Physics Letters B 637 (2006) 85–89 www.elsevier.com/locate/physletb 3–3–1 models at electroweak scale Alex G. Dias a,∗, J.C. Montero b, V. Pleitez b a Instituto de Física, Universidade de São Paulo, Caixa Postal 66.318, 05315-970 São Paulo, SP, Brazil b Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 São Paulo, SP, Brazil Received 10 January 2006; received in revised form 26 February 2006; accepted 6 April 2006 Available online 24 April 2006 Editor: M. Cvetič Abstract We show that in 3–3–1 models there exist a natural relation among the SU(3)L coupling constant g, the electroweak mixing angle θW , the mass of the W , and one of the vacuum expectation values, which implies that those models can be realized at low energy scales and, in particular, even at the electroweak scale. So that, being that symmetries realized in Nature, new physics may be really just around the corner. © 2006 Elsevier B.V. All rights reserved. PACS: 12.10.Dm; 12.10.Kt; 14.80.Mz Keywords: 3–3–1 model; Neutral currents 1. Introduction Many of the extension of the Standard Model (SM) implies the existence of at least one extra neutral vector boson, say Z′, which should have a mass of the order of few TeV in order to be consistent with present phenomenology. This is the case, for instance, in left–right models [1], any grand unified theories with symmetries larger than SU(5) as SO(10) and E6 [2], li- ttle Higgs scenarios [3], and models with extra dimensions [4]. This makes the search for extra neutral gauge bosons one of the main goals of the next collider experiments [5]. Usually, the interactions involving Z′ are parametrized (besides the pure kinetic term) as [6,7] LNC(Z′) = − sin ξ 2 F ′ μνF μν + M2 Z′Z′ μZ′μ + δM2Z′ μZμ (1)− g 2cW ∑ i ψ̄iγ μ ( f i V − f i Aγ 5)ψiZ ′ μ, where Z, which is the would be neutral vector boson of the SM, and Z′ are not yet mass eigenstates, having a mixing defined by the angle tan 2φ = δM2/(M2 Z′ − M2 Z); cW ≡ cos θW (and for * Corresponding author. E-mail address: alexdias@fma.if.usp.br (A.G. Dias). 0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.04.015 future use sW ≡ sin θW ) the usual parameter defined through the electroweak mixing angle θW . If Z1 and Z2 denote the mass eigenstates, then in most of the models MZ2 � MZ1 ≈ MZ . In this situation the vector and axial-vector couplings, g (SM) V and g (SM) A , respectively, of the SM Z boson with the known fermions are modified at tree level as follows: (2)gi V = g i(SM) V cφ + f i V sφ, gi A = g i(SM) A cφ + f i Asφ, where g i(SM) V = T i 3 − 2Qis 2 W and g i(SM) A = T i 3 , being T i 3 = ±1/2 and Qi the electric charge of the fermion i; we have used the notation cφ(sφ) = cosφ(sinφ). The coefficients f i V,A in Eq. (2) are not in general the same for all particles of the same electric charge, thus, Z′ induces flavor changing neutral currents (FCNC) which imply strong constraints coming from experimental data such as MK and other | S| = 2 processes. These constraints imply a small value for the mixing angle φ or, similarly, a large value to the energy scale, generically de- noted by Λ, related with the larger symmetry. If sφ = 0 is im- posed such constraints could be avoided, however in most of the models with Z′ this usually implies a fine tuning among U(1) charges and vacuum expectation values, that is far from being natural [8]. Here we will show that there are models in which, at the tree level, it is possible that: (i) there is no mixing between Z and Z′, http://www.elsevier.com/locate/physletb mailto:alexdias@fma.if.usp.br http://dx.doi.org/10.1016/j.physletb.2006.04.015 86 A.G. Dias et al. / Physics Letters B 637 (2006) 85–89 and the latter boson may have a mass even below the TeV scale; (ii) ρ0 = 1 since MZ1 = MZ , and (iii) the couplings of Z1 with fermions, gi V,A, being exactly those of the SM, g i(SM) V ,A , no mat- ter how large is Λ. This is implied not by a fine tuning but by a condition which can be verified experimentally involving the parameters of the model, g, MW , sW and one vacuum expecta- tion value (VEV). 2. The model The so-called 3–3–1 models are interesting extensions of the standard model [9–11] in which it is possible to explain the number of generations and they are also very predictive con- cerning new theoretical ideas as extra dimensions [12] and the little Higgs mechanism [13]. Those models also include an ex- tra neutral vector boson so that there is, in general, a mixing of Z, the vector boson of SU(2)L ⊂ SU(3)L, and Z′ the gauge bo- son related to the SU(3)L symmetry. Working in the Z,Z′ basis (the parameterization in Eq. (1) is also valid but in these mod- els there is no mixing in the kinetic term i.e., sin ξ = 0) it means that the condition sinφ � 1 can be obtained if in this case the energy scale Λ ≡ vχ , with vχ related to the SU(3)L symmetry, is above the TeV scale. Hence, it is usually believed that only approximately we can have that Z1 ≈ Z, even at the tree level. The same happens with the neutral current couplings, gi V,A, which only approximately coincide with g i(SM) V ,A . This is true since the corrections to the Z mass and gi V,A in these models, assuming vχ � vW 246 GeV, are proportional to (vW/vχ)2 and for vχ → ∞ we recover exactly the SM with all its degrees of freedom, with the heavier ones introduced by the SU(3)L symmetry decoupled. However, we expect that vχ should not be extremely large if new physics is predicted to show up in the near future experiments. In practice, measurements of the ρ0 parameter, and FCNC processes like MK , should impose constraints upon the vχ scale at which the SU(3)L symmetry arises. Let us consider, for instance, the model of Ref. [9] in which the electric charge operator is defined as Q= (T3 −√ 3T8)+X, where Ti are the usual SU(3) generators and X the charge as- signed to the Abelian factor U(1)X . Thus, the SM fermionic content is embedded in the extended group according to the multiplets transforming under SU(3)L ⊗U(1)X as: for leptons, ΨaL = (νa, l − a , (l−)ca) T L ∼ (3,0), a = e,μ, τ (the superscript c means charge conjugation operation); and for quarks, QmL = (dm,um, jm)TL ∼ (3∗,−1/3); m = 1,2; Q3L = (u3, d3, J )TL ∼ (3,2/3), uαR ∼ (1,2/3), dαR ∼ (1,−1/3), α = 1,2,3, jmR ∼ (1,−4/3), and JR ∼ (1,5/3). Here jm and J are new quarks needed to complete the representations. To generate masses for all these fields through spontaneous symmetry breaking three triplets of Higgs scalars and a sextet are introduced; they are η = (η0, η− 1 , η+ 2 )T ∼ (3,0), ρ = (ρ+, ρ0, ρ++)T ∼ (3,+1), χ = (χ−, χ−−, χ0)T ∼ (3,−1) and (3)S = ⎛ ⎝ σ 0 1 h− 1 h+ 2 h− 1 H−− 1 σ 0 2 h+ 2 σ 0 2 H++ 2 ⎞ ⎠ ∼ (6,0). The VEVs in the neutral components of the scalar multiplets are defined as 〈η0 1〉 = vη/ √ 2, 〈ρ0 1〉 = vρ/ √ 2, 〈χ0 1 〉 = vχ/ √ 2 and 〈σ 0 2 〉 = vs/ √ 2. It is also possible to have 〈σ 0 1 〉 �= 0 giving Majorana mass to the neutrinos, but we will not be concerned with this here. The VEV 〈χ0 1 〉 reduces the symmetry to the SM SU(2)L ⊗ U(1)Y symmetry and the other VEVs further reduce it to the electromagnetic U(1)Q factor. From the kinetic terms for the scalar fields, constructed with the covariant derivatives Dμϕ = ∂μϕ − ig �Wμ · �T ϕ − igXXϕBμ, (4)DμS = ∂μS − ig [ �Wμ · �T S + S �Wμ · �T T ] , where gX denotes the U(1)X gauge coupling constant and ϕ = η,ρ,χ , we obtain the mass matrices for the vector bosons. Besides W± there are two other charged vector bosons, V ± and U±±. The masses of these charged vector bosons are given exactly by M2 W = (g2/4)v2 W , M2 V = (g2/4)(v2 η + 2v2 s + v2 χ ) and M2 U = (g2/4)(v2 ρ + 2v2 s + v2 χ ), where we have defined v2 W = v2 η + v2 ρ + 2v2 s (in models where there are heavy leptons transforming nontrivially under SU(3)L ⊗ U(1)X there is not the contribution of the sextet and the above equations are still valid simply doing vs = 0 [10]). For the mass square matrix of the neutral vector bosons in this model we have the follow- ing form, after defining the dimensionless ratios v̄ρ = vρ/vχ , v̄W = vW/vχ and the parameter t2 = g2 X/g2 = s2 W/(1 − 4s2 W), M2 (5) = g2 4 v2 χ ⎛ ⎜⎜⎝ v̄2 W 1√ 3 (v̄2 W − 2v̄2 ρ) −2t v̄2 ρ 1√ 3 (v̄2 W − 2v̄2 ρ) 1 3 (v̄2 W + 4) 2√ 3 t (v̄2 ρ + 2) −2t v̄2 ρ 2√ 3 t (v̄2 ρ + 2) 4t2(v̄2 ρ + 1) ⎞ ⎟⎟⎠ , in the (W 3 μ,W 8 μ,Bμ) basis. This matrix has a zero eigenvalue corresponding to the photon and two nonzero ones which are given by (6)M2 Z1 = g2v2 χ 6 [ 3t2(v̄2 ρ + 1 ) + 1 + v̄2 W ] (1 − R), (7)M2 Z2 = g2v2 χ 6 [ 3t2(v̄2 ρ + 1 ) + 1 + v̄2 W ] (1 + R), with (8)R = [ 1 − 3(4t2 + 1)(v̄2 W(v̄2 ρ + 1) − v̄4 ρ) (3t2(v̄2 ρ + 1) + 1 + v̄2 W)2 ]1/2 . 3. ρ1 and ρ0 parameters In order to analyze the condition which allows to iden- tify Z1 of the 3–3–1 model with the Z of the SM, let us in- troduce a dimensionless ρ1-parameter defined at the tree level as ρ1 = c2 WM2 Z1 /M2 W . As we can see from Eqs. (6) and (7) both mass eigenvalues, MZ1 and MZ2 , have a complicate depen- dence on the VEVs but we observe from Eq. (6) that ρ1 � 1 (or, MZ1 � MZ) is a prediction of the model. Next, we can search for the conditions under which we have ρ1 ≡ ρ0 = 1, where A.G. Dias et al. / Physics Letters B 637 (2006) 85–89 87 ρ0 = c2 WM2 Z/M2 W is the respective parameter in the SM. This is equivalent to the condition that MZ1 ≡ MZ at the tree level. The equation ρ1 = 1 has besides the solution vχ → ∞, another less trivial one which can be obtained using Eq. (6) above: (9)v̄2 ρ = 1 − 4s2 W 2c2 W v̄2 W . The condition in Eq. (9) implies, using the definition of vW given above also v2 η + 2v2 s = [(1 + 2s2 W)/2c2 W ]v2 W . We recall that the U(1)X quantum number of the η and S fields are differ- ent from that of the ρ field, so there is no symmetry among the respective VEVs. We have verified that (9) is stable in the fol- lowing sense: small deviations from it implies small deviations from ρ1 = 1. With s2 W = 0.2312 [7] we obtain vρ ≈ 54 GeV and √ v2 η + 2v2 s ≈ 240 GeV. Notice that Eq. (9) is independent of the vχ scale. Hence, all consequences of it will be also in- dependent of vχ as claimed above in the Introduction. The fact that vχ does not need to have a large value to be consistent with the present phenomenology is interesting in the models of Refs. [9,10] since these models have a Landau-like pole at the TeV scale [14]. If we substitute Eq. (9) in Eqs. (6) and (7) we obtain M2 Z1 = (g2/4c2 W)v2 W ≡ MZ and (10) M2 Z2 ≡ M2 Z′ = g2v2 W 2 (1 − 2s2 W)(4 + v̄2 W) + s4 W(4 − v̄4 W) 6c2 W(1 − 4s2 W) v2 χ . Thus, assuming that Eq. (9) is valid we shall not distinguish between Z1 and Z and between Z2 and Z′ unless stated ex- plicitly. Moreover, the mass of Z′ can be large even if vχ is of the order of the electroweak scale. In fact, from Eq. (10) we see that for v̄W = 1 (the electroweak scale is equal to the 3–3–1 scale) we obtain MZ′ = 3.77MW . Of course for lower values of v̄W , Z′ is heavier, for instance for v̄W = 0.25 we have MZ′ = 18.36MW . We recall that since vχ does not con- tribute to the W mass it is not constrained by the 246 GeV upper bound. Thus, independently if v̄2 W is larger, smaller or equal to 1, the charged vector boson V is heavier than U , being M = √ M2 V − M2 U = 75.96 GeV, when vs = 0. 4. Neutral current couplings We have also obtained the full analytical exact expressions for the neutral current couplings gi V,A and f i V,A, and verified that they also depend on the VEVs in a complicated way. But when Eq. (9) is used in those expressions we obtain for the case of the known fermions gi V,A ≡ g i(SM) V ,A , and f i V,A = f i V,A(sW ), i.e., these couplings depend only on the electroweak mixing an- gle. For the lepton couplings with Z′, also after using Eq. (9) in the general expressions, we obtain f ν V = f ν A = f l V = −f l A = − √ 3(1 − 4s2 W)/6(≈ −0.07). We see that the couplings for all leptons with Z′ are leptophobic [15]. In particular, the cou- plings of Z to the exotic quarks jm and J are given by g jm V = (8/3)s2 , gJ = −(10/3)s2 and g jm = gJ = 0. Notice also that W V W A A the exotic quarks have pure vectorial couplings with Z. The couplings of Z′ in the quark sector are given by: f um V = 1 2 √ 3 1 − 6s2 W√ 1 − 4s2 W , f um A = 1 2 √ 3 1 + 2s2 W√ 1 − 4s2 W , f u3 V = − 1 2 √ 3 1 + 4s2 W√ 1 − 4s2 W , f u3 A = − 1√ 3 √ 1 − 4s2 W, f dm V = 1 2 √ 3 √ 1 − 4s2 W , f dm A = √ 1 − 4s2 W 2 √ 3 , f d3 V = − 1 2 √ 3 1 − 2s2 W√ 1 − 4s2 W , f d3 A = − 1 2 √ 3 1 + 2s2 W√ 1 − 4s2 W , f jm V = − 1√ 3 1 − 9s2 W√ 1 − 4s2 W , f jm A = − 1√ 3 c2 W√ 1 − 4s2 W , (11)f J V = 1√ 3 1 − 11s2 W√ 1 − 4s2 W , f J A = 1√ 3 c2 W√ 1 − 4s2 W . In literature [9,16] these couplings were considered as an ap- proximation of the exact couplings. Notice that, all these cou- plings refer to fermions which are still symmetry eigenstates, thus we see that in the leptonic sector there are not FCNCs nei- ther with Z nor with Z′ and, in the quark sector there are FCNC only coupled to Z′ as can be see from Eq. (11). However, FCNC mediated by the Z′ depend only on its mass, but these FCNC are not necessary large since there are also contributions in the scalar sector (see below). The main feature introduced by the validity of the condition Eq. (9), which we would like to stress, is the fact that all cou- plings between the already known particles are exactly those of the SM, regardless the value of the vχ scale. Hence, vχ is not required to be large to recover those observed couplings (until now vχ → ∞ was the usual approach to do that). In this way, the 3–3–1 gauge symmetry could be realized, for instance, at the electroweak scale (vχ = vW ) allowing the extra particles in- troduced by the SU(3)L to be light enough to not decouple and be discovered in the near future experiments. 5. A Goldberger–Treiman-like relation We can rewrite Eq. (9) as (12)g vρ√ 2 = √ 1 − 4s2 W cW MW. This is like the Goldberger–Treiman relation [17] in the sense that its validity implies a larger symmetry of the model (see below) and all quantities appearing in it can be measured in- dependently of each other. In fact, all but vρ , are already well known. However, cross sections of several processes, for in- stance e+e− → ZH where H is a neutral Higgs scalar trans- forming as doublet of SU(2), are sensitive to the value of vη (or vρ ) [18]. So, in principle it is possible to verified if Eq. (9), 88 A.G. Dias et al. / Physics Letters B 637 (2006) 85–89 or equivalently Eq. (12), is satisfied and if the 3–3–1 symmetry can be implemented near the weak scale. 6. Custodial symmetries and the oblique T parameter We can understand the physical meaning of Eq. (9) in the following way. The 3–3–1 models have an approximate SU(2) custodial symmetry. This is broken by the mixing between Z and Z′. In general we have a mixture between these neutral bosons in such a way that the mass eigenstates Z1 and Z2 can be written as [19] Z1 = Zcφ − Z′sφ and Z2 = Zsφ + Z′cφ , and the condition in Eq. (9) is equivalent to put φ = 0 i.e., no mix- ing at all between Z and Z′. There is also an approximate SU(3) custodial symmetry because when both (9) and sin θW = 0 are used, we have MU/MZ′ = 1. However this symmetry is badly broken. We stress that the alternative approach used in liter- ature [9,19,20] is that the condition sinφ � 1 is obtained by assuming that vχ � vW . This is of course still a possibility if the relation (9) is not confirmed experimentally. However, we have shown above that it is possible that φ = 0 even if vχ = vW . Of course, in any case radiative corrections will induce a mixing among Z and Z′, i.e., a finite contribution to φ. This should imply small deviations from ρ0 = 1. The oblique T pa- rameter constraints this deviations since ρ0 − 1 αT and it is given, for the 3–3–1 models, in Ref. [21]. Using the expres- sions of Ref. [21] but without the mixing at the tree level (φ = 0 in Eq. (4.1) of [21]), we obtain for example T = −0.1225 for v̄W = 1 and T = −0.012 for v̄W = 0.25, with T → 0 as v̄W → 0 (vχ → ∞), and all T values calculated with v̄W � 1 are within the allowed interval [7]. This implies that the con- dition Eq. (9) is not significantly disturbed by radiative cor- rections. It means that the natural value of sinφ, arisen only through radiative corrections, is small because the symmetry of the model is augmented when this parameter vanishes. The important thing is that even if Eq. (9) or equivalently Eq. (12) are valid only approximately, we will have that again MZ1 ≈ MZ and also the neutral current couplings of Z1 only approximately coincide with those of the SM, but now this is valid almost independently of the value of vχ . That is, the 3–3–1 symmetry still can be implemented at an energy scale near the electroweak scale. 7. Experimental constraints on the SU(3)L scale Once the vχ scale is arbitrary when Eq. (9) is satisfied, we can ask ourselves what about the experimental limit upon the masses of the extra particles that appear in the model. After all they depend mainly on vχ , the scale at which the SU(3)L symmetry is supposed to be valid. Firstly, let us con- sider the Z′ vector boson. It contributes to the MK at the tree level [22]. This parameter imposes constraints over the quantity (Od L)3d(Od L)3s(MZ/MZ′), which must be of the or- der of 10−4 to have compatibility with the measured MK . This can be achieved with MZ′ ∼ 4 TeV if we assume that the mixing matrix have a Fritzsch-structure Od Lij = √ mj/mi [23] or, it is possible that the product of the mixing angles sat- urates the value 10−4 [22], in this case Z′ can have a mass near the electroweak scale. More important is the fact that there are also in this model FCNC mediated by the neutral Higgs scalar which contributes to MK . These contributions depend on the mixing matrix in the right-hand d-quark sector, Od R , and also on some Yukawa couplings, Γ d , i.e., the interactions are of the form d̄L(Od L)d3Γ d 3α(OR)αsR . Thus, their contributions to MK may have opposite sing relative to that of the contri- bution of Z′. A realistic calculation of the MK in the context of 3–3–1 models has to take into account these extra contribu- tions as well. Muonium–antimuonium transitions would imply a lower bound of 850 GeV on the masses of the doubly charged gauge bileptons, U−− [24]. However this bound depends on assumptions on the mixing matrix in the lepton charged cur- rents coupled to U−− and also it does not take into account that there are in the model doubly charged scalar bileptons which also contribute to that transition [25]. Concerning these dou- bly charged scalars, the lower limit for their masses are only of the order of 100 GeV [26]. From fermion pair production at LEP and lepton flavor violating effects suggest a lower bound of 750 GeV for the mass MU but again it depends on assump- tions on the mixing matrix and on the assumption that those processes are induced only by the U−− boson [27]. Other phe- nomenological analysis in e+e−, eγ and γ γ colliders assume bileptons with masses between 500 GeV and 1 TeV [28,29]. The muonium fine structure only implies MU/g > 215 GeV [30] but also ignores the contributions of the doubly charged scalars. Concerning the exotic quark masses there is no lower limit for them but if they are in the range of 200–600 GeV they may be discovered at the LHC [31]. Direct search for quarks with Q = (4/3)e imply that they are excluded if their mass is in the interval 50–140 GeV [32] but only if they are stable. Sim- ilarly, most of the searches for extra neutral gauge bosons are based on models that do not have the couplings with the known leptons and quarks as those of the 3–3–1 model [6], anyway we have seen that even if vχ = vW the Z′ has a mass of the order of 300 GeV. Finally, rare processes like μ− → e−νeν c μ, induced by the extra particles are also not much restrictive. We may con- clude that there are not yet definitive experimental bounds on the masses of the extra degrees of freedom of the 3–3–1 models. 8. Conclusions Summarizing, we have shown that if the condition in Eq. (9) is realized, concerning the already known particles, the 3–3–1 model and the SM are indistinguishable from each other at the tree level and, as suggested by the value of the T -parameter obtained above, it is possible that this happens even at the one loop level. The models can be confirmed or ruled out by search- ing directly for the effects of their new particles, for instance in left–right asymmetries in lepton–lepton scattering. An asym- metry of this sort only recently has begun to be measured in an electron–electron fixed target experiment [33], but the effects of these asymmetries could be more evident in collider experi- ments [34,35]. From all we have discussed above, it is clear that new physics may be really just around the corner. We have also verified that in 3–3–1 models with heavy lep- tons [10] and with right-handed neutrinos transforming nontriv- A.G. Dias et al. / Physics Letters B 637 (2006) 85–89 89 ially under the 3–3–1 gauge symmetry [11] a similar situation occurs, but, in the later model, the equivalent of the relation in Eq. (9) is given by v̄2 ρ = [(1 − 2s2 W)/2c2 W ]v̄2 W [36]. 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