Searching for new physics effects in future W mass and sin 2 θ W ( Q 2 ) determinations

We investigate the phenomenology of the dark Z boson, Z d , which is associated with a new Abelian gauge symmetry and couples to the standard model particles via kinetic mixing ε and mass mixing ε Z . We examine two cases: (i) Z d is lighter than the Z boson, and (ii) Z d is heavier than that. In the first case, it is known that Z d causes a deviation in the weak mixing angle at low energies from the standard model prediction. We study the prediction in the model and compare it with the latest experimental data. In the second case, the Z - Z d mixing enhances the W boson mass. We investigate the effect of Z d on various electroweak observables including the W boson mass using the S , T , and U parameters. We point out an interesting feature: in the limit ε → 0, the equation S = − U holds independently of the mass of Z d and the size of ε Z , while | S | ≫ | U | in many new physics models. We find that the dark Z boson with a mass of O (100) GeV with a relatively large mass mixing can reproduce the CDF result within 2 σ while avoiding all other experimental constraints. Such dark Z bosons are expected to be tested at future high-energy colliders.


I. INTRODUCTION
The standard model (SM) is excellent in describing the behavior of the particles and various experimental results [1].However, there exist some mysterious phenomena which are unexplained in the SM such as neutrino oscillations [2,3], dark matter [4], baryon asymmetry of the Universe [4,5], and some experimental anomalies in the measurements of the g µ − 2 [6], the W boson mass [7], and so on.They suggest new physics beyond the SM.Thus, it is important to investigate how such new physics models can be tested in current and future experiments.
In Ref. [8], a new physics model with a new dark gauge symmetry U (1) d has been proposed, which is called the dark Z model.Although the SM particles do not have a dark charge, they can interact with the dark gauge boson Z d via kinetic and mass mixing.In this model, in addition to the gauge sector, the Higgs sector is also extended by adding the second Higgs doublet Φ 2 and the dark singlet Φ d which carry the dark charge.Z d acquires the mass from the vacuum expectation values (VEVs) of Φ 2 and Φ d , which is a quite different feature from the typical dark photon model of the kinetic mixing [9], where only the VEV of Φ d induces the mass of the dark gauge boson.The VEV of Φ 2 causes the mass mixing which is independent of the kinetic mixing and provides a new source of the parity violation in four fermion processes [8].The phenomenology of the dark Z model has been studied in Refs.[8,[10][11][12][13][14][15][16][17][18][19].
In this paper, we examine the phenomenological impact of the model on the measurements of the running weak mixing angle and the W boson mass.First, we focus on the dark Z bosons whose mass is smaller than the Z boson mass.As studied in the previous works [8,10,16], such light dark Z bosons can make a deviation in the weak mixing angle at low energies.We update results from previous works with the latest experimental data.
Second, as a main new part of this paper, we consider the dark Z bosons which are heavier than the Z boson.Such dark Z bosons can enhance the prediction of the W boson mass via the deviations in the mass and the gauge couplings of the Z boson induced by the kinetic mixing ε and the mass mixing ε Z [20][21][22].The deviation in the SM gauge sector is described by the S, T , and U paramters [23].We derive the S, T , and U parameters in the model and discuss the effect of Z d on various electroweak observables including the W boson mass.We investigate their behavior in detail and find a remarkable fact that the equation S = −U holds in the limit ε → 0 independently of the mass of Z d and the size of the mass mixing ε Z , while |S| ≫ |U | in many new physics models [24].We show that the effect of heavy Z d can be large enough to explain the W boson mass anomaly reported by the CDF collaboration [7] while avoiding the constraint from electroweak global fit.Such dark Z bosons are expected to have a mass of O(100) GeV and relatively large mass mixing.We also discuss the direct searches of Z d at LHC.The model can explain the W boson anomaly, consistent with the current LHC data in some mass regions of Z d .Such a dark Z boson is expected to be tested at future high-energy colliders.
The rest of this paper is organized as follows.In Sec.II, we overview the dark Z model.In Sec.III, we review the effect of the dark Z boson on the running weak mixing angle with the latest experimental results.In Sec.IV, the W boson mass in the dark Z model is discussed.We will show that the latest CDF-II result, which is significantly different from the SM prediction, can be explained in the model under the current constraints from the electroweak precision measurement and collider searches.The summary and conclusion are presented in Sec.V. Some formulas and discussions of the running weak mixing angle are shown in Appendix A. The effect of a heavy Z d on the S, T , and U parameters is discussed in Appendix B using higher-dimensional operators.

II. DARK Z MODEL
In this section, we review the dark Z model proposed in Ref. [8] and investigated in Refs.[8,[10][11][12][13][14][15][16][17][18].This model has a new Abelian gauge symmetry denoted by U (1) d in addition to the gauge symmetries in the SM, SU (3) C × SU (2) L × U (1) Y .Fermionic fields in the model and their quantum numbers are the same as those in the SM, i.e. the quarks and leptons do not carry the charge of U (1) d , which is denoted by Q d in the following.
Although the SM fermions do not carry the U (1) d charge, they interact with a new gauge boson via mixings including a kinetic mixing [9].The kinetic terms for the Abelian gauge bosons are given by where Fµν = ∂ µ Fν −∂ ν Fµ with F = B or Z d , and Bµ and Ẑdµ are gauge bosons associated with U (1) Y and U (1) d symmetries, respectively.The angle θ W is the weak mixing angle defined as in the SM; where g and g ′ are coupling constants of the SU (2) L and U (1) Y gauge interactions, respectively.The kinetic mixing is described by the dimensionless free parameter ε, whose normalization follows Ref. [8].The kinetic terms can be diagonalized by the following GL(2, R) transformation; where c W = cos θ W .
The discussion so far is common with the dark photon [9].However, the nature of the electroweak symmetry breaking in the dark Z model is quite different from that in the dark photon model.The dark Z model includes three scalar fields: two isospin doublets with the hypercharge Y = 1/2 (Φ 1 and Φ 2 ) and an isospin singlet with Y = 0 (Φ d ).They are color singlets, and their U (1) d charges are given by Q They acquire VEVs as follows; where i = 1, 2. The electroweak symmetry is broken by v 1 and v 2 , while the U (1) d symmetry is broken by v 2 and v d .The dark photon model corresponds to the limit v 2 → 0. For a detailed study of the Higgs sector in the model, see Ref. [11]. 1he VEVs v 1 , v 2 , and v d give masses to the electroweak and dark gauge bosons.Their mass terms are given by where ) being the gauge fields of the SU (2) L symmetry.W ± µ correspond to the W bosons with the mass mW = gv/2 = 80.4 GeV, where The squared masses of the neutral gauge bosons are given by a 2 × 2 symmetric matrix M2 V , whose elements are given by with m2 Z and m2 where and g d is the coupling constant of the U (1) d gauge interaction.Off-diagonal terms are proportional to a mass mixing parameter ε Z given by where the angle β satisfies tan β = v 2 /v 1 .The mass mixing ε Z is independent of the kinetic mixing ε.Therefore, ε Z provides a new source of the interactions between the SM fermions and the additional gauge boson.M 2 V can be diagonalized by an appropriate SO(2) rotation with an angle ξ; We identify Z µ and Z dµ as the observed Z boson and the dark Z boson, respectively.The mixing angle ξ then satisfies where m Z ≃ 91.2 GeV and m Z d are the masses of the Z boson and the dark Z boson, respectively.
In the following, we assume ε and ε Z are small and use perturbative expansions.Up to the quadratic order, m 2 Z , m 2 Z d , sin ξ and cos ξ are given by where r = mZ d / mZ , and r = m Z d /m Z .These formulas give a good approximation in the case that both |ε Z | and |εt W | are sufficiently smaller than one and |1 − r2 |.We extended the results of the previous works, which used only the leading order terms, to include the higher order terms 2 as they may be crucial in discussing the W boson mass studied in Sec.IV.In the limit ε Z → 0, Eqs. ( 13)-( 16) reproduce the results in the simplest dark photon model [20,25].One should note that r is different from r at the quadratic order.While r is the ratio of the physical masses m Z and m Z d , r has a physical meaning only in the no mixing limit, ε Z → 0 and ε → 0. The relation between them is given by r2 The kinetic and mass mixings lead to the interaction between the dark Z boson and the SM fermions as follows; where s ξ = sin ξ, c ξ = cos ξ, e is the coupling constant of the electromagnetic interaction, and J µ em and J µ NC are the electromagnetic and the weak neutral currents, respectively.At the leading order of the mixing parameters, the current interactions can be approximated by The ε Z induces an additional parity-violating source compared to the dark photon model (ε Z → 0).It gives a particularly important effect when m Z d ≪ m Z .In that case, L d is approximately given by 2 Equation ( 14) coincides with the result in Ref. [11].
Although the parity-violating effect induced by ε is much suppressed by r 2 = m 2 Z d /m 2 Z ≪ 1, that proportional to ε Z does not have such a suppression.Since ε Z is independent of ε, the dark Z model can provide a larger parity violation compared to the dark photon model, with a light gauge boson.Such an effect can be tested by precisely measuring the weak mixing angle at low-energy experiments as discussed in Refs.[8,10,16] and the next section.
Before closing this section, we describe a parameter δ satisfying which allows smooth m Z d → 0 behavior for ε Z -induced amplitudes involving Z d [8].By using δ, the neutral current interaction is given by −(g Z /2)rδ ′ J µ NC Z dµ , where in the case of m Z d not too small compared to m Z [16].
The parameter δ ′ is particularly convenient when we investigate the phenomenology of the light dark Z boson.

III. RUNNING WEAK MIXING ANGLE AND THE LIGHT DARK Z BOSON
As discussed in previous works [8,10,16], the light dark Z boson can shift the value of the weak mixing angle sin 2 θ W from the SM prediction at low energies Such a deviation can be tested by precise measurements of the running of the weak mixing angle.In this section, we show the latest experimental values of the weak mixing angle and compare them with the SM prediction at m Z and also using the running weak mixing angle, which provides an update of previous work [16], using the latest experimental data.

A. Running weak mixing angle
First, we shortly review the current situation of the weak mixing angle measurements.There are several ways to define the weak mixing angle [1].We employ the one defined in the modified minimal subtraction (MS) scheme; where ê(µ) and ĝ(µ) are the gauge couplings of the electromagnetic and SU (2) L gauge interactions, respectively, evaluated at the relevant mass scale µ.The renormalization scale µ is usually set to be m Z , and the SM prediction is given by [1] (SM) sin 2 θW (m Z ) = 0.23122(04).
The value of sin 2 θW at the Z pole is measured by high-energy colliders, LEP [26], SLC [26], Tevatron [27][28][29] and LHC [30][31][32][33].In these experiments, the value of the effective angle for leptons sin 2 θ lept eff [34] was measured.We note that the value of sin 2 θ lept eff is different from that of sin 2 θW (m Z ) although both are defined at the Z pole.The relation between them was investigated in Ref. [35] and is numerically given by [1] sin 2 θ lept eff ≃ sin 2 θW (m Z ) + 0.00032.(25) We use this equation to derive the value of sin 2 θW (m Z ) corresponding to the observed sin 2 θ lept eff values in each experiment.
At LEP and SLC, various asymmetries at final states were precisely measured, which are sensitive to parity violation.The average values of sin 2  θW in leptonic and hadronic (semileptonic) processes are given by (leptonic) sin 2  θW (m Z ) = 0.23081( 21), (hadronic) sin 2 θW (m Z ) = 0.23190( 27), (27) respectively [26].The former is derived from observations of the lepton forward-backward (FB) asymmetry of a charged lepton ℓ (A 0,ℓ FB ) and the τ polarization (P τ ) at LEP and the lepton asymmetry parameter (A ℓ ) at SLC.The latter is derived from A 0,b FB , A 0,c FB and the hadronic charge asymmetry Q had FB at LEP. Results from each observable are shown in Fig. 7.6 of Ref. [26].The intriguing point is that the average values in leptonic and hadronic processes differ by 3.2σ [26].A new physics scenario to explain this tension was proposed in Ref. [36].However, the cause of it is still unclear.
At Tevatron and LHC, the weak mixing angle was measured in the FB asymmetry of e + e − and µ + µ − .Although their center-of-mass beam energy is much higher than m Z , the observed quantity is at the Z pole because the invariant mass of the final state is dominated by values around the Z pole.Their results are given by (Tevatron) sin 2  θW (m Z ) = 0.23116(33), (LHC) sin 2 θW (m Z ) = 0.23097 (33), (29) where the former is the combined result of CDF [27] and D0 [28] experiments given in Ref. [29], and the latter is the average of ATLAS [30,33], CMS [32] and LHCb [31] experiments given in Ref. [1].
The weak mixing angle also has been measured in low-energy experiments such as atomic parity violation (APV) in 133 Cs [37,38] and low-energy accelerators: Qweak (e − − p elastic scattering) [39], E158 (Møller scattering) [40], and PVDIS (deep inelastic scattering of e − and deuteron) [41]; (Qweak) sin 2 θW (m Z ) = 0.2308 (11), (PVDIS) sin 2 θW (m Z ) = 0.2299 (43), (33) where the relevant energy scales of each experiment are given by Q ≃ 2.4 MeV [42], 157 MeV [39], 161 MeV [40], and 1.38 GeV [41], respectively. 3Although there are other low-energy measurements of the weak mixing angle, we do not show their results here because of their relatively large uncertainties [1].In Figs.1(a) and 1(b), we summarize the experimental values of sin 2  θW (m Z ) and the SM prediction.In  (26) [Eq.(27)] with 1σ uncertainties.The dashed lines in the bands represent the central values.The green solid line in both figures shows the SM prediction in Eq. ( 24) with a very small 1σ uncertainty.All experimental results are consistent with the SM prediction.However, it is interesting that the leptonic and hadronic average values deviate from the SM prediction in opposite directions.
Another way to compare the SM prediction and the experimental values is to use the running of the weak mixing angle [43][44][45][46][47][48]; where κ(q 2 ) is the form factor including the electroweak radiative corrections evaluated in the MS scheme.We employ κ(q 2 ) discussed in Refs.[45,46], which is defined in the context of the radiative correction in Møller scattering [49].Then, the main radiative correction comes from the γ-Z self-energy and the anapole moment [45,46,49].The concrete formula for κ(q 2 ) is shown in Appendix.A. We also provide a careful comparison of the running weak mixing angle formula we adopted and another formula based on the pinch technique [47] in the Appendix A.
By using Eq. ( 34) and the formula of κ(q 2 ), we can evaluate the SM prediction of the running as a function of q 2 , the momentum transfer of neutral current processes.We can also evaluate sin 2 θW (q 2 exp ) corresponding to the measured sin 2  θW (m Z ) values, where q 2 exp is typical energy scale of each experiment.
In Fig. 2, we show the SM prediction and the experimental values of the running sin 2 θW (q 2 ).The black solid (dashed) lines are the SM prediction for spacelike (timelike) momenta.The black points are experimental values corresponding to sin 2 θW (m Z ) shown above with 1σ uncertainties.The point "lep" ("had") represents the LEP and SLC average values in Eq. (26) [Eq.(27)].The blue arrows mean that the points "Tevatron" and "LHC" are  26) [Eq.( 27)] with 1σ uncertainties.The SM prediction is also shown with the green line.
data at the Z pole.The points at the Z pole are obtained from measurements of processes with timelike momentum transfer (q 2 > 0).On the other hand, the points at low energies are data from processes with spacelike momentum transfer (q 2 < 0).
In Fig. 2, we also show the anticipated sensitivity in future experiments: APV in Ra + [50], P2 with the option of the proton target and the carbon target [51], Moller [52], SoLID [53], SuperKEKB with a polarized electron beam [54], and Electron Ion Col-lider (EIC) [55,56] with the red points "APV(Ra + )," "P2(ep)," "P2(eC)," "Moller," "SOLID," "SKEKB," and "EIC," respectively.The EIC has several beam options [55,56] Z d [8,16].Such a deviation is expected to be tested in future low-energy experiments shown in Fig. 2. In this section, we are interested in low-energy experiments, and we consider neutral current processes with spacelike momentum transfers As explained in Sec.II, the dark Z boson interacts with the SM fermions via kinetic and mass mixing.Thus, it modifies the weak neutral current amplitudes and the weak mixing angle.The modification in the weak mixing angle is described by a factor κ d (Q 2 ) [8,16] given by By using κ d , the running weak mixing angle in the dark Z model is given by Eq. (34).At The dark Z boson effect on the running weak mixing angle is proportional to εδ ′ .Thus, we consider the experimental bound for the combination of the mixing parameters εδ ′ .The kinetic mixing is constrained by the electroweak precision tests [8].Following Refs.[57,58], we employ the bound As the constraint on δ ′ -m Z d plane, Refs.[8,12,16] investigated the rare Higgs decay ).We revisit this constraint with the latest result of the rare Higgs decay searches at the ATLAS [59] and CMS [60] experiments.The branching ratio of H → ZZ d is given by where The mixing angle α is defined in the context of two Higgs doublet models and diagonalizes neutral CP-even scalar states [61]. 4We here assume that the heavier CPeven scalar boson H is the SM Higgs boson with the mass 125 GeV.In the formula of BR[H → ZZ d ] shown in the previous works [8,12,16], s β−α /t β = 1 is implicitly assumed, while it is retained Eq. (38), which is consistent with that in Ref. [11].In the following, we consider the constraint in the case that s β−α /t β = 1 according to Refs.[8,12,16].However, it is interesting to note that the branching ratio for H → ZZ d vanishes at tree level in the alignment limit cos(β − α) = 1, where the Higgs boson couplings coincide with the SM prediction at tree level.
The branching ratio Br(H → ZZ d → 4ℓ) is constrained by the current data at the ATLAS [59] and CMS [60] experiments.The CMS result gives a slightly stronger bound on the δ ′ -m Z d plane.By using Br(Z d → e + e − ) + Br(Z d → µ + µ − ) ≃ 0.3 [62], the current bound on δ ′ with s β−α /t β = 1 is given by for 15 GeV < m Z d < 35 GeV.For lighter dark Z bosons, for example, m Z d = 10 GeV, there is almost no constraint from H → ZZ d because of contamination from heavy quarkonia [59,60].We note that this constraint becomes weaker if we consider a dark decay channel of Although we have obtained the constraint for |ε| [Eq.(37)] and |δ ′ | [Eq.(39)] for m Z d = O(10) GeV, we will comment on other conceivable experimental constraints in the following.
Refs. [8,19] investigated the rare meson decays K → πZ d and B → KZ d .These are generated at one-loop level [8] because the tree-level flavor changing neutral current via the additional Higgs bosons is prohibited by U (1) d symmetry [63].These processes give an ef- GeV [8].However, in the relevant mass region (m Z d = O(10) GeV), the constraint is expected to be inefficient.
For the relevant mass region, a strong bound comes from the dimuon resonance search of Z d at CMS [64] and LHCb [65] experiments.In the pure dark photon case (ε Z = 0), it is given by for m Z d = O(10) GeV if Z d decays only into the SM particles.This bound is stronger than that from the electroweak precision measurements in Eq. ( 37).However, this can be diluted if a decay channel into new light dark sector particles χ opens.Since the decay rate Z d → χ χ is not suppressed by the kinetic and mass mixing, it can be a dominant decay mode, and BR[Z d → µ + µ − ] can be small enough to avoid the constraint.
The light Z d can also be generated via the Z boson decay resulting in the four-lepton final state: Z → Z d ℓ + ℓ − → 4ℓ [66].If Z d decays only into the SM particles, the prediction on the decay obtains a constraint from Eqs. (39) and (40).For m Z d = 10 GeV, it is given by BR . If Z d predominantly decays into χ χ, the constraints in Eqs.(37) and (39) On the other hand, the current data for the four-lepton decay of the Z boson is BR(Z → 4ℓ) = (4.55 ± 0.17) × 10 −6 [1].Thus, in both cases, the signal from the on-shell Z d is hidden in the uncertainties, and we expect no constraint from this decay channel.
In the following, we consider constraints available in the case that Z d predominantly decays into χ χ, where the severe constraint in Eq. ( 40) can be avoided.Then, the exotic Higgs decays H → Z(γ)Z d → Z(γ) E are induced, where χ χ are identified as missing energy.H → ZZ d is generated at tree level as shown in Eq. ( 38), and we expect that the experimental constraint from this channel is weaker than Eq.(39) is the branching ratio calculated in the SM.Here, we have neglected the interference terms because they are not expected to drastically change the result.The experimental upper limit of BR[H → γZ d ] for Z d up to 40 GeV is set to be 2.7-3.1% at 95% CL [68].Thus, this process also does not give a constraint on the relevant parameter region.
The light dark sector particle χ may cause a deviation in the invisible decay rate of the Z boson, which is strongly constrained [26], through the Z-Z d mixing.However, we found that this constraint is weaker than the above constraints. 5Such a light dark sector particle could be searched for in beam dump experiments such as the proposed BDX [69].
As a result, we consider the case of BR(Z d → χ χ) ≃ 1 in the following and employ the bounds in Eqs. ( 37) and (39) for δ ′ and ε, respectively.We thus have the following current experimental bound on εδ ′ ; We note that this bound is not effective for m Z d < ∼ 10 GeV because of no constraint from BR[H → ZZ d ] in this mass region.
We perform the χ 2 fitting of εδ ′ and the measured sin 2 θW (−Q 2 ) at low energy experiments: APV in 133 Cs, E158, Qweak and PVDIS. 6Input data are shown in Table I.As a benchmark point, we consider two cases m Z d = 15 GeV and m Z d = 10 GeV.We found the following best-fit values of εδ ′ εδ ′ = −0.00025(29) with χ 2 = 3.4 in both cases.The fitted values are consistent with the SM prediction (εδ ′ = 0) within 1σ because the experimental values do not deviate significantly from the SM prediction.However, with future improved accuracy, it may be possible to observe the effects of such light dark Z bosons unless εδ ′ is too small.
We show the results of the fitting for m Z d = 15 GeV and 10 GeV in Figs.3(a) and 3(b), respectively.Here, the black solid curves represent the SM prediction for spacelike momenta.The effect of the dark Z boson with the fitted εδ ′ is shown with the blue bands with 1σ uncertainties.The dashed curves in the bands are the result with the best-fit value of εδ ′ .The darker blue bands represent points within the 1σ regions of the fitting which satisfy the experimental bound in Eq. (41).On the other hand, points in the lighter blue band are within the 1σ region but excluded by Eq. ( 41).We note that there is no light blue (excluded) region in Fig. 3 Before closing this section, we briefly comment on the constraints on the Higgs potential in the model.Because we assume no mixing between the doublet scalars and the singlet scalar, the latter does not couple to the SM fermions and gauge bosons at tree level.If the singlet scalar is lighter than m H /2, it can induce the Higgs invisible decay, which is constrained by the current LHC data [1].This constraint can be avoided by considering a singlet scalar heavier than m H /2, or a sufficiently small Higgs portal coupling.
The above constraints can be avoided by appropriately choosing the parameters in the Higgs potential.In particular, in the alignment limit, we can take the decoupling limit of the new particles [91], where all of their effects are described by higher-dimensional operators suppressed by the new physics scale.In such a limit, heavy additional Higgs bosons can naturally avoid all the constraints.In addition, the 125 GeV Higgs couplings coincide with the SM ones in this limit.Although the deviation from the alignment limit is severely constrained by the current LHC data, there still exist allowed parameter regions near (but not) the alignment limit [72,74].Since the above analysis on the running weak mixing angle is almost independent of the parameters in the Higgs potential, 7 we can always choose the allowed parameters without changing sin 2 θ(q 2 ).Higgs phenomenology in the model is further discussed in Ref. [11].

IV. W BOSON MASS AND A HEAVY DARK Z BOSON
In this section, we discuss the effect of a dark Z boson on the W boson mass.The mixing between Z and 7 Although the mass mixing is proportional to sin 2 β, the choice of it does not conflict with the constraints as long as |ε Z | is small.The Yukawa couplings of the charged and the CP-odd Higgs bosons are proportional to tan β.That of the additional CP-even Higgs boson is also approximately proportional to tan β near the alignment limit.Thus, smaller sin β makes experimental constraints weaker.Our definition of tan β is the inverse of the typical definition in the two Higgs doublet models in Ref. [11].
Z d induces deviations in the SM gauge sectors.Thus, predictions of the masses and the couplings of the gauge bosons change from the SM ones.In particular, as shown below, heavy dark Z bosons enhance the W boson mass, and it can explain the anomaly found by the CDF collaboration [7].
A. Anomaly in the W boson mass measurement In this section, we briefly review the current situation of the W boson mass measurements, in particular, the anomaly reported by the CDF collaboration [7].We also briefly discuss the new physics interpretation of this anomaly.
The W boson mass has been measured in both e + e − and hadron collider experiments.Before April 2022, the world average value was given by m World Ave.W = 80.377 (12) GeV. ( The above is in good agreement with the SM prediction from the electroweak global fit [1]; In April 2022, the CDF collaboration reported the result of the W boson mass measurement using the full Run-II data [7] as follows; This value is significantly different from both the world average value and the SM prediction.Because of the reduced uncertainty in the CDF-II result, it is a 7.0σ deviation from the SM prediction [7].Although the tension is reported by only one group, the deviation is considerably large, making it a potentially interesting hint for new physics.
Using the oblique parameters S, T and U [23], we can generally discuss the new physics effect on the W boson mass.The deviation in the squared W boson mass from the SM one is given by [23] ∆m where s 2 W = sin 2 θ W .In order to reproduce the central value of the CDF-II result, is required.Since S, T , and U also cause deviations in other electroweak observables, we have a constraint on them to keep the other observables consistent with experimental data.As a result of the electroweak global fit using the CDF-II result [92,93], the following results are obtained [92] 8 ; S = 0.06 (10), T = 0.11 (12), U = 0.14(09).(49) In previous works [25,94,95], it has been revealed that the simple dark photon model cannot reproduce the CDF-II result within 2σ because of tight constraints from electroweak precision measurements in Eq. ( 49).On the other hand, it has been shown that the effect of mass mixing can be sizable enough to reproduce the CDF-II result [96], where the authors investigate an extension of the dark Z model with additional vector-like leptons [97].However, they consider only the ρ parameter, or equivalently the T parameter.The S and U parameters are ignored, and the constraint on the dark Z boson from the electroweak precision measurements is not investigated in a detailed fashion.In addition, only a limited parameter space is studied in Ref. [96]; they assume ε = 0 and show the results for a few benchmark values of m Z d .
In this paper, we consider all the S, T , and U parameters and discuss their behavior in detail.The electroweak-global-fit constraint on these parameters is found by using Eq.(49).Considering this and collider bounds on Z d , we exhaustively examine parameter space where the W boson mass anomaly can be explained while avoiding all the experimental constraints.

B. Effect of dark Z bosons on the electroweak observables
In this section, we discuss the dark Z boson effect on various electroweak observables using the S, T , and U parameters.We give formulas for the S, T , and U paramters and discuss their behavior in detail.
In the dark Z model, kinetic mixing and mass mixing induce shifts in the mass and the current interaction of the Z boson from those in the SM at tree level.The effect of such shifts on various observables can be described in terms of the S, T and U parameters [20].In the following, we do not consider other new physics effects in four-fermion processes for simplicity, for example, oblique corrections induced by additional Higgs bosons and the dark Z mediation induced by the current interaction in Eq. (19).Since the former is a 1-loop effect, we can consider a parameter region such that their contribution is subleading and small enough. 9Although the latter is a 8 When U = 0 is assumed, the result is given by S = 0.15(08) and T = 0.27(06) in Table III of Ref. [92].Since the U parameter can be as large as the S and T parameters in the dark Z model, we employ the result in Eq. ( 49). 9 Large mass differences among the additional Higgs bosons can enhance the S and T parameters large enough to explain the W boson mass anomaly [92,[98][99][100][101].Here, we do not consider such parameter regions to investigate the effect of the kinetic and mass mixing.We choose the parameters in the Higgs potential such that all the constraints on them can be avoided as explained at the end of Sec.III B.
tree-level effect, it is expected to have little impact on observables at the Z pole unless m Z d is very close to m Z [95].
In the dark Z model, the mass terms and the current interactions of the SM electroweak gauge bosons are given by Deviations from the SM are described by ∆ 1 , ∆ 2 and ∆ 3 .Up to the quadratic order of ε (ε = ε or ε Z ), they are evaluated as These expansions are valid as long as |εt W | and |ε Z | are sufficiently smaller than one and |1 − r 2 |.In Eq. ( 51), we have replaced r2 in m 2 Z given by Eq. ( 13) with r 2 because the difference is higher-order of ε as one can see in Eq. (17).
The S, T and U parameters can be found by using ∆ 1 , ∆ 2 , and ∆ 3 as follows [20,21]; Equation ( 55) is consistent with a formula for the ρ parameter induced by the mass and kinetic mixings in Refs.[97,102].We also note that the U parameter in Eq. ( 56) is always positive which is preferred to explain W boson mass anomaly [92,93] [see Eq. ( 49)].
In order to investigate the behavior of the S, T , and U parameters, we consider two significant cases: the dark photon (DP) limit (ε Z → 0) and the pure dark Z (DZ) limit (ε → 0).First, we discuss the DP limit.Then, S, T , and U are given by where a subscript DP means that the quantities are evaluated in the DP limit.These formulas are consistent with those in the dark photon model in Refs.[20,25]. 10 We see that S DP , T DP , and U DP are all proportional to (ε/(1 − r 2 )) 2 .In Fig. 4(a), we show the behavior of S DP , T DP , and U DP normalized by the positive common factor α −1 (ε/(1 − r 2 )) 2 .T DP is always negative, while U DP is positive.We note that the negative T parameter is not favored to explain the W boson mass anomaly [See Eq. ( 49)].The sign of S DP depends on r.When r is larger (smaller) than c W ≃ 0.88, S DP is negative (positive).We note that the above behavior is correct only when |ε/(1 − r 2 )| ≪ 1 is satisfied because we use the perturbative expansion.The suppression of U DP is understood in terms of the standard model effective field theory (SMEFT).In the SMEFT, the S and T parameters are generated by dimension-six operators, while the U parameter is generated by a dimension-eight operator [24].Thus, U is generally expected to have a relative suppression factor Λ −2 10 Although Ref. [95] also studies the S, T , and U parameters in the dark photon model, their formulas do not coincide with Eqs. ( 57)-( 59) even accounting for differences in notation.This is because Ref. [95] takes the formulas from Ref. [22], which uses perturbative expansions by sin ξ ≃ ξ, not ε, up to the linear order including a part of quadratic terms.By using this method, the S, T , and U parameters in the dark Z model are given by αS ), and αU ≃ 0. However, these formulas are valid only in the case of r 2 ≫ 1 (m Z d ≫ m Z ) because some terms proportional to ξ 2 are dropped while the term (r 2 − 1)ξ 2 remains in the calculation.
compared to S and T , where Λ is the cut-off scale.This is the case for the decoupling limit of many new physics models including the dark photon model.In Appendix B, we present a detailed discussion of higher-dimensional operators and the S, T , and U parameters in the DP limit of the model.
Next, we discuss the pure DZ limit.In this limit, the S, T , and U parameters are given by where a subscript DZ means that the quantities are evaluated in the pure DZ limit.S DZ , T DZ , and In Fig. 4(b), we show the behavior of S DZ , T DZ , and U DZ normalized by the positive common factor α −1 (ε Z /(1 − r 2 )) 2 .We can see that it is quite different compared to that in the DP limit.S DZ is always negative, while U DZ is positive.The sign of T DZ depends on r.When r is larger (smaller) than √ 2 ≃ 1.4, T DZ is positive (negative).Thus, heavy Z d is favored to satisfy the constraint in Eq. ( 49).We note that the above behavior is correct only when |ε Z /(1 − r 2 )| ≪ 1 is satisfied like in the case of the DP limit.
In addition, we find an interesting relation, in the pure DZ limit.This is not a result of perturbative expansions.For ε = 0, Bµ = Bµ and Ẑdµ = Zdµ hold.Then, Zµ couples to only the neutral current as in the SM, and Zdµ does not couples to the SM fermions.Since ε Z induces the mixing between Zµ and Zdµ , the term ∆ 3 J µ em Z µ is not generated by ε Z .Thus, ∆ 3 is zero at all orders of ε Z in the pure DZ limit.This yields αS DZ = −αU DZ = 8s 2 W c 2 W ∆ 2 .The relation between S DZ and U DZ leads to a unique behavior for them in the decoupling limit of Z d (r 2 ≫ 1).In this limit, we obtain T DZ and U DZ are proportional to r −2 and r −4 , respectively, which is the same as in the DP limit.On the other hand, S DZ is proportional to r −4 not r −2 .It is suppressed by r −2 compared to S DP .In terms of the SMEFT, it indicates that S DZ is generated by a dimension-eight operator not a dimension-six one.This is an intriguing feature not seen in many other new physics models.The absence of a dimension-six operator which is a source of the S parameter is due to the fact that Zdµ has no couplings with the SM fermions at tree level, as discussed in detail in Appendix B.

C.
The W boson anomaly in the dark Z model In this section, we discuss the W boson mass in the dark Z model using the S, T , and U parameters derived in the previous section.We will show the dark Z model can explain the W boson mass anomaly under the constraint from the electroweak global fit in Eq. ( 49) and constraints from the direct search for Z d .
By using S, T and U parameters in Eqs ( 54)-( 56) and Eq. ( 47), ∆m 2  W is given by (66) We note that the sign of ∆m 2 W is determined by whether r 2 = (m Z d /m Z ) 2 is larger than one or not.The deviation ∆m 2  W is positive with heavy dark Z bosons (r 2 > 1) and negative with light dark Z bosons (r 2 < 1).As mentioned earlier, in order to explain the W boson mass anomaly, ∆m 2 W ≃ 12.5 GeV 2 > 0 is required.Therefore, light dark Z bosons cannot explain the W boson mass anomaly, while heavy dark Z bosons can.This is already known in the context of the dark photon model [25,94,95].
Using m Z = 91.1876GeV and s 2 W = 0.23122 [1], we can estimate the values of m Z d , ε, and ε Z required to reproduce the CDF-II result (∆m 2 W = 12.5 GeV 2 ) by the following equation11 ;  In Fig. 5, we show regions in ε-ε Z plane where the constraint on S, T and U parameters are satisfied within 2σ uncertainties with the colors green, blue and yellow, respectively in the case of m Z d = 200 GeV as a benchmark point.In the red region, the CDF-II results can be reproduced within 2σ.
There are two red bands because ∆m 2 W is proportional to (ε Z + εt W ) 2 .All colored regions are invariant by the simultaneous change of the sign ε Z → −ε Z and ε → −ε because the S, T , and U parameters and ∆m 2 W are given by the quadratics of the mixing parameters.We can see that there are overlapping regions of all the colored regions for small |ε|, where the CDF-II result can be reproduced under the constraint from the electroweak global fit.
In Fig. 6, we show the same plots as Fig. 5 but in the log scale.parameter |ε Z | is very small (|ε Z | < ∼ 0.01), the behavior of the S, T and U parameters and the W boson mass is almost the same with those in the dark photon model (ε Z → 0).In such regions, the constraints from the electroweak precision measurements cannot be satisfied on the red band within 2σ. 12 It is the same situation in the dark photon model [25,94,95].On the other hand, if the mass mixing parameter is O(0.1), there are regions where the CDF-II result can be reproduced while satisfying the constraints.Therefore, the W boson mass anomaly can be explained in the dark Z model with relatively large mass mixing and heavy dark Z bosons.
As shown in Figs. 5 and 6, the kinetic mixing ε is not important to explain the W boson mass anomaly while maintaining agreement with precision electroweak data.In Fig. 7, we thus show the allowed regions of S, T and U parameters and the CDF-II result within 2σ in the pure DZ limit (ε → 0) by using the same colors as those in Figs. 5 and 6.In the overlapping regions, the CDF-II result can be reproduced under the constraint from the electroweak global fit.We can see that the minimum value of m Z d to explain the W boson mass anomaly is given by about 130 GeV with |ε Z | ≃ 0.03.Larger mass mixing is required in the case of heavier dark Z bosons, for example, TeV.However, the perturbative expansion in ε Z becomes worse for such a where the lower limit is to avoid the Z boson peak region.Although the dijet resonance search [104] is also conceivable, it gives a weaker constraint in the relevant mass region [105].In Fig. 8(a) [Fig.8(b)], we show the upper bounds at 95% CL on the mixing parameters from the dilepton resonance search in the case of εε Z > 0 (εε Z < 0).As benchmark points, we consider the cases of m Z d = 225, 300, and 500 GeV, and contours of the upper bound are shown by the red, blue and black dashed curves, respectively.In evaluating the bound, we used FeynRules [106] and MadGraph5 aMC@NLO [67].In the corresponding colored regions, the W boson mass anomaly can be explained while avoiding the constraint from electroweak global fit within 2σ in each case.We can see that all the red and blue regions are excluded by the current dilepton resonance search.On the other hand, there is an allowed region with small kinetic mixing ε ≃ 10 −3 in the case of m Z d = 500 GeV.
We note that this collider bound can be relaxed by considering a new dark decay mode of the dark Z boson.For example, we consider the dark fermion χ carrying a unit dark charge Q d = 1.Then, the dark decay Z d → χ χ can be a dominant decay mode if it is kinematically allowed, and the constraint is relaxed as discussed in We can see that allowed regions appear in the case of m Z d = 225 GeV (the red region) and 300 GeV (the blue region).In the case of m Z d = 500 GeV, the allowed region is extended.These allowed regions are expected to be tested by future dilepton resonance searches.

V. SUMMARY AND CONCLUSION
In this paper, we have investigated the phenomenological impacts of the dark Z boson on the running weak mixing angle and the W boson mass measurements.In the first part, we have briefly reviewed the effect of a light dark Z boson on the running weak mixing angle and updated some results from previous works with the latest experimental data.
In the latter part of our work, we have investigated whether the dark Z model can explain the W boson mass anomaly reported by the CDF experiment along with the constraints from other electroweak observables and collider dilepton resonance searches.We have investigated the effect of Z d on various electroweak observables including the W boson mass by using the S, T , and U parameters.We have found that in the pure DZ limit, the equation S = −U holds independently of the mass of Z d and the size of ε Z .This is an intriguing feature of the dark Z model not common to many other new physics models including the dark photon model.It has revealed that heavy dark Z bosons with mass m Z d > m Z are required to resolve the W boson mass anomaly.The result of the electroweak global fit and the CDF-II result can be compatible in parameter regions with m Z d larger than 130 GeV and relatively large mass mixing ε Z > O(0.01).Although the current dilepton resonance searches strongly constrain such parameter regions, we have viable regions for heavy dark Z, m Z d > ∼ 500 GeV.By allowing dark decay channels of Z d , this constraint is relaxed, and the allowed regions appear even for lighter Z d and is extended for heavy Z d .Future resonance searches at high-energy colliders would be effective to search for such a dark Z boson.Λ 2 (x) is given by Now, we discuss the PT form.In Ref. [47], the running weak mixing angle is discussed with a gauge-invariant and process-independent form factor κPT , which is defined using the PT [111][112][113][114] as follows; where µ is the renormalization scale, and a γZ is a PT γ-Z self-energy given by [114] a γZ (q 2 , µ) = A γZ (q 2 , µ) − 2e 2 ĉŝ (2q 2 ĉ2 − m 2 W )I W W (q 2 , µ), (A8) where A γZ is the conventional γ-Z self-energy evaluated in the 't Hooft-Feynman gauge, and I W W is the pinch term from vertex and box diagrams.In Eq. (A8), divergent terms in the self-energy and the pinch term are subtracted using the MS prescription.
Reference [47] defines the running weak mixing angle as follows; An analytic formula for the real part of the pinch term in the MS scheme is given by [114] Re[ at µ = m Z .In the γ-Z self-energy, the real parts of the fermionic contribution A f γZ [115] and the bosonic contribution A b γZ [108] are given by Re respectively.In Eqs.(A11) and (A12), the divergent terms are subtracted using the MS prescription.
The difference between the two definitions sin 2 θW (q 2 ) and sin 2 θPT W (q 2 ) is caused by the bosonic contribution: vertex and box contributions mediated by the W boson [47].The fermionic contribution is the same because it comes from the same γ-Z self-energy diagrams.Using all the formulas above, the difference ∆ŝ 2 (q 2 ) is evaluated by At low energies (q 2 → 0 and |z| → ∞), In Fig. 9, the two definitions of the weak mixing angle are compared.The black-solid (red-solid) and black-dashed (red-dashed) curves show sin 2 θW (q 2 ) [sin 2 θPT W (q 2 )] in the spacelike and timelike domains, respectively.The curves for timelike momenta (dashed curves) are shown only in a domain |q 2 | > 20 GeV to avoid spikes from the (unphysical) effective quark masses (m q ) around the region q 2 ≃ 4m 2 q .Except for these spikes, the behavior in the time-like domain is in good agreement with that in the spacelike domain at low energies.In Ref. [47], the running of the electromagnetic coupling α [44] and the complex mass of the W boson [116], which soften the behavior around the W -W threshold, were employed in the evaluation of sin 2 θPT W .However, we do not employ them here in order to compare it to sin 2 θW (q 2 ) which does not use these prescriptions.We can see that the difference is larger at low energies and it seems to converge to a certain value, while it vanishes at high energies.These behaviors are what is expected from Eqs. (A16) and (A17).In this appendix, we discuss the S, T , and U parameters in the decoupling limit of the dark Z model (m Z d ≫ m Z ).They are described by higher-dimensional operators.As shown in Sec.IV B, the source of S is a dimension-six operator in the DP limit, while it is a dimension-eight operator in the pure DZ limit.The purpose of this appendix is to identify the cause of this difference.
Before considering higher-dimensional operators in the model, we review the general discussion in Ref. [21] of the effect of deviations in the gauge sector on electroweak observables.The deviation in the gauge sector is parametrized as follows; where F µν , W µν , and Z µν are the field strengths for the photon A µ , the W boson W µ , and the Z boson Z µ , respectively, and mW and mZ are the masses of the W and Z bosons in the SM.Canonical kinetic terms are given by redefining the gauge fields up to linear order corrections in small parameters A, B, C, and G.Then, the gauge sector is given by and the current interactions are given by where tildes on e, s W , and c W mean that they are defined by using tree-level gauge coupling as in the SM.They are not physical quantities because of the deviations.The physical quantities e, s W , and c W satisfy where s 2 W and c 2 W are defined using α, G F , and m Z as By using these deviations, the S, T , and U parameters are generally given by and the other deviations are zero [See Eq. ( 50)].We note that in this case, the difference between S and U is caused by G, and T is generated by only z. Next, we discuss higher-dimensional operators in the dark Z model.We can take the decoupling limit of Z d by considering v d ≫ v. Zdµ obtains a large mass proportional to v d , while other gauge bosons remain massless because Ẑdµ = η Zdµ , which does not include the electroweak gauge boson Bµ .In such a case, Zdµ represents Z d , a heavy mode that should be integrated out at low energies, while Zµ represents the Z boson.Then, at the electroweak scale, the effect of Z d is described by higherdimensional operators including the fields in the SM and Φ 2 .In the following, we employ the unitary gauge for Z d to avoid the nonphysical mode associated with the U (1) d breaking.
First, we consider higher-dimensional operators in the DP limit (ε Z → 0), which is realized by taking v 2 = 0.Then, Φ 2 is irrelevant in the present discussion, and we neglect it here.Zd has the following interactions; is the covariant derivative in the SM, Y is the hypercharge, and f is the SM fermions with a definite chirality. 13Therefore, at the leading order in 1/m 2 Z d , the following effective interactions are generated via Z dmediated tree-level diagrams L with We note that the coefficient of O (6) φ includes a symmetry factor 1/2.Although four-fermion interactions are also generated at O(m 2 Z d ), we neglect them because they do not affect the gauge sector.
After Φ 1 acquires a VEV (⟨Φ φ and O DP = η 2 ε 2 t W r −2 , z DP are caused by the deviation in the interaction between the Z boson and the hypercharge current.Since the hypercharge current is given by Here, we note that dimension-six operators do not contribute to the U parameter.However, a deviation in the hypercharge current interaction does not induce the U parameter, regardless of the dimension of the operator generating the deviation.At the leading order, the U parameter is generated by a dimension-eight operator whose Feynman diagram is the same as that for O where q µ is the momentum carried by Z d .The first term yields O O (8)  φ , (B25) where O φ is a dimension-eight operator given by The deviation in the hypercharge current interaction Z µ f f γ µ Y f is also generated by dimension-eight operators and induces C and G at O(m −4 Z d ).However, it does not contribute to U as shown in Eq. (B23).
Consequently, in the dark photon limit, the leading terms of the S, T , and U parameters in the decoupling limit are induced by O which coincides with Eqs.(60) when η ≃ 1 is taken.Next, we consider higher-dimensional operators in the pure DZ limit.In this limit, the kinetic mixing ε vanishes, and Ẑµ

Fig. 1 (
Fig. 1(a) [Fig.1(b)], sin 2 θW (m Z ) in leptonic (hadronic) processes are shown with error bars of 1σ.The red (blue) band in Fig. 1(a) [Fig.1(b)] shows the LEP and SLC average value in Eq. (26) [Eq.(27)] with 1σ uncertainties.The dashed lines in the bands represent the central values.The green solid line in both figures shows the SM prediction in Eq. (24) with a very small 1σ uncertainty.All experimental results are consistent with the SM prediction.However, it is interesting that the leptonic and hadronic average values deviate from the SM prediction in opposite directions.Another way to compare the SM prediction and the experimental values is to use the running of the weak mixing angle[43][44][45][46][47][48];

FIG. 1 .
FIG. 1.The weak mixing angle measured in (a) leptonic and (b) hadronic processes.The red (blue) band represents the average value in Eq. (26) [Eq.(27)] with 1σ uncertainties.The SM prediction is also shown with the green line.
, and it is represented by the error bar for |q 2 | at the point "EIC."B. Effects of a light dark Z boson In this section, we discuss the effect of a light dark Z boson (m Z d = O(10) GeV) on the running weak mixing angle.A light dark Z boson can change the running from the SM prediction at low momentum transfer |q 2 | < ∼ m 2

where sin 2
FIG. 2. Comparing the experimental values and the SM prediction of the running weak mixing angle in the MS scheme [Eq.(34)].The black solid (dashed) lines are the SM prediction for spacelike (timelike) momenta.The black points are existing experimental data, and the red points are future anticipated sensitivities.The curves for timelike momenta are shown only in a domain |q 2 | > 20 GeV.(See Appendix A.).
(b) (m Z d = 10 GeV) because of the aforementioned absence of constraint on δ ′ .

FIG. 3 .
FIG. 3. The effect of the dark Z boson on the running weak mixing angle in the cases that (a) m Z d = 15 GeV and (b) m Z d = 10 GeV.The black solid curve is the SM prediction in the MS scheme.The light and dark blue bands represent the result of the χ 2 fitting with 1σ uncertainties.The dashed curve is the prediction with the best-fit value of εδ ′ .Points on the light blue band are excluded by the experimental bound in Eq. (41).
For heavy Z d , |S DP | and |T DP | are much larger than |U DP |.It is because for r 2 ≫ 1, they behave as ) Therefore, |U DP | is suppressed compared to |S DP | and |T DP | in the case of heavy Z d .In addition, for r 2 ≫ 1, we have a simple relation between S DP and T DP ; S DP /T DP ≃ 4c 2 W [95].

FIG. 4 .
FIG. 4. The behavior of the S, T , and U parameters in (a) the DP limit and (b) the pure DZ limit with small mixing, |ε/(1 − r 2 )| ≪ 1.

( 67 )
For example, in the case of m Z d = 200 GeV, the CDF-II result can be explained by satisfying |ε Z + 0.55ε| ≃ 0.07.For a heavier dark Z boson, a larger value of |ε Z + εt W | is required.Thus, relatively large mixing, |ε Z | and/or |ε| ≃ O(0.1), are necessary to explain the W boson mass anomaly.

Figure 6 (
FIG. 5. Allowed regions for the S (green), T (blue) and U (yellow) parameters within 2σ in the case of mZ d = 200 GeV.On the two red bands (not including the region between the bands), the CDF-II result can be reproduced within 2σ.The dashed line represents εZ = 0.

FIG. 6 .
FIG. 6.The same as Fig. 5 but in the log scale in the cases of (a) εε Z > 0 and (b) εε Z < 0.

FIG. 7 .
FIG. 7. The allowed regions of the mass mZ d and the mass mixing εZ for ε = 0.The color code is the same as Fig. 5.The large |εZ | region of O(1) may not be valid as the perturbation in εZ is used.
Sec. III B. Here, we consider the case that m χ = 50 GeV and g d = 0.1.We note that such a dark fermion does not change the decay mode of the Z boson.The upper bounds on |ε| and |ε Z | are shown in Figs.8(a) and 8(b) by the solid lines of the same colors as the dashed lines.

FIG. 8 .
FIG. 8.The bound from the dilepton resonance search at LHC in the cases of (a) εε Z > 0 and (b) εεZ < 0. In the colored regions, the W boson mass anomaly can be explained under the constraint from the electroweak global fit.The solid (dashed) curves are the upper bounds when we do (do not) assume a dark decay channel Z d → χ χ of g d = 0.1 and mχ = 50 GeV.Strong constraints from the dilepton resonance searches are diluted when there is a dominant invisible decay mode and sizable parameter regions survive.
φψ are dimension-six operators defined as

( 6 )DP = 2 (
φψ induce deviations in the mass term and the current interactions of the Z boson, respectively.They are given by ∆L ηεt W ) 2 r −2 , G

( 6 ) 6 )DP = −4s 2 W η 2 ε 2 r − 2
DP = (ηεt W ) 2 r −2 , (B20) where r = m Z d /m Z .The S, T , and U parameters at O(m −2 Z d ) are given by αS (are canceled, and the U paramter is not generated at O(m −2 Z d ).This cancelation is due to the fact that C B22) the C and G generated by the deviation in the hypercharge current interaction, which are denoted by C Y and G Y , satisfy the relation C Y = 2t W G Y .Then, U Y , which is the U parameter induced by C Y and G Y , is given by

( 6 )
φ .The propagator of Z d in the unitary gauge is approximately given by Zµ d Zν d 0

TABLE I .
because the final state Z E includes the missing energy.On the other hand, H → γZ d Input data for the χ 2 fitting of εδ ′ . is generated at one-loop level via the current and gauge interactions as H → γγ or H → γZ in the SM.If Z d is much lighter than H, the branching ratio is approximately evaluated by BR[H → γZ d