cpcChinese Physics CChin. Phys. C1674-1137Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltdcpc_42_12_12310610.1088/1674-1137/42/12/12310642/12/123106PaperParticles and fieldsA model with flavor-dependent gauged U(1)B−L1×U(1)B−L2−L3 symmetry*
Supported by National Center for Theoretical Sciences and MoST (MoST-104-2112-M-007-003-MY3 and MoST-107-2119-M-007-013-MY3)
We propose a new model with flavor-dependent gauged U(1)B−L1×U(1)B−L2−L3 symmetry in addition to the flavor-blind symmetry in the Standard Model. The model contains three right-handed neutrinos to cancel gauge anomalies and several Higgs bosons to construct the measured fermion masses. We show the generic features of the model and explore its phenomenology. In particular, we discuss the current bounds on the extra gauge bosons from the K and B meson mixings as well as the LEP and LHC data, and focus on their contributions to the lepton flavor violating processes of ℓi+1→ℓiγ(i=1,2).
flavor-dependent gauged Abelian symmetriesmodel buildingnumerical analysis of gauge boson masses and coupling12.60.Cn12.60.Fr
Article funded by SCOAP^{3}
arxivppt1803.05633Introduction
Two vector U(1) gauge bosons often appear in grand unified theories (GUTs) such as SO(10) gauged symmetry [1] when it spontaneously breaks down, when a flavor-blind gauged U(1)B−L can be naturally induced along with right-handed neutrinos. On the other hand, flavor-dependent U(1) gauged symmetries are one of the promising scenarios to explain several anomalies beyond the Standard Model (SM), such as semi-leptonic decays involving b→sℓℓ¯, the muon anomalous magnetic moment, and so on [2].
In this paper, we propose a new model which contains two extra flavor-dependent gauge symmetries: U(1)B−L1×U(1)B−L2−L3, with the subscript numbers representing family indices besides the flavor-blind SM one. This type of the extension of the SM is, of course, difficult to embed into a larger group such as GUTs. But, due to the flavor dependence, there exist flavor changing processes via vector gauge bosons, resulting in slightly different signatures from typical gauged symmetries such as the flavor-blind U(1)B−L models.
This paper is organized as follows. In Section 2, we first construct our model by showing its field contents and their charge assignments, and then give the concrete renormalizable Lagrangian with scalar and vector gauge boson sectors. After that, we discuss the phenomenology, including the interaction terms, the bounds from the K and B meson mixings, the LEP [3] and LHC [4] experiments, and lepton flavor violations (LFVs). In Section 3, we perform numerical analysis. In Section 4, we extend our model to explain several anomalies indicated by current experiments. Finally, we conclude in Section 5 with some discussion.
Model setup and phenomenology
First of all, we impose two additional U(1)B−L1×U(1)B−L2−L3 gauge symmetries by including three right-handed neutral fermions NR1,2,3, with the subscripts representing the family indices. The field contents of the fermions and scalar bosons are given in Tables 1 and 2 respectively. Then, the anomaly cancellations among U(1)B−L13, U(1)B−L1, U(1)B−L2−L33, U(1)B−L2−L3, U(1)B−L12U(1)Y, U(1)B−L1U(1)Y2, U(1)B−L2−L32U(1)Y, U(1)B−L2−L3U(1)Y2 are the same as the typical single flavor-independent gauged U(1)B−L symmetry, while those from U(1)B−L1×U(1)B−L2−L32 and U(1)B−L12×U(1)B−L2−L3 are automatically cancelled because the two additional charge assignments are orthogonal to each other. In Table 2, H_{1} is expected to be the SM Higgs, while H_{2} is another isospin doublet scalar boson, which plays a role in providing the mixings of the 1-2 and 1-3 components in the CKM matrix, as we will see below. Under these symmetries, the renormalizable Lagrangian for the quark and lepton sectors and scalar potential are given by:−=yuQ¯L1H∼1uR1+yuijQ¯LiH∼1uRj+yui1′Q¯LiH∼2uR1+ydQ¯L1H1dR1+ydijQ¯LiH1dRj+yd1j′Q¯L1H2dRj+yνL¯L1H∼1NR1+yνijL¯LiH∼1NRj+yℓL¯L1H1eR1+yℓijL¯LiH1eRj+12yNijφ1N¯RiCNRj+12yNi′φ2(N¯R1CNRi+N¯RiCNR1)+h.c.,V=μH122|H1|2+μH22|H2|2+μφ12|φ1|2+μφ22|φ2|2+λH1|H1|4+λH2|H2|4+λφ1|φ1|4+λφ2|φ2|4+λH1H2|H1|2|H2|2+λH1H2′|H1†H2|2+λH1φ1|H1|2|φ1|2+λH1φ2|H1|2|φ2|2+λH2φ1|H2|2|φ1|2+λH2φ2|H2|2|φ2|2,respectively, where H∼≡(iσ2)H*, with σ2 being the second Pauli matrix, and i runs over 2 to 3.
Field contents of fermions and their charge assignments under SU(3)C×SU(2)L×U(1)Y×U(1)B−L1×U(1)B−L2−L3, where the subscripts 1 and i=2,3 correspond to the family indices.
Fermions
QL1
QLi
uR1
uRi
dR1
dRi
LL1
LLi
eR1
eRi
NR1
NRi
SU(3)_{C}
3
3
3
3
3
3
1
1
1
1
1
1
SU(2)_{L}
2
2
1
1
1
1
2
2
1
1
1
1
U(1)_{Y}
16
16
23
23
−13
−13
−12
−12
−1
−1
0
0
U(1)B−L1
13
0
13
0
13
0
−1
0
−1
0
−1
0
U(1)B−L2−L3
0
13
0
13
0
13
0
−1
0
−1
0
−1
Field contents of scalar bosons and their charge assignments under SU(3)C×SU(2)L×U(1)Y×U(1)B−L1×U(1)B−L2−L3.
Bosons
H_{1}
H_{2}
φ_{1}
φ_{2}
SU(3)_{C}
1
1
1
1
SU(2)_{L}
2
2
1
1
U(1)_{Y}
12
12
0
0
U(1)B−L1
0
13
0
1
U(1)B−L2−L3
0
−13
2
1
Scalar sector:
The scalar fields are parameterized asHi=wi+vi+hi+izi2,φi=vi′+φRi+izφi2,(i=1,2),with all four CP-odd bosons z1,2,φ1,φ2 massless, in which three of them are absorbed by vector gauged bosons ZSM, Z′ and Z″, respectively, where ZSM≡(g12+g22)v/4 with v≡v12+v22≈246 GeV, and Z′(Z″) arises from U(1)B−L1(U(1)B−L2−L3).^{1)} The features of the singly charged bosons are the same as in the typical two-Higgs doublet model. Consequently, the mass-squared, mixing and eigenvalue-squared matrices are found to beMC2=λH1H2′2v22v1v2v1v2v12,OC=cβsβ−sβcβ,DC2=Diag0,λH1H2′v22,respectively, where the above massless eigenstate is absorbed by the SM gauge boson W±, and cβ(sβ)=cosβ(sinβ) with tanβ≡v1/v2. As for the CP-even sector in the basis of [h1,h2,φR1,φR2]t, we get a four-by-four mass matrix squared MR2, which can be diagonalized by the mixing matrix O_{R} as D[H1,H2,H3,H4]≡ORMR2ORT, leading to [h1,h2,φR1,φR2]t=ORT[H1,H2,H3,H4]t. Here, we identify H1≡hSM.
Fermion sector:
The SM Dirac fermions are diagonalized by bi-unitary mixing matrices as Du,d,e=(Uu,d,e)Lmu(Uu,d,e†)R, and the active neutrinos are derived by an unitary mixing matrix as Dν=Uν*mνUν†, while the observed mixing matrices can be defined by VCKM≡UuL†UdL, and VMSN≡Uν†UeL, respectively [5]. However, we impose UuL=1 for simplicity. Hence, we reduce the formula to VCKM≡UdL. In the lepton sector, we classify the case of VMSN≈Uν† or VMSN≈UeL below. Here, the neutrino mass matrix m_{ν} is induced via the canonical seesaw mechanism in Eq. (1).
Neutral gauge boson sector
ZSM−Z′−Z″ mixing: Since H_{2} and φ1,2 have nonzero U(1)B−L1 and U(1)B−L2−L3 charges, there are mixings among ZSM, Z′ and Z″. The resulting mass matrix in the basis of (ZSM,Z′,Z″) is given bymZSM,Z′,Z″2=g2v24−16g1′gv2216g2′gv22−16g1′gv2219g1′2(v22+9v2′2)−19g1′g2′(v22−9v2′2)16g2′gv22−19g1′g2′(v22−9v2′2)19g2′2[v22+9(4v1′2+v2′2)],where g2≡g12+g22, mZSM≡g12+g22v2≈91.18GeV, and g_{1}, g_{2}, g1′ and g2′ are the gauge couplings of U(1)Y, SU(2)L, U(1)B−L1 and U(1)B−L2−L3, respectively. Here, we can identify the mass of Z_{1} as the SM one, since we expect v2<<v1<v1,2′ in order to reproduce the SM fermion masses and the LEP measurement of mZ1∼mZSM. This approximation is in good agreement with the current experimental data, as the mass difference between mZSM and mZ1 should be less than O(10−3)GeV.
The other part can be reduced tomZ′,Z″2∼g1′2v2′2g1′g2′v2′2g1′g2′v2′2g2′2(4v1′2+v2′2),which is diagonalized by the two-by-two mixing matrix V_{G} as VGmZ′,Z″2VGT≡Diag(mZ1′2,mZ2′2), withmZ1′2=12g2′2(4v1′2+v2′2)+g1′2v2′2−g2′4(4v1′2+v2′2)2+g1′4v2′4+2g1′2g2′2v2′2(−4v1′2+v2′2),mZ2′2=12g2′2(4v1′2+v2′2)+g1′2v2′2+g2′4(4v1′2+v2′2)2+g1′4v2′4+2g1′2g2′2v2′2(−4v1′2+v2′2),VG=cθsθ−sθcθ,sθ=121+g2′2(4v1′2+v2′2)−g1′2v2′2mZ2′2−mZ1′2.
Note here that we have to satisfy the following condition:16g1′2g2′2v1′2v2′2≤[g1′2v2′2+g2′2(4v1′2+v2′2)]2,that arises from the need for the vector boson masses to be positive real.
Here, we evaluate the typical scale of v_{2} that should be suppressed by the deviation of mZ1 from mZSM at the next leading order, δmZ≡|mZ1−mZSM|, approximately given byδmZ2∼g2v2472|g1′1−X+g2′1+X|2mZSM2−mZ1′2+|g1′1+X−g2′1−X|2mZSM2−mZ2′2,X=g2′2(4v1′2+v2′2)−g1′2v2′2mZ2′2−mZ1′2,where δmZ should satisfy δmZ≲2.1×10−3GeV from the electroweak precision test. As a result, we find,e.g., v2≲19.5GeV for v1,2′∼105GeV and g1,2′∼10−3.
Interacting Lagrangian: The interactions in the kinetic term between the neutral vector bosons and quarks in terms of the mass eigenstates are given by:q=−13(g1′cθ+g2′sθ)u¯γμuZ1μ′+(−g1′sθ+g2′cθ)∑i=c,tu¯iγμuiZ2μ′−13d¯iγμ(g1′cθOdZ′+g2′sθOdZ″)ijdjZ1μ′+d¯iγμ(−g1′sθOdZ′+g2′cθOdZ″)ijdjZ2μ′,OdZ′=VCKMdiag(1,0,0)VCKM†≈0.95−0.220.013+0.0032i−0.220.0509−0.0030−0.00075i0.013−0.0032i−0.0030+0.00075i0.00019,OdZ″=VCKMdiag(0,1,1)VCKM†≈0.0510.22−0.00014i−0.0082−0.0033i0.22+0.00014i0.95−0.0030−0.00075i−0.0082+0.0033i−0.0030+0.00075i1.0,where we have used the central values for the CKM elements in VCKM [6]. The interactions between the neutral vector bosons and charged-leptons depend on the parameterizations of V_{MNS}, given by:VMNS≈Uν†:ℓ(1)=(g1′cθ+g2′sθ)e¯γμeZ1μ′+(−g1′sθ+g2′cθ)∑i=μ,τℓ¯iγμℓiZ2μ′,VMNS≈UeL:ℓ(2)=ℓ¯iγμ(g1′cθOℓZ′+g2′sθOℓZ″)ijℓjZ1μ′+ℓ¯iγμ(−g1′sθOℓZ′+g2′cθOℓZ″)ijℓjZ2μ′,with ℓi,j=(e,μ,τ), where OℓZ′,ℓZ″ are derived asOℓZ′=VMNSdiag(1,0,0)VMNS†≈0.69−0.31−0.060i0.33−0.068i−0.31+0.060i0.14−0.14+0.060i0.33+0.068i−0.14−0.060i0.17,andOℓZ″=VMNSdiag(0,1,1)VMNS†≈0.310.31+0.060i0.33+0.068i0.31−0.060i0.860.14−0.060i0.33−0.068i0.14+0.060i0.83,respectively, by taking the best fitted results in Ref. [6] for V_{MNS}.
Phenomenology
Since Z1,2′ interact with the SM fermions in a non-universal manner, as discussed before, the constraints are unlikely to be the same as those in the typical U(1)B−L models. Here, we will examine the bounds on the extra gauge bosons from the K and B meson mixings as well as the LEP data, and discuss the lepton flavor violating processes of ℓi+1→ℓiγ (i = 1 and 2).
1. M−M¯ meson mixings
The extra gauge bosons induce the neutral meson (M)-antimeson (M¯) mixings with M=(K0,Bd,Bs), such as K0−K¯0, Bd−B¯d, and Bs−B¯s, at the tree level. The formulas for the mass splittings are given by [7]ΔmM≈|(g1′cθOdZ′+g2′sθOdZ″)21|2mZ1′2+|(−g1′sθOdZ′g2′cθOdZ″)21|2mZ2′2×mMfM2512−14mMmq+mq′2,for M=(K0,Bd,Bs) with qq′=(ds,db,ds), which should be less than the experimental values of (3.48×10−4,3.33×10−2,1.17)×10−11GeV [6], where fM=(156,191,200)MeV and mM=(0.498,5.280,5.367)GeV.
2. Bounds on Z1,2′ from LEP and LHC
From Eqs. (18) and (19), we obtain the effective Lagrangians as:VMNS≈Uν†:eff(1)=12G12mZ1′2(e¯γμe)(e¯γμe)+G1(V1d)dd3mZ1′2(e¯γμe)(d¯γμd)+G123mZ1′2(e¯γμe)(u¯γμu),VMNS≈UeL:eff(2)=∑i=1,212(Veeℓ)i2mZi′2(e¯γμe)(e¯γμe)+∑ℓ′=μ,τ(Veeℓ)i(Vℓ′ℓ′ℓ)imZi′2(e¯γμe)(ℓ¯′γμℓ′)+∑q′=d,s,b(Veeℓ)i(Vq′q′d)i3mZi′2(e¯γμe)(q¯′γμq′)+(Veeℓ)iGi3mZi′2(e¯γμe)(u¯γμu),respectively, where (Vijd(ℓ))1≡(g1′cθOd(ℓ)Z′+g2′sθOd(ℓ)Z″), (Vijd(ℓ))2≡(−g1′sθOd(ℓ)Z′+g2′cθOd(ℓ)Z″), G1≡g1′cθ+g2′sθ, and G2≡−g1′sθ+g2′cθ. As a results, the bounds for Z1,2′ from the measurements of e+e−→ff¯ at LEP [3] and qq¯→ee¯(μμ¯) at LHC [4] are found to beVMNS≈Uν†:(20.6TeV)28π≲mZ1′2G12,(11.4TeV)212π≲mZ1′2G1(V1d)ddforLEP;(37TeV)212π≲mZ1′2G12+(V1d)ddG1,(30TeV)212π≲mZ1′2G22+(V2d)ddG2forLHC,andVMNS≈UeL:(20.6TeV)28π≲∑i=1,2mZi′2(Veeℓ)i2,(18.9TeV)24π≲∑i=1,2mZi′2(Veeℓ)i(Vμμℓ)i,(15.8TeV)24π≲∑i=1,2mZi′2(Veeℓ)i(Vττℓ)i,(11.4TeV)212π≲∑i=1,2mZi′2(Veeℓ)i(Vddd)i,(16.2TeV)212π≲∑i=1,2mZi′2(Veeℓ)iGiforLEP;(37TeV)212π≲∑i=1,2mZi′2(Veeℓ)i[(Vddd)i+Gi],(30TeV)212π≲∑i=1,2mZi′2(Vμμℓ)i[(Vddd)i+Gi]forLHC,where f=e,μ,τ,d and u. It is worth mentioning that these neutral gauge boson searches will be carried out by experiments such as the International Linear Collider (ILC) [8], and more stringent constraints should be obtained in the near future.
3. Lepton flavor violating processes
For VMNS≈Uν†, one does not need to consider the lepton flavor violations from the Z1,2′ mediations, because the charged leptons are diagonal from the beginning. On the other hand, if VMNS≈UeL, the lepton flavor violating processes due to Z1,2′ can be induced. In this case, we getBR(ℓb→ℓaγ)≈48π3αemCba(4π)4GF2×18π2∑k=e,μ,τ∑i=1,2(Vℓaℓkℓ)i(Vℓkℓbℓ†)iFIImℓk2mZi′22,FII(r)=∫012rx(1−x)2r(1−x)+x,where G_{F} and αem are the Fermi and fine structure constants, respectively, while Cμe≈1, Cτe≈0.1784 and Cτμ≈0.1736. The current experimental limits are given by [9, 10]:BR(μ→eγ)≲4.2×10−13,BR(τ→eγ)≲4.4×10−8,BR(τ→μγ)≲3.3×10−8.These constraints are imposed in the numerical analysis below.^{1)}
Numerical analysis
In our numerical analysis, we explore the allowed gauge parameters of g1,2′ and mZ1,2′ by taking sθ=0 and sθ=1/2. We scan the parameter regions as follows:v1,2′∈[103,106]GeV,g1,2′∈[10−5,1].
VMNS≈Uν†:
In Fig. 1, we show the allowed parameter points in the planes of g1′-g2′ and mZ1′−mZ2′, with the left-hand and right-hand plots showing sθ=0 and 1/2, respectively. The top left plot suggests a wide allowed range of values for g1,2′ for sθ=0, whereas g1′ and g2′ should be degenerate for sθ=1/2. The bottom left plot indicates that any values with mZ1′≤mZ2′ are permitted for sθ=0, whereas the allowed parameter spaces for both mZ1′ and mZ2′ should be narrow within 10GeV≤mZ1,2′≤106 GeV for sθ=1/2.
(color online) Allowed regions in the planes of g1′-g2′ and mZ1′-mZ2′, where the left-hand and right-hand plots represent sθ=0 and 1/2, respectively, with VMNS≈Uν†.
VMNS≈UeL:
In Fig. 2, similar to Fig. 1, we illustrate the corresponding results for the case of VMNS≈UeL by including the plots for BR(τ→μγ)−BR(μ→eγ) at the bottom. The generic features for the first two plots from the top left in Fig. 2 are similar to those in Fig. 1. While g2′ is restricted to be g2′≤0.2, the allowed regions for g1′-g2′ and mZ1′-mZ2′ are more degenerate than the case of VMNS≈Uν† for sθ=1/2. For the lepton flavor violating processes at the bottom in Fig. 2, we see that BR(μ→eγ) reaches the current experimental bound in Eq. (31), which is clearly testable in the near future for both cases of s_{θ}. However, BR(τ→μγ) is much lower than the limit in Eq. (31). Note here that BR(τ→eγ)≈BR(τ→μγ). We remark that the current bounds on the masses of the extra gauge bosons are around 3 TeV, from the LHC experiments [11],^{2)} consistent with all the cases in our analyses with g1′=g2′=gZ≈0.72.
(color online) Allowed regions in the planes of g1′-g2′, mZ1′-mZ2′, and BR(τ→μγ)−BR(μ→eγ), where the left-hand and right-hand plots represent sθ=0 and 1/2, respectively, with VMNS≈UeL.
Finally, we also mention that the muon anomalous magnetic moment cannot be explained in our present model due to the constraint of the trident production via Z′ [12]. Moreover, the new contributions to the semi-leptonic decays of b→sℓ+ℓ− from the Z1,2′ mediations are negligibly small, so that our model sheds no light on the recent anomalies in B→K(*)μ+μ−, unlike the models with an extra Z′ in the literature [2]. Thus, we minimally extend our model to explain these issues in the next section.
An extension
We now extend our model by introducing two extra vector-like fermions: QL/R′=(3,2,1/6,1/2,−2/3) and LL/R′=(1,2,−1/2,1/2,−2), along with a neutral inert complex scalar S=(1,1,0,−1/2,1) under SU(3)C×SU(2)L×U(1)Y×U(1)B−L1×U(1)B−L2−L3 [13], resulting in the following additional Lagrangian:=fiL¯LiLR′S+giQ¯LiQR′S+MQ′Q¯′Q′+ML′L¯′L′+h.c.,where i = 2,3. Here, we have assumed the mass eigenstates for the above down-quark and charged-lepton sectors in the SM, and f3<<f2. As a result, the b→see¯ excess is negligible, which is consistent with the current experimental data, while τ→μγ at one-loop level is also suppressed to avoid the current experimental bound. Note that S is a complex boson that is assured by the charge assignment under U(1)B−L1×U(1)B−L2−L3, and its mass is denoted by m_{S}.
Muon anomalous magnetic moment:
The muon anomalous magnetic moment is formulated by :Δaμ=|f2|28π2∫01dxx2(1−x)x(x−1)+rL′x+(1−x)rS,where rL′≡(ML′/mμ)2 and rS≡(mS/mμ)2. The experimental deviation from the SM at 3.3σ C.L. is given by [14]Δaμ=(26.1±8.0(16.0))×10−10.
B→K*μ¯μ anomaly: The effective Hamiltonian for the b→sμ+μ− transition is induced via the box diagram [15], given byeff(b→sμ+μ−)=(g2g3*)|f2|2(4π)2Fbox(mS,MQ′,ML′)×(s¯γρPLb)(μ¯γρμ−μ¯γργ5μ)+h.c.≡−CSMC9O9−C10O10+h.c.,whereFbox(mS,MQ′,ML′)≈12∫01dx1∫01−x1dx2x1x1mS2+x2MQ′2+(1−x1−x2)ML′2,CSM≡VtbVts*GFαem2π,with Vtb∼0.9991 and Vts∼−0.0403 being the CKM matrix elements [6]. Here, we take C9=−C10, which is one of the promising relations to explain the anomaly [16], and the experimental result is given by[−0.85,−0.50]at1σ,[−1.22,−0.18]at2σ,where the best fit value is −0.68.
Neutral meson mixing: The neutral meson mixing gives the bounds on g_{i} and MQ′ at low energy, where our valid process is the Bs−B¯s mixing in our case. Similar to the B→K*μμ¯ anomaly, the formula is derived by [7]:ΔmBs:(g3g2*)(g2g3*)Fbox(mS,MQ′,MQ′)≲1.17×10−11×24π2mBsfBs2GeV,where the above parameters are found to be fBs=0.200GeV [7], and mBs=5.367GeV [6].
Dark matter candidate: We suppose that S is a DM candidate. First, we assume that any annihilation modes coming from the Higgs potential are negligibly small. This is a reasonable assumption, because we can avoid the strong constraint coming from the spin independent scattering cross section reported by several direct DM detection experiments, such as LUX [17]. Second, we do not consider the modes through Z1,2′ coming from the kinetic term, since this is suppressed enough by the masses of mZ1,2′. We comment here that there are two resonant solutions around the points of mZ1′=2mS and mZ2′=2mS. Subsequently, the dominant contribution to the thermal relic density comes from f and g, and the cross section is approximately given by [18](σvrel)≈mS216π|f2|46(mS2+ML′2)2+∑i=2,3|gigj|2(mS2+MQ′2)2vrel2+(vrel4),in the limit of massless final-state leptons and mQi,j2/MQ′2<<1. Here, the approximate formula is obtained by expanding the cross section in powers of the relative velocity; vrel: σvrel≈aeff+beffvrel2, where aeff=0. The resulting relic density is found to beΩh2≈1.07×109xf23g*(xf)MPLbeff,where the present relic density is 0.1199 ± 0.0108 [19], g*(xf≈25)≈100 counts the degrees of freedom for relativistic particles, and MPL≈1.22×1019GeV is the Planck mass.
Numerical analysis:
We now perform a numerical analysis to satisfy the anomalies of the muon g−2, B→K*μ¯μ, the constraints of the correct relic density, and the neutral meson mixing, as discussed above. We randomly select the input parameters as follows:f2=[−1,4π],g2,3=±[0.01,0],mS=(10,1000)TeV,(MQ′,ML′)=(1.2ms,5000)GeV,where 1.2mS is used to avoid the coannihilation processes among Q′,L′ and S, for simplicity. We show the allowed regions in Fig. 3, where the left(right)-hand figure represents the f2(mS)−−C9 plane, and the blue(red) points satisfy the muon g−2 in the range of (26.1±8.0(16.0))×10−10 in Eq. (35). The yellow(green) region denotes the experimentally allowed region [-1.22(-0.85),-0.18(-0.50)] at 1(2)σ in Eq. (38), where the black horizontal line inside the green region corresponds to the best fit value (BF). The left-hand plot suggests that f_{2} is restricted to [0.5,4π] for both blue and red points. The right-hand plot implies that m_{S} is limited to [10,170(90)] GeV for red(blue) points.
(color online) Allowed regions, where the left(right)-hand figure represents the f2(mS)−−C9 plane, and the blue(red) points satisfy muon g−2 in the range of (26.1±8.0(16.0))×10−10. The yellow(green) region denotes the experimental allowed region [-1.22(-0.85),-0.18(-0.50)] at 1(2)σ, where the black horizontal line inside the green region shows the best fit (BF).
Discussion and conclusions
We have proposed a new model with two flavor-dependent gauge symmetries: U(1)B−L1 and U(1)B−L2−L3, in addition to the SM one, along with introducing three right-handed neutrinos to cancel gauge anomalies and several scalars to construct the measured fermion masses. We have examined the experimental bounds on the extra gauge bosons by considering the K and B meson mixings as well as the LEP and LHC experiments. The allowed parameter spaces for the masses and couplings of Z1,2′ have been given. Even though all the regions are within the current exclusion bounds (∼3 TeV) from the LHC [11], more stringent constraints or their discoveries will be found at ILC, with its sensitivity to the cut-off scale being around 50-100 TeV, which is stronger than the LEP constraints.
In addition, the possible effects on the flavor violating processes have been explored. Particularly, we have shown that the branching ratio of μ→eγ for the case of VMNS≈UeL can be large, which is testable by future experiments.
Finally, we have discussed the possibility of explaining the muon g−2, B→K(*)μ¯μ, and dark matter candidate, by introducing vector-like fermions Q′,L′ and an inert complex boson S with appropriate charge assignments under SU(3)C×SU(2)L×U(1)Y×U(1)B−L1×U(1)B−L2−L3. We have also shown the allowed regions to satisfy all the anomalies and constraints, and found 0.5≲f2≲4π for both blue and red points, and 10≲mS≲170(90)GeV for red(blue) points. It is worth mentioning that the Z boson decay modes of Z→fif¯j at one-loop level could restrict our parameter space, where f_{i} represent all the SM fermions. It is expected that the sensitivities of these modes will further increase at future experiments, such as the CEPC [20], by several orders of magnitude.
This research was supported by the Ministry of Science, ICT and Future Planning, Gyeongsangbuk-do and Pohang City (H.O.).
We remark that the dangerous physical Goldstone boson from H_{2} can be evaded by introducing an isospin singlet boson φ_{3} of (-1/3,1/3) under U(1)B−L1×U(1)B−L2−L3, resulting in additional terms (H1†H2)φ3 and φ1*φ2φ33/Λ that give the non-vanishing CP-odd mass. Here, Λ is the cut-off scale, expected to be (100) TeV at most. Then, the CP-odd Higgs mass with (100)GeV is found. Even though φ_{3} affects the vector gauge boson masses, we neglect the contribution hereafter, by assuming vφ3′<<vφ1,2′. Note here that φ_{3} does not contribute to the fermion masses.
One can consider the anomalous magnetic moment because of the evading of the stringent constraint of the trident production via the Z′ boson (flavor eigenstate) [12]. In our case, its value is of the order 10−14, which is much smaller than the experimental value.
The LHC bounds are typically stronger than the LEP ones in case of a simple gauged U(1)B−L model.
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