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KIAS-P17002 A Three-loop Neutrino Model with Leptoquark Triplet Scalars Kingman Cheung,1,2,3, Takaaki Nomura,4, and Hiroshi Okada1, ∗ † ‡ 1Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300 2Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan 3Division of Quantum Phases and Devices, School of Physics, 7 Konkuk University, Seoul 143-701, Republic of Korea 1 0 2 4School of Physics, KIAS, Seoul 130-722, Korea r a (Dated: March 14, 2017) M Abstract 3 1 We propose a three-loop neutrino mass model with a few leptoquark scalars in SU(2) -triplet L ] h form, through which we can explain the anomaly of B K( )µ+µ , a sizable muon g 2 and ∗ − p → − - a bosonic dark matter candidate, and at the same time satisfying all the constraints from lepton p e h flavor violations. We perform global numerical analyses and show the allowed regions, in which we [ find somewhat restricted parameter space, such as the mass of dark matter candidate and various 2 v components of the Yukawa couplings in the model. 0 8 0 Keywords: 1 0 . 1 0 7 1 : v i X r a ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] 1 I. INTRODUCTION Recently, there was an 2.6σ anomaly in lepton-universality violation in the ratio R K ≡ B(B Kµµ)/B(B Kee) = 0.745+0.090 0.036 by the LHCb Collaboration [1]. In → → −0.074 ± addition, sizable deviations were observed in angular distributions of B K µµ [2]. The ∗ → results can be interpreted by a large negative contribution to the Wilson coefficient C of the 9 semileptonic operator O , and also contributions to other Wilson coefficients, in particular 9 to C [3–6]. 9′ The discrepancy between the theoretical prediction and experimental value on the muon anomalous magnetic dipole moment has been a long-standing problem, which stands at 3.6σ level with the deviation from the SM prediction at [7]. ∆a = aexp aSM = 288(63)(49) 10 11. µ µ − µ × − If one insists on fulfilling the muon g 2 within 1σ 2σ of the experimental value in any − − models, it puts a strong constraint on the parameter space. For example, it requires a relatively light spectrum in the supersymmetric particles in the MSSM in order to bring the prediction to be within 1σ 2σ of the experimental value. A number of leptoquark models − have been proposed to solve the B K( )µµ anomaly, but however it is very hard to satisfy ∗ → simultaneously the muon g 2: see for example Ref. [8]. − Inthiswork, weproposeathree-loopneutrinomassmodelwithafewleptoquarkscalarsin SU(2) -tripletform. WeattempttousethemodeltoexplaintheanomalyofB( ) Kµ+µ , L ∗ − → to achieve a sizable muon g 2, and to provide a bosonic dark matter candidate, and at the − same time satisfying all the constraints from lepton flavor violations. The concrete model is based on the SM symmetry and a Z symmetry as SU(3) SU(2) U(1) Z . The 2 C L Y 2 × × × model consists of the SM fields, 3 additional leptoquark triplet fields ∆a , and one colorless 1,2,3 doublet scalar field η. These fields are assigned different Z parities and hypercharges in 2 such a way that each of the Yukawa-type couplings contributes to either neutrino mass, B K( )µµ anomaly, muon g 2, or the dark matter interactions. In this way, although ∗ → − the model contains more parameter, it can however explain all the above anomalies. The achievements of the model can be summarized in the following. 1. The neutrino mass pattern and oscillation can be accommodated with the Yukawa 2 Quarks Leptons Vector Fermions Qa ua da L e L Li Ri Ri Li Ri ′i SU(3) 3 3 3 1 1 1 C SU(2) 2 1 1 2 1 2 L U(1) 1 2 1 1 1 1 Y 6 3 −3 −2 − −2 Z + + + + + 2 − TABLE I: Field contents of fermions and their charge assignments under SU(3) SU(2) C L × × U(1) Z , where the superscript (subscript) index a(i) = 1 3 represents the color (flavor). Y 2 × − coupling terms f,g,h in three-loop diagrams 1. 2. The Yukawa coupling term f can give useful contributions to the Wilson coefficients C in such a way that it can explain successfully the B K( )µµ anomaly. 9,10 ∗ → 3. The muon g 2 receives a large contribution from the Yukawa coupling term r. With − some adjustment of the parameters a level of 10 9 is possible. − 4. It provides a dark matter (DM) candidate η , the real part of the neutral component R of the η field with correct relic density. 5. The model can satisfy all the existing constraints from the lepton-flavor violations (LFVs), meson mixings, and rare B decays. This paper is organized as follows. In Sec. II, we describe the neutrino mass matrix and the solution to the anomaly in b sµµ¯. In Sec. III, we discuss various constraints of the → model, including lepton-flavor violations, FCNC’s, oblique parameters, and dark matter. In Sec. IV, we present the numerical analysis and allowed parameter space, followed by the discussion on collider phenomenology. Sec. IV is devoted for conclusions and discussion. II. THE MODEL In this section, we describe the model setup, derive the formulas for the active neutrino mass matrix, and calculate the contributions to b sµµ¯. → 1 See refs. [9–11] for representative three loop neutrino mass models 3 Φ η ∆a ∆a ∆a 1 2 3 SU(3) 1 1 3 3¯ 3¯ C SU(2) 2 2 3 3 3 L 1 1 2 1 1 U(1) Y 2 2 3 3 3 Z2 + + − − − TABLEII:FieldcontentsofbosonsandtheirchargeassignmentsunderSU(3) SU(2) U(1) C L Y × × × Z , where the superscript index a = 1 3 represents the color. 2 − A. Model setup We show all the field contents and their charge assignments in Table I for the fermionic sector and in Table II for the bosonic sector. 2 Under this framework, the relevant part of the renormalizable Lagrangian and Higgs potential related to the neutrino masses are given by −L = yℓiL¯LiΦeRi +fijL¯Li∆†3(iσ2)QcLj +gijL¯′Ri∆†1QLj +hijL¯′Li∆†2QcLj +rijL¯′LiηeRj +M L¯ L λ η ∆ ∆ Φ λ η ∆ ∆ Φ λ (η Φ)2 +h.c., (II.1) i ′Li ′Ri − 0 † 3 1 ∗ − ′0 † 3 ∗2 − 5 † where we have defined L [N,E]T, σ is the second Pauli matrix and we have abbreviated ′ 2 ≡ the trivial terms for the Higgs potential. The scalar fields can be parameterized as 0 η+ δ2(1/)3 δ(1) Φ = , η = , ∆ = √2 5/3 ,  v+φ   ηR+iηI  1 δ(1) δ2(1/)3  √2 √2 1/3 − √2 −       δ1(2/)3 δ(2) δ1(3/)3 δ(3) ∆ = √2 4/3 , ∆ = √2 4/3 , (II.2) 2 δ(2) δ1(2/)3  3 δ(3) δ1(3/)3  2/3 − √2 2/3 − √2 − −     where the subscript next to the each field represents the electric charge of the field, v = 246 GeV,andΦiswrittenintheformaftertheGoldstonefieldsareaboserbedasthelongitudinal components of W and Z bosons. Notice here that each of the components of ∆ and η is 3 2 The same contents of the field are found in the systematic analysis in the last part of Table 3 of ref. [12]. 4 in mass eigenstate, since there are no mixing terms that are assured by the Z and U(1) 2 Y symmetries. On the other hand, components of ∆ and ∆ can mix via Φ Φ ∆ ∆ term. 1 2 ∗ ∗ 1 2 In the following analysis, we ignore such mixing effects assuming the relevant coupling is small. Oblique parameters: Each of the mass components among ∆ is strongly restricted by the i oblique parameters. In order to evade such a strong constraint, we simply assume that each of the components should be of the same mass [13]. Thus, we define m as the mass for ∆i the components of ∆ . On the other hand, each component of η cannot have the same mass, i because the neutrino mass is proportional to the mass difference between the components of η, as you shall see later. Hence, we consider the oblique parameter constraints on η, which are characterized by ∆T and ∆S. Their formulae are given by [14] F[η ,η ]+F[η ,η ] F[η ,η ] 1 1 xm2 +(1 x)m2 ∆T = ± I ± R − I R , ∆S = x(1 x)ln ηR − ηI , 32π2α v2 2π − m2 em Z0 " η± # (II.3) where α 1/137 is the fine structure constant, and em ≈ m2 +m2 m2m2 m2 F[a,b] = a b a b ln a , m = m . (II.4) 2 − m2 m2 m2 a 6 b a − b (cid:20) b(cid:21) The experimental bounds are given by [7] (0.05 0.09) ∆S (0.05+0.09), (0.08 0.07) ∆T (0.08+0.07). (II.5) − ≤ ≤ − ≤ ≤ We consider these constraints in the numerical analysis. Active neutrino mass matrix: The neutrino mass matrix is induced at three-loop level as shown in Fig. 1, and its formula is generally given by = d + u +tr., [ u = 2 d(d u,δ(i) δ(i) )], (II.6) Mνij Mνij Mνij Mν Mν → 1/3 → 2/3 32λ λ (m2 m2)v2 3 d = 0 ′0 R − I f gT M h fTF [r ,r ,r ,r ,r ,r ,r ,r ], Mνab 2√2(4π)6M4 ia ab b ∗bc cj III ∆1 ∆2 ∆3 b R I dc da max (a,b,c)=1 X (II.7) where we used the shorthand notation m m , and define M R/I η Max ≡ R/I ≡ Max[M ,m ,m ,m ], r m2/M2 , and the three-loop function F is given in the b ∆i R I f ≡ f Max III Appendix. Here we adopt an assumption M = m , and require 1TeV . m (which max ∆3 ∆i 5 FIG. 1: Neutrino mass matrix at the three-loop level, where we have two kind of diagrams that are running up-quarks and down quarks inside the loop. suggests x ,x 0), which is required by the direct bound on leptoquarks [13]. In this d u ≈ case, the neutrino mass matrix can be simplified as 33λ λ (m2 m2)v2 0 ′0 R − I fgT(MF )h fT +tr., (II.8) Mνij ≈ 2√2(4π)6m4 III ∗ ij ∆3 (cid:2) (cid:3) where we have abbreviated the symbol of summation and the argument of F . Then III we derive the Yukawa coupling in terms of the experimental values and the parameters by introducing anarbitraryanti-symmetric matrixwith complex values A[15], that isAT+A = 0, as follows: 1 g = R 1(h ) 1 f 1V D VT (fT) 1 +A T , (II.9) 2 − † − − MNS ν MNS − (cid:2) (cid:3) or 1 h = 2R∗−1(g†)−1 f−1VMNSDνVMTNS(fT)−1 +A ∗, (II.10) (cid:2) (cid:3) where we shall adopt the former formula in the numerical analysis below, and we define D VT V and parametrize as ν ≡ MNSMν MNS 0 a a 12 13 33λ λ (m2 m2)v2MF R = 0 ′0 R − I III, A  a 0 a . (II.11) 2√2(4π)6m4 ≡ − 12 23 ∆3   a a 0  13 23   − −    Here we assume one massless neutrino (with normal ordering) in the numerical analysis below. On the term f: Thenewphysics contributionstoaccountfortheB K( )µµanomaly[2] ∗ → can be interpreted as the shifts in the Wilson coefficients C . In our model, the relevant 9,10 6 Wilson coefficients can be calculated as follows [13]: 1 f f V V G α (C )µµ = (C )µµ = bµ sµ, C tb t∗s F em, (II.12) 9 − 10 −C 4m2 SM ≡ √2π SM ∆3 where m m , G 1.17 10 5 GeV 2 is the Fermi constant. We can then compare ∆3 ≡ δ(3) F ≈ × − − 4/3 them to the best-fit values of C from a global analysis based on the LHCb data in Ref. [3] 9,10 as C = C : 0.68 . (II.13) 9 10 − − Here we also have to work within the 0.75 . C . 0.35 in order to satisfy the the LHCb 9 − − measurement of R = B(B+ K+µ+µ )/B(B+ K+e+e ) = 0.745+0.090 0.036, which K → − → − −0.074± shows a 2.6σ deviation from the SM prediction [13]. Notice here that various constraints arising from the f term include B ℓ+ℓ , ℓ ℓ γ, which were given in Refs. [13] d/s − i j → → and [8], and we consider these constraints in the current numerical analysis. Although the muon g 2 is also induced from this term, the typical order is 10 12 10 13 with a negative − − − ∼ sign [13]. Thus, we neglect this contribution to the muon g 2. − On the terms g and h: The main constraint on g and h comes from B(b sγ). The → partial decay width for b sγ is given by → 2 Γ(b → sγ) ≈ 34α(e4mπm)45b g2†a2ga3Fbsγ[δ2(1/)3,a]−h†3aha2 35Fbsγ[a,δ1(2/)3]+Fbsγ[δ1(2/)3,a] , (cid:12) (cid:12) (cid:12) (cid:18) (cid:19)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(II.14) (cid:12) (cid:12) 2m6 +3m4m2 6m2m4 +m6 +12m4m2ln[m /m ] F [a,b] = a a b − a b b a b b a , (II.15) bsγ 12(m2 m2)4 a − b then the branching ratio and its experimental bound [16] are given by Γ(b sγ) B(b sγ) → . 3.29 10 4 , (II.16) − → ≈ Γ × tot. where Γ 4.02 10 13 GeV is the total decay width of the bottom quark. In our tot. − ≈ × numerical analysis, we consider this constraint only for the g and h terms. On the term r: This term is very important in our model because it can induce a large contribution to the muon g 2 and explain the relic density of dark matter (DM) if we − assume the η to be the DM candidate. First of all, let us consider the LFVs processes, R ℓ ℓ γ, via one-loop diagrams. The branching ratio is given by a b → 48π3C α B(ℓ ℓ γ) = ab em( (a ) 2 + (a ) 2), (II.17) a → b G2m2 | R ab| | L ab| F a 7 where m is the mass for the charged-lepton eigenstate, C (1,0.1784,0.1736) for a(b) ab ≈ (a,b) = (2,1),(3,1),(3,2), and a is simply given by L(R) 3 (a ) rb†iriama FL(R)[N ,η ] 1FL(R)[η ,E ] 1FL(R)[η ,E ] , (II.18) L(R) ab ≈ (4π)2 lfv i ± − 2 lfv I i − 2 lfv R i i=1 (cid:18) (cid:19) X m4 +2m2m2 +2m2(m2 +m2)ln m2b m2 +m2ln m2b a a b b a b m2+m2 a b m2+m2 FL [a,b] = a b , FR [a,b] = a b , lfv 6m6 (cid:16) (cid:17) lfv 6m4(cid:16) (cid:17) a a (II.19) where the mass of E(N) is defined by M . Current experimental upper bounds are a E(N)a given by [17, 18] B(µ eγ) 4.2 10 13, B(τ µγ) 4.4 10 8, B(τ eγ) 3.3 10 8 . (II.20) − − − → ≤ × → ≤ × → ≤ × Muon g 2: The muon anomalous magnetic moment is simply given by ∆a m [a + µ µ L − ≈ − a ] in Eq. (II.19). Experimentally, it has been measured with a high precision, and its R 22 deviation from the SM prediction is ∆a = (10 9) [19]. It would be worth mentioning a µ − O new contribution to the leptonic decay of the Z boson. In our case, the Z boson can decay into a pair ofcharged leptons witha correction at one-looplevel, and it isproportional to the Yukawa couplings related to the muon g 2. Therefore it can be enhanced due to the large − Yukawa couplings. However, we have checked that this mode is within the experimental bound: B(Z ℓℓ¯) . 3 10 2. − → × 8 Q Q¯ mixing: The forms of K0 K¯0, B0 B¯0, and D0 D¯0 mixings are, respectively, − − d − d − given by ∆mK ≈ (41π)2 3 gi2g1†ig1†jgj2 FbKox[Ni,Nj,δ1(1/)3]+ FbKox[Ei,4Ej,δ2(1/)3] " ! i,j=1 X FK [ν ,ν ,δ ] + fi†1f2if2jfj†1 FbKox[ℓi,ℓj,δ4(3/)3]+ box i4 j 1/3 (cid:18) (cid:19) FK [N ,N ,δ(2)] + h†i1h2ih2jh†j1 FbKox[Ei,Ej,δ4(2/)3]+ box i4 j 1/3 . 3.48×10−15[GeV], !# (II.21) ∆mBd ≈ (41π)2 3 gi3g1†ig1†jgj3 FbBox[Ni,Nj,δ1(1/)3]+ FbBox[Ei,4Ej,δ2(1/)3] " ! i,j=1 X FB [ν ,ν ,δ ] + fi†3f1if1jfj†3 FbBox[ℓi,ℓj,δ4(3/)3]+ box i4 j 1/3 (cid:18) (cid:19) FB [N ,N ,δ(2)] + h†i3h1ih1jh†j3 FbBox[Ei,Ej,δ4(2/)3]+ box i4 j 1/3 . 3.36×10−13[GeV], !# (II.22) ∆mD ≈ (41π)2 3 gi2g1†ig1†jgj2 FbDox[Ei,Ej,δ5(1/)3]+ FbDox[Ni,4Nj,δ2(1/)3] " ! i,j=1 X FD [ν ,ν ,δ ] + fi†1f2if2jfj†1 FbDox[ℓi,ℓj,δ4(3/)3]+ box i4 j 1/3 (cid:18) (cid:19) FD [E ,E ,δ(2)] + h†i1h2ih2jh†j1 FbDox[Ni,Nj,δ2(2/)3]+ box i4 j 1/3 . 6.25×10−15[GeV], !# (II.23) 5m f2 m 2 δ(1 a b c d)dadbdcdd FQ (x,y,z) = Q Q Q − − − − , (II.24) box 24 m +m [am2 +bm2 +(c+d)m2]2 (cid:18) q1 q2(cid:19) Z x y z where (q ,q ) are respectively (d,s) for K, (b,d) for B, and (u,c) for D. Each of the 1 2 last inequalities in Eqs.(II.21 – II.23) represents the upper bound on the corresponding experimental value [7]. Here we used f 0.156 GeV, f 0.191 GeV, m 0.498 GeV, K B K ≈ ≈ ≈ and m 5.280 GeV. 3 B ≈ DarkMatter: Hereweidentifyη astheDMcandidate, anddenoteitsmassbym M . R R X ≡ Direct detection: We have a Higgs portal contribution to the DM-nucleon scattering process, 3 Since we assume that one of the neutrino masses to be zero with normal ordering that leads to the first column in g to be almost zero, i.e., (g)11,12,13 0, and so these constraints can easily be evaded. ≈ 9 which is constrained by direct detection search such as the LUX experiment [20]. Its spin independent cross section is simply given by [21] 2 (λ +λ +2λ )v σ 2.12 10 42 3 4 5 [cm]2, (II.25) N − ≈ × × M (cid:18) X (cid:19) where λ and λ are quartic couplings proportional to (η η)(Φ Φ) and (η Φ)(Φ η), respec- 3 4 † † † † tively. The current experimental minimal bound is σ . 2 10 46 cm2 at M 50 GeV. N − X × ≈ Once we apply this bound on our model, we obtain λ + λ + 2λ . 2 10 3. Hence we 3 4 5 − × assume that all the Higgs couplings are small enough to satisfy the constraint, and we neglect DM annihilation modes via Higgs portal in estimating the relic density below. Notice here that photon and Z boson fields transform as V Vµ under charge-parity (CP) µ → − conjugation, while X is CP-even. Thus X X γ(Z) couplings are not allowed because − − they violates the CP invariance. Relic density: We consider parameter region in which the DM annihilation cross section is d-wave dominant and the dark matter particles annihilate into a pair of charged-leptons, via ¯ the process η η ℓ ℓ with an E exchange. Notice that there exist annihilation modes R R i j a → such as η η W+W /2Z arising from the kinetic term. These modes require a DM R R − → mass heavier than at least 500 GeV [22] in order to obtain the correct relic density where coannihilation processes should be included. This case is, however, not in favor of explaining the muon g 2 anomaly. Thus we assume that M . 80 GeV and η η W+W /2Z X R R − − → processes are not kinematically allowed. 4 Then the relic density is simply given by Ωh2 1.70×107x3f , d 3 |riara†,j|2MX6 , (II.26) ≈ √g M d [GeV] eff ≈ 120π(M2 +M2)4 ∗ P eff (i,j,a)=1 Ea X X where g 100, M 1.22 1019, x 25. In our numerical analysis below, we use the ∗ P f ≈ ≈ × ≈ current experimental range for the relic density: 0.11 Ωh2 0.13 [23]. ≤ ≤ 4 Here we impose the condition m /2( 41GeV) . m +m to forbid the invisible decay of Z boson in Z R I ≈ our numerical analysis, although the invisible decay of SM Higgs is automatically suppressed in the limit of zero couplings in the Higgs potential. 10