A "Vector-like chiral" fourth family to explain muon anomalies
Stuart Raby Andreas Trautner
hep-ph







 
 


 
 
 




 

 



 

 

 
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    ^1Department of Physics
        The Ohio State University
        191 W. Woodruff Ave, Columbus, OH 43210, USA^2Bethe Center for Theoretical Physics and
        Physikalisches Institut der Universität Bonn,
        Nussallee 12, 53115 Bonn, GermanyA “Vector-like chiral” fourth family to explain muon anomalies
            Andreas Trautner^2[atrautner@uni-bonn.de]
            December 30, 2023
        ==============================================================
        
        emptygobble
        
        
         0pt 5pt
        The Standard Model (SM) is amended by one generation of quarks and leptons which are vector-like (VL) under the SM gauge group but chiral with respect to a new 1_3-4 gauge symmetry.
        We show that this model can simultaneously explain the deviation of the muon g-2 as well as the observed anomalies in b→ sμ^+μ^- transitions
        without conflicting with the data on Higgs decays, lepton flavor violation, or B_s-B̅_s mixing.
        The model is string theory motivated and GUT compatible, i.e. UV complete,
        and fits the data predicting VL quarks, leptons and a massive Z' at the TeV scale,
        as well as τ→3μ and τ→μγ within reach of future experiments.
        The Higgs couplings to SM generations are automatically aligned in flavor space.
        
        
        
        arabic
        
        § INTRODUCTION
        
        The Standard Model (SM) is a highly successful theory in predicting and fitting many experimental measurements, with few exceptions.
        One of the discrepancies between the SM prediction and experimental measurement that has been known for a long time, is the muon anomalous magnetic moment.
        The discrepancy between the measured value and the SM prediction is <cit.>
            Δ a_μ
              = a_μ^exp-a_μ^SM
              = 288(63)(49)×10^-11 .
        
        More recently, there appeared deviations from the SM predictions in b→ sμ^+μ^- transitions related to
        tests of lepton flavor universality in the observables R(K) and R(K^*)<cit.>, semi-leptonic branching ratios
        <cit.>,
        and angular distributions <cit.>.
        
        Most interestingly, all of the more recent anomalies can simultaneously be explained 
        <cit.>
        by specific deviations from the SM in one or more of the Wilson coefficients C_9, C'_9, C_10 and/or C'_10 of the
        effective Hamiltonian <cit.> (see <cit.> for a possible role of additional tensor operators)
        
            ℋ_eff= -4 G_F/√(2) V_tbV_ts^* e^2/16π^2∑_j=9,10(C_j𝒪_j + C'_j𝒪'_j)+h.c. ,
        
        where
        
            𝒪_9    = (s̅γ_μ P_L b)(μ̅γ^μμ) ,   𝒪'_9    = (s̅γ_μ P_R b)(μ̅γ^μμ) ,   
            𝒪_10   = (s̅γ_μ P_L b)(μ̅γ^μγ_5  μ) ,    𝒪'_10   = (s̅γ_μ P_R b)(μ̅γ^μγ_5  μ) .
        
        A simple extension of the SM that can explain the discrepancy of the muon g-2 are VL leptons that couple exclusively to muons <cit.>.
        On the other hand, the anomalies in b→ sμ^+μ^- transitions can be explained by a new massive vector boson of a spontaneously broken U(1)_μ-τ gauge symmetry
        and the introduction of VL quarks <cit.> (see <cit.> for a generalization of the new gauge symmetry).
        Indeed, it has been shown that combining an additional Z', VL leptons, and VL quarks
        one can successfully address both the muon g-2 and the anomalous B physics observables simultaneously <cit.>.
        Typically these models predict significant deviations of the SM in h→μμ<cit.>, h→μτ<cit.>
        and have an upper bound on the Z' mass by keeping B_s-B̅_s oscillations close to their SM value <cit.>.
        
        In the present paper we suggest a holistic way of solving the discrepancies in (g-2)_μ and b→ sμ^+μ^-.
        We amend the SM by one complete family of fermions, i.e. a full spinor representation of SO(10), which is VL with respect to the SM but chiral with respect to a new
        spontaneously broken “U(1)_3-4” gauge symmetry. Under the new gauge group the third SM family and the left-handed part of the new “VL” family have
        charges +1 and -1, respectively, while all other fermions are neutral. Our model is motivated by heterotic string orbifold constructions <cit.>,
        which, in addition to the full MSSM spectrum, typically contain myriads of states which are VL with respect to the SM gauge group, but chiral under new U(1)' gauge symmetries.
        In addition, there are many SM singlet scalars that break the additional gauge symmetries, thus giving mass to the vector-like states and the extra gauge bosons.
        While in earlier constructions these extra states were lifted to the string scale, our model is a prototype of what can happen if at least one extra generation is kept light,
        i.e. at the TeV scale. Our analysis is not supersymmetric, but it could easily be extended to a supersymmetric model in which case gauge coupling
        unification is maintained.
        
        We find that the model can simultaneously fit the observed quark and lepton masses, as well as the (g-2)_μ and b→ sμ^+μ^- anomalies without violating
        bounds from electroweak precision observables, lepton flavor violating (LFV) decays or B_s-B̅_s mixing.
        Interestingly, the electroweak singlet Higgs boson couplings in our model are automatically aligned with the SM values to a very high degree.
        Contrary to <cit.> there is no upper bound on the Z' mass from the B_s-B̅_s mixing constraint,
        simply because the “VL” fermions and the Z' simultaneously obtain mass of the order of the U(1)_3-4 breaking scale.
        
        To substantiate our arguments we present two data points that can fit all measured observables
        while predicting others. The masses of new quarks and leptons, as well as of the new Z' are all at the TeV scale.
        The Z' in our example has very suppressed couplings to the first family, meaning that Z' production at the LHC is suppressed.
        B_s-B̅_s mixing is predicted to deviate from the SM at the level of a few percent. There are significant enhancements
        in BR(B_s→ K^(*)ττ̅) and BR(B_s→ϕττ̅), while R_K^(*)^νν̅ is suppressed.
        Furthermore, our best fit points predict BR(τ→μγ) and BR(τ→3μ) in reach of upcoming experiments.
        
        
        
        § MODEL
        
        The model under investigation is the SM with three right-handed neutrinos extended by a complete extra generation of left-chiral fields and a complete extra generation of right-chiral fields.
        Furthermore, we introduce a new “U(1)_3-4” gauge symmetry under which the third SM generation as well as the left-chiral part of the fourth generation of particles is charged.
        The U(1)_3-4 gauge symmetry is spontaneously broken by the vacuum expectation value (VEV) of the new scalar Φ.
        All fields and their corresponding quantum numbers are summarized in Table <ref>. The relevant part of the Lagrangian for this study is given by
        
            ℒ⊃ℒ_3,H+ℒ_VL,H+ℒ_3,Φ+ℒ_VL,Φ+ℒ_12,φ+ℒ_Maj ,
        
        with[
        In our notation P_L N_L=N_L is a two component Dirac spinor which can be written
        in terms of the two component Weyl spinor, η,  as N_L =(η,0)^T.
        Then N_L^𝒞=(0,σ_2η^*)^T.
        ]
            ℒ_3,H:=   
             -y_b q̅_L^3H d_R^3 - y_τ ℓ̅_L^3 H e_R^3 - y_ν ℓ̅_L^3 Hν_R^3 +h.c. ,
            ℒ_3,Φ:=   -λ_3 Φ(q̅_L^3 Q_R + 
                                     d̅_R^3 D_L +  ℓ̅_L^3 L_R + e̅_R^3 E_L )+h.c. ,  
            ℒ_VL,H:=   -λ_LR(
                                             Q̅_L H D_R + L̅_L H E_R + L̅_L H N_R)+h.c.
               -λ_RL(
                                             Q̅_R H D_L + L̅_R H E_L + L̅_R H N_L )+h.c. ,  
            ℒ_VL,Φ:=    -Φ^*( λ_Q Q̅_L Q_R 
             + λ_D D̅_R D_L + λ_L L̅_L L_R + λ_E E̅_R E_L + λ_N N̅_R N_L )+h.c. , 
            ℒ_12,φ:=   -λ_2 φ^a (q̅_L^a Q_R + 
                                     d̅_R^a D_L +  ℓ̅_L^a L_R + e̅_R^a E_L )+h.c. .  
            ℒ_Maj:=   -1/2M_L N_L^𝒞 N_L -1/2 M_R^ab(ν_R^a)^𝒞ν_R^b - (M_R N_R^𝒞ν_R^3+h.c.) .
        
        We take all couplings to be real and – in some GUT spirit – set many of them alike.
        Couplings to the up quark sector electroweak singlets (u^a_R, u^3_R, U_R, and U_L) are not displayed because they will not be constrained by our analysis. 
        It is summed over the repeated indices a=1,2 of the 2⊕1 flavor structure of the SM families which can, for example,
        originate from a 4 flavor symmetry <cit.>.
        The first and second families are distinguished by the direction of the 4
        breaking VEV ⟨φ_a⟩=δ_a2 v_φ.
        We assume this alignment to happen at a high-scale M (one should imagine M ∼ M_string or M ∼ M_GUT), and the corresponding effective operator coefficient, thus,
        should be imagined as v_φ≡⟨Φ̃⟩⟨φ_2⟩/M where Φ̃ is a SM and 1_3-4 neutral scalar that gets a VEV around the weak scale.
        This justifies our assumption here that the first family does not directly mix with the VL states. We will focus on the
        flavor structure of the second and third generations in this study,
        remarking that the first family can always be fit in.
        A more detailed analysis should include all three families and their flavor physics, but that is beyond the scope of the present paper.
        
        
        
          §.§.§ Charged Lepton and Down Quark Masses
        
        The charged lepton mass terms are given by
        
            e̅_L^A  ℳ^ℓ_AB e_R^B≡[   E̅_L;  E̅'_L; e̅_L^3; e̅_L^2 ]^T[   λ_RL v  λ_E v_Φ  λ_3 v_Φ  λ_2 v_φ;  λ_L v_Φ   λ_LR v        0        0; λ_3  v_Φ        0    y_τ v        0;  λ_2 v_φ        0        0    y_μ v;          ][  E'_R;   E_R; e_R^3; e_R^2 ] ,
        
        where A,B=1,..,4 and the scalar VEVs and couplings are all assumed to be real.
        Analogously, the down quark mass terms are given by
        
            d̅_L^A  ℳ^d_AB d_R^B  ,
        
        with
        
            d_L   := ( D_L,  D'_L,  d^3_L,  d^2_L ) 
            d_R   := ( D'_R,  D_R,  d^3_R,  d^2_R )  .
        
        The matrix ℳ^d has exactly the same structure as ℳ^ℓ with the replacements λ_E→λ_D, λ_L→λ_Q,
        y_τ→ y_b, and y_μ→ y_s.
        Let U^ℓ,d_L and U^ℓ,d_R be unitary matrices that diagonalize the respective mass matrix,
        
            (U^ℓ_L)^† ℳ^ℓ  U^ℓ_R   =(ℳ^ℓ)^diag≡diag( m_E, m_L, m_τ, m_μ) ,
            
              (U^d_L)^† ℳ^d   U^d_R   =(ℳ^d)^diag≡diag( m_D, m_Q, m_b, m_s) .
        
        The physical fields in the mass basis are then given by
        
            [ê_L,R]^A =[(U^ℓ_L,R)^†]^A B[e_L,R]^B  and  [d̂_L,R]^A =[(U^d_L,R)^†]^A B[d_L,R]^B .
        
        
          §.§.§ Neutrino Masses
        
        Defining the vectors
        
            ν_L:= ( N_L,  N'_L, ν^3_L, ν^2_L )   and  ν_R   := ( N'_R,  N_R, ν^3_R, ν^2_R )  ,
        
        the neutrino masses can be written as
        
            [   ν̅_L; ν̅^𝒞_R ]^T[   ℳ_L   ℳ_D; ℳ^T_D   ℳ_R ][ ν^𝒞_L;   ν_R ]≡N̅_L^α ℳ^ν_αβ N_R^β ,
        
        where α,β=1,..,8 and N_L^𝒞=N_R.
        The Dirac mass terms ℳ_D have the same structure as ℳ^ℓ with the replacements
        λ_E→λ_N, y_τ→ y_ν_1, and y_μ→ y_ν_2. The Majorana
        mass terms ℳ_L,R have non-zero elements 2[ℳ_L]_11= M_L,
        [ℳ_R]_23=[ℳ_R]_32=M_R and 2 [ℳ_R]_44= M_R^11
        with all other elements being zero. Assuming the hierarchy M∼ M_L,R≫ v_Φ,φ≫ v the neutrino mass matrix can be analytically diagonalized
        and we give details about that in app:NeutrinoMasses. The physical states are
        
            N̂^α_L =[(U^ν)^T]^αβN^β_L ,    N̂^α_R =[(U^ν)^†]^αβN^β_R ,
        
        with corresponding masses
        
            (U^ν)^T ℳ^ν  U^ν =(ℳ^ν)^diag≈diag( M, M, M/2, M/2, M_D, M_D, 0 , 0  ) ,
        
        up to corrections of the order v_(Φ,φ)^2/M.
        There are four sterile neutrinos with mass at the high scale.
        Furthermore, there are two light active neutrinos with mass of order M_W^2/M_GUT and one (mostly) Dirac neutrino with a TeV scale mass (cf. app:NeutrinoMasses)
        
            M_D=√((λ_L v_Φ)^2 + (λ_3 v_Φ)^2 + (λ_2 v_φ)^2) .
        
        Adding the first generation back in gives one additional high scale sterile neutrino and one additional light active neutrino.
        
        
        
          §.§.§ Z-Lepton Couplings
        
        The Z-lepton couplings in the mass basis are
        
            ℒ⊃ Z_μ(ê̅̂_L^A γ^μ[ĝ_L^Z]_A B ê_L^B + ê̅̂_R^A  γ^μ[ĝ_R^Z]_A B ê_R^B) ,
        
        with coupling matrices
        
            ĝ_L,R^Z = (U^ℓ_L,R)^† g_L,R^Z U^ℓ_L,R .
        
        The un-hatted coupling matrices are in the gauge basis and given by
        
            g^Z_L,R=g/c_W[1 g^Z,SM_L,R±diag(1/2,0,0,0)] ,
        
        where g^Z,SM_L=(-1/2+s_W^2), g^Z,SM_R=s_W^2, and we use the abbreviations s_W=sinθ_W, c_W=cosθ_W.
        Since these matrices are not proportional to the identity matrix, the Z-lepton couplings are not diagonal in the mass basis.
        Hence, this model has LFV Z boson decays, which are, however, only effective amongst the heavy VL quarks and leptons.
        
        
        
          §.§.§ W-Lepton Couplings
        
        The W-lepton couplings in the mass basis are
        
            ℒ⊃ W_μ^+(𝐍̂̅̂_L^αγ^μ[ĝ^W_L]_α Bê_L^B +
              𝐍̂̅̂^α_R γ^μ[ĝ^W_R]_α Bê_R^B) + h.c. ,
        
        where
        
            ĝ_L^W    = g/√(2)[(U^ν)^Tg^W_L U^ℓ_L]      and   ĝ_R^W    = g/√(2)[(U^ν)^† g^W_R U^ℓ_R] ,
        
        with the 8×4 coupling matrices of the gauge basis
        
            g^W_L=[ diag(0,1,1,1);         0_4×4 ]  and  
             g^W_R=[         0_4×4; diag(1,0,0,0) ] .
        
        
          §.§.§ Z' Couplings
        
        The Z' couplings to charged leptons in the mass basis are
        
            ℒ⊃ g' Z'_μ(ê̅̂_L^A γ^μ[ĝ^ℓ_L]_ABê_L^B +
              ê̅̂_R^A γ^μ[ĝ^ℓ_R]_ABê_R^B )  ,
        
        where
        
            ĝ^ℓ_L,R = (U^ℓ_L,R)^† g_L,R U^ℓ_L,R ,
        
        with the 1_3-4 charge matrices
        
            g_L=g_R=diag(0,-1,1,0) .
        
        The  Z' couplings here are not left-right symmetric, recall the charge assignment tab:SM4,
        and our skewed definition of the right-handed states in (<ref>) and (<ref>).
        The Z'-down quark couplings in the mass basis are completely analogously given by
        
            ℒ⊃ g' Z'_μ(d̂̅̂_L^A γ^μ[ĝ^d_L]_ABd̂_L^B +
              d̂̅̂_R^A γ^μ[ĝ^d_R]_ABd̂_R^B )  ,
        
        with
        
            ĝ^d_L,R = (U^d_L,R)^† g_L,R U^d_L,R .
        
        
        The Z' mediated flavor changing neutral currents (FCNC) between the SM 2↔ 3 generations are naturally suppressed because they only
        arise from the mixing with the heavy VL states.
        
        The Z' couplings to neutrinos in the mass basis can be written as
        
            ℒ⊃ g' Z'_μ(𝐍̂̅̂_L^αγ^μ[ĝ^n]_αβ𝐍̂_L^β)  ,
        
        with the coupling
        
            ĝ^n=(U^ν)^T g^n (U^ν)^* ,
        
        and the gauge basis charge matrix
        
            g^n=diag(0,-1,1,0,0,1,-1,0) .
        
        
          §.§.§ Higgs-Lepton Couplings
        
        The couplings between the physical Higgs boson, h, and the charged leptons in the mass basis are
        
            ℒ⊃ -1/√(2)  h ê̅̂_L^A  Ŷ_AB^ℓ ê_R^B + h.c. ,
        
        where
        
            Ŷ^ℓ = (U^ℓ_L)^†  Y^ℓ  U^ℓ_R  ,
        
        with the gauge basis couplings
        
            Y^ℓ =
              [ λ_RL    0    0    0;    0 λ_LR    0    0;    0    0  y_τ    0;    0    0    0  y_μ;      ] .
        
        A very interesting feature of this model is that the masses of the SM families are to a very high accuracy
        linear in the Higgs VEV. Thus, the Higgs couplings in the mass basis, Ŷ^ℓ, are to a high precision
        diagonal in the lower 2×2 block.
        Hence, the Higgs couplings to the SM states are very much SM-like and there are no significant flavor violating Higgs couplings among the SM states.
        We give an analytic proof of this feature in app:HiggsCouplings.
        Flavor off-diagonal couplings of the VL states (also to the SM states) can be sizable.
        
        
        
        § OBSERVABLES
        
        
          §.§.§ Lepton Non-Universality
        
        
        Generally, our model gives rise to lepton non-universality in the operators 𝒪^(')_i=9,10 by tree-level Z' exchange.
        The corresponding effective contributions to the Wilson coefficients are
        
            C^('),NP_i=-√(2)/4 G_F1/V_tb V_ts^*16π^2/e^21/2 v_Φ^2 g^(')_eff,i ,
        
        with the couplings
        
            g_eff,9   =[ĝ^d_L]_43[ĝ^ℓ_R + ĝ^ℓ_L]_44 ,    g_eff,10   =[ĝ^d_L]_43[ĝ^ℓ_R - ĝ^ℓ_L]_44 ,    
            
             g^'_eff,9   =[ĝ^d_R]_43[ĝ^ℓ_R + ĝ^ℓ_L]_44 ,    g^'_eff,10   =[ĝ^d_R]_43[ĝ^ℓ_R - ĝ^ℓ_L]_44 .
        
        These couplings are expressible solely through mixing matrix-elements, for example
        
            g_eff,9=   ([U^d †_L]_43[U^d_L]_33-[U^d †_L]_42[U^d_L]_23)×
               ([U^ℓ †_L]_43[U^ℓ_L]_34-[U^ℓ †_L]_42[U^ℓ_L]_24+
             [U^ℓ †_R]_43[U^ℓ_R]_34-[U^ℓ †_R]_42[U^ℓ_R]_24) .
        
        While we focus on the Z' coupling to muons in order to explain the observed anomalies, our model also modifies the
        effective Wilson coefficients C^('),ττ_9,10 and C^('),νν_9,10 leading to lepton non-universality
        also in BR(B_s→ K^(*)ττ̅) and  BR(B_s→ϕττ̅).
        Quantitative results for these observables have been obtained using the formulas given in <cit.> from
        where we also adopt the SM prediction (cf. also <cit.>).
        The NP contributions to the Wilson coefficients C^('),νν_9,10 affect the SM prediction for BR(B_s→ K^(*)νν̅)
        and we have followed <cit.> to quantify these effects in our model.
        
        
        
          §.§.§ Muon Anomalous Magnetic Moment
        
        The W, Z and h contributions to the anomalous magnetic moment of the muon are very close to their SM values and we do not detail them here.
        On the other hand, the Z' contribution can be sizeable, despite its TeV-scale mass.
        This is due to off-diagonal muon-Z' couplings to VL leptons, allowing them to significantly
        contribute to the g-2 loop.
        Since the VL leptons and Z' masses are of the same scale, the leading order contribution has one power of m_μ/M_Z' less than
        the naive flavor diagonal Z' contribution (cf. e.g. the discussion in <cit.>).
        Parametrically, the dominant modification of (g-2)_μ in our model is of the size
        
            δ a^Z'_μ≃m_μ/16π^2 v_Φ∑_a∈VL[ĝ^ℓ_L]_4a[ĝ^ℓ_R]_4a ,
        
        where the sum goes over the VL leptons.
        Naively this points to a scale v_Φ∼10^2 TeV, but the FC couplings to the VL leptons can easily be 𝒪(0.1).
        In addition, the contributions of individual VL leptons can partly cancel against one another and we will see
        this effect to be at work in our numerical analysis below. We give detailed formulas for δ a^Z'_μ in app:aMuZp.
        
        
        
          §.§.§ B_s-B̅_s Mixing
        
        
        There is a new tree-level contribution to B_s-B̅_s mixing due to Z' exchange.
        We adopt the results and numerical factors from <cit.> and estimate the relative change of the mixing matrix element
        
            δ M_12≃(g^2/16π^2 M_W^2(V_tsV_tb)^2 2.3)^-11/2 v_Φ^2(|[ĝ^d_L]_34|^2+|[ĝ^d_R]_34|^2+9.7 Re([ĝ^d_L]_34[ĝ^d,*_R]_34)) .
        
        The most recently updated theoretical uncertainty shows that a deviation from the SM of δ M_12≲6% can currently not be excluded <cit.>.
        This gives an important constraint on the down sector flavor changing Z' couplings.
        In our model the b-s coupling is suppressed
        because it only arises from the mixing with the heavy VL states such that 
        the Z' can be kept at the TeV scale consistent with the bound derived in <cit.>.
        
        
        
          §.§.§ Lepton Flavor Violating τ Decays
        
        Tree-level Z' exchange also induces the decay τ→3μ. We follow Ref. <cit.>
        to estimate
        
            BR(τ→3μ)≈   1/Γ_τm_τ^5/1536 π^31/4 v_Φ^4×
               (2|[ĝ^ℓ_L]_43[ĝ^ℓ_L]_44|^2 + 2|[ĝ^ℓ_R]_43[ĝ^ℓ_R]_44|^2 +
             |[ĝ^ℓ_L]_43[ĝ^ℓ_R]_44|^2 + |[ĝ^ℓ_R]_43[ĝ^ℓ_L]_44|^2) .
        
        There is also a new contribution to τ→μγ due to Z' and the new VL leptons in the loop, which is enhanced by a power of the VL mass.
        Using the results of  <cit.> (cf. also <cit.>) we estimate the leading order contribution to be
        
            BR(τ→μγ)≃1/Γ_τα m_τ^3/1024 π^41/4 v_Φ^2{|∑_a∈VL[ĝ^ℓ_L]_4a[ĝ^ℓ_R]_a3|^2+|∑_a∈VL[ĝ^ℓ_R]_4a[ĝ^ℓ_L]_a3|^2} ,
        
        where the sum is over internal VL leptons. For the numerical analysis we have used a more detailed result which we present in app:taumugamma.
        
        
        
          §.§.§ Other Observables
        
        Flavor violating couplings of the Z or the SM scalar h to the SM families are generally suppressed far
        below their experimental thresholds. In contrast, flavor changing couplings to the heavy VL
        leptons can be large.
        Since our Z' is heavy, neutrino trident production <cit.>
        does not give any important constraints. Lepton unitarity bounds (cf. e.g. <cit.>) are easily fulfilled.
        In addition, there are no constraints coming from the branching ratios for h →γγ or h → g g via loop diagrams, since these contributions are suppressed
        by factors of  (v/M_VL)^2 (see e.g. <cit.>).
        
        
        
        
        
        
        § ANALYSIS
        
        
          §.§.§ Strategy
        
        
        This model can explain the anomalies in the muon g-2 and b→ sμ^+μ^- transitions
        without obviously conflicting with other experimental data. To demonstrate this, we have constructed a χ^2-function including the appropriate errors
        to simultaneously fit the anomaly in (g-2)_μ with the value given in (<ref>), and to reproduce
        two of the best-fit values of <cit.> 
        (cf. also <cit.>)
        for a consistent explanation of b→ sμ^+μ^- anomalies:
        
            I.    :    C^NP_9   ≈-1.21±0.2 ,    C'_9   ≈ C^NP_10≈ C'_10≈0 ,   
            or  II.    :    C^NP_9   ≈-1.25±0.2 ,    C'_9   ≈0.59±0.2 ,     C^NP_10   ≈ C'_10≈0 .
        
        Furthermore we fit the masses m_d, m_s, m_μ, and m_τ at the weak scale <cit.> while requiring δ M_12≲15%
        to be consistent with the data and theoretical error of B_s-B̅_s mixing.
        All other observables are not constrained in the fit, i.e. they arise as predictions of our best fit points.
        However, there are currently more parameters than observables and so it is not excluded that there are more points that fit the data well
        with different predictions. The first family could always be included in the analysis in a straightforward way without affecting our conclusions.[
        Including the first family in the most straightforward way the Yukawa couplings and hence resulting mass matrices are extended to (and similarly for neutrinos and down quarks) 
        
            ℳ^ℓ   =
             [   λ_RL v  λ_E v_Φ  λ_3 v_Φ  λ_2 v_φ        0;  λ_L v_Φ   λ_LR v        0        0        0; λ_3  v_Φ        0    y_τ v        0        0;  λ_2 v_φ        0        0   y_22 v   y_21 v;        0        0        0   y_12 v   y_11 v;          ] .
        ]
        
          §.§.§ Results
        
        We have found two points which give a very good fit for the cases (I.) and (II.), they are listed in tab:bestfit (and in tab:bestfitII in tab:bestfitII).
        We cannot find a good fit for C^NP_9≃-C^NP_10.
        
        
        
        The predictions of the best fit points are listed in tab:results together with current experimental bounds.
        Effects on other observables such as h→μτ, h→γγ,
        h→ gg, neutrino trident production or PMNS unitarity violation have also been considered, but they are robustly suppressed
        in this model and so we do not discuss them in detail. While Z→μτ is practically absent at tree level,
        the dominant contribution arises from a one-loop diagram involving
        Z' and VL leptons in the loop. A rough estimate of this shows that BR(Z→μτ)
        nonetheless comes out well below the current bound of 1.2×10^-5<cit.>.
        
        
        
        The loop contributions to (g - 2)_μ from Z, W, and h are very close to their SM values while the FC Z' exchange with VL leptons in the loop completely accounts for the anomaly.
        The Z' contribution to B_s-B̅_s is mixing suppressed while larger flavor diagonal couplings to μμ̅ can explain a sizable C^NP_9.
        The fit prefers a corner of the parameter space where λ_E∼λ_L which makes the
        Z' couplings to leptons approximately left-right (anti-)symmetric (the couplings are LR symmetric or LR anti-symmetric depending on the specific coupling and 
        we display the coupling matrices for one of the best fit points in app:bestfit).
        The approximately LR symmetric couplings to the mu and tau leptons leads to an enhanced contribution to C^NP_9 (and possibly also C^'_9)
        and a dramatic cancellation in the axial vector couplings to leptons C^(')_10.
        By contrast, the approximately equal but opposite LR (anti-)symmetric contributions of individual 
        VL leptons to the muon g-2 and τ→μγ cancel only to an order of magnitude.
        There are no cancellations in τ→3μ where the couplings, in fact, add up constructively.
        Nevertheless, the tree level process τ→3μ is suppressed by the naturally small flavor off-diagonal
        couplings of the Z' to the SM fermion generations. The best-fit predictions for BR(τ→μγ) and BR(τ→3μ) fall
        close to regions that can be probed by future experiments <cit.>, cf. fig:a.
        A positive value of C^'_9 allows for a negative shift of δ M_12, thereby cushioning a
        1.8σ tension between SM and experiment <cit.>. 14pt
        This happens to be the case for our best fit point (II.).
        
        
        
        
        The NP contributions to the Wilson coefficients C^('),ττ_9,10 qualitatively follows the patterns in (<ref>) but with reversed signs.
        BR(B_s→ Kττ̅) and  BR(B_s→ϕττ̅) are significantly increased compared to the SM, cf. fig:b.
        By contrast R_K^(*)^νν̅ is suppressed compared to the SM.
        The suppression of R_K^(*)^νν̅ together with a Z' explanation of b→ sμ^+μ^- is not in contradiction with the results of <cit.>.
        In our model, we find that the enhancement of R_K^(*)^νν̅ from the second family <cit.> is counter-acted by a vast suppression of the third family contributions.
        
        We do not find any observable that causes problems. We stress again that the Z'
        does, by construction, only very feebly couple to the SM first generation.
        The gauge coupling g' is only very mildly constrained by the fit with a
        χ^2<3 region of g'∈-[5.33(5.08),0.32(0.39)] for point I.(II.).
        Bounds for family specific Z' bosons can be obtained from the LHC √(s)=13 TeV searches <cit.>
        and range from M_Z'≳1.3-2.0 TeV<cit.>, while some models can survive with M_Z' as low as
        ≳500 GeV<cit.>. Deriving a robust limit for our model requires to specify the details of the first family couplings which is beyond the scope of this 
        work. Nevertheless, our Z' mass comes out right in the ballpark of current limits and so it can be searched for at the LHC in dimuon final states. 
        If not with the full run 2 data, our best fit parameter regions will conclusively be tested by a high-luminosity run of the LHC.
        
        We have scattered one thousand points randomly within the χ^2<1(3) regions around the best fit values.
        There are strong correlations among the masses of the pair of VL leptons, and between the VL lepton masses
        and the prediction for BR(τ→μγ)fig:c.
        Also BR(B_s→ Kττ̅) and  BR(B_s→ϕττ̅)
        are strongly correlated fig:b. All other predicted observables are largely uncorrelated, see e.g. fig:d.
        
        Given the parameters of our best fit points we can also evaluate the effective Higgs couplings of
        the SM quarks and leptons (see app:HiggsCouplings for details).
        The resulting Higgs couplings are diagonal in the mass basis and,
        moreover, proportional to the masses, just like in the Standard Model.
        
        For model building it is interesting to compare the (gauge basis) Higgs Yukawa couplings
        to the mixing induced contributions to masses of the SM fermion generations.
        The muon and strange quark obtain mass predominantly due to mixing effects,
        and this is also reflected by their large off–diagonal couplings to the VL states.
        The tau lepton and b quark, by contrast, obtain almost all of their mass from the direct Higgs Yukawa
        coupling and mixing effects are miniscule.
        
        
        
        
        
        § CONCLUSION
        
        
        We have studied the Standard Model extended by one complete family of “VL” fermions, including right-handed neutrinos,
        which are vector-like with respect to the Standard Model gauge group.
        In addition we have introduced a new “1_3-4” gauge symmetry under which the third SM family of quarks and leptons have
        1_3-4 charge 1 while the left-chiral part of the new 4th family has charge -1.
        Hence, the model is free of gauge anomalies.
        In our analysis, we have only considered the mixing of the VL states with the second and third SM families,
        while the absence of mixing to the first family is motivated by a high scale flavor symmetry.
        The first family could always be included in a straightforward way without affecting our conclusions, and this should be done in the future
        in order to study the collider phenomenology of this model in more detail.
        
        We have shown that this model can fit the anomalies in the muon g-2 and b→ sμ^+μ^- transitions
        without conflicting with other experimental data.
        The best fit points predict (cf. tab:results) new quarks, leptons and a family specific Z' at the TeV scale, as well as testable effects
        in the lepton flavor violating processes τ→μγ and τ→3μ.
        Furthermore, we find a significant enhancement of BR(B_s→ K^(*)ττ̅) and BR(B_s→ϕττ̅), while R_K^(*)^νν̅
        is suppressed relative to the SM.
        Effects on other observables such as Z→μτ, h→μτ, h→γγ, h→ gg, neutrino trident production,
        or PMNS unitarity violation are all robustly suppressed.
        The Higgs couplings in our model are to a very high degree SM-like and we have given an analytic proof of that.
        
        
        
        
        
        § NEUTRINO SEE-SAW
        
        The neutrino mass matrix is described in (<ref>).
        In the limit M∼ M_L,R≫ v_Φ,φ≫ v the eigenvectors of ℳ^ν are to a very good approximation given by
        
            N_1,2=   (0,0,0,0,0,1/√(2),±1/√(2),0) , 
            
             N_3,4=   (1/√(2),0,0,0,0,0,0,±1/√(2)) , 
            
             N_5,6=   1/√(2)(0,λ_L v_Φ/M_D,λ_3 v_Φ/M_D,λ_2 v_φ/M_D,±1,0,0,0) , 
            
             N_7=   (0,λ_3/√(λ_3^2+λ_L^2),-λ_L/√(λ_3^2+λ_L^2),0,0,0,0,0) , 
            
             N_8=   1/𝒩_8(0,1,-λ_3/λ_L,-v_Φ/v_φ(λ_3^2+λ_L^2)/λ_2 λ_L,0,0,0,0) .
        
        Here we have used
        
            M_D:=   √((λ_L v_Φ)^2 + (λ_3 v_Φ)^2 + (λ_2 v_φ)^2) , 
            𝒩_8:=   √(1+(λ_3/λ_L)^2+[v_Φ/v_φ(λ_3^2+λ_L^2)/λ_2 λ_L]^2) .
        
        It is then straightforward to construct the diagonalization matrix U^ν from the eigenvectors.
        
        
        
        § HIGGS COUPLING DIAGONALIZATION
        
        We wish to show that the Higgs couplings (<ref>) are diagonal in the mass basis.
        We will focus on the charged leptons here, but the whole analysis of this section fully applies also to the d-type quarks by formally replacing
        λ_E→λ_D, λ_L→λ_Q, y_τ→ y_b, and y_μ→ y_s.
        Phenomenologically we are led to the relations v≪ v_Φ∼ v_φ and λ_LR,y_τ,y_μ≪1.
        Therefore, we will treat the lower 3×3 block of ℳ^ℓ in (<ref>) as perturbation.
        The leading order mass matrix, hence, is given by
        
            ℳ^ℓ≈ℳ^ℓ≡ v_Φ[ λ_RL ξ    λ_E    λ_3    λ_2;    λ_L      0      0      0;    λ_3      0      0      0;    λ_2      0      0      0;        ] ,
        
        where ξ:=v/v_Φ. The left- and right-singular vectors of this matrix are given by
        
            u^ℓ_L,1=   1/N^1/2_L,1(-u^L_-,λ_L/λ_2,λ_3/λ_2,1) ,    u^ℓ_R,1=   1/N^1/2_R,1(u^R_-,-λ_E/λ_2,-λ_3/λ_2,-1) ,   
            
            u^ℓ_L,2=   1/N^1/2_L,2(u^L_+,-λ_L/λ_2,-λ_3/λ_2,-1) ,    u^ℓ_R,2=   1/N^1/2_R,2(u^R_+,-λ_E/λ_2,-λ_3/λ_2,-1) ,   
            
            u^ℓ_L,3=   1/N^1/2_L,3(0,-λ_3/λ_L,1,0) ,    u^ℓ_R,3=   1/N^1/2_R,3(0,-λ_3/λ_E,1,0) ,   
            
            u^ℓ_L,4=   1/N^1/2_L,4(0,-λ_2/λ_L,0,1) ,    u^ℓ_R,4=   1/N^1/2_R,4(0,-λ_2/λ_E,0,1) ,
        
        where
        
            1.15u^L(R)_∓:=(-)(λ_L^2-λ_E^2)-ξ^2λ_RL^2∓√((λ_E^2-λ_L^2)^2+4 ξ^2 λ_2^2 λ_RL^2 + 4 ξ^2 λ_3^2 λ_RL^2 + 2ξ^2λ_Eλ_RL^2+2 ξ^2λ_L^2 λ_RL^2+ξ^4 λ_RL^4)/2 ξ λ_2 λ_RL ,
        
        and the normalization factors N_L(R),i are defined by the requirement |u^ℓ_L(R),i|^2=1.
        The matrices U^ℓ_L,R have these vectors as columns.
        The four corresponding singular values are given by
        
            (U^ℓ_L)^† ℳ^ℓ U^ℓ_R=diag( m_E, m_L,0, 0) ,
        
        where one can obtain analytic expressions for m_E,L but we do not need them here and so they are not displayed.
        Numerically, U^ℓ_L,R≈ U^ℓ_L,R and m_E≈m_E and m_L≈m_L hold at the percent level.
        Now regard the perturbation
        
            ℳ^ℓ=ℳ^ℓ+ξ diag( 0, λ_LR, y_τ, y_μ) .
        
        To leading order in perturbation theory the eigenvalues of the perturbed matrix are obtained by diagonalizing it with the
        zeroth-order left- and right-singular matrices.
        Following this procedure we find the small singular values of ℳ^ℓ to be
        
            m_τ   =v/N_3|λ_τ+λ_3^2 λ_LR/λ_E λ_L| ,      and   m_μ   =v/N_2|λ_μ+λ_2^2 λ_LR/λ_E λ_L| ,
        
        with
        
            N_2(3):=√([1+(λ_2(3)/λ_E)^2][1+(λ_2(3)/λ_L)^2]) .
        
        These values agree with the numerical values to 𝒪(1%). There are off-diagonal residuals which are given by
        
            [(U^ℓ_L)^† ℳ^ℓ U^ℓ_R]_34 =    δ m_τμ ≡ v λ_2 λ_3 λ_LR/√((λ_2^2+λ^2_E)(λ_3^2+λ^2_L)) , 
            [(U^ℓ_L)^† ℳ^ℓ U^ℓ_R]_43 =    δ m_μτ ≡ v λ_2 λ_3 λ_LR/√((λ_3^2+λ^2_E)(λ_2^2+λ^2_L)) .
        
        They are numerically subdominant compared to the diagonal entries.
        
        The crucial point is that the Higgs couplings in the gauge basis (<ref>)
        are, at least with regard to the lower 2×2 block, exactly of the same form as the perturbations to the mass matrix.
        Approximately diagonalizing the mass matrices with U^ℓ_L,R, hence, results in the Higgs couplings
        
            Y^ℓ_light = 1/v[    m_τ δ m_τμ; δ m_μτ    m_μ ] ,
        
        which are directly proportional to the mass matrices.
        There are slight deviations to the proportionality arising at higher order, but they are numerically not relevant.
        
        Evaluating the effective Higgs couplings in the mass basis numerically for our best-fit point (I.) (cf. tab:bestfit) one finds
        
            Ŷ^ℓ_light   = GeV/v[       1.74618 3.46843×10^-7; 3.47055×10^-7      0.102717;               ] .   
            Ŷ^d_light   = GeV/v[       2.85391 7.71892×10^-7; 1.41639×10^-6     0.0543716;               ] .
        
        Clearly the off-diagonal corrections are negligible. Thus, the Higgs coupling to quarks and leptons
        is diagonal in the mass basis and, moreover, proportional to the masses, just like in the Standard Model.
        
        
        
          §.§.§ Integrating out the VL Fermions
        
        The fact that the Higgs couplings are very much SM like also holds for the effective mass matrix of the light (SM) fermions
        which is obtained after integrating out the heavy VL fermions, and we wish to show this analytically.
        The 4 × 4 mass matrix (<ref>) can be rotated to a basis where there is a heavy 2 × 2 block.
        This is given by the transformation
        
            M^ℓ = (X_L)^† ℳ^ℓ  X_R ,
        
        with
        
            X_L = [                         1                         0                       - 0                         0;                         0              N_2L λ_L v_Φ             -N_3L λ_3 v_Φ               N_4Lλ_L v_Φ;                         0              N_2L λ_3 v_Φ             -N_3L λ_L v_Φ               N_4Lλ_3 v_Φ;                         0              N_2L λ_2 v_φ                       - 0 - N_4L (λ_L^2 + λ_3^2)v_Φ ] ,
        
        and
        
            X_R = [                         1                         0                         0                         0;                         0              N_2R λ_E v_Φ             -N_2R λ_3 v_Φ              N_2R λ_2 v_φ;                         0              N_3R λ_3 v_Φ            - N_3R λ_E v_Φ                         0;                         0              N_4R λ_E v_Φ             -N_4R λ_3 v_Φ -N_4R  (λ_E^2 + λ_3^2)v_Φ ] ,
        
        where
        
            N_2L(R) :=    [ (λ_L(E)^2 +  λ_3^2 ) v_Φ^2 +  λ_2^2  v_φ^2]^-1/2 ,   
            
             N_3L(R) :=    [ (λ_L(E)^2 +  λ_3^2 ) v_Φ^2 ]^-1/2 ,   
            
             N_4L(R) :=    [ (λ_L(E)^2 + λ_3^2) λ_2^2v_φ^2 + (λ_L(E)^2 + λ_3^2)^2v_Φ^2]^-1/2 .
        
        In terms of 2 × 2 blocks M^ℓ is then given by
          
            M^ℓ =
              [ M_heavy     V_R;     V_L M_light ] .
        
        Upon integrating out the heavy states perturbatively we obtain the effective 2×2 mass matrix for the light states given by
        
            M^eff_light = M_light - V_L   M_heavy^-1 V_R .
        
        In particular one finds that the effective light masses are to very high accuracy linear in the Higgs VEV.
        There are higher order corrections in v but they are numerically irrelevant.
        Thus when one diagonalizes the fermion mass matrix one simultaneously diagonalizes the coupling of the Higgs field to fermions.
        Hence there are no significant flavor violating Higgs couplings.
        
        
        
        § DETAILS OF Δ A^Z'_Μ
        
        The leading order contribution to the muon g-2 arising from the Z' coupling to leptons as in (<ref>)
        is given by (see e.g. <cit.>)
        
            δ a^Z'_μ=-g^' 2 m_μ^2/8π^2 M_Z'^2∑_a[ ( |[ĝ^ℓ_L]_4a|^2+|[ĝ^ℓ_R]_4a|^2)F(x_a) +
                          Re([ĝ^ℓ_L]_4a[ĝ^ℓ,*_R]_4a) m_a/m_μ G(x_a)] ,
        
        where a runs over all leptons with mass m_a in the loop, x_a:=(m_a/M_Z')^2, and the loop functions
        
            F(x)   :=(5 x^4 - 14 x^3 + 39 x^2 - 38 x - 18 x^2 ln x + 8)/[12 (1 - x)^4] , 
            
             G(x)   :=(x^3 + 3 x - 6 x ln x - 4)/[2 (1 - x)^3] .
        
        
        § DETAILS OF Τ→ΜΓ
        
        The leading order contribution to τ→μγ arises from the flavor off-diagonal Z' couplings
        between τ-VL and μ-VL. We have used the general results given in <cit.> to find the leading order contributions to
        the partial width
        
            Γ(τ→μγ)=α g^' 4/1024 π^4m_τ^5/M_Z'^4(|σ̃_L|^2+|σ̃_R|^2) ,
        
        with
        
            σ̃_L=   ∑_a ([ĝ^ℓ_L]_4a [ĝ^ℓ_L]_a3 F(x_a)+ m_a/m_τ [ĝ^ℓ_L]_4a [ĝ^ℓ_R]_a3 G(x_a) ) , 
            σ̃_R=   ∑_a ([ĝ^ℓ_R]_4a [ĝ^ℓ_R]_a3 F(x_a)+ m_a/m_τ [ĝ^ℓ_R]_4a [ĝ^ℓ_L]_a3 G(x_a) ) ,
        
        where a runs over all leptons with mass m_a in the loop, x_a:=(m_a/M_Z')^2, and the loop functions are the same as in (<ref>) and (<ref>).
        
        
        
        § Z' COUPLINGS AT THE BEST FIT POINT(S)
        
        
        Below we show the coupling matrices of Z' in the mass basis for our best fit point (I.). The analogous
        couplings for point (II.) are qualitatively the same.
        
            ĝ^ℓ_L =
            	(
            [  -0.2669   0.2798  0.01800   0.3425;   0.2798  -0.2934 -0.01890  -0.3591;  0.01800 -0.01890   0.9996  -0.0100;   0.3425  -0.3591  -0.0100  -0.4393;          ])
        
            ĝ^ℓ_R =
            	(
            [  -0.2674  -0.2799 -0.01800  -0.3427;  -0.2799  -0.2930 -0.01890  -0.3588; -0.01800 -0.01890   0.9996  -0.0100;  -0.3427  -0.3588  -0.0100  -0.4392;          ])
        
            ĝ^d_L =
               (
            [ -0.000100 -0.004400 0.0007000   0.01030; -0.004400   -0.1539   0.02570    0.3613; 0.0007000   0.02570    0.9994 -0.006300;   0.01030    0.3613 -0.006300   -0.8454;           ])
        
            ĝ^d_R =
              (
            [     -0.8418     0.05670     0.01670      0.3604;     0.05670   -0.003800   -0.001100    -0.02430;     0.01670   -0.001100      0.9998 5.546×10^-6;      0.3604    -0.02430 5.546×10^-6     -0.1542;             ])
        
        
        
        
        § ACKNOWLEDGMENTS
        
        A.T. is grateful to J. Heeck, M. E. Krauss, and A. Vicente for useful discussions.
        S.R. acknowledges that this material is based upon work supported by the Department of Energy under Award Number DE-SC0011726.
        This work has partially been supported by the German Science Foundation (DFG) within the SFB-Transregio TR33 “The Dark Universe".
        S.R. acknowledges partial support from the Alexander von Humboldt Foundation and the Bethe Center of Bonn University where this work was begun, 
        and the Kobayashi-Maskawa Institute, Nagoya University where this work was completed.
        
        
        
        
        
        
        utphys
