A tale of two portals: testing light, hidden new physics at future e + e − colliders

Journal of High Energy Physics, Jun 2017

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 − → 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.

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A tale of two portals: testing light, hidden new physics at future e + e − colliders

JHE physics at future e+e Jia Liu 0 1 Xiao-Ping Wang 0 1 Felix Yu 0 1 0 Johannes Gutenberg University , 55099 Mainz , Germany 1 PRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics 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 modi es the Standard Model Higgsstrahlung channel, e+e 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 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 machines, whose invisible decays can be tagged from the recoil mass method. Beyond Standard Model; Cosmology of Theories beyond the SM; Higgs - HJEP06(217) Physics 1 Introduction 2 Overview of the double dark portal model: simultaneous kinetic mixing and scalar portal couplings 2.1 2.2 2.3 Neutral vector boson mixing Scalar boson mixing Dark matter interactions 4.1 4.2 4.3 5.1 5.2 5.3 4.3.1 4.3.2 5.3.1 5.3.2 5.3.3 5.3.4 3 4 Direct detection and indirect detection phenomenology and constraints 3.1 Direct detection and relic abundance 3.2 Indirect constraints from CMB, Gamma-ray and e measurements Collider phenomenology of the Double Dark Portal model and current constraints from LEP and LHC Recoil mass method for probing new, light, hidden states Modi cations to electroweak precision 4.2.1 LEP-I and LEP-II constraints Modi cations to Higgs physics and LHC constraints Modi cations to Drell-Yan processes Radiative return processes and dark matter production at the LHC 5 Prospects for future colliders Electroweak precision tests, Higgsstrahlung, and invisible Higgs decays at future e+e colliders Production of new, light states at future e+e colliders Testing and sin with new particle production Z~K~ production A~K~ production Z~S production Z~H0, H0 ! K~ Z~ exotic decay 5.4 Summary 6 Conclusion A Two limiting cases for K~ , Z~, and A~ mixing B Cancellation e ect in multiple kinetic mixing terms C Annihilation cross sections 2 3 Introduction Searches for new, light, hidden states are strongly motivated from the overriding question of determining the particle nature of dark matter. The possible couplings to such light states, however, remain highly model-dependent. Because higher dimension operators are expected to be suppressed in scattering processes at low energies, the most promising couplings give marginal Lagrangian operators at dimension four. Along these lines, two well-studied couplings are a new kinetic mixing term between a new, light, hidden photon and the hypercharge gauge boson and a new quartic Higgs portal coupling HP between a hidden charged scalar eld and the Standard Model Higgs eld. In this work, we argue and demonstrate that both marginal couplings can be simultaneously probed in future measurements of a high energy e+e collider. Such a collider is, of course, very strongly motivated by a rich and diverse set of possible Higgs measurements, with leading sensitivity to the total Higgs width, Higgs couplings to Standard Model (SM) particles, exotic Higgs decays, and additional precision measurements of the top quark mass and exotic Z boson decays if additional running conditions are a orded [1{4]. We highlight that such a machine also has leading sensitivity to new, weakly coupled, hidden sectors, which can be probed via both radiative return processes and exotic invisible and semi-visible Higgs decays. We will show that these measurements are enabled because of the expected high precision photon resolution in the electromagnetic calorimeter, the exquisite reconstruction of charged leptons, and clean discrimination of exotic signals from SM background processes. Both of these marginal operators have been studied autonomously at electron colliders in the hidden photon context [5{17] and the hidden scalar context [18{23]. Some works study both operators in tandem [24{26] or adopt an e ective operator approach [27]. The current status of light, sub-GeV hidden photon searches and future prospects is summarized in ref. [28]. In contrast with previous studies, we focus on higher mass hidden photons beyond the reach of B-physics experiments and beam-dump experiments. For illustrative purposes, we show our projections to dark photons as light as 1 GeV to demonstrate the complementarity with recent results from B-physics experiments such as BaBar [29]. 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{33], increasingly strong constraints on thermal WIMPs from indirect detection (ID) experiments [34{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 su ers 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{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 { 2 { 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 section 2, we review the theoretical framework for the Double Dark Portal model, which uni es the kinetic mixing portal and the scalar Higgs portal into a minimal setup with dark matter. In section 3, 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 section 4. We then present the prospects for exploring new, light hidden states at a future e+e machine in section 5 and conclude in section 6. In appendix A, we o er some detailed discussion of limiting cases in our Double Dark Portal model for pedagogical clarity, and we discuss a cancellation e ect in scattering processes via kinetic mixing in appendix B. We also present the dark matter annihilation cross sections for charged SM nal states in appendix C. 2 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, is the dark matter and a SM gauge singlet fermion 2H > 0 and 2D > 0, which trigger spontaneous symmetry breaking of the SM electroweak symmetry and the U( 1 )D dark gauge symmetry, respectively. The W parameter is the tree-level SM weak mixing angle, W = tan 1(g0=g). The nonzero Higgs portal coupling, HP , induces mass mixing between the h and scalars, which results in mass eigenstates H0 and S. Simultaneously, the kinetic mixing result in an e ective mass mixing between the SM Z gauge boson and the K dark gauge boson, which results in the mass eigenstates Z~ and K~ . The two marginal couplings, will and HP , are commonly referred to as vector and scalar portals, respectively [46]. Because the phenomenology of such portal couplings changes signi cantly when a light dark matter particle is added, we call the Lagrangian in eq. (2.1) 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), = p (vD + ) ; H = p (vH + h) ; 1 1 2 2 { 3 { (2.1) (2.2) (2.3) 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 B K )T as L 1 2 W 3 B K g0g H g02 H 0 B B B B B CCCC BBB B K U( 1 )D ! U( 1 )em, the resulting eld strength tensors of the individual neutral vectors corresponding to the gauge eigenstates W 3, B, and K all have Abelian eld strengths, while non-Abelian vector interactions are inherited from the SU(2)L gauge boson eld strength tensor. We will not explicitly write the non-Abelian vector interactions in the following, but instead understand that they are correspondingly modi ed when we perform the rescaling needed to canonically normalize the Abelian eld strengths of the neutral vectors.1 2.1 Neutral vector boson mixing To simplify the Lagrangian in the broken phase, we rst rotate by the tree-level SM weak mixing angle, which reduces the mass matrix to rank 2 and correspondingly modi es the kinetic mixing between the Abelian eld strengths. Explicitly, we sandwich R W RTW twice in eq. (2.4), with cW = cos W and sW = sin W , which gives sW cW 0 C ; 0 0 0 C B A ; SM CA ; K (2.5) (2.6) 0 tW B where tW = tan W , m2Z; SM = (g2 + g02)vH2 =4 is the tree-level SM Z-boson mass, and m2K = gD2vD2 is the tree-level U( 1 )D gauge boson mass. To canonically normalize the 1We remark that the Stueckelberg mechanism [47, 48] provides an alternative mass generation for K~ , kinetic terms for the neutral gauge bosons, we use the successive transformations 0 2cW2 0 3tW (1 2)(1 2cW2) 0 1 (2.7) HJEP06(217) (2.8) (2.9) (2.10) ; (2.11) where the kinetic terms are now canonically normalized and only one further unitary rotation is needed to diagonalize the mass matrix. We remark that j j < cW is required to ensure the kinetic mixing matrix in eq. (2.6) has a positive de nite determinant, which allows U2 to remain non-singular. The nal Jacobi rotation required is 1 0 C 1 A for cM = cos M and sM = sin M and M de ned by tan M = m2Z; SM(1 and the corresponding neutral vector basis is RMT U 1 mark that these are exact expressions valid for arbitrary . For 1, we provide compact expressions for the masses and the corresponding gauge elds in the mass basis. To O( 3), m2K~ = m2K + m2K cW2 2(m2Z; SMcW 2 = BB 0 B B B B B 0 Z ; SM 1 K tW m2K 1 is insu cient for mK ! 0 or mK ! mZ; SM. These two limits are discussed in appendix A. Given that is small, the masses of K~ and Z~ 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: m2K JZ + e Jem 2m2K m2Z; SM + m4K )cW2 2 2(m2Z; SM m2K )2 J D + A~ eJem : (2.15) m2Z; SM(m2Z; SM 2m2K )t2W 2 2(m2K m2Z; SM)2 J Z Again, the situation for mK ! 0 or mK ! mZ; SM is discussed in appendix A. From these U( 1 )D sector correspondingly receives an O( ) dark charge mediated by Z~ . expressions, we see explicitly that SM fermions, encoded via Jem and JZ , obtain an O( ) electric charge and an O( ) neutral weak charge mediated by K~ . Matter charged in the { 6 { 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. (2.2) and eq. (2.3), we have 2D = 2H = DvD2 + H vH2 + 1 1 2 HP vH2 ; 2 HP vD2 : S ! H0 = cos sin sin cos ! h ! ; tan 2 = HP vH vD DvD2 H vH2 The scalar mass eigenstates are then is the scalar mixing angle. The scalar masses are m2S; H0 = H vH2 + DvD2 q ( H vH2 DvD2)2 + HP vH2 vD2 : We can thus reparametrize the scalar Lagrangian couplings mS, mH0 , vD, vH , and . The reparametrizations for 2 , 2H and D D, H , D, H , HP as HP are given above, while the reparametrization for D and H are H = D = 1 1 4vH2 4vD2 m2H0 + m2S + (m2H0 m2H0 + m2S (m2H0 m2S) cos 2 m2S) cos 2 ; : We also calculate the scalar interactions in the mass eigenstate basis H0 and S. The cubic scalar interactions are L S3m2S vH cos3 + vD sin3 2vDvH + H0S2 m2H0 + 2m2S (vH cos 4vDvH H02S 2m2H0 + m2S (vD cos 4vDvH H03m2H0 vD cos3 vD sin ) sin 2 vH sin3 2vDvH + vH sin ) sin 2 : (2.23) j HP j > p We have, of course, mH0 = 125 GeV and vH = 246 GeV, but the other observables are free parameters. We will restrict HP > 0 in our analysis, recognizing that HP < 0 and H D can cause tree-level destabilization of the electroweak vacuum. { 7 { Lastly, the scalar-vector-vector interactions of K~ , Z~, S and H0 in the mass basis to O( 2) are L m2Z;SM + 2 tW + m2K + m2Z;SM + 2 tW + m2K cos vH m2K m2Z; SM (m2Z; SM m2K ) sin vD sin vH tW (m2K tW (m2K m2K m2Z; SM (m2Z; SM m2K ) tW (m2K sin cos vH + sin vD m2K m2Z; SM m2K m2Z; SM Z~ K~ H0 m2Z; SM)2 vH m2Z; SM)2 vD cos cos ! ! K~ K~ H0 Z~ Z~ S cos vD + sin vH Z~ K~ S cos vD tW (m2K m2K m2Z; SM sin ! m2Z; SM)2 vH K~ K~ S : (2.24) bosons is changed not only by cos coupling proportional to sin and also 2 cos . with S ! K~ K~ ! 4 are theoretical parameters that must be constrained by data, and hence a particular hierarchy between and would re ect model-dependent assumptions. As a result, eq. (2.24) forms a consistent basis for determining the sensitivity to and simultaneously. We can characterize the changes in the phenomenology of the Higgs-like H0 state as a combination of modi ed SM-like production and decay modes and the opening of new exotic production and decay channels. One main e ect of is to suppress all of the SM fermion couplings of the H0 state by cos , while the S state acquires Higgs-like couplings to SM fermions proportional to sin . This feature also applies to the loop-induced couplings to gluons and photons for H0 and S. On the other hand, the coupling between H0 to Z~ but also by 2 sin , while the S state acquires a Z~ In addition, if kinematically open, the H0 state can decay to pairs of S or pairs of K~ , as possible subsequent decays. These Higgs invisible decays are also mimicked by the exotic H0 ! Z~K~ decay, when Z~ ! total invisible width of H0 is sensitive to a combination of di erent couplings and masses in eq. (2.23) and eq. (2.24), further demonstrating the viability of the Double Dark Portal model as a self-consistent theoretical framework for constraining Higgs observables. We remark that we have not added a direct Yukawa coupling between and , e.g. if has charge +2 under U( 1 )D, we would introduce a direct decay from S to dark matter and also . As a result, the split the Dirac dark matter into Majorana fermions [50]. 2.3 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 { 8 { couplings to SM particles via the JD current shown in eq. (2.15). Explicitly, the dark matter particle Lagrangian reads L m2K Direct detection and indirect detection phenomenology and constraints The Double Dark Portal model presented in eq. (2.1) o ers many phenomenological opportunities, including dark matter signals at direct detection, indirect detection, and collider experiments and modi cations of electroweak precision and Higgs physics at colliders. We remark that aside from the vacuum stability requirement on HP and upper bound on j j, the theory parameter space of the Double Dark Portal model is wide open and subject only to experimental constraints. This vast parameter space has been extremely useful in motivating searches for light, hidden mediators at high intensity, beam-dump experiments, as reviewed in refs. [28, 46]. Our focus, however, is the O(10 its accompanying Higgs partner S, which will both dominantly decay to the dark matter particle . This is readily motivated by considering m K~ has an on-shell two-body decay to charged particles. Moreover, for mK~ < mS=2 and sin and an 2=gD2 suppressed branching ratio to SM < mK~ =2 and gD , so that gD, the SM gauge singlet scalar S dominantly decays to pairs of K~ and only have sin2 =gD2 suppressed rates to SM pairs. All of these choices, however, can be reversed to give markedly di erent phenomenology. If m > mK~ =2, for example, then the total width of mK~ scales as 2 [10] and K~ decays to pairs of SM charged fermions, as long as it is heavier than 2me. For very small , however, the K~ lifetime can be long, leading to either displaced vertex signatures or missing energy signatures. The lifetime and decay length of K~ can be estimated to be 100 GeV) scale for the K~ vector mediator and 1 = = 0:9 = 60 (as from a Higgs two-body decay) and N~F is the e ective factor for kinematically open charge-weighted two-body SM nal states. If mS, mK~ > mH0 =2, then any possible exotic decay of the SM-like Higgs will be strongly suppressed by multi-body phase space and a combination of gD, sin , or HP . We remark that choosing mS < mH0 =2 already gives an exotic Higgs decay, H0 ! SS, which is sensitive directly to HP . Given our mass hierarchy, the dominant collider signature from production of either K~ or S is missing energy from escaping particles, while the relic density of 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 section 4 and section 5. { 9 { D 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 of Z~ and K~ , as evident from the J D and Jem interactions shown in eq. (2.15). Because of the relative sign between the K~ and Z~ terms, dark matter scattering proportional to g g 2 2 2 is naturally suppressed by extra 2 or Q2=m2K D factors, where Q is the momentum transfer scale, and the leading contribution is hence proportional to e2g2 2 . This cancellation between K~ and Z~ mediators is generic, and we outline the details in appendix B. As a result, the dominant DM-nucleon interaction for direct detection is mainly from DM-proton scattering. With the SM and DM currents from eq. (2.15), the DM-proton scattering cross-section is p ' D p is the reduced mass of the dark matter and the proton and e = p4 =137. The cross-section p is calculated at leading order in and vin, the incoming DM velocity, and agrees with previous results when DM only interacts via t-channel K~ 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(Q2=m2V ) suppressed after summing all the vector boson contributions, where mV is the smaller of either gauge boson mass. For DM direct detection, the momentum transfer is about Q2 (m vin) 2 m2K~ ;Z~, hence the contribution induced by the JZ current cancels and we arrive at the same result in ref. [41]. channel insensitive to . We can also motivate particular contours in the vs. mK~ plane by considering rst the requirement 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 rst calculate the annihilation cross sections for where f denotes a SM fermion. We focus on the region m ! f f , W +W , < mK~ , since the annihilation ! K~ K~ opens up otherwise and the dominant self-annihilation cross section is In this setup, the annihilation cross section will be proportional to g2 2 D . 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 , mZ~ = 2m and mK~ = mZ~. The rst two resonances are from the s-channel resonant exchange of K~ and Z~, while the last one is due to maximal mixing between K~ and Z~ when mK is close to mZ, SM, as discussed in appendix A. In gure 1, we show the direct detection constraints in the vs. mK 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 gD = e, 0:1, and 0:01, and the dark matter mass xed to 0:2mK , 0:495mK , 0:6 GeV, or 30 GeV. 10-1600 ϵ10-3 10-5 100 10-1 10-2 ϵ10-3 10-4 10-5 10-1600 10-1 10-2 ϵ10-3 10-4 10-5 100 10-1 10-2 ϵ10-3 10-4 10-5 gD = 0:1 (solid), and gD = 0:01 (dashed), and we overlay black contours to mark the relic density requirement from the Planck collaboration [34]. Note that mK is approximately the mK~ mass eigenvalue according to eq. (2.11). The thermal relic abundance limit on is given in gure 1 using the h 2 = 0:12 requirement from the Planck collaboration [34]. The dip around mK mZ, SM re ects increasing mixing between K~ and Z~. While for mK is enhanced by the s-channel K~ resonance, thus the required 2m , the annihilation cross section is very small. When 2m mK~ < Tf , where Tf section is enhanced by (mK~ =Tf )2 due to thermal averaging. When 2m thermal average will not bene t the resonance e ect anymore. m =25 is the DM freeze-out temperature, the annihilation cross mK~ > 0, the In recasting the direct detection limits, we recognize that the experiments assume that the local DM density is xed to 0:3 GeV=cm3. 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 xed mK and varying 6= relic, with relic corresponding to h v i = 0:3 GeV=cm3, 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 xed to 0:3 GeV=cm3 regardless of , in order to determine the sensitivity to the direct detection cross section. For large , when the local DM relic density predicted in the DDP model is generally underabundant, extra dark matter particles beyond the DDP model are needed, while for small , the DM relic density is generally overabundant and extra annihilation channels are typically needed. Hence, the direct detection exclusion contours in each panel simply illustrate the fractional relic density, relative to h 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 is required to satisfy the 0:3 GeV=cm3 assumption, and is excluded by DD experiments as shown in the red shaded region. In particular, for xed mK , 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 2 p=m2K~ suppression in eq. (3.3). For m constraint on p . We see that for heavy K~ and light , the direct detection sensitivity can / mK~ , the sensitivity on generally follows the experimental be weak, leaving signi cant parameter space to be probed by colliders. Interesting parameter space also exists for m . mK~ =2, which will be discussed further in the next section. 3.2 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{57]. The constraint is calculated using the energy deposition yield, fei , where i denotes a particular annihilation or decay channel and fe describes the e ciency of energy absorption by the CMB from the energy released by DM in particular channel. The constraint is expressed as pann = 1 m X f i i e h vii ; (3.4) where the Planck experiment has constrained pann < 4:1 we sum all the SM fermion pair f f and W +W parameters are plotted in gure 2 as shaded purple regions. 10 28 cm3 s 1 GeV 1 [34], and channels in annihilation. The excluded 10-1600 10-1 10-2 ϵ10-3 10-4 10-5 100 10-1 10-2 ϵ10-3 10-4 10-5 10-1600 F Dw rem fra -iIG C M B C M B 101 ments from dwarf galaxies [35, 36] and the inner galactic region [37] and e+ ux measurement from AMS-02 [38]. The constraints are shown in the vs. mK plane for m = 0:2 mK (top left), 0:495 mK (top right), 0:6 GeV (bottom left), and 30 GeV (bottom right), with gD = e (dotted), 0:1 (solid), and 0:01 (dashed). Note that mK is approximately the mK~ mass eigenvalue according to eq. (2.11). 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 , + , uu, bb and W +W nal states, while in ref. [36], MAGIC has made a combined analysis with Fermi-LAT and presented constraints on + , + , bb and W +W . The computation of constraints on our model is straightforward, since we can calculate each individual limit on for each channel and take the most stringent constraint for each m mass. The excluded parameter region is shaded by cyan in gure 2. Similarly, we consider the gamma ray constraints from the inner Milky way [37]. This analysis sets conservative constraints on various SM nal states by using the inclusive photon spectrum observed by HJEP06(217) the Fermi-LAT satellite. We apply their results by calculating the most stringent annihilation pro le, assuming the Navarro-Frenk-White pro le for the DM density distribution in the galactic center [58]. We can see in gure 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 nal states, which mainly derive from the observed positron ux. 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 masses in gure 2. To summarize, in gure 2, we see that the CMB constraint is strongest at small m , HJEP06(217) while AMS-02 is strongest at higher m . The constraint from gamma ray observations in dwarf galaxies is very close to the CMB constraint. Meanwhile, the dips in gure 2 nicely show the two s-channel resonances of K~ and Z~ as well as the maximal mixing peak between K~ and Z~. 4 Collider phenomenology of the Double Dark Portal model and current constraints from LEP and LHC In this section, we give an overview of the possible probes of the Double Dark Portal Model at both lepton and hadron colliders. While many separate searches have been performed at LEP and LHC experiments in the context of either kinetic mixing or Higgs mixing scenarios, we highlight the fact that a future e+e machine must synthesize both e ects in any given search. Hence, the Double Dark Portal model is a natural framework to study light, hidden physics at a future e+e machine. The Double Dark Portal model motivates observable deviations in measurements of both the SM-like H0 and the Z~ bosons, which test the scalar mixing angle as well as the kinetic mixing parameter . Notably, the primary SM Higgsstrahlung workhorse process at an e+e for nonzero H0Z~ Z~ Higgs factory, e+e ! Zh, can deviate signi cantly from the SM expectation or . For instance, nonzero causes a well-known cos suppression of the vertex, but nonzero gives an additional diagram with intermediate K~ , which captured by a simple cos rescaling of the H0Z~ Z~ vertex. becomes on-shell when mK~ > mZ~ + mH0 . We remark that these e ects are not generically In gure 3, we show the new possibilities for SM-like and dark scalar Higgsstrahlung from the intermediate massive vector bosons K~ and Z~. We also show the radiative return process for e+e ! A~K~ or A~Z~, and the diboson process e+e ! K~ Z~. All of these processes give di erent signals at a future e+e machine. If we also consider the possibility of Z-pole measurements and Drell-Yan processes probing eq. (2.15), then we can categorize the collider phenomenology of the Double Dark Portal model into four groups: electroweak precision and Z-pole observables, Higgs measurements, Drell-Yan measurements, and radiative return processes. We point out, however, that e+e machines o er unique opportunities for probing new, light, hidden particles by virtue of the recoil mass method, which we discuss rst. A˜, Z˜ K˜ , Z˜ χ, f χ¯, f¯ e− e + ˜ K ˜ K K˜ , Z˜ χ χ¯ χ¯ χ K˜ , Z˜ S, H0 e− e + ˜ Z ˜ K S H0 ˜ K vector production and (bottom row) example new decay processes in the Double Dark Portal model sensitive to the kinetic mixing and scalar Higgs HP portal couplings. Note Z~, A~, and H0 are the mass eigenstates corresponding to the SM-like Z, photon, and Higgs bosons, respectively. Recoil mass method for probing new, light, hidden states S and K~ masses. This p new physics signal in the recoil mass distribution. As long as they are kinematically accessible, both S and K~ 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 and , according to the interactions from eq. (2.15) and eq. (2.24). This is familiar from the leading e+e sistent with a 125 GeV recoil mass gives a rate dependent only on the H0Z~ Z~ Higgsstrahlung production process, where the reconstruction of the Z~ ! `+` decay concoupling. We emphasize (see also ref. [60]) that this generalizes to any scattering process at an e+e ! Z~H0 machine if visible SM states are produced in association with a new, light, hidden particle. Moreover, sensitivity to the hidden states S and K~ can be improved by scanning over ps, where the various production modes of Z~S, K~ , and Z~K~ can be optimized for the di erent s adjustment would be immediately motivated, for example, by a The recoil mass method uses the knowledge that the center-of-mass frame for the e+e collision is xed to be (ps; 0; 0; 0) in the lab frame, where p s is the energy of the collider. Hence, for an invisibly decaying nal state particle X produced in association with a SM state Y , four-momentum conservation requires or equivalently, EY = p s 2 + m2Y p 2 s m2X ; mX = q s + m2Y 2EY p s : (4.1) (4.2) If there are multiple visible states Yi, this generalizes to X EYi = p s 2 + (P pi)2 p 2 s m2X ; r mX = s + X pi 2 2 X EYi ps ; (4.3) where (P pi)2 is the total invariant mass of the Yi system. We see that studying the di erential distribution of EY will show a characteristic excess at a given EY when X is produced. Identifying this monochromatic peak is formally equivalent to nding a peak in the recoil mass distribution, but we emphasize that these two distributions reconstructed di erently at e+e colliders. Speci cally, the recoil mass distribution uses both the energy and total four-momentum of each detected SM particle, which the di erential energy Higgsstrahlung process, the recoil mass method distribution only requires calorimeter information. In particular, for the Z~H0, Z~ ! `+` requires measurements of each individual lepton four-momentum and the event-by-event invariant mass m``. The resulting di erential distribution also includes o -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 and K~ have dominant decay widths to the dark matter , which does not leave tracks or calorimeter energy deposits as it escapes. The recoil mass technique, however, also readily probes both the K~ and S masses in numerous production modes, when we produce K~ or S in association with a visible SM nal state. For example, while the SM-like Z~ boson is a canonical choice to study Z~H0 events, we can use the recoil mass technique in the radiative return process for A~K~ production to identify the invisible decay of K~ . An even more striking possibility is to use the SM-like Higgs boson, H0, as the recoil mass particle to probe K~ H0 production. 4.2 Modi cations to electroweak precision We now consider the four categories of collider processes in turn. The rst set of observables we consider are those from electroweak precision tests. In the Double Dark Portal model, Z-pole observables will show deviations according to the new decay channel Z~ ! Z~ ! SK~ , sensitive to , shifts in the Z~ mass from the mixing with K~ , and deviations in the weak mixing angle from the mixing between K~ , A~, and Z~. In particular, identifying or the Z~ 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. 4.2.1 LEP-I and LEP-II constraints At LEP-II, contact operators (4 = 2)e ef f were used to test for new physics, analogous to angular distributions in dijet studies at the LHC. In the e+e ! `+` channel, the corresponding bound on because of maximal mixing, and again around p s Because the K~ decay width is dominated by K~ suppressed the branching ratio of K~ ! `+` . ! p constraint on s = 200 GeV, we can place a constraint on by matching the coe cient of the contact operator at tree-level to an intermediate K~ mediator, gK2~ ``=(q(s m2~ )2 + m2~ 2K~ ). The K K is & 20 TeV [63]. Since the majority of the dataset was taken at a xed is around O(0:1). There is sharper sensitivity for mK mZ 200 GeV from resonant production. , the sensitivity at resonance is and gure 7 as \LEP-EWPT." invisibly, e+e ! K~ , K~ ! As mentioned above, the mixing between ZSM and K leads to shifts in the Z~ mass and couplings to SM fermions, leading to a constraint of < 0:03 for mK~ < mZ using a combination of electroweak precision observables [64]. The constraint is weakened for mK~ > mZ where the limit on is about 0:1 at mK~ = 200 GeV [64], and is shown in gure 6 Recently, the BaBar collaboration has published constraints on dark photons decaying [29]. Their constraints directly map to our parameter space and place strong limits on . 0:001 for masses between 1 GeV to 10 GeV. These are reproduced in gure 6 and gure 7, and labeled as \BaBar." Although the canonical SM Higgs production channel e+e LEP-II, the scalar mixing angle sin can still be probed by the e+e tion mode when S is kinematically accessible by LEP-II. For mS < 114 GeV, the nonobservation of Higgs-like scalar decays constrains sin2 the S ! K~ K~ decay is turned o . < O(0:01 0:1) [22], as long as decay, with Z? ! `+` The LEP experiments have also searched for a low mass Higgs in the exotic Z ! HZ? and H decaying invisibly, which excludes mH < 66:7 GeV if the ! Zh was ine ectual at ! Z~S producinvisible branching fraction is 100% [18]. The ZH Higgsstrahlung process is also used to push the mass exclusion to 114:4 GeV [19{21], although the intermediate mass range between these two limits are not comprehensively covered. In our model, S ! K~ K~ is the dominant decay when gD the production cross section (Z~S) are hence sin2 sin , and the decay branching fraction Z~ ! SZ~? and Z~H0 Higgsstrahlung process in Therefore, these limits apply to S as bounds on sin in section 5. Note the constraint from the exotic Z~ ! SZ~? decay is much stronger than gure 8 due to the high statistics of Z decays, and in the suppressed compared to the SM rate. and mS, which we will show in gure 8 and BR(K~ ! ). calculation we accounted for the subsequent decay branching fractions of BR(S ! K~ K~ ) 4.3 Modi cations to Higgs physics and LHC constraints H0 ! K~ K~ decays, have also given constraints on With the era of precision Higgs characterization underway after the discovery of a Higgs-like boson [65, 66], the ATLAS and CMS collaborations have provided the strongest constraints on the possible mixing of the SM Higgs boson with a new gauge singlet . In addition, searches for an invisible decay of the 125 GeV Higgs boson, sensitive to H0 ! SS or HP , sin , and . The growing Higgs dataset at the LHC continues to show no signi cant deviations from the SM expectation, but the current sensitivity of the LHC experiments to our proposed signals is limited. The most important constraint comes from the search for an invisible decay of the H0 ! K~ Z~ decay widths at leading order in and are 125 GeV Higgs, where the Run 1 combination of ATLAS and CMS data constrains BR(h ! inv) 0:23 [67, 69]. We highlight, however, that this limit requires that the Higgs is produced in the Z~H0 and vector boson fusion processes at SM rates, which is violated in the DDP model. Moreover, in the DDP model, there are two possible direct invisible decay H0 ! Z~K~ ! decays, H0 ! SS ! 4K~ ! 8 and H0 ! 2K~ ! 4 , in addition to the possible exotic , which is often semi-visible. The H0 ! SS, H0 ! K~ K~ and (H0 ! SS) = gD2 sin2 (H0 ! K~ K~ ) = gD2 sin2 (H0 ! K~ Z~) = 2 2 t W 16 m3H0 1 s v u mH0 32 32 mH0 ut1 cos vH m2K r + svinD The rst two decay widths are proportional to mK2 while the last one is proportional to m2K , therefore the last one is usually much smaller comparing with the rst two when mK is light. In the Higgs invisible studies, the experiments will constrain the rate for Higgs invisible decays in the DDP model, BR(H0 ! inv) (H0 ! SS)BR2(S ! K~ K~ )BR4(K~ ! ) + (H0 ! K~ K~ )BR2(K~ ! ) + (H0 ! K~ Z~)BR(K~ ! )BR(Z~ ! BRienv = (Z~H0) SM(Zh) = cos2 1 H0; tot where K~ ! H0; tot = cos2 widths K~ ! f f , W +W is h; SM + (H0 ! SS) + (H0 ! K~ K~ ) + (H0 ! K~ Z~). The decay can be found in the appendix of ref. [70], while the decay width ) ; (4.7) (4.8) (K~ ! ) = g D2 q 12 m2~ K 4m2 1 + 2m2 ! m2~ K : The prediction for the invisible decay branching fraction of the 125 GeV Higgs is shown in the left and middle panels of gure 4 in the sin vs. plane for mS = 50 GeV, mK = 20 GeV, and gD = e, 0:01. The current constraint of BRinv < 0:23 is adopted from s 10-2 n i 10-3 10-3 10-4 10-5 10-6 10-7 mS = 50 GeV mK = 20 GeV mχ = 0 GeV 10-2 BR n i s 10-3 10-3 10-140-4 BR(K~ and BR(K~ ! ! plane, setting mS = 50 GeV, mK = 20 GeV, and gD = e (left) and 0:01 (middle). (Right panel) Exclusion regions in the sin vs. mK plane from the search for an invisible decay of the 125 GeV Higgs by ATLAS and CMS giving BRinv < 0:23 [67, 69], and projected reach from a s = 240 GeV data using unpolarized beams in ref. [3]. This prospective using 10 ab 1 of p refs. [67, 69], while the prospective sensitivity of BRinv < 0:005 is adopted from the estimate limit can be lowered in combined ts, with more luminosity, or with other assumptions about detector performance to the O(0:001) level [1, 2, 4]. In gure 4, we see that when gD is large, the sensitivity to sin is much stronger than , because the decay widths for H0 ! SS and H0 ! K~ K~ are much larger than H0 ! K~ Z~ due to light mK , as discussed previously. More importantly, for large gD, BR(S ! K~ K~ ) ) are close to 100%. When gD < sin or gD < , BR(S ! K~ K~ ) and ) will both be subdominant and result in the decrease of BRienv as in the middle panel of gure 4. As shown in the right panel of gure 4, the constraint on sin from invisible Higgs decays can be relaxed by making gD smaller. These exotic decays can also give fully visible and semi-visible signatures [8, 71{75] when the K~ or S particle decays to SM nal states, which provide additional handles for Higgs collider phenomenology. Those references concentrate on the scenario where the decay to SM nal states are dominant, e.g. mK~ < 2m . Therefore, such constraints should be modi ed by the relevant branching ratios in the DDP model because the DDP model includes a DM decay. If visible decays of K~ dominate, though, the search for h ! 2a ! 4 [76] constrains the process H0 ! K~ K~ , and bounds 0HP . 0:01 for mK~ . 10 GeV [8], while the bound strengthens to 0HP . 0:001 [8] for mK~ & 10 GeV from recasting the di erential distributions in 8 TeV h ! ZZ ! 4` data [77]. The coupling 0HP = HP m2H0 =jm2H0 m2Sj, is roughly the same as HP if mS is not close to or much larger than mH0 . The high luminosity LHC (HL-LHC) with 3 ab 1 of 14 TeV luminosity is expected to be sensitive to 0HP . (few) the mK~ mass. decay H0 ! Z~K~ , which gives sensitivity to data is weak, with the strongest sensitivity for mK~ improvement at the HL-LHC is expected to reach The four lepton nal state has also been used to constrain the exotic from (2.24). The current bound using 8 TeV 30 GeV giving . 0:05, while the . 0:01 [8]. These gains are mainly limited by the statistics a orded by Higgs production rates. We remark that for very small and m > mK~ =2, as discussed in section 3, 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]. Modi cations to Drell-Yan processes The K~ decay to SM forbidden or gD nal states can be dominant, if the decay to DM pairs is kinematically . In this case, at the LHC, the Drell-Yan process pp ! Z~; K~ ! `+` Drell-Yan data, while high mass K~ can be probed at the 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 [78, 79], which restricts . 0:01 at mK~ = 200 GeV and weakens to . 0:05 at mK~ = 1000 GeV [80, 81]. For mK~ between 10 and 80 GeV, the Drell-Yan search using 7 TeV data by CMS [82] and the corresponding sensitivity using 8 TeV data gives . 0:005, stronger than the current electroweak precision constraints [8, 81]. The HL-LHC is expected to constrain . 0:001 for mK~ between 10 and 80 GeV using 0:002 level for mK~ = 200 GeV K ! and the 0:01 level for mK~ = 1000 GeV [8, 80, 81]. The recent 13 TeV, 3:2 fb 1 search for high mass dilepton resonances by ATLAS [83] also constrains the kinetic mixing parameter . 0:04 for mK~ > 100 GeV [84], but this result is hampered by the small statistics. We see that as long as K~ has an appreciable branching fraction to SM nal 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 gD= 1, the decay 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 gure 6 and gure 7. 4.3.2 Radiative return processes and dark matter production at the LHC at xed p The radiative return process, e+e ! 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 decays K~ ! `+` 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 that K~ ! is the dominant decay channel, leading to an overall 2=gD2 suppression in the above visible decay rates. Both ATLAS and CMS have searches for dark matter production using 8 TeV data [85, 86], sensitive to mediator masses as low as 10 GeV [85]. The corresponding 13 TeV searches [39, 40] have yet to achieve the same sensivity at low masses. In the DDP model, the K~ mediator is produced on-shell and decays dominantly to , and calculating the results for on-shell mediator production at the LHC, we obtain HJEP06(217) . 0:07, similar to previous studies [87{89]. It is also possible to search for the dark bremsstrahlung of K~ from the DM pair [90, 91], as a probe of , although these rates are negligible in our model. 5 Prospects for future colliders We have established that signi cant room remains to be explored in both the and portal couplings. We will now demonstrate that a future e+e collider, currently envisioned HP as a Higgs factory, will have leading sensitivity to probing both couplings simultaneously through the production of new, light, hidden states K~ and S. The primary motivation 250 GeV center-of-mass energy of such a collider is to optimize the ! 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 states K~ and S from radiative return processes, exotic Higgs decays, and exotic Higgsstrahlung diagrams. A few di erent variations exist for next-generation e+e machines, namely the Internaa p tional Linear Collider (ILC) [2], an e+e Future Circular Collider (FCC-ee), which shares the physics we discuss will only depend very mildly on the particular p strong overlap with TLEP [3], or a Circular Electron-Positron Collider (CEPC) [4]. Since s of the future machine and possible polarization of the incoming electron and positron beams, we will adopt 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 p s = 500 GeV machine with L = 5 ab 1 total integrated luminosity. Our work will complement and extend previous sensitivity estimates made for various speci c collider environments, which we review rst. Electroweak precision tests, Higgsstrahlung, and invisible Higgs decays at colliders Because the SM Z and the dark vector K mix, the Z~ mass eigenstate develops a new invisible decay channel, Z~ ! be accurately measured at a future e+e , if kinematically allowed. This invisible decay width can machine. At FCC-ee, for example, running on the W +W threshold using the radiative return process e+e complemented by additional runs at p s = 240 GeV (Lint = 10:44 ab 1) and p ! Z (Lint = 15:2 ab 1), s = 350 GeV (Lint = 0:42 ab 1), can constrain the number of active neutrino species, assuming statistical uncertainties, down to N 0:001 [92, 93]. This leads to the possible constraint < 0:01 for mK~ < mZ~=2. Measurements directly on the Z-pole are not expected to compete with this constraint because of theory uncertainties on the small-angle Bhabha-scattering cross section remain too large [92]. This constraint also applies to the Z~ ! K~ S ! 6 exotic decay, if mK~ + mS < 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 although the constraint is much weaker for mK~ > mZ~ [8]. . 0:004 for mK~ < mZ~, 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, 94]. These rates imply that the scalar mixing angle is probed to sin . 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 subsection 4.3, the current constraint on the Higgs invisible decay branching ratio is BRinv < 0:23 [67, 69], while the limit at FCC-ee is expected to be BRinv < 0:005 [3]. We have discussed the two and H0 ! K~ K~ ! 4 , as well as the irreducible in subsection 4.3. We also show the corresponding sensivity vs. mK~ planes in gure 4. We again emphasize that these limits can be stronger or weaker because of their dependence on gD, as seen in gure 4. Production of new, light states at future e+e colliders As outlined in section 4 and shown in gure 3, many new possibilities open up for production of new, light, hidden particles in the Double Dark Portal model. We can classify the new physics processes into vector + scalar, radiative return, and massive diboson production topologies. For small and scalar mixing angle , the leading processes are: tional contribution from intermediate K~ that can interfere with Z~ exchange. ! Z~H0 : the usual Higgsstrahlung diagram is suppressed by cos2 , with an addi! Z~S : this new process can be probed by the usual recoil mass method for wellreconstructed Z~ decays, studying the entire recoil mass di erential distribution. ! K~ S : this exotic production process involves two non-standard objects, and is dominantly produced via K~ , with a rate proportional to 2 cos2 . Since K~ and S dominantly decay to dark matter, though, we would require an additional photon or a visible decay of K~ or S in order to tag the event. ! A~K~ : the radiative return process produces K~ in association with a hard photon , giving direct sensitivity to . We remark that the K~ been studied for mK~ between 10 GeV to 240 GeV at an p ! `+` decay has s = 250 GeV machine with 10 ab 1 [10], giving . 5 10 4, if the decay K~ ! `+` is assumed dominant. ! Z~K~ : the massive diboson pair production process also provides direct sensitivity to , but measuring the rate precisely will pay leptonic branching fractions of the Z~. ! H0K~ : this very interesting scenario can be probed by using the 125 GeV SM-like Higgs as a recoil candidate for the K~ heavy vector. The total rate gives sensitivity to both and and highlights the power of considering the SM-like Higgs as a signal probe for new physics. Having identi ed the main production modes for the K~ and S states, we can match them to decay topologies illustrated in the bottom row of gure 3. We also include the underexplored decay H0 ! K~ Z~, which gives an exotic decay of the SM-like Higgs into the SM-like Z~ boson and the hidden photon K~ sensitive to . As mentioned in section 3, we focus on S ! K~ K~ ! 4 and K~ ! , 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 of K~ in the Z~K~ and K~ processes as well as the fully inclusive recoil mass distribution targeting K~ production. Since the workhorse SM Higgsstrahlung process has been studied extensively [2{4], we use these previous results to recast the sensitivity for and . We also ignore the K~ S production mode, since the dominant signature has nothing visible to tag the event. The H0K~ process is interesting to consider for future work, but it requires optimizing the H0 decay channel to gain maximum sensitivity to the recoil mass of the rest of the event. This leaves the Z~H0, Z~S, K~ and Z~K~ processes as new opportunities to revisit or study. We simulate each process using MadGraph5 v2.4.3 [95], Pythia v6.4 [96] for showering and hadronization, and Delphes v3.2 [97] for detector simulation. Detector performance parameters were taken from the preliminary validated CEPC Delphes card [98]. Backgrounds for each process are generated including up to one additional photon to account for initial state and nal state radiation e ects. Events are required to pass preselection cuts of j j < 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 as K~ and S give missing energy signatures. with new particle production Z~K~ production and K~ ! by g2 = 2. We also study Z~ ! `+` D The cross section for Z~K~ production is shown in gure 5 for various choices of mK and . We see that Z~K~ production grows with 2, as expected. We consider both K~ ! `+` decays, for ` = e or , where the missing energy branching ratio dominates with the SM branching fraction of 6:8% [99]. We show the background cross sections for the corresponding 2`2 and 4` nal states after the preselection cuts described in subsection 5.2 in table 1. The 2`2 background includes a combination of Z attributed to ZZ=Z , ZZ and W +W production. processes, while the 4` background is mainly For the 2`+E= nal state, we require a Z-candidate with jm`` mZ j < 10 GeV and then look for a peak in the recoil mass distribution, see eq. (4.2). For the 4` nal state, we identify the Z-candidate from the opposite-sign, same- avor dilepton pair whose invariant mass is closest to the Z mass and then study the invariant mass distribution of the remaining dilepton pair. The K~ 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 plane for an integrated luminosity L = 5 ab 1 in gure 6 and gure 7. The relative weight nal states is xed by choosing gD = e 0:3. We see that the fully visible 4` nal state performs worse than the 2`2 signal selection, simply re ecting the dominant signal statistics in the missing energy channel. HJEP06(217) 100 10-3 10-4 10-5 ˜K˜A 10-1 σ 10-3 10-4 10-150-3 Z p s panel) . Solid lines correspond to e+e beams, while dashed lines correspond to p machines operating at p s = 250 GeV with unpolarized s = 500 GeV. The mass of K~ is derived from mK using eq. (2.11). ! A~K~ , for K~ ! , `+` , and inclusive A~K~ production We study the radiative return process, e+e decays. While each search will use the same observable, namely a monochromatic peak in the photon energy as in eq. (4.1), the di erent contributions of SM backgrounds in each event selection will result in the best sensitivity for the K~ ! and signal regions are shown in table 1. For the inclusive decay of K~ , the background f f decay. Background rates is generated where f is a SM fermion, including neutrinos. As mentioned in subsection 5.2, 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 gure 6 and gure 7, we see that the most sensitive decay channel is K~ ! , again re ecting the dominant statistics in this nal state and the a ordable reduction of SM backgrounds by the E= and mono-chromatic photon requirements. The single photon in the background generally comes from initial state radiation and hence tends to be soft except when produced in the on-shell Z~ ! long as mK 6= mZ~, however, the signal peak will not run overlap the background peak, and process. As thus we have a at sensitivity to sensitivity. The rst is for mK due to maximal K~ Z~ mixing, and the second is for mK ps, when the production is when mK < mZ~. There are two spikes in exclusion mZ~, when the production cross section is greatly enhanced enhanced by soft, infrared divergent photon emission. Note the exclusion can only reach 5 GeV because of the preselection cut on the photon energy. The next process we consider is the exotic Higgs decay, H0 ! K~ Z~, with K~ ! ! `+` . This Higgs exotic decay partial width, from eq. (4.6), is proportional to and Parameter Signal process Background (pb) Signal region Z~K~ A~K~ Z~H0 Z~S Z~ ! ``, K~ ! Z~ ! ``, K~ ! `` K~ inclusive decay K~ ! `` K~ ! H0 ! K~ Z~ with K~ ! , Z~ ! `` Z~ ! `` S ! K~ K~ ! 4 0.929 (250 GeV) and jmrecoil mK~ j < 2:5 GeV jm`` and jm`` mZj < 10 GeV, mK~ j < 2:5 GeV N 1, and ( p2s 2mp2K~s )j < 2:5 GeV ( p2s 2m1p,2K~sN)j` < 2:5 GeV, N 2, and jm`` mK~ j < 5 GeV ( p2s 2mp2K~s )j < 2:5 GeV, and E= > 50 GeV jm`` mZj < 10 GeV, and and jmrecoil mK~ j < 2:5 GeV jm`` mZj < 10 GeV, jmrecoil mSj < 2:5 GeV sin given for p given in the main text. along with the most salient cuts to identify the individual signals. All background processes include up to one additional photon to account for initial and nal state radiation. Background rates are s = 250 GeV or 500 GeV, and visible particles are required to satisfy preselection cuts 2 cos2 , as long as mK~ . 34 GeV and sin candidates balancing an invisible K~ particle, which we identify from the peak in the recoil mass distribution. Our event selection cuts, summarized in table 1, require two pairs of opposite sign and same avor lepton with invariant masses in a window around mZ~, and the recoil mass from the four visible charged leptons should be in a window around the test variable mK~ . The resulting sensitivity, as seen in with the other K~ production processes, given the limited Higgs production statistics and the suppression of the small leptomic decay branching ratio of Z~. We remark that this decay can also be probed via H0 ! invisible searches using the SM rate for Z ! , which gure 6 and gure 7, is not competitive is neglected. The signal process thus has 2 Z was discussed in subsection 4.3. 5.3.4 Z~S production Lastly, we can also probe the scalar mixing angle sin in Z~S production. This search is exactly analogous to the previous search at LEP-II for a purely invisible decaying Higgs [18], where the visible Z~ ! `+` Z~S cross section is proportional to sin2 decay is used to construct the recoil mass distribution. The if we neglect , and Z~S is shown in gure 8 for HJEP06(217) 10-1 10-2 10-3 10-4 K˜˜Z→2 2 gD = e 2 2 0 H ˜ ˜ ˜ ˜ K Z→ ˜ ˜ K A→ 2 ˜ ˜ K A inclusive s = 250 GeV, 5 ab-1 LHC-DY (8 TeV) LHC-DY (HL-LHC) 50 100 500 of K~ production. Solid lines enclose expected exclusion regions with L = 5 ab 1 of p operator search, LEP electroweak precision tests (LEP-EWPT), BaBar K~ invisible decay search machine data. Dashed lines indicate existing limits from the LEP e e+ ! ` `+ contact (BaBar) and LHC Drell-Yan constraints (LHC-DY). The 3 ab 1 HL-LHC projection for DrellYan constraints is also shown as a solid line. Note mK is approximately the mK~ mass eigenvalue according to eq. (2.11). sin = 0:1 and 0:01 at p s = 250 GeV and p study Z~ ! `+` and S decaying invisibly. The signal region is summarized in table 1 and light mK using L = 5 ab 1 luminosity for p focuses on selecting a dilepton Z candidate and reconstructing the recoil mass distribution to identify the S peak. From this analysis, we nd that sin = 0:03 can be probed for s = 250 GeV, as shown in gure 8. This result would signi cantly improve on the current global t to Higgs data by ATLAS and CMS, which constrains sin < 0:33 [68]. This sensitivity also exceeds the projected LHC reach of sin < 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 p s of the collider to maximize the (e+e ! Z~S) rate for the test S mass (see also ref. [60]). s = 500 GeV. To maximize sensitivity to , we 100 10-1 10-3 10-4 1 ˜ ˜ K Z→ 2 2 ˜ ˜ 10-2 K A inclusive BaBar s = 500 GeV, 5 ab-1 LHC-DY (8 TeV) 2 2 0 H ˜ ˜ ˜ ˜ K Z→ ˜ ˜ K A→ 2 HJEP06(217) 5 10 LHC-DY (HL-LHC) 50 100 500 in di erent channels at a future e+e collider running s = 250 (500) GeV with L = 5 ab 1 in gure 9, and we compare the collider searches with constraints from direct detection and indirect detection experiments. In gure 9, the dark green shaded region is the exclusion limit from the strongest of the e+e collider searches presented in gure 6 and gure 7. We also show the strongest limit from direct detection and indirect detection experiments from gure 1 and gure 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 is kinematically allowed and gD . We note that for mK around mZ~, the best limit comes from the inclusive A~K~ search, which is insensitive to gD, while for mK 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 signi cantly with dark matter mass. When m = 0:495mK , the dark matter resonantly annihilates, improving the reach for indirect searches and dramatically lowering 10-2 10-6 10-7 100 10-1 10-2 10-3 10-4 10-1500 10-1200 100 10-1 10-2 10-3 10-4 10-1500 Z~S, Z~ ! `+` search in the recoil mass distribution for invisible S decays in the sin 500 GeV as a function of mS, with sin = 0:1 and 0:01. (Right panel) Exclusion reach from the vs. mS plane s = 250 GeV or 500 GeV. We also show comparisons to the current t, sin < 0:33 [68], future LHC projections of 0.28 (0.20) using 300 fb 1 (3 ab 1) luminosity [1], and precision (Zh) measurements constraining 0.084 (0.055) using 5 ab 1 (10 ab 1) [3, 4, 94]. We plot the excluded region from LEP searches for invisible low mass Higgs in ZS channel in ! Z~S process at p s = 250 GeV and e-e+ Collider, 5 ab-1 ϵ ϵ P lan c k ID S e a BaBar rch LEP-EWPT h rc a e S D D s =250 GeV vs. mK plane. We choose gD = 0:01, m = 0:2mK (left panel) and m the Planck collaboration [34] as black dashed lines. The collider constraint is adapted from gure 6 and gure 7, taking into account the changes in the K~ branching fractions. We also include existing constraints from LEP electroweak precision searches (LEP-EWPT) and the BaBar search for the K~ invisible decay (BaBar). from the required to satisfy the relic density measurement. During thermal freeze-out, the nite temperature of the velocity distribution gives a strong boost to the annihilation cross section, and thus only very small is needed. For m = 0:2mK , 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 , since the nuclear recoil spectrum is too soft to pass the ducial energy threshold. In addition, the decreasing sensitivity for heavy m comes from the fall o in the scattering cross section scaling as 2 p=m2~ , see eq. (3.3). K We also emphasize that the collider constraint is not sensitive to varying gD as long as , which ensures the invisible decay of K~ dominates. Hence, the collider constraints gure 6 and gure 7 and gure 9 are essentially unchanged, since changing gD from e to 0:01 does not signi cantly change the invisible branching fraction, except for the trade o between inclusive K~ decays and invisible K~ decays around mK mZ~. On the other hand, the direct detection and indirect detection rates scale with gD2, and thus collider searches will have better sensitivity for small gD. From gure 9, we see that the prospective collider limits, corresponding to the radiative return process e+e ! A~K~ , are expected to overtake the current bounds from direct detection and indirect detection experiments. In the case where dark matter mass is light, m = 0:2mK , 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 = 0:495mK , the thermal relic abundance measured by Planck [34] o ers an attractive target parameter space for experimental probes. The projected e+e sensitivity exceeds the current experimental sensitivity around mK 10 GeV and mK > 100 GeV, and while improvements in the dark matter experiments will also challenge the open parameter space for mK 10 GeV, the striking sensitivity of e+e radiative return processes for mK > 100 GeV is expected to be unmatched. Thus, results from a future 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. 6 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 . 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 , 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 matter and LHC experiments. machine to produce new particles, which are not probed with the current dark HJEP06(217) decay, K~ decay, K~ ! largely unexplored. We work out the interactions in the mass eigenstate basis of neutral vector bosons and Higgses. 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 ! `+` , and leaving the prospects and sensitivity estimates for the invisible 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 both K~ and S decays are dominantly invisible, the recoil mass method a orded by an e+e machine is crucial. We also highlight that the recoil mass method can be simpli ed to a monochromatic photon study in the case that the new particle is produced in the radiative return process, which simpli es 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 1, and we obtain the best sensitivity on Higgsstrahlung process Z~S, respectively. from the radiative return process A~K~ and In comparing prospects, we analyzed the future collider reach with direct detection, indirect detection and relic abundance sensitivities. The collider prospects are less a ected by DM mass m , and surpass the other experimental probes for small gD. Since K~ 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 process A~K~ reach. Therefore, a future e+e complementary sensitivity test of the DDP model. collider provides an important and projection is weaker than the direct Z~S search. For sin , the best constraints come from studying the singlet bremsstrahlung process Z~S, the Higgs invisible decay rate, and precision measurements of SM Higgs production rates. We studied the Z~S process with S decaying invisibly for a future e+e and estimated the sensitivity to be sin 0:03. This compares favorably with earlier LEP studies for light mS, and readily provides leading sensitivity for heavy S. We also recasted bounds using the Higgs invisible decay channel, where the current LHC constraint is BRinv < 0:23 [67, 69] and the future e+e collider reach is BRinv < 0:005 [3]. In the DDP model, these bounds simultaneously constrain the three exotic processes, H0 ! SS, H0 ! K~ K~ , and H0 ! Z~K~ when Z~ decays to neutrinos. While the constraints on sin can be strong, these limits also depend sensitively on gD and are insigni cant for small gD. The future ! Zh) precision measurement readily constrains cos2 , but this In summary, the Double Dark Portal model predicts new dark sector particles, K~ , S, and , 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 nd the invisible decays of the K~ and S mediators. An additional bene t of the e+e search strategies discussed in this work is that, in the event of a discovery, the K~ 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. Acknowledgments This research is supported by the Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA-EXC 1098). FY would like to thank the hospitality of the CERN theory group while this work was being completed. The work of JL and XPW is also supported by the German Research Foundation (DFG) under Grants No. KO 4820/1-1, and No. FOR 2239, and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant No. 637506, \ Directions"). A Two limiting cases for K~ , Z~, and A~ mixing From subsection 2.1, we decompose the gauge eigenstate vectors into their mass eigenstate components according to 0 Z ; SM 1 K 0 Z~ 1 where the expressions for U1, U2 and RM have been given in eq. (2.7), and eq. (2.9), respectively. We will consider the two limiting cases, mK ! 0 and mK ! mZ;SM, and study the corresponding changes for the kinetic and mass mixing matrices. For mK ! 0, the gauge boson masses are mA~ = mK~ = 0 ; m2Z~ = m2Z; SM 1 and the eld rede nition is U1U2 = BB 0 B B q(1 2)(1 2cW2) q(1 2)(1 2cW2) tW 1 0 1 0 2 2cW2 0 1 C p m2Z; SM 1 + 2 2 tW + O( 3) ; 0 1 + 12 2t2 W B B 2tW tW 0 1 0 0 1 + 12 2 1 A CC + O( 3) : (A.1) (A.2) (A.3) The Jacobi rotation RM , from eq. (2.9), is now ill-de ned in the lower right two-by-two block, since A~ and K~ can be rotated into each other keeping both the kinetic terms and masses unchanged. This simply re ects the residual unbroken U( 1 )em metry. For RM = I3, the currents are L + K~ gDtW J D + gDJD + e Jem + 1 2 gt2W 2 J 1 2 gD 2 J Z D + A~ eJem ; etW 2 Jem but under a unitary rotation UX where (A~0; K~ 0)T = UX (A~; K~ )T , the dark matter the SM fermions will generally have nonzero charges mediated by both A~0 and K~ 0, leading to photon and dark photon-mediated electric and dark millicharges. For mK ! mZ; SM, the masses of the three vector bosons are (A.5) and (A.6) (A.7) (B.1) (B.2) mA~ = 0; m2K~ ; Z~ = m2Z; SM 1 tW + 2 1 2 1 + 2t2W ; and the eld rede nition required, to O( 2), is U1U2RM = p 4 (tW1 2tW ) 1 + 4 (tW1 2tW ) 1 0 1 4 (tW1 + 2tW ) 0 1 4 (tW1 + 2tW ) CC ; A where the top and bottom signs correspond to mK ! mZ; SM. We see that the mixing between Z and K is nearly maximal, 45 , while the discontinuous behavior for mK below and above mZ; SM re ects the level crossing in the mass eigenvalues. We remark that as long as 6= 0, this maximal mixing feature remains, dictated by the structure of the symmetric mass matrix in eq. (2.8). If = 0 and mK = mZ , then the rotation matrix in eq. (2.9) becomes ill-de ned and the maximal mixing feature is lost. B Cancellation e ect in multiple kinetic mixing terms We observe that the Z~ and K~ mediated couplings in eq. (2.15) show a cancellation e ect 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 Kab = ab + O( )(1 ab) is the kinetic mixing matrix and M 2 is the diagonal mass eigenstates, V~ = U 1V . Moreover, the gauge currents now become matrix, with a, b as indices. Then, we de ne the eld rede nition matrix U such that KU = I, which also gives M~ 2 = U T M 2 U as the mass matrix corresponding to the mass KV + 2 1 V T M 2V ; giV iJ i = giUikV~k; Ji ; in the mass basis. As a result, scattering rates between two currents Ja and Jb (which represent the corresponding fermion bilinears) are schematically M / (gaJa ) (gbJb ) UakUbk (gaJa ) (gbJb ) Uak(U T )kb " " g q q =m2~ # Vk Q2 g + g Q2 !# O( m4~ ) Vk : (B.3) Vk The g =m2~ term in the parentheses, however, vanishes, when including the sum over Uak(U T )kb, because these transformations are controlled by the diagonalization requirement of the two mass matrices, speci cally U M~ 2 U T = M 2. The leading contribution in the In this section, we present the annihilation cross sections for the processes W +W , where f is a SM fermion. We focus on the case with m the direct annihilation of dark matter to dark vectors K~ K~ opens up and does not depend on . In this setup, the annihilation cross section is proportional to gD2 2. The diagrams include s-channel K~ and Z~ exchange. The annihilation cross sections before thermal averaging are < mK~ , since otherwise ! f f , ! `+` e2 2g2 q D s 4m`2 2m2 + s 48 s3=2c4W m2~ Z 5s+7m`2 s2(m2~ Z m2~ )2 +m2~m2K~ (mK~ Z~ K Z mZ~ K~ ) 2 12c2W (s+2m`2)m2~ s(m2~ Z Z m4K~ +m2Z~(m2~ + 2 ) 2mZ~mK~ Z~ K~ +m2~ ( 2m2~ + 2~ ) i Z Z~ K Z K ; (C.1) v ( ! uu) = 144 s3=2c4W m2~ Z m2~ )2 +m2~m2K~ (mK~ Z~ K Z mZ~ K~ ) 2 40c2W (s+2m2u)m2~ s(m2~ Z Z m2~ )2 +mZ~mK~ ( mK~ Z~ +mZ~ K~ )( mZ~ Z~ +mK~ K~ ) K + 32c4W (s+2m2u)m4~ Z m4K~ +m2Z~(m2~ + 2~) 2mZ~mK~ Z~ K~ +m2~ ( 2m2~ + 2~ ) i Z Z K Z K ; ! dd = e2 2g2 q s2(m2~ Z m2~ )2 +m2~m2K~ (mK~ Z~ K Z mZ~ K~ ) 2 4c2W (s+2m2d)m2~ s(m2~ Z Z + 8c4W (s + 2m2d)m4~ Z m4K~ + m2Z~(m2Z~ + 2 ) ~ Z 2mZ~mK~ Z~ K~ + m2~ ( 2m2Z~ + K ; 2~ ) i K ; (C.3) (C.4) (C.5) v ( ! ` `) = ! 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Jia Liu, Xiao-Ping Wang, Felix Yu. A tale of two portals: testing light, hidden new physics at future e + e − colliders, Journal of High Energy Physics, 2017, 77, DOI: 10.1007/JHEP06(2017)077