Semi-visible dark photon in a model with vector-like leptons for the ( g − 2) e,µ and W -boson mass anomalies

We propose a model realizes that a semi-visible dark photon which can contribute to the anomalous magnetic moment ( g − 2) of both electron and muon. In this model, the electron g − 2 is deviated from the Standard Model (SM) prediction by the 1-loop diagrams involving the vector-like leptons, while that of muon is deviated due to a non-vanishing gauge kinetic mixing with photons. We also argue that the W -boson mass can be deviated from the SM prediction due to the vector-like lepton loops, so that the value obtained by the CDF II experiment can be explained. Thus, this model simultaneously explains the recent three anomalies in g − 2 of electron and muon as well as the W -boson mass. The constraints on the O (1) GeV dark photon can be avoided because of the semi-visible decay of the dark photon, A ′ → 2 N → 2 ν 2 χ → 2 ν 4 e , where N is a SM singlet vector-like neutrino and χ is a CP-even Higgs boson of the U (1) ′ gauge symmetry.

The model with a U (1) ′ gauge symmetry and the vector-like fourth family is studied in Refs.[45,46] 1 , to explain the muon g−2 and another anomaly in the b → sℓℓ process [59][60][61][62][63][64][65][66][67][68] 2 .In these works, the U (1) ′ gauge boson is assumed to be heavier than 100 GeV, so the gauge boson is called a Z ′ -boson.The muon g−2 is explained by the 1-loop diagrams involving the vector-like leptons via mixing with muons.In this case, however, the electron g−2 can not be explained simultaneously because it causes the lepton flavor violations if the mixing with electrons is introduced.In Ref. [70], it has been shown that the W -boson mass measured by the CDF II [71], m CDF W = 80.4335 (94) GeV, (1.3) which is larger than the previous measurements m PDG W = 80.379 (12) GeV and the SM prediction m SM W = 80.361 (6) GeV [72], can be explained by the 1-loop diagrams involving the vector-like leptons lighter than about 200 GeV.
In this work, we study a new parameter space of the model proposed in Ref. [45,46], where the U (1) ′ gauge boson is much lighter than the Z-boson mass and therefore we call it a dark photon A ′ throughout this work.In such a scenario, the dark photon can explain ∆a µ if it is lighter than O (1) GeV and its gauge kinetic mixing with the photon is O (10 −5 − 10 −2 ) depending on the dark photon mass [73].Note that the dark photon contribution from the gauge kinetic mixing can not explain the negative shift of the electron g − 2 in Eq. (1.2),

Gauge Symmetry ℓ
Table 1: Quantum numbers of the scalars and leptons in the model under the gauge symmetry SU (2) L ×U (1) Y ×U (1) ′ .The index i = 1, 2, 3 runs over the three generations of the SM leptons.
since it is predicted to be positive.In this model, we can explain ∆a e by the 1-loop diagrams involving the vector-like leptons as for ∆a µ in the heavy Z ′ scenario [45,46], without lepton flavor violations.We also point out that the W -boson mass measured by the CDF II can be explained in the same manner as in Ref. [70].Altogether, we study the light dark photon region of the model in Ref. [45,46] in order to explain both electron and muon g−2, as well as m W measured by the CDF II experiment without extending the model.The dark photon explaining ∆a µ is excluded by the experiments if it decays dominantly to e + e − [74][75][76] or invisible particles [77,78].This limit will be relaxed and the dark photon explanation is still viable if the dark photon decays to both visible and invisible particles [79][80][81][82][83], namely if the dark photon is semi-visible.Interestingly, in this model, the SM singlet vector-like neutrino N can be lighter than the dark photon, and then N can decay to the U (1) ′ breaking Higgs boson χ whose dominant decay mode is e + e − .Thus, the decay of the dark photon A ′ proceeds as A ′ → 2N → 2ν 2χ → 2ν 4e which is a semi-visible decay.
The paper is organized as follows.In Sec. 2, we briefly review the model with particular interests in the gauge kinetic mixing.We study the observables, including ∆a e , ∆a µ and m W in Sec. 3, and then discuss signals from the dark photon in Sec. 4. Finally, we draw our conclusions in Sec. 5.The details of the model and the loop functions for the oblique parameters are respectively in Appendices A and B.

The model
We review the model proposed in Refs.[45,46] in which the SM is extended by a U (1) ′ gauge symmetry and a family of vector-like leptons.The matter contents of the model is summarized in Table 1.

Gauge boson sector
Unlike the studies in Refs.[45,46], we explicitly introduce the gauge kinetic mixing of the U (1) ′ and U (1) Y symmetries.The gauge kinetic terms are given by where F µν , F ′ µν and G a µν are the gauge field strengths of U (1) Y , U (1) ′ and SU (2) L , respectively.Here, ϵ is the gauge kinetic mixing factor.We denote the neutral vector fields of U (1) Y , U (1) ′ and SU (2) L by B µ , V µ and W 3 µ , respectively.After the symmetry breaking by the SM Higgs boson and the U (1) ′ breaking scalar Φ, the mass squared matrix for (W 3 µ , B µ , V µ ) is given by where t W := g 1 /g 2 and t Here, g 1 , g 2 and g ′ are respectively the gauge coupling constants of U (1) Y , SU (2) L and U (1) ′ .The canonically normalized mass basis of the gauge bosons are defined as ).In this limit, where The explicit form of these matrices are shown in Appendix A.

Fermion sector
In the gauge basis, the relevant part of the Lagrangian specifying the mass terms of the vectorlike leptons and their Yukawa interactions are given by Here, H = iσ 2 H * and i, j = 1, 2, 3 label the SM generations.After the symmetry breaking via non-zero vacuum expectation values (VEVs) of the scalar fields, v Φ and v H , the mass matrices for e In this work, we do not explicitly introduce the right-handed neutrinos and treat neutrinos as massless particles.As shown in Ref. [46], the phenomenology will not be changed up to O (v H /M Maj ), when we introduce the heavy right-handed neutrinos with Majorana mass M Maj ∼ 10 10 GeV.The mass matrices are diagonalized as where U e L,R and U n L (U n R ) are 5 × 5 (2 × 2) unitary matrices.The leptons in the mass basis are defined as The Dirac fermions are defined as where [n R ] j = 0 for j = 1, 2, 3. Throughout this work, we assume that the U (1) ′ breaking scalar Φ exclusively couples to the first generation, i.e.
so that the lepton flavor violations are not induced from the mixing.As we shall study the dark photon of O (1) GeV, the VEV of Φ is expected to be in this order, which is much smaller than that studied in Refs.[45,46].In this regime, with omitting the mixing with the second and third generations, the diagonalization matrices are approximately given by where (2.12) The first matrices diagonalize the right-lower 2 × 2 block of M e and their analytical forms, as well as the diagonalization of the neutrino mass matrix, are shown in Appendix A. The second matrices approximately diagonalize the small off-diagonal elements of the electron and the vector-like leptons up to the second order in η :

Fermion interactions
The gauge interactions of the leptons with the neutral gauge bosons in the mass basis are given by where with P R := diag(0, 0, 0, 0, 1) =: 1 − P L and P ′ := diag(0, 0, 0, 1, 1).The electric coupling constant is defined as e = g 1 g 2 / g 2 1 + g 2 2 , and the electric charged are Q e = −1 and Q n = 0.The W -boson couplings are given by where (2.16) The U (1) ′ Higgs boson Φ is expanded as where a χ is the Nambu-Goldstone boson absorbed by the dark photon A ′ .The Yukawa interactions of the CP-even Higgs χ are given by where Up to O (ϵ 2 ), the gauge couplings are given by As explicitly shown in Appendix A, we find where will appear in ∆a e expression in Sec. 3.For the U (1) ′ boson couplings, Hence, the Z-boson couplings to the SM leptons are shifted at O (ϵ 2 , η 2 ) and those of the dark photon A ′ appears at ϵ with the sub-dominant contributions at O (ϵ 2 , η 2 ).The off-diagonal couplings of the SM leptons and the vector-like ones are induced at O (η).The structures are similar for the couplings involving the neutral leptons.The Yukawa couplings of the χ boson is approximately given by (2.25) 3 Anomalous magnetic moments and W -boson mass

Anomalous magnetic moments
The 1-loop contribution to the anomalous magnetic moment of the lepton ℓ = e, µ via the neutral gauge boson X = Z, A ′ and the charged leptons is given by where Here, m e B is the mass of the B-th generation charged lepton, with flavor index B = 1, . . ., 5. The index i ℓ = 1, 2 for ℓ = e, µ.The loop functions F Z (x), G Z (x) are  2.
defined in Appendix B. The 1-loop contribution from the χ scalar to ∆a ℓ is given by [84,85] where, y χ e B := m 2 e B /m 2 χ .Also, the loop functions F S (x), G S (x) are defined in Appendix B. Altogether, the new physics contribution to the anomalous magnetic moment is given by where the SM contribution via the Z-boson loop, is subtracted.The contributions from the Z, W and Higgs bosons are negligible because the off-diagonal couplings in the mass basis are suppressed.The Feynman diagrams dominantly contribute to ∆a e and ∆a µ are shown in Fig. 1.
Let us estimate the sizes of ∆a ℓ in our model.From Eq. (2.20), the dark photon contribution to ∆a µ is approximately given by It is turned out that ∆a e is dominantly from the 1-loop diagrams involving the vectorlike leptons along with the dark photon or the χ boson, because of the chiral enhancement proportional to the vector-like lepton masses.From Eqs. (2.20) and (2.24), we find and η e is approximately given by for v H ≪ m E .Thus, the vector-like mass around the TeV-scale can explain the deviation in ∆a e for the Yukawa coupling constants of O (0.1) and v Φ ∼ O (1) GeV.Note that the contribution from the gauge kinetic mixing will be sub-dominant when ∆a µ is explained because the coupling induced by the kinetic mixing is flavor universal and it is estimated as (3.9) For η e ∼ 10 −7 , the Z-boson couplings of the SM leptons are very close to the SM one since the deviation is at O (η 2 e ), see Eq. (2.22).Fig. 2 shows the values of ∆a µ (left) and ∆a e (right) based on our numerical analysis.We see that ∆a µ is explained for ϵ ∼ 0.02 for the 1 GeV dark photon as expected from Eq. (3.5).For (ϵ, m A ′ ) = (0.02, 1 GeV), ∆a e is explained by the vector-like lepton loops if the vectorlike lepton masses are 1.5 TeV (500 GeV) with λ e = 0.1 (0.01), as expected from Eqs. (3.7) and (3.8).Thus, our model provides a unified explanation for both ∆a e and ∆a µ without introducing lepton flavor violations.

W -boson mass
As shown in Refs.[70,86], the W -boson mass shift can be explained by the 1-loop effects of the fourth family vector-like leptons.The T parameter [87,88] has a dominant contribution to this shift compared to the S, U parameters and the T parameter is given by [89,90] where the indices a, b (α, β) run over the neutral (charged) leptons, and The formula of 2πS can be obtained by replacing θ ± → ψ ± (θ ± → χ ± ) in the first line (the second and third lines), while by replacing θ ± → χ ± the formula of −2πU can be obtained.The loop functions are defined in Appendix B. The W -boson mass is given by [91,92] where is the tree-level contribution from the Z-boson mass squared shift ∆m to the kinetic mixing and the W -boson coupling to the SM leptons ∆h L eν := 1−[h L ] 11 ∼ O (η 2 ) 3 .The tree-level contributions are too small to explain the shift in the W -boson mass, and hence T ∼ O (0.1) is necessary to explain the CDF II measurement.In fact, the limit on the dark-photon contributions to the EW precision data is ϵ < 2.7 × 10 −2 for m A ′ ≪ 10 GeV [95], where the most important effect is from the shift of the Z-boson mass which results the shift of the W -boson mass.
The T parameter is approximately given by where we assume The first term in the parenthesis comes from the W -boson contributions involving N 2 and E 1 which are sensitive to the mass difference in the doublet-like states.Since the second term is negative due to the logarithmic term, the T parameter slightly increases as it is suppressed by m E .For m L ≪ m E , the T parameter is estimated as Thus, the shift of the W -boson mass suggested by the CDF II measurement can be explained if 100 ≲ m L ≲ 300 GeV and λ ′ n ∼ 1, so that the mass split between the neutral and charged doublet-like states is sizable 4 .
On the left panel of Fig. 3, we plot the region where the W -boson mass is shifted due to the vector-like lepton loops.The values favored by the CDF II and PDG are explained in the 1σ (2σ) range in the darker (lighter) red and blue regions, respectively.In this plot, the input parameters except m L , m E and λ L = λ E are set to the values at the BP-B shown in Table 2.The value of λ L = λ E are chosen to explain ∆a e ≃ −8.7 × 10 −13 based on the approximated formula in Eq. (3.7), and hence both ∆a e and ∆a µ are explained everywhere on the (m L , m E ) plane.The CDF II value is explained if the doublet-like vector-like lepton is about 200 GeV, while that of the PDG is explained at m L ∼ 500 GeV depending on the singlet mass m E .We shall briefly discuss about the LHC signals of the light vector-like charged leptons in the next section.
Table 2 shows the three benchmark points (BPs) which explain both ∆a e and ∆a µ .At the all points, ϵ = 0.02 and m A ′ = 1 GeV for ∆a µ ∼ 2 × 10 −9 .The Yukawa couplings and vectorlike masses are set to explain ∆a e .As discussed in the next section, we assume the spectrum m χ < m N 1 /2 < m A ′ to realize the semi-visible dark photon compatible with the current limits.We also assume λ n ∼ 0 to keep m N 1 of O (1) GeV.At the BP-A, the vector-like leptons are about 1.5 TeV, and hence the W -boson mass is very close to the SM value.At the BP-B (BP-C), the lightest charged lepton mass is about 300 (500) GeV, so that the W -boson mass favored by the CDF II (PDG) data is explained.We see that the W mass shift is dominantly explained by the T parameter, and the other oblique parameters, S and U , are much smaller.

Signals of light particles 4.1 Semi-visible dark photon
The experiments exclude the dark photon responsible for the muon anomalous magnetic moment if it decays to a pair of electrons or invisible particles [74][75][76][77][78].The invisible dark photons are also searched in meson decays [96][97][98][99].There are limits from deep inelastic scatterings independently to decays of the dark photon, and the current limit for O (1) GeV dark photon is ϵ ≲ 0.035 [100][101][102][103][104], which is larger than our benchmark points ϵ = 0.02.However, the experiments lose sensitivity for the other semi-visible dark photon decay modes, as discussed in Refs.[79][80][81][82].There is the experimental analysis searching for such dark photon at the fixed-target experiment NA64 [83].According to Refs.[82,83], the dark photon explanation for ∆a µ is viable for m A ′ ∼ O (0.1 − 1) GeV if the decay of heavy neutral fermion is fast enough.In our model, the dark photon will dominantly decay to a pair of vector-like neutrinos N 1 if 2m N < m A ′ .Then the vector-like neutrino N 1 will decay to the CP-even Higgs boson χ in the U (1) ′ breaking scalar Φ.The scalar χ subsequently decays to a pair of electrons.Altogether, the decay chain of the dark photon is shown in Fig. 4: which is kinematically allowed if m A ′ /2 > m N 1 > m χ > 2m e .There are two pairs of electrons in the final state accompanied with two neutrinos.Thus, the signal at the experiments will be semi-visible if these decays happen inside detectors whose size is O (1 m).
The first decay A ′ → N 1 N 1 occurs promptly because N 1 ∼ N has the U (1) ′ charge and there is the coupling without suppression from η.The second decay N 1 → χν is relatively long, but is enough short since the coupling is suppressed only by v H /m L .Note that the decay width of N 1 is too small if the scalar χ is much heavier than N 1 so that there is only three-body decays via A ′ or the SM bosons.The decay width of the scalar χ is approximately given by Interestingly, this is directly related to the approximated formula of ∆a e in Eq. (3.7), so that the length of flight of χ is estimated as Thus, the scalar χ decays before reaching or inside the detectors if |∆a e | ∼ O (10 −13 ), whereas the decay can not be detected and thus the signal is invisible if |∆a e | ≪ 10 −13 .The decay widths of A ′ , N 1 and χ as well as the corresponding branching fractions at the BPs are shown in Table 2.We see that the lifetime of A ′ and N 1 are (much) less than O (cm) and these dominantly decay to N 1 N 1 and χν, respectively.Here, we calculated the two-body decays of A ′ to two leptons and that of N 1 to νχ on top of the three body-decays via the gauge bosons which are negligibly small because of the suppressed couplings and the kinetic suppression.Thus, we confirmed that the dark photon decay can be dominated by If χ only decays to two electrons, the length of flight is O (1) cm, and hence this will be detected as prompt decay or displaced vertices depending on the detector design.It is also possible that the χ scalar decays to two pions if there are couplings in the quark sector as for the electrons.
In this case, the lifetime would be shorter.In Ref. [82], the dark photon decay proceeds as where ψ i 's are neutral exotic fermion and ψ 1 is considered to be stable, so that it can be the dark matter.In this scenario, the neutral fermion ψ i decays to three particles via off-shell dark photon, and thus their lifetimes tend to be longer than our case in which the decay chain N 1 → νχ, χ → ee proceeds via only two-body decays.Furthermore, the energy deposits from the χ decay will be larger than those from the decays of ψ i because of the larger phase space.Therefore, the signals form our dark photon will more easily evade from the experimental limits searching for invisible dark photons.We expect that the dark photon of O (0.1 − 1) GeV in our case will not be excluded by the current data.The simulation as done in Ref. [82] is beyond the scope of this work, but the simulation would confirm that the semi-visible dark photon responsible for the lepton magnetic moments would not be excluded by the experiments.

The light vector-like neutrino and U (1) ′ scalar
In the realization of the semi-visible dark photon, the vector-like neutrino N 1 and the U (1) ′ scalar χ should also be O (0.1 GeV).The light vector-like neutrino N 1 mixes with the SM neutrinos through the mixing induced by v Φ and v H . Using the results in Appendix A, the mixing between the light vector-like neutrino and the electron neutrino is approximately given by where h L is defined in Eq. (2.15), and thus this mixing is O (10 −6 ) for our model.This is safely below the current experimental limits on the active-sterile mixing for m N 1 ∼ O (0.1 GeV), see Fig. 6 in Ref. [105].

Vector-like lepton search at the LHC
We briefly discuss the LHC limits for the charged vector-like lepton E 1 , which is expected to be light particularly to explain the W -boson mass shift.The vector-like leptons might be excluded by the LHC limits.For the doublet-like leptons, the mass below 800 GeV is excluded if it decays to the SM particles [115,116].In our model, however, the vector-like lepton E 1 decays to W N 1 , A ′ e and/or χe, as discussed in Refs.[70].The branching fractions of these decay modes of our BPs are shown in Table 2.For the BPs, the dominant decay mode E 1 → W N 1 , followed by N 1 → χν → eeν, has at least two electrons in the final states.This case might be covered by the same search studied in Ref. [70], but there is no study for searching for the cascade decay.Thus, we can not exclude this possibility.In addition, due to the many-body decay cascade, the phase space of the decay E 1 → W N 1 is small and thus the many leptons in the final state are relatively soft.The sub-dominant decay modes E 1 → χe → eee and E 1 → A ′ e → eee have three electrons in the final state.These signals are similar to those from E 1 → Z ′ µ → µµµ, studied in Ref. [70], which excludes the vector-like lepton masses up to 500 GeV for Br(E 1 → eee) ∼ 10%.For our BP-A, Br(E 1 → eee) ≃ 12% and m E 1 ≃ 1.5 TeV, which is safely above this limit.On the other hand, the limit for branching fractions less than 10% are not visible, therefore the BP-B and BP-C whose Br(E 1 → eee) ≃ 5%, may be allowed.We also note that this will not be the case if χ dominantly decays to quarks5 .

Conclusions
In this work, we proposed a scenario in which both anomalies in electron and muon anomalous magnetic moments are explained without extending the model proposed in Refs.[45,46].The discrepancy for electron, ∆a e , is explained by the 1-loop diagrams involving the dark photon and the vector-like leptons, whereas that for muon, ∆a µ is explained by the 1-loop diagrams induced by the gauge kinetic mixing with photons.Since the latter effect is always positive, we can not consider the opposite case in which ∆a e < 0 is explained by the gauge kinetic mixing.Since two discrepancies are explained by the different origins, there is no lepton flavor violations induced by the new particles in the model.We also showed that the W -boson mass measured at the CDF II can be explained if the vector-like lepton is below 300 GeV.Such a light vector-like lepton would be excluded by the high-multiplicity lepton channels at the LHC, depending on its decay modes, as discussed in Sec.4.3.If the light vector-like lepton is not excluded by the LHC, this model can address the three anomalies simultaneously.
The dark photon explanation of ∆a µ is severely constrained by the experiments in the simplest setups.In our model, however, the dark photon can decay to a pair of vector-like neutrinos, A ′ → N 1 N 1 , followed by the decays N 1 → χ(→ ee)ν, so that the dark photon becomes semi-visible which is not excluded by the dark photon searches.We also find that the lifetime of the χ field is directly related to the new physics contribution to ∆a e , and thus our resolution to avoid the invisible dark photons search works only if |∆a e | ≳ 10 −14 .This scenario would be probed by the direct searches for the semi-visible dark photons, or pair productions of the charged vector-like leptons at the LHC, which are subjects of our future works.Our model provides an explicit example of the semi-visible dark photon relying only on two-body decays which are qualitatively different from those considered in the literature.

A Details of the model A.1 Diagonalization of vector boson mass matrix
We show the explicit form of the diagonalization matrix to obtain the canonically normalized mass basis of the vector bosons.We decompose the diagonalization matrix as where E canonically normalizes the kinetic terms, R 1 block diagonalize the massless photon and the others, and R 2 diagonalize the 2 × 2 block of the massive bosons.Their explicit forms are given by where η ± := 1/ √ 1 ± ϵ and Altogether, the diagonalization matrix has the form The masses after diagonalization are given by Up to the second order in ϵ and t V , (A.9)

A.2 Diagonalization of the fermion mass matrices
We show the diagonalization matrices of the leptons, Here, we omit the second and third generations under the assumption of Eq. (2.10).We also assume that m L , m E , m N > 0. The diagonalization matrices of the charged leptons are given by where The second matrices diagonalize the mixing between the first generation and the vector-like lepton.The singular values are given by 6 where For the neutrinos, the diagonalization matrices are given by where The first matrix is to rotate away the (2, 1) element, and then the (1, 2) element is rotated away by the second matrix.The angles in the last matrix, 6 The diagonal elements after the rotation by the first matrix are given by, where µ E1,2 are, in general, not positive.Under the assumption, m L , m E > 0 and v H ≪ m E , µ Ea > 0, and thus µ Ea = m Ea given by Eq. (A.17).
c n L,R and s n L,R , are given by formally replacing and s e L,R shown in Eq. (A.12).The singular values m N 1 and m N 2 are respectively obtained by the same replacement from m E 1 and m E 2 in Eq. (A.17).Note that the diagonalization for M n is exact, not relying on any approximation.
For v H ≪ m E and m L < m E , the mixing angles are approximately given by The masses of the vector-like leptons are given by GeV is necessary to make the vector-like neutrino N 1 light so that the dark photon can decay.We shall assume λ n = 0 for simplicity.The neutrino mixing angles are approximately given by and the vector-like neutrino masses are given by (A.24)

A.3 Lepton couplings
The approximate forms of E A , N A and h A are given by up to O (η 2 ) and O s 2 L 2 .Here, we take s L 1 = 0 and the sub-dominant contributions in the lower-right 2 × 2 block are omitted.For the Z ′ -boson couplings, The Yukawa couplings are given by

B Loop functions
The loop functions for ∆a ℓ are given by The loop functions for the oblique parameters are given by θ + (y Here, the function f is defined as

Figure 1 :
Figure 1: The Feynman diagrams dominantly contribute to the ∆a e (left and middle) and ∆a µ (right).

Figure 2 :
Figure 2: The left panel shows ϵ versus ∆a µ with m A ′ = 0.2, 1, 5, 25 GeV.The dark (light) purple region is the 1σ (2σ) range.The right panel shows m L = m E versus −∆a e with λ e = 0.001, 0.01, 0.1, 1.0.The dark (light) green region is the 1σ (2σ) range.The input parameters other than m L = m E are chosen as the BP-A shown in Table2.

. 14 )Figure 3 :
Figure 3: On the left panel, m W is explained the CDF II and PDG result within the 1σ (2σ) range in the darker (lighter) red and blue regions, respectively.The solid line are the masses of the lightest charged exotic lepton m E 1 = 100, 200 and 500 GeV from bottom to top (left to right).The value of λ L = λ E chosen to explain ∆a e .On the right panel, ∆a e = −8.7 × 10 −13 on the green line, and it is within the 1σ (2σ) range in the darker (lighter) green region.The red lines are the length of flight of χ.The inputs are those at the BP-B in Table 2 except (m L , m E ) (and λ L = λ E ) on the left (right) panel.

E 2 L
s e L η e /λ E −c e L η e /λ E s e L η e /λ E c 2 e L c e L s e L −c e L η e /λ E c e L s e L s 2 c e R η e /λ L −s e R η e /λ L c e R η e /λ L c 2 e R −c e R s e R −s e R η e /λ L −s e R c e R s 2

Table 2 :
Values of the inputs and the outputs at the benchmark points.At the all points, the other inputs not shown in the table are set to ϵ = 0.0203, λ n = 0 , (m V , m χ , m N 1 ) = (1.0,0.3, 0.4) GeV and v Φ = 2 √ 2 GeV.The mass parameters are in the unit of GeV unless it is specified.