]>NUPHB14590S0550-3213(19)30064-110.1016/j.nuclphysb.2019.03.002The AuthorsHigh Energy Physics – PhenomenologyFig. 1Two-loop diagram to induce neutrino mass matrix. Here the blobs indicate the scalar mixing between η0 and χ.Fig. 1Fig. 2LFV diagrams.Fig. 2Fig. 3The diagram for the lepton flavor-changing/conserving Z boson decay.Fig. 3Fig. 4Scatter plots of BR(τ → eγ) (red) and BR(τ → μγ) (blue) in terms of BR(μ → eγ). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)Fig. 4Fig. 5Scatter plots of BR(Z → eτ) (red) and BR(Z → μτ) (blue) in terms of BR(Z → eμ).Fig. 5Table 1Particle contents and charge assignments of leptons and new particles under SU(2)L × U(1)Y × Z3. Here ω ≡ e2πi/3.Table 1Lepton fieldsScalar fields

LLeRNL/RΦηχχ+

SU(2)L2112211

U(1)Y−12−10121201

Z311ω1ωωω

Table 2Summary for the μ − e conversion in various nuclei: Z, Zeff, F(q), Γcapt, and the bounds on the capture rate R.Table 2Nucleus NZAZeff|F(−mμ2)||Γcapt (106sec−1)Experimental bound (Future bound)Y≡RBR(μ→eγ)

A1327l11.50.640.7054(RAl≲10−16) [50]0.25

T2248i17.60.542.59RTi≲4.3 × 10−12 [51] (≲10−18 [35])0.44

A79197u33.50.1613.07RAu≲7 × 10−13 [52]0.36

P82208b340.1513.45RPb≲4.6 × 10−11 [53]0.34

A two loop radiative neutrino modelSeungwonBaekasbaek@korea.ac.krHiroshiOkadab⁎hiroshi.okada@apctp.orgYutaOrikasacYuta.Orikasa@utef.cvut.czaDepartment of Physics, Korea University, Seoul 02841, Republic of KoreaDepartment of PhysicsKorea UniversitySeoul02841Republic of KoreabAsia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Republic of KoreaAsia Pacific Center for Theoretical PhysicsPohangGyeongbuk790-784Republic of KoreacInstitute of Experimental and Applied Physics, Czech Technical University, Prague 12800, Czech RepublicInstitute of Experimental and Applied PhysicsCzech Technical UniversityPrague12800Czech Republic⁎Corresponding author.Editor: Hong-Jian HeAbstractWe explore the possibility to explain a bosonic dark matter candidate with a gauge singlet inside the loop to generate the neutrino mass matrix at two-loop level. The mass matrix is suppressed by a small mixing that comes from the bound on direct detection experiments of the dark matter, and equivalent of the three-loop neutrino model due to the small mixing between neutral inert bosons. Here, our setup is the Zee-Babu type scenario with Z3 discrete symmetry, in which we consider the neutrino oscillation data, lepton flavor violations, muon g−2, μ−e conversion rate, lepton flavor-changing and conserving Z boson decay and bosonic dark matter candidate.1IntroductionRadiatively induced neutrino masses is one of the promising scenarios which make strong correlations between neutrinos and any fields that are introduced inside loops. If a dark matter (DM) candidate is introduced in the model, its testability is enhanced due to the fact that parameter space is strongly constrained by neutrino data. Especially two-loop induced models that we will focus on in this paper have been widely studied in various aspects [1–34].In this paper, we study the muon anomalous magnetic moment, various lepton flavor violations (LFVs), and DM phenomenology in the framework of Zee-Babu type of neutrino model, emphasizing μ−e conversion rate in Titanium nuclei that will be tested in the near future experiment such as PRISM/PRIME [35]. Despite a two-loop model, we will also show that the scale of neutrino mass in our model is equivalent to a three-loop model. It is due to the small mixing between neutral bosons dictated by the direct detection bound of the DM candidate.This paper is organized as follows. In Sec. 2, we introduce our model, including neutrino sector, LFVs, muon anomalous magnetic moment and lepton flavor-changing and conserving Z boson decay. In Sec. 3, we present our numerical analysis and identify regions consistent with the current experiments. We conclude and discuss in Sec. 4.2Model setupWe introduce three families of iso-spin singlet vector-like neutral fermions Ni (i=1,2,3), iso-spin doublet scalar η, an isospin singlet neutral scalar χ and charged scalars χ± in addition to the SM fields. We impose a discrete Z3 symmetry on all the new particles (N,η,χ,χ±) in order to assure the stability of DM (χ in our case). The particle contents and their charge assignments under SU(2)L×U(1)Y×Z3 are shown in Table 1.11Although there are many other possible symmetries to realize our model, Z3 is a minimal symmetry. Thus we expect that only the SM-like Higgs Φ has a vacuum expectation value (VEV), which is denoted by 〈Φ0〉=v/2 [21]. Then the relevant Lagrangian and scalar potential respecting the symmetries are given by(2.1)−LY=(yℓ)ijL¯LiΦeRj+(yη)ijL¯Li(iσ2)η⁎NRj+yNRijN¯RicNRjχ+yNLijN¯LicNLjχ+(yχ)ijN¯LieRjχ++MNiN¯LiNRi+h.c.,(2.2)V=mΦ2Φ†Φ+mη2|η|2+mχ2|χ|2+mχ±2|χ+|2+μ(ηT(iσ2)Φχ−+h.c.)+μηχ(Φ†ηχ⁎+h.c.)+μχ(χ3+h.c.)+λΦ(Φ†Φ)2+λ0(Φ†ηχ2+h.c.)+λη(η†η)2+λχ(χ⁎χ)2+λχ±(χ+χ−)2+λΦη(Φ†Φ)(η†η)+λΦη′|Φ†η|2+λΦχ(Φ†Φ)χ⁎χ+ληχ(η†η)χ⁎χ+λΦχ±(Φ†Φ)χ+χ−+ληχ±(η†η)χ+χ−+λχχ±(χ⁎χ)χ+χ−, where σ2 is the second Pauli matrix, i,j=1−3, and the first term of LY can generate the SM charged-lepton masses mℓ≡yℓv/2 (ℓ=1−3) after the electroweak symmetry breaking. Both Ni and ei can be considered as the mass eigenstates without loss of generality. For simplicity we assume all the parameters in (2.1) are real and positive.The scalar fields can be parameterized as follows:(2.3)Φ=[0v+ϕ2],η=[η+η0],χ,χ±, where η0 and χ are complex inert neutral bosons, η± and χ± are the singly charged bosons, v≃246 GeV is VEV of the Higgs doublet, and ϕ is the SM Higgs boson with mass mϕ≈125.5 GeV. To ensure the stability of DM, the following condition should be at least satisfied:(2.4)|μ+μηχ+μχ|<Λ(mΦ2+mη2+mχ2+mχ±2)12,Λ≡∑i=all quartic couplingsλi.Notice here that we have mixing between inert bosons through μηχ and μ, the resulting mass eigenvalues and their rotation matrices are obtained by(2.5)OHTMHOH=[mH100mH2],[χη0]=OH[H1H2]=[cosαsinα−sinαcosα][H1H2],(2.6)VCTMH±VC=[mH1±00mH2±],[χ±η±]=VC[H1±H2±]=[cosβsinβ−sinβcosβ][H1±H2±], where (H1(2),H1(2)±) and (α, β) can be written in terms of the parameters in the scalar potential, and we use the short hand notation sα(β) and cα(β) for sinα(β) and cosα(β) below.22See ref. [21] for scalar mass spectra in more details.DM candidate: The lightest neutral scalar H1 is our DM candidate. Here we consider constraints on H1. As for the direct detection experiment, the dominant elastic scattering cross section comes from Z-boson portal through mixing and found as(2.7)σSI≃(mH1mpmH1+mp)22GF2sα4π(1−4sw2)2, where mp is a proton mass, sw2≈0.23 is the Weinberg angle and GF is the Fermi constant. Note here that sα4 comes from the kinetic term of η, Dμη†Dμη, where the covariant derivative Dμ includes the SM gauge boson Z. Since Z-boson couples only to the isospin doublet scalar, in the effective H1−H1−Z coupling only η component of H1 couples to Z-boson. This leads to sα2 suppression in the effective coupling and sα4 suppression in the cross section for the DM scattering off the nuclei. In the experiment of LUX [36], the typical upper bound on the cross section is σSI≲10−45 cm2 at mH1=O(100) GeV. Then the required condition on α is given by33If we consider the contribution of the Higgs portal [37,39] to the direct detection, the constraint is given by 2λΦχcα2−2μηχvsαcα+(λΦη+λΦη′)sα2≲10−2. Thus one can satisfy the bound of direct detection by tuning the Higgs trilinear and quartic couplings.(2.8)|sα|≲5×10−2. It implies that the dominant component of the DM candidate is the gauge singlet boson χ. Hereafter we will neglect any terms proportional to sα4. To explain the relic density, we rely on the resonant effect via s-channel of the SM-Higgs, and we consider only the annihilation processes, since our DM is gauge singlet.44If our DM is dominated by SU(2)L gauge doublet, then coannihilation is important and the allowed mass is at around 535 GeV [38]. In this case, to satisfy the relic density Ωh2≈0.12, the DM mass should be around the Higgs resonance region mH1≈mϕ/2≈63 GeV [39].2.1Neutrino massesThe active neutrino mass matrix mν is generated at two-loop level as shown in Fig. 1, and its formula is given by(2.9)(mν)ij=122μχsα2cα2(yη)ia(yN)ab(yη)bjT(4π)4FII≡(yη)ia(RN)ab(yη)Tbj,(2.10)FII=∫[dx]z−1∫[da][cα2[ln(Δ111Δ112)−ln(Δ211Δ212)]+sα2[ln(Δ222Δ221)−ln(Δ122Δ121)]],(2.11)Δℓmn=−axMNb2+ymHn2+zmHm2z2−z+bMNa2+cmHℓ2, where RN is a parameter with a mass dimension and depends on the parameters (yN,μχ,sα,FII), [dx]≡dxdydzδ(x+y+z−1), [da]≡dadbdcδ(a+b+c−1), and we assume yN≡yNR≈yNL. As we mentioned in Section 1, although they are generated at two-loop level, the neutrino masses scale like three-loop model due to sα suppression. Then the active neutrino mass matrix (mν)ij can be diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix VMNS [40] as(2.12)(mν)ij=(VMNS⁎DνVMNS†)ij,Dν≡diag(mν1,mν2,mν3),(2.13)VMNS=[c13c12c13s12s13e−iδ−c23s12−s23s13c12eiδc23c12−s23s13s12eiδs23c13s23s12−c23s13c12eiδ−s23c12−c23s13s12eiδc23c13]. We assume the neutrino masses are normal ordered, neglect the Majorana phases, and fix the Dirac phase δ=−π/2 in the numerical analysis for simplicity. Then we apply the generalized Casas-Ibarra parametrization55In our case, the central matrix is not diagonal but the symmetric matrix which is proportional to yN. to our analysis which use the observed neutrino oscillation data with global fit [41]. We impose ∑i=1−3mνi<0.12 eV at 95% C.L. as reported by Planck collaboration [42]. The Yukawa coupling yη can be rewritten in terms of the following parameters:(2.14)yη≈VMNS⁎DνO(θi)(RNCh)−1, where O(θi) is an arbitrary orthogonal 3×3 matrix with three complex values θi (i=1-3) that satisfy OOT=OTO=Diag(1,1,1)66In case the family number of NR is two, O(θ) is an arbitrary 3×2 matrix with a complex value θ that satisfies OOT=Diag(0,1,1) and OTO=Diag(1,1). The numerical result for the three families does not change much from this case. and RNCh is Cholesky decomposed matrix. This matrix is a lower triangular matrix and satisfies the following relation (see appendix A),(2.15)RN=RNCh(RNCh)T.2.2Lepton flavor violations and muon (g−2)Lepton flavor violations (LFVs) at one-loop level arise from the terms with yη and yχ as shown in the left panel of Fig. 2.77Recently sophisticated analysis has been done by refs. [44,45]. Between two classes of LFV modes ℓi→ℓjγ and ℓi→ℓjℓkℓℓ, the former tends to give more stringent bounds on the related couplings and masses [43]. Thus we consider only this mode below. The model prediction for the radiative decay channel in our case is given by(2.16)BR(ℓi→ℓjγ)=48π3CijαemGF2(|(aRηχ)ij+aRijη+ϵijaRijχ|2+|(aLηχ)ij+ϵijaLijη+aLijχ|2),(2.17)(aRηχ)ij=(aLηχ)ij†=−sβcβ(4π)2∑k=1,2,3MNkmℓi(yη)jk(yχ)ki(F1(MNk,mH1±)−F1(MNk,mH2±)),(2.18)aRijη=aLijη=∑k=1,2,3(yη)jk(yη†)ki(4π)2[sβ2Flfv(MNk,mH1±)+cβ2Flfv(MNk,mH2±)],(2.19)aRijχ=aLijχ=∑k=1,2,3(yχ†)jk(yχ)ki(4π)2[cβ2Flfv(MNk,mH1±)+sβ2Flfv(MNk,mH2±)],(2.20)F1(m1,m2)=m12+m222(m12−m22)2−m12m22(m12−m22)3logm12m22,(2.21)Flfv(ma,mb)=2ma6+3ma4mb2−6ma2mb4+6mb6+12ma4mb2ln(mb/ma)12(ma2−mb2)4, where ϵij≡(mℓj/mℓi)(≪1), η± is the singly charged component of η, GF≈1.17×10−5 [GeV]−2 is the Fermi constant, αem≈1/137 is the fine structure constant, C21≈1, C31≈0.1784, and C32≈0.1736. Note that in the limit mℓi→0 Eq. (2.17) is divergent, but this mass comes from the total decay rate and mℓi→0 limit is unphysical. Experimental upper bounds are BR(μ→eγ)≲4.2×10−13, BR(τ→eγ)≲3.3×10−8, and BR(τ→μγ)≲4.4×10−8 [46,47].The μ−e conversion rate can be expressed by using aR/L defined in Eqs. (2.17), (2.18), and (2.19). The Feynman diagram is shown in the right panel of Fig. 2, and its capture rate R is obtained approximately to be88We neglect the contribution from the Z-penguin diagram due to suppression factor (mℓ/MN)2.(2.22)R≈Cμe|Z|2Γcap(|(aRηχ)μe+aRμeη+ϵμeaRμeχ|2+|(aLηχ)μe+ϵμeaLμeη+aLμeχ|2),Cμe≈4αem5Zeff4|F(q)|2mμ5Z, where Z, Zeff, F(q), and R are given in Table 2.Here let us define Y≡RBR(μ→eγ), since their flavor structures are same. Then it is given by(2.23)Y≈1.22×10−24(ZZeff4|F(q)|2Γcap). Depending on the nuclei, Y≈O(0.1), as listed in Table 2. It suggests that the constraint from the μ−e conversion is always satisfied once we satisfy the constraint of μ→eγ. We will discuss the sensitivity of future experiments, RTi and RAl, in the numerical analysis.New contribution to the muon anomalous magnetic moment (muon g−2), whose diagram is displayed in the left panel of Fig. 2, is given by(2.24)Δaμ≈−mμ2[(aRηχ)μμ+aRμμη+ϵμμaRμμχ+(aLηχ)μμ+ϵμμaLμμη+aLμμχ], where only the terms (aR(L)ηχ)μμ are positive contributions to the muon g−2. Thus we expect aR(L)μμη,aR(L)μμχ≪(aR/Lηχ)μμ to explain the discrepancy between the experimental results and the theoretical predictions which is of order of O(10−9) [48].Lepton Flavor-Changing/Conserving Z Boson Decay: Here, we consider the flavor changing/conserving Z boson decay Z→ℓi−ℓj+ as shown in Fig. 3, whose branching fractions have been measured or restricted by experiments as [46],(2.25)BR(Z→e−e+)=(3.363±0.004)%,(2.26)BR(Z→μ−μ+)=(3.366±0.007)%,(2.27)BR(Z→τ−τ+)=(3.370±0.008)%(2.28)BR(Z→e∓μ±)≲7.5×10−7,(2.29)BR(Z→e∓τ±)≲9.8×10−6(2.30)BR(Z→τ∓μ±)≲1.2×10−5. They will be improved by future experiments Giga-Z, ILC, and CEPC. The model prediction for the branching fraction is(2.31)BR(Z→ℓi−ℓj+)=Γ(Z→ℓi−ℓj+)Γtot=mZ24πΓtot(|ΓLij|2+|ΓRij|2),(2.32)ΓLij≈g2cw(−12+sw2)[δij+∑a=1−3yηiayηaj†(4π)2G(MNa,mH2±)],(2.33)ΓRij≈g2sw2cw[δij+∑a=1−3yχia†yχaj(4π)2G(MNa,mH1±)], where we have neglected terms proportional to sα2 and/or (mℓ/mZ)2, Γtot≈(2.4952±0.0023) GeV, and defined [49]G(ma,mb)≈ma4−4ma2mb2+3mb4−4ma4ln[ma]+8ma2mb2ln[ma]−4mb4ln[mb]4(ma2−mb2)2. They are constrained by Eqs. (2.25) −(2.30).3Numerical analysisFor the numerical analysis, we generate input parameters randomly in the following ranges:(3.1)(yNij,|yχ|)∈[10−8,0.1],θ1,2,3∈[10−3i,2π+100i],sα∈[10−5,10−3],sβ∈[−1,1],(μ,μχ,μηχ)∈[103]GeV,mH1,2±∈[80,103]GeV,(mH2,MN1,MN2,MN3)∈[200,103]GeV, where i,j=1,2,3, yN is a symmetric matrix, θ1,2,3 are arbitrary complex values in the Casas-Ibarra parametrization. We fixed mH1=mϕ/2. The lower bound on H1,2±, 80 GeV, comes from the LEP experiment [46,54]. In addition, the LHC gives a mass bound for the charged boson. Especially, the SU(2)L originated charged boson would have a feature similar to the slepton in the supersymmetric model, since it decays into a charged lepton and missing energy. The lower bound of the mass from the CMS collaboration is 450 GeV [55]. Hence we might apply this bound for our case, although the detail analysis is beyond our scope of this paper. Notice here yη should satisfy the perturbative limit; yη≲4π.In Fig. 4, we show scatter plots BR(τ→eγ) (red) and BR(τ→μγ) (blue) as a function of BR(μ→eγ). It suggests that BR(τ→eγ) and BR(τ→μγ) are much less than the upper bounds of experiments, while the maximum value of BR(μ→eγ) reaches the experimental upper bound. In Fig. 5, we show scatter plots of BR(Z→eτ) (red) and BR(Z→μτ) (blue) as a function of BR(Z→eμ). It suggests that BR(Z→eτ) and BR(Z→μτ) are much less than the upper bounds of experiments, while the maximum value of BR(Z→eμ) is close to the experimental upper bound. Thus BR(Z→eμ) could be tested in the future experiments. As for muon g−2, the maximum value is at most 5×10−15, which is much smaller than the current discrepancy. This is because the positive contribution comes from the mixing term between yη and yχ only.4Summaries and discussionsWe have explored the possibility to explain bosonic dark matter candidate with a gauge singlet inside the loop to generate the neutrino mass matrix at two-loop level. Here, our setup is the Zee-Babu type scenario with Z3 discrete symmetry, in which we have considered the neutrino oscillation data, DM, and lepton flavor violations.First of all, we have found the upper bound on sα to be of the order 10−2 from the direct detection experiment. Thus the only solution to satisfy the observed relic density is to use the SM Higgs resonance with the DM mass around the half of the Higgs mass, mH1≈mϕ/2.Second, the neutrino mass matrix is reduced by not only the two-loop suppression but also sα2≈10−3 suppression that comes from the direct detection bound of the DM. As a result, the scale of the neutrino masses is equivalent to that of the three-loop neutrino model.We have found that the positive muon g−2 thanks to χ±. But its typical value O(10−14) in our global analysis is too small to explain the ∼3σ discrepancy of the muon g−2 between the experiment and the SM.We briefly mention the possibility to detect our new particles at the LHC or the ILC. In these kinds of radiative seesaw models, they tend to have large Yukawa couplings in the lepton sector. Therefore, the effect of LFV processes can be large. On the other hand, the bounds from the LHC experiments are typically weaker than the LFV constraints, since new scalar bosons do not couple to quarks. Hence, in this paper we consider only LFV effects.AcknowledgementsThis work was supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT), Grant No. NRF-2018R1A2A3075605 (S.B.). H. O. is sincerely grateful for all the KIAS members in Korea. This research is supported by the Ministry of Science, ICT and Future Planning, Gyeongsangbuk-do and Pohang City (H.O.). The work was supported from European Regional Development Fund-Project Engineering Applications of Microworld Physics (No. CZ.02.1.01/0.0/0.0/16-019/0000766) (Y.O.).Appendix ACholesky decompositionA symmetric matrix M can be factorized by Cholesky decomposition. The decomposition is as follows:(A.1)M=[m11m12m13m12m22m23m13m23m33]=LLT, where L is a lower triangular matrix. 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