A Tale of Two Portals: Testing Light, Hidden New Physics at Future $e^+ e^-$ Colliders

We investigate the prospects for producing new, light, hidden states at a future $e^+ e^-$ collider in a Higgsed dark $U(1)_D$ model, which we call the Double Dark Portal model. The simultaneous presence of both vector and scalar portal couplings immediately modifies the Standard Model Higgsstrahlung channel, $e^+ e^- \to Zh$, at leading order in each coupling. In addition, each portal leads to complementary signals which can be probed at direct and indirect detection dark matter experiments. After accounting for current constraints from LEP and LHC, we demonstrate that a future $e^+ e^-$ Higgs factory will have unique and leading sensitivity to the two portal couplings by studying a host of new production, decay, and radiative return processes. Besides the possibility of exotic Higgs decays, we highlight the importance of direct dark vector and dark scalar production at $e^+ e^-$ machines, whose invisible decays can be tagged from the recoil mass method.

. In addition, we will emphasize the unique capability of e + e − machines to reconstruct invisible decays, which is a marked improvement over the reconstruction prospects at hadron colliders.
The lack of evidence for weakly interacting massive particles (WIMPs) in direct detection (DD) experiments [30][31][32][33], increasingly strong constraints on thermal WIMPs from indirect detection (ID) experiments [34][35][36][37][38], and non-observation of beyond the Standard Model (BSM) missing transverse energy signatures at the LHC [39,40], combine to an increasing unease with the standard WIMP miracle paradigm. On the other hand, dark matter coupled to kinetically mixed hidden photons suffers from strong direct detection constraints (see, e.g., [41]). A consistent dark matter model must hence simultaneously address the relic density mechanism and non-observation in the current experimental probes, and thus minimal models either require nonthermal dark matter production in the early universe, coannihilation channels [42][43][44], or resonant dark matter annihilation in order to divorce the early universe dynamics from collider processes (see, e.g., [45]).
Moreover, while the nuclear recoil energy spectrum at direct detection experiments requires the dark matter mass as input, colliders instead probe mediator masses if they are on-shell, which shows the complementarity between both approaches. In our work, we will further demonstrate these complementary aspects between dark matter experiments and hadron and lepton colliders in the context of our dark matter model.
In Sec. II, we review the theoretical framework for the Double Dark Portal model, which unifies the kinetic mixing portal and the scalar Higgs portal into a minimal setup with dark matter.
In Sec. III, we detail the phenomenology of the dark matter for direct detection and indirect detection experiments. We discuss the extensive collider phenomenology of the model and review the current constraints from experiments at the Large Electron-Positron (LEP) collider and the Large Hadron Collider (LHC) in Sec. IV. We then present the prospects for exploring new, light hidden states at a future e + e − machine in Sec. V and conclude in Sec. VI. In Appendix A, we offer some detailed discussion of limiting cases in our Double Dark Portal model for pedagogical clarity, and we discuss a cancellation effect in scattering processes via kinetic mixing in Appendix B. We also present the dark matter annihilation cross sections for charged SM final states in Appendix C.

II. OVERVIEW OF THE DOUBLE DARK PORTAL MODEL: SIMULTANEOUS KINETIC MIXING AND SCALAR PORTAL COUPLINGS
We begin with the Lagrangian of the Double Dark Portal Model, where K µν is the field strength tensor for the U (1) D gauge boson, Φ is a dark Higgs scalar field with charge +1 under U (1) D , and χ is the dark matter and a SM gauge singlet fermion with charge +1 under U (1) D . We take µ 2 H > 0 and µ 2 D > 0, which trigger spontaneous symmetry breaking of the SM electroweak symmetry and the U (1) D dark gauge symmetry, respectively. The θ W parameter is the tree-level SM weak mixing angle, θ W = tan −1 (g /g). The nonzero Higgs portal coupling, λ HP , induces mass mixing between the h and φ scalars, which results in mass eigenstates H 0 and S. Simultaneously, the kinetic mixing will result in an effective mass mixing between the SM Z gauge boson and the K dark gauge boson, which results in the mass eigenstatesZ andK.
The two marginal couplings, and λ HP , are commonly referred to as vector and scalar portals, respectively [46]. Because the phenomenology of such portal couplings changes significantly when a light dark matter particle is added, we call the Lagrangian in Eq. (1) the Double Dark Portal (DDP) model.
We solve the Lagrangian in the broken phase after the Higgs and the dark Higgs obtain their vacuum expectation values (vevs), by diagonalizing and canonically normalizing the kinetic terms for the electrically neutral gauge bosons and diagonalizing their mass matrix. We can rewrite the Lagrangian using matrix notation, with mass terms acting on the gauge basis vector ( W 3 µ B µ K µ ) T as In this breaking of SU (2) L × U (1) Y × U (1) D → U (1) em , the resulting field strength tensors of the individual neutral vectors corresponding to the gauge eigenstates W 3 , B, and K all have Abelian field strengths, while non-Abelian vector interactions are inherited from the SU (2) L gauge boson field strength tensor. We will not explicitly write the non-Abelian vector interactions in the following, but instead understand that they are correspondingly modified when we perform the rescaling needed to canonically normalize the Abelian field strengths of the neutral vectors. 1

A. Neutral vector boson mixing
To simplify the Lagrangian in the broken phase, we first rotate by the tree-level SM weak mixing angle, which reduces the mass matrix to rank 2 and correspondingly modifies the kinetic mixing between the Abelian field strengths. Explicitly, we sandwich R θW R T θW twice in Eq. (4), with c W = cos θ W and s W = sin θ W , which gives where t W = tan θ W , m 2 Z, SM = (g 2 +g 2 )v 2 H /4 is the tree-level SM Z-boson mass, and m 2 K = g 2 D v 2 D is the tree-level U (1) D gauge boson mass. To canonically normalize the kinetic terms for the neutral 1 We remark that the Stueckelberg mechanism [47,48] provides an alternative mass generation forK, which we do not employ here. The collider phenomenology of a dark neutral gauge boson with mass arising from the Stueckelberg mechanism is presented in Ref. [48]. gauge bosons, we use the successive transformations which give where the kinetic terms are now canonically normalized and only one further unitary rotation is needed to diagonalize the mass matrix. We remark that | | < c W is required to ensure the kinetic mixing matrix in Eq. (6) has a positive definite determinant, which allows U 2 to remain non-singular. The final Jacobi rotation required is for c M = cos θ M and s M = sin θ M and θ M defined by tan θ M = 1 and the upper (lower) sign in tan θ M corresponds to m Z, SM > m K (m Z, SM < m K ). The resulting non-zero mass eigenvalues are and the corresponding neutral vector basis is We remark that these are exact expressions valid for arbitrary . and We note that this expansion for 1 is insufficient for m K → 0 or m K → m Z, SM . These two limits are discussed in Appendix A. Given that is small, the masses ofK andZ are altered only at the 2 level.
With the O( 3 ) expressions for the mass eigenstate vectors with canonically normalized kinetic terms, we can now write down the corresponding currents associated with the mass eigenstate vectors: Again, the situation for m K → 0 or m K → m Z, SM is discussed in Appendix A. From these expressions, we see explicitly that SM fermions, encoded via J µ em and J µ Z , obtain an O( ) electric charge and an O( ) neutral weak charge mediated byK µ . Matter charged in the U (1) D sector correspondingly receives an O( ) dark charge mediated byZ µ .

B. Scalar boson mixing
The analysis of the scalar sector is simpler and follows previous discussions of scalar Higgs portals in the literature (see, e.g. [49]). From Eq. (2) and Eq. (3), we have The scalar mass eigenstates are then where is the scalar mixing angle. The scalar masses are We can thus reparametrize the scalar Lagrangian couplings µ D , µ H , λ D , λ H , λ HP as m S , m H 0 , v D , v H , and α. The reparametrizations for µ 2 D , µ 2 H and λ HP are given above, while the reparametrization for λ D and λ H are We also calculate the scalar interactions in the mass eigenstate basis H 0 and S. The cubic scalar interactions are We have, of course, m H 0 = 125 GeV and v H = 246 GeV, but the other observables are free parameters. We will restrict λ HP > 0 in our analysis, recognizing that λ HP < 0 and |λ HP | > √ λ H λ D can cause tree-level destabilization of the electroweak vacuum.
Lastly, the scalar-vector-vector interactions ofK,Z, S and H 0 in the mass basis to O( 2 ) are We reiterate that both α and are theoretical parameters that must be constrained by data, and hence a particular hierarchy between α and would reflect model-dependent assumptions. As a result, Eq. (24) forms a consistent basis for determining the sensitivity to α and simultaneously.
We can characterize the changes in the phenomenology of the Higgs-like H 0 state as a combination of modified SM-like production and decay modes and the opening of new exotic production and decay channels. One main effect of α is to suppress all of the SM fermion couplings of the H 0 state by cos α, while the S state acquires Higgs-like couplings to SM fermions proportional to sin α. This feature also applies to the loop-induced couplings to gluons and photons for H 0 and S.
On the other hand, the coupling between H 0 toZ bosons is changed not only by cos α but also by 2 sin α, while the S state acquires aZ coupling proportional to sin α and also 2 cos α.  [50].

C. Dark matter interactions
Finally, we will consider the DM interactions with the mass eigenstates of the gauge bosons and scalars. The main observations can be obtained by recognizing that DM inherits its couplings to SM particles via the J D current shown in Eq. (15). Explicitly, the dark matter particle Lagrangian  [28,46].
Our focus, however, is the O(10−100 GeV) scale for theK vector mediator and its accompanying Higgs partner S, which will both dominantly decay to the dark matter particle χ. This is readily motivated by considering m χ < mK/2 and g D , so thatK has an on-shell two-body decay tō χχ and an 2 /g 2 D suppressed branching ratio to SM charged particles. Moreover, for mK < m S /2 and sin α g D , the SM gauge singlet scalar S dominantly decays to pairs ofK and only have sin 2 α/g 2 D suppressed rates to SM pairs. All of these choices, however, can be reversed to give markedly different phenomenology. If m χ > mK/2, for example, then the total width of mK scales as 2 [10] andK decays to pairs of SM charged fermions, as long as it is heavier than 2m e . For very small , however, theK lifetime can be long, leading to either displaced vertex signatures or missing energy signatures. The lifetime and decay length ofK can be estimated to be τ = 1 Γ = 0.9 × 10 −2 ps 1 GeV mK L = γcτ ≈ 150 × 10 −6 m 1 GeV mK for γ = 60 (as from a Higgs two-body decay) andÑ F is the effective factor for kinematically open charge-weighted two-body SM final states. If m S , mK > m H 0 /2, then any possible exotic decay of the SM-like Higgs will be strongly suppressed by multi-body phase space and a combination of g D , sin α, or λ HP . We remark that choosing m S < m H 0 /2 already gives an exotic Higgs decay, H 0 → SS, which is sensitive directly to λ HP .
Given our mass hierarchy, the dominant collider signature from production of eitherK or S is missing energy from escaping χ particles, while the relic density of χ in our local dark matter halo can be probed via nuclear recoils in terrestrial direct detection experiments or through their annihilation products in satellite indirect detection experiments. We will discuss the direct and indirect constraints from dark matter searches in the remainder of this section and focus on the collider signatures for vector and scalar mediator production in Sec. IV and Sec. V.

A. Direct detection and relic abundance
Dark matter direct detection experiments search for anomalous nuclear recoil events consistent with the scattering of the dark matter halo surrounding Earth. Direct detection scattering occurs via t-channel exchange ofZ andK, as evident from the J µ D and J µ em interactions shown in Eq. (15).
Because of the relative sign between theK andZ terms, dark matter scattering proportional to is naturally suppressed by extra 2 or Q 2 /m 2 K factors, where Q is the momentum transfer scale, and the leading contribution is hence proportional to e 2 g 2 where µ χp is the reduced mass of the dark matter χ and the proton and e = 4π/137. The cross-section σ p is calculated at leading order in and v in , the incoming DM velocity, and agrees with previous results when DM only interacts via t-channelK exchange with strength proportional to the SM electromagnetic current [41].
Given that the momentum transfer in the propagator is smaller than gauge boson masses mK and mZ, then the scattering amplitude is O(Q 2 /m 2 V ) suppressed after summing all the vector boson contributions, where m V is the smaller of either gauge boson mass. For DM direct detection, the momentum transfer is about Q 2 ∼ (m χ v in ) 2 m 2K ,Z , hence the contribution induced by the J Z current cancels and we arrive at the same result in Ref. [41].
We can also motivate particular contours in the vs. mK plane by considering first the require-ment that the DM obtains the correct relic density and second the constraints on the possible rates for DM annihilation to SM particles coming from indirect detection experiments. We first calculate the annihilation cross sections for χχ → ff , W + W − , where f denotes a SM fermion. We focus on the region m χ < mK, since the annihilation channel χχ →KK opens up otherwise and the dominant self-annihilation cross section is insensitive to .
In this setup, the annihilation cross section will be proportional to g 2 D 2 . We calculate the annihilation in center of mass frame, and give the annihilation cross sections before thermal averaging in Appendix C. We perform the thermal averaging of the annihilation cross section numerically according to Ref. [51]. The annihilation cross section generally has three physical resonances, mK = 2m χ , mZ = 2m χ and mK = mZ. The first two resonances are from the s-channel resonant exchange ofK andZ, while the last one is due to maximal mixing betweenK andZ when m K is close to m Z, SM , as discussed in Appendix A.
In Fig. 1, we show the direct detection constraints in the vs. m K plane from experiments LUX [33] with data from 2013 to 2016, PANDAX-II [32], and CRESST-II [31] as well as CDMSlite [30] for low mass DM. Each panel shows choices of g D = e, 0.1, and 0.01, and the dark matter mass fixed to 0.2m K , 0.495m K , 0.6 GeV, or 30 GeV.
The thermal relic abundance limit on is given in Fig. 1 using the Ωh 2 = 0.12 requirement from the Planck collaboration [34]. The dip around m K ∼ m Z, SM reflects increasing mixing betweeñ K andZ. While for m K ∼ 2m χ , the annihilation cross section is enhanced by the s-channel K resonance, thus the required is very small. When 2m χ − mK < T f , where T f ≈ m χ /25 is the DM freeze-out temperature, the annihilation cross section is enhanced by (mK/T f ) 2 due to thermal averaging. When 2m χ − mK > 0, the thermal average will not benefit the resonance effect anymore.
In recasting the direct detection limits, we recognize that the experiments assume that the local DM density is fixed to 0.3 GeV/cm 3 . Hence, the respective constraints are identically meaningful only when the DDP model parameters give this assumed local relic density. For other parameter space points, in particular for fixed m K and varying = relic , with relic corresponding to σv = 0.3 GeV/cm 3 , the predicted rate of direct detection scattering events will be independent of . This is because the predicted local DM relic density will scale with ( / relic ) 2 while the scattering cross section will scale with ( relic / ) 2 , leaving the product, and thus the predicted direct detection rate, insensitive to .
In our recasting, however, we keep the local DM relic density fixed to 0.3 GeV/cm 3 regardless of , in order to determine the sensitivity to the direct detection cross section. For large , when the Hence, the direct detection exclusion contours in each panel simply illustrate the fractional χ relic density, relative to Ωh 2 = 0.12, that is excluded by the direct detection constraint. When the DD contours are weaker than the relic density contours, the model only minimally requires extra inert dark matter to make up the absent relic abundance. When the DD contours are stronger than the relic density contours, an extra contribution to the thermal relic annihilation cross section for χ is required to satisfy the 0.3 GeV/cm 3 assumption, and is excluded by DD experiments as shown in the red shaded region. In particular, for fixed m K , the strengthening to the annihilation cross section can be parametrized by the squared ratio of from the blue contour to at the red contour.
We see that light DM masses are much less constrained, because of the µ 2 χp /m 2K suppression in Eq. (28). For m χ ∝ mK, the sensitivity on generally follows the experimental constraint on σ p . We see that for heavyK and light χ, the direct detection sensitivity can be weak, leaving significant parameter space to be probed by colliders. Interesting parameter space also exists for m χ mK/2, which will be discussed further in the next section.

B. Indirect constraints from CMB, Gamma-ray and e ± measurements
After the relic abundance constraint, we next consider the constraints from cosmic microwave background (CMB) observations. Measurements of the CMB generally give constraints on DM annihilation or decay processes, which inject extra energy into the CMB and thus delay recombination [52][53][54][55][56][57]. The constraint is calculated using the energy deposition yield, f i eff , where i denotes a particular annihilation or decay channel and f eff describes the efficiency of energy absorption by the CMB from the energy released by DM in particular channel. The constraint is expressed as where the Planck experiment has constrained p ann < 4.1 × 10 −28 cm 3 s −1 GeV −1 [34], and we sum all the SM fermion pairf f and W + W − channels in annihilation. The excluded parameters are plotted in Fig. 2 as shaded purple regions.
The next constraints we consider are the gamma ray observations from Fermi-LAT and MAGIC in dwarf galaxies [35,36]. In Ref. [35], Fermi-LAT gives constraints on e + e − , µ + µ − , τ + τ − ,ūu,bb and W + W − final states, while in Ref. [36], MAGIC has made a combined analysis with Fermi-LAT and presented constraints on µ + µ − , τ + τ − ,bb and W + W − . The computation of constraints on our model is straightforward, since we can calculate each individual limit on for each channel   [34], gamma-ray measurements from dwarf galaxies [35,36] and the inner galactic region [37] and e + flux measurement from AMS-02 [38]. The and take the most stringent constraint for each m χ mass. The excluded parameter region is shaded by cyan in Fig. 2. Similarly, we consider the gamma ray constraints from the inner Milky way [37]. This analysis sets conservative constraints on various SM final states by using the inclusive photon spectrum observed by the Fermi-LAT satellite. We apply their results by calculating the most stringent annihilation profile, assuming the Navarro-Frenk-White profile for the DM density distribution in the galactic center [58]. We can see in Fig. 2 that the constraint from galactic center region, shaded in orange, is much weaker than that from dwarf galaxies.
The last indirect detection constraint is based on e + and e − data from the AMS-02 satellite [38].
We use the constraints from Ref. [59] to set bounds on various SM final states, which mainly derive from the observed positron flux. We again adopt the limits from the strongest channel to constrain for each mass parameter choice. We can see the constraint from AMS-02 is the strongest at the largest m χ masses in Fig. 2.
To summarize, in Fig. 2, we see that the CMB constraint is strongest at small m χ , while AMS-02 is strongest at higher m χ . The constraint from gamma ray observations in dwarf galaxies is very close to the CMB constraint. Meanwhile, the dips in Fig. 2 nicely show the two s-channel resonances ofK andZ as well as the maximal mixing peak betweenK andZ. The Double Dark Portal model motivates observable deviations in measurements of both the SM-like H 0 and theZ bosons, which test the scalar mixing angle α as well as the kinetic mixing parameter . Notably, the primary SM Higgsstrahlung workhorse process at an e + e − Higgs factory, e + e − → Zh, can deviate significantly from the SM expectation for nonzero or α. For instance, nonzero α causes a well-known cos α suppression of the H 0ZµZ µ vertex, but nonzero gives an additional diagram with intermediateK, which becomes on-shell when mK > mZ + m H 0 . We remark that these effects are not generically captured by a simple cos α rescaling of the H 0ZµZ µ vertex.
In that e + e − machines offer unique opportunities for probing new, light, hidden particles by virtue of the recoil mass method, which we discuss first.
A. Recoil mass method for probing new, light, hidden states As long as they are kinematically accessible, both S andK can be produced in e + e − collisions in association with SM particles. Hence, even if they decay invisibly, the recoil mass method can be used to probe the couplings sin α and , according to the interactions from Eq. (15) and Eq. (24). This is familiar from the leading e + e − →ZH 0 Higgsstrahlung production process, where the reconstruction of theZ → + − decay consistent with a 125 GeV recoil mass gives a rate dependent only on the H 0ZµZ µ coupling. We emphasize (see also Ref. [60]) that this generalizes to any scattering process at an e + e − machine if visible SM states are produced in association with a new, light, hidden particle. Moreover, sensitivity to the hidden states S andK can be improved by scanning over √ s, where the various production modes ofZS, γK, andZK can be optimized for the different S andK masses. This √ s adjustment would be immediately motivated, for example, by a new physics signal in the recoil mass distribution.
The recoil mass method uses the knowledge that the center-of-mass frame for the e + e − collision is fixed to be ( √ s, 0, 0, 0) in the lab frame, where √ s is the energy of the collider. Hence, for an invisibly decaying final state particle X produced in association with a SM state Y , four-momentum conservation requires or equivalently, If there are multiple visible states Y i , this generalizes to where ( p i ) 2 is the total invariant mass of the Y i system. We see that studying the differential distribution of E Y will show a characteristic excess at a given E Y when X is produced. Identifying this monochromatic peak is formally equivalent to finding a peak in the recoil mass distribution, but we emphasize that these two distributions reconstructed differently at e + e − colliders. Specifically, the recoil mass distribution uses both the energy and total four-momentum of each detected SM particle, which the differential energy distribution only requires calorimeter information.
In particular, for theZH 0 ,Z → + − Higgsstrahlung process, the recoil mass method requires measurements of each individual lepton four-momentum and the event-by-event invariant mass m . The resulting differential distribution also includes off-shell contributions and interference, giving a smeared peak in the recoil mass distribution whose width is dominated by experimental resolution and not the intrinsic Higgs width. On the other hand, in radiative return processes, both the recoil mass distribution and the photon energy spectrum are only limited by the possible width of the recoiling new physics particle and the photon energy resolution (see also [61,62]).
For our studies, we assume both S andK have dominant decay widths to the dark matter χ, which does not leave tracks or calorimeter energy deposits as it escapes. The recoil mass technique, however, also readily probes both theK and S masses in numerous production modes, when we produceK or S in association with a visible SM final state. For example, while the SM-likeZ boson is a canonical choice to studyZH 0 events, we can use the recoil mass technique in the radiative return process forÃK production to identify the invisible decay ofK. An even more striking possibility is to use the SM-like Higgs boson, H 0 , as the recoil mass particle to probeKH 0 production.

B. Modifications to electroweak precision
We now consider the four categories of collider processes in turn. The first set of observables we consider are those from electroweak precision tests. In the Double Dark Portal model, Zpole observables will show deviations according to the new decay channelZ →χχ orZ → SK, sensitive to , shifts in theZ mass from the mixing withK, and deviations in the weak mixing angle from the mixing betweenK,Ã, andZ. In particular, identifying theZ mass eigenstate of the DDP as the 91.2 GeV Z boson studied by LEP, measurements of the Z mass, total width, and the invisible decay to SM neutrinos give strong constraints on and the possibility of exotic decays. For mK < 10 GeV, both the visible and invisible channels can be constrained by various experiments, as reviewed in Ref. [28]. We thus focus on the status and prospects for mK > 10 GeV.

LEP-I and LEP-II constraints
At LEP-II, contact operators (4π/Λ 2 )ēγ µ ef γ µ f were used to test for new physics, analogous to angular distributions in dijet studies at the LHC. In the e + e − → + − channel, the constraint on Λ is 20 TeV [63]. Since the majority of the dataset was taken at a fixed √ s = 200 GeV, we can place a constraint on by matching the coefficient of the contact operator at tree-level to an intermediateK mediator, . The corresponding bound on is around O(0.1). There is sharper sensitivity for m K ≈ m Z because of maximal mixing, and again around √ s ∼ 200 GeV from resonant production. Because theK decay width is dominated byK →χχ, the sensitivity at resonance is suppressed the branching ratio ofK → + − .
As mentioned above, the mixing between Z SM and K leads to shifts in theZ mass and couplings to SM fermions, leading to a constraint of < 0.03 for mK < m Z using a combination of electroweak precision observables [64]. The constraint is weakened for mK > m Z where the limit on is about 0.1 at mK = 200 GeV [64], and is shown in Fig. 6 and Fig. 7 as "LEP-EWPT." Recently, the BaBar collaboration has published constraints on dark photons decaying invisibly, e + e − → γK,K →χχ [29]. Their constraints directly map to our vs. m K parameter space and place strong limits on 0.001 for masses between 1 GeV to 10 GeV. These are reproduced in Fig. 6 and Fig. 7, and labeled as "BaBar." Although the canonical SM Higgs production channel e + e − → Zh was ineffectual at LEP-II, the scalar mixing angle sin α can still be probed by the e + e − →ZS production mode when S is kinematically accessible by LEP-II. For m S < 114 GeV, the non-observation of Higgs-like scalar decays constrains sin 2 α < O(0.01 − 0.1) [22], as long as the S →KK decay is turned off.
The LEP experiments have also searched for a low mass Higgs in the exotic Z → HZ decay, with Z → + − and H decaying invisibly, which excludes m H < 66.7 GeV if the invisible branching fraction is 100% [18]. The ZH Higgsstrahlung process is also used to push the mass exclusion to 114.4 GeV [19][20][21], although the intermediate mass range between these two limits are not comprehensively covered. In our model, S →KK is the dominant decay when g D sin α, and the decay branching fractionZ → SZ and the production cross section σ(ZS) are hence sin 2 α suppressed compared to the SM rate. Therefore, these limits apply to S as bounds on sin α and m S , which we will show in Fig. 8 in Sec. V. Note the constraint from the exoticZ → SZ decay is much stronger thanZH 0 Higgsstrahlung process in Fig. 8 due to the high statistics of Z decays, and in the calculation we accounted for the subsequent decay branching fractions of BR(S →KK) and BR(K →χχ).

C. Modifications to Higgs physics and LHC constraints
With the era of precision Higgs characterization underway after the discovery of a Higgs-like boson [65,66], The first two decay widths are proportional to m −2 K while the last one is proportional to m 2 K , therefore the last one is usually much smaller comparing with the first two when m K is light. In the Higgs invisible studies, the experiments will constrain the rate for Higgs invisible decays in the DDP model, where Γ H 0 , tot = cos 2 αΓ h, SM + Γ(H 0 → SS) + Γ(H 0 →KK) + Γ(H 0 →KZ). The decay widths K →f f , W + W − can be found in the appendix of Ref. [69], while the decay widthK →χχ is The prediction for the invisible decay branching fraction of the 125 GeV Higgs is shown in the left and middle panels of Fig. 4 in the sin α vs. plane for m S = 50 GeV, m K = 20 GeV, and g D = e, 0.01. The current constraint of BR inv < 0.23 is adopted from Refs. [67,68], while the prospective sensitivity of BR inv < 0.005 is adopted from the estimate using 10 ab −1 of √ s = 240 GeV data using unpolarized beams in Ref. [3]. This prospective limit can be lowered in combined fits, with more luminosity, or with other assumptions about detector performance to the O(0.001) level [1,2,4].
In Fig. 4, we see that when g D is large, the sensitivity to sin α is much stronger than , because the decay widths for H 0 → SS and H 0 →KK are much larger than H 0 →KZ due to light m K , as discussed previously. More importantly, for large g D , BR(S →KK) and BR(K →χχ) are close to 100%. When g D < sin α or g D < , BR(S →KK) and BR(K →χχ) will both be subdominant and result in the decrease of BR eff inv as in the middle panel of Fig. 4. As shown in the right panel of Fig. 4, the constraint on sin α from invisible Higgs decays can be relaxed by making g D smaller. These exotic decays can also give fully visible and semi-visible signatures [ The coupling λ HP = λ HP m 2 H 0 /|m 2 H 0 − m 2 S |, is roughly the same as λ HP if m S is not close to or much larger than m H 0 . The high luminosity LHC (HL-LHC) with 3 ab −1 of 14 TeV luminosity is expected to be sensitive to λ HP (few) × 10 −5 in this same channel, depending on the mK mass. The four lepton final state has also been used to constrain the exotic decay H 0 →ZK, which gives sensitivity to from (24). The current bound using 8 TeV data is weak, with the strongest sensitivity for mK ≈ 30 GeV giving 0.05, while the improvement at the HL-LHC is expected to reach 0.01 [8]. These gains are mainly limited by the statistics afforded by Higgs production rates. We remark that for very small and m χ > mK/2, as discussed in Sec. III, the hidden photon will have a displaced decay to SM states, which provides a new set of challenges to trigger and detect at colliders. Current exclusions and future prospects for displaced decays can be found, e.g., in Refs. [8,28,46].

Modifications to Drell-Yan processes
TheK decay to SM final states can be dominant, if the decay to DM pairs is kinematically forbidden or g D . In this case, at the LHC, the Drell-Yan process pp →Z,K → + − can be used to constrain the kinetic mixing parameter , since this process has been studied with exquisite precision by the ATLAS and CMS experiments. Both ATLAS and CMS have searched for dilepton resonances at high mass, mK 200 GeV, using 20 fb −1 of 8 TeV data [77,78], which restricts 0.01 at mK = 200 GeV and weakens to 0.05 at mK = 1000 GeV [79,80].
For mK between 10 and 80 GeV, the Drell-Yan search using 7 TeV data by CMS [81] and the corresponding sensitivity using 8 TeV data gives 0.005, stronger than the current electroweak precision constraints [8,80].
The HL-LHC is expected to constrain 0.001 for mK between 10 and 80 GeV using Drell-Yan data, while high massK can be probed at the ∼ 0.002 level for mK = 200 GeV and the ∼ 0.01 level for mK = 1000 GeV [8,79,80]. The recent 13 TeV, 3.2 fb −1 search for high mass dilepton resonances by ATLAS [82] also constrains the kinetic mixing parameter 0.04 for mK > 100 GeV [83], but this result is hampered by the small statistics. We see that as long as K has an appreciable branching fraction to SM final states, in particular leptons, the Drell-Yan process at the LHC and HL-LHC will provide stronger sensitivity to compared to electroweak precision observables. For g D / 1, the decayK →χχ is dominant and the situation reverses, and then Drell-Yan constraints will not compete with the electroweak precision observables. After rescaling by the appropriate visible branching ratio, we plot these Drell-Yan constraints as the "LHC-DY" contours in Fig. 6 and Fig. 7.

Radiative return processes and dark matter production at the LHC
The radiative return process, e + e − → γX, enables on-shell production of new particles at fixed √ s colliders by using an extra radiated photon to conserve four-momentum. At hadron colliders, since the colliding objects are composite, dark matter production via radiative return is more commonly known as monojet or monophoton processes, recognizing the fact that the partonic center of mass energy is not constant on an event-by-event basis.
As a result, the visible decaysK → + − discussed in the Higgs to four leptons and the Drell-Yan contexts are complemented by the LHC searches for dark matter production in monojet and monophoton processes. We remark that in our DDP model, we will assume thatK →χχ is the dominant decay channel, leading to an overall 2 /g 2 D suppression in the above visible decay rates. Both ATLAS and CMS have searches for dark matter production using 8 TeV data [84,85], sensitive to mediator masses as low as 10 GeV [84]. The corresponding 13 TeV searches [39,40] have yet to achieve the same sensivity at low masses. In the DDP model, theK mediator is produced on-shell and decays dominantly toχχ, and calculating the results for on-shell mediator production at the LHC, we obtain 0.07, similar to previous studies [86][87][88]. It is also possible to search for the dark bremsstrahlung ofK from the DM pair [89,90], as a probe of , although these rates are negligible in our model.

V. PROSPECTS FOR FUTURE COLLIDERS
We have established that significant room remains to be explored in both the and λ HP portal couplings. We will now demonstrate that a future e + e − collider, currently envisioned as a Higgs factory, will have leading sensitivity to probing both couplings simultaneously through the production of new, light, hidden statesK and S. The primary motivation for the √ s ∼ 240 − 250 GeV centerof-mass energy of such a collider is to optimize the expected σ(e + e − → Zh) SM Higgsstrahlung cross section, taking into account the possible polarization of the incoming electron-positron beams.
Such high energies, however, also enable production of the new statesK and S from radiative return processes, exotic Higgs decays, and exotic Higgsstrahlung diagrams.
A few different variations exist for next-generation e + e − machines, namely the International Linear Collider (ILC) [2], an e + e − Future Circular Collider (FCC-ee), which shares strong overlap with TLEP [3], or a Circular Electron-Positron Collider (CEPC) [4]. Since the physics we discuss will only depend very mildly on the particular √ s of the future machine and possible polarization of the incoming electron and positron beams, we will adopt a √ s = 250 GeV machine colliding unpolarized e + and e − beams as our reference machine with a total integrated luminosity of L = 5 ab −1 in our collider studies. For comparison, we also show future expectations for a possible √ s = 500 GeV machine with L = 5 ab −1 total integrated luminosity. Our work will complement and extend previous and sin α sensitivity estimates made for various specific collider environments, which we review first. This leads to the possible constraint < 0.01 for mK < mZ/2. Measurements directly on the Zpole are not expected to compete with this constraint because of theory uncertainties on the small-angle Bhabha-scattering cross section remain too large [91]. This constraint also applies to theZ →KS → 6χ exotic decay, if mK + m S < mZ. For a lepton collider running on the Z-pole, though, other electroweak precision observables will have greatly enhanced precision. The combination of improved electroweak precision observables can constrain 0.004 for mK < mZ, although the constraint is much weaker for mK > mZ [8].
The e + e − Higgs factory is expected to have a precision measurement of the Higgsstrahlung process e + e − → Zh, with accuracies ranging from O(0.3% − 0.7%) expected, using 5 − 10 ab −1 of luminosity [3,4,93]. These rates imply that the scalar mixing angle is probed to sin α 0.055 − 0.084, simply from the observation of the Higgsstrahlung process.
Aside from precision Higgs measurements, a future e + e − machine will have leading sensitivity to an invisible decay of the 125 GeV Higgs. As reviewed in Subsec. IV C, the current constraint on the Higgs invisible decay branching ratio is BR inv < 0.23 [67,68], while the limit at FCC-ee is expected to be BR inv < 0.005 [3]. We have discussed the two main invisible decays, H 0 → SS → 8χ and H 0 →KK → 4χ, as well as the irreducible signal from H 0 →ZK →ννχχ in Subsec. IV C.
We also show the corresponding sensivity in the sin α vs. and sin α vs. mK planes in Fig. 4. We again emphasize that these limits can be stronger or weaker because of their dependence on g D , as seen in Fig. 4. e + e − →ZH 0 : The usual Higgsstrahlung diagram is suppressed by cos 2 α, with an additional contribution from intermediateK * that can interfere withZ * exchange.
e + e − →ZS: This new process can be probed by the usual recoil mass method for well-reconstructed Z decays, studying the entire recoil mass differential distribution.
e + e − →KS: This exotic production process involves two non-standard objects, and is dominantly produced viaK * , with a rate proportional to 2 cos 2 α. SinceK and S dominantly decay to dark matter, though, we would require an additional photon or a visible decay ofK or S in order to tag the event.
e + e − →ÃK: The radiative return process producesK in association with a hard photonÃ ∼ γ, giving direct sensitivity to . We remark that theK → + − decay has been studied for mK between 10 GeV to 240 GeV at an √ s = 250 GeV machine with 10 ab −1 [10], giving e + e − →ZK: The massive diboson pair production process also provides direct sensitivity to , but measuring the rate precisely will pay leptonic branching fractions of theZ.
e + e − → H 0K : This very interesting scenario can be probed by using the 125 GeV SM-like Higgs as a recoil candidate for theK heavy vector. The total rate gives sensitivity to both and α and highlights the power of considering the SM-like Higgs as a signal probe for new physics.
Having identified the main production modes for theK and S states, we can match them to decay topologies illustrated in the bottom row of Fig. 3. We also include the underexplored decay H 0 →KZ, which gives an exotic decay of the SM-like Higgs into the SM-likeZ boson and the hidden photonK sensitive to . As mentioned in Sec. III, we focus on S →KK → 4χ andK →χχ, and thus the dark portal couplings must be tested by recoil mass techniques or mono-energetic photon spectra searches. We also demonstrate the importance of these missing energy searches by explicitly considering leptonic decays ofK in theZK and γK processes as well as the fully inclusive recoil mass distribution targeting γK production.
Since the workhorse SM Higgsstrahlung process has been studied extensively [2][3][4], we use these previous results to recast the sensitivity for and α. We also ignore theKS production mode, since the dominant signature has nothing visible to tag the event. The H 0K process is interesting to consider for future work, but it requires optimizing the H 0 decay channel to gain maximum sensitivity to the recoil mass of the rest of the event.
This leaves theZH 0 ,ZS, γK andZK processes as new opportunities to revisit or study. We simulate each process using MadGraph5 v2.4.3 [94], Pythia v6.4 [95] for showering and hadronization, and Delphes v3.2 [96] for detector simulation. Detector performance parameters were taken from the preliminary validated CEPC Delphes card [97]. Backgrounds for each process are generated including up to one additional photon to account for initial state and final state radiation effects. Events are required to pass preselection cuts of |η| < 2.3 for all visible particles, while photons and charged leptons must have E > 5 GeV, jets must have E > 10 GeV, and missing transverse energy must satisfy / E > 10 GeV. Our analysis is insensitive to the dark matter mass as long asK and S give missing energy signatures.
C. Testing and sin α with new particle production

1.ZK production
The cross section forZK production is shown in Fig. 5 for various choices of m K and . We see thatZK production grows with 2 , as expected. We consider bothK → + − andK →χχ decays, for = e or µ, where the missing energy branching ratio dominates by g 2 D / 2 . We also studỹ Z → + − with the SM branching fraction of 6.8% [98]. We show the background cross sections for the corresponding 2 2ν and 4 final states after the preselection cuts described in Subsec. V B in Table I. The 2 2ν background includes a combination of Zνν, ZZ and W + W − processes, while the 4 background is mainly attributed to ZZ/Zγ * production.
For the 2 + / E final state, we require a Z-candidate with |m − m Z | < 10 GeV and then look for a peak in the recoil mass distribution, see Eq. (31). For the 4 final state, we identify the Z-candidate from the opposite-sign, same-flavor dilepton pair whose invariant mass is closest to the Z mass and then study the invariant mass distribution of the remaining dilepton pair. TheK signal is tested for each signal mass point in the corresponding mass distributions, and we draw 95% C.L. exclusion regions for each channel in the vs. m K plane for an integrated luminosity L = 5 ab −1 in Fig. 6 and Fig. 7. The relative weight between the 2 2χ and 4 final states is fixed by choosing g D = e ≈ 0.3. We see that the fully visible 4 final state performs worse than the 2 2ν signal selection, simply reflecting the dominant signal statistics in the missing energy channel. Solid lines correspond to e + e − machines operating at √ s = 250 GeV with unpolarized beams, while dashed lines correspond to √ s = 500 GeV. The mass ofK is derived from m K using Eq. (11).
While each search will use the same observable, namely a monochromatic peak in the photon energy as in Eq. (30), the different contributions of SM backgrounds in each event selection will result in the best sensitivity for theK →χχ decay. Background rates and signal regions are shown in Table I. For the inclusive decay ofK, the background γf f is generated where f is a SM fermion, including neutrinos. As mentioned in Subsec. V B, the visible energy distribution is technically equivalent to the recoil mass distribution, and this equivalence is sharpest when the visible SM state is a single photon.
From the results in Fig. 6 and Fig. 7, we see that the most sensitive decay channel isK →χχ, again reflecting the dominant statistics in this final state and the affordable reduction of SM backgrounds by the / E and mono-chromatic photon requirements.
The single photon in the background γνν generally comes from initial state radiation and hence tends to be soft except when produced in the on-shell γZ → γνν process. As long as m K = mZ, however, the signal peak will not run overlap the background peak, and thus we have a flat sensitivity to when m K < mZ. There are two spikes in exclusion sensitivity. The first is for m K ∼ mZ, when the production cross section is greatly enhanced due to maximalK −Z mixing, The next process we consider is the exotic Higgs decay, H 0 →KZ, withK →χχ andZ → + − .
This Higgs exotic decay partial width, from Eq. (35), is proportional to 2 cos 2 α, as long as mK 34 GeV and sin α is neglected. The signal process thus has 2 Z candidates balancing an invisibleK particle, which we identify from the peak in the recoil mass distribution. Our event selection cuts, summarized in Table I  leptons should be in a window around the test variable mK. The resulting sensitivity, as seen in Fig. 6 and Fig. 7, is not competitive with the otherK production processes, given the limited Higgs production statistics and the suppression of the small leptomic decay branching ratio ofZ.
We remark that this decay can also be probed via H 0 → invisible searches using the SM rate for Z →νν, which was discussed in Subsec. IV C.

4.ZS production
Lastly, we can also probe the scalar mixing angle sin α inZS production. This search is exactly analogous to the previous search at LEP-II for a purely invisible decaying Higgs [18], where the visibleZ → + − decay is used to construct the recoil mass distribution. TheZS cross section is proportional to sin 2 α if we neglect , and σZ S is shown in Fig. 8 for sin α = 0.1 and 0.01 at √ s = 250 GeV and √ s = 500 GeV. To maximize sensitivity to α, we studyZ → + − and S decaying invisibly. The signal region is summarized in Table I and focuses on selecting a dilepton Z candidate and reconstructing the recoil mass distribution to identify the S peak. From this analysis, we find that sin α = 0.03 can be probed for light m K using L = 5 ab −1 luminosity for √ s = 250 GeV, as shown in Fig. 8. This result would significantly improve on the current global fit to Higgs data by ATLAS, which constrains sin α < 0.33 [99]. This sensitivity also exceeds the data at √ s = 250 GeV or 500 GeV. We also show comparisons to the current fit, sin α < 0.33 [99], future LHC projections of 0.28 (0.20) using 300 fb −1 (3 ab −1 ) luminosity [1], and precision δσ(Zh) measurements constraining 0.084 (0.055) using 5 ab −1 (10 ab −1 ) [3,4,93]. We plot the excluded region from LEP searches for invisible low mass Higgs in ZS channel in cyan [18][19][20][21].
projected LHC reach of sin α < 0.28 (0.20) using 300 fb −1 (3 ab −1 ) data and critical reductions in theoretical uncertainties [1]. We remark that improved sensivity can be obtained by varying the √ s of the collider to maximize the σ(e + e − →ZS) rate for the test S mass (see also Ref. [60]).

D. Summary
We summarize the sensitivity to in different channels at a future e + e − collider running at √ s = 250 (500) GeV with L = 5 ab −1 in Fig. 9, and we compare the collider searches with constraints from direct detection and indirect detection experiments. In Fig. 9, the dark green shaded region is the exclusion limit from the strongest of the e + e − collider searches presented in Fig. 6 and Fig. 7. We also show the strongest limit from direct detection and indirect detection experiments from Fig. 1 and Fig. 2, as well as the contour satisfying the correct dark matter relic abundance measured by Planck [34]. While the constraints from dark matter detection experiments depend sensitively on the dark matter mass, the collider prospects are insensitive to the dark matter mass, as long as the decay to χ is kinematically allowed and g D . We note that for m K around mZ, the best limit comes from the inclusiveÃK search, which is insensitive to g D , while for m K larger or smaller than mZ, the best sensitivity comes from the monochromatic photon search with / E.
On the other hand, the indirect detection sensitivity and the relic abundance contour both change significantly with dark matter mass. When m χ = 0.495m K , the dark matter resonantly annihilates, improving the reach for indirect searches and dramatically lowering the required to satisfy the relic density measurement. During thermal freeze-out, the finite temperature of the χ velocity distribution gives a strong boost to the annihilation cross section, and thus only very small is needed. For m χ = 0.2m K , however, the limits from indirect detection exclude the relic abundance contour, and the parameter space is instead characterized by an overabundance of the dark matter relic density. For this region to satisfy the Planck bound, additional mediators or new dark matter dynamics controlling the freeze-out behavior are needed. Direct detection experiments also lose sensitivity to dark matter signals for light m χ , since the nuclear recoil spectrum is too soft to pass the fiducial energy threshold. In addition, the decreasing sensitivity for heavy m χ comes from the fall off in the scattering cross section scaling as µ 2 χp /m 2K , see Eq. (28).
We also emphasize that the collider constraint is not sensitive to varying g D as long as g D , which ensures the invisible decay ofK dominates. Hence, the collider constraints from Fig. 6 and Fig. 7 and Fig. 9 are essentially unchanged, since changing g D from e to 0.01 does not significantly change the invisible branching fraction, except for the trade off between inclusiveK decays and invisibleK decays around m K ≈ mZ. On the other hand, the direct detection and indirect detection rates scale with g 2 D , and thus collider searches will have better sensitivity for small g D . From Fig. 9, we see that the prospective collider limits, corresponding to the radiative return process e + e − →ÃK, are expected to overtake the current bounds from direct detection and indirect detection experiments. In the case where dark matter mass is light, m χ = 0.2m K , the collider limits are typically at least one order of magnitude stronger than the current limits, especially in the high mass region, and hence out of the reach of next generation 1-ton scale direct detection experiments.
For dark matter close to half the mediator mass, m χ = 0.495m K , the thermal relic abundance measured by Planck [34] offers an attractive target parameter space for experimental probes. The projected e + e − sensitivity exceeds the current experimental sensitivity around m K ∼ 10 GeV and m K > 100 GeV, and while improvements in the dark matter experiments will also challenge the open parameter space for m K ∼ 10 GeV, the striking sensitivity of e + e − radiative return processes for m K > 100 GeV is expected to be unmatched. Thus, results from a future e + e − collider will both complement and supersede the reach from dark matter searches, stemming from its ability to produce directly the mediators of dark matter interactions.

VI. CONCLUSION
We have presented a comprehensive discussion of the phenomenology of the Double Dark Portal model, which addresses the simultaneous possibility of a kinetic mixing parameter with a scalar Higgs portal λ. We emphasize that these Lagrangian parameters are generic in any U (1) extension of the SM when the additional gauge symmetry is Higgsed. An additional motivation for considering such a U (1) extension is the fact that such a symmetry readily stabilizes the lightest dark sector fermion χ, making this model a natural framework to study possible dark matter interactions in tandem with updated precision Z and Higgs constraints anticipated at future colliders. This study also demonstrates the ability of a future e + e − machine to produce new particles, which are not probed with the current dark matter and LHC experiments.
We work out the interactions in the mass eigenstate basis of neutral vector bosons and Hig-gses. The direct detection limits for this model have been studied, along with indirect detection constraints from CMB measurements, gamma ray measurements, and e ± measurements, where we have explored both the non-resonant and resonant dark matter parameter regions. For collider constraints, we discussed the existing bounds from by electroweak precision and Z-pole observables, Higgs measurements, Drell-Yan measurements, and radiative return processes. Previous constraints have mostly focused on the visible decay,K → + − , and leaving the prospects and sensitivity estimates for the invisible decay,K →χχ largely unexplored.
We studied both the Higgs bremsstrahlung and radiative return processes for a future e + e − collider, emphasizing that a future lepton collider not only has vital Higgs precision capabilities but also new possibilities for producing light new particles,K and S. Since bothK and S decays are dominantly invisible, the recoil mass method afforded by an e + e − machine is crucial. We also highlight that the recoil mass method can be simplified to a monochromatic photon study in the case that the new particle is produced in the radiative return process, which simplifies the search procedure and enhances the importance for upcoming calorimeters to have a precise, high-resolution energy determination for photons. The various Higgsstrahlung and radiative return processes we study are listed in Table I, and we obtain the best sensitivity on and sin α from the radiative return processÃK and Higgsstrahlung processZS, respectively.
In comparing prospects, we analyzed the future collider reach with direct detection, indirect detection and relic abundance sensitivities. The collider prospects are less affected by DM mass m χ , and surpass the other experimental probes for small g D . SinceK decays invisibly, the most relevant current constraints are from electroweak precision measurements and LHC mono-jet searches, but they are not as strong as the radiative return processÃK reach. Therefore, a future e + e − collider provides an important and complementary sensitivity test of the DDP model.
For sin α, the best constraints come from studying the singlet bremsstrahlung processZS, the Higgs invisible decay rate, and precision measurements of SM Higgs production rates. We studied theZS process with S decaying invisibly for a future e + e − collider and estimated the sensitivity to be sin α ∼ 0.03. This compares favorably with earlier LEP studies for light m S , and readily provides leading sensitivity for heavy S. We also recasted bounds using the Higgs invisible decay channel, where the current LHC constraint is BR inv < 0.23 [67,68] and the future e + e − collider reach is BR inv < 0.005 [3]. In the DDP model, these bounds simultaneously constrain the three exotic processes, H 0 → SS, H 0 →KK, and H 0 →ZK whenZ decays to neutrinos. While the constraints on sin α can be strong, these limits also depend sensitively on g D and are insignificant for small g D . The future σ(e + e − → Zh) precision measurement readily constrains cos 2 α, but this projection is weaker than the directZS search.
In summary, the Double Dark Portal model predicts new dark sector particles,K, S, and χ, whose vector and scalar portal interactions with the Standard Model can be uniquely tested at a future e + e − collider. We explicitly propose and study radiative return and Higgsstrahlung processes to find the invisible decays of theK and S mediators. An additional benefit of the e + e − search strategies discussed in this work is that, in the event of a discovery, theK or S mass is immediately measured in the recoil mass distribution. Hence, a future e + e − collider not only has exciting prospects for determining the precise properties of the 125 GeV Higgs boson, but also has a unique and promising new physics program founded on the production of new, light, hidden particles.
where the expressions for U 1 , U 2 and R M have been given in Eq. (7), and Eq. (9), respectively.
We will consider the two limiting cases, m K → 0 and m K → m Z,SM , and study the corresponding changes for the kinetic and mass mixing matrices.
For m K → 0, the gauge boson masses are mÃ = mK = 0 , (A2) and the field redefinition is The Jacobi rotation R M , from Eq. (9), is now ill-defined in the lower right two-by-two block, sincẽ A andK can be rotated into each other keeping both the kinetic terms and masses unchanged.
This simply reflects the residual unbroken U (1) em × U (1) D gauge symmetry. For R M = I 3 , the currents are but under a unitary rotation U X where (Ã ,K ) T = U X (Ã,K) T , the dark matter χ and the SM fermions will generally have nonzero charges mediated by bothÃ andK , leading to photon and dark photon-mediated electric and dark millicharges.
For m K → m Z, SM , the masses of the three vector bosons are and the field redefinition required, to O( 2 ), is where the top and bottom signs correspond to m K → m ∓ Z, SM . We see that the mixing between Z µ and K µ is nearly maximal, 45 • , while the discontinuous behavior for m K below and above m Z, SM reflects the level crossing in the mass eigenvalues. We remark that as long as = 0, this maximal mixing feature remains, dictated by the structure of the symmetric mass matrix in Eq. (8). If = 0 and m K = m Z , then the rotation matrix in Eq. (9) becomes ill-defined and the maximal mixing feature is lost.
Appendix B: Cancellation effect in multiple kinetic mixing terms We observe that theZ andK mediated couplings in Eq. (15) show a cancellation effect when mediating DM interactions with SM fermions. This feature can be generalized to the situation with multiple U (1) gauge groups with multiple kinetic mixing terms between each other. Explicitly, we analyze the Lagrangian where K ab = δ ab + O( )(1 − δ ab ) is the kinetic mixing matrix and M 2 is the diagonal mass matrix, with a, b as indices. Then, we define the field redefinition matrix U such that U T KU = I, which also givesM 2 = U T M 2 U as the mass matrix corresponding to the mass eigenstates,Ṽ = U −1 V .
Moreover, the gauge currents now become in the mass basis. As a result, scattering rates between two currents J a and J b (which represent the corresponding fermion bilinears) are schematically .
The −g µν /m 2 V k term in the parentheses, however, vanishes, when including the sum over U ak (U T ) kb , because these transformations are controlled by the diagonalization requirement of the two mass matrices, specifically UM −2 U T = M −2 . The leading contribution in the amplitude is then proportional to Q 2 /m 2 V .

Appendix C: Annihilation cross sections
In this section, we present the annihilation cross sections for the processesχχ →f f , W + W − , where f is a SM fermion. We focus on the case with m χ < mK, since otherwise the direct annihilation of dark matter to dark vectorsKK opens up and does not depend on . In this setup, the annihilation cross section is proportional to g 2 D 2 . The diagrams include s-channelK andZ exchange. The annihilation cross sections before thermal averaging are σv χχ → + − = e 2 2 g 2 D s − 4m 2 2m 2 χ + s σv (χχ →ūu) = e 2 2 g 2 D s − 4m 2 u 2m 2 χ + s σv χχ →dd = e 2 2 g 2 D s − 4m 2 d 2m 2 χ + s σv χχ → W + W − = e 2 2 g 2 D 2m 2 χ + s s − 4m 2 W 3/2 20sm 2 W + 12m 4 W + s 2