PHYSICAL REVIEW D 100, 015004 (2019)
Some phenomenological aspects of the 3-3-1 model with the Cárcamo-Kovalenko-Schmidt mechanism
† ‡ ∥ H. N. Long,1,2,* N. V. Hop,3, L. T. Hue,4,5, N. H. Thao,6,§ and A. E. Cárcamo Hernández7, 1Theoretical Particle Physics and Cosmology Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam 2Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam 3Department of Physics, Can Tho University, 3/2 Street, Can Tho 900000, Vietnam 4Institute for Research and Development, Duy Tan University, Da Nang City 550000, Vietnam 5Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi 100000, Vietnam 6Department of Physics, Hanoi Pedagogical University 2, Phuc Yen, Vinh Phuc 280000, Vietnam 7Universidad T´ecnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile
(Received 18 April 2019; published 8 July 2019)
We perform a comprehensive analysis of several phenomenological aspects of the renormalizable extension of the inert 3-3-1 model with sequentially loop-generated Standard Model fermion mass hierarchy. Special attention is paid to the study of the constraints arising from the experimental data on the ρ parameter, as well as those ones resulting from the charged lepton flavor violating process μ → eγ and dark matter. We also study the single Z0 production via Drell-Yan mechanism at the LHC. We have found that Z0 gauge bosons heavier than about 4 TeV comply with the experimental constraints on the oblique ρ parameter as well as with the collider constraints. In addition, we have found that the constraint on the charged lepton flavor violating decay μ → eγ sets the sterile neutrino masses to be lighter than about 1.12 TeV. In addition the model allows charged lepton flavor violating processes within reach of the forthcoming experiments. The scalar potential and the gauge sector of the model are analyzed and discussed in detail. Our model successfully accommodates the observed dark matter relic density.
DOI: 10.1103/PhysRevD.100.015004
I. INTRODUCTION the Froggatt-Nielsen (FN) mechanism [1]. According to the FM mechanism, the mass differences between generations Despite its great successes, the Standard Model (SM) of fermions arise from suppression factors depending on does not explain the observed mass and mixing hierarchies the FN charges of the particles. It has been noticed that in in the fermion sector, which remain without a compelling order to implement the aforementioned mechanism, the explanation. It is known that in the SM, the masses of the effective Yukawa interactions have to be introduced, thus matter fields are generated from the Yukawa interactions. In – – making this theory nonrenormalizable. From this point of addition, the Cabibbo Kobayashi Maskawa quark mixing view, the recent mechanism proposed by Cárcamo, matrix is also constructed from the same Yukawa cou- Kovalenko, and Schmidt [2] (called by CKS mechanism) plings. To solve these puzzles, some mechanisms have been based on sequential loop suppression mechanism, is more proposed. To the best of our knowledge, the first attempt to natural since its suppression factor arises from the loop explain the huge differences in the SM fermion masses is factor l ≈ ð1=4πÞ2. One of the main purposes of the models based on the gauge group SUð3Þ × SUð3Þ × Uð1Þ (for short, 3-3-1 *[email protected] C L X † model) [3–10] is concerned with the search of an explan- [email protected] ‡ [email protected] ation for the number of generations of fermions. Combined §[email protected] with the QCD asymptotic freedom, the 3-3-1 models ∥ [email protected] provide an explanation for the number of fermion gen- 1 erations. These models have nonuniversal Uð ÞX gauge Published by the American Physical Society under the terms of assignments for the left handed quarks fields, thus implying the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to that the cancellation of chiral anomalies is fulfilled when 3 the author(s) and the published article’s title, journal citation, the number of SUð ÞL fermionic triplets is equal to the 3 3 and DOI. Funded by SCOAP . number of SUð ÞL fermionic antitriplets, which happens
2470-0010=2019=100(1)=015004(21) 015004-1 Published by the American Physical Society LONG, HOP, HUE, THAO, and CÁRCAMO HERNÁNDEZ PHYS. REV. D 100, 015004 (2019) when the number of fermion families is a multiple of three. Section III is devoted to gauge boson mass and mixing. Some other advantages of the 3-3-1 models are: (i) they Taking into account of data on the ρ parameter, and if solve the electric charge quantization [11,12], (ii) they only contributions of the gauge bosons are mentioned, we contain several sources of CP violation [13,14], and will show that the mass of the heavy neutral boson Z0 (iii) they have a natural Peccei-Quinn symmetry, which will be constrained nearly to the excluded regions derived solves the strong-CP problem [15–18]. from other experimental data such as LHC searches, In the framework of the 3-3-1 models, most of the and K, D, and B meson mixing. The Higgs sector is research is focused on radiative seesaw mechanisms, and considered in Sec. IV. The Higgs sector consists of two but some involving nonrenormalizable interactions intro- parts: the first part contains lepton number conserving duced to explain the SM fermion mass and mixing pattern terms and the second one is lepton number violating. We (see references in Ref. [19]). study in details the first part and show that the Higgs The FN mechanism was implemented in the 3-3-1 sector has all necessary ingredients. The ρ parameter will models in Ref. [20]. It is interesting to note that the FN be investigated including Higgs contributions. In Sec. V, mechanism does not produce a new scale since the scale of lepton flavor violating decays of the charged leptons are the flavor breaking is the same as the symmetry breaking discussed, where sterile neutral lepton masses are con- scale of the model. strained. Section VI is devoted to the production of the 0 The CKS mechanism has been implemented for the first heavy Z and the heavy neutral scalar H4. In Sec. VII,we time in thepffiffiffi 3-3-1 model without exotic electric charges deal with the DM relic density. We make conclusions in (β ¼ −1= 3) in Ref. [19]. The implementation of the CKS Sec. VIII. The scalar potential of the model is given in mechanism in the 3-3-1 model leads to viable renormaliz- Appendix. able 3-3-1 model that provides a dynamical explanation for the observed SM fermion mass spectrum and mixing parameters consistent with the SM low energy fermion II. REVIEW OF THE MODEL flavor data [19]. It is worth mentioning that the extension of To implement the CKS mechanism, only the heaviest the inert 331 model of Ref. [19] contains a residual discrete particles such as the exotic fermions and the top quark get ðLgÞ — Z2 lepton number symmetry arising from the sponta- masses at tree level. The next medium ones: bottom, neous breaking of the global U 1 symmetry. Under this charm quarks, tau, and muon get masses at one-loop level. ð ÞLg residual symmetry, the leptons are charged and the other Finally, the lightest particles: up, down, strange quarks and particles are neutral [19]. the electron acquire masses at two-loop level. To forbid the However, in the mentioned work, the authors have just usual Yukawa interactions, the discrete symmetries should focused on the data concerning fermions (both quarks and be implemented. Hence, the full symmetry of the model leptons including neutrino mass and mixing), but some under consideration is questions are open for the future study. SUð3Þ × SUð3Þ × Uð1Þ × Z4 × Z2 × Uð1Þ ; ð1Þ The purpose of this work is to study several phenom- C L X Lg enological aspects of the renormalizable extension of the inert 3-3-1 model with sequentially loop-generated SM where Lg is the generalized lepton number defined in fermion mass hierarchy. In particular, the constraints Refs. [19,21]. It is interesting to note that, in this model, arising from the experimental data on the ρ parameter, as the light active neutrinos get their masses from a combi- well as those ones resulting from the charged lepton flavor nation of linear and inverse seesaw mechanisms at two- violating process μ → eγ and dark matter. Furthermore our loop level. work discusses the Z0 production at proton-proton collider As in the ordinary 3-3-1 model without exotic electric 3 via quark-antiquark annihilation. To determine the oblique charges, the quark sector contains the following SUð ÞC × ρ 3 1 SUð3Þ × Uð1Þ representations [19] parameter constraints on the SUð ÞL × Uð ÞX symmetry L X breaking scale vχ, which will be used to constrain the heavy T 0 Q ¼ðD ; −U ;J Þ ∼ ð3; 3 ; 0Þ; Z gauge boson mass, we proceed to study in detail the nL n n n L gauge and Higgs sectors of the model. In addition we 1 T ∼ 3 3 1 2 determine the constraints imposed by the charged lepton Q3L ¼ðU3;D3;TÞL ; ; 3 ;n¼ ; ; μ → γ flavor violating process e and dark matter on the 1 2 model parameter space. In what regards the scalar potential ∼ 3 1 − ∼ 3 1 1 2 3 DiR ; ; 3 ;UiR ; ; 3 ;i¼ ; ; ; of the model, due to the implemented symmetries, the Higgs sector is rather simple and can be completely solved. 1 2 ∼ 3 1 − ∼ 3 1 All Goldstone bosons and the SM like Higgs boson are JnR ; ; 3 ;TR ; ; 3 ; defined. ˜ 2 1 The further content of this paper is as follows. In Sec. II, TL;R ∼ 3; 1; ;BL;R ∼ 3; 1; − ; ð2Þ we briefly present particle content and SSB of the model. 3 3
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∼ 3 where denotes the quantum numbers for the three above In the leptonic sector, besides the usual SUð ÞL lepton 3 subgroups, respectively. Note that the SUð ÞL singlet exotic triplets, the model contains extra three charged leptons ˜ 1 up type quarks TL;R, down type quarks BL;R in the last line EjðL;RÞ (j ¼ , 2, 3) and four neutral leptons, i.e., NjR and of Eq. (2) are added to the quark spectrum of the ordinary ΨR (j ¼ 1, 2, 3). The leptonic fields have the following 3 3 1 3-3-1 model in order to implement the CKS mechanism. SUð ÞC × SUð ÞL × Uð ÞX assignments:
1 ν νc T ∼ 1 3 − ∼ 1 1 −1 1 2 3 LiL ¼ð i;ei; i ÞL ; ; 3 ;eiR ð ; ; Þ;i¼ ; ; ; ð3Þ
E1L ∼ ð1; 1; −1Þ;E2L ∼ ð1; 1; −1Þ;E3L ∼ ð1; 1; −1Þ;
E1R ∼ ð1; 1; −1Þ;E2R ∼ ð1; 1; −1Þ;E3R ∼ ð1; 1; −1Þ;
N1R ∼ ð1; 1; 0Þ;N2R ∼ ð1; 1; 0Þ;N3R ∼ ð1; 1; 0Þ; ΨR ∼ ð1; 1; 0Þ; ð4Þ
ν νc ≡ νc μ τ where iL; R and eiL (eL, L, L) are the neutral and charged lepton families, respectively. 0 0 0 þ þ þ þ The Higgs sector contains three scalar triplets: χ, η, and ρ and seven singlets φ1, φ2, ξ , ϕ1 , ϕ2 , ϕ3 , and ϕ4 . Hence, the scalar spectrum of the model is composed of the following fields 1 χ χ χ0 ∼ 1 3 − ¼h iþ ; ; 3 ; ð5Þ
χ T 1 T v 0 0 − hχi¼ 0; 0; pffiffiffi ; χ ¼ χ1; χ2 ; pffiffiffi ðRχ0 − iIχ0 Þ ; 2 2 3 3 1 T 2 þ þ ρ ¼ ρ ; pffiffiffi ðRρ − iIρÞ; ρ ∼ 1; 3; ; 1 2 3 3 1 η η η0 ∼ 1 3 − ¼h iþ ; ; 3 ; η T 1 T v 0 − 0 hηi¼ pffiffiffi ; 0; 0 ; η ¼ pffiffiffi ðRη0 − iIη0 Þ; η2 ; η3 ; 2 2 1 1 0 0 φ1 ∼ ð1; 1; 0Þ; φ2 ∼ ð1; 1; 0Þ; þ þ þ þ ϕ1 ∼ ð1; 1; 1Þ; ϕ2 ∼ ð1; 1; 1Þ; ϕ3 ∼ ð1; 1; 1Þ; ϕ4 ∼ ð1; 1; 1Þ; 1 0 0 00 0 vξ 00 ξ ¼hξ iþξ ; hξ i¼pffiffiffi ; ξ ¼ pffiffiffi ðRξ0 − iIξ0 Þ ∼ ð1; 1; 0Þ: ð6Þ 2 2
0 0 The Z4 × Z2 assignments of scalar the fields are shown (VEVs) of the χ3 and ξ scalar fields. At this step, all new in Table I. extra fermions, non-SM gauge bosons, as well as the The fields with nonzero lepton number are presented in electrically neutral gauge singlet lepton ΨR gain masses. Table II. Note that the three gauge singlet neutral leptons In addition, the entries of the neutral lepton mass matrices NiR as well as the elements in the third component of the with negative lepton number (−1) also get values propor- νc −1 lepton triplets, namely iL have lepton number equal . tional to vξ. At this step, the initial group breaks down to In the model under consideration, the spontaneous the direct product of the SM gauge group and the Z4 × symmetry breaking (SSB) occurs by two steps [19]. The ðLgÞ first step is triggered by the vacuum expectation values Z2 discrete group. The second step is triggered by vη providing masses for the top quark as well as for the W and 3 1 TABLE I. Scalar assignments under Z4 × Z2. Z gauge bosons and leaving the SUð ÞC × Uð ÞQ × Z4 × ðLgÞ ðLgÞ χηρ 0 0 þ þ þ þ 0 Z2 symmetry preserved. Here Z2 is residual symmetry φ1 φ2 ϕ1 ϕ2 ϕ3 ϕ4 ξ where only leptons are charged, thus forbidding inter- Z4 11−1 −1 ii−1 −1 11 −1 −1 −1 −1 actions having an odd number of leptons. This is crucial to Z2 11111 1 guarantee the proton stability [19]. Thus
015004-3 LONG, HOP, HUE, THAO, and CÁRCAMO HERNÁNDEZ PHYS. REV. D 100, 015004 (2019)
TABLE II. Nonzero lepton number L of fields.
νc Ψ 0 χþ 0 ρþ ϕþ ϕþ ϕþ 0 1 TL;R J1L;R J2L;R iL eiL;R EiL;R NiR R χ1 2 η3 3 2 3 4 ξ i ¼ ,2,3 L −2 22−1 11−1 122−2 −2 −2 −2 −2 −2
SU 3 × SU 3 × U 1 × Z4 × Z2 × U 1 tree-level mass for the right handed Majorana ð ÞC ð ÞL ð ÞX ð ÞLg neutrino Ψ . It is crucial for generating two loop- vχ ;vξ R ⟶ 3 2 1 ðLgÞ SUð ÞC × SUð ÞL × Uð ÞY × Z4 × Z2 level masses for the down and strange quarks. vη (2) The SUð3Þ singlet exotic down type quark, i.e., B, ⟶ 3 1 ðLgÞ L SUð ÞC × Uð ÞQ × Z4 × Z2 : ð7Þ is crucial for the implementation of the one loop level radiative seesaw mechanism that generate the A consequence of the chain in (7) is bottom quark mass. The SUð3Þ singlet exotic up ˜ ˜ L type quarks, i.e., T1 and T2, are needed to generate a 246 ≪ ∼ ∼ O 10 vη ¼ v ¼ GeV vχ vξ ð Þ TeV: ð8Þ one loop level charm quark mass as well as two loop level down and strange quark masses. The three The corresponding Majoron associated to the sponta- SUð3Þ singlet exotic charged leptons, i.e., Ej neous breaking of the Uð1Þ global symmetry is a gauge- L Lg (j ¼ 1, 2, 3), are required in order to provide the singlet scalar and, therefore, unobservable. ∼ ∼ O 10 radiative seesaw mechanisms that generate one loop An explanation for the relation vχ vξ ð Þ TeV is level tau and muon masses and two loop level provided in the following. The present lower limits on the electron mass. The four right handed Majorana Z0 gauge boson mass in 3-3-1 models arising from LHC neutrinos, i.e., N (j ¼ 1, 2, 3), Ψ , are crucial searches reach around 2.5 TeV [22]. These bounds can be jR R for the implementation of the two loop level linear translated into limits of about 6.3 TeV on the SUð3Þ × C and inverse seesaw mechanisms that give rise to the SU 3 × U 1 gauge symmetry breaking scale vχ. ð ÞL ð ÞX light active neutrino masses. Furthermore, electroweak data from the decays Bs;d → þ − þ − 0 μ μ and Bd → K ðKÞμ μ set lower bounds on the Z gauge boson mass ranging from 1 TeVup to 3 TeV [23–27]. III. GAUGE BOSONS Furthermore, as shown in Ref. [28], the experimental data A. Gauge boson masses and mixing on K, D, and B meson mixings set a lower bound of about 4 TeV for the Z0 gauge boson mass in 3-3-1 models, which After SSB, the gauge bosons get masses arising from the η χ 3 3 kinetic terms for the and SUð ÞL scalar triplets, as translates in a lower limit of about 10 TeV for the SUð ÞL × 1 follows: Uð ÞX gauge symmetry breaking scale vχ. Finally, to close this section we provide a justification of gauge χ † μ χ η † μ η the role of the different particles of our model: Lmass ¼ðDμh iÞ D h iþðDμh iÞ D h i; ð9Þ 3 ϕþ (1) The presence of the SUð ÞL scalar singlet 3 ,is needed to generate two loop level down and strange with the covariant derivative for triplet defined as quark masses, as shown in Ref. [19]. Besides that, in λ λ order to implement a two loop level radiative seesaw a 9 Dμ ¼ ∂μ − igAμa − igXX Bμ; ð10Þ mechanism for the generation of the up, down, and 2 2 strange quark masses as well as the electron mass, 3 φ0 φ0 ϕþ where g and gX are the gauge coupling constants of the the Z4 charged SUð ÞL scalar singlets 1, 2, 1 , þ SUð3Þ and Uð1Þ groups, respectively. Here, λ9 ¼ ϕ2 (which do not acquire a vacuum expectation pffiffiffiffiffiffiffiffiL X value) are also required in the scalar sector. The Z4 2=3 diagð1; 1; 1Þ is defined such that Trðλ9λ9Þ¼2, 3 φ0 λ 1 charged SUð ÞL scalar singlet 1 is also needed to similarly as the usual Gell-Mann matrix a, a ¼ ; generate one loop level masses for the charm and 2; 3; …; 8. By matching gauge the coupling constants at 3 1 bottom quarks as well as for the tau and muon the SUð ÞL × Uð ÞX symmetry breaking scale, the follow- 3 φ0 ing relation is obtained [9] leptons. The Z4 charged SUð ÞL scalar singlets 2 ϕþ 3 ϕþ and 3 as well as the SUð ÞL scalar singlet 4 , pffiffiffi neutral under Z4 are also crucial for the implemen- g 3 2 sin θ ðM 0 Þ t ≡ X ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW Z : ð11Þ tation of two loop level linear and inverse seesaw 2 g 3 − 4sin θWðMZ0 Þ mechanisms that give rise to the light active neutrino 3 ξ0 masses. The SUð ÞL scalar singlet is introduced Let us provide the definition of the Weinberg angle θW. to spontaneously break the Uð1Þ generalized 0 θ 0 Lg As in the SM, one puts g ¼ g tan W, where g is gauge 1 lepton number symmetry and thus giving rise to a coupling of the Uð ÞY subgroup satisfying the relation [9]
015004-4 SOME PHENOMENOLOGICAL ASPECTS OF THE 3-3-1 MODEL … PHYS. REV. D 100, 015004 (2019) pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi ! 3 2 0 ggX tW tW g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð12Þ Aμ ¼ s Aμ3 þ c − pffiffiffi Aμ8 þ 1 − Bμ ; 18 2 − 2 W W 3 3 g gX rffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 tW tW Thus Zμ ¼ c Aμ3 − s − pffiffiffi Aμ8 þ 1 − Bμ ; W W 3 3 pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigX t t tan θW ¼ : ð13Þ 0 1 − W Wffiffiffi 18 2 − 2 Zμ ¼ Aμ8 þ p Bμ; ð19Þ g gX 3 3 θ θ Denoting where we have denoted sW ¼ sin W, cW ¼ cos W, tW ¼ tan θW. The coupling of the photon Aμ gives e ¼ gsW. 1 1 After the first step, the squared mass matrix is a block dia- Wμ ¼ pffiffiffi ðAμ1 ∓ iAμ2Þ;Yμ ¼ pffiffiffi ðAμ6 iAμ7Þ; 0 2 2 gonal one in the new basis (Aμ, Zμ, Zμ), where the entry in the 1 top is zero (due to the masslessness of the photon), while the 0 2 2 0 Xμ ¼ pffiffiffi ðAμ4 − iAμ5Þ; ð14Þ × matrix for (Zμ, Zμ) in the bottom has the form 2 ! 2 2 2 MZ MZZ0 and substituting (10) and (14) into (9) one gets the M 2×2 ¼ 2 2 : ð20Þ ð Þ M M following squared masses for the charged/non-Hermitian ZZ0 Z0 gauge bosons The matrix elements in (20) are given by
2 2 2 2 2 2 2 g 2 2 g 2 2 2 g 2 2 g vη mW m ¼ vη;M0 ¼ ðvχ þ vηÞ;M¼ vχ; M ¼ ¼ ; W 4 X 4 Y 4 Z 4 2 2 cW cW ð15Þ 2 2 pgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 M 0 ¼ vηð1 − 2s Þ; ZZ 4 2 3 − 4 2 W cW sW where vη ¼ v ¼ 246 GeV, as expected. 2 2 2 1 − 2 2 2 From Eq. (15) we find the following gauge boson mass 2 g cW 2 vηð sWÞ M 0 4vχ : 21 Z ¼ 4 3 − 4 2 þ 4 ð Þ squared splitting ð sWÞ cW 2 2 2 2 Note that our formula of MZ0 is consistent with that given M 0 − M ¼ m : ð16Þ X Y W in [23]. In the last step of diagonalization, the Z − Z0 mixing For neutral gauge bosons, the squared mass mixing angle ϕ and mass eigenstates Z1;2 are determined as matrix has the form 2 2M 0 tan 2ϕ ¼ ZZ ; ð22Þ 1 2 2 2 ngauge T M 0 − M Lmass ¼ 2 V MngaugeV; ð17Þ Z Z 1 0 Zμ ¼ Zμ cos ϕ − Zμ sin ϕ; T where V ¼ðAμ3;Aμ8;BμÞ and 2 0 Zμ ¼ Zμ sin ϕ þ Zμ cos ϕ: ð23Þ 0 1 2 2 vη 2 2 pffiffi − ptffiffi Our definition of ϕ is consistent with that in Ref. [29] vη 3 3 6 vη 2 B C needed to study the ρ parameter. 2 g B 1 2 2 2 2 2 C M ¼ B 4 ptffiffi 2 − C: ð18Þ ngauge 4 @ 3 ð vχ þ vηÞ 9 2 ð vχ vηÞ A The masses of physical neutral gauge bosons are 2t2 2 2 determined as 27 ðvχ þ vηÞ
1 1 2 2 2 2 2 2 2 2 2 M 1 ¼ fM 0 þM −½ðM 0 −M Þ þ4ðM 0 Þ g; The down-left entries in (18) are not written, due to the fact Z 2 Z Z Z Z ZZ that the above matrix is symmetric. 1 1 2 2 2 2 2 2 2 2 2 M 2 ¼ fM 0 þM þ½ðM 0 −M Þ þ4ðM 0 Þ g: ð24Þ The matrix in (18) has vanishing determinant, thus Z 2 Z Z Z Z ZZ giving rise to a massless gauge boson, which corresponds 2 ≫ 2 to the photon. The diagonalization of the squared mass In the limit vχ vη, one approximates matrix for neutral gauge bosons of Eq. (18) is divided 2 2 4 ðM 0 Þ vη in two steps. In the first step, the massive fields are 2 ≃ 2 − ZZ 2 O MZ1 MZ 2 þ MZ × 4 ; ð25Þ identified as MZ0 vχ
015004-5 LONG, HOP, HUE, THAO, and CÁRCAMO HERNÁNDEZ PHYS. REV. D 100, 015004 (2019)
FIG. 1. Left-panel: ρ parameter as a function of vχ , upper and lower horizontal lines are upper and lower limits in (31). Right-panel: 2 Relation between vχ and MZ2 , upper horizontal lines are an upper and a lower limits of vχ , respectively.
2 2 4 Consequently, the validity of our model depends on the ðM 0 Þ vη 2 ≃ 2 ZZ 2 O ≃ 2 MZ2 MZ0 þ 2 þ MZ × 4 MZ0 : ð26Þ condition that the non-SM particles do not contradict M vχ Z0 those experimental results. Let us note that one of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ 2 2 2 most important observables in the SM is the parameter ð1 − 2s Þ 3 − 4s vη ϕ ≃ W W : defined as tan 4 4 2 ð27Þ cW vχ m2 ρ ¼ W : ð28Þ c2 M2 B. Limit on Z0 mass from the ρ parameter W Z The presence of the non-SM particles modifies the For the model under consideration, one-loop contributions oblique corrections of the SM, the values of which of the new heavy gauge bosons to the oblique correction lead have been extracted from high precision experiments. to the following form of the ρ parameter [29]
ffiffiffi 2 p 2 2 2 M 0 3 2G 2M M0 M0 ρ − 1 ≃ tan2ϕ Z − 1 þ F M2 þ M2 þ þ ln m2 16π2 þ 0 M2 − M2 M2 Z þ 0 þ 2 2 αðm Þ M ε ðM ;M0Þ − Z t2 0 þ O ε3 M ;M ; 4π 2 W ln 2 þ 2 þ ð ð þ 0ÞÞ ð29Þ sW Mþ
M2−m2 0 þ ε ≡ where M0 ¼ MX ;Mþ ¼ MY and ðM; mÞ m2 . Combining with Eq. (16), one gets ffiffiffi 2 p 2 2 2 2 2 M 0 3 2G 2M þ ðM þ þ m Þ ðM þ þ m Þ ρ − 1 ≃ 2ϕ Z − 1 F 2 2 2 − Y Y W Y W tan 2 þ 2 MYþ þ mW 2 ln 2 m 16π m M þ Z W Y 2 2 4 α m ðM þ þ m Þ m − ð ZÞ 2 Y W W 4π 2 tW ln 2 þ 2 2 2 2 ; ð30Þ sW MYþ ðMYþ þ mWÞ
1 3 57 ≤ ≤ 6 09 where αðmZÞ ≈ 128 [30]. . TeV vχ . TeV: ð32Þ 2 0 23122 Taking into account sW ¼ . [30] and Substituting (32) into (26) and evaluating in Fig. 1 (the right-panel) we get a bound on the Z0 mass as follows ρ ¼ 1.00039 0.00019; ð31Þ 1 42 ≤ 2 ≤ 2 42 . TeV MZ . TeV: ð33Þ we have plotted Δρ as a function of vχ in Fig. 1 (the left- Then, the bilepton gauge boson mass is constrained to be panel). From Fig. 1 (the left-panel), it follows in the range:
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465 GeV ≤ MY ≤ 960 GeV; ð34Þ the scalar potential minimization conditions at tree level as follows where mW ¼ 80.379 GeV [30]. The above limit is stronger than the one obtained from the wrong muon decay MY ≥ w3 ¼ 0; ð35Þ 230 GeV [31]. 1 1 It is worth mentioning that the second term in (30) is 2 2 2 2 −μχ ¼ vχλ13 þ vηλ5 þ λχξv ; much smaller the first one. Consequently, the limit derived 2 2 ξ 1 1 from the tree level contribution is very close to the one −μ2 2λ 2λ λ 2 obtained when we consider the radiative corrections arising η ¼ vη 17 þ 2 vχ 5 þ 2 ηξvξ; from heavy vector exchange. 1 1 −μ2 λ 2 λ 2 λ 2 From LHC searches, it follows that the lower bound ξ ¼ 2 χξvχ þ 2 ηξvη þ ξvξ: ð36Þ on the Z0 boson mass in 3-3-1 models ranges from 2.5 to þ − From the analysis of the scalar potential, taking into 3TeV[22,32]. From the decays Bs;d → μ μ and Bd → − account the constraint conditions of Eq. (35), we find that K ðKÞμþμ [23–27], the lower limit on the Z0 boson mass ranges from 1 TeV to 3 TeV. We will show that when scalar the charged scalar sector is composed of two massless ηþ χþ ρ fields, i.e., 2 and 2 which are the Goldstone bosons eaten contributions to the parameter are included, there will þ þ exist allowed m values larger than the range given in (33), by the longitudinal components of the W and Y gauge Z2 bosons, respectively. The other massive electrically charged so that they satisfy the recent lower bounds concerned from ϕþ ϕþ ϕþ LHC searches. fields are 1 ; 2 and 4 whose masses are respectively For conventional notation, hereafter we will call Z1 and given by: 2 Z by Z and Z0, respectively. 1 2 μ2 2λχϕ 2ληϕ 2λϕξ Now we turn to the main subject—the Higgs sector. mϕþ ¼ ϕþ þ ½vχ 1 þ vη 1 þ vξ 1 ; 1 1 2 1 2 μ2 2λχϕ 2ληϕ 2λϕξ IV. ANALYSIS OF THE LEPTON mϕþ ¼ ϕþ þ ½vχ 2 þ vη 2 þ vξ 2 ; 2 2 2 NUMBER CONSERVING PART 1 OF THE SCALAR POTENTIAL 2 μ2 2λχϕ 2ληϕ 2λϕξ mϕþ ¼ ϕþ þ ½vχ 4 þ vη 4 þ vξ 4 : ð37Þ 4 4 2 Below we present lepton number conserving part VLNC þ þ þ of the scalar potential of the model shown in Appendix. In addition, the basis (ρ1 ; ρ3 ; ϕ3 ) corresponds to the Expanding the Higgs potential around the VEVs, one gets following squared mass matrix
0 1 2 1 1 A þ 2 vηðλ6 þ λ9Þ 0 2 vηvξλ3 B 1 2 2 1ffiffi C 2 B 0 A þ ðvχλ7 þ vηλ6Þ p vχw2 C Mcharged ¼ @ 2 2 A; ð38Þ 1 λ 1ffiffi μ2 2 vηvξ 3 p vχw2 þ þ B3 2 ϕ3
2 2 0 m ¼ μφ þ B2; ð41Þ where we have used the following notations Iφ2 2 1 ≡ μ2 2λ λ 2 where A ρ þ 2 ½vχ 18 þ ρξvξ ; 1 2 χφ 2 ηφ 2 φξ 1 χϕ ηϕ ϕξ 0 ≡ λ λ λ 1 2 ≡ 2λ 2λ 2λ 1 2 3 4 Bn 2 ðvχ n þ vη n þ vξ n Þ;n¼ ; : ð42Þ Bi 2 ðvχ i þ vη i þ vξ i Þ;i¼ ; ; ; : ð39Þ
þ The squared mass matrix of the four remaining CP-odd From (38), it follows that in the limit vη ≪ vξ, ρ is a 1 Higgs fields separate into two block diagonal submatrices physical field with mass corresponding to the two original base I 0 ;I 0 and ð χ1 η3 Þ 1 ðIφ ;IρÞ, namely 2 2 λ λ 1 mρþ ¼ A þ vηð 6 þ 9Þ; ð40Þ 1 2 ! 2 − þ þ 2 λ8 vη vχvη while the two massive bilepton scalars ρ3 and ϕ3 mix with m ; CPodd1 ¼ 2 2 each other. −vχvη vχ ! Now we turn into CP-odd Higgs sector. There are three μ2 0 − 1 λ − λ φ1 þ B1 C 2 vχvηð 1 2Þ 0 0 0 2 massless fields: Iχ ;Iη , and Iξ . The field Iφ2 is a physical m 2 ¼ ; ð43Þ 3 1 CPodd 1 λ − λ λ6 2 state itself with squared mass 2 vχvηð 1 2Þ A þ 2 vη
015004-7 LONG, HOP, HUE, THAO, and CÁRCAMO HERNÁNDEZ PHYS. REV. D 100, 015004 (2019) where The physical states relating to the second matrix in (43) are 2 2 2 C ≡ vχλ22 þ vηλ24 þ vξλ25 ð44Þ θ θ The first matrix in (43) provides two mass eigenstates, A2 cos ρ sin ρ Iφ1 ¼ ; ð47Þ where one of them is massless, A3 − sin θρ cos θρ Iρ
G1 cos θ Iχ0 sin θ Iη0 ;m0; ¼ a 1 þ a 3 G1 ¼ where the mixing angle is given by λ 2 2 8vχ A1 − sin θ I 0 cos θ I 0 ;m ; 45 ¼ a χ1 þ a η3 A1 ¼ 2 ð Þ 2cos θa vχvηðλ1 − λ2Þ tan 2θρ ¼ : ð48Þ where μ2 − 0 − − λ6 2 ð φ1 C þ B1 A 2 vηÞ vη tan θa ¼ : ð46Þ vχ Their squared masses are
n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio 2 1 2 2 2 2 2 2 m ¼ A þ D1 ∓ ðA − D1Þ þ vη½2ðA − D1Þλ6 þ vηλ6 þ vχðλ13 − λ14Þ ; ð49Þ A2;3 2
2 0 1 2 where D1 μφ B − C vηλ6. corresponding to the two original base Rχ0 ;Rη0 and ¼ 1 þ 1 þ 2 ð 1 3 Þ Next, the CP-even scalar sector is our task. We find that ðRρ;Rφ1 Þ, respectively. The physical states of the first Rφ 2 is physical with mass matrix in (52) are determined as follows,
2 2 2 1 2 χφ 2 ηφ 2 φξ m ¼ m ¼ μφ þ ðvχλ2 þ vηλ2 þ vξλ2 Þ: ð50Þ Rφ2 Iφ2 2 2 R ¼ cos θ Rχ0 þ sin θ Rη0 ;m¼ 0; G1 a 1 a 3 RG1 φ0 λ 2 As mentioned in Ref. [19], the lightest scalar 2 is a 2 2 8vχ H1 ¼ − sin θ Rχ0 þ cos θ Rη0 ;m¼ m ¼ : possible DM candidate with light mass smaller than 1 TeV. a 1 a 3 H1 A1 2 2cos θa Therefore, Eq. (50) suggests a reasonable assumption ð53Þ
2 1 2 χφ 2 φξ μφ ¼ − ðvχλ þ v λ Þ: ð51Þ 2 2 2 ξ 2 The physical states of the second matrix in (52) are 0 In this case, the model contains the complex scalar DM φ2 2 2 1 ληφ 2 with mass mRφ ¼ mIφ ¼ 2 2 vη. 2 2 H2 cos θr sin θr Rρ There are other seven CP-even Higgs components which ¼ ; ð54Þ H3 − sin θ cos θ Rφ the squared mass matrix separates into two 2 × 2 and one r r 1 3 × 3 independent matrices. The 2 × 2 matrices are ! 2 where the mixing angle is 2 λ8 vη vχvη mCPeven1 ¼ 2 ; 2 vχvη vχ ! vχvηðλ1 þ λ2Þ λ6 2 1 tan 2θ ¼ ð55Þ A þ vη − vχvηðλ1 þ λ2Þ r 2 0 λ6 2 2 2 2 ðμφ þ C þ B − A − vηÞ m 2 ¼ ; 1 1 2 CPeven − 1 λ λ μ2 0 2 vχvηð 1 þ 2Þ φ1 þ C þ B1 ð52Þ and their squared masses are
n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio 2 1 2 2 2 2 2 2 m ¼ A þ D2 ∓ ðA − D2Þ þ vη½2ðA − D2Þλ6 þ vηλ6 þ vχðλ13 þ λ14Þ ; ð56Þ H2;3 2
μ2 0 1 2λ where D2 ¼ φ1 þ B1 þ C þ 2 vη 6.
015004-8 SOME PHENOMENOLOGICAL ASPECTS OF THE 3-3-1 MODEL … PHYS. REV. D 100, 015004 (2019)
The squared mass matrix corresponding to the basis Combination of G1 and RG1 is Goldstone boson for 0 (Rχ0 ;Rη0 ;Rξ0 )is 0 3 1 neutral bilepton gauge boson X , namely GX ¼ 1ffiffi 0 1 p ðR − iG1Þ. The massive fields are Rφ ;H1;H2 2 2 G1 2 2vχλ13 vχvηλ5 λχξvχvξ B C and three massive Rχ, Rη, Rξ0 and the SM-like Higgs 2 B λ 2 2λ λ C mCPeven3 ¼ @ vχvη 5 vη 17 ηξvηvξ A; ð57Þ boson h. Note that there exists degeneracy in 2 Eqs. (50) and (53) when the contribution arising λχξvχvξ ληξvηvξ 2λξvξ from Z2 × Z4 soft breaking scalar interactions is not considered. Thus, the complex scalar φ2 has mass which contains a SM-like Higgs boson found by LHC. The given by Eq. (50), which is consistent with the mass eigenstates will be discussed using simplified prediction in Ref. [19]. To be a DM candidate, the conditions. condition (51) can be used to eliminate the terms Let us summarize the Higgs content: with large VEVs such as vχ and vξ. As a result, the (1) In the charged scalar sector: there are two Goldstone − − − mass of the DM candidate is bosons η and χ eaten by the gauge bosons W − and Y . Three massive charged Higgs bosons are 2 1 2 ηφ þ þ þ þ þ mφ ¼ vηλ2 : ð58Þ ϕ1 ; ϕ2 , and ϕ4 . The remaining fields ρ1 ; ϕ3 , and 2 2 þ ρ3 are mixing. (2) In the CP-odd scalar sector: there is one massless According [30], the weakly interacting massive particles (WIMP) candidate has mass around Majoron scalar Iξ0 which is denoted by GM. For- ηφ 10 GeV, implying that λ2 ≈ 0.04. To get the second tunately, it is a gauge singlet, therefore, is phenom- 0 DM candidate, namely, φ1, we have to carefully enologically harmless. Two massless scalars Iη0 and 1 choose conditions. Iχ0 are Goldstone bosons for the gauge bosons Z and 3 Equations (45) and (53) result in a new complex w 0 Z , respectively. There exists another massless state defined as follows denoted by G1, its role will be discussed below. Here we just mention that in the limit vη ≪ vχ, this field is 1 λ 2 ffiffiffi 2 8vχ 0 ω ¼ p ðH1 − iA1Þ;mω ¼ : ð59Þ Iχ . The massive CP-odd field are Iφ2 , A1 and other 2 1 2 2cos θa two Iφ1 ;Iρ are mixing. (3) In the CP-even scalar sector: There is one massless Let us rewrite the Higgs content in terms of the mass field: R , and in the limit vη ≪ vχ, it tends to Rχ0 . eigenstates mentioned above: G2 1
0 1 0 þ 1 0 1 1 G 0 ρ pffiffi − X 1 2 ðvη þ h iGZÞ B C B 1 C B C − ffiffi χ ≃ @ G A ρ @ p Rρ − iIρ A η ≃ @ − A Y ; ¼ 2 ð Þ ; GW ; 1ffiffi p ðvχ þ Rχ0 − iG 0 Þ þ ω 2 3 Z ρ3
0 1 φ ¼ pffiffiffi ðRφ − iIφ Þ ∼ ð1; 1; 0;i;1; 0Þ ∼ DM candidate; 2 2 2 2
0 1 ξ ¼ pffiffiffi ðvξ þ Rξ0 − iG Þ ∼ ð1; 1; 0Þ: ð60Þ 2 M
A. Simplified solutions λ1 ¼ λ2; λ15 ¼ λ16; λ19 ¼ λ20;w1 ¼ w4: ð61Þ We have shown that the mass eigenstates of scalars have been determined explicitly, except those relating to the two In the next steps, we just pay attention to find the masses 3 × 3 matrices (38) and (57). By introducing some further and mass eigenstates of the two matrices (38) and (57). The constrained assumptions to simplify these matrices so that other will be summarized if necessary. the physical states can be found, we will point out that the parameter space of the model under consideration contains 1. The CP-odd Higgs bosons valid regions, which are consistent with the experiment data including the ρ parameter. To reduce the arbitrary of the Under the assumption (61), the CP-odd scalar sector consists of four massless fields I 0 ;Iη;G ;G1 and four unknown Higgs couplings in the potential (A1), the f χ3 M g following relations are assumed first massive fields fA1;A2;A3Iφ2 g, as summarized in Table III.
015004-9 LONG, HOP, HUE, THAO, and CÁRCAMO HERNÁNDEZ PHYS. REV. D 100, 015004 (2019)
TABLE III. Squared mass of CP-odd scalars under condition in (61) and vχ ≫ vη.
Fields Iχ0 G1 ∈ G 0 Iχ0 G 0 Iη0 G Iη0 A1 Iρ A2 Iφ0 A3 Iφ0 DM Iξ0 G 1 ¼ X 3 ¼ Z 1 ¼ Z 3 ¼ ¼ 1 ¼ 2 ¼ ¼ M Squared mass 0 0 0 m2 m2 m2 m2 0 A1 A2 A3 Iφ0 2
TABLE IV. Squared masses of CP-even scalars under condition in (62) and vχ ≫ vη.
Fields Rχ0 ∈ G 0 Rχ0 ≃ H4 Rη0 hRη0 H1 Rρ H2 Rφ0 H3 Rφ0 DM Rξ0 ≃ H5 1 X 3 1 ¼ 3 ¼ ¼ 1 ¼ 2 ¼ 2 4 2 2 2 2 2 2 2 2 Squared mass 0 λvχ λvη m ¼ m m m m ¼ m 3λvχ 3 H1 A1 H2 H3 Rφ0 Iφ0 2 2
2. The CP-even and SM-like Higgs bosons where h is the SM-like Higgs boson with mass 126 GeV Now we turn to the sector where the SM Higgs boson identified with that found by LHC, whereas H4 and H5 are physical heavy scalars acquiring masses at the breaking exists, i.e., the matrix in the basis (R 0 ;R 0 ;R 0 ) is given by χ3 η1 ξ scale of the SUð3Þ × Uð1Þ × Z4 × Z2 × Uð1Þ sym- Eq. (57). Let us assume a simplified scenario worth to be L X Lg considered is characterized by the following relations: metry. Thus, we find that h has couplings very close to SM expectation with small deviations of the order of 2 vη −3 λ λ λ λ λ λ λ 2 ∼ Oð10 Þ. In addition, the squared masses of the 5 ¼ 13 ¼ 17 ¼ ξ ¼ χξ ¼ ηξ ¼ ;vξ ¼ vχ: ð62Þ vχ physical scalars h and H4;5 are given in (64). In this scenario, the squared matrix (57) takes the simple Now, the content of the CP-even scalar sector is form: summarized in Table IV. 0 1 Taking into account mass of the SM Higgs boson equal 2 2x xx 126 GeV, from Table IV we obtain λ ≈ 0.187, which can be 2 B C 2 vη λ@ 21A θ used to calculate masses of the H4 5 once vχ is fixed. mCPeven3 ¼ x vχ;x¼ ¼ tan a: ; vχ x 12 3. The charged Higgs bosons ð63Þ The charged scalar sector contains two massless fields:
Because vχ ≫ vη, the matrix (63) can be perturbatively GWþ and GYþ which are Goldstone bosons eaten by the þ þ diagonalized as follows: longitudinal components of the W and Y gauge bosons, þ þ þ respectively. The other massive fields are ϕ1 ; ϕ2 , and ϕ4 0 1 4 λ 2 00 with respective masses given in (39). 3 vη þ þ þ B C In the basis (ρ1 ; ρ3 ; ϕ3 ), the squared mass matrix is T 2 ≃ @ 0 λ 2 0 A RCPeven3mCPeven3RCPeven3 vχ ; given in (38). Let us make effort to simplify this matrix. 2 μ2 μ2 μ2 003λvχ Note that χ, η, and ξ can be derived using relations (35) 0 ffiffi 1 2 p and (62). In addition, it is reasonable to assume −1 þ x 0 2 x B 9 3 C 2 2 B qffiffiffiffi q C vχ vχ χϕ ϕξ B 1 1 C μ2 − λ λ ≈ μ2 μ2 − λ λ ≃ B x − C ρ ¼ ð 18 þ ρξÞ η; ϕþ ¼ ð 2 þ 2 Þ; RCPeven3 B 3 2 2 C; 2 3 2 @ qffiffiffiffi q A x 1 1 ð66Þ 3 2 2 we obtain the simple form of the squared mass matrix of the ð64Þ charged Higgs bosons, 0 1 1 2 λ3 Thus, we find that the physical scalars included in the 2 vηðλ6 þ λ9Þ 0 2 vηvχ 2 B C matrix m 3 are 1 2 2 1ffiffi CPeven 2 B 0 ðvχλ7 þ λ6vηÞ p vχw2 C Mchargeds ¼ @ 2 2 A: 0 2 1 λ 1 1 2 ηϕ 0 1 −1 x x x 0 1 3 pffiffi λ þ 9 3 3 2 vηvχ 2 vχw2 2 vη 2 Rη0 h B qffiffiffiffi q C 1 B C B 0 − 1 1 CB C ð67Þ @ H4 A ≃ B 2 2 C@ Rχ0 A; ð65Þ @ A 3 pffiffi qffiffiffiffi q H5 2 1 1 Rξ0 The matrix (67) predicts that there may exist two light 3 x 2 2 þ charged Higgs bosons H1;2 with masses at the electroweak
015004-10 SOME PHENOMENOLOGICAL ASPECTS OF THE 3-3-1 MODEL … PHYS. REV. D 100, 015004 (2019)
TABLE V. Squared mass of charged scalars under condition in (66) and vχ ≫ vη.
þ þ þ þ þ þ þ þ Fields η2 ¼ GWþ χ2 ¼ GYþ H1 H3 H2 ϕ1 ϕ2 ϕ4 2 2 2 2 2 2 Squared mass 0 0 m þ m þ m þ mϕþ mϕþ mϕþ H1 H3 H2 1 2 4
þ þ scale and the mass of H3 which is mainly composed of ρ3 the decoupling of the SM-like Higgs boson with other þ is around 3.5 TeV. In addition, the Higgs boson H1 almost CP-even neutral Higgs bosons. As a consequence, the one- does not carry lepton number, whereas the others two do. loop contribution of the SM-like Higgs boson to the ρ Generally, the Higgs potential always contains mass parameter is the same as in the SM. Excepting for the terms which mix VEVs. However, these terms must be components of the scalar triplet ρ, the other heavy CP-even small enough to avoid high order divergences (for exam- neutral Higgs bosons do not couple with the SM gauge ples, see Refs. [33,34]) and provide baryon asymmetry bosons W and Z and thus they do not provide contributions of Universe by the strong electroweak phase transition to the ρ parameter. Contributions of the remaining Higgs (EWPT). bosons can be calculated using the results given in Ignoring the mixing term containing λ3 in (67) does not Ref. [38]. In particular, contributions of any Higgs bosons affect other physical aspects, since the above-mentioned in our case to Δρ are determined as follows term just increases or decreases a small amount of the Π ð0Þ Π ð0Þ charged Higgs bosons. Therefore, without loss of general- Δρ ¼ WW − ZZ ; ð69Þ M2 M2 ity, neglecting the term with λ3 satisfies other aims such W Z as EWPT. where Π ð0Þ and Π ð0Þ are the coefficients of −igμν in λ WW ZZ Hence, in the matrix of (67), the coefficient 3 is the vacuum-polarization amplitudes of charged and neutral reasonably assumed to be zero. Therefore we get immedi- W bosons and Z gauge bosons, respectively. Our case ρþ ately one physical field 1 with mass given by relates with only the contribution of “non-Higgs scalars” ϕ 1 1;2 with masses m1;2 and coupling 2 2 λ λ mρþ ¼ vηð 6 þ 9Þ: ð68Þ ↔ 1 2 μ μ icϕ1 ∂ μϕ2V ≡ ic½ϕ1∂μϕ2 − ð∂μϕ1Þϕ2 V ; ðV ¼ W;ZÞ; The other fields mix by submatrix given at the bottom of þ þ ð70Þ (67). The limit ρ1 ¼ H1 when λ3 ¼ 0 is very interesting for discussion of the Higgs contribution to the ρ parameter. The corresponding contribution is The content of the charged scalar sector is summarized in 2 Table V. It is worth mentioning that the masses of three jcj ΠðscalarÞ¼ f ðm1;m2Þ; ð71Þ ϕþ 1 16π2 s charged scalars i , i ¼ , 2, 4 are still not fixed. The potential including lepton number violations, i.e., where Vfull ¼ VLNC þ VLNV is quite similar to the previous one. 2 2 2 There are some differences: m1m2 m2 (1) The masses of the fields receive some new contri- fsðm1;m2Þ¼fsðm2;m1Þ¼ 2 2 ln 2 m1 − m2 m1 butions. 1 (2) The complex scalar φ0 has the same mass in 2 2 2 þ 2 ðm1 þ m2Þ both cases. 1 (3) Majoron does not exist and its mass only arises from 2 2 x lnðxÞ þ x ¼ m f ðxÞ¼m þ ; lepton number violating scalar interactions. 1 s 1 1 − x 2 (4) The mixing of the CP-even scalar fields is more 2 ≡ m2 complicated. x21 2 : ð72Þ m1
B. Scalar contributions to the ρ parameter The function in Eq. (72) satisfies fsðm1;m1Þ¼limm2→m1 × 0 0 ≠ The new Higgs bosons may give contribution to the ρ fsðm1;m2Þ¼ and fsðm1;m2Þ > with m1 m2.Asa ϕ parameter at one-loop level, as shown in many models consequence, the charged Higgs bosons 1;2;4 having beyond the SM, such as the simplified models [35], the two vanishing Higgs-gauge couplings with other Higgs bosons Higgs doublet models [36,37], and the supersymmetric give vanishing contributions to the ρ parameter. Nonvani- version of the SM [38]. In the 3-3-1 CKS model, we will shing contributions now may arise from the charged Higgs ρ ρ ϕ consider the effect of the Higgs contributions to the ρ bosons H1;2;3 corresponding to the basis ( 1 , 3 , 3 ) and parameter at one-loop level. These contributions will be the CP-odd neutral Higgs relating with Iρ. The relevant determined in the limit of the suppressed Z − Z0 mixing and Lagrangian is
015004-11 LONG, HOP, HUE, THAO, and CÁRCAMO HERNÁNDEZ PHYS. REV. D 100, 015004 (2019)
FIG. 2. Contour plots of the ρ parameter (dotted-dashed curves) and M 0 (black curves) as functions of vχ and m þ . The green regions Z H1 are excluded by the recent experimental constraint of the ρ.
† μ þ μ þ L ¼ðDμρÞ ðD ρÞþðDμϕ Þ ðD ϕ Þ VHH 3 3 1 − 2 2 ↔ −2 2 ↔ − ↔ ig μ sW − þ sW − þ i → Z ðρ ∂ μρ Þþ ðρ ∂ μρ Þþ ðIρ ∂ μRρÞ 2 c 1 1 c 3 3 c W W W − 2 ↔ ↔ ↔ igsW μ − þ ig þμ − þ Z ðϕ3 ∂ μϕ3 Þþ − W ρ1 ð∂ μRρ − i∂ μIρÞþH:c: : ð73Þ cW 2
015004-12 SOME PHENOMENOLOGICAL ASPECTS OF THE 3-3-1 MODEL … PHYS. REV. D 100, 015004 (2019)
þ In the scalar-gauge interactions of Eq (73), only the last relation (75), the mass of the charged Higgs boson H1 term contributes to ΠWWð0Þ, thus giving rise to a non- should also be at the TeV scale. From Fig. 2, we can see ρ 2 negative contribution to the parameter, which may make that MZ0 ≥ 4 TeV is allowed if Δm is large enough, for 2 2 the allowed MZ0 mass to move outside the excluded region example Δm ≥ ð0.246 TeVÞ . recently reported by LHC searches [39]. On the contrary, all of the remaining terms contributing to ΠZZð0Þ,give nonpositive contributions to the ρ parameter. For illustra- V. CHARGED LEPTON FLAVOR VIOLATING tion, we will consider a simple case where only positive DECAY CONSTRAINTS ρ ρ ≡ contributions to the parameter are kept, namely 1;3 In this section we will determine the constraints that the ϕ ≡ ≡ μ → γ τ → μγ H1;3, 3 H2 , Iρ A2 and Rρ are mass eigenstates. Then, charged lepton flavor violating decays e , and τ → γ all contributions to ΠZZð0Þ arising from the charged Higgs e imposed on the parameter space of our model. As 0 ρ bosons are proportional to fsðms;msÞ¼ with s ¼ 1;3; mentionedinRef.[19], the sterile neutrino spectrum of the ϕ3 . In addition, the simplified condition (61) with λ1 ≪ 1 model is composed of two almost degenerate neutrinos with results in m2 ¼ m2 , leading to a vanishing neutral Higgs masses at the Fermi scale and four nearly degenerate neutrinos Iρ Rρ with TeV scale masses. These sterile neutrinos together with boson contribution to Π ð0Þ∶ f ðm ;m Þ¼0. The only ZZ s Iρ Rρ the heavy W0 gauge boson induce the l → l γ decay at one nonzero contribution has the form i j loop level, whose branching ratio is given by: [44–46]: pffiffiffi 2 2 H g GF 3 2 5 2 2 2 Δρ þ þ α s m m m ¼ 2 2 fsðmH ;mRρ Þ¼ 2 fsðmH ;mRρ Þ; W W li N1 N2 16π 1 16π 1 Br l → l γ 2G 4G ; mW ð i j Þ¼256π2 4 Γ 2 þ 2 mW0 i mW0 mW0 ð74Þ 2x3 þ 5x2 − x 3x3 GðxÞ¼− − ln x: ð76Þ where 4ð1 − xÞ2 2ð1 − xÞ4 λ 2 2 2 9vη 2 m 100 Δ ≡ − þ − ∼ O In our numerical analysis we have fixed N1 ¼ GeV and m mRρ mH ¼ ðvηÞ: ð75Þ 1 2 we have varied the W0 gauge boson mass in the range 0 Allowed regions of the parameter space for some specific 4 TeV ≲ mW0 ≲ 5 TeV. We consider neutral heavy Z gauge values of Δm2 are shown in Fig. 2. It can be seen that the boson masses larger than 4 TeV to fulfill the bound arising large values of vχ are still allowed, thus implying that no from the experimental data on K, D and B meson mixings upper bounds are required. The allowed values of vχ and [28].Figure3 shows the allowed parameter space in the − 0 mW0 mN plane consistent with the constraints arising from MZ strongly depend on the lower bound of mH and mRρ , 1 charged lepton flavor violating decays. As seen from Fig. 3, which may have previously reported from LHC searches. μ → γ Unfortunately, the Higgs triplets ρ containing the neutral theobtainedvaluesforthebranchingratioof e decay are 4 2 10−13 components ρ0 with zero VEV, hence all of the three below its experimental upper limit of . × since these 2 3 10−13 ≲ μ → γ ≲ components of ρ do not couple to the two SM gauge bosons values are located in the range × Brð e Þ 4 10−13 Z and W. This Higgs triplet also does not contribute to the × , for sterile neutrino masses mN2 lower than about SM-like Higgs boson. As a result, all of the Higgs bosons 1.12 TeV.In the same region of parameter space, the obtained τ → μγ τ → γ H1 , Rρ, and Iρ are not affected by the following decays branching ratios for the and e decays can reach searched by LHC: H1 → W Z, W h and Rρ, Iρ → WþW−; ZZ; Zh. These Higgs bosons do not couple with the SM quarks [19], cannot be produced at LHC from the recent channel searching [40]. Only the allowed tree level decays to two SM fermions are leptonic decays: H1 → ν2;3τ; ν2;3μ and Rρ, Iρ → e¯iei (i ¼ 1, 2, 3), which are also searched by LHC, but the couplings of these Higgs with SM quarks are necessary to produce these Higgs bosons. Because the above Higgs bosons have couplings with many new charged particles in the model under consid- eration, their one-loop level decays to photons may appear, thus making them interesting channels for their search at the LHC, in particular these charged Higgs bosons feature the following decay modes H1 → W γ, Rρ, Iρ → Zγ [41], and Rρ, Iρ → γγ [42,43]. The heavy neutral Higgs boson masses are predicted to be at the TeV scale, which is FIG. 3. Allowed parameter space in the mW0 − mN plane outside the LHC excluded regions. Combined with the consistent with the LFV constraints.
015004-13 LONG, HOP, HUE, THAO, and CÁRCAMO HERNÁNDEZ PHYS. REV. D 100, 015004 (2019) values of the order of 10−13, which is below their correspond- A. Phenomenology of Z0 gauge boson −9 −9 ing upper experimental bounds of 4.4 × 10 and 3.3 × 10 , In what follows we proceed to compute the total cross respectively. Consequently, our model is compatible with the section for the production of a heavy Z0 gauge boson at the charged lepton flavor violating decay constraints provided LHC via Drell-Yan mechanism. In our computation for the that the sterile neutrinos are lighter than about 1.12 TeV. total cross section we consider the dominant contribution due to the parton distribution functions of the light up, VI. SEARCH FOR Z0 AT LHC down and strange quarks, so that the total cross section for the production of a Z0 via quark antiquark annihilation in In this section, we present two typical effects of the LHC, pffiffiffi namely, production of single a particle in proton-proton proton proton collisions with center of mass energy S collisions. takes the form:
8 qffiffiffiffiffi sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi 2 ! ! Z m 2 < − Z0 2 2 g π ln S m 0 m 0 σðDrellYanÞ 0 2 0 2 qffiffiffiffiffi Z y μ2 Z −y μ2 → 0 ðSÞ¼ 2 ½ðg Þ þðg Þ 2 fp=u e ; fp=u¯ e ; dy pp Z 6 : uL uR m cWS Z0 S S ln S qffiffiffiffiffi sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi 2 ! ! Z m − Z0 2 2 ln S m m 0 2 0 2 qffiffiffiffiffi Z0 y μ2 Z0 −y μ2 þ½ðg Þ þðg Þ 2 f e ; f ¯ e ; dy dL dR m p=d p=d Z0 S S ln S qffiffiffiffiffi sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi 9 2 ! ! Z m − Z0 2 2 = ln S m m 0 2 0 2 qffiffiffiffiffi Z0 y μ2 Z0 −y μ2 þ½ðg Þ þðg Þ 2 fp=s e ; fp=s¯ e ; dy dL dR m ; Z0 S S ln S
Figure 4 displays the Z0 total production cross section at In particular, in Ref. [47] it has been shown the Z0 decays pffiffiffi the LHC via Drell-Yan mechanism at the LHC for S ¼ into a lepton pair in 3-3-1 models have branching ratios of 13 0 the order of 10−2, which implies that the total LHC cross TeV and as a function of the Z mass, which is taken to pffiffiffi range from 4 TeV up to 5 TeV. We consider neutral heavy Z0 section for the pp → Z0 → lþl− resonant production at S ¼ gauge boson masses larger than 4 TeV to fulfill the bound 13 TeV will be of the order of 1 fb for a 4 TeV Z0 gauge arising from the experimental data on K, D,andB meson boson, which is below its corresponding lower experimental mixings [28]. For such as a region of Z0 masses we find that limit arising from LHC searches [39]. On the other hand, at the total production cross section ranges from 85 fb up to the proposed energy upgrade of the LHC at 28 TeV center of 10 fb. The heavy neutral Z0 gauge boson after being pro- mass energy, the total cross section for the Drell-Yan pro- duced it will decay into pair of SM particles, with dominant duction of a heavy Z0 neutral gauge boson gets significantly decay mode into quark-antiquark pairs as shown in detail in enhanced reaching values ranging from 2.5 pb up to 0.7 pb, 0 as indicated in Fig. 5. Consequently, the LHC cross section Refs. [24,47]. The two body decays of the Z gauge boson pffiffiffi in 3-3-1 models have been studied in details in Refs. [47]. for the pp → Z0 → lþl− resonant production at S ¼ 28 TeV
FIG. 4. Total cross section for the Z0 production via Drell-Yan FIG. 5. Total cross section for the Z0 production via Drell-Yan pffiffiffi 13 mechanism at the proposed energy upgrade of the LHC with mechanism at the LHC for S ¼ TeV and as a function of the pffiffiffi Z0 mass. S ¼ 28 TeV as a function of the Z0 mass.
015004-14 SOME PHENOMENOLOGICAL ASPECTS OF THE 3-3-1 MODEL … PHYS. REV. D 100, 015004 (2019)
−2 will be of the order of 10 pb for a 4 TeV Z0 gauge boson, Let us note that the singly heavy scalar H4 is mainly which corresponds to the order of magnitude of its correspond- produced via gluon fusion mechanism mediated by a ing lower experimental limit arising from LHC searches [39]. triangular loop of the heavy exotic quarks T, J1, and J2. Thus, the total cross section for the production of the heavy scalar H4 through gluon fusion mechanism in proton B. Phenomenology of H4 heavy Higgs boson pffiffiffi In what follows we proceed to compute the LHC proton collisions with center of mass energy S takes production cross section of the singly heavy scalar H4. the form:
α2 2 2 2 2 2 SmH4 jðRCPeven3Þ22j mH4 mH4 mH4 σ → → ðSÞ¼ I þ I þ I pp gg H4 64π 2 2 2 2 vχS mT mJ1 mJ2 qffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi 2 m ! ! Z H 2 2 − ln 4 m m qffiffiffiffiffiffi S H4 y 2 H4 −y 2 × 2 f e ; μ f e ; μ dy m p=g p=g H4 S S ln S
μ2 μ2 4 4 ≲ ≲ 8 where fp=gðx1; Þ and fp=gðx2; Þ are the distributions of field H4 with mass in the range . TeV MH4 TeV, gluons in the proton which carry momentum fractions x1 based on Ref. [48], it is reasonable to assume that its μ ¯ and x2 of the proton, respectively. Furthermore ¼ mH4 is dominant decay mode will be on tt pair. Furthermore, in the the factorization scale and IðzÞ is given by: region of H4 masses considered in our analysis, the H4 Z Z decay into exotic quark pairs will be kinematically for- 1 1−x 1 − 4xy bidden for exotic quark Yukawa couplings of order unity. IðzÞ¼ dx dy : ð77Þ 0 0 1 − zxy Note that we have chosen values for vχ larger than 10 TeV, which corresponds to a Z0 gauge boson heavier than 4 TeV, which is required to guarantee the consistency of 331 Figure 6 displays the H4 total production cross section at pffiffiffi models with the experimental data on K, D, and B meson the LHC via gluon fusion mechanism for S ¼ 13 TeV, as mixings [28]. Here, for the sake of simplicity we have a function of the SUð3Þ × Uð1Þ symmetry breaking scale L X restricted to the simplified scenario described by Eq. (62) vχ, which is taken to range from 10 TeV up to 20 TeV. The and we have chosen the exotic quark Yukawa couplings aforementioned range of values for the SUð3Þ × Uð1Þ L X equal to unity, i.e., yðTÞ ¼ yðJ1Þ ¼ yðJ2Þ ¼ 1. In addition, symmetry breaking scale vχ corresponds to a heavy scalar the top quark mass has been taken to be equal to mass mH4 varying between 4.4 TeV and 8.9 TeV. Consid- mt ¼ 173 GeV. We find that the total cross section for ering the mass of the heavy scalar field H4 in the range the production of the H4 scalar at the LHC takes a value 8 TeV ≲ M ≲ 8.9 TeV, it is reasonable to assume that it −4 H4 close to about 10 fb for the lower bound of 10 TeVof the will have dominant decay modes into W0W0 and Z0Z0 heavy gauge boson pairs. On the other hand, for a heavy scalar
FIG. 7. Total cross section for the H4 production via gluon FIG. 6. Total cross section for the H4 production via gluon fusion mechanism at the proposed energy upgrade of the LHC pffiffiffi pffiffiffi 13 28 3 1 fusion mechanism at the LHC for S ¼ TeV and as a with S ¼ TeV as a function of the SUð ÞL × Uð ÞX 3 1 function of the SUð ÞL × Uð ÞX symmetry breaking scale vχ symmetry breaking scale vχ for the simplified scenario described for the simplified scenario described in Eq. (62). in Eq. (62).
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3 1 SUð ÞL × Uð ÞX symmetry breaking scale vχ arising from estimate of the DM relic density in our model, under some the experimental data on K, D, and B meson mixings [28] simplifying assumptions motivated by the large number of and decreases when vχ takes larger values. We see that the scalar fields of the model. We do not intend to provide a total cross section at the LHC for the H4 production via sophisticated analysis of the DM constraints of the model gluon fusion mechanism is small to give rise to a signal under consideration, which is beyond the scope of the present 3 1 for the allowed values of the SUð ÞL × Uð ÞX symmetry paper. We just intend to show that our model can accom- breaking scale vχ. A similar situation happens at the pro- modate the observed value of the DM relic density, by having pffiffiffi posed energy upgrade of the LHC with S ¼ 28 TeV, a scalar DM candidate with a mass in the TeV range and a where this total cross section takes a value of 1.6 × 10−2 fb quartic scalar coupling of the order unity, within the pertur- bative regime. We start by surveying the possible scalar DM for vχ ¼ 10 TeV as shown in Fig. 7. Because of the very candidates in the model. Considering that the Z4 symmetry is small H4 production cross section, we do not perform a detailed study of its decay modes. It is worth mentioning preserved and taking into account the scalar assignments that the smoking gun signatures of the model under under this symmetry, given by Eq. (1), we can assign this role 3 φ0 consideration will be the Z0 production and the charged to either any of the SUð ÞL scalar singlets, i.e., Re n and φ0 1 φ φ0 lepton flavor violating decay μ → eγ, whose observation Im n (n ¼ , 2). In this work we assume that the I ¼ Im 1 φ0 φ0 1 will be crucial to assess to viability of this model. is the lightest among the Re n and Im n (n ¼ ,2)scalar fields and also lighter than the exotic charged fermions, as VII. DARK MATTER RELIC DENSITY well as lighter than ΨR, thus implying that its tree-level decays are kinematically forbidden. Consequently, in this In this section we provide a discussion of the implications 0 mass range the Imφ scalar field is stable. of our model for DM, assuming that the DM candidate is a 1 The relic density is given by (cf. Ref. [30,49]) scalar. Let us recall that our goal in this section is to provide an
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 ffiffiffi ∞ X s s − 4mφ p T 2 σ φφ → ¯ s 4 gp vrel ð ppÞK1ð Þds 32π 2 p¼W;Z;t;b;h 2 0.1pb A 4mφ T Ωh2 ¼ ; hσvi¼ ¼ P ; ð78Þ σ 2 T 2 mφ 2 h vi neq ð2π2 p¼W;Z;t;b;hgpmφK2ð T ÞÞ where hσvi is the thermally averaged annihilation cross the Higgs field. In addition we assume that the decoupling section, A is the total annihilation rate per unit volume at of the nonrelativistic WIMP of our model is supposed temperature T, and neq is the equilibrium value of the to happen at a very low temperature. Because of this particle density. Furthermore, K1 and K2 are modified reason, for the computation of the relic density, we take Bessel functions of the second kind and order 1 and 2, T ¼ mφ=20 as in Ref. [49], corresponding to a typical respectively [49] and mφ ¼ mImφ. Let us note that we freeze-out temperature. We assume that our DM candidate assume that our scalar DM candidate is a stable weakly φ annihilates mainly into WW, ZZ, tt¯, bb¯, and hh, interacting particle (WIMP) with annihilation cross sec- with annihilation cross sections given by the following tions mediated by electroweak interactions mainly through relations [50]:
12m4 4m2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 1 W − W 2 2φ2 s þ s2 s 4m v σðφ φ → WWÞ¼ h 1 − W; rel I I 8π ðs − m2Þ2 þ m2Γ2 s h h h 12m4 4m2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 1 Z − Z 2 2φ2 s þ s2 s 4m v σ φ φ → ZZ h 1 − Z; rel ð I I Þ¼ 16π − 2 2 2Γ2 ðs mhÞ þ mh h s rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4m2 3 λ2 2 1 − f Nc 2φ2 mq s v σ φ φ → qq¯ h ; rel ð I I Þ¼ 4π − 2 2 2Γ2 ðs mhÞ þ mh h rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 2 2 2 2 2 2 3 4λ 2 2 v 4 σ φ φ → h φ 1 mh − h φ 1 − mh vrel ð I I hhÞ¼16π þ − 2 − 2 2 ; ð79Þ s s mh s mh s
015004-16 SOME PHENOMENOLOGICAL ASPECTS OF THE 3-3-1 MODEL … PHYS. REV. D 100, 015004 (2019)
0.5
0.4
0.3 2 h 0.2
0.1
0.0 200 300 400 500 600 m [GeV]
2 FIG. 9. Correlation between the quartic scalar coupling and the FIG. 8. Relic density Ωh , as a function of the mass mφ of the φ mass mφ of the scalar DM candidate φ, consistent with the scalar field, for several values of the quartic scalar coupling λ 2φ2 . h experimental value Ωh2 ¼ 0.1198 for the relic density. λ 2 2 0 5 The curves from top to bottom correspond to h φ ¼ . , 0.7, 0.8, 0.9, 1, respectively. The horizontal line shows the observed value Ωh2 ¼ 0.1198 [54] for the relic density. The vertical lines the extrapolation of our model at high energy scales as well correspond to the obtained lower and upper limits 300 GeV and as the preservation of perturbativity at one loop level. 570 GeV, respectively, of the mass mφ of the scalar dark matter candidate consistent with the experimental measurement of the VIII. CONCLUSIONS dark matter relic density. We have studied some phenomenological aspects of the pffiffiffi extended inert 331 model, which incorporates the mecha- where s is the center-of-mass energy, N ¼ 3 is the color c nism of sequential loop-generation of the SM fermion factor, m ¼ 125.7 GeV and Γ ¼ 4.1 MeV are the SM h h masses, explaining the observed strong hierarchies between Higgs boson h mass and its total decay width, respectively. them as well as the corresponding mixing parameters. A Note that we have worked on the decoupling limit where particular emphasis has been made on analyzing the the couplings of the 126 GeV Higgs boson to SM particles constraints arising from the experimental data on the ρ and its self-couplings correspond to the SM expectation. parameter, as well as those ones resulting from the charged The vacuum stability and tree level unitarity constraints lepton flavor violating process μ → eγ and dark matter. of the scalar potential are [51–53] Furthermore, we have studied the production of the heavy Z0 gauge boson in proton-proton collisions via the Drell- 2 2 λ 4 > 0; λφ4 > 0; λ 2 2 < λ 4 λφ4 : ð80Þ Yan mechanism. We found that the corresponding total h h φ 3 h cross section at the LHC ranges from 85 fb up to 10 fb when the Z0 gauge boson mass is varied within a 4–5TeV λ 4 8π λ 2 2 4π φ < ; h φ < : ð81Þ interval. The Z0 production cross section gets significantly enhanced at the proposed energy upgrade of the LHC with pffiffiffi The dark matter relic density as a function of the mass mφ 28 – φ S ¼ TeV reaching the typical values of 2.5 0.7 pb. of the scalar field I is shown in Fig. 8, for several values of 0 þ − 2 From these results we find that the pp → Z → l l the quartic scalar coupling λ 2 2 , set to be equal to 0.7, 0.8 h φ resonant production cross section reachffiffiffi values of about and 0.9 (from top to bottom). The horizontal line corre- −2 p 1 fb and 10 pb at M 0 ¼ 4 TeV, for S ¼ 14 TeV and Ω 2 0 1198 pffiffiffi Z sponds to the experimental value h ¼ . for the S ¼ 28 TeV, respectively. These obtained values for the relic density. We found that the DM relic density constraint pp → Z0 → lþl− resonant production cross sections are gives rise to a linear correlation between the quartic scalar below and of the same order of magnitude of its corre- λ 2 2 coupling h φ and the mass mφ of the scalar DM candidate sponding lower experimental limit arising from LHC φ pffiffiffi pffiffiffi I, as indicated in Fig. 9. searches, for S ¼ 13 TeV and S ¼ 28 TeV, respec- We find that we can reproduce the experimental value tively. Besides that, we have found that Z0 gauge bosons Ω 2 0 1198 0 0026 h ¼ . . [54] of the DM relic density, when heavier than about 4 TeV comply with the experimental φ the mass mφ of the scalar field I is in the range constraints on the oblique ρ parameter as well as with the 300 GeV ≲ mφ ≲ 570 GeV, for a quartic scalar coupling collider constraints. In addition, we have found that the λ 2 2 0 5 ≲ λ 2 2 ≲ 1 h φ in the window . h φ , which is consistent constraint on the charged lepton flavor violating decay with the vacuum stability and unitarity constraints shown μ → eγ set the sterile neutrino masses to be lighter than in Eqs. (80) and (81). Note that our range of values about 1.12 TeV. We have found that the obtained values of λ 2 2 μ → γ chosen for the quartic scalar coupling h φ also allow the branching ratio for the e decay are located in the
015004-17 LONG, HOP, HUE, THAO, and CÁRCAMO HERNÁNDEZ PHYS. REV. D 100, 015004 (2019) range 3 × 10−13 ≲ Brðμ → eγÞ ≲ 4 × 10−13, whereas the fixed. The scalar potential contains a Majoron but it is obtained branching ratios for the τ → μγ and τ → eγ decays harmless, because it is a scalar singlet. Due to the unbroken −13 can reach values of the order of 10 . Consequently, our Z4 symmetry our model has the stable scalar dark matter 0 0 model predicts charged lepton flavor violating decays candidates Reφn and Imφn (n ¼ 1, 2) and the fermionic within the reach of future experimental sensitivity. We dark matter candidate ΨR. In this work we assume that φ φ0 φ0 φ0 found that the total cross sectionffiffiffi for the production of the I ¼ Im 1 is the lightest among the Re n and Im n p 1 H4 scalar at the LHC with S ¼ 13 TeV takes a value (n ¼ , 2) scalar fields and also lighter than the exotic −4 Ψ close to about 10 fb for the lower bound of 10 TeVof the charged fermions and than R, which implies that it is 3 1 stable and thus it is the dark matter candidate considered in SUð ÞL × Uð ÞX symmetry breaking scale vχ required by the consistency of the ρ parameter with the experimental this work. To reproduce the dark matter relic density, the −2 mass of the scalar dark matter candidate has to be in the data. This value is increased to 1.6 × 10ffiffiffi fb at the pro- p range 300 GeV ≲ mφ ≲ 570 GeV, for a quartic scalar posed energy upgrade of the LHC with S ¼ 28 TeV, thus coupling λ 2φ2 in the window 0.5 ≲ λ 2φ2 ≲ 1. In addition, implying that the total cross section at the LHC for the H4 h h production via gluon fusion mechanism is very small to it has been shown in Ref. [19] that requiring that the DM 3 candidate φ0 lifetime be greater than the universe lifetime give rise to a signal for the allowed values of the SUð ÞL × τ ≈ 13 8 0 ∼ 1 Uð1Þ symmetry breaking scale vχ even at the proposed u . Gyr and assuming mφ TeV, we estimate the X 10 energy upgrade of the LHC. We also analyzed in detail the cutoff scale of our model Λ > 3 × 10 GeV. Thus we scalar potential and the gauge sector of the model. conclude that under the above specified conditions the 0 The general Higgs sector is separated into two parts. The model contains viable fermionic ΨR and scalar φ DM first part consists of lepton number conserving terms and candidates. A sophisticated analysis of the DM constraints the second one contains lepton number violating couplings. of our model is beyond the scope of the present paper and is The first part of potential was considered in details and the left for future studies. SM Higgs boson was derived and as expected, mainly 0 arises from η1. We have shown that the whole scalar ACKNOWLEDGMENTS potential, excepting its CP-even sector, has a quite similar This research has been financially supported by Fondecyt situation since the resulting physical scalar mass spectrum (Chile), Grant No. 1170803, CONICYT PIA/Basal FB0821, is similar in both cases. The scalar spectrum contains the Vietnam National Foundation for Science and Techno- enough number of Goldstone bosons for massive gauge logy Development (NAFOSTED) under Grant No. 103.01- bosons. In the CP-odd scalar sector, there are four massive 2017.356. H. N. L. is very grateful to the Bogoliubov bosons and one of them is a DM candidate. The CP-even Laboratory for Theoretical Physics, JINR, Dubna, Russia scalar mass spectrum consists of seven massive fields for the warm hospitality during his visit. including the SM Higgs boson and a DM candidate. The singly electrically charged Higgs boson sector contains six APPENDIX: THE SCALAR POTENTIAL massive fields. Two of them have masses at the electroweak scale and the remaining one has a mass around 3.5 TeV. The The renormalizable potential contain three parts: the first ϕþ 1 G masses for the three charged bosons i ;i¼ , 2, 4 are not one invariant under group in (1) is given by
X4 X2 μ2χ†χ μ2ρ†ρ μ2η†η μ2 ϕþϕ− μ2 φ0φ0 μ2ξ0 ξ0 VLNC ¼ χ þ ρ þ η þ ϕþ i i þ φi i i þ ξ i i¼1 i¼1 † † † † † † † † 2 þ χ χðλ13χ χ þ λ18ρ ρ þ λ5η ηÞþρ ρðλ14ρ ρ þ λ6η ηÞþλ17ðη ηÞ † † † † † † þ λ7ðχ ρÞðρ χÞþλ8ðχ ηÞðη χÞþλ9ðρ ηÞðη ρÞ X4 X2 χ†χ λχϕϕþϕ− λχφφ0φ0 λ ξ0 ξ0 þ i i i þ i i i þ χξ i¼1 i¼1 X4 X2 ρ†ρ λρϕϕþϕ− λρφφ0φ0 λ ξ0 ξ0 þ i i i þ i i i þ ρξ i¼1 i¼1 X4 X2 η†η ληϕϕþϕ− ληφφ0φ0 λ ξ0 ξ0 þ i i i þ i i i þ ηξ i¼1 i¼1
015004-18 SOME PHENOMENOLOGICAL ASPECTS OF THE 3-3-1 MODEL … PHYS. REV. D 100, 015004 (2019) X4 X4 X2 ϕþϕ− λϕϕϕþϕ− λϕφφ0φ0 λϕξξ0 ξ0 þ i i ij j j þ ij j j þ i i¼1 j¼1 j¼1 X2 X2 φ0φ0 λφφφ0φ0 λφξξ0 ξ0 λ ξ0 ξ0 2 þ i i ij j j þ i þ ξð Þ 1 1 i¼ j¼ þ 2 − 2 þ 2 − 2 þ 2 − 2 0 2 0 † − † 0 þ λ10ðϕ2 Þ ðϕ3 Þ þ λ11ðϕ2 Þ ðϕ4 Þ þ λ12ðϕ3 Þ ðϕ4 Þ þ w1ðφ2Þ φ1 þ w2χ ρϕ3 þ w3η χξ
0 2 0 þ − 0 þ − 0 0 0 † − 0 0 þ w4ðφ2Þ φ1 þ w5ϕ3 ϕ4 φ1 þ w6ϕ3 ϕ4 φ1 þ χρηðλ1φ1 þ λ2φ1 Þþχ ρϕ4 ðλ15φ1 þ λ16φ1 Þ † − 0 þ − 0 0 − þ þ − 0 2 0 3 0 þ λ3η ρϕ3 ξ þ λ4ϕ1 ϕ2 φ2ξ þðλ19ϕ3 ϕ4 þ λ20ϕ3 ϕ4 Þðφ2Þ þ λ21ðφ1Þ φ1 X4 X2 λ χ†χ λ ρ†ρ λ η†η λ ϕþϕ− λ φ0φ0 λ ξ0 ξ0 φ0 2 þð 22 þ 23 þ 24 þ 61i i i þ 62i i i þ 25 Þð 1Þ þ H:c: : ðA1Þ i¼1 i¼1
The second part is a lepton number violating one (the subgroup U 1 is violated) ð ÞLg μ2 χ†η η†χ λ χ†χ λ ρ†ρ λ η†η χ†η η†χ VLNV ¼ χηð þ Þþ½ 26ð Þþ 27ð Þþ 28ð Þ ð þ Þ † 2 † 2 † † † † þ λ29½ðχ ηÞ þðη χÞ þλ30½ðη ρÞðρ χÞþðχ ρÞðρ ηÞ X4 X2 ξ0 χ†χ ρ†ρ η†η ϕþϕ− φ0φ0 ξ0 ξ0 þ w7 þ w8 þ w9 þ w2i i i þ w3i i i þ w10 i¼1 i¼1 0 † 0 2 0 2 0 2 † − − þ 0 þ ξ ½w11χ η þ w12ðφ1Þ þ w13ðφ1 Þ þ w14ðξ Þ þw15η ρϕ3 þ w16ϕ2 ϕ1 φ2 X4 X2 ξ0 2 λ χ†χ λ ρ†ρ λ η†η λ ϕþϕ− λ φ0φ0 λ ξ0 ξ0 λ φ0 2 λ φ0 2 þð Þ 31 þ 32 þ 33 þ 63i i i þ 64i i i þ 34 þ 35ð 1Þ þ 36ð 1 Þ i¼1 i¼1 X4 X2 χ†η λ ϕþϕ− λ φ0φ0 λ ξ0 ξ0 λ φ0 2 λ φ0 2 λ ξ0 2 λ ξ0 2 þ 65i i i þ 66i i i þ 37 þ 38ð 1Þ þ 39ð 1 Þ þ 40ð Þ þ 41ð Þ i¼1 i¼1 † − 0 − 0 − 0 † þ 0 0 þ η ρðλ42ϕ4 φ1 þ λ43ϕ4 φ1 þ λ44ϕ3 ξ Þþρ χϕ3 ðλ45ξ þ λ46ξ Þ þ 2 − − þ − 0 0 0 0 0 0 þ λ47ðϕ1 Þ ϕ3 ϕ4 þ ϕ1 ϕ2 ðλ48φ1φ2 þ λ49φ1 φ2 þ λ50φ2ξ Þ þ − 0 0 0 0 0 0 0 0 þ ϕ ϕ ðλ51φ ξ þ λ52φ ξ þ λ53φ ξ þ λ54φ ξ Þ 3 4 1 1 1 1 0 2 0 0 0 0 0 0 0 0 þðφ2Þ ðλ55φ1ξ þ λ56φ1ξ þ λ57φ1 ξ þ λ58φ1 ξ ÞþH:c: ðA2Þ
The last part which breaks softly Z4 × Z2,isgivenby For the neutrino mass generation, beside the first term in (A5), the additional part is given as scalars μ2φ0φ0 μ2φ0φ0 μ2 φ0 2 μ2ϕþϕ− Lgsoft ¼ 4 1 2 þ 6 1 2 þ 1ð 2Þ þ 2 2 3 μ2ϕþϕ− μ2ϕþϕ− λ χ†χ 2 λ χ†χ η†η þ 5 2 4 þ 3 3 4 þ H:c: ðA3Þ LHiggsneutrino ¼ 13ð Þ þ 5ð Þð Þ † † † 2 − þ λ27 ρ ρ χ η η χ μ ϕ ϕ : : The total potential is composed of three above mentioned þ½ ð Þð þ Þþ 3 4 3 þ H c parts ðA6Þ
V ¼ V þ V þ Lscalars: ðA4Þ LNC LNV soft It is worth mentioning that for the generation of masses The scalar interactions needed for quark and charged lepton for quark and charged lepton, only terms in the conserving mass generation, read as follows part VLNC are enough, while for the generation of the light active neutrino masses, one needs the lepton number λ χρηφ0 λ η†ρϕ−ξ0 λ ϕþϕ−φ0ξ0 violating scalar interactions of VLNV as well as the softly LHiggsqcl ¼ 1 1 þ 3 3 þ 4 1 2 2 breaking part Lscalars [the last term in (A6)] of the scalar φ0 2φ0 χ†ρϕ− soft þ w1ð 2Þ 1 þ w2 3 þ H:c: ðA5Þ potential.
015004-19 LONG, HOP, HUE, THAO, and CÁRCAMO HERNÁNDEZ PHYS. REV. D 100, 015004 (2019)
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