b→s μ + μ − anomalies and related phenomenology in U(1 ) B 3 − x μ L μ − x τ L τ flavor gauge models
PPrepared for submission to JHEP b → sµ + µ − anomalies and related phenomenologyin U (1) B − x µ L µ − x τ L τ flavor gauge models P. Ko a,b
Takaaki Nomura a Chaehyun Yu c a School of Physics, KIAS, Seoul 02455, Korea b Quantum Universe Center, KIAS, Seoul 02455, Korea c Department of Physics, Korea University, Anam-ro 145, Sungbuk-gu, Seoul 02841, Korea
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We propose a generation dependent lepton/baryon gauge symmetry, U (1) B − x µ L µ − x τ L τ ≡ U (1) X (with x µ + x τ = 1 for anomaly cancellation), as a possible solution for the b → sµ + µ − anomalies. By introducing two Higgs doublet fields, we can reproduce the observed CKMmatrix, and generate flavor changing Z (cid:48) interactions in the quark sector. Thus one can ex-plain observed anomalies in b → s(cid:96) + (cid:96) − decay with the lepton non-universal U (1) X chargeassignments. We show the minimal setup explaining b → s(cid:96) + (cid:96) − anomalies, neutrino massesand mixings and dark matter candidate, taking into account experimental constraints offlavor physics such as charged lepton flavor violations and the B s – ¯ B s mixing. Finally wediscuss collider physics focusing on Z (cid:48) production at the Large Hadron Collider and relicdensity of our dark matter candidate. a r X i v : . [ h e p - ph ] M a y ontents x µ , x τ ) case 32.1.1 Z (cid:48) interactions with SM fermions 42.1.2 Effective interaction for b → sµ + µ − Z (cid:48) contribution to muon g − B s – ¯ B s mixing 133.6 Prediction on B → K ( ∗ ) τ + τ − Z (cid:48) production at the LHC 154.2 Dark matter 16 Although the standard model (SM) of particle physics is very successful we still do nothave clear understanding of the physics regarding the flavors; namely the origin of fermionmasses and mixing patterns. Then it is interesting to construct a model describing fla-vor physics with some symmetry as a guiding principle. One of the attractive possibilityis an introduction of flavor dependent U (1) gauge symmetry which can constrain struc-ture of Yukawa couplings generating masses for quarks, charged leptons and neutrinos. Inthis kind of approaches to the flavor problem, these models may generate flavor chang-ing neutral current (FCNC) processes through Z (cid:48) boson exchange, which will induce richphenomenology.Recently there have been some indication of anomalies in B physics measurements for b → s(cid:96) + (cid:96) − process. The angular observable P (cid:48) in decay of B meson, B → K ∗ µ + µ − [1], in-dicates 3 . σ deviations from the data with integrated luminosity of 3.0 fb − at the LHCb [2],confirming an earlier observation with 3 . σ deviations [3]. In addition, 2 . σ deviations were– 1 –eported for the same observable by Belle [4, 5]. Furthermore, an anomaly in the measure-ment of lepton flavor universality by the ratio R K = BR ( B + → K + µ + µ − ) /BR ( B + → K + e + e − ) [6, 7] at the LHCb shows 2 . σ deviations from the SM prediction [8]. More-over the LHCb collaboration also reported an anomaly in the ratio R K ∗ = BR ( B → K ∗ µ + µ − ) /BR ( B → K ∗ e + e − ) where the observed values are deviated from the SM pre-diction by ∼ . σ as R K ∗ = 0 . +0 . − . ± . . +0 . − . ± . m µ ) < q < . (1.1 GeV < q < ) [9].These anomalies in the b → s(cid:96) + (cid:96) − channels (with (cid:96) = e, µ ) can be explained by fla-vor dependent Z (cid:48) interactions inducing effective operator of (¯ bγ α s )(¯ µγ α µ ), if new physicscontribution to the corresponding Wilson coefficient C µ is roughly ∆ C µ ∼ − Z (cid:48) inter-actions [14–44].In this paper, motivated by b → s(cid:96) + (cid:96) − anomalies, we propose a model based on flavordependent Abelian gauge symmetry U (1) B − x µ L µ − x τ L τ , which is anomaly-free for x µ + x τ =1. In this model we introduce two Higgs doublet fields to generate the realistic CKM matrix,where small mixings associated with third generation quarks can be obtained naturally asshown in Ref. [14]. In the reference it is also shown that Z (cid:48) bs interaction is induced afterelectroweak symmetry breaking in a model with flavor dependent U (1) L µ − L τ − a ( B + B − B ) gauge symmetry where a can be arbitrary real number. Then, b → s(cid:96) + (cid:96) − anomaliescan be explained by the effective operator induced by exchange of a TeV scale Z (cid:48) boson.Following the same mechanism to induce Z (cid:48) bs interaction we can explain the anomaliesby our flavor dependent U (1) gauge symmetry if x µ has negative value to get ∆ C µ ∼ − b → s(cid:96) + (cid:96) − anomalies and generatingnon-zero neutrino masses in which two SM singlet scalar fields are introduced. Also weintroduce Dirac fermionic dark matter (DM) candidate in order to account for the darkmatter of the Universe. In addition to ∆ C µ , we formulate neutrino mass matrix, leptonflavor violations (LFVs) and B s – ¯ B s mixing, and experimental constraints from them aretaken into account. Then we discuss collider physics regarding Z (cid:48) production at the LargeHadron Collider (LHC) and relic density of our DM candidate.This paper is organized as follows. In Sec. II, we introduce our model and discuss quarkmass, ∆ C µ by Z (cid:48) and scalar masses in the minimal case. In Sec.III we discuss neutrino massmatrix, charged lepton flavor violations and B s – ¯ B s mixing taking into account experimentalconstraints. The numerical analysis is carried out in Sec. IV to discuss collider physics for Z (cid:48) production at the LHC and relic density of DM candidate showing allowed parameterregion. Finally summary and discussion are given in Sec. V. In this section we introduce our model based on flavor dependent U (1) B − x µ L µ − x τ L τ gaugesymmetry that we denote simply U (1) X in the following . The SM fermions with 3 right-handed (RH) neutrinos are charged under the U (1) X as shown in Table. 1. The gauge In our analysis we ignore kinetic mixing between U (1) Y and U (1) X assuming it is sufficiently small. – 2 – ermions Q iL u iR d iR Q L t R b R L L L L L L e R µ R τ R ν R ν R ν R SU (3) C SU (2) L U (1) Y
16 23 −
13 16 23 − − − − − − − U (1) X
13 13 13 − x µ − x τ − x µ − x τ − x µ − x τ Table 1 . Charge assignment for the SM fermions and right-handed neutrinos where the indices i = 1 , anomalies are cancelled when the U (1) X charges of fermions satisfy the condition x µ + x τ = 1 , (2.1)which we will always assume in the following. In Sec. 2.1, we first discuss the case withgeneral x µ,τ and investigate an explanation of b → s(cid:96) + (cid:96) − anomalies via flavor-changing Z (cid:48) interactions. Then the minimal model with x µ = − / ( x µ , x τ ) case Firstly we consider quark sector which does not depend on our choice of x µ and x τ = 1 − x µ .In this model we have to introduce at least two Higgs doublets in order to induce the realisticCKM mixing matrix:Φ : ( , )(1 / , − / , Φ : ( , )(1 / , , ( SU (3) C , SU (2) L )( U (1) Y , U (1) X ) (2.2)Then the Yukawa couplings for quarks are given by −L Q = y uij ¯ Q iL ˜Φ u jR + y dij ¯ Q iL Φ d jR + y u ¯ Q L ˜Φ t R + y d ¯ Q L Φ b R + ˜ y u i ¯ Q L ˜Φ u iR + ˜ y di ¯ Q iL Φ b R + h . c ., (2.3)where i = 1 , i = iσ Φ ∗ i . Φ is the Higgs doublet with vanishing U (1) X charge,and is the SM-like Higgs doublet. After two Higgs doublet fields get the non-zero vacuumexpectation values (VEVs) (cid:104) Φ , (cid:105) = (0 v , / √ T , we obtain the following forms of quarkmass matrices: M u = 1 √ v y u v y u v y u v y u
00 0 v y u + ξ u ) ( ξ u ) ,M d = 1 √ v y d v y d v y d v y d
00 0 v y d + ξ d ) ξ d ) . (2.4)Note that the matrices ( ξ u,d ) ij ≡ ˜ y u,dij v / √ ξ u,d are small perturbation effects– 3 –enerating realistic 3 × v y u ( d )33 ∼ √ m t ( b ) following the discussion in Ref. [14].As in the SM, the quark mass matrices are diagonalized by unitary matrices U L,R and D L,R which change quark fields from interaction basis to mass basis: u L,R → U † L,R u L,R ( d L,R → D † L,R d L,R ). Then the CKM matrix is given by V CKM = U † L D L . Thus we obtain relationbetween mass matrices M u,d and diagonalized ones as follows: M d = D L m d diag D † R , M u = U L m u diag U † R , (2.5)where diagonal mass matrices are given by m d diag = diag( m d , m s , m b ) and m u diag = diag( m u , m c , m t ).Then U L [ R ] and D L [ R ] are associated with diagonalization of M u ( M u ) † [( M u ) † M u ] and M d ( M d ) † [( M d ) † M d ] by M u ( M u ) † (cid:104) ( M u ) † M u (cid:105) = U † L ( m u diag ) U L (cid:104) U † R ( m u diag ) U R (cid:105) ,M d ( M d ) † (cid:104) ( M d ) † M d (cid:105) = D † L ( m u diag ) D L (cid:104) D † R ( m u diag ) D R (cid:105) . (2.6)The structures of mass matrices in Eq. (2.4) indicate that the off-diagonal elements asso-ciated with 3rd generations are more suppressed for M u ( M u ) † and ( M d ) † M d than thosein ( M u ) † M u and M d ( M d ) † . More specifically, we find that (cid:16) M u ( M u ) † (cid:17) , , , (cid:20)(cid:16) ( M d ) † M d (cid:17) , , , (cid:21) ∝ v √ y ij ξ k [ k , (cid:16) ( M u ) † M u (cid:17) , , , (cid:20)(cid:16) M d ( M d ) † (cid:17) , , , (cid:21) ∝ v √ y ξ k [ k , (2.7)where { i, j, k } = 1 ,
2. Then we can approximate U L and D R to be close to unity matrixsince they are associated with diagonalizaition of M u ( M u ) † and ( M d ) † M d , respectively,where mixing angles in D R ( U L ) generated by ξ parameters are suppressed by m d,s ( u,c ) /m b ( t ) to those in D L ( U R ). Therefore CKM matrix can be approximated as V CKM (cid:39) D L , and D R (cid:39) , as obtained in Ref. [14]. Taking D L = V CKM , we can obtain sizes of ( ξ d ) and( ξ d ) from Eq. (2.6) applying mass eigenvalues of down-type quarks. We thus obtain | ( ξ d ) | ∼ .
034 GeV , | ( ξ d ) | ∼ .
18 GeV (2.8)with y v / √ (cid:39) m b (cid:39) . D L (cid:39) V CKM taking these values for ξ d (values of y ij are chosen to fit m d and m s ). In addition, the values of ξ u tend to be smaller than ξ d due to mass relation m b (cid:28) m t . Z (cid:48) interactions with SM fermions The Z (cid:48) couplings to the SM fermions are written as L ⊃ − g X ( x µ ¯ µγ µ µ + x τ ¯ τ γ µ τ + x µ ¯ ν µ γ µ P L ν µ + x τ ¯ ν τ γ µ P L ν τ + x µ ¯ ν γ µ P R ν + x τ ¯ ν γ µ P R ν ) Z (cid:48) µ + g X tγ µ tZ (cid:48) µ + g X (cid:16) ¯ d α γ µ P L d β Γ d L αβ + ¯ d α γ µ P R d β Γ d R αβ (cid:17) Z (cid:48) µ , (2.9)– 4 –here g X is the gauge coupling constant associated with the U (1) X and the lepton sectoris given in the flavor basis here. The coupling matrices Γ d R and Γ d L for down-type quarksare given approximately byΓ d L (cid:39) | V td | V ts V ∗ td V tb V ∗ td V td V ∗ ts | V ts | V tb V ∗ ts V td V ∗ tb V ts V ∗ tb | V tb | , Γ d R (cid:39) , (2.10)where V qq (cid:48) ’s are the CKM matrix elements. We have applied the relation V CKM (cid:39) D L , aswe discussed above. In our model the Z (cid:48) mass, m Z (cid:48) , is dominantly given by the VEV ofSM singlet scalar field as discussed below.At this point, x µ is an arbitrary parameter requiring only anomaly cancellation con-dition Eq. (2.1). This value will be fixed to obtain negative ∆ C µ and to realize minimalscalar sector. The mass of Z (cid:48) can be a free parameter since it is given by new gaugecoupling g X and scalar singlet VEV where we have freedom to chose the VEV even if thegauge coupling is fixed. b → sµ + µ − Gauge interactions in Eq. (2.9) induce the effective Hamiltonian for b → sµ + µ − processsuch that∆ H eff = − x µ g X V tb V ∗ ts m Z (cid:48) (¯ sγ µ P L b )(¯ µγ µ µ ) + h.c. = x µ g X m Z (cid:48) (cid:32) √ πG F α em (cid:33) (cid:18) − G F √ α em π V tb V ∗ ts (cid:19) (¯ sγ µ P L b )(¯ µγ µ µ ) + h.c., (2.11)where G F is the Fermi constant and α em is the electromagnetic fine structure constant.We thus obtain the Z (cid:48) contribution to Wilson coefficient ∆ C µ as∆ C µ = x µ g X m Z (cid:48) (cid:32) √ πG F α em (cid:33) (cid:39) . × x µ (cid:16) g X . (cid:17) (cid:18) . m Z (cid:48) (cid:19) . (2.12)In order to obtain ∆ C µ ∼ − x µ should be negative and g X is required to be ∼ . m Z (cid:48) = 1 . x µ = − . Figure 1 shows the contour of ∆ C µ in the ( m Z (cid:48) , g X ) planewhere we took x µ = − where the yellow(light-yellow) region corresponds to 1 σ (2 σ ) regionfrom global fit in Ref. [11]. Here we consider the minimal cases for choosing U (1) X charges of leptons as x µ = − , x τ = 43 . (2.13)In this case we add two SU (2) L singlet scalar fields: ϕ : ( , )(0 , / , ϕ : ( , )(0 , , (2.14)– 5 – - - - m Z ' @ GeV D g X D C Μ Figure 1 . The contours showing Z (cid:48) contribution to ∆ C µ on the m Z (cid:48) - g X plane with x µ = − where yellow(light-yellow) region corresponds to 1 σ (2 σ ) region from global fit in Ref. [11]. Fields Φ Φ ϕ ϕ χSU (2) L U (1) Y
12 12 U (1) X − Table 2 . Scalar fields and extra fermion χ in the minimal model and their representation under SU (2) × U (1) Y × U (1) X where these fields are color singlet. where ϕ is also necessary to induce Φ † Φ terms , while ϕ is added for generating the23(32) element of Majorana mass matrix of right-handed neutrino. Note that we obtain amassless Goldstone boson from two Higgs doublet sector without ϕ due to an additionalglobal symmetry. In addition we introduce additional Dirac fermion χ of mass m X with U (1) X charge 5 /
6, which can be our DM candidate since its stability is guaranteed due tofractional charge assignment under U (1) X . Note that the stability of Dirac fermion DM χ is guaranteed by remnant Z symmetry after U (1) X symmetry breaking: particles with U (1) X charge 2 n/ n is integer) are Z even and those with U (1) X charge (2 n + 1) / Z odd, since U (1) X symmetry is broken by VEVs of scalar fields ϕ , ϕ and Φ whosecharges correspond to 2 n/ Note that we need one more scalar singlet to generate neutrino mass when x µ (cid:54) = − / – 6 –n our set up, the full scalar potential for scalar fields in our model is given by V = − µ (Φ † Φ ϕ ∗ + h . c . ) + µ | Φ | + µ | Φ | + µ ϕ | ϕ | + µ ϕ | ϕ | + λ | Φ | + λ | Φ | + λ | Φ | | Φ | + λ | Φ † Φ | + λ ϕ | ϕ | + λ ϕ | ϕ | + λ Φ ϕ | Φ | | ϕ | + λ Φ ϕ | Φ | | ϕ | + λ Φ ϕ | Φ | | ϕ | + λ Φ ϕ | Φ | | ϕ | + λ ϕ ϕ | ϕ | | ϕ | − λ X ( ϕ ϕ ∗ + h.c. ) , (2.15)where we assumed all the coupling constants are real for simplicity. The VEVs of singletscalar fields are written by √ (cid:104) ϕ (cid:105) = v ϕ and √ (cid:104) ϕ (cid:105) = v ϕ . In our scenario, we assume v ϕ (cid:29) v ϕ (cid:29) v , and U (1) X symmetry is spontaneously broken at a scale higher thanthe electroweak scale. We then approximately obtain VEVs of ϕ , from the condition ∂V /∂v ϕ , = 0: v ϕ (cid:39) (cid:115) − µ ϕ λ ϕ , v ϕ (cid:39) λ X v ϕ µ ϕ + 2 λ ϕ ϕ v ϕ , (2.16)where the above assumption for VEV hierarchy can be consistent requiring λ X v ϕ (cid:28) µ ϕ .Then the mass of the Z (cid:48) boson is approximately given by m Z (cid:48) (cid:39) g X v ϕ . (2.17)Then a typical value of the ϕ VEV is v ϕ (cid:39) . × ( m Z (cid:48) / . . /g X ) TeV in ourscenario. Note that the Z – Z (cid:48) mass mixing is highly suppressed by v /v ϕ factor which is ∼ − for tan β = v /v = 10 and v ϕ = 7 . .After U (1) X symmetry breaking, we obtain two-Higgs doublet potential effectively : V T HDM = m | Φ | + m | Φ | − ( m Φ † Φ + h.c. )+ λ | Φ | + λ | Φ | + λ | Φ | | Φ | + λ | Φ † Φ | , (2.18) m = µ + 12 λ Φ ϕ v ϕ + 12 λ Φ ϕ v ϕ , m = 1 √ µv ϕ . (2.19)Here we write Φ i ( i = 1 ,
2) as Φ i = (cid:32) w + i √ ( v i + h i + iz i ) (cid:33) . (2.20) The Z – Z (cid:48) mixing effect is constrained by precision measurements of Z ¯ f SM f SM coupling at the LEPexperiments where the upper bound of the mixing θ ZZ (cid:48) is around ∼ − − − [45, 46]. Thus our mixingangle is sufficiently smaller than the bound. Here we do not consider scalar bosons from ϕ , since they are assumed to be much heavier than thosefrom Higgs doublets and mixing among singlet and doublet scalars will be small. – 7 –s in the two-Higgs-doublet model (THDM), we obtain mass eigenstate { H, h, A, H ± } inthe two Higgs doublet sector: (cid:32) z ( w +1 ) z ( w +2 ) (cid:33) = (cid:32) cos β − sin β sin β cos β (cid:33) (cid:32) G Z ( G + ) A ( H + ) (cid:33) , (2.21) (cid:32) h h (cid:33) = (cid:32) cos α − sin α sin α cos α (cid:33) (cid:32) Hh (cid:33) , (2.22)where tan β = v /v , G Z ( G + ) is a Nambu-Goldstone boson (NG) absorbed by the Z ( W + )boson, and h is the SM-like Higgs boson. The masses of H ± and A are given as in THDM: m H ± = m sin β cos β − v λ , m A = m sin β cos β . (2.23)Mass eigenvalues of CP-even scalar bosons are also obtained by m H,h = 12 (cid:18) M + M ± (cid:113) ( M − M ) + 4 M (cid:19) , (2.24) M = v ( λ cos β + λ sin β ) + v λ sin β, (2.25) M = m sin β cos β + v sin β cos β ( λ + λ − λ ) , (2.26) M = v β ( − λ cos β + λ sin β ) + v λ sin 2 β cos 2 β, (2.27)where ¯ λ = λ + λ and lighter mass eigenvalue m h is identified as the SM-like Higgs mass.Note that Higgs bosons in doublet interact with Z (cid:48) and three point couplings can beobtained such that( D µ H ) † ( D µ H ) ⊃ i g X Z (cid:48) µ ( w +1 ∂ µ w − − w − ∂ µ w +1 ) + 2 g X Z (cid:48) µ ( h ∂ µ z − z ∂ µ h ⊃ i g X sin β Z (cid:48) µ ( H + ∂ µ H − − H − ∂ µ H + ) + 2 g X sin β sin α Z (cid:48) µ ( h∂ µ A − A∂ µ h )+ 2 g X sin β cos α Z (cid:48) µ ( A∂ µ H − H∂ µ A ) . (2.28)Thus Z (cid:48) can decay into HA , hA and H + H − pair.Here we briefly comment on deviation in the couplings of the SM-like Higgs h andconstraint in the scalar sector in the model. The Yukawa interactions with h are given byEq. (5.1) in the Appendix. In particular, we have flavor violating interaction associatedwith ξ u,d coupling. In our analysis, we assume the interactions are SM-like that can berealized taking large tan β and alignment limit of cos( α − β ) (cid:39)
0. Note also that new scalarbosons do not contribute to explanation of b → sµ + µ − anomalies in our scenario except forrelaxing the constraint from B s – ¯ B s mixing as we discuss below; we can fit the data withthe mass value of ∼
500 to ∼ Neutrino mass and flavor constraints
In this section we formulate neutrino mass matrices (both Dirac and Majorana mass ma-trices), and explore constraints from flavor physics such as µ → eγ , µ → e conversion and B s – ¯ B s mixing. The Yukawa interactions for leptons are given by −L ⊃ y eaa ¯ L aL e aR Φ + y νaa ¯ L aL ν aR ˜Φ + ˜ y e ¯ L L µ R Φ + ˜ y ν ¯ L L ν R ˜Φ + M ¯ ν c R ν R + Y ¯ ν c R ν R ϕ ∗ + Y ¯ ν c R ν R ϕ ∗ + h.c., (3.1)where a = 1 , , Y ab = Y ba . After the symmetry breaking, Dirac and Majorana massmatrices for neutrinos have the structure of M D = ( M D ) M D ) ( M D )
00 0 ( M D ) , M ν R = ( M ν R ) ( M ν R ) M ν R ) M ν R ) M ν R ) , (3.2)where the elements of the mass matrices are given by( M D ) aa = 1 √ y νaa v , ( M D ) = 1 √ y v , ( M ν R ) = M, ( M ν R ) = 1 √ Y v ϕ , ( M ν R ) = 1 √ Y v ϕ . (3.3)The active neutrino mass matrix is given by type-I seesaw mechanism: m ν (cid:39) − M D M − ν R M TD = ( M D ) ( M νR ) ( M D ) ( M D ) ( M νR ) − ( M D ) ( M D ) ( M νR ) ( M νR ) ( M νR ) ( M D ) ( M D ) ( M νR ) ( M D ) ( M νR ) ( M D ) ( M D ) ( M νR ) (cid:16) − ( M D ) ( M νR ) ( M νR ) ( M D ) (cid:17) − ( M D ) ( M D ) ( M νR ) ( M νR ) ( M νR ) ( M D ) ( M D ) ( M νR ) (cid:16) − ( M D ) ( M νR ) ( M νR ) ( M D ) (cid:17) ( M D ) ( M νR ) ( M νR ) ( M νR ) . (3.4)Note that our neutrino mass matrix does not have zero structure and neutrino oscilla-tion data can be easily fit. Here we do not carry out further analysis of the neutrinophenomenology in this paper. The charged lepton mass matrix is given by M e = 1 √ y e v ˜ y e v y e v
00 0 y e v ≡ m e δm e m e
00 0 m e . (3.5)– 9 – igure 2 . One loop diagrams inducing µ → eγ process. For δm e (cid:28) m e , the mass matrix can be diagonalized in good approximation as m e m µ
00 0 m τ (cid:39) V eL M e ( V eR ) † , (3.6) V eR (cid:39) , V eL (cid:39) − (cid:15) (cid:15) , (3.7)where (cid:15) = δm e /m e we also find m e (cid:39) m e , m µ (cid:39) m e and m e = m τ . Here we consider charged lepton flavor violation (cLFV) in the model associated with Z (cid:48) .The Z (cid:48) gauge interactions for mass eigenstates of charged leptons are given by L ⊃ − g X (cid:96) i γ µ V eL − V e † L ij P L (cid:96) j Z (cid:48) µ − g X (cid:96) i γ µ − ij P R (cid:96) j Z (cid:48) µ , (3.8)where the flavor violating structure for left-handed charged lepton currents is given by V eL − V e † L (cid:39) − (cid:15) (cid:15) (cid:15) − . (3.9)Thus we have LFV interaction for e and µ . Then we first consider µ → eγ process inducedby Z (cid:48) loop in Fig. 2 where the left diagram gives dominant contribution due to suppressionby (cid:15) . Estimating the loop diagram we obtain dominant contribution to the decay widthfor the µ → eγ process such thatΓ µ → eγ (cid:39) e m µ π | a R | , (3.10) a R (cid:39) e(cid:15)g X m µ π (cid:90) dxdydzδ (1 − x − y − z ) 2 x (1 + y )[( x − x ) + xz + y + z ] m µ + xm Z (cid:48) . (3.11)– 10 – Z ' = ´ - - - - - - - - - - g X l og È Ε È BR H Μ® e Γ L m Z ' = ´ - - - - - - - - - - g X l og È Ε È BR H Μ® e Γ L Figure 3 . BR ( µ → eγ ) as a function of { g X , log | (cid:15) |} fixing m Z (cid:48) = 1 . .
0) TeV for left(right) plotwhere the shaded regions are excluded.
Branching ratio for the LFV process is given by BR ( µ → eγ ) = Γ µ → eγ Γ µ → e ¯ ν e ν µ (cid:39) αG F m µ | a R | , (3.12)where G F (cid:39) . × − GeV − is the Fermi constant and α (cid:39) /
137 is the fine structureconstant. In Fig. 3, we show BR ( µ → eγ ) on { g X , log | (cid:15) |} plane fixing m Z (cid:48) = 1 . .
0) TeVwhere the shaded regions are excluded by the current constraint BR ( µ → eγ ) (cid:46) . × − by the MEG experiment [49]. Further parameter region will be explored in future withimproved sensitivity [50].Here we also discuss µ → e conversion via Z (cid:48) exchange. In our case, the relevanteffective Lagrangian for the process is derived as follows [51–53] L eff = − G F √ (cid:88) N = p,n (cid:2) C NNV L ¯ eγ α P L µ ¯ N γ α N + C NNAL ¯ eγ α P L µ ¯ N γ α γ N (cid:3) , (3.13)where the corresponding coefficients are given by C pp ( nn ) V L = − C pp ( nn ) AL = (2) √ (cid:15)g X | V td | G F m Z (cid:48) . (3.14)Then we obtain the spin-independent contribution to the BR for µ → e conversion on anucleus such that BR ( µ → e ) = 32 G F m µ Γ cap (cid:12)(cid:12)(cid:12) C ppV L V ( p ) + C nnV L V ( n ) (cid:12)(cid:12)(cid:12) , (3.15)– 11 –ucleus AZ N V p V n Γ capt [10 sec − ] Al 0 . . . Au 0 . .
146 13 . Table 3 . A summary of parameters for the µ − e conversion formula for Al and
Au nuclei [52,54]. - - - - - - m Z ' = - - - - g X l og È Ε È BR H Μ® e L@ Al D - - - - - - m Z ' = - - - - g X l og È Ε È BR H Μ® e L@ Al D Figure 4 . BR ( µ → e ) on Al as a function of { g X , log | (cid:15) |} fixing m Z (cid:48) = 1 . .
0) TeV forleft(right) plot where gray(light-gray) shaded region is excluded by current µ → eγ BR ( µ → e BRon
Au [55]) constraints. where Γ cap is the rate for the muon to transform to a neutrino by capture on the nucleus,and V ( p,n ) is the integral over the nucleus for lepton wavefunctions with correspondingnucleon density. The values of Γ cap and V ( n,p ) depend on target nucleus and those for Au and
Al are given in Table. 3 [52, 54]. In Fig. 4, we show BR ( µ → e ) for Alon { g X , log | (cid:15) |} plane fixing m Z (cid:48) = 1 . .
0) TeV in left(right)-panel where gray(light-gray)shaded region is excluded by current µ → eγ BR ( µ → e BR on
Au [55]) constraints.We find that large parameter region can be explored by µ → e conversion measurementsince its sensitivity will reach ∼ − on Al nucleus in future experiments [56, 57].We next consider the LFV B decay B s → µ ± e ∓ which is related to C µ above. It isbecause that the process is induced from C µe which is obtained as C µe = − (cid:15) ∆ C µ in themodel. The branching ratio can be given by BR ( B s → µe ) = (cid:12)(cid:12)(cid:12)(cid:12) C µe C SM (cid:12)(cid:12)(cid:12)(cid:12) BR ( B s → µ + µ − ) SM (cid:39) | . × (cid:15) ∆ C µ | BR ( B s → µ + µ − ) SM , (3.16)where we used C SM ( µ b ) (cid:39) − . BR ( B s → µ + µ − ) SM = (3 . ± . × − is the– 12 –M prediction for the BR of B s → µ + µ − . We find that BR ( B s → µe ) < − in theparameter region satisfying the constraint from BR ( µ → eγ ) which is well below the currentconstraint.Here we also discuss the branching ratio for B → K ( ∗ ) µe through lepton flavor violating Z (cid:48) coupling. It is suppressed compared to BR ( B → K ( ∗ ) µµ ) by a factor of | (cid:15) ∆ C µ /C µ | ∼ − for ∆ C µ = − (cid:15) = 0 .
1. Thus the BR is small as order of 10 − − − and it iswell below current bound and challenging to search for the signal at the future experimentssuch as (upgraded) LHCb [58] and Belle II [59]. Z (cid:48) contribution to muon g − U (1) X gauge coupling and Z (cid:48) mass are constrained by the neutrino trident process νN → νN µ + µ − where N is a nucleon [60]. The bound is approximately given by m Z (cid:48) /g X (cid:38)
550 GeV for m Z (cid:48) > { m Z (cid:48) , g X } satisfying thisbound.The observed muon magnetic dipole moment is deviated from the SM prediction as∆ a µ = (26 . ± . × − [61] (muon g − Z (cid:48) boson can contribute to muon g − a Z (cid:48) µ ≈ g X x µ π (cid:90) da ra (1 − a ) r (1 − a ) + a , (3.17)where r ≡ ( m µ /M Z (cid:48) ) . We find that the Z (cid:48) contribution is small for the parameter regionproviding ∆ C ∼ −
1; for example ∆ a Z (cid:48) µ ∼ . × − with m Z (cid:48) = 1500 GeV and g X = 0 . B s – ¯ B s mixing In our model, Z (cid:48) and neutral scalar bosons induce flavor changing neutral current (FCNC)interactions. Here we consider constraints from B s – ¯ B s mixing where other ∆ F = 2 pro-cesses are more suppressed by CKM factors.The effective Hamiltonian for the B s – ¯ B s mixing is given by H eff = C (¯ sγ µ P L b )(¯ sγ µ P L b ) + C (cid:48) (¯ sP R b )(¯ sP R b ) . (3.18)The relevant Wilson coefficients are C = 12 g X m Z (cid:48) (Γ d L sb ) , C (cid:48) = (cid:88) η = h,H,A − m η (Γ ηsb ) , (3.19)where Γ ηqq (cid:48) is couplings for η ¯ qq (cid:48) interactions ( η = h, H, A ), the explicit expressions of whichare given in the Appendix. Using these Wilson coefficients we obtain ratio between ∆ m B s – 13 –n our model and the SM prediction ∆ m SMB s , under large tan β and small α , such that R B s = ∆ m B s ∆ m SMB s (cid:39) g X ( V tb V ∗ ts ) m Z (cid:48) (8 . × − TeV − ) − + (cid:20) .
12 cos ( α − β ) tan β + 0 .
19 tan β (cid:18) (200 GeV) m H − (200 GeV) m A (cid:19)(cid:21) , (3.20)where the first and second terms in the right-hand side corresponds to contributions from Z (cid:48) and scalars, respectively [14, 62, 63]. The allowed range of R B s is estimated by [62, 63]0 . < R B s < . . (3.21)We find that R B s will be deviated from the allowed range by Z (cid:48) contribution when ∆ C µ (cid:39)− Z (cid:48) and scalar contribution is necessary to satisfythe experimental constraint . Here we derive allowed parameter region on { m H , m A − m H } plane satisfying B s – ¯ B s constraints when we fit C µ to explain b → s(cid:96) + (cid:96) − anomalies choosingtan β = 10 and cos( α − β ) ∼ B → K ( ∗ ) τ + τ − Here we discuss B → K ( ∗ ) τ + τ − process in our model. The branching ratios are given byWilson coefficient C associated with τ such that [68]10 × BR ( B → Kτ + τ − ) [15 , =(1 .
20 + 0 . C τ + 0 . C τ ) ) ± (0 .
12 + 0 . C τ ) , (3.22)10 × BR ( B → K ∗ τ + τ − ) [15 , =(0 .
98 + 0 . C τ + 0 . C τ ) ) ± (0 .
09 + 0 . C τ + 0 . C τ ) ) , (3.23)where the superscript indicates the q range for the dilepton invariant mass in unit of[GeV ]. For the b → sτ + τ − channel, we obtain ∆ C τ = − C µ from our charge assignments,and the BRs are slightly enhanced from the SM prediction by factor ∼ .
5. However currentupper bounds of the BRs are much larger than the prediction as BR ( B → Kτ + τ − ) < . × − [69]. Therefore it is difficult to test the enhancement effect. In this section we explore collider physics focusing on Z (cid:48) production at the LHC and esti-mate relic density of our DM candidate searching for parameter region providing observedvalue. Similar phenomena were also observed in the flavor gauge model where U (1) (cid:48) gauge interaction couplesonly to the right-handed top quark in the interaction basis in the context of the top forward-backwardasymmetry and the same sign top pair productions at hadron colliders [64–67]. In that model, cancellationbetween the amplitudes with t -channel exchanges of vector and (pseudo)scalar bosons occur in the samesign top pair production through u R u R → t R t R , which saves the U (1) flavor model from the stringentconstraints from the same sign top pair production at the Tevatron and the LHC. – 14 – an Β = H Α - Β L ~
400 500 600 700 80050100150200 m H @ GeV D m A - m H @ G e V D Figure 5 . The allowed region on { m H , m A − m H } plane satisfying B s – ¯ B s constraints with fitting C to explain b → s(cid:96) + (cid:96) − anomalies where the yellow(light yellow) region corresponds to that inFig. 1. Here we take tan β = 10 and cos( α − β ) ∼ Z (cid:48) production at the LHC Here we discuss Z (cid:48) production at the LHC 13 TeV where Z (cid:48) can be produced via interactionin Eq. (2.9), followed by decay modes of Z (cid:48) → µ + µ − and Z (cid:48) → τ + τ − [Drell-Yan (DY)productions]. In this model Z (cid:48) mainly decays into τ + τ − model with BR ( Z (cid:48) → τ + τ − ) ∼ . µ + µ − mode is suppressed by factor of 1 /
16. The production cross section isestimated by CalcHEP [70] using the CTEQ6 parton distribution functions (PDFs) [71].In Fig. 6, we show σ ( pp → Z (cid:48) ) BR ( Z (cid:48) → (cid:96) + (cid:96) − /τ + τ − ) ( (cid:96) = e, µ ) as a function of the Z (cid:48) massfor several values of g X . The cross sections are compared with constraints from LHC data;from Refs. [72] and [73] for (cid:96) + (cid:96) − and τ + τ − modes. We thus find that (cid:96) + (cid:96) − mode (mostly µ + µ − ) provides more strict bound although BR ( Z (cid:48) → µ + µ − ) : BR ( Z (cid:48) → τ + τ − ) = 1 : 16.Here we set masses of H , A and H ± as 400 GeV and apply tan β = 10 and cos( α − β ) = 0where the effects of the Z (cid:48) decays into scalar bosons are small. Also right-handed neutrinosand DM χ are taken to be heavier than m Z (cid:48) / Z (cid:48) does not decay into on-shell right-handed neutrinos and DM.Our Z (cid:48) boson also decays into neutrinos with BR value of BR ( Z (cid:48) → ν τ ¯ ν τ ) = 16 BR ( Z (cid:48) → ν µ ¯ ν µ ) (cid:39) .
25. Thus we can also test our model by pp → Z (cid:48) g → ν ¯ νg process at the LHC ex-periments searching for signal with mono-jet plus missing transverse momentum. The crosssection of pp → Z (cid:48) g → ν ¯ νg process is, for example, ∼ g X = 0 . m Z (cid:48) = 1500GeV estimated by CalcHEP with p T >
25 GeV cut. We thus need large integrated lumi-nosity to analyze the signal [74] and it will be tested in future LHC experiments.– 15 –
HC limit g X =
500 1000 1500 2000 2500 300010 - - m Z ' @ GeV D Σ × BR H Z ' ™ l + l - L @ pb D LHC limit g X =
500 1000 1500 2000 2500 300010 - m Z ' @ GeV D Σ × BR H Z ' ™ Τ + Τ - L @ pb D Figure 6 . Left(right) plot: σ ( pp → Z (cid:48) ) BR ( Z (cid:48) → (cid:96) + (cid:96) − ( τ + τ − )) with (cid:96) = e, µ for several values of g X compared with LHC limit; from Refs. [72] and [73] for (cid:96) + (cid:96) − and τ + τ − modes. We consider a Dirac fermion χ as our DM candidate, and the relic density is determined bythe DM annihilation process χ ¯ χ → Z (cid:48) → f SM ¯ f SM /HA/H + H − where f SM is a SM fermionand/or χ ¯ χ → Z (cid:48) Z (cid:48) depending on kinematic condition. Then we estimate relic density ofour DM using micrOMEGAs 4.3.5 [75] implementing relevant interactions. Fig. 7 showsthe relic density Ω h as a function of DM mass m X where we apply several values of g X and m Z (cid:48) = 1 . h value by horizontaldashed line [76]. We see that the relic density drops at around m Z (cid:48) ∼ m X due to resonantenhancement of the annihilation cross section.In addition, we scan parameters in the range of m X ∈ [200 , , m Z (cid:48) ∈ [500 , , g X ∈ [0 . , . , (4.1)with assuming that tan β = 10 and cos( α − β ) = 0 as reference values. We note that theeffects of scalar bosons are subdominant. The left panel of Fig. 8 shows the parameterregion which accommodates the observed relic density of DM, Ω h = 0 . ± . σ range of observed value by the Planck collaboration [76]. Moreover the rightpanel of the figure indicates the region in which both observed relic density and b → s(cid:96) + (cid:96) − anomalies are explained within 2 σ . Notice that the allowed region with m Z (cid:48) < m X ispartly excluded by or close to LHC constraint shown in Fig. 6 and will be explored infuture LHC experiments. In addition DM-nucleon scattering cross section by Z (cid:48) exchangeis suppressed by CKM factor and the allowed region is not constrained by the DM directdetection experiments.Before closing this section we discuss possibility of indirect detection of our DM. In thismodel DM pair annihilates mainly through χ ¯ χ → Z (cid:48) → τ + τ − and/or χ ¯ χ → Z (cid:48) Z (cid:48) → τ + τ − – 16 – h = m Z ' = g X =
500 1000 1500 200010 - m X W h Figure 7 . Relic density of DM as function of DM masses for different values of U (1) X gaugecouplings, g X = 0 . , . .
3. We have fixed m Z (cid:48) = 1 . Figure 8 . (Left): parameter region which accommodates the observed DM relic density. (Right):parameter region which explains both DM relic density and b → s(cid:96) + (cid:96) − anomalies. and gamma-ray search gives the strongest constraint on the annihilation cross sectionby Fermi-LAT observation [77, 78]. In our parameter region of m Z (cid:48) >
500 GeV, DMannihilation cross section explaining the relic density is well below the constraint for the τ + τ − dominant case [77, 78] unless there is large enhancement factor; constraint on crosssection for four τ mode would be similar. Thus our model is safe from indirect detectioncross section and will be tested with larger amount of data in future.– 17 – Summary and discussions
We have discussed a flavor model based on U (1) B − x µ L µ − x τ L τ ( ≡ U (1) X ) gauge symmetryin which two Higgs doublet fields are introduced to obtain the observed CKM matrix.Flavor changing Z (cid:48) interactions with the SM quarks are obtained after diagonalizing quarkmass matrix, and b → s(cid:96) + (cid:96) − anomalies can be explained due to lepton flavor non-universalcharge assignment when x µ is taken to be negative value. Then we have considered minimalset up explaining b → s(cid:96) + (cid:96) − anomalies and generated neutrino mass matrix where two SMsinglet scalar fields and Dirac fermionic DM candidate are introduced.We have computed the Z (cid:48) contribution to the Wilson coefficient C µ relevant for b → sµ + µ − , as wel as neutrino mass matrices, charged lepton flavor violations and the B s – ¯ B s mixing, including the relevant experimental constraints. We have found that cancellationbetween Z (cid:48) and scalar bosons contributions to B s – ¯ B s is required to satisfy experimentalconstraint, while explaining b → s(cid:96) + (cid:96) − anomalies. In addition, we have shown constraintsfrom lepton flavor violation process µ → eγ and future prospects for µ → e conversionmeasurements.Then collider physics regarding Z (cid:48) production at the LHC and relic density of DM areexplored. We have shown cross sections for the DY processes, pp → Z (cid:48) → µ + µ − ( τ + τ − ),where constraints on the { m Z (cid:48) , g X } parameter space dominantly come from the data of di-muon resonance search at the LHC. The relic density of DM further constrains { m Z (cid:48) , g X } parameter space since the relic density is determined by DM pair annihilation processvia Z (cid:48) interactions. The preferred parameter region can be further tested in future LHCexperiments and observations for flavor physics such as LFVs. Acknowledgments
The work of CY was supported by the National Research Foundation of Korea(NRF) grantfunded by the Korea government(MSIT), NRF-2017R1A2B4011946 and NRF-2017R1E1A1A01074699.
Appendix: Yukawa interactions
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