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JID:PLB AID:32515 /SCO Doctopic: Phenomenology [m5Gv1.3; v1.194; Prn:28/12/2016; 9:09] P.1(1-6) Letters B ••• (••••) •••–•••

1 Contents lists available at ScienceDirect 66 2 67 3 68 4 Physics Letters B 69 5 70 6 71 7 www.elsevier.com/locate/physletb 72 8 73 9 74 10 75 11 76 12 Unifying left–right symmetry and 331 electroweak theories 77 13 78 14 Mario Reig, José W.F. Valle, C.A. Vaquera-Araujo 79 15 80 AHEP Group, Institut de Física Corpuscular – C.S.I.C./Universitat de València, Parc Científic de Paterna, C/Catedrático José Beltrán, 2, E-46980 Paterna (Valencia), 16 81 Spain 17 82 18 83 19 a r t i c l e i n f o a b s t r a c t 84 20 85 21 86 Article history: We propose a realistic theory based on the SU(3)c ⊗ SU(3)L ⊗ SU(3)R ⊗ U(1)X gauge group which requires 22 Received 8 November 2016 the number of families to match the number of colors. In the simplest realization masses arise 87 23 Received in revised form 19 December 2016 from the canonical seesaw mechanism and their smallness correlates with the observed V-A nature of 88 Accepted 22 December 2016 24 the weak force. Depending on the symmetry breaking path to the one recovers either 89 Available online xxxx a left–right symmetric theory or one based on the SU(3) ⊗ SU(3) ⊗ U(1) symmetry as the “next” step 25 Editor: A. Ringwald c L 90 26 towards new physics. 91 © 27 2016 Published by Elsevier B.V. This is an open access article under the CC BY license 92 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 28 93 29 94 30 95 31 96 1. Introduction the seesaw mechanism [13–17], their explicit chiral structure pre- 32 97 vents a dynamical understanding of parity and its possible relation 33 98 to the smallness of neutrino mass, precluding a deeper under- 34 Despite its great success, the Standard Model (SM) is an in- 99 standing of the meaning of the hypercharge quantum number. 35 complete theory, since it fails to account for some fundamental 100 In this paper we will address some of these issues jointly, 36 phenomena such as the existence of neutrino masses, the under- 101 suggesting that they are deeply related. Our framework will be 37 lying dynamics responsible for their smallness, the existence three 102 an extended left–right symmetric model which implies the ex- 38 families, the role of parity as a fundamental symmetry, as well 103 istence of mirror gauge bosons, i.e. in addition to weak gauge 39 as many other issues associated to cosmology and the inclusion 104 bosons we have right-handed gauge bosons so as to restore par- 40 of gravity. Here we take the first three of these shortcomings as 105 ity at high energies. We propose a unified description of left– 41 valuable clues in determining the next step in the route towards 106 physics Beyond the Standard Model. right symmetry and 331 electroweak theories in terms of the 42 ⊗ ⊗ ⊗ 107 43 One unaesthetic feature of the Standard Model is that the chiral extended SU(3)c SU(3)L SU(3)R U(1)X gauge group as a com- 108 44 nature of the weak interactions is put in by hand, through ex- mon ancestor: Depending on the spontaneous symmetry break- 109 45 plicit violation of parity at the fundamental level. Moreover the ing path towards the Standard Model one recovers either con- 110 ⊗ ⊗ ⊗ 46 Adler–Bell–Jackiw anomalies [1,2] are canceled miraculously and ventional SU(3)c SU(2)L SU(2)R U(1)B−L symmetry or the 111 ⊗ ⊗ 47 thanks to the ad-hoc choice of hypercharge assignments. Left–right SU(3)c SU(3)L U(1) symmetry as the missing link on the road to 112 48 symmetric schemes such as Pati–Salam [3] or the left–right sym- physics beyond the Standard Model. Other constructions adopting 113 49 metric models can be made to include parity and offer a solution these symmetries have been already mentioned in the literature. 114 50 to neutrino masses through seesaw mechanism [4–8] and a way to In [18,19] a model for neutrino mass generation through dimen- 115 51 “understand” hypercharge [6]. However in this case the number of sion 5 operators is studied, and in [20,21] models for the diphoton 116 52 fermion families is a free parameter. anomaly and were presented. 117 This work is organized as follows. We first construct a new left– 53 Conversely, SU(3)c ⊗ SU(3)L ⊗ U(1)X schemes provide an ex- 118 54 planation to the family number as a consequence of the quantum right symmetric theory showing how the gauge structure is deeply 119 55 consistency of the theory [9,10], but are manifestly chiral, giving related both to anomaly cancellation as well as the presence of 120 56 no dynamical meaning to parity. Even if these models allow for parity at the fundamental level. In the next sections we build 121 57 many ways to understand the smallness of neutrino mass either a minimal model where neutrino masses naturally emerge from 122 58 through radiative corrections [11,12] or through various variants of the seesaw mechanism. Finally we study the symmetry breaking 123 59 sector, identifying different patterns of symmetry breaking and 124 60 showing how they are realized by different hierarchies of the rel- 125 61 E-mail addresses: [email protected] (M. Reig), valle@ific.uv.es (J.W.F. Valle), evant vacuum expectation values. In the appendix we outline the 126 62 vaquera@ific.uv.es (C.A. Vaquera-Araujo). anomaly cancellation in the model. 127 63 128 http://dx.doi.org/10.1016/j.physletb.2016.12.049 64 129 0370-2693/© 2016 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 65 130 JID:PLB AID:32515 /SCO Doctopic: Phenomenology [m5Gv1.3; v1.194; Prn:28/12/2016; 9:09] P.2(1-6) 2 M. Reig et al. / Physics Letters B ••• (••••) •••–•••

1 Table 1 66 2 Particle content of the model, with a = 1, 2, 3and α = 1, 2. See text for the definition of q. 67

3   α α 3 3 68 ψaL ψaR Q L Q R Q L Q R φ ρ L R 4 69 SU(3)c 113333111 1 5 ∗ 70 SU(3)L 313131336 1 ∗ ∗ 6 SU(3)R 131313331 6 71 q−1 q−1 − q − q q+1 q+1 2q+1 2(q−1) 2(q−1) 7 U(1)X 3 3 3 3 3 3 0 3 3 3 72 8 73 9 74 2. The model In this setup, the mechanism behind anomaly cancellation is anal- 10 75 ogous to the one characterizing 331 models, as detailed in Ap- 11 76 In this paper we propose a class of manifest left–right sym- pendix A. Thus the first interesting result in this framework is the 12 77 metric models based on the extended electroweak gauge group fact that quantum consistency requires that the number of triplets 13 78 SU 3 ⊗ SU 3 ⊗ SU 3 ⊗ U 1 in which the electric charge gen- must be equal to the number of antitriplets. This can be achieved 14 ( )c ( )L ( )R ( )X 79 erator is written in terms of the diagonal generators of SU 3 if two multiplets transform as triplets whereas the third one 15 ( )L,R 80 transforms as an antitriplet, which in turn implies that the num- 16 and the U(1)X charge as 81 ber of generations must coincide with the number of colors, an 17 82 Q = T 3 + T 3 + β(T 8 + T 8 ) + X, (1) appealing property of 331 models [9]. 18 L R L R 83 The scalar sector needed for spontaneous symmetry breaking 19 where β is a free parameter that determines the electric charge of 84 and fermion mass generation contains a bitriplet 20 the exotic fields of the model [22–24], and its value is restricted 85 21 by the SU(3) and U(1) coupling constants g = g = g and g ⎛ + − ⎞ 86 L,R X L R X φ0 φ φ q 22 to comply with the relation 11 12 13 87 ⎜ − − ⎟ 23 = − 0 q 1 ∼ ∗ 88 φ ⎝ φ φ φ ⎠ (3L, 3R), (5) 2 2 21 22 23 24 g sin θ q q+1 89 X = W 0 25 , (2) φ31 φ32 φ33 90 g2 1 − 2(1 + β2) sin2 θ 26 W 91 a bi-fundamental field 27 92 with θW as the electroweak mixing angle [18]. This relation im- 2 ⎛ ⎞ 28 plies that β2 < −1 + 1/(2 sin θ ), consistent with the original + + 93 W ρ ρ0 ρq 1 29 models [9,10]1 Notice that special features may arise for specific 11 12 13 94 ⎜ − q ⎟ 30 = ⎝ 0 ⎠ ∼ 95 choices, such as β = 0, which contains fractionally charged lep- ρ ρ21 ρ22 ρ23 (3L, 3R), (6) 31 q+1 q 2q+1 96 tons [25]. In a general SU(N)L ⊗ SU(M)R ⊗ U(1)X theory β = 0 ρ31 ρ32 ρ33 32 always implies that the charge X becomes proportional to B − L. 97 33 98 Moreover, for N, M > 2the number of families must match the as well as two sextets L ∼ (6L , 1R ), R ∼ (1L , 6R ) with compo- 34 number of colors in order to cancel the anomaly, since two quark nents 99 35 families transform as the fundamental representation and one as 100 ⎛ − ⎞ 36 anti-fundamental. In order to illustrate the peculiar features of  q 101 0 √12 √13 37 this class of models, that combine inherent aspects of both left– ⎜ 11 2 2 ⎟ 102 ⎜ − − ⎟ 38  −− q 1 103 right symmetric models and 331 gauge structures, we will consider ⎜ √12 √23 ⎟ L,R = ⎜  ⎟ . (7) 39 throughout this work, the general case where β is not fixed. ⎝ 2 22 2 ⎠ 104 − 40 q q 1 105 The particle content and the transformation properties of the √13 √23 2q 41 fields are summarized in Table 1. We assume manifest left–right 2 2 33 L,R 106 42 107 symmetry, implemented by an additional Z2 symmetry that acts 43 → † → T → 108 as parity, interchanging SU(3)L and SU(3)R and transforming the The above fields transform as φ U L φ U R , ρ U L ρ U R , L 44 a ↔ a α ↔ α 3 ↔ 3 ↔ T ↔ † T → T ⊗ 109 fields as ψL ψR , Q L Q R , Q L Q R , ρ ρ , φ φ and U L L U L and R U R R U R under SU(3)L SU(3)R. The sym- 45 110 L ↔ R . metry breaking pattern in the scalar sector is assumed to be driven 46 Fermion fields are arranged in chiral multiplets by 111 47 112 ⎛ ⎞ ⎛ ⎞α 48 a d 113 ν = √1 = √1 49 a ⎝ − ⎠ α ⎜ ⎟ φ diag(k1,k2,n) L diag(v L, 0, 0), 114 ψ =  , Q = ⎝ −u ⎠ , 2 2 50 L,R L,R 115 q −q− 1 χ J 3 1 51 L,R L,R = √ 116 ⎛ ⎞ (3) R diag(v R , 0, 0), 52 3 117 u 2 (8) 53 ⎜ ⎟ ⎛ ⎞ 118 54 Q 3 = ⎝ d ⎠ , 0 k3 0 119 L,R ⎝ ⎠ 55 q+ 2 ρ = 000 , 120 J 3 56 L,R 000 121 57 122 transforming as triplets or antitriplets under both SU(3)L,R groups. 58 a where the vacuum expectation values (VEV) n and v R set the 123 The electric charge of the third components of ψL,R determined by 59 q is related to the parameter β through scale of symmetry breaking down to the standard model one, and 124 60 √ subsequently k1, k2, k3 and v L are responsible for the SM elec- 125 61 β =−(2q + 1)/ 3. (4) troweak spontaneous symmetry breaking. Thus, for consistency, 126 62 n , v R k1 , k2 , k3 , v L . In what follows we will explore sponta- 127 63 neous symmetry breaking patterns determined by the value of 128 64 1 129 The model √is unique up to exotic fermion charge assignments. Unfortunately, n/v R , as well as the natural expected hierarchy for the remain- 65 the choice β = 3is excluded by consistency of the model. ing VEVs k1 , k2 , k3 , v L . 130 JID:PLB AID:32515 /SCO Doctopic: Phenomenology [m5Gv1.3; v1.194; Prn:28/12/2016; 9:09] P.3(1-6) M. Reig et al. / Physics Letters B ••• (••••) •••–••• 3

g g 1 = − L − R − 66 3. Particle masses Dμ ∂μ i Wμ i Wμ igX XBμ , (17) 2 2 2 67 3 We now turn to the Yukawa interactions of the theory. These where the vector bosons are expressed as a 68 4 are similar to the ones present in the most popular left–right sym- 69 5 metric models, namely 8 70 L,R = i 6 Wμ W Lμ i 71 2 = 7 α ∗ β i 1 72 L = hQ Q φ Q ⎛ ⎞ 8 y αβ L R 3 + √1 8 + −q 73 = W W W V 9 α,β 1 ⎜ 3 ⎟ 74 ⎜ − 3 √1 8 −q−1 ⎟ 10 = W −W + W V , (18) 75 2 ⎝ 3 ⎠ α ∗ 3 3 11 + Q 3 + Q α + Q 3 q  q+1 √2 8 76 hα3 Q L ρ Q R h3α Q L ρ Q R h33 Q L φ Q R V V − W 12 3 L,R 77 α=1 (9) 13 78 3 with i as the Gell-Mann matrices. There are in total 17 gauge 14 79 +  a b + ca † b + ca † b bosons in the physical basis, the photon: γ , four electrically neu- 15 habψ L φψR fab ψ L L ψL ψ R R ψR   ± ± 80 tral states: Z L , Z R , Z , Z , four charged bosons: W , W , four 16 a,b=1 L R L R 81 + ±(1+q) ±(1+q) 17 states with charge q 1: XL , X R , and four with charge 82 + h.c. ±q ±q 18 q: Y L , Y R . One can determine the mass matrices and diagonalize 83

19 with hQ = (hQ )† and h = (h)†. After spontaneous symmetry them (assuming the VEV hierarchy v L k1,2,3 v R , n) to obtain 84 20 breaking, the first line in Eq. (9) produces the following Dirac mass the gauge boson masses 85 21 matrices for the Standard Model and exotic : 86 g2 22 ⎛ ⎞ m2 ≈ (k2 + k2 + k2 + 2v2), 87 Q Q W L 1 2 3 L 23 h k2 h k2 0 2 88 1 ⎜ 11 12 ⎟ 24 Mu = √ ⎝ Q Q ⎠ 2 89 h21k2 h22k2 0 , 2 g 2 2 2 2 25 m ≈ (k + k + k + 4v ), 90 2 Q Q Q Z L 2 1 2 3 L − − 2cos θW 26 h31k3 h32k3 h33k1 91 ⎛ ⎞ (10) 2 2 27 hQ k hQ k hQ k m  ∼ O (n ), 92 11 1 12 1 13 3 Z L 28 1 ⎜ ⎟ 93 d = √ Q Q Q M ⎝ h k1 h k1 h k3 ⎠ , 2 29 21 22 23 2 g 2 2 2 2 94 2 Q m ≈ (k + k + k + 2v ), 30 W R 1 2 3 R (19) 95 00h33k2 2 31 96 Q Q 2 −q− 1 1 h nhn q+ 2 g 32 J 3 = √ 11 12 J 3 = Q m2 ≈ (k2 + k2 + k2 + 4v2 ), 97 M Q Q , M h33n. (11) Z R 2 1 2 3 R 33 2cos θW 98 2 h21nh22n 34 2 ∼ O 2 + 2 99 Notice that in the absence of , the quark mass matrices are block m  (n v R ) 35 ρ Z R 100 diagonal and the ratio between the electroweak scales k and k 36 1 2 2 ≈ 2 ≈ 2 ∼ O 2 101 = mX mY mY (n ), 37 is fixed by the bottom and top masses k2/k1 mb/mt . Moreover, L L R 102 the upper 2 ×2 blocks in Mu and Md are proportional. This implies 38 m2 ∼ O(n2 + v2 ). 103 X R R 39 that, in this limit the CKM matrix is trivial. As a result in our model 104 ρ generates all entries of the quark mixing matrix, the Cabibbo 40 4. Scalar potential 105 41 angle, V ub as well as V cb. 106 42 For the situation is qualitatively different. The charged 107 mass matrix is simply given by The most general CP conserving scalar potential compatible 43 with all the symmetries of the model is 108 44 k 109  = √2  = + + + 45 mab hab , (12) V V  V φ Vρ V mix, (20) 110 2 46 111 q with 47 112 and the new leptons χL,R form heavy Dirac pairs with masses 48 2 † † 113 n V  = μ Tr(L  ) + Tr(R  ) 49 χ = √   L R 114 mab hab . (13) 50 2 + † 2 + † 2 115 λ1 Tr(L L ) Tr(R R ) 51 Concerning , their mass matrix can be written as 116 52 † † † † 117 + λ2 Tr(L  L  ) + Tr(R  R  ) , 53 ML mD L L R R 118 m = , (14) 54 ν T 2 † † 2 † † 119 mD M R V φ = μ Tr(φφ ) + λ3Tr(φφ ) + λ4Tr(φφ φφ ) 55 φ 120 56 where + fφ(φφφ + h.c.), 121 57 = =  = 2 † † 2 † † 122 ML 2 fab v L , mD habk1, M R 2 fab v R . (15) Vρ = μ Tr(ρρ ) + λ5Tr(ρρ ) + λ6Tr(ρρ ρρ ), 58 ρ 123 59 Thus we obtain a combination of type I and type II seesaw mech- † † † † † T ∗ 124 V mix = λ7Tr(φφ )Tr(ρρ ) + λ8 Tr(φφ ρρ ) + Tr(φ φρ ρ ) 60 anisms, the same situation than in the SU(2)L ⊗ SU(2)R models: 125 † † 61 −1 T + 126 m ≈ M − m M m , m ≈ M . (16) λ9Tr(R R )Tr(L L ), (21) 62 1 L D R D 2 R 127 † † † 63 + + 128 We now turn to the gauge boson masses which come as usual λ10Tr(φφ ) Tr(R R ) Tr(L L ) 64 from their couplings with the scalars present in the theory. The 129 † † 65 + † + † 130 relevant covariant derivative is defined as λ11 Tr(φ φR R ) Tr(φφ L L ) JID:PLB AID:32515 /SCO Doctopic: Phenomenology [m5Gv1.3; v1.194; Prn:28/12/2016; 9:09] P.4(1-6) 4 M. Reig et al. / Physics Letters B ••• (••••) •••–••• ∗ 1 + T + 66 λ12 Tr(φR φ L ) h.c. 2 67 3 + † † + † 68 λ13Tr(ρρ ) Tr(R R ) Tr(L L ) 4 69 5 + T ∗ † + † † 70 λ14 Tr(ρ ρ R R ) Tr(ρρ LL ) 6 71 7 T ∗ † 72 + λ15Tr(φρ φ ρ ). 8 73 9 The extremum conditions can be solved in terms of the dimen- 74 10 sionful parameters of the potential 75 11 76 1 12 2 =− 2 + + 2 + 2 + 2 + 2 + 2 77 μ 2v R (λ1 λ2) v L λ9 (n k1 k2)λ10 k1λ11 13 2 78 14 v 79 + L 2 + 2 15 k1λ12 k3λ13 , 80 v R 16 81 1 17 2 =− 2 + 2 + 2 + 2 + 2 + 2 + 2 82 μφ (v L v R )λ10 2(n k1 k2)λ3 (n 2k2)λ4 18 2 83 19 84 k2k2 20 2 2 3 85 + k λ7 − λ8 , (22) 3 2 − 2 21 n k2 86 22 87 2 1 2 2 2 2 2 2 23 μ =− 2k (λ5 + λ6) + (n + k + k )λ7 + (k + k )λ8 88 ρ 2 3 1 2 1 2 24 89 25 90 + 2 + 2 + 2 Fig. 1. Phase diagram of the SU(3)c ⊗ SU(3)L ⊗ SU(3)R ⊗ U(1)X electroweak theory, (v R v L )λ13 v L λ14 , 26 discussed in Sec. 5, see also Fig. 2. The Standard Model singlet VEVs n and v R are 91 in GeV units and their ratio is determined by Eq. (25). 27 2 − 2 − 2 92 nk2 2(n k2)λ4 k3λ8 28 fφ = √ , 93 2 − 2 29 6 2k1(n k2) 5. Spontaneous symmetry breaking 94 30 95 together with the conditions: 31 In order to recover the Standard Model at low energies we 96 32 need to break the SU(3)c ⊗ SU(3)L ⊗ SU(3)R ⊗ U(1)X symmetry. The 97 k2 33 + − − 3 − 2 = breaking of the gauge structure can be achieved in several ways 98 v L v R 2(λ1 λ2) λ9 λ14 λ12k1 0 , 34 2 − 2 (see Fig. 2) depending on the value of n/v . For n > v k 99 v R v L R R 1,2,3 35 the symmetry breaking pattern is: 100 2 − 2 36 n k2 2 2 λ11 2 2 101 (k − k )λ4 − (v + v ) − λ12 v L v R ⊗ ⊗ 37 2 1 2 2 L R SU(3)L SU(3)R U(1)X 102 k1 (23) 38 −→n ⊗ ⊗ ⊗ 103 2 2 2 2 SU(2)L SU(2)R U(1)B−L U(1)A 39 n k (k − k ) 104 3 1 2 v − λ8 = 0 , −−→R ⊗ 40 2 2 − 2 SU(2)L U(1)Y . (26) 105 2k1(n k2) 41 = √1 8 106 At this stage one has φ diag(0, 0, n) breaking the T L,R gen- 42 λ15 = 0 . 2 107 8 + 8 43 erators but preserving T L T R , and since φ carries no X charge, 108 k2 ⊗ ⊗ ⊗ 44 Assuming  ≡ 3 1 and natural values for the quartic cou- the resulting gauge group is SU(2)L SU(2)R U(1)B−L U(1)A, 109 v2 −v2 45 R L with 110 plings, the first condition leads to the well-known VEV seesaw 46 B − L 111 relation = 8 + 8 + 47 β(T L T R ) X , 112 2 (27) 48 2 113 λ12k1 = 8 + 8 − 49 v L v R = , (24) A β(T L T R ) X . 114 2(λ1 + λ2) − λ9 − λ14 50 It is important to notice that A is not involved in the electric 115 51 which characterizes dynamically the seesaw mechanism. This is charge definition since it reads 116 52 consistent with the hierarchy between the VEVs v k v and 117 R 1 L B − L 53 consequently, the second condition reduces to = 3 + 3 + 8 + 8 + ≡ 3 + 3 + 118 Q T R T L β(T L T R ) X T R T L . (28) 54 2 119 55 (k2 − k2) λ =     120 2 1 2 11 2 The potential acquires the form V V (φ , ρ , R , L , ...) where λ4n ≈ v (25) 56 2 2 R 121 k1 + 57  ∼ ∗  ∼ − 4q 2 122 φ (2L, 2R, 0, 0), ρ (2L, 2R, 0, ), 58 at leading order in the Standard Model singlet VEVs n and v R . The 3 123 59 latter shows clearly, as seen in Fig. 1, that the primordial SU(3) ⊗ 124 c  1 − 4q  1 − 4q 60 SU(3) ⊗ SU(3) ⊗ U(1) theory can break either directly to the  ∼ (3L, 1R, −1, ),  ∼ (1L, 3R, −1, ), 125 L R X L 3 R 3 61 Standard Model (central part of the plot) or through the interme- 126 (29) 62 diate SU(3)c ⊗ SU(2)L ⊗ SU(2)R ⊗ U(1)B−L or SU(3)c ⊗ SU(3)L ⊗ U(1) 127 63 phases, corresponding to the upper and lower regions, respectively. are the 2 × 2 upper left sub-matrices contained in the original 128 64 This behavior is mainly controlled by the quartic parameters λ4 SU(3)c ⊗ SU(3)L ⊗ SU(3)R ⊗ U(1)X scalar multiplets in notation 129 − 65 B L 130 and λ11 in the scalar potential. More details in the next section. (SU(2)L, SU(2)R, 2 , A), and the dots stand for the extra scalars. JID:PLB AID:32515 /SCO Doctopic: Phenomenology [m5Gv1.3; v1.194; Prn:28/12/2016; 9:09] P.5(1-6) M. Reig et al. / Physics Letters B ••• (••••) •••–••• 5

1 66 2 67 3 68 4 69 5 70 6 71 7 72 8 73 9 74 10 75 11 76 12 77 13 78 Fig. 2. Spontaneous symmetry breaking paths in the SU(3) ⊗ SU(3) ⊗ SU(3) ⊗ U(1) electroweak theory. See also Fig. 1 and the VEV seesaw relation in Eq. (24) as well as 14 c L R X 79 the breaking pattern determining condition in Eq. (25). 15 80 16 81 A √1 T Apart from the existence of the scalar multiplet ρ and the In a second step the triplet φ = (0, 0, n) breaks the SU(3)L ⊗ 17 2 82 extra symmetry U(1) the situation here resembles the popular   18 A SU(2)R ⊗ U(1)X symmetry down to the Standard Model. For this 83 SU(3) ⊗ SU(2) ⊗ SU(2) ⊗ U(1) − electroweak model in [6]. The 19 c L R B L VEV hierarchy, signals associated to exotic fermions are expected 84 second step of the spontaneous symmetry breaking is carried out 20 at intermediate energies, while new physics related to left–right 85 √1 3 by R = diag(v R , 0, 0). We note here that the generators T 21 2 R symmetry, like neutrino masses, emerges at higher energies. 86 22 and A are broken at this stage, though the combination Finally, a third situation in which n ∼ v R k1,2,3 is also possi- 87 ⊗ ⊗ ⊗ 23 − ble. The SU(3)c SU(3)L SU(3)R U(1)X gauge group in this case 88 3 B L 24 T + = Y , (30) is broken directly to the that of the Standard Model: 89 R 2 25 90 ⊗ ⊗ −n−−, v→R ⊗ 26 remains unbroken and hence can be identified with the Standard SU(3)L SU(3)R U(1)X SU(2)L U(1)Y . (34) 91 Model hypercharge. In this scenario, an SU(2)L ⊗ SU(2)R ⊗ U(1)B−L 27 In this scenario one expects new physics associated to left–right 92 28 structure emerges as the effective theory at lower scales. Moreover, 93 in this case, at energy scales above v , we expect to observe new and 331 symmetries at comparable energy scales. 29 R 94 physics associated to a 331 model, such as virtual effects associ- 30 6. Discussion and conclusion 95 ated with exotic quarks and leptons and new gauge bosons, even 31 96 if these particles are too heavy to show up directly. 32 We have proposed a fully realistic scheme based on the 97 Alternatively, if the VEV hierarchy is v R > n k1,2,3, the sym- 33 SU(3) ⊗ SU(3) ⊗ SU(3) ⊗ U(1) gauge group. Quantum con- 98 metry breakdown follows a different route2 c L R X 34 sistency requires that the number of families must match the 99 35 v R 100   number of colors, hence predicting the number of generations. In SU(3)L ⊗ SU(3)R ⊗ U(1)X −−→ SU(3)L ⊗ U(1)X ⊗ SU(2)R 36 the simplest realization neutrino masses arise from a dynamical 101 −→n ⊗ 37 SU(2)L U(1)Y . (31) seesaw mechanism in which the smallness of neutrino masses is 102 38 √1 correlated with the observed V-A nature of the weak interaction. 103 Now R = diag(v R , 0, 0) breaks the SU(3)R group down to 39 2 √ 104 6 7 1 8 3 Depending on the symmetry breaking path to the Standard Model 40 SU(2)R , generated by {T , T , ( 3T − T )}. Simultaneously, 105 R R 2 R R √ (see Figs. 1 and 2) one recovers either a SU(3)c ⊗SU(2)L ⊗SU(2)R ⊗  β+ 3 41 = 8 + U(1) − theory or one based on the SU(3) ⊗SU(3) ⊗U(1)  gauge 106 √U(1)X is broken by v R but the combination X 4 (T R B L c L X 42 3 + symmetry. This illustrates the versatility of the theory since, de- 107 3T R ) X is preserved so our theory becomes an effective 331 43 pending on a rather simple input parameter combination, it can 108 model with an additional SU(2)R symmetry at intermediate ener- 44 109 gies. In terms of the generators of the intermediate symmetries, mimic either of two apparently irreconcilable pictures of nature: 45 110 electric charge reads one based upon left–right symmetry and another characterized by 46 111 the SU(3)c ⊗ SU(3)L ⊗ U(1)X gauge group. Either of these could be 47 = 3 + 3 + 8 + 8 + 112 Q T L T R β(T L T R ) X the “next” step in the quest for new physics. 48 √ 113 3β − 1 √ 49 ≡ 3 + 8 + 8 − 3 +  Acknowledgements 114 T L βT L ( 3T R T R ) X . (32) 50 4 115 51 The potential in this case can be written as V = V (φ A , φ B , ρ A , ρ B , We thank Renato Fonseca, Martin Hirsch, P.V. Dong and D.T. 116 52  Huong for useful discussions. Work supported by Spanish grants 117 L , ...), where the relevant fields transform under (SU(3)L, 53  118 SU(2)R , X ) as FPA2014-58183-P, Multidark CSD2009-00064, SEV-2014-0398 54 − − (MINECO) and PROMETEOII/2014/084 (Generalitat Valenciana). 119 55 A q 1 B 1 q 120 φ ∼ (3L, 1R , ), φ ∼ (3L, 2R , ), C.A.V-A. acknowledges support from CONACYT (Mexico), grant 56 3 6 274397. M. R. was supported by JAEINT-16-00831. 121 57 + + 122 A q 2 B 5q 1 ρ ∼ (3L, 1R , ), ρ ∼ (3L, 2R , ), (33) 58 3 6 Appendix A. Anomaly cancellation in 123 59 124 SU(3)c ⊗ SU(3)L ⊗ SU(3)R ⊗ U(1)X  2(q − 1) 60 ∼  125 L (6L, 1R , ). 61 3 In this section we outline the anomaly cancellation in the 126 62 SU(3)c ⊗ SU(3)L ⊗ SU(3)R ⊗ U(1)X model. The potential anoma- 127 2 3 3 2 63  lies are [SU(3) ] ⊗ U(1) , [SU(3) ] , [SU(3) ] , [SU(3) ] ⊗ U(1) , 128 2 Notice that this extra SU(2) group comes from the fact we keep β arbitrary: c X L R L X 64 R [ ]2 ⊗ [ ]2 ⊗ [ ]3 129 it is easy to see that one can recover the usual 331 [9] electroweak group for q = 0, SU(3)R U(1)X, Grav U(1)X, U(1)X . First notice that the 65 2 130 which allows 33 to have a non-zero vev. [SU(3)c] ⊗ U(1)X anomaly cancels straightforwardly JID:PLB AID:32515 /SCO Doctopic: Phenomenology [m5Gv1.3; v1.194; Prn:28/12/2016; 9:09] P.6(1-6) 6 M. Reig et al. / Physics Letters B ••• (••••) •••–••• 1 XqL − XqR Notice that L-R symmetry plays an important role in anomaly can- 66 2 67 quarks quarks cellation since it automatically implies that the fermion content 3 68 q q + 1 of the model satisfy XL = X R for every multiplet and all parti- 4 = − × 3 × 3 × 2 + × 3 × 3 69 3 3 cles are arranged in chiral multiplets. We also remark that in our 5 ⊗ ⊗ ⊗ 70 q q + 1 SU(3)c SU(3)L SU(3)R U(1)X theory the anomaly cancellation 6 − − × 3 × 3 × 2 − × 3 × 3 = 0 . (A.1) holds irrespective of the value of the β parameter of the electric 71 7 3 3 72 charge generator in Eq. (1). 8 Since triplets and anti-triplets contribute with opposite sign to the 73 9 [ ]3 [ ]3 74 SU(3)L and SU(3)R anomalies, these are only canceled if the References 10 number of triplets is equal to the number of anti-triplets, which 75 11 3 3 76 is the case in our model. Thus from [SU(3)R] and [SU(3)L] we 12 [1] S.L. Adler, Axial vector vertex in electrodynamics, Phys. Rev. 177 (1969) 77 conclude that if f is the number of lepton families, n is the num- 2426–2438. 13 78 ber of quark triplets and m is the number of quark anti-triplets the [2] J. Bell, R. Jackiw, A PCAC puzzle: π 0 → γγ in the sigma model, Nuovo Cimento 14 following equation must hold: A 60 (1969) 47–61. 79 15 × + × × = × × → = − [3] J.C. Pati, A. Salam, Lepton number as the fourth color, Phys. Rev. D 10 (1974) 80 16 3 f 3 3 n 3 3 m f 3(m n). (A.2) 275–289. 81 2 2 [4] M. Gell-Mann, P. Ramond, R. Slansky, Complex and unified theories, 17 Next, we consider [SU(3)L] ⊗ U(1)X and [SU(3)R] ⊗ U(1)X 82 Conf. Proc. C790927 (1979) 315–321, arXiv:1306.4669 [hep-th]. 18 which are canceled independently because the sum of all X 83 [5] T. Yanagida, Horizontal symmetry and masses of neutrinos, Conf. Proc. 19 charges is equal to zero: 84 C7902131 (1979) 95–99. 20 85 q − 1 q [6] R.N. Mohapatra, G. Senjanovic, Neutrino mass and spontaneous parity noncon- 21 XL = × 3 × 3 + − × 3 × 3 × 2 servation, Phys. Rev. Lett. 44 (1980) 91. 86 22 3 3 [7] J. Schechter, J. Valle, Neutrino masses in SU(2) × U(1) theories, Phys. Rev. D 22 87 fermions (1980) 2227. 23 q + 1 88 [8] G. Lazarides, Q. Shafi, C. Wetterich, Proton lifetime and fermion masses in an 24 + × 3 × 3 = 0 , 89 3 SO(10) model, Nucl. Phys. B 181 (1981) 287. 25 90 (A.3) [9] M. Singer, J. Valle, J. Schechter, Canonical neutral current predictions from the 26 q − 1 q weak electromagnetic gauge group SU(3) × u(1), Phys. Rev. D 22 (1980) 738. 91 X R = × 3 × 3 + − × 3 × 3 × 2 27 3 3 [10] J.W.F. Valle, M. Singer, Lepton number violation with quasi Dirac neutrinos, 92 28 fermions Phys. Rev. D 28 (1983) 540. 93 29 q + 1 [11] S.M. Boucenna, S. Morisi, J.W.F. Valle, Radiative neutrino mass in 3-3-1 scheme, 94 + × 3 × 3 = 0 . Phys. Rev. D 90 (1) (2014) 013005, arXiv:1405.2332 [hep-ph]. 30 3 95 [12] S.M. Boucenna, R.M. Fonseca, F. Gonzalez-Canales, J.W.F. Valle, Small neutrino 31 masses and gauge coupling unification, Phys. Rev. D 91 (3) (2015) 031702, 96 32 In more general terms, this condition is the equivalent to arXiv:1411.0566 [hep-ph]. 97 33 q − 1 q [13] J.C. Montero, C.A. de S. Pires, V. Pleitez, Neutrino masses through the see- 98 34 XL = × 3 × f + − × 3 × 3 × m saw mechanism in 3-3-1 models, Phys. Rev. D 65 (2002) 095001, arXiv:hep- 99 3 3 ph/0112246. 35 fermions 100 [14] M.E. Catano, R. Martinez, F. Ochoa, Neutrino masses in a 331 model with right- 36 q + 1 101 + × 3 × 3 × n = 0 . (A.4) handed neutrinos without doubly charged Higgs bosons via inverse and double 37 3 seesaw mechanisms, Phys. Rev. D 86 (2012) 073015, arXiv:1206.1966 [hep-ph]. 102 38 [15] A. Addazi, J.W.F. Valle, C.A. Vaquera-Araujo, String completion of an SU(3)c ⊗ 103 Plugging the previous result f = 3(m − n) into the above relation ⊗ 39 SU(3)L U(1)X electroweak model, Phys. Lett. B 759 (2016) 471–478, arXiv: 104 implies that m = 2n and f = 3n, independently of the value of q. 1604.02117 [hep-ph]. 40 105 This condition is complemented by QCD asymptotic freedom, that [16] J.W.F. Valle, C.A. Vaquera-Araujo, Dynamical seesaw mechanism for Dirac neu- 41 106 requires the number of quark flavors to be less than or equal to 16 trinos, Phys. Lett. B 755 (2016) 363–366, arXiv:1601.05237 [hep-ph]. 42 ⊗ ⊗ 107 leading to 3 × (m + n) = 9n ≤ 16, leaving only one positive integer [17] M. Reig, J.W.F. Valle, C.A. Vaquera-Araujo, Realistic SU(3)c SU(3)L U(1)X 43 model with a type II Dirac neutrino seesaw mechanism, Phys. Rev. D 94 (3) 108 solution n = 1, hence we must have f = 3 generations. (2016) 033012, arXiv:1606.08499 [hep-ph]. 44 2 109 Finally, the theory is free from gravitational anomaly [Grav] ⊗ × 45 [18] A.G. Dias, C.A. de S. Pires, P.S. Rodrigues da Silva, The left–right SU(3)(L) 110 U(1)X if SU(3)(R) × U(1)(X) model with light, keV and heavy neutrinos, Phys. Rev. D 82 46 111 (2010) 035013, arXiv:1003.3260 [hep-ph]. 47 112 XL = X R , (A.5) [19] C.P. Ferreira, M.M. Guzzo, P.C. de Holanda, Cosmological bounds of sterile neu- 48 ⊗ ⊗ ⊗ 113 fermions fermions trinos in a SU(3)C SU(3)L SU(3)R U (1)N model as dark matter candi- 49 dates, Braz. J. Phys. 46 (4) (2016) 453–461, arXiv:1509.02977 [hep-ph]. 114 [ ]3 50 which is trivially satisfied, while the cancellation of the U(1)X [20] D.T. Huong, P.V. Dong, Left–right asymmetry and 750 GeV diphoton excess, 115 Phys. Rev. D 93 (9) (2016) 095019, arXiv:1603.05146 [hep-ph]. 51 anomaly follows from 116 [21] P.V. Dong, D.T. Huong, Left–right model for dark matter, arXiv:1610.02642 52 117 X3 − X3 [hep-ph]. L R  53 [22] A.J. Buras, F. De Fazio, J. Girrbach-Noe, Z-Z mixing and Z-mediated FCNCs in 118 f L ,qL f R ,qR 54 SU(3)C × SU(3)L × U(1)X models, J. High Energy Phys. 1408 (2014) 039, arXiv: 119 3 3 55 q − 1 q 1405.3850 [hep-ph]. 120 = × 3 × 3 + − × 3 × 3 × 2 56 3 3 [23] R.M. Fonseca, M. Hirsch, A flipped 331 model, J. High Energy Phys. 08 (2016) 121 003, arXiv:1606.01109 [hep-ph]. 57 3 3 122 q + 1 q − 1 [24] R.M. Fonseca, M. Hirsch, Lepton number violation in 331 models, arXiv:1607. 58 + × 3 × 3 − × 3 × 3 123 3 3 06328 [hep-ph]. 59 [25] P. Langacker, G. Steigman, Requiem for an FCHAMP? Fractionally CHArged, Mas- 124 3 60 q 3 q + 1 sive Particle, Phys. Rev. D 84 (2011) 065040, arXiv:1107.3131 [hep-ph]. 125 − − × 3 × 3 × 2 − × 3 × 3 = 0(A.6) 61 3 3 126 62 127 63 128 64 129 65 130