Complete Vector-like Fourth Family and new $\mathrm{U(1)^\prime}$ for Muon Anomalies

We consider the Standard Model (SM) with the addition of a $\mathrm{U(1)^\prime}$ gauge symmetry and a complete fourth family of quarks and leptons which are vector-like with respect to the full $\mathrm{SU(3)_C}\times \mathrm{SU(2)_L} \times \mathrm{U(1)_Y}\times \mathrm{U(1)^\prime}$ gauge symmetry. The model provides a unified explanation of experimental anomalies in $(g - 2)_\mu$ as well as $b \rightarrow s \ell^+ \ell^-$ 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 $Z^\prime$ gauge boson, a $\mathrm{U(1)^\prime}$-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.

In this paper, we propose a model with a complete fourth family of fermions which are VL under both the SM and a U(1) gauge symmetry. Similar models with chiral U(1) gauge symmetry were considered in Refs. [40,46]. In these models, the SM families typically have sizable couplings to the Z gauge boson in the gauge basis. A VL U(1) has been studied where a new singlet scalar [55] or the singlet VL neutrino [56] are dark matter candidates. However, the parameter space there is very restricted, so that ∆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, ∆a µ and the b → s + − 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 2, then we discuss the most relevant observables for the muon anomalies in Section 3. In Section 4 we show the best fit points of our χ 2 analysis and study their phenomenology. Section 5 is devoted to our conclusions. Values of the input parameters and all observables calculated in this analysis are listed in the Appendix.

Matter Content and Masses
In this paper, we study a model with a complete VL fourth family and U(1) gauge symmetry. The quantum numbers of all fields are listed in Tables 1 and 2 The model is trivially anomaly-free since the U(1) charge assignment is completely vectorlike.
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 where L SM := u Ri y u ij q L jH + d Ri y d ij q L j H + e Ri y e ij l Lj H + ν Ri y n ij l LjH , (2.2) 3) Here, we have usedH := 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 where A, B = 1, . . . , 5. We define the mass basis via with unitary matrices that satisfy 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 M e by formally replacing e → u, E → U , or e → d, E → D, respectively.
As a consequence of the U(1) charges, only the three standard generations of righthanded (RH) neutrinos have Majorana masses, The neutrino Dirac mass matrix is obtained from M 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.
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 Here we have introduced the effective quartic couplings λ χ and λ 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. λ L V and λ E V , 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 12) where, in the gauge basis, 13) and completely analogously for the quarks. The Yukawa coupling matrices in the mass basis are given byŶ (2.14) Combining LH and RH fieldsà la , the W boson couplings are given by where we have used the flavor space projectors P 5 := diag(0, 0, 0, 0, 1), and P 5 := 1 5 − P 5 . (2.16) The couplings in the mass basis are 1 Note that there are also right-handed charged current interactions unlike in the SM. The extended CKM matrix is a 5 × 5 matrix, 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 matrixV CKM is not unitary. The Z boson couplings are given by 2 The couplings for a fermion f = u, d, e, n in the mass basis are given bŷ Finally, the couplings to the Z boson are given by 1 Note that here and in the following we neglect effects of O v 2 H /M Maj , implying that we can treat the left-and right-handed neutrino rotations separately. 2 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.
In the mass basis,ĝ 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 [57]. 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 2 f /M 2 VL ) coefficients implying that they are generally suppressed. We prove this analytically in Appendix A. 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 U(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 This gives rise to a scale of the Landau pole for g , where µ Z ∼ 1 TeV is the typical scale of the model. Figure 1 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, ∆a µ is explained by 1-loop contributions involving the Z boson and VL leptons. NP contributions to C ( )µ 9,10 are provided by tree-level Z exchange. NP contributions will also affect observables which are currently consistent with the SM such as Br (µ → eγ), Br (τ → µγ), Br (τ → µµµ), 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.

∆a µ and U(1) Charge Assignment
The dominant Z boson contribution to ∆a µ is given by (see e.g. [39,58]), where x a := m 2 Ea /m 2 Z and the loop function is given by The dominant contribution of the scalar χ is given by where y a := m 2 Ea /m 2 χ and the loop function is There are also new contributions from loops involving the SM bosons and the VL fermions, but these are negligible. Figure 2 shows typical values of the muon mass m 2 and the Z contribution to δa µ . For illustration, GeV and y e 22 v H = 0.1 GeV have been fixed while all other couplings except λ e and λ e are set to zero. We see that λ e 0.03 and λ e 10 −3 are required in order to obtain ∆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 λ e 10 −3 .
We see that the Higgs coupling λ e must exist in the model to explain ∆a µ . This explains the non-universal charge assignment in Table 2: The U(1) charges of VL-fermions needs to be opposite for SU(2) L doublets and singlets in order to allow the coupling λ e . For this reason the U(1) gauge symmetry is incompatible with SU(5) unification. However, it is still compatible with the Pati-Salam gauge group SU(4) × SU(2) L × SU(2) R .

Br( i → j γ)
The branching fraction of i → j γ is given by [59] Br where m i and Γ i are the mass and total decay width of the lepton i , while α e is the electromagnetic fine-structure constant. The dominant contributions arise from Z or χ exchange and they are given by and σ R which is given by formally replacing L → R andŶ χ e → Ŷ χ e † in the above expression. Other contributions, involving the SM bosons or σ, only amount to subpercent corrections at our best fit points.

Wilson Coefficients for
The Wilson coefficients defined in Eqs. (1.2) are given by where i = 1, 2, 3 for = e, µ, τ , respectively. We refer to the recent two-dimensional analysis of Ref. [29] and adopt the best fit values of the Wilson coefficients as 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 → eγ. Although it has been pointed out that some flavor universal contributions seem to be favored [29,30,33], we do not discuss this possibility in the present paper.

Neutral Meson Mixing
There are strong constraints on neutral meson mixing [60,61]. The relevant effective Hamiltonian is given by The four-fermi operators are defined as Here α and β are color indices and (F, 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 b → s + − anomalies. The Wilson coefficients induced by Z or neutral scalar exchange, including O(α s ) QCD corrections are given by [62] C VLL are obtained by formally replacing L → R. The off-diagonal element of the B q (q = d, s) meson mass matrix is given by where the first term is the SM contribution and m Bq is the meson's physical mass. The SM contribution for B q -B q mixing is given by [64,65] quantifies the short distance radiative corrections, while f Bq andB Bq 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 3. 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 BqB Bq have been taken from Ref. [66], while those for are taken from Refs. [67,68] and Ref. [69], respectively. The QCD running between the respective lattice scales and µ = 1 TeV has been calculated based on the anomalous dimensions shown in Ref. [70].
The observables for B q -B q mixing are defined as The mass differences ∆M d and ∆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 ∆M d (∆M s ). Note that unlike the analyses in e.g. Refs [61,71], 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 b → s + − (cf. Eq. (3.12)) is disfavored, because this would cause large Z contributions to ∆M s . A large negative contribution Re C µ 9 together with a positive Re C µ 9 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 negative, this would imply a large and positive left-right contribution to ∆M s . However, as the current SM prediction is already larger than the experimental value, therefore the Z coupling with Re C µ 9 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 µ µµ off a nucleus is a rare process that has been observed at a rate consistent with SM expectations [72][73][74]. This process can also be mediated by Z exchange and therefore constitutes an important bound on NP scenarios [40,[75][76][77][78][79]. The ratio of the cross section including NP at the CCFR experiment can be estimated as [79] with a current experimental limit of σ/σ SM = 0.82 ± 0.28 at 95% C.L.. The effective four-fermi couplings ∆g V,A µµµµ in our model are given by where g Z ν νµνµ is defined in the flavor basis, 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 U(1) Y gauge boson, namely the Z of the SM. The Z-Z mixing parameter ε is estimated as where m F (F = L, E, Q, U, D) are the VL mass terms for the VL fermions and g Y is the U(1) Y gauge coupling constant. Current experimental limits are summarized in Ref. [80]. Values of ε ∼ 0.05 cannot be ruled out if the Z is heavier than a few 100 GeV.

χ 2 Fitting
We minimize the χ 2 function where x is a point in the parameter space, while y I (x) is the value of observable I with central value y 0 I 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, ∆a µ , C ( )l 9,10 , and some others. In total we consider 98 observables and they are all listed in appendices B and C.
In total there are 65 input parameters represented by x in our analysis. The bosonic sector has 5 parameters, 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 2, 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  Table 4. Masses and dominant decay modes of new particles are summarized in Tables 5 and 6.
At both best fit points, deviations from the SM in ∆a µ and C µ 9 , C µ 10 are explained by NP contributions. Besides a dominant positive loop contribution to ∆a µ involving the Z , there is a slight cancellation from a negative contribution of the scalar χ at both

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 b → s + − anomalies are studied in Refs. [88,89]. The most stringent bound from the LHC on our model comes from resonance searches in the dimuon channel, 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 5 and 6. Exclusion bounds are given in Ref. [ . This is roughly 4 (6) times smaller than the experimental limit for the respective Z masses [90]. 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 ε ∼ 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 3.2, loops involving Z or the scalar χ together with VL leptons can lead to chirally enhanced contributions to 1 → 2 γ processes. Furthermore, also LFV tree-level Z exchange is possible which could induce → 1 2 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 λ L,E 2 λ L,E 3 , λ 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 Br(µ → eγ) is close to its experimental upper bound. Just like for ∆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. [39,44,94]. 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 A this comes about because mixing between the SM families and the VL family are only induced by the U(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 b → s + − anomalies and B s -B s mixing since these are induced by common operators, see e.g. [61]. The recent lattice results [66,68,95,96] imply that the SM contribution to ∆M s is slightly larger than the experimental central value. The Z -boson contribution to C µ 9 also gives a constructive correction to ∆M s , so that ∆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 b → s + − 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 ∆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 B and C. Very recently it has been argued that experiments may be in favor of CKM non-unitarity [97]. 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 ∆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 [98][99][100][101][102][103][104]. At point A, the lightest VL lepton E 1 is approximately a weak singlet and it decays to hµ, Zµ, and W ν with branching fractions of about 70%, 15%, and 15%, respectively. Given the analysis of Ref. [99], 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. [100]. 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 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, 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. [105,106] 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 [106]. 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, 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 U(1) gauge symmetry. We find parameter points at which the experimentally observed deviations in the muon anomalous magnetic moment ∆a µ and b → s + − 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 U(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 ∆a µ is that the U(1) charge assignment for the VL family is not compatible with an SU(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.

A Analytical Analysis
We discuss the analytical expressions for the couplings in the mass basis. We diagonalize the mass matrix in Eq. (2.6) perturbatively by exploiting m M , where m andM represent the typical mass scales of charged leptons and VL leptons, respectively.
Let us define the unitary matrices, with the four-component vectors and z E i , z L i , which obey the conditions Here,M The rotated mass matrix is In this matrix,ỹ e ij ,ỹ L i ,ỹ R i , andλ e are of the order O(m /v H ), whileM L ,M E ∼M . Here we assume λ e v H m µ , in order for the muon mass to be explained without fine-tuning. Note that λ e v H can be as large as the VL lepton masses if λ 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 z L i = [u L ] ij z L j and z E i = [u E ] ij z E j , for arbitrary 3 × 3 unitary matrices u L and u E , also satisfy the conditions in Eq. (A.3), we can always find a set of vectors z L i , z 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 byỹ L andỹ 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 λ L,E 2 λ L,E 1,3 . In this case, We can show that the unitary matrices block-diagonalize the mass matrix as (A.10) 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 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, Altogether, the fields in the mass basisê L ,ê R can be written as We can now use this in order to study the the scalar and gauge-boson couplings in the mass basis. Using Eqs. (A.1) and (A.8), one can show that and The sizes of the perturbative corrections are estimated as 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. [1], neutrino-nucleon scattering [57] 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 /M q , which only becomes 0.1 if VL quark masses exceed ∼ 1.5 TeV. However, in particular for the up-type quarks we may alternatively assume a hierarchy of couplings,

B.2 Observables
In Tables 7-10 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.