Supersymmetric One-family Model without Higgsinos Jesús M. Mira,1 William A. Ponce,1 Diego A. Restrepo,1 and Luis A. Sánchez1, 2 2 1 Instituto de Fı́sica, Universidad de Antioquia, A.A. 1226, Medellı́n, Colombia Escuela de Fı́sica, Universidad Nacional de Colombia, A.A. 3840, Medellı́n, Colombia arXiv:hep-ph/0301088v2 24 Jan 2003 The Higgs potential and the mass spectrum of the N = 1 supersymmetric extension of a recently proposed one family model based on the local gauge group SU (3)C ⊗ SU (3)L ⊗ U (1)X , which is a subgroup of the electroweak-strong unification group E6 , is analyzed. In this model the slepton multiplets play the role of the Higgs scalars and no Higgsinos are needed, with the consequence that the sneutrino, the selectron and six other sleptons play the role of the Goldstone bosons. We show how the µ problem is successfully addressed in the context of this model which also predicts the existence of a light CP-odd scalar. PACS numbers: 12.60.Jv, 12.60.Fr, 12.60.Cn I. INTRODUCTION In spite of the remarkable experimental success of our leading theory of fundamental interactions, the so-called Standard Model (SM) based on the local gauge group SU (3)c ⊗ SU (2)L ⊗ U (1)Y [1], it fails to explain several issues like hierarchical fermion masses and mixing angles, charge quantization, CP violation, replication of families, among others. These well known theoretical puzzles of the SM have led to the strong belief that the model is still incomplete and that it must be regarded as a lowenergy effective field theory originating from a more fundamental one. Among the unsolved questions of the SM, the elucidation of the nature of the electroweak symmetry breaking remains one of the most challenging issues. If the electroweak symmetry is spontaneously broken by Higgs scalars, the determination of the value of the Higgs mass MH becomes a key ingredient of the model. By direct search, LEP-II has set an experimental lower bound of MH ≥ 114.4 GeV [2]. Close after the proposal of the SM many scenarios for a more fundamental theory have been advocated in several attempts for answering the various open questions of the model. All those scenarios introduce theoretically well motivated ideas associated to physics beyond the SM [3]. Supersymmetry (SUSY) is considered as a leading candidate for new physics. Even though SUSY does not solve many of the open questions, it has several attractive features, the most important one being that it protects the electroweak scale from destabilizing divergences, that is, SUSY provides an answer to why the scalars remain massless down to the electroweak scale when there is no symmetry protecting them (the “hierarchy problem”). This has motivated the construction of the Minimal Supersymmetric Standard Model (MSSM) [4], the supersymmetric extension of the SM, that is defined by the minimal field content and minimal superpotential necessary to account for the known Yukawa mass terms of the SM. At present, however, there is no experimental evidence for Nature to be supersymmetric. In the MSSM it is not enough to add the Higgsino to construct the left chiral Higgs supermultiplet. Because of the holomorphicity of the superpotential and the requirement of anomaly cancellation, a second Higgs doublet together with its superpartner must be introduced. The two Higgs doublets mix via a mass parameter (the so called µ-parameter) whose magnitude remains to be explained. Besides, since the quartic Higgs self-couplings are determined by the gauge couplings, the mass of the lightest Higgs boson h is constrained very stringently; in fact, the upper limit mh ≤ 128 GeV has been established [5] (the tree level limit is mh ≤ mZ , the mass of the SM neutral gauge boson [6]). Since at present there are not many experimental facts pointing toward what lies beyond the SM, the best approach may be to depart from it as little as possible. In this regard, SU (3)L ⊗ U (1)X as a flavor group has been considered several times in the literature; first as a family independent theory [7], and then with a family structure [8, 9]. Some versions of the family structure provide a solution to the problem of the number N of families, in the sense that anomaly cancellation is achieved when N is a multiple of three; further, from the condition of SU (3)c asymptotic freedom which is valid only if the number of families is less than five, it follows that in those models N is equal to 3 [8]. Over the last decade two three family models based on the SU (3)c ⊗ SU (3)L ⊗ U (1)X local gauge group (hereafter the 3-3-1 structure) have received special attention. In one of them the three known left-handed lepton components for each family are associated to three SU (3)L + triplets [8] as (νl , l− , l+ )L , where lL is related to the right− handed isospin singlet of the charged lepton lL in the SM. In the other model the three SU (3)L lepton triplets c are of the form (νl , l− , νlc )L where νlL is related to the right-handed component of the neutrino field νlL [9]. In the first model anomaly cancellation implies quarks with exotic electric charges −4/3 and 5/3, while in the second one the exotic particles have only ordinary electric charges. All possible 3-3-1 models without exotic electric charges are presented in Ref. [10], where it is shown that there are just a few anomaly free models for one or three families, all of which have in common the same 2 gauge-boson content. In this paper we are going to present the supersymmetric version of the one-family 3-3-1 model introduced in Ref. [11]. The non-SUSY version has the feature that the fermion states in the model are just the 27 states in the fundamental representation of the electroweakstrong unification group E6 [12]. Besides, the scale of new physics for the non-SUSY version of this model is in the range of 1-5 TeV [11, 13], so it is just natural to link this new scale with the SUSY scale. Our main motivation lies in the fact that in the nonSUSY model, the three left-handed lepton triplets and the three Higgs scalars (needed to break the symmetry down to SU (3)c ⊗ U (1)Q in two steps) transform as the 3̄ representation of SU (3)L and have the same quantum numbers under the 3-3-1 structure. This becomes interesting when the supersymmetric N = 1 version of the model is constructed, because the existing scalars and leptons in the model can play the role of superpartners of each other. As a result four main consequences follow: first, the reduction of the number of free parameters in the model as compared to supersymmetric versions of other 3-3-1 models in the literature [14]; second, the result that the sneutrino, selectron and six other sleptons do not acquire masses in the context of the model constructed playing the role of the Goldstone bosons; third, the absence of the µ-problem, in the sense that the µterm is absent at tree level, arising only as a result of the symmetry breaking, and fourth, the existence of a light CP-odd scalar which may have escaped experimental detection [15]. This paper is organized as follows: in Sec. II we briefly review the non-supersymmetric version of the model; in Sec. III we comment on its supersymmetric extension and calculate the superpotential; in Sec. IV we calculate the mass spectrum (excluding the squark sector) and in Sec. V we present our conclusions. II. THE NON-SUPERSYMMETRIC MODEL Let us start by describing the fermion content, the scalar sector and the gauge boson sector of the nonsupersymmetric one-family 3-3-1 model in Ref. [11]. We assume that the electroweak gauge group is SU (3)L ⊗U (1)X ⊃ SU (2)L ⊗U (1)Y , that the left handed quarks (color triplets) and left-handed leptons (color singlets) transform as the 3 and 3̄ representations of SU (3)L respectively, that SU (3)c is vectorlike, and that anomaly cancellation takes place family by family as in the SM. If we begin with QTL = (u, d, D)L , where (u, d)L is the usual isospin doublet of quarks in the SM and DL is an isospin singlet exotic down quark of electric charge −1/3, then the restriction of having particles without exotic electric charges and the condition of anomaly cancellation, produce the following multiplet structure for one family [11]:   u QL =  d  ∼ (3, 3, 0), D L 2 ucL ∼ (3̄, 1, − ), 3 1 1 c dcL ∼ (3̄, 1, ), DL ∼ (3̄, 1, ), 3 3  −  e 1 L1L =  νe  ∼ (1, 3̄, − ), 3 N10  − L E 1 L2L =  N20  ∼ (1, 3̄, − ), 3 N30  0 L N4 2 L3L =  E +  ∼ (1, 3̄, ), 3 e+ L (1) where N10 and N30 are SU (2)L singlet exotic leptons of electric charge zero, and (E − , N20 )L ∪ (N40 , E + )L is an SU (2)L doublet of exotic leptons, vectorlike with 0 respect to the SM as far as we identify N4L = 0c N2L . The numbers inside the parenthesis refer to the (SU (3)c , SU (3)L , U (1)X ) quantum numbers respectively. In order to break the symmetry following the pattern SU (3)c ⊗ SU (3)L ⊗ U (1)X −→ SU (3)c ⊗ SU (2)L ⊗ U (1)Y −→ SU (3)c ⊗ U (1)Q , (2) and give, at the same time, masses to the fermion fields in the non-supersymmetric model, the following set of Higgs scalars is introduced [11]  − φ1 1 φ1 =  φ01  ∼ (1, 3̄, − ), ′ 3 φ10  − φ2 1 φ2 =  φ02  ∼ (1, 3̄, − ), ′ 3 φ20  0  φ3 2 + (3) φ3 =  φ3  ∼ (1, 3̄, ); ′ 3 + φ3 with Vacuum Expectation Values (VEV) given by hφ1 iT = (0, 0, W ), hφ2 iT = (0, v, 0), T ′ hφ3 i = (v , 0, 0), (4) with the hierarchy W > v ∼ v ′ ∼ 174 GeV, the electroweak breaking scale. From Eqs. (1) and (3) we can see that the three left-handed lepton triplets and the three Higgs scalars have the same quantum numbers under the 3-3-1 gauge group, so they can play the role of superpartners. Also, the isospin doublet in φ2 plays the role of φd and the isospin doublet in φ3 plays the role of φu in extensions of the SM with two Higgs doublets (2HDM), in which φd couples only to down type quarks and φu couples only to up type quarks (2HDM Type II). 3 There are a total of 17 gauge bosons in this 3-3-1 model. One gauge field B µ associated with U (1)X , the 8 gluon fields Gµ associated with SU (3)c which remain massless after breaking the symmetry, and another 8 gauge fields Aµ associated with SU (3)L and that we write for convenience in the following way:   D1µ W +µ K +µ 1 1 µ λα Aµα = √  W −µ D2 K 0µ  , 2 −µ 0µ 2 K K̄ Dµ 3 √ √ √ √ where D1µ = Aµ3 / 2√+ Aµ8 / 6, D2µ = −Aµ3 / 2 + Aµ8 / 6, and D3µ = −2Aµ8 / 6. λi , i = 1, 2, ..., 8 are the eight Gell-Mann matrices normalized as T r(λi λj ) = 2δij . The covariant derivative for this model is given by the α α expression: Dµ = ∂ µ − i(g3 /2)λα Gα µ − i(g2 /2)λ Aµ − ig1 XB µ , where gi , i = 1, 2, 3 are the gauge coupling constants for U (1)X , SU (3)L and SU (3)c respectively. The sine of the electroweak mixing angle is given by 2 SW = 3g12 /(3g22 + 4g12 ). The photon field is thus:   p TW Aµ0 = SW Aµ3 + CW √ Aµ8 + (1 − TW /3)B µ , (5) 3 where CW and TW are the cosine and tangent of the electroweak mixing angle. Finally, the two neutral currents in the model are defined as:   p TW Z0µ = CW Aµ3 − SW √ Aµ8 + (1 − TW /3)B µ , 3 p TW (6) Z0′µ = − (1 − TW /3)Aµ8 + √ B µ , 3 where Z µ coincides with the weak neutral current of the SM, with the gauge boson associated with the Y hypercharge given by:   TW µ p µ µ Y = √ A8 + (1 − TW /3)B . 3 The consistency of the model requires the existence of eight Goldstone bosons in the scalar spectrum, out of which four are charged and four are neutral (one CPeven state and three CP-odd) [13] in order to provide with masses for W ± , K ± , K 0 , K̄ 0 , Z 0 and Z ′0 . III. THE SUPERSYMMETRIC EXTENSION When we introduce supersymmetry in the SM, the entire spectrum of particles is doubled as we must introduce the superpartners of the known fields, besides two scalar doublets φu and φd must be used in order to cancel the triangle anomalies; then the superfields φ̂u , and φ̂d , related to the two scalars, may couple via a term of the form µφ̂u φ̂d which is gauge and supersymmetric invariant, and thus the natural value for µ is expected to be much larger than the electroweak and supersymmetry breaking scales. This is the so-called µ problem. However, in a non supersymmetric model as the one presented in the former section, in which the Higgs fields and the lepton fields transform identically under the symmetry group, we can have (as far as we take proper care of the mass generation and the symmetry breaking pattern) the three lepton triplets and the three Higgs triplets as the superpartners of each other. Consequently, we can construct the supersymmetric version of our model without the introduction of Higgsinos, with the supersymmetric extension automatically free of chiral anomalies. For one family we thus end up with the following seven ˆ D̂, L̂1 , L̂2 , and L̂3 , plus gauge chiral superfields: Q̂, û, d, bosons and gauginos. The identification of the gauge bosons eigenstates in the SUSY version follows the nonSUSY analysis as we will show below. A. The Superpotential Let us now write the most general SU (3)c ⊗ SU (3)L ⊗ U (1)X invariant superpotential X U = (hu Q̂a ûL̂a3 + λ(1) Q̂a dˆL̂a1 + hd Q̂a dˆL̂a2 a (2) + λ Q̂a D̂L̂a1 + hD Q̂a D̂L̂a2 ) + λ(3) ûdˆD̂ X + ǫabc (he L̂a1 L̂b2 L̂c3 + λ(4) Q̂a Q̂b Q̂c ), (7) abc where a, b, c = 1, 2, 3 are SU (3)L tensor indices and the chirality and color indices have been omitted. Notice the absence of terms bilinear in the superfields, so a bare µ term is absent in the superpotential U , but it can be generated, after symmetry breaking, by one of the terms in Eq.(7); as a matter of fact it is proportional to he (hÑ10 iÑ20 +hÑ30 iν̃)Ñ40 , where h...i stands for the VEV of the neutral scalar field inside the brackets and the tilde denotes the superpartner of the respective field. This effective µ term is at most of the order of the supersymmetry breaking scale, but as we will show in the next section he ≈ 0 in order to have a consistent supersymmetric model. This is the way how the Supersymmetric µ problem is avoided in the context of the model in this paper. The ûdˆD̂ and Q̂Q̂Q̂ terms violate baryon-number and can possibly lead to rapid proton decay. We may forbid these interactions by introducing the following baryonparity ˆ D̂) → −(Q̂, û, d, ˆ D̂), (Q̂, û, d, (L̂1 , L̂2 , L̂3 ) → +(L̂1 , L̂2 , L̂3 ). (8) This protects the model from too fast proton decay, but the superpotential still contains operators inducing lepton number violation. This is desirable if we want to describe Majorana masses for the neutrinos in our model. 4 † −|L̃†1 L̃2 |2 − |L˜1 L̃3 |2 − |L̃†2 L̃3 |2 ). Another discrete symmetry worth considering is L1L ↔ L2L , which implies he = 0, λ(1) = hd and λ(2) = hD . As we will see in the next section, a very small value of he is mandatory for having a neutrino with a very small tree-level mass. VD = B. The scalar potential VSP = VF + VD + Vsoft , (9) where the first two terms come from the exact SUSY sector, while the last one is the sector of the theory that breaks SUSY explicitly. We now display the different terms in Eq. (9): X ∂U ∂φi i D α = g2 3 X 8 X L∗i,a ( i=1 a,b=1 −λα ∗ )ab Li,b 2 (α = 1, ..., 8), and D = g1 2 3 X 8 X L∗i,a X(L)Li,a i=1 a=1 (a, b = 1, 2, . . . 8 are SU (3)L tensor indices). Then we have = |he |2 (|L̃1 |2 |L̃2 |2 + |L̃1 |2 |L̃3 |2 + |L̃2 |2 |L̃3 |2 VD = 1 2 1  † g (L̃1 L̃1 )2 + (L̃†2 L̃2 )2 + (L̃†3 L̃3 )2 − (L̃†1 L̃1 )(L̃†2 L̃2 ) 2 2 3  − (L̃†1 L̃1 )(L̃†3 L̃3 ) − (L̃†2 L̃2 )(L̃†3 L̃3 ) + |L̃†1 L̃2 |2 + |L̃†1 L̃3 |2 + |L̃†2 L̃3 |2 1  + g12 L̃†1 L̃1 )2 + (L̃†2 L̃2 )2 + 4(L̃†3 L̃3 )2 + 2(L̃†1 L̃1 )(L̃†2 L̃2 ) 18  − 4(L̃†1 L̃1 )(L̃†3 L̃3 ) − 4(L̃†2 L̃2 )(L̃†3 L̃3 ) (On deriving VF and VD we have used the identities 2 α ǫijk ǫilm = δjl δkm − δjm δkl , and λα ij λkl = 2δil δjk − 3 δij δkl ). Finally, the soft SUSY-breaking potential is given by Vsoft = m2L1 L̃†1 L̃1 + m2L2 L̃†2 L̃2 + m2L3 L̃†3 L̃3 + m2L1 L2 Re (L̃†1 L̃2 ) + h′ Re (ǫabc L̃a L̃b L̃c ) + 1 α α 1 2 D D + D , 2 2 where The scalar potential is written as VF = (10) M1 0 0 M2 B̃ B̃ + 2 2 8 X Ãa Ãa + . . . , (12) (11) where M1 is the soft mass parameter of the U (1)X gaugino and M2 refers to the soft mass parameter of the SU (3)L gauginos. After redefining (Ẽ − , Ñ2 ) as φd and (Ñ4 , Ẽ + ) as φu , the parts of V = VF + VD containing the sleptons are given by a=1 V = δ[(φ†d φd + Ñ3† Ñ3 )2 + (ẽ+ ẽ− + ν̃ † ν̃ + Ñ1† Ñ1 )2 ] + η(φ†u φu + ẽ+ ẽ− )2 +γ(φ†d φu + Ñ3† ẽ+ )(φ†u φd + ẽ− Ñ3 ) + β(φ†d φd + Ñ3† Ñ3+ )(φ†u φu + ẽ+ ẽ− ) +α(φ†d φd + Ñ3† Ñ3 )(|ẽ|2 + |ν̃|2 + |Ñ1 |2 ) +β(φ†u φu + ẽ+ ẽ− )(|ẽ|2 + |ν̃|2 + |Ñ1 |2 ) +γ(ẽ+ Ẽ − + ν̃ † Ñ2 + Ñ1† Ñ3 )(Ẽ + ẽ− + Ñ2† ν̃2 + Ñ3† Ñ1 ) +γ(ẽ+ Ñ4− + ν̃ † Ẽ + + Ñ1† ẽ+ )(Ñ4† ẽ− + Ẽ − ν̃ + ẽ− Ñ1 ), where δ =  g22 6 + g12 18  ,η=  g22 6 + 2g12 9  ,γ=  g22 2  − |he |2 ,  β = |he |2 − g22 6 − (13) 2g12 9   , and α = |he |2 − g22 6 + g12 9  . 5 IV. scalar potential, which at tree level are: MASS SPECTRUM Masses for the particles are generated in this model from the VEV of the scalar fields and from the soft terms in the superpotential. For simplicity we assume that the VEVs are real, which means that spontaneous CP violation through the scalar exchange is not considered in this work. Now, for convenience in reading we rewrite the expansion of the scalar fields acquiring VEVs as: 0 0 Ñ1R + iÑ1I √ , 2 0 Ñ 0 + iÑ2I , hÑ20 i + 2R √ 2 0 Ñ 0 + iÑ3I hÑ30 i + 3R √ , 2 0 Ñ 0 + iÑ4I , hÑ40 i + 4R √ 2 ν̃R + iν̃I √ , hν̃i + 2 hÑ20 ihν̃i = −hÑ30 ihÑ1 i, m2L1 L2 = h′ hÑ40 i m2L1 = −α(hÑ20 i2 + hÑ30 i2 ) − βhÑ40 i2 − 2δ(hν̃i2 +hÑ10 i2 ) − h′ Ñ30 = Ñ40 = ν̃e = hÑ40 ihÑ30 i , 2hν̃i m2L2 = −βhÑ40 i2 − α(hν̃i2 + hÑ10 i2 ) − 2δ(hÑ20 i2 Ñ10 = hÑ10 i + Ñ20 = hÑ30 ihÑ10 i + hν̃ihÑ20 i = 0, hÑ30 ihν̃i − hÑ20 ihÑ10 i +hÑ30 i2 ) − h′ hÑ40 ihÑ10 i , 2hÑ20 i m2L3 = −β(hν̃i2 + hÑ10 i2 + hÑ20 i2 + hÑ30 i2 ) −2ηhÑ40 i2 − h′ (14) in an obvious notation taken from Eq.(1), where 0 0 ÑiR (ν̃R ) and ÑiI (ν̃I ) i = 1, 2, 3, 4 refer, respectively, to the real sector and to the imaginary sector of the sleptons. In general, hν̃e i and hÑi0 i, i = 1, 2, 3, 4 can be all different from zero, but as we will see in the following analysis there are some constraints relating them. Also hν̃i ≤ 0.2 TeV, hÑi i ≤ 0.2 TeV, for i = 2, 4 in order to respect the SM phenomenology, and hÑj i ≥ 1 TeV, for j = 1, 3 in order to respect the phenomenology of the 3-3-1 model in Refs. [11, 13]. Our approach will be to look for consistency in the sense that the mass spectrum must include a light spin 1/2 neutral particle (the neutrino) with the other spin 1/2 neutral particles having masses larger than or equal to half of the Z 0 mass, to be in agreement with experimental bounds. Also we need eight spin zero Goldstone bosons, four charged and four neutral ones, out of which one neutral must be related to the real sector of the sleptons and three neutrals to the imaginary sector, in order to produce masses for the gauge bosons after the breaking of the symmetry. As we will show in this section, a consistent set of VEV is provided by hν̃e i = v, hÑ30 i = V, hÑ20 i = vd and hÑ40 i = vu , with the hierarchy V > vu ∼ vd ∼ v, and the constraint hÑ1 i = −vvd /V . This situation implies a symmetry breaking pattern of the form SU (3)c ⊗ SU (3)L ⊗ U (1)X −→ SU (3)c ⊗ U (1)Q , instead of the chain in Eq.(2). So, we can not claim that the MSSM is an effective theory of the model presented here; rather the model here is an alternative to the MSSM so well analyzed in the literature [4, 5, 6]. Playing with the VEV and the other parameters in the superpotential, special attention must be paid to the several constraints coming from the minimization of the hÑ30 ihν̃i − hÑ10 ihÑ20 i , 2hÑ40 i (15) where α, β, δ and η were defined above. The result mL1 L2 = 0 comes from the first constraint and has important consequences as we will see in what follows. A. Spectrum in the Gauge Boson Sector With the most general VEV structure presented in Eq. (14), the charged gauge bosons Wµ± and Kµ± mix up and the diagonalization of the corresponding squared-mass matrix yields the masses [13] g22 (hÑ40 i2 + hν̃i2 + hÑ10 i2 ), 2 g2 = 2 (hÑ30 i2 + hÑ40 i2 + hÑ20 i2 ), 2 2 MW ′ = 2 MK ′ (16) ′ related to the physical fields Wµ = η(hÑ20 iKµ − hÑ30 iWµ ) ′ and Kµ = η(hÑ30 iKµ + hÑ20 iWµ ) associated with the ′ ′ known charged current Wµ± , and the new one Kµ± predicted in the context of this model (η −2 = hÑ20 i2 + hÑ30 i2 is a normalization factor). Notice that with the hierarchy hÑ30 i >> hÑ20 i ∼ hÑ40 i ∼ hν̃i, the mixing between Wµ± and Kµ± is well under control due to fact that the physical W ′± is mainly the W ± of the weak basis, with a small component along K ± of the order of hÑ20 i/hÑ30 i. The expression for the W ′± mass combined with the minimization conditions in Eq.(15) implies (hÑ40 i2 + hν̃i2 + hν̃i2 hÑ20 i2 /hÑ30 i2 )1/2 ≈ 174 GeV. For the five electrically neutral gauge bosons we get first, that the imaginary part of Kµ0 decouples from the other four electrically neutral gauge bosons, acquiring 2 2 2 0 2 + hÑ30 i2 + hÑ40 i2 ) a mass MK 0 = (g2 /4)(hν̃i + hÑ2 i I 0µ [13]. Now, in the basis (B µ , Aµ3 , Aµ8 , KR ), the obtained squared-mass matrix has determinant equal to zero which implies that there is a zero eigenvalue associated to the photon field with eigenvector Aµ0 as given in Eq. (5). 6 The mass matrix for the neutral gauge boson sector ′ 0µ ), where can now be written in the basis (Z0µ , Z0µ , KR ′ µ µ the fields Z0 and Z0 have been defined in Eqs. (6). We can diagonalize this mass matrix in order to obtain the physical fields, but the mathematical results are not very illuminating. Since hÑ30 i >> hÑ20 i ∼ hÑ40 i ∼ hν̃i ∼ 174 GeV, we perform a perturbation analysis for the particular case hÑ20 i = hÑ40 i = hν̃i ≡ v using q = v/hÑ30 i as the expansion parameter. In this way we obtain one eigenvalue of the form   1 2 2 −2 2 2 2 4 MZ1 ≈ g2 CW v 1 + q (7 + 6TW − 9TW ) , (17) 8 superpotential produces for the up type quark a mass mt = hu hÑ40 i = 174 GeV, which implies hÑ40 i ≈ 102 GeV and hu ∼ 1, while for the down type quarks the second to fifth terms generate, in the basis (d, D) [(dR , DR ) column and (dL , DL ) row], the mass matrix and other two of the order hÑ30 i2 [13]. So, we have a neutral current associated to a mass scale v ≃ 174 GeV which may be identified with the known SM neutral current, and two new electrically neutral currents associated to a mass scale hÑ30 i >> v. Now, using the expressions for MW ′ and MZ1 we obtain for the ρ parameter at tree-level[16] Using the former results and the expression for the W ± mass it follows that hÑ40 i ≈ hν̃i ≈ hÑ20 i ≈ 102 GeV. MdD =  λ(1) hν̃i + hd hÑ20 i λ(1) hÑ10 i + hd hÑ30 i λ(2) hν̃i + hD hÑ20 i λ(2) hÑ10 i + hD hÑ30 i  , (19) which produces a mass of the order of hÑ30 i for the exotic quark D, and for the ordinary quark d a mass of the order of (hν̃i+hÑ20 i), suppressed by differences of Yukawa couplings (it is zero for λ(1) = hd and λ(2) = hD ). It is worth noticing that the isospin doublet in L̃3L couples only to up type quarks, while the isospin doublets in L̃1L and L̃2L couple only to down type quarks. 3 2 2 2 2 2 4 ρ = MW q (1 + 2TW − 3TW ), (18) ′ /(MZ CW ) ≃ 1 − 1 8 so that the global fit ρ = 1.0012+0.0023 −0.0014 [17] provides us with the lower limit hÑ30 i ≥ 8.7 TeV (where we are using 2 SW = 0.23113 [18] and neglecting loop corrections which depend on the splitting of the SU (2)L doublets). This result justifies both the imposition of the hierarchy hÑ30 i >> hÑ20 i ∼ hÑ40 i ∼ hν̃i and the existence of the expansion parameter q ≤ 0.02. This in turn shows,first  q that the small component hÑ20 i/ hÑ20 i2 + hÑ30 i2 Kµ of the eigenstate Wµ′ will contaminate tree-level physical processes at most at the level of 2% (by the way, such a mixing can contribute to the ∆I = 1/2 enhancement in nonleptonic weak processes), and second that the estimated order of the masses of the new charged and neutral gauge bosons in the model are not in conflict neither with constraints on their mass scale calculated from a global fit of data relevant to electron-quark contact interactions [19], nor with the bounds obtained in pp̄ collisions at the Tevatron [20]. B. Masses for the Quark Sector Let us assume in the following analysis that we are working with the third family. The first term in the C. Masses for neutralinos The neutralinos are linear combinations of neutral gauginos and neutral leptons (there are not Higgsinos). For this model and in the basis ˜ 0 ), their mass ma(ν , N , N , N , N , B̃ 0 , Ã , Ã , K̃ 0 , K̄ e 1 2 3 4 3 8 trix is given by Mntns =  T MN MgN MgN Mg  , (20) where MN is the matrix  0 0 0 hÑ40 i hÑ30 i  0 0 −hÑ40 i 0 −hÑ20 i   he   0 −hÑ 0 i MN = 0 0 −hÑ10 i  4 ,  2  0 hÑ4 i 0 0 0 hν̃i  0 0 0 hÑ3 i −hÑ2 i −hÑ1 i hν̃i 0 (21) MgN is given by  7 MgN  √ √ and from the soft terms in the superpotential we read Mg = Diag(M1 , M2 , M2 , A2×2 ), where A2×2 is a 2 × 2 matrix with entries zero in the main diagonal and M2 in the secondary diagonal. Now, in order to have a consistent model, one of the eigenvalues of this mass matrix must be very small (corresponding to the neutrino field), with the other eigenvalues larger than half of the Z 0 mass. It is clear that for he very small and simultaneously Mi , i = 1, 2 very large, we have a see-saw type mass matrix; but Mi , i = 1, 2 very large is inconvenient because it restores the hierarchy problem. A detailed analysis shows that Mntns contains two Dirac neutrinos and six Majorana neutral fields, and that for Mi ≤ 10 TeV, i = 1, 2 we have a mass spectrum consistent with the low energy phenomenology only if he ≈ 0. By imposing he = 0, a zero tree-level Majorana mass for the neutrino is obtained, with the hope that the radiative corrections should produce a small mass. (The symmetry L1L ↔ L2L implies he = 0). To diagonalize Mntns analytically is a hopeless task, so we propose a controlled numerical analysis using fixed values for some parameters as suggested by the low energy phenomenology (for example g1 (TeV)≈ 0.38 and g2 (TeV)≈ 0.65) and leaving free other parameters, but in a range of values bounded by theoretical and experimental restrictions. With this in mind we use 0.1 TeV ≤ Mi ≤ 10 TeV, i = 1, 2 (in order to avoid the hierarchy problem) and he ≈ 0 (in order to have a consistent mass spectrum). The random numerical analysis with the constraints stated above shows that for M1 ≈ 0.35 TeV, M2 ≈ 3.1 TeV, hÑ30 i ≈ 10 TeV, hÑ40 i ≈ 150 GeV, hÑ20 i = hν̃i ≈ 80 2γvu2 − h′ vu V /v 2γhÑ10 ivu + h′ vd  2γhÑ 0 iv + h′ v 2γ(hÑ 0 i + V 2 ) + h′ hÑ10 ivd −vV d  1 u 1 vu  0 2γvu V − h′ v  γ(vvd + hÑ1 iV )  2γvvu − h′ V √ √ √ −g1 32 hν̃i −g1 32 hÑ10 i −g1 32 hÑ20 i −g1 32 hÑ30 i g1 2 3 2 hÑ40 i  g √1 hν̃i 0 g2 √12 hÑ20 i 0 −g2 √12 hÑ40 i  2 2  =  −g2 √1 hν̃i g2 √2 hÑ10 i −g2 √1 hÑ20 i g2 √2 hÑ30 i −g2 √1 hÑ40 i 6 6 6 6 6   −g2 hÑ 0 i 0 −g2 hÑ30 i 0 0 1 0 −g2 hν̃i 0 −g2 hÑ20 i 0 2γ(vhÑ10 i + vd V ) The analysis shows that only for h′ = 0 this matrix has two eigenvalues equal to zero which correspond to the four Goldstone bosons needed to produce masses for W ± and K ± . So, h′ = 0 is mandatory (h′ = 0 is a conse-     ,   (22) GeV, hÑ10 i calculated from the constraints coming from the minimum of the scalar potential (see Eq. (15)), and he ≈ 0, we get a neutrino mass of a few electron volts, while all the other neutral fields acquire masses above 45 GeV as desired. Also, the analysis is quite insensitive to the variation of the parameters, with the peculiarity that an increase in M1 and M2 implies an increase in hÑ30 i. We are going to use from now on the notation hν̃i = v, hÑ4 i = vu , hÑ20 i = vd , hÑ30 i = V , with hÑ10 i = −vd v/V as constrained by the minimization conditions in Eq. (15). Another possibility with he 6= 0 but very small demands for hν̃i = hÑ10 i = 0, and produces a lightest neutralino only in the KeV scale, which may be adequate for the second and third family, but not for the first one. The advantage of this particular case is that it reduces to the study of the scalar potential presented in Ref.[13] for the non-supersymmetric case, with an analysis of the mass spectrum similar to the one in that paper. D. Masses for the scalar sector For the scalars we have three sectors, one charged and two neutrals (one real and the other one imaginary) which do not mix, so we can consider them separately. 1. The charged scalars sector − − − For the charged scalars in the basis (ẽ− 1 , ẽ2 , Ẽ1 , Ẽ2 ), we get the squared-mass matrix: γ(vvd + hÑ10 iV ) 2γvu V − h′ v 2γvu2 2γvvu − h′ V 2γ(vhÑ10 i + vd V ) 2γvu vd + h′ hÑ10 i 2γvd vu + h′ hÑ10 i 2γ(v 2 + vd2 ) + h′ vd hÑ10 i−vV vu    .  quence of the symmetry L1L ↔ L2L ). For the other two eigenvalues one is in the TeV scale and the other one at the electroweak mass scale. 8 2. where the submatrices are: The neutral real sector For the neutral real sector and in the basis (ν̃R , Ñ1R , Ñ2R , Ñ3R , Ñ4R ) we get the following mass matrix:   M2×2 M2×3 2 , (23) Mreal = T M2×3 M3×3 M2×3 = =  ′ ′ M2×2 M2×3 ′T ′ M2×3 M3×3  , (27) where the submatrices are ′ ′ M2×3 = (26)  ′ vvu γv 2 − h 2V γvhÑ10 i h′ hÑ10 i/2 ′   vvu ′ h′ v/2 M3×3 =  γvhÑ10 i+ γhÑ10 i2 − h2V . 0 h Ñ iv −vV d ′ 0 ′ ′ 1 h hÑ1 i/2 −h v/2 h 2vu (30) Using the constraints in Eqs. (15), this mass matrix has three eigenvalues equal to zero which identify three real Goldstone bosons (two of them CP-odd), needed to 0µ produce masses for Z0µ , Z0′µ and KR . In the limit h′ = 0, this mass matrix has one eigenvalue in the TeV scale and four eigenvalues equal to zero that correspond to the three Goldstone bosons identified for the case h′ 6= 0, plus an extra CP-odd scalar of zero mass at tree level. The neutral imaginary sector 2 Mimag  (25)  E. = ,  ′ vvu γv 2 + 4δvd2 − h2V γvhÑ10 i + 4δvd V 2βvu vd − h′ hÑ10 i/2 ′  vvu 2βvu V + h′ v/2  =  γvhÑ10 i + 4δvd V γhÑ10 i2 + 4δV 2 − h2V . hÑ 0 iv −vV 2βvu vd − h′ hÑ10 i/2 2βvu V + h′ v/2 4ηvu2 + h′ 1 2vdu For the neutral imaginary sector and in the basis (ν̃I , Ñ1I , Ñ2I , Ñ3I , Ñ4I ) we get the following mass matrix: ′ M2×2   Using the constraints in Eqs. (15), this mass matrix has one eigenvalue equal to zero which identifies one real Goldstone boson needed to produce a mass for KI0µ . Now, using he ≈ 0, h′ = 0 and with the other values as given before, we get for the remaining four eigenvalues that two of them are in the TeV scale, other one is at the electroweak mass scale, while for the lightest CPeven scalar h we get a tree-level mass smaller than the one obtained in the MSSM. This result, which is strongly dependent on the value of he , is not realistic due to the fact that the radiative corrections have not been taken into account, but such analysis is not in the scope of the present work. 3. M2×2 = vd v(4δ − γ) 2αvV + γvd hÑ10 i + h′ vu /2 2βvvu + h′ V /2 0 ′ 2αhÑ1 ivd + γvV − h vu /2 (γ − 4δ)vvd 2βhÑ10 ivu − h′ vd /2  M3×3 ! ′ vu V γvd V + 4δvhÑ10 i γvd2 + 4δv 2 − h 2v , ′ vu V γvd V + 4δvhÑ10 i γV 2 + 4δhÑ10 i2 − h 2v (24) vu V γvd V γvd2 − h 2v ′ vu V 2 γvd V γV − h 2v ! , γhÑ10 iV −γhÑ10 ivd − h′ v2u −h′ V2 ′ vu −γvV + h 2 −γhÑ10 iV h′ v2d (28) The charginos in the model are linear combinations of the charged leptons and charged gauginos. In the gauge eigenstate basis ψ ± = + + − − − (e1 , E1 , W̃ + , K̃ + , e− , E , W̃ , K̃ ) the chargino mass 1 1 terms in the Lagrangian are of the form (ψ ± )T M ψ ± , where   0 MCT , M= MC 0 and  he vd −he v 0 −g2 vu  −he V he hÑ 0 i −g2 vu 0  1 . MC =   −g2 v −g2 vd M2 0  −g2 hÑ10 i −g2 V 0 M2   , (29) Masses for Charginos (31) In the limit he = 0 and M2 very large, this mass matrix 9 is a see-saw type matrix. The numerical evaluation using the parameters as stated before produces a tree-level mass for the τ lepton of the order of 1 GeV, with all the other masses above 90 GeV. V. GENERAL REMARKS AND CONCLUSIONS We have built the complete supersymmetric version of the 3-3-1 model in Ref. [11] which, like the MSSM, has two Higgs doublets at the electroweak energy scale (the isospin doublets in L̃1L and L̃3L ). Since the MSSM is not an effective theory of the model constructed, exploring the Higgs sector at the electroweak energy scale it is important to realize that, the MSSM is not the only possibility for two low energy Higgs doublets. For the model presented here the slepton multiplets play the role of the Higgs scalars and no Higgsinos are required, which implies a reduction of the number of free parameters compared to other models in the literature [14]. The absence of bilinear terms in the bare superpotential avoids the presence of possible unwanted µ terms; in this way the so called µ problem is absent in the construction developed in this paper. The sneutrino, selectron and other six sleptons do not acquire masses in the context of the model, and they play the role of the Goldstone bosons needed to produce masses for the gauge fields. The right number of Goldstone bosons is obtained by demanding h′ = mL1 L2 = 0 in Vsoft . h′ = 0 in Vsoft has as a consequence the existence of a zero mass CP-odd Higgs scalar at tree level. Once radiative corrections are taken into account we expect it acquires a mass of a few (several?) GeV, which in any case is not troublesome because, as discussed in Ref. [15], a light CP-odd Higgs scalar not only is very difficult to be detected experimentally, but also it has been found that in the two Higgs doublet model type II and when a two-loop calculation is used, a very light (∼ 10 GeV) CP-odd scalar A0 can still be compatible with precision data such as the ρ parameter, BR(b → sγ), Rb , Ab , and BR(Υ → A0 γ)[21]. he = 0 or very small is a necessary condition in order to have a consistent model, in the sense that it must include a very light neutrino, with masses for the other spin 1/2 neutral particles larger than half the Z 0 mass. There is not problem with this constraint, because due to the existence of heavy leptons in the model, he is not the only parameter controlling the charged lepton masses. We have also analyzed the mass value at tree-level for h, the lightest CP-even Higgs scalar in this model, which is smaller than the lower bound of the lightest CP-even Higgs scalar in the MSSM, although strongly dependent on the radiative corrections. This fact is not in conflict with experimental results due to the point that the coupling hZZ and hA0 Z are suppressed because of the mixing of the SU (2)L doublet sleptons with the singlets Ñ10 and Ñ30 . The recent experimental results announced by the Muon (g − 2) collaboration[22] show a small discrepancy between the SM prediction and the measured value of the muon anomalous spin precession frequency, which only under special circumstances may be identified with the muon’s anomalous magnetic moment aµ [23], a quantity related to loop corrections. Immediately following the experimental results a number of papers appeared analyzing the reported value, in terms of various forms of new physics, starting with the simplest extension of the SM to two Higgs doublets[21], or by using supersymmetric extensions, technicolor models, leptoquarks, exotic fermions, extra gauge bosons, extra dimensions, etc., in some cases extending the analysis even at two loops (for a complete bibliography see the various references in [24]). More challenging, although not in complete agreement between the different authors, are the analyses presented in Refs.[25] and [26] where it is shown how the MSSM parameter space gets constrained by the experimental results. Our model, even though different from the MSSM shares with it the property that very heavy superpartners decouple from the aµ value yielding a negligible contribution. Nevertheless, the model in this paper includes many interesting new features that may be used for explaining the measured value of the muon’s anomalous precession frequency, as for example a light CP-odd and a light CPeven scalars which get very small masses at tree level, but that the loop radiative corrections may rise these masses up to values ranging from a few GeV to the electroweak mass scale. But an analysis similar to the one presented in Refs.[25] and [26] is outside the scope of the present study, because in our case it depends crucially on the predicted values of the Higgs scalar masses, an obcan scure matter in supersymmetry. (For example, aexp µ be understood in the context of our model if the CP-odd scalar has a mass of the order of a few GeV[21], with all the other scalars and supersymmetric particles acquiring masses larger than the electroweak mass scale. Similarly, the light CP-even Higgs boson h with enough suppressed hZZ and hA0 Z couplings can contribute significantly to aµ [24]). The idea of using sleptons as Goldstone bosons is not new in the literature [27], but as far as we know there are just a few papers where this idea is developed in the context of specific models, all of them related to one family structures [28]. The model can be extended to three families, but the price is high since nine SU (3)L triplets of leptons with their corresponding sleptons are needed, which implies the presence of nine SU (2)L doublets of Higgs scalars. An alternative is to work with the three family structures presented in Refs.[8, 9]. In conclusion, the present model has a rich phenomenology and it deserves to be studied in more detail. 10 ACKNOWLEDGMENTS Work partially supported by Colciencias in Colombia and by CODI in the U. de Antioquia. L.A.S. acknowl- [1] For an excellent compendium of the SM see: J.F. Donoghue, E. Golowich, and B. 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