]>PLB34264S03702693(18)30896710.1016/j.physletb.2018.11.049The AuthorPhenomenologyFig. 1λσR versus mσ for σ relic abundance.Fig. 1Table 1Basic fermion content of model with unconventional SU(2)R.Table 1FermionSU(3)CSU(2)LSU(2)RU(1)XZ5
(ν,e)L121−1/21
νR11101

(N,e)R112−1/2ω−1
NL1110ω3

(u,d)L3211/6ω
dR311−1/3ω

(u,h)R3121/6ω2
hL311−1/3ω−2
Table 2Scalar content defining the model of baryonlepton duplicity.Table 2ScalarSU(3)CSU(2)LSU(2)RU(1)XZ5
(ϕL+,ϕL0)1211/21
(ϕR+,ϕR0)1121/2ω−1

η1220ω

ζ311−1/3ω−2

σ1110ω
Baryonlepton duplicity as the progenitor of longlived dark matterErnestMaab⁎ma@phyun8.ucr.eduaPhysics and Astronomy Department, University of California, Riverside, CA 92521, USAPhysics and Astronomy DepartmentUniversity of CaliforniaRiversideCA92521USAbJockey Club Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong, ChinaJockey Club Institute for Advanced StudyHong Kong University of Science and TechnologyHong Kong, China⁎Correspondence to: Physics and Astronomy Department, University of California, Riverside, CA 92521, USA.Physics and Astronomy DepartmentUniversity of CaliforniaRiversideCA92521USAEditor: A. RingwaldAbstractIn an SU(2)R extension of the standard model, it is shown how the neutral fermion N in the doublet (N,e)R may be assigned baryon number B=1, in contrast to its SU(2)L counterpart ν in the doublet (ν,e)L which has lepton number L=1. This baryonlepton duplicity allows a scalar σ which couples to NLNL to be longlived dark matter.1IntroductionIn the conventional SU(2)R extension of the standard model (SM) of quarks and leptons, the neutral fermion N in the doublet (N,e)R is identified with the Dirac mass partner of ν in the SU(2)L doublet (ν,e)L. Hence N has lepton number L=1. However, it is also possible that N is not the mass partner of ν, and that it has L=0 [1] or L=2 [2]. As such N may be considered a darkmatter candidate using [3] (−1)L+2j as the stabilizing dark symmetry. In the following, it will be shown how N may be assigned baryon number B=1 instead [4], in which case a scalar σ coupling to NLNL may become longlived dark matter.2ModelThe basic framework for considering (N,e)R differently from (ν,e)R is originally inspired by E6 models with the decomposition E6→SU(3)C×SU(3)L×SU(3)R→SU(3)C×SU(2)L×SU(2)R×U(1)X, where the SU(2)R is not [5] the one contained in SO(10)→SU(5) in the conventional left–right model. Consider the fermion particle content of the basic model given in Table 1.The electric charge is Q=I3L+I3R+X. The discrete Z5 symmetry (ω5=1) serves to forbid the terms N¯LνR and h¯LdR and others to be discussed. The scalar particle content of the proposed model of baryonlepton duplicity is given in Table 2.In the above, η is a bidoublet, with SU(2)L acting vertically and SU(2)R acting horizontally, i.e.(1)η=(η10η2+η1−η20),(2)η˜=σ2η⁎σ2=(η¯20−η1+−η2−η¯10). The Z5 symmetry distinguishes η from η˜. The resulting Yukawa interactions are(3)LY=fν(ν¯Lϕ¯L0−e¯LϕL−)νR+fd(u¯LϕL++d¯LϕL0)dR+fN(N¯Rϕ¯R0−e¯RϕR−)NL+fh(u¯RϕR++h¯RϕR0)hL+fe[(ν¯Lη10+e¯Lη1−)NR+(ν¯Lη2++e¯Lη20)eR]+fu[(u¯Lη¯20−d¯Lη2−)uR+(−u¯Lη1++d¯Lη¯10)hR]+f1σ⁎NLNL+f2N¯LdRζ⁎+f3ν¯RhLζ⁎+f4ϵijk(uiLdjL−diLujL)ζk+H.c., where each f is a 3×3 matrix for the three families of quarks and leptons and the last term is the product of three color triplets. Note that ν and d masses come from 〈ϕL0〉, e and u masses come from 〈η20〉, N and h masses come from 〈ϕR0〉. This structure guarantees the absence of treelevel flavorchanging neutral currents.If the scalar color triplet ζ and singlet σ are absent, the model of Ref. [2] is recovered with L=2 for N and L=−1 for h. As it is, a very different outcome is obtained with B=1 for N and B=2 for σ as well as B=−2/3 for h, as shown below. The first thing to realize is that even though the input symmetry is Z5, the Lagrangian of Eq. (3) actually has a larger symmetry due to the chosen particle content under the gauge symmetry. It is an U(1) symmetry S under which(4)(u,d)L,dR∼1/3,hL∼−2/3,(u,h)R∼−1/6,NL∼1,(N,e)R∼1/2,(5)ΦR∼1/2,η∼−1/2,σ∼2,ζ∼−2/3. This S symmetry is broken by 〈ϕR0〉 as well as 〈η20〉, but not the combination S+I3R. Indeed this residual symmetry is just baryon number, i.e. 1/3 for the known quarks and zero for the known leptons. There is another residual symmetry, i.e. lepton parity under which the known leptons are odd. Note that νR is allowed a Majorana mass, hence the canonical seesaw mechanism for neutrino mass is applicable.As for the new particles beyond the SM, their baryon number and lepton parity assignments are(6)ζ∼(−2/3,+),N∼(1,+),σ∼(2,+),h∼(−2/3,−),(7)WR±∼(±1,−),Z′∼(0,+),(η10,η1−)∼(−1,−). Hence ζ is a scalar diquark, h is a fermion diquark with odd lepton parity, N is a fundamental B=1 fermion, σ is a fundamental B=2 scalar, WR+ is a fundamental B=1 vector boson with odd lepton parity, and (η1+,−η¯10) is a fundamental B=1 scalar SU(2)L doublet with odd lepton parity. Underlying this exotic scenario is the duplicity between N and ν in their SU(2)R/SU(2)L interactions. Two symmetries are conserved: baryon number and lepton parity. The lightest lepton, i.e. the lightest neutrino, is stable. The lightest baryon, i.e. the proton, is stable. However, just as the heavier neutrinos are very longlived, the heavier B=2 σ may also be very longlived and become dark matter.3Gauge boson masses and interactionsLet(8)〈ϕL0〉=v1,〈η20〉=v2,〈ϕR0〉=vR, then the SU(3)C×SU(2)L×SU(2)R×U(1)X gauge symmetry is broken to SU(3)C×U(1)Q, with residual baryon number and lepton parity as discussed in the previous section. Consider now the masses of the gauge bosons. The charged ones, WL± and WR±, do not mix because of B and (−1)L, as in the original alternative left–right models. Their masses are given by(9)MWL2=12gL2(v12+v22),MWR2=12gR2(vR2+v22). Since Q=I3L+I3R+X, the photon is given by(10)A=egLW3L+egRW3R+egXX, where e−2=gL−2+gR−2+gX−2. Let(11)Z=(gL2+gY2)−1/2(gLW3L−gY2gRW3R−gY2gXX),(12)Z′=(gR2+gX2)−1/2(gRW3R−gXX), where gY−2=gR−2+gX−2, then the 2×2 masssquared matrix spanning (Z,Z′) has the entries:(13)MZZ2=12(gL2+gY2)(v12+v22),(14)MZ′Z′2=12(gR2+gX2)vR2+gX4v12+gR4v222(gR2+gX2),(15)MZZ′2=gL2+gY22gR2+gX2(gX2v12−gR2v22). Their neutralcurrent interactions are given by(16)LNC=eAμjQμ+gZZμ(j3Lμ−sin2θWjQμ)+(gR2+gX2)−1/2Zμ′(gR2j3Rμ−gX2jμX), where gZ2=gL2+gY2 and sin2θW=gY2/gZ2. Assuming also that gR=gL, then gX2/gZ2=sin2θWcos2θW/cos2θW. In that case, setting v12/v22=cos2θW/sin2θW would result in zero Z−Z′ mixing which is constrained by precision data to be less than a few times 10−4.The present bound on MZ′ from the Large Hadron Collider (LHC) is about 4 TeV. However, if the lightest N is considered as dark matter, then its gauge interaction through Z′ with quarks would constrain MZ′ to be above 10 TeV or higher from directsearch experiments, depending on mN. Here it will be assumed that σ is dark matter and since it does not couple to Z′, this constraint is not applicable.4Scalar sectorConsider the most general scalar potential consisting of ΦL,R and η, i.e.(17)V=−μL2ΦL†ΦL−μR2ΦR†ΦR−μη2Tr(η†η)+[μ3ΦL†ηΦR+H.c.]+12λL(ΦL†ΦL)2+12λR(ΦR†ΦR)2+12λη[Tr(η†η)]2+12λη′Tr(η†ηη†η)+λLR(ΦL†ΦL)(ΦR†ΦR)+λLηΦL†ηη†ΦL+λLη′ΦL†η˜η˜†ΦL+λRηΦR†η†ηΦR+λRη′ΦR†η˜†η˜ΦR. Note that(18)2det(η)2=[Tr(η†η)]2−Tr(η†ηη†η),(19)(ΦL†ΦL)Tr(η†η)=ΦL†ηη†ΦL+ΦL†η˜η˜†ΦL,(20)(ΦR†ΦR)Tr(η†η)=ΦR†η†ηΦR+ΦR†η˜†η˜ΦR. The minimum of V satisfies the conditions(21)μL2=λLv12+λLηv22+λLRvR2+μ3v2vR/v1,(22)μη2=(λη+λη′)v22+λLηv12+λRηvR2+μ3v1vR/v2,(23)μR2=λRvR2+λLRv12+λRηv22+μ3v1v2/vR. The 3×3 masssquared matrix spanning 2Im(ϕL0,η20,ϕR0) is then given by(24)MI2=μ3(−v2vR/v1vRv2vR−v1vR/v2−v1v2−v1−v1v2/vR), and that spanning 2Re(ϕL0,η20,ϕR0) is(25)MR2=μ3(−v2vR/v1vRv2vR−v1vR/v2v1v2v1−v1v2/vR)+2(λLv12λLηv1v2λLRv1vRλLηv1v2(λη+λη′)v22λRηv2vRλLRv1vRλRηv2vRλRvR2). Hence there are two zero eigenvalues in MI2 with one nonzero eigenvalue −μ3[v1v2/vR+vR(v12+v22)/v1v2] corresponding to the eigenstate A=(−v1−1,v2−1,vR−1)/v1−2+v2−2+vR−2. In MR2, the linear combination H=(v1,v2,0)/v12+v22, is the standardmodel Higgs boson, with(26)mH2=2[λLv14+(λη+λη′)v24+2λLηv12v22]/(v12+v22). The other two scalar bosons, i.e. H′=(v2,−v1,0)v12+v22 and HR=(0,0,1) are assumed not to mix with H and each other by finetuning the λ parameters to avoid further experimental constraints.There are four charged scalars in the Higgs potential of Eq. (17). Two have B=1, i.e. ϕR+,η1+. One linear combination becomes the longitudinal component of WR+. The orthogonal combination, i.e. (vRη1+−v2ϕR+)/vR2+v22 has a mass given by(27)m2=(λRη′−λRη)(vR2+v22)−μ3v1(vR2+v22)vRv2. The other two charged scalars have B=0, i.e. ϕL+,η2+. One linear combination becomes the longitudinal component of WL+. The orthogonal combination, i.e. (v1η2+−v2ϕL+)/v12+v22 has a mass given by(28)m2=(λLη′−λLη)(v12+v22)−μ3vR(v12+v22)v1v2. The two physical charged scalars do not mix, for the same reason that WR and WL do not mix, because they have different B values. Note that in the limit v1,2<<vR, the B=0 charged scalar has the same mass as the B=0 scalar H′ and the B=0 pseudoscalar A, as expected. As for the B=1 neutral scalar η10, it has a mass given by(29)m2(η10)=(λRη′−λRη)vR2+(λLη′−λLη)v12−λη′v22−μ3(v1/v2)vR. Note again that in the limit v1,2<<vR, the B=1 charged and neutral scalars have the same mass, as expected.5Diquark connection to dark matterThe scalar diquark ζ is crucial in assigning B=1 to N and σ. The decay of NL to dR and a virtual ζ⁎ which converts to uLdL means that N is longlived if mζ is very large. The current LHC bound on mζ is about 2.5 TeV. On the other hand, if mσ<mN, then the former's decay is even more suppressed. It will be shown how σ may indeed be suitable as longlived dark matter.Of the new particles with B≠0, ζ is assumed to be the heaviest and N to be the lightest except σ. Now N decays to udd with a decay rate given by [6](30)Γ(N→udd)=f22f42mN58(4π)3mζ4∫01dλ2λ2(1−λ2)2=f22f42mN596(4π)3mζ4. As an example, a lifetime of(31)τN=(5×1024s)(0.01f2)2(0.01f4)2(300GeVmN)5(mζ1012GeV)4 is obtained. Note that the age of the Universe is 4.35×1017 s, but the bound on decaying dark matter [7] is much greater, say about 1025 s, from the constraint of the cosmic microwave background (CMB). Hence N may be longlived enough for it to be dark matter, with some adjustment of parameter values. However, because its interactions through the new gauge boson Z′ are constrained by directsearch data as pointed out already, its resulting annihilation cross section is too small and would result in a thermal relic abundance exceeding what is observed. Hence it will be assumed from now on that N decays quickly, using for example mζ=105 GeV so that τN=5×10−4 s which is certainly short enough not to disturb Big Bang Nucleosynthesis (BBN).Excepting σ, the other new particles with B≠0 all decay quickly to N. The vector gauge boson WR− decays to e−N¯. The fermion diquark h decays to uWR− if mh>MWR, or to ue−N¯ if mh<MWR. The B=−1 scalar doublet (η10,η1−) decays to the B=0 scalar doublet (η2+,η20) through WR−, again converting to e−N¯. In all these decays, N would appear as missing energy because its lifetime is long enough to escape detection in the experimental apparatus.To estimate the decay rate of σ→NN, let p1,2 be the sum of the fourmomenta of the three quark jets from each N. Then(32)Γσ∼mσ16π[f1f22f4296(4π)3mζ4]2∫dp12dp22(mσ−p12−p22)2p14p24(p12−mN2)2(p22−mN2)2, where p1,22>0 and p12+p22<mσ2. Letting p12=p22=mσ2 in the denominator, the integral is bounded from above by(33)1(mN2−mσ2)4∫0mσ2p14dp12∫0mσ2−p12p24(mσ2−p12−p22)2dp22=mσ165040(mN2−mσ2)4. Hence(34)Γσ<mσ1735π(mN2−mσ2)4[f1f22f429(32π)3mζ4]2. For mζ=105 GeV, f1,2,4=0.01, mN=300 GeV, and mσ=250 GeV, τσ>6×1028 s is obtained. This shows that σ is longlived enough to be dark matter.6Relic abundance of σThe quartic interaction coupling λσH with the SM Higgs boson must be small (<10−3) to avoid the constraint of directsearch experiments. This means that the σσ⁎→HH cross section is not large enough to obtain the correct thermal relic abundance for σ as dark matter. However, if the HR boson is lighter than σ, the latter's annihilation to the former (which does not couple to SM fermions) is a possible mechanism. Using(35)σann×vrel=λσR2r1−r64πmHR2[1+3r4−r−(λσRλR)r2−r]2, where r=mHR2/mσ2 and assuming as an example λR=2×10−4 with vR=10 TeV, so that mHR=200 GeV, the allowed range of λσR is plotted versus mσ=200GeV/r in Fig. 1 for σann×vrel=4.4×10−26cm2/s.7Concluding remarksIn an SU(2)R extension of the SM, where an input Z5 discrete symmetry is imposed with the particle content of Table 1 and Table 2, it has been shown that two conserved symmetries emerge. One is lepton parity (−1)L so that the known leptons are odd and other SM particles are even. Neutrino masses are obtained through the usual canonical seesaw mechanism. The other is baryon number B with the usual assignment of 1/3 for the SM quarks. The conservation of B and (−1)L separately implies that the proton is stable.What is new and unconventional in this model is the nature of the neutral fermion N in the SU(2)R doublet (N,e)R. It is not the Dirac mass partner of the neutrino ν in the SU(2)L doublet (ν,e)L. Instead of having L=1, it actually has B=1, as explained in the text because of the Z5 symmetry and the chosen particle content. This baryonlepton duplicity allows new particles to have nonzero B as well as odd (−1)L. Whereas N itself decays into three quark jets and may have a long lifetime, a scalar σ with B=2 and mσ<mN is proposed instead as dark matter with a lifetime many orders of magnitude exceeding the age of the Universe. It has a correct thermal relic abundance from its interaction with the Higgs boson HR associated with SU(2)R symmetry breaking. Its interaction with the SU(2)L Higgs boson H is however adjustable, so that present directsearch bounds are obeyed, but may reveal itself in the future if a positive signal is measured.To test this model, the SU(2)R gauge sector has to be probed. If WR or Z′ can be produced, then N is predicted as a decay product. It will appear as an invisible massive particle. Another prediction is the existence of the fermion diquark h with odd lepton parity. It is produced readily by gluon interactions at the LHC and decays to ue−N¯ which looks like a fourthfamily quark but again N appears as an invisible massive particle and not the expected light neutrino. As the LHC gathers more data, these processes may be searched for.AcknowledgementThis work was supported in part by the U.S. Department of Energy Grant No. DESC0008541.References[1]S.KhalilH.S.LeeE.MaPhys. Rev. D792009041701[2]S.KhalilH.S.LeeE.MaPhys. Rev. D812010051702[3]E.MaPhys. Rev. Lett.1152015011801[4]E.MaPhys. Rev. 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