13) α−1 U=α−1 2L(MZ)−b2L 2πlnMX MZ−bX 3L 2πlnM8 MX−bΩ 3L 2πlnMU M8,(6.14) α−1 U=α−1 3C(MZ)−b3C 2πlnMX MZ−bX 3C 2πlnMU MX,(6.15) where we note that the fermionic octets only affect the evolution of α3Lfrom M8to MU. Hence in the above we denote the beta coefficients for the running of SU(3)Lfrom MXto – 16 – JHEP03(2022)034 M8by bX 3Land M8to MUby bΩ 3L, respectively. From eqs. (6.14) and (6.15) one obtains the unification scale MUas a function of MXand M8as MU(MX, M8) = M bX 3C bX 3C−bΩ 3L X M bΩ 3L bX 3C−bΩ 3L 8 M8 MXbX 3L bX 3C−bΩ 3LMX MZb2L−b3C bX 3C−bΩ 3Lexp "2πα−1 3C(MZ)−α−1 2L(MZ) bX 3C−b3L#. (6.16) Using eqs. (6.12) and (6.13) the normalization n2 Ycan be obtained as n2 Y=1 3+α−1 em (MZ)cos2θw(MZ)−1 3α−1 2L(MZ)−bUN Y−1 3b2L 2πlnMX MZ+bUN X 2πlnMU(MX,M8) MX ×α−1 2L(MZ)−b2L 2πlnMX MZ−bX 3L 2πlnM8 MX−bΩ 3L 2πlnMU(MX,M8) M8−1 ,(6.17) where MU(MX, M8)is given by eq. (6.16). The one-loop beta coefficients relevant for the running between different scales are collected in table 2. In figure 3left plot, we show the unification scale as a function of the 3-3-3-1 symmetry breaking scale MX(cf. eq. (6.16)), for three different benchmark choices M8=MX(solid curve), M8= 3MX(dashed curve) and M8= 10MX(dot-dashed curve). The blue band indicates the range for the unification scale consistent with the current experimental limit on proton decay lifetime if the 3-3-1-1 gauge group is embedded in a unified gauge group. However, we note that a dynamical gauge coupling unification achieved with an anomalyfree set (under the 3-3-1-1 gauge group) of fields [20–26], is not subject to such a constraint. The tilted red line corresponds to the asymptotic limit MX=MU. In figure 3right plot, we show the corresponding hypercharge normalization as a function of the 3-3-1-1 symmetry breaking scale MX, for the benchmark choices described above. The red line shows the standard hypercharge normalization for a reference SU(5) unified theory. Note that for a given benchmark M8value, the relevant (solid, dashed or dot-dashed) curves in the left and right panels of figure 3correspond to the SU(3)c×SU(3)L×U(1)Xunification scale MUand the relevant hypercharge normalization required for a successful unification of gauge couplings. Therefore, such a curve represents a family of gauge coupling unification possibilities, with each point corresponding to a particular choice of the 3-3-1-1 symmetry breaking scale MX. In figure 4we show such an example point in the dashed curves in figure 3with the 3-3-1-1 symmetry breaking scale MX= 10 TeV and M8= 3MX= 30 TeV, demonstrating a successful SU(3)c×SU(3)L×U(1)Xunification. Having discussed the SU(3)c×SU(3)L×U(1)Xunification, we now explore the possibility of U(1)Nunification with SU(3)c×SU(3)L×U(1)X. First we note that, a priori, it is a valid theoretical possibility that SU(3)c×SU(3)L×U(1)Xcan first unify into a larger gauge group G3−3−1, independent of U(1)N, and at some higher energy scale the unification of U(1)Nand G3−3−1takes place. However, for the sake of simplicity, we will only consider the case of U(1)Nunifying at the same scale of G3−3−1unification. We recall that U(1)Ndoes not contribute to the electric charge and therefore the initial value for αNat MXand its normalization nNdefined in eq. (6.8) are not fixed by the hypercharge and electroweak input parameters. Therefore, in order to derive the viable – 17 – JHEP03(2022)034 Figure 3. (Left) unification scale MUas a function of the 3-3-1-1 symmetry breaking scale MX, for three benchmark choices M8=MX(solid curve), M8= 3MX(dashed curve) and M8= 10MX (dot-dashed curve). (Right) hypercharge normalization n2 Yas a function of MX, for the same benchmark choices as the left panel. Figure 4. An example of SU(3)c×SU(3)L×U(1)Xunification for the 3-3-1-1 symmetry breaking scale MX= 10 TeV and M8= 3MX= 30 TeV, corresponding to the dashed curves in figure 3. Figure 5. The left panel shows the α−1 Ncontours in the U(1)Nnormalisation (n2 N) vs the 3-3-1-1 symmetry breaking scale MXplane, for a benchmark choice M8=MX, while the right one is the same as the left, but with M8= 10MX. – 18 – JHEP03(2022)034 Figure 6. An example of SU(3)c×SU(3)L×U(1)X×U(1)Nunification for a phenomenologically accessible 3-3-1-1 symmetry breaking scale MX= 10 TeV and M8= 3MX= 30 TeV, corresponding to the dashed curves in figure 3. initial value for αNat the 3-3-1-1 symmetry breaking scale MXand the normalisation nN, we require that the U(1)Ncoupling must unify with the remaining gauge groups associated to SU(3)C,SU(3)L,U(1)Xat the same scale MU, leading to the relation α−1 U=α−1 3C(MU) = α−1 3L(MU) = αN X(MU)−1=αN N(MU)−1,(6.18) subject to the bcoefficients shown in table 2. This in turn yields α−1 N(MX) = n2 Nα−1 U+bUN N 2πln MU MX,(6.19) where α−1 Uis obtained using any of the eqs. (6.10), (6.11), or (6.12), and MU(MX, M8) is given by eq. (6.14). In figure 5we show the contours for α−1 N(MX)in the U(1)N normalisation(n2 N) vs 3-3-1-1 symmetry breaking scale MXplane (predicted by the requirement that the U(1)Ncoupling must unify with the remaining gauge groups associated to SU(3)C,SU(3)L,U(1)Xat the same scale MU), for two benchmark choices: in the left plot M8=MXand in the right one M8= 10MX. In order to explicitly give an example of a unification scenario, in figure 6we show SU(3)c⊗SU(3)L⊗U(1)X⊗U(1)Nunification for a 3-3-1-1 symmetry breaking scale MX= 10 TeV and M8= 3MX= 30 TeV, with the corresponding input for α−1 N(MX)computed using eq. (6.16). Intriguingly, we notice from figures 3and 5that successful gauge coupling unification can occur for a 3-3-1-1 symmetry breaking scale MXand fermionic octet mass scale M8 around O(10) TeV, accessible at the current and future collider experiments. Therefore, our 3-3-1-1 model provides a very exciting phenomenological alternative for having new physics at an energy scale around O(10) TeV in the form of the gauge bosons associated with 3-3-1-1 symmetry breaking, as well as the fermionic octet. Such mass scales can not only be explored at collider experiments [27], but also lead to interesting charged lepton flavour violation signals [28–31]. Moreover they may also be probed in low-energy neutrino experiments, e.g. neutrinoless double beta decay searches [32]. To conclude this section we comment on the possible embeddings of the 3-3-1-1 gauge group. One of the minimal possibilities is to unify the SU(3)c⊗SU(3)L⊗U(1)X⊗U(1)N – 19 – JHEP03(2022)034 gauge group inside SU(6) ⊗U(1)N. In this case the SU(3)c⊗SU(3)L⊗U(1)Xpart of the 3-3-1-1 gauge group unifies into SU(6) [33,34]. The fermion content of the model (except for the octets) can be embedded into the anomaly-free combination of SU(6) representations ¯ 6 + ¯ 6+15+20, while the octets can be embedded in the 35 multiplet of SU(6) which does not contribute to the anomaly. Given that the specific multiplicity of the fermionic triplets in the 3-3-1-1 model is dictated by the number of families, the required combination of anomaly-free SU(6) multiplets to accomodate such structure requires additional fields. In order to identify the required multiplets in the unified theory it may be useful to make use of flux breaking tools implemented through the Hosotani mechanism [35]. As seen above, by normalizing the fundamental representation of SU(6) the hypercharge and U(1)Xnormalizations can be fixed to nY=p5/3and nX= 2/√3 respectively. Moreover, the SU(6) multiplets required for 3-3-1-1 unification can be further embedded in a E(6) theory with one of its maximal subgroups being SU(6) ⊗SU(2). The 27 representation of E(6) can break into ¯ 6and 15 representations of SU(6) and the 78 can break into 35,20 and 1representations of SU(6). The E(6) embedding can be particularly interesting from the perspective of E(6) F-theories [36–40]. Finally, note that a 3-3-1-1 embedding into a unified SU(6) ⊗U(1)Ngroup would lead to further constraints on the unification scale, due to the fact that the SU(6) gauge bosons can mediate a proton decay mode such as p→e+π0. Experimental searches for the latter lead to a stringent limit MU&1015.5GeV [1]. We would like to note that there are different variants of the standard minimal 3-3-1 model. These have different assignments of the quarks and leptons, e.g. the SVS [9], the flipped (e.g. [41]) and the sequential (e.g. [33]) variants of the 3-3-1 model. While all of them are in principle consistent with gauge coupling unification [33] only those with large leptonic multiplets (e.g. leptonic octets) allow for a 3-3-1 symmetry breaking scale within reach of upcoming collider experiments. 7 Summary and outlook As a follow-up of our previous work we have now proposed a scotogenic scheme where dark matter stability is ensured by a gauged matter parity symmetry, eq. (1.1). The same physics responsible for neutrino mass generation drives the unification of the fundamental gauge couplings. A crucial role is played by the leptonic octets in the model: they are responsible for generating the light active neutrino masses through a Scotogenic mechanism (see figure 1), while driving gauge coupling unification (see figures 3–6). Their masses can be accessible to experiments at O(10) TeV scale. Taking such dynamical unification approach as the guiding principle, we have used the unification of the electrically neutral U(1)Nsymmetry with SU(3)c⊗SU(3)L⊗U(1)Xto predict the initial value and normalization for the coupling strength of the new interaction associated with U(1)N(see figure 5). Indeed, while not exclusive, this approach is very attractive, as it naturally predicts the free parameters associated with the new interaction. The construction is suggestive of a plethora of new physics associated to the new gauge bosons and to the exotic states dictated by the 3-3-1-1 gauge symmetry, which can be probed at future collider and low-energy experiments. In addition to the possibility of observing the new exotic fermions at future – 20 – JHEP03(2022)034 collider searches, this model can also be probed at the upcoming experiments searching for charged Lepton Flavour Violation (cLFV). In particular, there are two distinct types of cLFV contributions that can arise in this model. One is the standard “scotogenic” diagram for µ→eγ, with dark particles running in the loop, as noted in [8]. Such a contribution is strongly suppressed, since it goes as the fourth power of the mixing angle between the electrically charged components of the SU(3)Lscalar triplets χand ρ. This mixing angle is of the order of the ratio between the electroweak scale and the 3-3-1-1 symmetry breaking scale, i.e. .O(10−2). As a result, apart from the loop suppression, the scotogenic contribution to the µ→eγ decay has another suppression factor of O(10−8). Taking reasonable values for the neutrino Yukawa couplings, the mass of the neutral heavy lepton and the electically charged scalar mediators, we find that such scotogenic contribution is several orders of magnitude below the current experimental upper limit of MEG. Another type of contribution in this model is the inverse-seesaw-type contribution [42], which in this case involves the leptonic octets and Wboson in the loop. Such a contribution is suppressed by the mixing between the light-neutrino and neutral-octet. As expected, taking M8∼w∼104GeV and Tr[yΩyΩ†]∼2×10−5, one finds that the inverse seesaw contribution gives Br[µ→eγ]∼8.7×10−14, which will be probed at MEG-II. To conclude, we stress once again that our model does not require supersymmetry, though it can me made supersymmetric, should one desire that route to address the gauge hierarchy problem instead of, say, the warping of extra spacetime dimensions. In contrast, our suggestion provides a potentially testable approach where other drawbacks of the SM are addressed in an interconnected manner, such as •number of the fermion families equals the number of colors, •WIMP dark matter mediates neutrino mass generation, •dark matter stability results from a residual gauge matter-parity, •dynamical unification of gauge couplings. In short, we have illustrated an idea which seems worth of a dedicated scrutiny of its potential implications. Acknowledgments Work supported by the Spanish grants PID2020-113775GB-I00 (AEI / 10.13039/501100011033) and PROMETEO/2018/165 (Generalitat Valenciana). A.E.C.H and S.K. are supported by ANID-Chile FONDECYT 1210378 and ANID-Chile FONDECYT 1190845 as well as by ANID PIA/APOYO AFB180002 and Milenio-ANIDICN2019_044. C.H. acknowledges support from the DFG Emmy Noether Grant No. HA 8555/1-1. CAV-A is supported by the Mexican Catedras CONACYT project 749 and SNI 58928. The relic abundance and direct detection constraints were calculated using the MicroOmegas package [43] at GuaCAL (Guanajuato Computational Astroparticle Lab). – 21 – JHEP03(2022)034 Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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