A renormalizable left-right symmetric model with low scale seesaw mechanisms
AA renormalizable left-right symmetric model with low scale seesaw mechanisms
A. E. C´arcamo Hern´andez ∗ and Ivan Schmidt † Universidad T´ecnica Federico Santa Mar´ıa and Centro Cient´ıfico-Tecnol´ogico de Valpara´ıso,Casilla 110-V, Valpara´ıso, Chile (Dated: January 11, 2021)We propose a low scale renormalizable left-right symmetric theory that successfully explains theobserved SM fermion mass hierarchy, the tiny values for the light active neutrino masses, the leptonand baryon asymmetries of the Universe, as well as the muon and electron anomalous magneticmoments. In the proposed model the top and exotic quarks obtain masses at tree level, whereasthe masses of the bottom, charm and strange quarks, tau and muon leptons are generated froma tree level Universal Seesaw mechanism, thanks to their mixings with the charged exotic vectorlike fermions. The masses for the first generation SM charged fermions arise from a radiativeseesaw mechanism at one loop level, mediated by charged vector like fermions and electricallyneutral scalars. The light active neutrino masses are produced from a one-loop level inverse seesawmechanism. Our model is also consistent with the experimental constraints arising from the Higgsdiphoton decay rate. We also discuss the Z (cid:48) and heavy scalar production at a proton-proton collider. I. INTRODUCTION
Despite the great success of the Standard Model (SM) as a theory of fundamental interactions, it features drawbackssuch as, for example, the lack of explanation of the SM flavor structure; in particular, the observed pattern of SMfermion masses and mixings, the origin of Dark Matter (DM), the source of parity violation in electroweak (EW)interactions, the lepton and baryon asymmetries of the Universe and the anomalous magnetic moments of the muonand electron. In order to address these issues, it is necessary to propose a possible more general higher energy theory.In this sense, left-right symmetric electroweak extensions of the Weinberg-Salam theory have many appealing features,foremost of which is to address the origin of parity violation as a low energy effect, a remanent of its breaking ata certain high energy scale. We are therefore proposing, as a possible explanation of the problems listed before,a minimal renormalizable Left-right symmetric theory [1, 2] based on the gauge symmetry SU (3) C × SU (2) L × SU (2) R × U (1) B − L , supplemented by the Z (1)4 × Z (2)4 discrete group, where the Z (1)4 symmetry is completely broken,whereas the Z (2)4 symmetry is broken down to the preserved Z , thus allowing the implementation of a radiativeinverse seesaw mechanism to generate the tiny masses of the light active neutrinos. In fact, left-right models areparticularly appealing to consider as extensions of the SM, since they have the possibility of explaining its parityfeature at low energy, having a symmetry between left and right particles at higher energies. In the proposed model,the top and exotic quarks obtain masses at tree level whereas the masses of the bottom, charm and strange quarks,tau and muon leptons arise from a tree level Universal Seesaw mechanism. The masses for the first generation SMcharged fermions are generated from a one loop level radiative seesaw mechanism mediated by charged vector likefermions and electrically neutral scalars.Some recent left-right symmetric models have been considered in Refs. [3–6]. Unlike the model of Ref [3], where nonrenormalizable Yukawa interactions are employed for the implementation of a Froggart Nielsen mechanism to producethe current SM fermion mass and mixing pattern, our proposed model is a fully renormalizable theory, with minimalparticle content and symmetries, where tree level Universal and a one-loop level radiative inverse seesaw mechanisms ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - ph ] J a n are combined to explain the observed hierarchy of SM fermion masses and fermionic mixing parameters. Furthermore,unlike Ref. [3] our model successfully explains the electron and muon anomalous magnetic moments and includes adiscussion about leptogenesis and collider signatures of heavy scalar and Z (cid:48) gauge bosons, which is not presented in[3]. In our current model, the charged vector-like leptons responsible for the tree level Universal and one-loop levelradiative seesaw mechanism that produces the SM charged fermion mass hierarchy, allows to reproduce the measuredvalues of the muon and electron anomalous magnetic moments, thus linking the fermion mass generation mechanismand the g − SU (2) L scalar triplet (transforming as a SU (2) R doublet) and one SU (2) R scalar triplet (transforming as a SU (2) L doublet), thus amounting to 32 scalar degrees offreedom, our current left-right model has one scalar bidoublet, one SU (2) L scalar doublet, one SU (2) R scalar doublet,two electrically neutral gauge singlet real scalars and two electrically neutral gauge singlet complex scalars, whichcorresponds to 22 scalar degrees of freedom. Another advantage of our proposal with respect to the one presented in[4] is that in the former a mechanism that naturally explains the SM fermion mass hierarchy is presented, whereasthe latter does not include such mechanism.The paper is organized as follows. In section II we outline the proposed model. The implications of our model in theSM fermion hierarchy is discussed in section III. The consequences of our model in leptogenesis are described in sectionIV, while the model scalar potential is analyzed in section V. The implications of our model in the Higgs diphotondecay are discussed in section VI, and in section VII we analyze its application to the muon and electron anomalousmagnetic moments. The Z (cid:48) and heavy scalar production at a proton-proton collider are discussed in sections VIIIand IX, respectively. We conclude in section X. II. AN EXTENDED LEFT-RIGHT SYMMETRIC MODEL
Before providing a detailed explanation of our left-right symmetric model, we will explain the reasoning behindintroducing extra scalars, fermions and symmetries, needed for implementing an interplay of tree level universal andradiative seesaw mechanism to explain the SM charged fermion mass hierarchy and one loop level inverse seesawmechanism to generate the tiny neutrino masses. It is worth mentioning that in our proposed model, the mass of thetop quark will be generated from a renormalizable Yukawa operator, with an order one Yukawa coupling, i.e. Q L Φ Q iR , i = 1 , , Q L and Q iR are SU (2) L and SU (2) R quark doublets, respectively, whereas Φ is a scalar bidoblet, with theVEV pattern (cid:104) Φ (cid:105) = (cid:32) v v (cid:33) , (2)where we have set v = 0 to prevent a bottom quark mass arising from the above given Yukawa interaction. Now, togenerate tree level masses via a Universal Seesaw mechanism for the bottom, strange and charm quarks, as well asfor the tau and muon leptons, one loop level masses for the first generation SM charged fermions and the tiny massesfor the light active neutrinos via a one loop level inverse seesaw mechanism, we need to forbid the operators: Q nL Φ Q iR , Q nL (cid:101) Φ Q iR , n = 1 , , i = 1 , , ,L iL (cid:101) Φ L jR , L iL (cid:101) χ L N jR , ( m N ) ij N iR N CjR , i, j = 1 , , . (3)and allow the operators: Q L χ L B R , Q nL χ L B R , B nL χ † R Q iR , B L ρB R , B L σB R ,Q nL (cid:101) χ L T R , T L (cid:101) χ † R Q iR , T L σT R , n = 1 , , i = 1 , , ,L iL χ L E nR , E nL χ † R L jR , E nL ηE nR ,L iL Φ L jR , N CiR (cid:101) χ † R L jR , Ω nR Ω CnR η, N nR Ω CkR ϕ, n, k = 1 , . (4)This requires to add Z (1)4 and Z (2)4 discrete symmetries, which are spontaneously broken, where the former is completelybroken, and the latter is broken down to the preserved Z symmetry. Such remaining conserved Z symmetry allowsto implement an inverse seesaw mechanism at one loop level to produce the tiny neutrino masses. Here L iL and L iR are SU (2) L and SU (2) R lepton doublets, respectively, while N iR ( i = 1 , ,
3) and Ω nR ( n = 1 ,
2) are gauge singletneutral leptons. Let us note that the gauge singlet neutral leptons Ω nR ( n = 1 ,
2) are crucial for generating theterm ( m N ) ij N iR N CjR ( i, j = 1 , ,
3) at one loop level, thus allowing the implementation of the one-loop level inverseseesaw mechanism. Additionally, the above mentioned exotic neutral lepton content is the minimal one required togenerate the masses for two light active neutrinos, as required from the neutrino oscillation experimental data. Besidesthat, the SM charged fermion sector has to be extended to include the following heavy fermions: up type quark T ,down type quarks B n and charged leptons E n ( n = 1 ,
2) in singlet representations under SU (2) L × SU (2) R . As aconsequence of the above mentioned exotic charged fermion spectrum, the SM charged fermion mass matrices willfeature a proportionality between rows and columns, thus implying that the first generation SM charged fermions willbe massless at tree level. The one loop level corrections to these matrices will break such proportionality, thus yieldingone-loop level masses for the up and down quarks as well as for the electron. Consequently, the aforementioned exoticcharged fermion spectrum is the minimal necessary so that no massless charged SM-fermions would appear in themodel, provided that one loop level corrections are taken into account.On the other hand, in what regards the scalar sector, it is worth mentioning that χ L and χ R are SU (2) L and SU (2) R scalar doublets, respectively, whereas η , σ and ϕ are gauge singlet scalars. Furthermore, the χ L ( χ R ) scalar is crucialfor generating mass mixing terms between left (right) handed SM charged fermions and right (left) handed exoticcharged fermions. Furthermore, the SU (2) R scalar doublet χ R is crucial for triggering the spontaneous breakingof the SU (2) L × SU (2) R × U (1) B − L symmetry down to the SM electroweak gauge group. Besides that, the η , σ and ϕ are gauge singlet scalars, whose inclusion is necessary for generating the masses of the exotic fermions.Moreover, the inclusion of the scalar bidoublet Φ is crucial to generate a tree level top quark mass, as well as theDirac neutrino submatrix, as will be shown below. The aforementioned scalar content is the minimal required for asuccessful implementation of the tree level universal and one loop level radiative seesaw mechanisms to explain theSM charged fermion mass hierarchy, as well as of the one loop level inverse seesaw mechanism to produce the tinyneutrino masses. By suitable charge assignments to be specified below, we can implement the aforementioned seesawmechanisms, useful for explaining the SM fermion mass hierarchy.Our proposed model is based on the gauge symmetry SU (3) C × SU (2) L × SU (2) R × U (1) B − L , supplemented by the Z (1)4 × Z (2)4 discrete group, where the full symmetry G exhibites the following breaking scheme: G = SU (3) C × SU (2) L × SU (2) R × U (1) B − L × Z (1)4 × Z (2)4 ⇓ v σ , v η , v ρ SU (3) C × SU (2) L × SU (2) R × U (1) B − L ⇓ v R SU (3) C × SU (2) L × U (1) Y × Z ⇓ vSU (3) C ⊗ U (1) Q × Z (5)Both Z (1)4 and Z (2)4 discrete groups are spontaneously broken, and are crucial for avoiding a tree level inverse seesawmechanism. The Z (1)4 symmetry is completely broken, whereas the Z (2)4 symmetry is broken down to the preserved Z symmetry. It is assumed that such discrete symmetries are broken at the scale much larger than the scale of breakingof the left-right symmetry. We further assume that the left-right symmetry breaking scale is about v R ∼ O (10) TeV.In addition, the Z (2)4 symmetry, which is spontaneously broken to the preserved Z , is crucial in order to forbid theappearance of the term ( m N ) ij N iR N CjR at tree level, thus allowing the implementation of the one loop level inverseseesaw mechanism that generates the light active neutrino masses. Besides that, the spontaneously broken Z (1)4 symmetry is crucial to prevent tree level Yukawa mass terms involving the scalar bidoublet and SM charged fermionslighter than the top quark. As we will see in the following, in the SM fermion sector only the top quark will acquire itsmass from a renormalizable Yukawa interaction with the scalar bidoublet, whereas the SM charged fermions lighterthan the top quark will get their masses from tree level Universal seesaw and radiative seesaw mechanisms.The fermion assignments under the SU (3) C × SU (2) L × SU (2) R × U (1) B − L group are: Q iL = (cid:32) u iL d iL (cid:33) ∼ (cid:18) , , , (cid:19) , Q iR = (cid:32) u iR d iR (cid:33) ∼ (cid:18) , , , (cid:19) , i = 1 , , ,L iL = (cid:32) ν iL e iL (cid:33) ∼ ( , , , − , L iR = (cid:32) ν iR e iR (cid:33) ∼ ( , , , − , i = 1 , , ,T R ∼ (cid:18) , , , (cid:19) , T L ∼ (cid:18) , , , (cid:19) , n = 1 , ,B nR ∼ (cid:18) , , , − (cid:19) , B nL ∼ (cid:18) , , , − (cid:19) , E nR ∼ ( , , , − , E nL ∼ ( , , , − ,N iR ∼ ( , , , , Ω nR ∼ ( , , , , n = 1 , . (6)Let us note that we have extended the fermion sector of the original left-right symmetric model model by introducingone exotic up type quark T , two exotic down type quarks B n ( n = 1 , E n and five Majorananeutrinos, i.e., N iR ( i = 1 , ,
3) and Ω nR ( n = 1 , SU (2) L × SU (2) R group. The above mentioned exotic fermion content is the minimal one required to generatetree level masses via a Universal seesaw mechanism for the bottom, charm and strange quarks, as well as the tauand muon, and one loop level masses for the first generation SM charged fermions, i.e., the up, down quarks, and theelectron. Q nL Q L Q iR L iL L iR T L T R B nL B R B R E nL E nR N iR Ω nR SU (3) C SU (2) L SU (2) R U (1) B − L
13 13 13 − −
43 43 − − − − − Z (1)4 − i − i − i − i − iZ (2)4 − − i − i − − − − i − i i SU (3) C × SU (2) L × SU (2) R × U (1) B − L × Z (1)4 × Z (2)4 . Here i = 1 , , n = 1 , The scalar assignments under the SU (3) C × SU (2) L × SU (2) R × U (1) B − L group are:Φ = (cid:32) √ (cid:0) v + φ R + iφ I (cid:1) φ +2 φ − √ (cid:0) v + φ R + iφ I (cid:1) (cid:33) ∼ ( , , , ,χ L = (cid:32) χ + L √ (cid:0) v L + Re χ L + i Im χ L (cid:1) (cid:33) ∼ ( , , , , χ R = (cid:32) χ + R √ (cid:0) v R + Re χ R + i Im χ R (cid:1) (cid:33) ∼ ( , , , σ ∼ ( , , , , ϕ ∼ ( , , , , η ∼ ( , , , , ρ ∼ ( , , , . (7)To implement the tree level Universal and radiative seesaw mechanisms we have introduced the scalars χ L , χ R whichare responsible for generating tree level mixings between the exotic and SM fermions. We have further introducedthe gauge singlet scalars σ and ϕ which are crucial for the implementation of the radiative inverse seesaw mechanismnecessary to produce the light active neutrino masses. Furthermore, the gauge singlet scalar σ provides tree levelmasses for the exotic T and B quarks. Besides that, the gauge singlet scalars ρ and η are included in the scalarspectrum in order to provide tree level masses for the exotic down type quark B and exotic leptons E nR andΩ nR ( n = 1 , Q L Φ Q iR ( i = 1 , , χ L and χ R are: (cid:104) Φ (cid:105) = (cid:32) v v (cid:33) , (cid:104) χ L (cid:105) = (cid:32) v L (cid:33) , (cid:104) χ R (cid:105) = (cid:32) v R (cid:33) , (8)where for the sake of simplicity we will set v = 0.The fermion assignments under Z (1)4 × Z (2)4 are: Q nL ∼ ( − , − , Q L ∼ ( i, , Q jR ∼ (1 , , T L ∼ (1 , , T R ∼ (1 , − ,B nL ∼ (1 , , B R ∼ ( − i, − , B R ∼ (1 , − , L jL ∼ (1 , − i ) , L jR ∼ ( − i, − i ) , (9) E nL ∼ (1 , − i ) , E nR ∼ ( − , − i ) , N jR ∼ ( i, i ) , Ω nR ∼ ( − i, , j = 1 , , , n = 1 , . The scalar fields have the following Z (1)4 × Z (2)4 assignments:Φ ∼ ( i, , χ L ∼ ( − , , χ R ∼ (1 , ϕ ∼ (1 , i ) , σ ∼ (1 , − , η ∼ ( − , , ρ ∼ ( i, − . (10)The fermion and scalar assignments under the SU (3) C × SU (2) L × SU (2) R × U (1) B − L × Z (1)4 × Z (2)4 symmetry areshown in Tables I and II, respectively.Let us note that all scalar fields acquire nonvanishing vacuum expectation values, excepting the scalar singlet ϕ ,whose Z (2)4 charge corresponds to a nontrivial charge under the preserved remnant Z symmetry. Furthermore, due Φ χ L χ R ϕ σ η ρSU (3) C SU (2) L SU (2) R U (1) B − L Z (1)4 i − − iZ (2)4 i − − SU (3) C × SU (2) L × SU (2) R × U (1) B − L × Z (1)4 × Z (2)4 . to such remnant Z symmetry, the real and imaginary components of the scalar singlet ϕ will not have mixings withthe remaining CP even and CP odd neutral scalar fields of the model, and thus Re ϕ and Im ϕ can be considered asphysical fields.It is worth mentioning that the preserved Z symmetry allows for stable scalar and fermionic dark matter candidates.The scalar dark matter candidate is the lightest among the CP-even and CP-odd neutral components of the gaugesinglet scalar ϕ . The fermionic dark matter candidate is the lightest among the right handed Majorana neutrinos N iR ( i = 1 , , W W , ZZ , tt , bb and h SM h SM via a Higgs portal scalar interaction. These annihilation channels will contribute to the DM relic density, whichcan be accommodated for appropriate values of the scalar DM mass and of the coupling of the Higgs portal scalarinteraction. Some studies of the dark matter constraints for the scenario of scalar singlet dark matter candidate areprovided in [7–9]. Thus, for the DM direct detection prospects, the scalar DM candidate would scatter off a nucleartarget in a detector via Higgs boson exchange in the t -channel, giving rise to a constraint on the Higgs portal scalarinteraction coupling. Regarding the scenario of fermionic DM candidate, the Dark matter relic abundance can beobtained through freeze-in, as shown in [8]. The resulting constraints can therefore be fulfilled for an appropriateregion of parameter space, along similar lines of Refs. [8, 10, 11]. A detailed study of the implications of our modelin dark matter is beyond the scope of this work and will be done elsewhere.With the above particle content, the following relevant Yukawa terms arise: −L Y = (cid:88) i =1 α i Q L Φ Q iR + (cid:88) n =1 x ( T ) n Q nL (cid:101) χ L T R + (cid:88) i =1 z ( T ) i T L (cid:101) χ † R Q iR + x ( B )3 Q L χ L B R + (cid:88) n =1 x ( B ) n Q nL χ L B R + (cid:88) n =1 3 (cid:88) i =1 z ( B ) ni B nL χ † R Q iR + y T T L σT R + y B B L ρB R + y B B L σB R + (cid:88) n =1 y E n E nL ηE nR + (cid:88) i =1 2 (cid:88) n =1 x ( E ) in L iL χ L E nR + (cid:88) n =1 3 (cid:88) i =1 z ( E ) nj E nL χ † R L jR + (cid:88) i =1 3 (cid:88) j =1 y ( L ) ij L iL Φ L jR + (cid:88) i =1 3 (cid:88) j =1 x ( N ) ij N CiR (cid:101) χ † R L jR + (cid:88) n =1 ( y Ω ) n Ω nR Ω CnR η + (cid:88) n =1 2 (cid:88) k =1 x ( S ) nk N nR Ω CkR ϕ + H.c, (11)To close this section, in the following we discuss the implications of our model for flavor changing neutral currents(FCNC). The FCNC in the down type quark sector are expected to be very suppressed since at energies belowthe scale v R of breaking of the left-right symmetry, only the SU (2) L scalar doublet χ L will appear in the downtype quark Yukawa terms. In what regards the up type quark sector, there would be FCNC at tree level, since atlow energies (below v R ), the bidoblet scalar Φ and the SU (2) L scalar doublet χ L participate in the up type quarkYukawa interactions. However, such FCNC can be suppressed by sufficiently small values of the α and α Yukawacouplings. In particular, setting α = α = 0 do not generate problems for the quark masses and mixing parameters.Furthermore, concerning the charged lepton sector, the corresponding FCNC can be suppressed by making the matrix y ( L ) ij diagonal. III. FERMION MASS MATRICES.
From the Yukawa interactions, we find that the mass matrices for SM charged fermions are given by: M U = ∆ U × A U × m t B U m T , A U = (cid:32) x ( T )1 x ( T )2 (cid:33) v L √ ,B U = (cid:16) z ( T )1 , z ( T )2 (cid:17) v R √ , m t = x ( Q )3 v √ , (12) M D = (cid:32) ∆ D A D B D M B (cid:33) , A D = x ( B )12 x ( B )22 x ( B )3 x ( B )32 v L √ ,B D = (cid:32) z ( B )11 , z ( B )12 , z ( B )13 z ( B )21 , z ( B )22 , z ( B )23 (cid:33) v R √ , M B = (cid:32) m B m B (cid:33) , (13) M E = (cid:32) ∆ E A E B E C E (cid:33) , A E = x ( E )11 x ( E )12 x ( E )21 x ( E )22 x ( E )31 x ( E )32 v L √ ,B D = (cid:32) z ( E )11 , z ( E )12 , z ( E )13 z ( E )21 , z ( E )22 , z ( E )23 (cid:33) v R √ , C E = (cid:32) m E m E (cid:33) , (14)As seen from Eqs. (12), (13) and (14), the exotic heavy vector-like fermions mix with the SM fermions lighter than topquark. The masses of these vector-like fermions are much larger than the scale of breaking of the left-right symmetry v R ∼ O (10) TeV, since the gauge singlet scalars η , σ and ρ are assumed to acquire vacuum expectation values muchlarger than this scale. Therefore, the charm, bottom and strange quarks, as well as the tau and muon leptons, acquiretheir masses from the tree-level Universal seesaw mechanism, whereas the first generation SM charged fermions, i.e.,the up, down quarks and the electron get one-loop level masses from a radiative seesaw mechanism. Thus, the SMcharged fermion mass matrices take the form: (cid:102) M U = (cid:32) ∆ U + A U M − (cid:101) T B U × × m t (cid:33) , (15) (cid:102) M D = ∆ D + A D M − B B D , (16) (cid:102) M E = ∆ E + A E M − E B E , (17)where ∆ U , ∆ D and ∆ E are the one loop level contributions to the SM charged fermion mass matrices arising fromthe one-loop Feynman diagrams of Figure 1. ¯ u L u iR × v ¯ u nL u iR T R ¯ T L × v σ × v L × v R ¯ d iL d jR B nR ¯ B nL × v ρ , v σ × v L × v R ¯ l iL l jR E nR ¯ E nL × v η × v L × v R ¯ u nL u iR T R ¯ T L × v σ Re χ L , Im χ L Re χ R , Im χ R × v L × v R ¯ d iL d jR B nR ¯ B nL × v ρ , v σ Re χ L , Im χ L Re χ R , Im χ R × v L × v R ¯ l iL l jR E nR ¯ E nL × v η Re χ L , Im χ L Re χ R , Im χ R × v L × v R Figure 1: Feynman diagrams contributing to the entries of the SM charged fermion mass matrices. Here, n = 1 , i, j = 1 , , × v σ N n N k Ω r Ω r Re ϕ , Im ϕ Re ϕ , Im ϕ × v η Figure 2: One-loop Feynman diagram contributing to the Majorana neutrino mass submatrix µ . Here, n, k = 1 , , r = 1 , Concerning the neutrino sector, we find that the neutrino Yukawa interactions give rise to the following neutrino massterms: − L ( ν ) mass = 12 (cid:16) ν CL ν R N R (cid:17) M ν ν L ν CR N CR + (cid:88) n =1 ( m Ω ) n Ω nR Ω CnR + H.c, (18)where the neutrino mass matrix reads: M ν = × m νD m TνD M M µ , (19)and the submatrices are given by:( m νD ) ij = y ( L ) ij v √ , M ij = x ( N ) ij v R √ , i, j = 1 , , , n, k, r = 1 , ,µ nk = (cid:88) r =1 x ( S ) nr x ( S ) kr m r π (cid:34) m ϕ R m ϕ R − m r ln (cid:32) m ϕ R m r (cid:33) − m ϕ I m ϕ I − m r ln (cid:32) m ϕ I m r (cid:33)(cid:35) , . (20)The µ block is generated at one loop level due to the exchange of Ω rR ( r = 1 ,
2) and ϕ in the internal lines, as shownin Figure 2. To close the corresponding one loop diagram, the following trilinear scalar interaction is needed: V µ = A ( ϕ ∗ ) σ, (21)The light active masses arise from a combination of inverse and linear seesaw mechanisms and the physical neutrinomass matrices are: (cid:102) M ν = m νD (cid:0) M T (cid:1) − µM − m TνD , (22) M (1) ν = − (cid:0) M + M T (cid:1) + 12 µ, (23) M (2) ν = 12 (cid:0) M + M T (cid:1) + 12 µ. (24)where M (1) ν corresponds to the mass matrix for light active neutrinos ( ν a ), whereas M (2) ν and M (3) ν are the massmatrices for sterile neutrinos ( N − a , N + a ) which are superpositions of mostly ν aR and N aR as N ± a ∼ √ ( ν aR ∓ N aR ).0In the limit µ →
0, which corresponds to unbroken lepton number, the light active neutrinos become massless. Thesmallness of the µ - parameter is responsible for a small mass splitting between the three pairs of sterile neutrinos,thus implying that the sterile neutrinos form pseudo-Dirac pairs. The full neutrino mass matrix given by Eq. (19)can be diagonalized by the following rotation matrix [12]: R = R ν R R (1) M R R (2) M − ( R † + R † ) √ R ν (1 − S ) √ R (1) M (1+ S ) √ R (2) M − ( R † − R † ) √ R ν ( − − S ) √ R (1) M (1 − S ) √ R (2) M , (25)where S = − M − µ, R (cid:39) R (cid:39) √ m ∗ νD M − . (26)Notice that the physical neutrino spectrum is composed of three light active neutrinos and six exotic neutrinos. Theexotic neutrinos are pseudo-Dirac, with masses ∼ ± (cid:0) M + M T (cid:1) and a small splitting µ . Furthermore, R ν , R (1) M and R (2) M are the rotation matrices which diagonalize (cid:102) M ν , M (1) ν and M (2) ν , respectively.On the other hand, using Eq. (25) we find that the neutrino fields ν L = ( ν L , ν L , ν L ) T , ν CR = (cid:0) ν C R , ν C R , ν C R (cid:1) and N CR = (cid:0) N C R , N C R , N C R (cid:1) are related with the physical neutrino fields by the following relations: ν L ν CR N CR = R Ω L (cid:39) R ν R R (1) M R R (2) M − ( R † + R † ) √ R ν (1 − S ) √ R (1) M (1+ S ) √ R (2) M − ( R † − R † ) √ R ν ( − − S ) √ R (1) M (1 − S ) √ R (2) M Ψ (1) L Ψ (2) L Ψ (3) L , Ψ L = Ψ (1) L Ψ (2) L Ψ (3) L , (27)where Ψ (1) jL , Ψ (2) jL = N + j and Ψ (3) jL = N − j ( j = 1 , ,
3) are the three active neutrinos and six exotic neutrinos, respectively.
IV. LEPTOGENESIS
In this section we will analyze the implications of our model in leptogenesis. To simplify our analysis we assume that y ( L ) and x ( N ) are diagonal matrices and we consider the case where (cid:12)(cid:12)(cid:12) y ( L )11 (cid:12)(cid:12)(cid:12) (cid:28) (cid:12)(cid:12)(cid:12) y ( L )22 (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) y ( L )33 (cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12) x ( N )11 (cid:12)(cid:12)(cid:12) (cid:28) (cid:12)(cid:12)(cid:12) x ( N )22 (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) x ( N )33 (cid:12)(cid:12)(cid:12) .It is worth mentioning that the scenario of diagonal y ( L ) matrix is crucial for suppressing tree level FCNC in thecharged lepton sector. In the above mentioned scenario only the first generation of N ± a can give the contribution tothe Baryon asymmetry of the Universe (BAU). We further assume that the exotic leptonic fields E nR and Ω nR areheavier than the lightest pseudo-Dirac fermions N ± = N ± , while in addition, for the sake of simplicity, we work inthe basis where the SM charged lepton mass matrix is diagonal. Then, the lepton asymmetry parameter, which isinduced by decay process of N ± , has the following form [13, 14]: ε ± = (cid:88) i =1 (cid:2) Γ (cid:0) N ± → ν i A (cid:1) − Γ (cid:0) N ± → ν i A (cid:1)(cid:3) (cid:88) i =1 [Γ ( N ± → ν i A ) + Γ ( N ± → ν i A )] (cid:39) Im (cid:26)(cid:18)(cid:104)(cid:0) y N + (cid:1) † (cid:0) y N − (cid:1)(cid:105) (cid:19) (cid:27) πA ± rr + Γ ± m N ± , (28)with: r = m N + − m N − m N + m N − , A ± = (cid:104)(cid:0) y N ± (cid:1) † y N ± (cid:105) , Γ ± = A ± m N ± π ,y N ± = y ( L ) √ ∓ S ) = y ( L ) √ (cid:18) ± M − µ (cid:19) (29)It is worth mentioning that CP violation in the lepton sector necessary to generate the lepton asymmetry parameter,and can arise from complex entries in y ( L ) , M or µ , as indicated by Eqs. (28) and (29).1If one neglects the interference terms involving the two different sterile neutrinos N ± , the washout parameter K N + + K N − is huge as mentioned in [15]. However, the small mass splitting between the pseudo-Dirac neutrinos leads to adestructive interference in the scattering process [16]. The washout parameter including the interference term is givenas follows: K eff (cid:39) (cid:0) K N + δ + K N − δ − (cid:1) , (30)where: δ ± = m N + − m N − Γ N ± , K N ± = Γ ± H ( T ) , H ( T ) = (cid:114) π g ∗ T M P (31)where g ∗ = 118 is the number of effective relativistic degrees of freedom, M P l = 1 . × GeV is the Planck constantand T = m N ± . In the weak and strong washout regimes, the baryon asymmetry is related to the lepton asymmetry[14] as follows Y ∆ B = n B − n B s = − (cid:15) + + (cid:15) − g ∗ , for K eff (cid:28) , (32) Y ∆ B = n B − n B s = − . (cid:15) + + (cid:15) − ) g ∗ K eff (ln K eff ) . , for K eff (cid:29) , (33)The correlation of the baryon asymmetry parameter Y B with the leptonic CP violating phase δ ( l ) CP for the weak andstrong washout regimes is shown in the left and right panels of Figure 3, respectively. The leptonic CP violating phase δ ( l ) CP is defined as follows: δ ( l ) CP = arcsin (cid:32) J ( l ) CP sin 2 θ sin 2 θ sin 2 θ cos θ (cid:33) , (34)where J ( l ) CP is the Jarlskog invariant of the lepton sector, defined as follows J ( l ) CP = Im [ U U ∗ U U ∗ ] , (35)where U is the PMNS leptonic mixing matrix given by U = R † (cid:96) R ν , being R (cid:96) and R ν the rotation matrices thatdiagonalize the mass matrices for the SM charged leptons and light active neutrinos, respectively. As shown fromthe above given relations, phases in the Majorana neutrino mass terms leading to complex entries in the µ block willgenerate complex entries in the PMNS leptonic mixing matrix, thus yielding a nonvanishing leptonic CP phase δ ( l ) CP .As shown in Figure 3, our model successfully accommodates the experimental value of the baryon asymmetry param-eter Y B : Y ∆ B = (0 . ± . × − (36) V. THE SIMPLIFIED SCALAR POTENTIAL
In order to simplify our analysis, we will consider a bechmark scenario where the singlet real scalar fields σ , η and ρ will not feature mixings with the neutral components of the Φ, χ L and χ R scalars. The justification of this benchmarkscenario arises from the fact that such gauge singlet scalars σ , η and ρ are assumed to acquire vacuum expectationvalues much larger than the scale of breaking of the left-right symmetry, thus allowing to neglect the mixings ofthese fields with the Φ, χ L and χ R scalars and to treat their scalar potentials independently. Let us note that themixing angles between those fields are suppressed by the ratios of their VEVs, as follows from the method of recursive2 Figure 3: Correlation of the baryon asymmetry parameter Y B with the leptonic CP violating phase δ ( l ) CP for the weak(left-plot) and strong (right-plot) washout regimes. expansion of Ref. [17]. Furthermore, the preserved remnant Z symmetry will forbid mixings of the Re ϕ and Im ϕ scalars with the remaining scalar fields of our model. The scalar potential for the Φ, χ L and χ R scalars takes theform: V = µ ( χ † L χ L ) + µ ( χ † R χ R ) + µ T r (Φ † Φ) + µ T r ( (cid:101) ΦΦ † + (cid:101) Φ † Φ) + λ ( χ † L χ L ) + λ ( χ † R χ R ) + λ ( χ † L χ L )( χ † R χ R )+ λ (cid:2) T r (Φ † Φ) (cid:3) + λ T r (cid:2) (Φ † Φ) (cid:3) + λ (cid:104) T r ( (cid:101) Φ (cid:101) Φ ∗ ) (cid:105) + λ T r (cid:104) ( (cid:101) Φ (cid:101) Φ ∗ ) (cid:105) + λ ( χ † L χ L ) T r (Φ † Φ) + λ ( χ † R χ R ) T r (Φ † Φ)+ λ ( χ † L χ L ) T r ( (cid:101) Φ (cid:101) Φ ∗ ) + λ ( χ † R χ R ) T r ( (cid:101) Φ (cid:101) Φ ∗ ) (37)where the term µ T r ( (cid:101) ΦΦ † + (cid:101) Φ † Φ) softly breaks the Z (1)4 symmetry.The minimization conditions of the scalar potential yields the following relations: µ = 12 (cid:0) − λ v L − λ v R − ( λ + λ ) v (cid:1) , (38) µ = 12 (cid:0) − λ v L − λ v R − ( λ + λ ) v (cid:1) , (39) µ = 12 (cid:0) − ( λ + λ ) v L − ( λ + λ ) v R − λ + λ + λ + λ ) v (cid:1) . (40)The squared mass matrix for the electrically charged scalars in the basis (cid:0) χ + L , χ + R , φ +1 I , φ +2 I (cid:1) − (cid:0) χ − L , χ − R , φ − I , φ − I (cid:1) takesthe form: M = (cid:0) − λ v L − λ v R − λ + λ ) v (cid:1) − µ − µ (cid:0) − λ v L − λ v R − λ + λ + λ ) v (cid:1) (41)where the massless scalar eigenstates χ ± L and χ ± R correspond to the Goldstone bosons associated with the longitudinalcomponents of the W ± and W (cid:48)± gauge bosons. Besides that, there are physical electrically charged scalars H ± and H ± , whose squared masses are given by: m H ± = 12 (cid:18) − λ v L − λ v R − (cid:113) µ + λ v − λ v − λ v − λ v (cid:19) , (42) m H ± = 12 (cid:18) − λ v L − λ v R + (cid:113) µ + λ v − λ v − λ v − λ v (cid:19) . (43)3The squared mass matrix for the CP-odd neutral scalar sector in the basis (cid:0) Im χ L , Im χ R , φ I , φ I (cid:1) M CP − odd = − µ − µ ( λ + λ ) (cid:0) − v (cid:1) (44)The massless scalar eigenstates Im χ L and Im χ R are associated with the Goldstone bosons associated with thelongitudinal components of the Z and Z (cid:48) gauge bosons. Furthermore, CP-odd neutral scalar sector contains twomassive CP odd scalars whose squared masses are given by: m A = 12 (cid:18) − (cid:113) µ + ( λ v + λ v ) − λ v − λ v (cid:19) , (45) m A = 12 (cid:18)(cid:113) µ + ( λ v + λ v ) − λ v − λ v (cid:19) . (46)The squared mass matrix for the CP-even neutral scalar sector in the basis (cid:0) φ R , Re χ L , φ R , Re χ R (cid:1) M CP − even = λ + λ + λ + λ ) v ( λ + λ ) v v L µ ( λ + λ ) v v R ( λ + λ ) v v L λ v L λ v L v R µ λ + λ ) (cid:0) − v (cid:1) λ + λ ) v v R λ v L v R λ v R (47)Correlations between the masses of the non SM scalars are shown in Figure 4 and indicates that there are a largenumber of solutions for the scalar masses consistent with experimental bounds. VI. HIGGS DIPHOTON DECAY RATE
The decay rate for the h → γγ process takes the form:Γ( h → γγ ) = α em m h π v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) f a hff N C Q f F / ( ρ f ) + a hW W F ( ρ W ) + (cid:88) k =1 , C hH ± k H ∓ k v m H ± k F ( ρ H ± k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (48)where ρ i are the mass ratios ρ i = m h M i with M i = m f , M W ; α em is the fine structure constant; N C is the color factor( N C = 1 for leptons and N C = 3 for quarks) and Q f is the electric charge of the fermion in the loop. From thefermion-loop contributions we only consider the dominant top quark term. Furthermore, C hH ± k H ∓ k is the trilinearcoupling between the SM-like Higgs and a pair of charged Higges, whereas a htt and a hW W are the deviation factorsfrom the SM Higgs-top quark coupling and the SM Higgs-W gauge boson coupling, respectively (in the SM thesefactors are unity). Such deviation factors are close to unity in our model, which is a consequence of the numericalanalysis of its scalar, Yukawa and gauge sectors.Furthermore, F / ( z ) and F ( z ) are the dimensionless loop factors for spin-1 / F / ( z ) = 2( z + ( z − f ( z )) z − , (49) F ( z ) = − z + 3 z + 3(2 z − f ( z )) z − , (50) F ( z ) = − ( z − f ( z )) z − , (51)4 Figure 4: Correlations between the non SM scalar masses. with f ( z ) = arcsin √ z ≤ − (cid:18) ln (cid:16) √ − z − −√ − z − − iπ (cid:17) (cid:19) for z > R γγ , which is defined as: R γγ = σ ( pp → h )Γ( h → γγ ) σ ( pp → h ) SM Γ( h → γγ ) SM (cid:39) a htt Γ( h → γγ )Γ( h → γγ ) SM . (53)That Higgs diphoton signal strength, normalizes the γγ signal predicted by our model in relation to the one given bythe SM. Here we have used the fact that in our model, single Higgs production is also dominated by gluon fusion asin the Standard Model.The ratio R γγ has been measured by CMS and ATLAS collaborations with the best fit signals [18, 19]: R CMSγγ = 1 . +0 . − . and R AT LASγγ = 0 . ± . . (54)The correlation of the Higgs diphoton signal strength with the charged scalar mass m H ± is shown in Figure 5, whichindicates that our model successfully accommodates the current Higgs diphoton decay rate constraints. Furthermore,as indicated by Figure 5, our model favours a Higgs diphoton decay rate lower than the SM expectation but insidethe 3 σ experimentally allowed range.5 Figure 5: Correlation of the Higgs diphoton signal strength with the a hWW deviation factor from the SM Higgs-W gaugeboson coupling. VII. MUON AND ELECTRON ANOMALOUS MAGNETIC MOMENTS
In this section we will analyze the implications of our model in the muon and electron anomalous magnetic moments.The leading contributions of the muon and electron anomalous magnetic moments arise from vertex diagrams involvingthe exchange of neutral scalars and vector like leptons running in the internal lines of the loop. Then, in our modelthe contributions to the muon and electron anomalous magnetic moments take the form:∆ a µ = (cid:88) k =1 β k γ k m µ π (cid:34)(cid:0) R TCP − even (cid:1) (cid:0) R TCP − even (cid:1) I ( µ ) S ( m E k , m h ) + (cid:88) i =1 (cid:0) R TCP − even (cid:1) i (cid:0) R TCP − even (cid:1) i I ( µ ) S (cid:16) m E k , m H i (cid:17)(cid:35) , ∆ a e = (cid:88) k =1 β k γ k m e π (cid:34)(cid:0) R TCP − even (cid:1) (cid:0) R TCP − even (cid:1) I ( e ) S ( m E k , m h ) + (cid:88) i =1 (cid:0) R TCP − even (cid:1) i (cid:0) R TCP − even (cid:1) i I ( e ) S (cid:16) m E k , m H i (cid:17)(cid:35) , (55)where the loop I S ( P ) ( m E , m ) has the form [20–22]: I ( e,µ ) S ( P ) ( m E , m S ) = (cid:90) x (cid:16) − x ± m E m e,µ (cid:17) m µ x + (cid:0) m E − m e,µ (cid:1) x + m S,P (1 − x ) dx (56)Considering that the muon and electron anomalous magnetic moments are constrained to be in the ranges [23–25]:(∆ a µ ) exp = (26 . ± × − (∆ a e ) exp = ( − . ± . × − . (57)We plot in Figure 6 the correlations of the muon and electron anomalous magnetic moments with the mass m E ofone of the charged exotic leptons (top plots) as well as the correlation between the electron and muon anomalousmagnetic moments (bottom plot), for fixed values of the CP even neutral scalar masses. We have checked that the6 Figure 6: Correlations of the muon and electron anomalous magnetic moments with the mass m E of one of the chargedexotic leptons (top plots). Correlation between the electron and muon anomalous magnetic moments (bottom plot). correlations of ∆ a µ,e with m E are very similar with the ones involving m E instead of m E . Here we have fixed m H = 400 GeV, m H = 500 GeV and m H = 12 TeV. We find that our model can successfully accommodates theexperimental values of the muon and electron anomalous magnetic moments. VIII. HEAVY SCALAR PRODUCTION AT THE LHC
In this section we discuss the singly heavy scalar H production at a proton-proton collider. Such productionmechanism at the LHC is dominated by the gluon fusion mechanism, which is a one-loop process mediated by the topquark. Thus, the total H production cross section in proton-proton collisions with center of mass energy √ S takesthe form: σ pp → gg → H ( S ) = α S a H t ¯ t m H πv S (cid:34) I (cid:32) m H m t (cid:33)(cid:35) (cid:90) − ln (cid:115) m H S ln (cid:115) m H S f p/g (cid:115) m H S e y , µ f p/g (cid:115) m H S e − y , µ dy, (58)where f p/g (cid:0) x , µ (cid:1) and f p/g (cid:0) x , µ (cid:1) are the distributions of gluons in the proton which carry momentum fractions x and x of the proton, respectively. Furthermore µ = m H is the factorization scale, whereas I ( z ) has the form: I ( z ) = (cid:90) dx (cid:90) − x dy − xy − zxy . (59)Figure 7 shows the H total production cross section at the LHC via gluon fusion mechanism for √ S = 14 TeV(left-plot) and √ S = 28 TeV (right-plot), as a function of the scalar mass m H , which is taken to range from 400 GeVup to 600 GeV. Furthermore, the coupling a H t ¯ t of the heavy scalar H with the top-antitop quark pair has beenset to be equal to 0 .
4, which is consistent with our numerical analysis of the scalar potential. In the aforementioned7
400 450 500 550 6000.40.60.81.01.2 m H [ GeV ] σ ( pp → H ) [ pb ]
400 450 500 550 6001.52.02.53.03.54.04.55.0 m H [ GeV ] σ ( pp → H ) [ pb ] Figure 7: Total cross section for the H production via gluon fusion mechanism at the LHC for √ s = 14 TeV (left-panel) and √ S = 28 (right-panel) TeV as a function of the heavy scalar mass m H . region of masses for the heavy H scalar, we find that the total production cross section ranges from 1 . . √ S = 28 TeV, the total cross section for the H isenhanced reaching values between 5 pb and 1 . H scalar, after being produced, will have dominant decay modes into top-antitopquark pairs, SM Higgs boson pairs as well as into a pair of SM gauge bosons, thus implying that the observation of anexcess of events in the multileptons or multijet final states over the SM background can be a smoking gun signatureof this model, whose observation will be crucial to assess its viability. IX. Z (cid:48) GAUGE BOSON PRODUCTION AT THE LHC
In this section we discuss the single heavy Z (cid:48) gauge boson via Drell-Yan mechanism at proton-proton collider. Weconsider the dominant contributions due to the parton distribution functions of the light up, down and strange quarks,so that the total cross section for the production of a Z (cid:48) via quark antiquark annihilation in proton-proton collisionswith center of mass energy √ S takes the form: σ ( DrellY an ) pp → Z (cid:48) ( S ) = g R π S (cid:90) − ln (cid:114) m Z (cid:48) S ln (cid:114) m Z (cid:48) S (cid:88) q = u,d,s f p/q (cid:32)(cid:114) m Z (cid:48) S e y , µ (cid:33) f p/q (cid:32)(cid:114) m Z (cid:48) S e − y , µ (cid:33) dy (60)where f p/u (cid:0) x , µ (cid:1) ( f p/u (cid:0) x , µ (cid:1) ), f p/d (cid:0) x , µ (cid:1) ( f p/d (cid:0) x , µ (cid:1) ) and f p/s (cid:0) x , µ (cid:1) ( f p/s (cid:0) x , µ (cid:1) ) are the distributionsof the light up, down and strange quarks (antiquarks), respectively, in the proton which carry momentum fractions x ( x ) of the proton. The factorization scale is taken to be µ = m Z (cid:48) . Fig. 8 displays the Z (cid:48) total production crosssection at the LHC via the Drell-Yan mechanism for √ S = 14 TeV (left panel) and √ S = 28 TeV (right panel) as afunction of the Z (cid:48) mass M Z (cid:48) in the range from 4 TeV up to 5 TeV. For this region of Z (cid:48) masses we find that the totalproduction cross section ranges from 13 fb up to 1 fb. The heavy neutral Z (cid:48) gauge boson, after being produced, willsubsequently decay into the pair of the SM fermion-antifermion pairs, thus implying that the observation of an excessof events in the dileptons or dijet final states over the SM background can be a signal of support of this model at theLHC. On the other hand, at the proposed energy upgrade of the LHC at 28 TeV center of mass energy, the total cross8 M Z' [ GeV ] σ ( pp → Z ' ) [ f b ] M Z' [ GeV ] σ ( pp → Z ' ) [ f b ] Figure 8: Total cross section for the Z (cid:48) production via Drell-Yan mechanism at a proton-proton collider for √ S = 14 TeV(left-panel) and √ S = 28 (right-panel) TeV as a function of the Z (cid:48) mass. section for the Drell-Yan production of a heavy Z (cid:48) neutral gauge boson gets significantly enhanced reaching valuesranging from 280 fb up to 120 fb, as indicated in the right panel of Fig. 8. X. CONCLUSIONS
We have built a renormalizable left-right symmetric theory with additional symmetry Z (1)4 × Z (2)4 consistent withthe observed SM fermion mass hierarchy, the tiny values for the light active neutrino masses, the lepton and baryonasymmetries of the Universe as well as the muon and electron anomalous magnetic moments. As the main appealingfeature of the proposed model, the top and exotic fermions get their masses at tree level whereas the masses of thebottom, charm and strange quarks, tau and muon leptons are generated from a tree level Universal Seesaw mechanismthanks to their mixings with charged exotic vector like fermions. The first generation SM charged fermions massesare produced from a radiative seesaw mechanism at one loop level mediated by charged vector like fermions andelectrically neutral scalars. The tiny masses of the light active neutrinos arise from an inverse seesaw mechanism atone-loop level. Furthermore, we have also shown that the proposed model successfully accommodates the currentHiggs diphoton decay rate constraints, yielding a Higgs diphoton decay rate lower than the SM expectation butinside the 3 σ experimentally allowed range. We also studied the heavy H scalar and Z (cid:48) gauge boson productionin a proton-proton collider at √ S = 14 TeV and √ S = 28 TeV, via the gluon fusion and Drell-Yan mechanisms,respectively. We found that the singly H scalar production cross section reach values of 1 . √ S = 14TeV and √ S = 28 TeV, respectively, for a 400 GeV heavy scalar mass. On the other hand, we found that the totalcross section for the Z (cid:48) gauge boson production takes the values of 13 fb and 280 fb at √ S = 14 TeV and √ S = 28TeV, respectively, for a 4 TeV Z (cid:48) gauge boson mass.9 Acknowledgments
A.E.C.H and I.S. are supported by ANID-Chile FONDECYT 1170803, ANID-Chile FONDECYT 1180232 and ANID-Chile FONDECYT 3150472, and by the project ANID PIA/APOYO AFB180002 (Chile) [1] J. C. Pati and A. Salam, “Lepton Number as the Fourth Color,”
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