Complete Vector-like Fourth Family and new $\mathrm{U(1)^\prime}$ for Muon Anomalies
Junichiro Kawamura Stuart Raby Andreas Trautner










1.35cm




Complete Vector-like Fourth Family and new 1 for Muon Anomalies



1.5cm

Junichiro Kawamura^a,b,[
mailto:kawamura.14@osu.edukawamura.14@osu.edu],
Stuart Raby^a,[
mailto:raby.1@osu.eduraby.1@osu.edu],
and
Andreas Trautner^c,[
mailto: trautner@mpi-hd.mpg.detrautner@mpi-hd.mpg.de]

0.8cm

^aDepartment of Physics, Ohio State University, Columbus, Ohio
 43210, USA


^bDepartment of Physics, Keio University, Yokohama 223-8522, Japan


^cMax-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany





1.5cm

 
We consider the Standard Model (SM) with the addition of a U(1)^' 
gauge symmetry and a complete fourth family of quarks and leptons which are vector-like with respect to 
the full SU(3)_C×SU(2)_L×U(1)_Y×U(1)^' gauge symmetry.   
The model provides a unified explanation of experimental anomalies in (g - 2)_μ as well as b → s ℓ^+ ℓ^- decays.
We find good fits to the
deviations from the SM, while at the same time fitting all other SM observables. 
The model includes a new Z^' gauge boson, a U(1)^'-breaking scalar, and vector-like leptons all with mass of order a few 100 GeV.  
It is consistent with all currently released high energy experimental data, 
however, it appears imminently testable with well designed future searches.
Also precision flavor experiments, especially more accurate direct determinations of CKM matrix elements,
would allow to probe the best fit points.










§ INTRODUCTION

The Standard Model (SM) is very successful,
explaining most experimental results.
However, there are experimental discrepancies with some SM predictions.
One is the anomalous magnetic moment of the muon, (g-2)_μ.
The current experimental measurement <cit.>
shows the discrepancy,

    Δ a_μ := a_μ^exp - a_μ^SM = 268 (63) (43) × 10^-11.

Other discrepancies are reported in observables related to
 processes.
The observables testing lepton flavor non-universality, R_K and R_K*,
deviate from the SM prediction <cit.>,
even though the most recent data is consistent with the SM at 2.5 σ <cit.>
or has large error bars <cit.>. There are also deviations from the SM predictions
for semi-leptonic branching ratios <cit.>
and angular distributions <cit.>.

It is interesting that all the observables for  can be explained
by New Physics (NP) contributions to
the effective Hamiltonian <cit.>,

    _eff^ℓ = -4G_F/√(2)α_e/4π V_tbV^*_ts∑_a=9,10(C_a^ℓ_a^ℓ+C_a^'ℓ_a^'ℓ),

where ℓ = e, μ, τ and
the operators are defined as

    _9^ℓ :=    [ s^μ P_L b  ]
                                                                 [ ℓ_μℓ],  _10^ℓ :=  [ s^μ P_L b  ]
                                                                 [ ℓ_μγ_5 ℓ], 
    _9^'ℓ :=    [ s^μ P_R b  ]
                                                                 [ ℓ_μℓ],  _10^'ℓ :=   [ s^μ P_R b  ]
                                                                 [ ℓ_μγ_5 ℓ].

The analyses before Moriond 2019 <cit.>
and after Moriond 2019 <cit.>
show that several patterns of NP contributions explain the discrepancies significantly better than the SM.
In all cases, there should be a sizable negative contribution to C_9^μ.

The muon anomalous magnetic moment Δ a_μ
can be explained by introducing vector-like (VL) leptons which exclusively couple
to muons <cit.>.
A way to address the  anomalies is to introduce a Z' gauge boson
which couples to muons and down-type quarks.
For instance, 1_μ-τ gauge symmetry and VL quarks (with 1_μ-τ charge) are introduced
to control the flavor dependent couplings of the Z' boson <cit.>
(see Ref. <cit.> for a more general discussion of 1' gauge symmetry).
It is shown in Refs. <cit.>
that both of Δ a_μ and  anomalies are successfully explained
in models with VL-leptons, VL-quarks and a Z' boson.

Box-diagram contributions involving new fermions and scalars can also account
for the  anomalies <cit.>. They may even include
particle candidates for dark matter <cit.>.
These extensions, however, typically induce deviations from the SM predictions
for Lepton Flavor Violating (LFV) decays, Higgs decays, and B_s-B_s mixing.

In this paper, we propose a model with a complete fourth family of fermions
which are VL under both the SM and a 1' gauge symmetry.
Similar models with chiral 1' gauge symmetry were considered
in Refs. <cit.>.
In these models, the SM families typically have sizable couplings to the Z' gauge boson in the gauge basis.
A VL 1' has been studied where a new singlet scalar <cit.> 
or the singlet VL neutrino <cit.> are dark matter candidates. 
However, the parameter space there is very restricted, so that Δ a_μ was not addressed.
In our present model, all Z' couplings to the SM fermions are controlled
by mixing of the SM families with the VL family.
We find that a certain pattern of mixings can simultaneously
address both, Δ a_μ and the  anomalies.

We analyze this model involving all three SM families.
This allows us to explicitly discuss both the CKM matrix
and exotic particle production from quarks and gluons at the Large Hadron Collider (LHC).
We find points in the parameter space which explain the muon anomalies
and all other observables by using a χ^2 fit.
The purpose of this paper is to demonstrate the existence of points
which are consistent with the anomalies, as well as all other SM observables,
and study the expected phenomenology at these points.
A more detailed analysis of the expected phenomenology in a wider parameter space
is delegated to future work.

The rest of this paper is organized as follows.
The model is introduced in Section <ref>,
then we discuss the most relevant observables for the muon anomalies in Section <ref>.
In Section <ref> we show the best fit points of our χ^2 analysis and study their phenomenology.
Section <ref> is devoted to our conclusions.
Values of the input parameters and all observables calculated
in this analysis are listed in the Appendix.



§ MODEL





 §.§ Matter Content and Masses



In this paper,
we study a model with a complete VL fourth family and 1' gauge symmetry.
The quantum numbers of all fields are listed in Tables <ref> and <ref>.
The SM 2_L doublets are defined as
q_L_i = (u_L_i, d_L_i), l_L_i=(ν_L_i, e_L_i), and H=(H_0, H_-).
The new doublets are Q_L = (U'_L, D'_L), L_L = (N'_L, E'_L), Q_R = (-D'_R, U'_R), and L_R = (-E'_R, N'_R).
The model is trivially anomaly-free since the 1' charge assignment is completely vector-like.

In the gauge basis, there are no couplings between the SM families and the Z' boson.
These are induced in the mass basis by mixing effects.
The Yukawa couplings in the gauge basis are given by

    _Yukawa =    _SM +_H +_ϕ +_Φ+h.c. ,

where

    _SM :=   u_R_i y^u_ijq_L_jH  +d_R_i y^d_ijq_L_j H +e_R_i y^e_ijl_L_j H
                        + ν_R_i  y^n_ijl_L_j H , 
    _H    :=    _uU_R Q_L H + _dD_R Q_L H+ _eE_R L_L H
                              + _nN_R L_L H
        + '_uQ_R H U_L - '_dQ_R H D_L - '_eL_R H E_L
                              + '_nL_R H N_L , 
    _ϕ  :=    ϕ( ^Q_VQ_R Q_L-^U_VU_R U_L-^D_VD_R D_L . 
        + .^L_VL_R L_L-^E_VE_R E_L-^N_VN_R N_L ) , 
    _Φ :=    Φ( ^Q_iQ_R q_L_i +^L_iL_R l_L_i ) 
        - Φ^* ( ^U_iu_R_i U_L  + ^D_id_R_i D_L
                                            + ^E_ie_R_i E_L + ^N_iν_R_i N_L
                                                )  .

Here, we have used H:=iσ_2 H^* = (H_-^*, -H_0^*) and i,j=1,2,3 run over the three SM generations.

The scalar fields acquire vacuum expectation values (VEVs) given by
v_H:=H, v_ϕ:=ϕ, and v_Φ:=Φ. The charged lepton mass matrix then is given by

    e^A_R ^e_AB e^B_L
    =
    [ e_R_i   E_R  E'_R ][ y^e_ij v_H        0_i   ^E_i v_Φ;        0_j     _e v_H    _V^Ev_ϕ;   ^L_j v_Φ   _V^L v_ϕ    '_e v_H ][ e_L_j;  E'_L;   E_L ] ,

where A, B = 1,…,5. We define the mass basis via

    [ê_L]_A := [(U^e_L)^†]_AB[e_L ]_B ,
      [ê_R]_A := [(U^e_R)^†]_AB[e_R ]_B ,

with unitary matrices that satisfy

    ( U_R^e )^†^e U_L^e = diag(m_e, m_μ, m_τ, m_E_1, m_E_2) .

Here m_E_1 and m_E_2 are masses for the extra charged leptons, which are predominantly the VL leptons of the gauge basis.
The mass matrices for the up and down quarks are obtained from ^e by formally replacing e→ u, E→ U, or e→ d, E→ D, respectively.

As a consequence of the 1' charges, only the three standard generations of right-handed (RH) neutrinos have Majorana masses,

    _Maj = - 1/2ν_R_i M^ij_Majν^c_R_j .

The neutrino Dirac mass matrix is obtained from ^e by formally replacing e→ n and E→ N.
The Majorana masses are assumed to be 𝒪(10^14) GeV, thereby explaining the tiny observed neutrino masses
via a standard type I see-saw mechanism.
As usual, three generations of left-handed (LH) neutrinos have tiny masses of 𝒪(v_H^2 / M_Maj)
while three generations of RH neutrinos have huge masses 𝒪(M_Maj).
Unlike in the standard type I see-saw case, there are 4 more degrees of freedom here, which form two Dirac fermions
with masses of 𝒪(TeV).
The mixing of those with the other neutrinos is negligible as it is suppressed by the Majorana mass terms.

The neutral scalar fields are expanded as

    H_0   =  v_H    +1/√(2)( h + i a_h),   Φ =  v_Φ +1/√(2)( χ + i a_χ),   ϕ =  v_ϕ + σ.

The CP-odd degrees of freedom, namely  a_h and a_χ,
get eaten by the gauge fields and only the real components h and χ are physical.
We assume that ϕ is a real scalar field.

We parametrize the masses for χ and σ as

    m_χ^2 = λ_χ v_Φ^2   and    m_σ^2 = λ_σ v_ϕ^2.

Here we have introduced the effective quartic couplings _χ and _σ.
The scalar χ should not be much heavier than the Z' boson,
as long as the effective quartic coupling stays perturbative and the new gauge coupling g' is not tiny.
Importantly, the couplings of χ are responsible for the mass mixing of SM particles with the VL families. Consequently,
to the extent that this mixing is necessary to fit the muon anomalies, χ contributes significantly in our fits.
On the other hand, v_ϕ could be very large compared with v_Φ as long as the Yukawa couplings to ϕ, e.g. _V^L and _V^E, are
small enough to prevent the VL fermions from decoupling much above the TeV scale.
The scalar σ, thus, can be heavy and therefore irrelevant for current observables.
Indeed, contributions from σ will be negligible at the best fit points shown below.



 §.§ Yukawa and Gauge Couplings

The real scalar fields couple to the charged leptons as
 
    -  _Yukawa =    1/√(2) h e_R Y^h_e  e_L
                                      + 1/√(2)χ e_R Y^χ_e  e_L
                                      + h.c. ,

where, in the gauge basis,

    Y^h_e =
    [ y^e_ij    0_i    0_i;    0_j     _e      0;    0_j      0    '_e ],  
     Y^χ_e =
    [ 0_ij  0_i ^E_i;  0_j    0    0; ^L_j    0    0 ],

and completely analogously for the quarks.
The Yukawa coupling matrices in the mass basis are given by

    Ŷ^S_e = (U_R^e)^† Y^S_e U_L^e,  where   S = h, χ .


Combining LH and RH fields à la f^A:=(f_L^A,f_R^A), the W boson couplings are given by

    _W =    g/√(2) W^+_μ[ uγ^μ( P_5  P_L + P_5 P_R ) d
                      +  νγ^μ(  P_5  P_L + P_5 P_R ) e
                                    ] + h.c. ,

where we have used the flavor space projectors

    P_5 := diag(0,0,0,0,1),  and   :=1_5-P_5.

The couplings in the mass basis are[
Note that here and in the following we neglect effects of 𝒪(v_H^2 / M_Maj),
implying that we can treat the left- and right-handed neutrino rotations separately.
]

    ĝ^W_q_L   = g/√(2)(U_L^u)^† P_5    U_L^d ,      ĝ^W_q_R   = g/√(2)(U_R^u)^† P_5         U_R^d , 
    ĝ^W_ℓ_L   =  g/√(2)(U_L^n)^† P_5 U_L^e ,     ĝ^W_ℓ_R   = g/√(2)(U_R^n)^† P_5      U_R^e .

Note that there are also right-handed charged current interactions unlike in the SM.

The extended CKM matrix is a 5× 5 matrix,

    V̂_CKM =  (U_L^u)^† P_5 U_L^d.

The 3× 3 CKM matrix for the three SM families is not unitary because of the mixing with the VL family.
We remark that also the 5× 5 matrix V̂_CKM is not unitary.



The Z boson couplings are given by[
Here we abbreviate the (co)sine of the weak mixing angle as s_W(c_W),
T^f_3 and Q_f are respectively the third component of weak isospin and the electromagnetic charge of the fermion f,
and five dimensional identity matrices 1_5 in flavor space are implicit where appropriate.]

    _Z =    g/c_W Z_μ∑_f=u,d,e,nfγ^μ[  ( T^f_3 P_5 - Q_f s^2_W ) P_L
                             + ( T^f_3 P_5       - Q_f s^2_W ) P_R
                       ] f.

The couplings for a fermion f=u,d,e,n in the mass basis are given by

    ĝ^Z_f_L =g/c_W[ (U_L^f)^† U_L^f -Q_f s_W^2 ] ,   ĝ^Z_f_R =g/c_W[ (U_R^f)^† P_5 U_R^f -Q_f s_W^2 ] .


Finally, the couplings to the Z' boson are given by

    _Z' =    g' Z'_μ∑_f=u,d,e,νfγ^μ(  Q'_f_L P_L   +  Q'_f_R  P_R ) f 
    
                =    Z'_μ∑_f=u,d,e,nf̂γ^μ( ^Z'_f_L P_L +  ^Z'_f_R P_R ) f̂ ,

where the fermions in the mass basis are denoted by f̂ and the charge matrices are

    Q'_f_L = Q'_f_R = diag(0,0,0,-1,-1) .

In the  mass basis,

    ĝ^Z'_f_L =g' (U^f_L)^†  Q'_f_L U_L^f   and  ĝ^Z'_f_R =g' (U^f_R)^†  Q'_f_R U_R^f ,

implying that all couplings between Z' and the SM fermions are controlled by the mixing matrices.

Altogether, we find that the model has non-unitary CKM mixing
and tree-level flavor changing neutral currents
mediated by Z, Z', the SM Higgs boson, as well as by the new boson χ.
In addition, W bosons also acquire couplings to the right-handed charged currents of SM fermions, which
are constrained by measurements such as neutrino-nucleon scattering <cit.>.
All of these effects are in general severely constrained by experiments.
However, we find that in our model all those effects are controlled by 𝒪(m_f^2/M_VL^2) coefficients
implying that they are generally suppressed. We prove this analytically in Appendix <ref>.
In agreement with this, both, the unitarity of the 3×3 CKM and PMNS matrix
as well as the absence of all
tree-level flavor violating effects for SM fermions,
are restored in the limit of a heavy VL family. That is, the model approaches the SM in the decoupling limit.



 §.§ RGE Evolution of the U(1)' Gauge Coupling




The 1' gauge coupling constant g' should be sufficiently small
at the TeV scale such that it stays perturbative under RGE running up to a scale where UV physics,
such as a Grand Unified Theory (GUT), emerges.
The 1-loop beta function for g' is given by

    d g'/d lnμ =  g'^3/16π^265/3 .

This gives rise to a scale of the Landau pole for g',

    Λ_g' = μ_Z'exp( 24 π^2/65  g'(μ_Z')^2),

where μ_Z'∼ 1 TeV is the typical scale of the model.
Figure <ref> shows the scale of the Landau pole in dependence of g'(μ_Z') at the
TeV scale.
For example, g'(1 TeV) ≲ 0.35 (0.48) is required for Λ_g'∼ 10^16 (10^10) GeV.
In our numerical analysis we focus on a situation where the model is correct up to a typical GUT scale of 10^16 GeV,
such that g'(1 TeV) < 0.35 is required.




§ OBSERVABLES


In this model, Δ a_μ is explained by 1-loop contributions involving the Z' boson and VL leptons.
NP contributions to C_9,10^(')μ are provided by tree-level Z' exchange.
NP contributions will also affect observables which are currently consistent with the SM such as
μeγ, τμγ, τμμμ, B_q-B_q mixing, etc.
The most relevant observables for the muon anomalies will be discussed in the following.
An in-depth discussion of these and further observables is postponed to future work.



 §.§ amu and 1' Charge Assignment






The dominant Z' boson contribution to Δ a_μ is given by (see e.g. <cit.>), 

    δ_Z' a_μ≃ -m_μ/8π^2m_Z'^2∑_a=1,2Re( [^Z'__L]_μ E_a[^Z'__R]^*_μ E_a)
            m_E_a G_Z(x_a) ,

where x_a := m_E_a^2/m_Z'^2 and the loop function is given by

    G_Z(x) :=     x^3+3x-6x ln(x)-4/2(1-x)^3 .

The dominant contribution of the scalar χ is given by

    δ_χ a_μ≃    -m_μ/32π^2 m_χ^2∑_a = 1,2[Ŷ^χ_e]_μ E_a[Ŷ^χ_e]_E_a μ
    m_E_a G_S(y_a)
     ,

where y_a := m_E_a^2/m_χ^2 and the loop function is

    G_S(x) := x^2-4x+2ln(x)+3/(1-x)^3 .

There are also new contributions from loops involving the SM bosons and the VL fermions,
but these are negligible.

Figure <ref> shows typical values of the muon mass m_2 and the Z' contribution to δ a_μ.
For illustration, g' = 0.25, m_Z' = ^L,E_2 v_Φ = ^L,E_V v_ϕ = 500 GeV and
y^e_22 v_H = 0.1 GeV have been fixed while all other couplings except _e and '_e are set to zero.
We see that '_e≳ 0.4 and _e ≲ 10^-3 are required in order to obtain Δ a_μ≳ 10^-9 and m_μ∼ 0.1 GeV.
This illustrates how the muon mass is affected by the mixing and enhanced above m_μ∼ 0.1 GeV for _e ≳ 10^-3.

We see that the Higgs coupling '_e must exist in the model to explain Δ a_μ.
This explains the non-universal charge assignment in Table <ref>:
The 1' charges of VL-fermions needs to be opposite for 2_L doublets and singlets
in order to allow the coupling '_e. For this reason the 1' gauge symmetry is incompatible with 5 unification.
However, it is still compatible with the Pati-Salam gauge group 4×2_L×2_R.



 §.§ BR(li->gamma lj)


The branching fraction of ℓ_i→ℓ_j γ is given by <cit.>

    ℓ_iℓ_jγ≃1/Γ_ℓ_iα_e  m^3_ℓ_i/1024π^4(σ_L^2 + σ_R ^2 ),

where m_ℓ_i and Γ_ℓ_i are the mass and total decay width of the lepton ℓ_i,
while α_e is the electromagnetic fine-structure constant.
The dominant contributions arise from Z' or χ exchange and they are given by

    σ_L ≃    ∑_a=1,2(m_E_a/m_Z'^2[^Z'_R ]_j E_a[^Z'_L ]_E_a i G_Z(x_a)
                 + m_E_a/4m_χ^2[^χ_e ]_j E_a[^χ_e ]_E_ai G_S(y_a)
                    ),

and σ_R which is given by formally replacing L→ R and ^χ_e →(^χ_e )^†
in the above expression.
Other contributions, involving the SM bosons or σ, only amount to sub-percent corrections at our 
best fit points.



 §.§ Wilson Coefficients for bsll

The Wilson coefficients defined in Eqs. (<ref>) are given by
 
    C^ℓ_9 =     -√(2)/4 G_F4π/α_e1/V_tbV_ts^*1/2m^2_Z'[^Z'_d_L]_23[^Z'_e_R+^Z'_e_L]_ii, 
    
     C^ℓ_10 =    -√(2)/4 G_F4π/α_e1/V_tbV_ts^*1/2m^2_Z'[^Z'_d_L]_23[^Z'_e_R-^Z'_e_L]_ii, 
    
     C^'ℓ_9 =     -√(2)/4 G_F4π/α_e1/V_tbV_ts^*1/2m^2_Z'[^Z'_d_R]_23[^Z'_e_R+^Z'_e_L]_ii, 
    
     C^'ℓ_10 =    -√(2)/4 G_F4π/α_e1/V_tbV_ts^*1/2m^2_Z'[^Z'_d_R]_23[^Z'_e_R-^Z'_e_L]_ii,

where i = 1,2,3 for ℓ = e,μ,τ, respectively.


We refer to the recent two-dimensional analysis of Ref. <cit.> and adopt the best fit values of the Wilson coefficients as

    (I)      Re  C_9^μ = -0.7 ± 0.3,      Re C^μ_10 = 0.4 ± 0.25 ;   
       (II)      Re C_9^μ = -1.04 ± 0.24,      Re C^'μ_9 = 0.48 ± 0.30 .

Note that the Z' should not introduce sizable Wilson coefficients for the electron
because that would generically also induce sizable violation of lepton flavor in μ→ e γ.
Although it has been pointed out that some flavor universal contributions seem to be favored <cit.>,
we do not discuss this possibility in the present paper.



 §.§ Neutral Meson Mixing



There are strong constraints on neutral meson mixing <cit.>.
The relevant effective Hamiltonian is given by

    ^Δ F=2_eff= ∑_i,a C_i^a  Q_i^a,

where (i,a) = (1,VLL), (1,VRR), (1,LR), (2,LR), (1,SLL), (2,SLL), (1,SRR), (2,SRR).
The four-fermi operators are defined as

    Q_1^VLL =    (F^αγ_μ P_L f^α)(F^βγ^μ P_L f^β) ,   
     Q_1^VRR =    (F^αγ_μ P_R f^α)(F^βγ^μ P_R f^β) ,   
    
     Q_1^LRR  =    (F^αγ_μ P_L f^α) (F^βγ^μ P_R f^β) ,   
     Q_2^LRR  =    (F^α P_L f^α) (F^β  P_R f^β) ,   
    
     Q_1^SLL =    (F^α P_L f^α)(F^β  P_L f^β) ,   
     Q_2^SLL =    (F^ασ_μν P_L f^α)(F^βσ^μν P_L f^β) ,   
    
     Q_1^SRR =    (F^α P_R f^α)(F^β  P_R f^β) ,   
     Q_2^SRR =    (F^ασ_μν P_R f^α)(F^βσ^μν P_R f^β) .

Here α and β are color indices and (F, f) = (b,d), (b,s), (s,d) or (c, u) for
B_d-B_d, B_s-B_s, K-K or D-D mixing, respectively.
We focus here on B_q-B_q (q=d,s) mixing since these are the most relevant for the  anomalies.

The Wilson coefficients induced by Z' or neutral scalar exchange, including 𝒪(α_s) QCD corrections are given by <cit.>

    C_1^VLL(μ)=    [ 1+α_s/4π(-2logm_Z'^2/μ^2 + 11/3) ]
                                    g^Z'_L g^Z'_L/2 m_Z'^2,  
    
    C_1^LR(μ)=     [1+α_s/4π(-logm_Z'^2/μ^2 - 1/6) ]
                                      g^Z'_L g^Z'_R/m_Z'^2
                                      - ( -3/2α_s/4π)
                                     y^χ_L y^χ_R/2 m_χ^2,  
    
    C_2^LR(μ)=     α_s/4π(-6logm_Z'^2/μ^2 - 1 )
                                  g^Z'_L g^Z'_R/m_Z'^2
                                  - (1-α_s/4π)
                                     y^χ_L y^χ_R/2 m_χ^2,  
    
    C_1^SLL(μ) =    - 
                         [1+α_s/4π(-3logm_χ^2/μ^2 + 9/2)]
                                            y^χ_L y^χ_L/4 m_χ^2 ,   
    
    C_2^SLL(μ)=     -
                                      α_s/4π(-1/12logm_χ^2/μ^2+1/8)
                                          y^χ_L y^χ_L/4 m_χ^2 ,

 where

    g_L^Z' =  [^Z'_d_L]_3k,  g_R^Z' = [^Z'_d_R]_3k,y_L^χ  = [^χ_d]_3k, y_R^χ  = [^χ_d]^*_k3,

and k=1(2) for B_d(s)-B_d(s) mixing, respectively.
The analogous Wilson coefficients C_1^VRR and C_1,2^SRR are obtained by formally replacing L→ R.
The off-diagonal element of the B_q(q=d,s) meson mass matrix is given by

    M_12^*(B_q) = M^SM*_12(B_q) + 1/2 m_B_q∑_i,a C_i^a(μ) ⟨B_q| Q^a_i(μ)|B_q⟩ ,

where the first term is the SM contribution and m_B_q is the meson's physical mass.
The SM contribution for B_q-B_q mixing is given by

    M_12^SM*(B_q) =
               G_F^2/12π^2  m_W^2  (λ^(q)_t)^2   S_0(x_t)  η_B  m_B_q  f^2_B_q B̂_B_q .

Here, _t^(q) = V_tb^* V_tq, S_0(x_t=m_t^2/M_W^2)≈ 2.32 is the Inami-Lim loop function <cit.>,
η_B=0.55±0.01 <cit.> quantifies the short distance radiative corrections, while
f_B_q and  B̂_B_q denote the corresponding decay constant and SM hadronic matrix element.
The SM and necessary BSM hadronic matrix elements are calculated by lattice collaborations,
and their values at 1 TeV according to our own evaluation are listed in Table <ref>.
Values for K-K and D-D mixing are also listed for completeness.
All hadronic matrix elements for Kaon oscillations and the value of
 f^2_B_qB̂_B_q
have been taken from Ref. <cit.>, while those for f_B_q^2 B^(2-5)_B_q and B_D^(1-5)
are taken from Refs. <cit.> and Ref. <cit.>, respectively.
The QCD running between the respective lattice scales and μ = 1 TeV
has been calculated based on the anomalous dimensions shown in Ref. <cit.>.

The observables for B_q-B_q mixing are defined as
 
    Δ M_d :=    2 M_12(B_d) ,   
    S_ψ K_s :=    sin(Arg[M_12(B_d)]) ,   
    Δ M_s :=    2 M_12(B_s) ,   
    S_ψϕ   :=    - sin(Arg[M_12(B_s)]) .

The mass differences Δ M_d and Δ M_s are measured with high accuracy
and theoretical uncertainties are prevailing.
The dominant theoretical uncertainties arise from the CKM elements, hadronic matrix elements, and NLO QCD corrections.
Altogether we find 15.6% (14.1%) relative uncertainty for Δ M_d (Δ M_s).
Note that unlike the analyses in e.g. Refs <cit.>, 
we cannot reduce the uncertainties by assuming exact unitarity of the CKM matrix here,
simply because CKM unitarity is not guaranteed in our model.
We therefore have to rely on the measured CKM matrix elements and their respective errors.
Despite the possible CKM non-unitarity, we still use formulas for the SM contributions which are obtained
under the implicit assumption of exact CKM unitarity (i.e. a working GIM mechanism).
This adds some additional theoretical uncertainty which is hard to quantify.
As the CKM matrix at our fit points is still approximately unitary to the observed degree we neglect this additional uncertainty.

We find that the pattern (II) of the Wilson coefficients for  (cf. Eq. (<ref>)) is disfavored,
because this would cause large Z' contributions to Δ M_s.
A large negative contribution Re C_9^μ
together with a positive Re C_9^'μ
requires a relative sign difference between
Re [g^Z'_d_L]_23 and
Re[g^Z'_d_R]_23.
Since the hadronic matrix element O^LR_1 is sizable and negative,
this would imply a large and positive left-right contribution to Δ M_s.
However, as the current SM prediction is already larger than the experimental value,
therefore the Z' coupling with Re C_9^'μ is strongly disfavored.
For this reason we could not find any good fit points for the pattern (II).



 §.§ Neutrino Trident Production

The so-called neutrino trident production ν_μ→ν_μμμ off a nucleus
is a rare process that has been observed at a rate consistent with SM expectations <cit.>.
This process can also be mediated by Z' exchange and therefore constitutes an important bound on NP scenarios <cit.>.
The ratio of the cross section including NP at the CCFR experiment can be estimated as <cit.>

    σ_CCFR/σ^SM_CCFR≃(1+4s_W^2 + Δ g^V_μμμμ)^2 +
    1.13 (1-Δ g^A_μμμμ)^2/(1+4s_W^2 )^2 + 1.13  ,

with a current experimental limit of σ/σ^SM=0.82±0.28 at 95% C.L.. 
The effective four-fermi couplings Δ g^V,A_μμμμ in our model are given by

    Δ g^V,A_μμμμ = 1/√(2)G_F m_Z'^2 [ g^Z'_ν]_ν_μν_μ([ g^Z'_e_R]_22±[ g^Z'_e_L]_22) ,

where [ g^Z'_ν]_ν_μν_μ is defined in the flavor basis,

    g^Z'_ν = g' U_e_L^† 
    Q'_n_L  U_e_L.

This constraint is particularly relevant for light Z''s and quickly becomes insensitive
to NP once the Z' is heavier than a few 100 GeV.



 §.§ Gauge Kinetic Mixing

A potentially light Z' boson can experience sizable gauge kinetic mixing
with the 1_Y gauge boson, namely the Z of the SM.
The Z-Z' mixing parameter  is estimated as

    ≃g_Y g'/6π^2log(m_E^2/m_L^2 m_Q^2 m_D^2/m_U^4)  ,

where m_F (F=L,E,Q,U,D) are the VL mass terms for the VL fermions
and g_Y is the 1_Y gauge coupling constant.
Current experimental limits are summarized in Ref. <cit.>.
Values of ∼ 0.05 cannot be ruled out if the Z'
is heavier than a few 100 GeV.



§ RESULTS




 §.§ chi2 Fitting


We minimize the χ^2 function

    χ^2(x) := ∑_I(y_I(x) - y_I^0)^2/σ_I^2,

where x is a point in the parameter space, while
y_I(x) is the value of observable I with central value y_I^0 and uncertainty σ_I.
Observables we fit to include the SM fermion masses, CKM matrix element absolute values and relative phases,
SM particle branching fractions (including flavor violating decays),
neutral meson mixing, Δ a_μ, C_9,10^(')l, and some others.
In total we consider 98 observables and they are all listed in appendices <ref> and <ref>.

In total there are 65 input parameters represented by x in our analysis.
The bosonic sector has 5 parameters,

    m_Z', v_ϕ, g', λ_χ, λ_σ,

which are the Z' mass, the VEV of ϕ, the 1' gauge coupling constant,
and the effective quartic couplings defined in Eq. (<ref>), respectively.
All other parameters are Yukawa couplings as defined in Eqs. (<ref>)-(<ref>).
For the neutrino sector we only consider the couplings λ_n^(') and λ_V^N
which are relevant for the new TeV-scale state.
The remaining Yukawa couplings y^n_ij, λ^N_j
as well as the heavy Majorana masses are not varied in our analysis,
because these are relevant only for the details of masses and mixings of the SM neutrinos
(and the RH Majorana neutrinos). Thus, effects suppressed by the Majorana masses are neglected.
We expect that the number of parameters in the neutrino sector is sufficient to explain
the observed neutrino mass squared differences and the PMNS matrix without
changing any observables studied in our analysis.
We assume that all Yukawa couplings are real except for y^u,d_13 and y^u,d_31.
Altogether there are then 60 real parameters for the Yukawa couplings.
All Yukawa couplings and effective quartic coupling
values are restricted to be smaller than unity.
Furthermore, as already discussed at the end of Section <ref>, g' < 0.35 is required
so that the gauge coupling g' stays perturbative up to ∼ 10^16 GeV.



 §.§ Best Fit Points

We find two best fit points A and B with χ^2 = 25.1 and χ^2 =24.9, respectively, for 98-65=33 d.o.f. .
The values of all observables and the corresponding input parameters are listed in appendices <ref> and <ref>.
The values of selected observables are shown in Table <ref>.
Masses and dominant decay modes of new particles are summarized in Tables <ref> and <ref>.




At both best fit points, deviations from the SM in Δ a_μ and C^μ_9, C^μ_10 are explained by NP contributions.
Besides a dominant positive loop contribution to Δ a_μ involving the Z',
there is a slight cancellation from a negative contribution of the scalar χ at both points.
All observables are fit within their 2σ ranges at both points,
except for Δ a_e which cannot be explained in the SM either, see e.g. <cit.>.
We could not find points which have Δ a_e near the experimental value while LFV processes, especially μeγ, are sufficiently suppressed.
The only observables with pulls exceeding 1σ are μeγ, Δ M_B_d, and some absolute values of CKM elements.
Interestingly, the NP correction to B_d-B_d from Z'-boson exchange is sizable,
while B_s-B_s mixing is within its 1σ range (taking into account the experimental uncertainties for V_ts and V_tb).

The 3× 3 CKM matrix of the SM families, 
[V_CKM]_ij := [V̂_CKM]_ij,  
is almost unitary with deviations smaller than

    |  V_CKM^† V_CKM - 1_3 | ≲
    5.1  (1.9) × 10^-8,

at the point A(B).
The full 5× 5 CKM matrices at the best fit points are shown 
in appendices <ref> and <ref>.



 §.§ Phenomenology


We now discuss the phenomenology of this model at the best fit points.
Since the Z' gauge boson and the VL leptons are comparatively light,
constraints from direct searches at the LHC and from muon flavor physics
are both very important.




  §.§.§ Z' Physics

The Z' boson mass is 494.7(377.1) GeV at the best fit point A(B).
There are strong limits on such a comparatively light Z' from direct searches at the LHC.
Other important constraints on the Z' mass arise from
the so-called neutrino trident production as well as from gauge kinetic mixing with the Z boson.
However, we will see that despite its relative lightness, the Z' boson is still sufficiently heavy to evade these bounds.

General LHC limits on Z' bosons responsible for  anomalies are studied in Refs. <cit.>.
The most stringent bound from the LHC on our model comes from resonance searches in the dimuon channel,

    p p → Z' →μ^+μ^- .

This is particularly pronounced in our model,
as the dominant decay modes of the Z' are Z'→μ^+μ^- and Z'→νν,
as shown in Tables <ref> and <ref>.
Exclusion bounds are given in Ref. <cit.> based on 139 fb^-1 of data.
We have calculated the fiducial cross section, as defined in Ref. <cit.>, using  <cit.> 
based on an  <cit.> model file generated with  <cit.>.
The fiducial cross section is 0.477(0.482) fb at the point A(B).
This is roughly 4 (6) times smaller than the experimental limit for the respective Z' masses <cit.>.
The production cross sections are very suppressed because the Z' couplings to the SM quarks are at most 𝒪(10^-3).
We stress that both of our fit points realize solutions to the observed anomalies where
the dimuon coupling of the Z' is maximal, while the Z'bs coupling is minimal.

The ratio of neutrino trident production at CCFR is 1.014  (1.006),
very close to the SM and well in agreement with the experimental constraints.
The gauge kinetic Z-Z' mixing at the best fit points is estimated
as ∼ 2.0× 10^-3 (7.6× 10^-4)
and therefore also well below the experimental bounds.



  §.§.§ Lepton Flavor Physics

In general, a variety of charged LFV processes are present in this model.
As discussed in Subsection <ref>, loops involving Z' or the scalar χ together with VL leptons
can lead to chirally enhanced contributions to ℓ_1 →ℓ_2 γ processes.
Furthermore, also LFV tree-level Z' exchange is possible which could induce ℓ→ℓ_1 ℓ_2 ℓ_3 processes.
The LFV couplings here arise from the mixing between the SM families and the VL family.
Although these LFV processes exist in principle,
they can easily be suppressed by certain patterns of Yukawa couplings such as ^L,E_2≫^L,E_3, ^L,E_1
and gauge eigenstates which are otherwise closely aligned with the mass eigenstates.
Specific textures of the Yukawa couplings like this could be explained by flavor symmetries.

At the best fit points we find that from all possible charged LFV processes only μeγ is close to its experimental upper bound.
Just like for Δ a_μ, the dominant contribution to μ→ eγ originates from the chirally enhanced Z'-loop with heavy leptons,
with an 𝒪( 10%) cancellation arising from the χ-loop contribution.

The SM boson decays may in general also be affected by mixing effects.
Models with VL leptons mixing to the SM families often affect the LFV Higgs boson decays such as
h→μτ, and also changes in the rates of lepton flavor conserving decays, see e.g. <cit.>.
However, in the generic parameter regions of our best fit points, there is no significant contribution from the mixing to these processes.
As analytically demonstrated in Appendix <ref> this comes about because
mixing between the SM families and the VL family are only induced by the 1' breaking scalar Φ instead of the Higgs boson.
Thus, the Higgs boson decays to SM generations are very much aligned with the SM.
The same is true for the couplings of the Z boson which are very SM-like for the three SM generations.



  §.§.§ Quark Flavor Physics

There is much literature discussing the correlation between the  anomalies
and B_s-B_s mixing since these are induced by common operators, see e.g. <cit.>.
The recent lattice results <cit.>
imply that the SM contribution to Δ M_s is slightly larger than the experimental central value.
The Z'-boson contribution to C^μ_9 also gives a constructive correction to Δ M_s,
so that Δ M_s tends to deviate from the central value even more.
However, the theoretical uncertainties are large, so this is currently not the tightest
bound on the model.

We stress that unlike the case for the charged leptons, all of the SM quark families must mix in the up and/or down quark sectors
to explain the observed CKM matrix. This implies that there can be sizable NP contributions not only to B_s-B_s mixing
– as commonly considered in the context of  anomalies –
but also to B_d-B_d, K-K, and D-D mixing.
Even if the Z' contributions to the latter are much smaller at face value than those to B_s-B_s mixing,
the NP effects can still be significant, as also the SM contributions are further suppressed.
This is demonstrated by our two best fit points where Δ M_d ∼ 0.6 ps^-1 is about 1σ
larger than the experimentally observed value.

In general we note that future, more precise determinations and over-constraining
of the CKM elements gives very important tests for this model, complementary to other probes.
Currently there are several tensions with current data at the ∼1σ level, cf. the tables in Appendices <ref> and <ref>.
Very recently it has been argued that experiments may be in favor of CKM non-unitarity <cit.>.
While it has to be carefully evaluated whether these hints hold up, we remark
that our model is in principle very well equipped to explain such effects.



  §.§.§ Collider Signals of Vector-Like Fermions

The VL leptons tend to be light in order to explain Δ a_μ.
If the VL lepton is lighter than both the Z' boson and the scalar χ, as for example
the lightest charged VL lepton E_1 at point A,
it decays to a SM boson and a SM lepton,
as usually considered as a signal for VL leptons <cit.>.
At point A, the lightest VL lepton E_1 is approximately a weak singlet
and it decays to hμ, Zμ, and Wν with branching fractions
of about 70%, 15%, and 15%, respectively.
Given the analysis of Ref. <cit.>, which is based on LHC run 1 data,
the VL lepton at point A is not excluded.
The LHC Run 2 data was studied to search for a weak doublet VL lepton decaying
to a SM boson and a tau lepton in Ref. <cit.>.
The limit from this analysis is expected to be much weaker for a weak singlet VL lepton
decaying to a muon and a SM boson.
We hope that a VL lepton of this type will be searched for by a dedicated analysis
based on LHC run 2 data.

In contrast, if the VL lepton is heavier than χ and/or Z' it tends to decay to them.
For example, at point B the lightest charged VL-lepton E_1 predominantly decays to χμ,
and χ subsequently decays to dimuons or di-tops, if kinematically allowed.
An expected signal in this case is

    p p → E^+_1 E^-_1 →μ^+ χ (→μ^+μ^-)+μ^-  χ (→μ^+μ^-).

This signal is very clean with 6 muons and two pairs of dimuon resonances.
Furthermore, (E_1, N_2) forms approximately a weak doublet,
such that the pair production cross section is enhanced compared to the weak singlet case.

In addition to the lightest VL leptons, also the heavier VL leptons produce distinctive signals.
These tend to decay to the lighter VL leptons, with the emission of a large number of light leptons.
For instance, at the point A, the pair produced E_2 gives a dramatic signal,

    p p → E_2^+ E_2^- → Z E_1^+ + Z E_1^- → Z μ^+  (h/Z) + Z μ^- (h/Z),

with up to 10 leptons in the final state.
These high-multiplicity lepton signals could provide a strong probe of this model.

The VL quarks are also detectable at the LHC.
Limits for VL quarks are studied in Refs. <cit.> using the LHC Run2 data,
but the decay patterns of the VL quarks in our model are much more complicated
than the ones assumed in these analyses.
Furthermore, even the lightest VL quark has a mass of 2.1(1.5) TeV at the point A(B),
which is heavier than the experimental lower bound of 1.4 TeV <cit.>.
In fact, the VL quarks are typically much heavier than both the Z' or χ.
They decay to a Z' or χ and a SM quark with comparable branching fractions.
The signal from the pair production of the VL quarks is thus two (top) jets together with two resonance signatures.
An interesting signal arises again for the case that a boson decays to dimuons,

    p p → Q Q→jet Z'(→μ^+μ^-) + jet  Z' (→μ^+μ^-),

where Q is one of the VL quarks. Again, this should give very clean signals at the LHC.
Finally, note that the VL quarks can also induce missing energy signals like squarks
when both of a pair of produced VL quarks decay as Q →jet Z' (→νν).




§ CONCLUSION



We have studied a model with a complete fourth family of vector-like fermions which are charged under a new 1' gauge symmetry.
We find parameter points at which the experimentally observed deviations in the muon anomalous magnetic moment Δ a_μ and  processes
are explained without altering too much those observables that are consistent with the SM predictions.

The model can be embedded into more unified pictures, like grand unification and/or string models,
and it has a straightforward supersymmetric extension.
To this extent, it is important that all observables can be consistently explained with small 1' gauge coupling g', such that
the coupling remains perturbative up to a typical GUT scale ∼ 10^16 GeV.
An important consequence of demanding a resolution to Δ a_μ is that the 1' charge assignment for the VL family
is not compatible with an 5 GUT, but instead with a Pati-Salam gauge symmetry.

In the present paper, we have displayed two good fit points
which demonstrate that this model can explain the muon anomalies
without spoiling other observables.
The explanation of the anomalies are correlated with other beyond the Standard Model predictions
for observables including lepton flavor violation, neutral meson mixing,
deviations from the SM CKM matrix and rare meson decays.
The CKM matrix in the model easily fulfills unitarity at the currently
observed level, but is in general non-unitary.
Hints for CKM non-unitarity found in a recent analysis, thus,
could easily be accommodated and would give a strong motivation to further consider this model.
Distinct signals at the LHC in Z'→μμ
and pair production of VL leptons and VL quarks together with
clean and distinct (resonant) multi-lepton final states
are predicted and provide important means to
test the considered parameter space.

In general, there are upper bounds on the VL fermions in order to explain the muon anomalies in this model.
It will, thus, be interesting to have a global study of how wide a parameter space is consistent with current and future experiments.
More details of our analysis and more global features of this model
will be discussed in an upcoming paper.



§ ACKNOWLEDGMENT

The authors are grateful to R. Dermisek for useful discussions about this vector-like model.
The work of J.K. and S.R. is supported in part by the Department of Energy (DOE) under Award No. DE-SC0011726.
The work of J.K. is supported in part by the Grant-in-Aid for Scientific Research from the
Ministry of Education, Science, Sports and Culture (MEXT), Japan No. 18K13534.
The work of A.T. was partly supported by a postdoc fellowship of the German Academic Exchange Service (DAAD).
A.T. is grateful to the Physics Department of Ohio State University and
Centro de Física Teórica de Partículas (CFTP) at Instituto Superior Técnico, Lisbon
for hospitality during parts of this work.



§ ANALYTICAL ANALYSIS


We discuss the analytical expressions for the couplings in the mass basis.
We diagonalize the mass matrix in Eq. (<ref>) perturbatively
by exploiting m_ℓ≪M̃_ℓ,
where m_ℓ and M̃_ℓ represent the typical mass scales
of charged leptons and VL leptons, respectively.

Let us define the unitary matrices,

    U_L^0 :=
    [ _L_j   _L     ;  0_j    0    1 ],  
     U_R^0 :=
    [ _E_j   _E     ;  0_j    0    1 ],

with the four-component vectors

    _L := 1/M̃_L[ ^L*_i v_Φ; _V^L* v_ϕ ],  _E := 1/M̃_E[ ^E_i v_Φ; _V^E v_ϕ ],

and _E_i, _L_i, which obey the conditions

    _L_i^†_L = _E_i^†_E = 0,  _L_i^†_L_j = _E_i^†_E_j = δ_ij.

Here,

    M̃_L := √(∑_i=1^3 ^L_i^2 v_Φ^2 + ^L_V^2 v_ϕ^2 ) ,  M̃_E := √(∑_i=1^3 ^E_i^2 v_Φ^2 + ^E_V^2 v_ϕ^2 ) .

The rotated mass matrix is

    ^e := U_R^0†^e U_L^0  =
    [ ỹ^e_ij v_H   ỹ_R_iv_H        0_i;   ỹ_L_jv_H     _e v_H       M̃_E;        0_j       M̃_L    '_e v_H ],

where

    ỹ^e_ij := _E_i^†ŷ^e  _L_j,  ỹ_R_i :=    _E_i^†ŷ^e  _L,   ỹ_L_j := _E^†ŷ^e  _L_j,  _e :=     _E^†ŷ^e  _L,

with

    ŷ^e :=
     [ y^e_ij    0_i;    0_j     _e ].

In this matrix,
ỹ^e_ij, ỹ_L_i, ỹ_R_i, and _e are of the order 𝒪(m_ℓ/v_H),
while M̃_L,M̃_E ∼M̃_ℓ. 
Here we assume _e v_H ≲ m_μ, in order for the muon mass to be explained without fine-tuning.
Note that '_e v_H can be as large as the VL lepton masses if '_e ∼ 1 and the VL leptons are lighter than ∼ 500 GeV.
Hence, it cannot be treated as an expansion parameter in general.

Since the vectors '_L_i=[u_L]_ij_L_j and '_E_i=[u_E]_ij_E_j,
for arbitrary 3× 3 unitary matrices u_L and u_E, also satisfy the conditions in Eq. (<ref>),
we can always find a set of vectors _L_i, _E_i that diagonalize the SM Yukawa matrices,
ỹ^e_ij = diag(y^e_1, y^e_2, y^e_3).
The mass matrix for the SM families then is almost diagonal
except for the mixing with the VL family induced by ỹ_L and ỹ_R.
In order to explain the muon anomalies,
there should be a sizable mixing among the muon and the VL leptons,
while the mixing with the electron and tau can be suppressed in order to
avoid lepton flavor violations which are strongly constrained by experiments.
The simplest way to achieve this is by imposing the hierarchy ^L,E_2 ≫^L,E_1,3.
In this case, ỹ_L_i,  ỹ_R_i∼m_μ/v_H is expected.

We can show that the unitary matrices

    U_L^1 =    1_5 + 1/M̃_E[                     0_ij - '_e ỹ_L_i^* v_H^2/M̃_L              ỹ_L_i^* v_H;     '_e ỹ_L_j v_H^2/M̃_L                        0                        0;               -ỹ_L_j v_H                        0                        0 ]
    + 𝒪( m_ℓ^2 /M̃_ℓ^2 ),
    
    U_R^1 =    1_5 + 1/M̃_L[                   0_ij - '_e ỹ_R_i v_H^2/M̃_E              ỹ_R_i v_H; '_e ỹ_R_i^* v_H^2/M̃_E                      0                      0;           -ỹ_R_j^* v_H                      0                      0 ]
    + 𝒪( m^2_ℓ/M̃_ℓ^2 ),

block-diagonalize the mass matrix as

    U_R^1†ℳ̃^e U_L^1  =
    diag[ ỹ^e_ijv_H + ỹ_R_iỹ_L_j v_H^2M̃_LM̃_E'_e v_H
    + 𝒪(  m^3_ℓM̃^2_ℓ),                            [  _e v_H    M̃_E;    M̃_L '_e v_H ] ].

Here, higher order corrections for the heavy states have been neglected.
The perturbative corrections to the mass matrix of the SM families are estimated as

    ỹ_R_iỹ_L_jv_H^2/M̃_LM̃_E'_e v_H
    ∼m_μ^2/M̃_ℓ^2'_e v_H
     =    2.2 × 10^-5 GeV×('_e v_H /174 GeV)
                                                                (300 GeV/M̃_ℓ)^2.

For typical parameters, this is much smaller than the electron mass.
Finally, we define unitary matrices U^2_L,R := diag(1_3,  u_L,R)
which diagonalize the mass matrix of the VL family,

    u_R^†[  _e v_H    M̃_E;    M̃_L '_e v_H ]
    u_L = diag(m_E_1, m_E_2).

Altogether, the fields in the mass basis ê_L, ê_R can be written as

    e_L = U^e_L ê_L := U_L^0 U_L^1 U_L^2 ê_L,  
     e_R = U^e_R ê_R := U_R^0 U_R^1 U_R^2 ê_R .

We can now use this in order to study the 
the scalar and gauge-boson couplings in the mass basis.
Using Eqs. (<ref>) and (<ref>), one can show that

    [ (U^e_R)^† Y_e^h  U^e_L ]_ij =  ỹ^e_ij +
    2'_e v_H^2/M̃_LM̃_Eỹ_R_iỹ_L_j
    + 𝒪(m_ℓ^2/M̃^2),

and

    [ (U^e_L)^†  U^e_L     ]_ij =  δ_ij
       + 𝒪(m_ℓ^2/M̃^2),  [ (U^e_R)^† P_5  U^e_R    ]_ij =  𝒪(m_ℓ^2/M̃^2).

The sizes of the perturbative corrections are estimated as

    '_e v_H^2/M̃_LM̃_Eỹ_R_iỹ_L_j∼'_e m_μ^2/M̃_ℓ^2    =  1.2 × 10^-7×('_e/1.0)( 300 GeV/M̃_ℓ)^2, 
     m_ℓ^2/M̃_ℓ^2≲m_τ^2/M̃_ℓ^2    = 3.5 × 10^-5×( 300 GeV/M̃_ℓ)^2.

Therefore, the Higgs boson coupling matrix is effectively diagonalized simultaneously with the mass matrix,
and the left-handed neutral current gauge interactions of the SM families in the mass basis are to a good accuracy proportional to the identity.
In principle, there are also right-handed current corrections of the Z/W boson couplings to the SM fermions
which are tested by precision measurements of Z and W boson properties, see e.g. <cit.>,
neutrino-nucleon scattering <cit.> and so on,
but their size is too small to be testable by these experiments.

The quark sector mass matrices can be diagonalized in a completely analogous fashion with the same 
conclusions. A significant difference with respect to the lepton sector can arise due to the heavy top quark mass.
The relevant expansion parameter of the perturbation then is m_t/M̃_q, which only
becomes ≲ 0.1 if VL quark masses exceed .
However, in particular for the up-type quarks we may alternatively assume a hierarchy of couplings, 
^Q_i v_Φ, ^U_i v_Φ≪^Q_V v_ϕ, ^U_V v_ϕ
which leads to the same conclusion.



§ DETAILS OF BEST FIT POINT A




 §.§ Input parameters

The input parameters for the boson sector are

    m_Z' = 494.696,  v_ϕ = 4356.63 ,  g' = 0.266224, _χ = 0.999975,  _σ = 0.997241  .

The mass matrices are

    0.85M_e =[     -0.000486577  1.51118 · 10^-6  1.99873 · 10^-6                0 -1.14426 · 10^-6; -1.98226 · 10^-6        -0.252886      -0.00083412                0         -179.053; -5.17686 · 10^-6     0.0000939331         -1.74617                0       -0.0268504;                0                0                0 -3.80325 · 10^-7          276.103;    -0.0000358853          314.243       -0.0370005         -173.903         -172.644;                  ] ,


    0.85M_n =
    [             0             0             0     -0.816618      -4356.63; -0.0000358853       314.243    -0.0370005      -173.903      0.181800;               ] ,


    0.85M_u =
     [             -0.00210998                0.118413  1.44892· e^-0.0260154i                       0                -5.43376;              0.00415738               -0.612239                0.701898                       0               -0.168839; 0.00168153· e^-1.90643i                0.132278                -174.104                       0                -30.2055;                       0                       0                       0               0.0122855                 4251.92;              -0.0725549                -34.1972                -344.810                -2091.71                -19.5774;                         ] ,


    0.85M_d =
    [           -0.000539915            -0.00944158   0.143965· e^1.62963i                      0              -0.465179;              0.0105179             0.00784947               -1.28759                      0                8.33584; 0.0124618 · e^-1.98605               0.130429               -2.58136                      0                17.9036;                      0                      0                      0           -0.000110584               -2946.22;             -0.0725549               -34.1972               -344.810               -2091.71                26.9254;                        ] .

Entries involving the three families of right-handed neutrinos are omitted here
because these are suppressed by the huge Majorana mass and are irrelevant for our analysis,
see Section <ref>.



 §.§ Observables

In Tables <ref>-<ref> we show results for observables at the best fit point A.
When quoted with reference, we have fitted the corresponding observable to experimental data.
Otherwise we have fitted to the tree level SM prediction which is indicated by  “Ref.”=“SM”.
Eight observables, namely the real and imaginary parts of C^e,(')_9,10, are not shown
because they are at most about 10^-10. 
More details on the fitting procedure will be given in our upcoming global analysis. 

For convenience we also state the extended CKM matrix at the best fit point A. It is given by 

    0.85V̂_CKM=
       0.85[                0.974475                0.224469   0.003594· e^-1.25247i                      0.                      0.;   0.224324· e^-3.14095i                0.973639                0.041311                      0.                      0.;  0.008827· e^-0.385988i   0.040516· e^-3.12266i                0.999139    0.001086· e^1.21990i    0.000008· e^3.11582i;    0.000010· e^2.75752i   0.000044· e^0.020825i   0.001084· e^-3.13970i    0.999902· e^1.22180i    0.013636· e^3.11681i;    0.000003· e^2.77910i   0.000012· e^0.045064i 0.000287 · e^-3.11555 i    0.003122· e^1.24590i   0.000043· e^3.14091 i;                         ].


       
       
       
    



§ DETAILS OF BEST FIT POINT B




 §.§ Input Parameters

The input parameters for the boson sector are

    m_Z' = 377.090,  v_ϕ = 2915.42,  g' = 0.305856, _χ = 0.155278, _σ = 1.00000  .

The mass matrices are

    0.85M_e =[      0.000486573  1.13824 · 10^-8  4.04521 · 10^-6                0 -4.59216 · 10^-6;  1.53811 · 10^-7         -0.13295        0.0515715                0         -360.594;    -0.0000578552      0.000174906         -1.74549                0        -0.142312;                0                0                0     -0.000263151          1099.09; -1.26999 · 10^-6          337.256        0.0262194          472.383         -174.012;                  ] ,


    0.85M_n =
    [                0                0                0          2.30874          280.271; -1.26999 · 10^-6          337.256        0.0262194          472.383         -16.3708;                  ] ,


    0.85M_u =
     [               0.0105038                0.494412 0.0290073· e^-0.532076i                       0                -0.30203;             -0.00667761               -0.384139               -0.110669                       0                 7.16682;   0.450446· e^-1.77965i                 8.95794                -171.499                       0                -13.1159;                       0                       0                       0             -0.00998379                -2915.41;              -0.0540921                -104.906                 71.9157                 1542.31                -4.45885;                         ] ,


    0.85M_d =
    [              -0.0111889               0.0531604 0.0128086· e^-0.481607i                       0               0.0871297;              0.00257029              0.00163007              -0.0514564                       0                -1.52119;  0.00240628· e^1.70936i               0.0308145                -2.85816                       0                -65.9271;                       0                       0                       0               0.0101523                 2901.32;              -0.0540921                -104.906                 71.9157                 1542.31                 3.43706;                         ] .





 §.§ Observables

Tables <ref>-<ref> show the observables at the best fit point B. 
The extended CKM matrix is given by 

    0.85V̂_CKM=
       0.85[               0.974460               0.224533  0.003631· e^-1.21489i                     0.                     0.;  0.224393· e^-3.14095i               0.973619               0.041392                     0.   0.000001· e^1.63235i; 0.008717· e^-0.389908i  0.040627· e^-3.12277i               0.999136  0.000623· e^-1.59708i   0.000022· e^1.63174i;  0.000005· e^-2.00027i   0.000025· e^1.54988i  0.000622· e^-1.61053i 0.999999· e^-0.066018i  0.000889· e^0.021202i;  0.000002· e^-2.01687i   0.000011· e^1.52851i  0.000265· e^-1.63172i   0.001107· e^3.05439i  0.000001· e^-3.14158i;                        ].




      
    
        
    


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