Exploring dark matter-gauge boson effective interactions at the current and future colliders

In this work, we investigate the collider constraints on effective interactions between Dark Matter (DM) particles and electroweak gauge bosons in a systematic way. We consider the simplified models in which scalar or Dirac fermion DM candidates only couple to electroweak gauge bosons through high dimensional effective operators. Taking into account the induced DM-quarks and DM-gluons operators from the Renormalization Group Evolution (RGE) running effect, we present comprehensive constraints on effective energy scale $\Lambda$ and Wilson coefficients $C_B(\Lambda),\,C_W(\Lambda)$ from direct detection, indirect detection, and collider searches. In particular, we present the corresponding sensitivity from the Large Hadron Electron Collider (LHeC) and the Future Circular Collider in electron-proton mode (FCC-ep) for the first time, update the mono-$j$ and mono-$\gamma$ search limits at the Large Hadron Collider (LHC), and derive the new limits at the Circular Electron Positron Collider (CEPC).


I. INTRODUCTION
Observations from astrophysics and cosmology have provided overwhelming evidence for the existence of DM [1].However, the microscopic properties of DM are still poorly known since all of such observations are based on its gravitational effects.The canonical DM candidate is the Weakly Interacting Massive Particles (WIMPs).Such a scenario is easy to fulfill relic abundance through thermal freeze-out mechanism (the "WIMP miracle") [2], and can be naturally embedded in many popular theoretical frameworks [3,4].Moreover, due to the property of weak interaction with the SM particles, it can be probed through three experimental prongs, i.e., collider, direct and indirect detections.
Typically, the interactions between DM and SM particles are model-dependent and it is a great challenge to explore the complete model landscape.Instead of exhausting all possible theoretical model parameter space, the Effective Field Theory (EFT) framework is extensively used, which allows us to capture some key features of the high-scale physics effects on the electroweak scale while significantly simplifying the analysis procedures.For EFT collider searches, various signal channels based on different SM portal effective operators have been extensively studied at the hadron and electron colliders .
Among them, the parameter space of DM-quark interaction which fulfills the correct relic density has a tension with the exclusion limits from direct detection, indirect detection, and collider searches.In this work, we consider the dimension-6 (7) DM-diboson effective operators for scalar (Dirac fermion) DM and examine their sensitivity at current LHC, future electron and electronproton colliders.The current limits of DM-diboson operators at the LHC are mainly constrained by mono-j and mono-γ signal channels, and mono-γ signature is a sensitive channel at the CEPC.
For electron-proton collider, LHeC and FCC-ep are the two most promising proposals at current.
LHeC is based on an economic LHC upgrade, plan to add a 60-140 GeV electron beam to collide with a 7 TeV proton beam in the LHC ring during the period of High Luminosity LHC (HL-LHC) run.FCC-ep is designed to collide a 60 GeV electron with the 50 TeV proton beam.DM pairs can be produced through the VBF process with a cleaner background, which corresponds to / E T + e − j signal.
The rest paper is organized as follows.In Sec.II, we firstly give a brief description of the DMdiboson effective operators.Then relevant collider phenomenology of / E T + e − j signal channel at the LHeC and FCC-ep is discussed in Sec.III, including event simulations, kinematical distribu-tions of final states and cut selections related to signals and backgrounds.In Sec.IV, we present the sensitivity at the CEPC with mono-γ signature.The limits from the LHC with current mono-j and mono-γ searches are updated in Sec.V.The constraints from current direct and indirect DM search experiments are also discussed in Sec.VI.Then we show the results in Sec.VII.Finally, we give a summary and conclusion in Sec.VIII.

II. FRAMEWORK OF DM-GAUGE BOSON EFFECTIVE INTERACTIONS
We consider the DM particle as a Dirac fermion (χ) or scalar (ϕ), which connects with SM sector through high-dimensional DM-gauge boson effective operators.At the lowest order, such operators allow DM couple to single gauge boson, which induce electromagnetic dipole and anapole interactions [35,36].However, these interactions lead to monochromatic gamma-rays due to the unsuppressed cross-section of DM annihilating into the γγ and γZ modes, which are severely constrained by current indirect detection experiments [37].We thus start from DM-diboson operators These operators are assumed to be generated at the energy scale Λ with other high energy resonances much heavier than Λ are all decoupled.Notice that in this scenario, the coupling between DM and the SM quarks can also be induced through the renormalization group evolution (RGE) from the Λ scale down to the electroweak scale µ EW .As a consequence, two extra dimension-6 (7) operators need to be taken into account.In above equation, y q = √ 2m q /v are the Yukawa couplings of SM quarks, and v is the vacuum expectation value (VEV) of the SM Higgs doublet H. Then the total effective Lagrangian at scale µ (µ EW < µ < Λ) is given by To the leading logarithmic (LL) order, the Wilson coefficients of the effective operators O y and where where F F = AA, AZ, ZZ, W W .The relevant matching conditions are with θ W is the Weinberg angle.
Integrating out the heavy top quarks, the effective DM-gluon operators can be generated, where G a µν denotes the gluon field strength.The matching condition at leading order is given by [39] When RGEs are evolved from the EW scale down to the hadronic scale µ had ≃ 1 GeV, O AA will contribute to O q through the exchanging of virtual photons.For scale µ with m q < µ < µ EW , one has [40] C q (µ) at LL order, where Q q is the electric charge of quark and the electromagnetic coupling constant At m b and m c threshold, one has to integrate out the corresponding heavy bottom and charm quarks by taking into account Eq. ( 11) again.Now at scale µ had ≃ 1 GeV, we get where the Wilson coefficient C AA (µ EW ) is determined by the first matching condition in Eq. ( 9).

III. SENSITIVITIES AT THE LHEC AND FCC-EP
In this section, we will investigate the sensitivity of the DM-gauge boson effective operators at future electron-proton colliders.

A. Signal channels and SM backgrounds
From the effective interactions in Eqs. ( 1) and ( 2), DM pairs can be produced at the electronproton collider through both the Charged Current (CC) and the Neutral Current (NC), which respectively correspond to W boson mode and γ/Z mode of VBF processes.In CC production, DM can be generated through the process of e − p → ν e jχ χ(ϕϕ * ) via W W fusion, resulting in a mono-jet plus missing energy signature.It is important to note that the mono-jet signature coincidentally aligns with the background of CC deeply inelastic scattering.Due to the absence of kinematic handles in the final state, distinguishing this signal channel from its primary background poses a significant challenge.Consequently, our study is primarily focused on NC productionspecifically the process e − p → e − jχ χ (ϕϕ * ), as depicted in Fig. 1.The corresponding signature is thereby characterized as the / E T + e − j channel.For SM background estimation, we consider both reducible background (RB) and irreducible background (IB).The IB mainly is due to the processes e − p → e − jν ℓ νℓ (ℓ = e, µ, τ ).Among them, both W/Z-boson bremsstrahlung processes e − p → e − jZ/ν e jW − (Z → ν e νe /W − → e − νe ) contribute to ν e final state, while for ν µ,τ , only bremsstrahlung process e − p → e − jZ (Z → ν µ,τ νµ,τ ) is relevant.For RB, a major contribution comes from e − p → e − jW ± with W 's leptonic decay W ± → ℓ ± ν ℓ .There then two possibilities that need to be taken into account.One is that electron and muon final states run out of the detector acceptance and τ final state fakes a hard jet in the detector.Another possibility is that the products of hadronic τ decays are too soft to be tagged [41,42].As a consequence, The SM background can be summarized as follows: • IB: e − p → e − jν ℓ νℓ , • RB: e − p → e − jℓ ± ν ℓ (ν ℓ ), where ℓ = e, µ, τ .

B. Event simulation and kinematical cuts
For the purpose of event simulation, we generate Universal FeynRules Output (UFO) [43] model file by using FEYNRULES [44].Which is fed into MADGRAPH@NLO [45] to generate parton-level events.For both signal and backgrounds at the parton level, we apply the following pre-selection cuts: where p T and η are the transverse momentum and pseudorapidity for the corresponding particles, and ∆R = ∆ϕ 2 + ∆y 2 is the separation in the azimuthal angle-rapidity (ϕ − y) plane.Notice that all of the cuts in Eq. ( 15) are defined in the lab frame.We then use PYTHIA6.4 [46] to implement parton shower and hadronization, and DELPHES3.4[47] to fast detector simulation according to the LHeC designed parameters [48].
In order to suppress RB, we apply the following veto criteria [42] as basic cuts: • events are required to contain exactly one hard electron, one hard jet, and / E T , • events containing any extra jets with p j T > 3 GeV and |η j | < 2 or leptons with p e,µ T > 5 GeV or tagged τ jets are voted.
In Fig. 2 and Fig. 3, we respectively display the normalized distributions of missing transverse energy / E T , the invariant mass of tagged electron and jet system M (e − , j) and inelasticity variable y for both signal and backgrounds after the basic cuts, at the 140 GeV ⊗ 7 TeV LHeC and 60 GeV ⊗ 50 TeV FCC-ep.For signal, we assumed the benchmark values m χ, ϕ = 1 GeV, C B, W = 1 and Λ = 500 GeV.For inelasticity variable y, we follow the definition in Ref. [41] which is given as where k P, e are the 4-momenta of the initial proton and electron, and p e is the 4-momenta of the final electron.One can observe the following features: In Table.At the LHeC, after all cuts, the fermion (scalar) DM signal can survive 13.2% (14.9%), while RB (IB) only survived 0.37% (3.5%).At the FCC-ep, the signal for fermion (scalar) DM survived 13.3% (14.6%), and RB (IB) survived 0.047% (0.65%) at the end.It is worth noting that the remaining cross-section of background at the FCC-ep are only 14% of ones that at the LHeC, while signal for fermion (scalar) DM production at the FCC-ep could reach 6 (1.5) times than that  For the monophoton signature at CEPC, an irreducible background arises from the neutrino production e + e − → ν l νl γ processes, where ν l = ν e , ν µ , ν τ .Due to the Breit-Wigner distribution of Z boson in the irreducible BG, there is a resonance in the final photon energy spectrum, which 1 We take √ s = 160 GeV for the W + W − threshold scan mode throughout our analysis.
is located at with a full-width-at-half-maximum as Γ Z γ = M Z Γ Z / √ s.In order to suppress the irreducible background contribution, we will veto the events within 5Γ Z γ at the Z resonance in the monophoton energy spectrum [27].The vetoing cut can be written as The major reducible SM backgrounds come from the e + e − → γ + / X processes, where only one photon is visible in the final state, and / X denotes the other undetectable particle(s) due to the limited detection capability of the detectors.In our analysis, the parameters for the EMC coverage: | cos θ| < 0.99 and E > 0.1 GeV, are adopted following CEPC CDR [50].The processes e + e − → f f γ and e + e − → γγγ provide dominate contributions to the reducible background when final f f and γγ are emitted with | cos θ| > 0.99.Due to the momentum conservation in the transverse direction and energy conservation, the maximum photon energy as a function of its polar angle can be obtained as [27] where the polar angle θ b corresponds to the boundary of the EMC with | cos θ| < 0.99.In order to suppress the monophoton events arising from the reducible background, we adopt the detector cut E γ > E m γ (θ γ ) on the final state photon in our analysis.

V. UPDATE LIMITS FROM THE LHC MONO-j AND MONO-γ SEARCHES
Due to the DM-diboson operators, the WIMP can be produced via the processes qq ′ → V ′ * → ϕϕ * ( χχ) + V (V = γ, Z, W ), which can exhibit a mono-photon ( / E T + γ) or a monojet ( / E T + W/Z (→ hardons)) signature, respectively.The representative Feynman diagrams are shown in Fig. 5. Besides, the loop-induced effective DM-quarks (O q ) and DM-gluons (O G ) operators, could also result in the mono-jet signature via q q → χχ + g, gg → χχ + g, and qg → χχ + q processes.The representative Feynman diagrams are shown in Fig. 6.
In this section, we consider the sensitivities from LHC mono-j and mono-γ searches.To simulate the signal from dark matter, the parton-level events are generated by MADGRAPH@NLO [45], and the following parton shower and hadronization are dealt with PYTHIA6.4 [46].We use Representative Feynman diagrams for a mono-photon (Left) and a mono-jet signature (Right) arising from DM-diboson operators.
FIG. 6. Representative Feynman diagrams for a mono-jet signature arising from loop-induced effective DM-quarks and DM-gluons operators.
DELPHES3.4 [47] to carry out a fast detector simulation for the ATLAS or CMS detector with the corresponding parameter setup.We follow the procedure of the mono-photon [51] and mono-jet [52] latest analysis by the ATLAS Collaboration with an integrated luminosity of 139 fb −1 at a center-of-mass energy of 13 TeV.
In the case of / E T + γ search channel, events in our analysis are required to have a leading γ > 1200 GeV provides the best limits.At 95% C.L. the bound on the corresponding fiducial cross section is given by σ fid < 0.3 fb.
In Tab.III, we present the ratio of the number of events for mono-jet signature at LHC from the process pp → χ χ + W/Z (→ hardons) via dimension-7 DM-diboson operators with the Feynman diagrams shown in Fig. 5 to one from the process pp → χ χ + q/g via loop-induced DMquarks and DM-gluons operators with the Feynman diagrams shown in Fig. 6 with Λ=1000 GeV.
There we consider three cases, i.e., C B = C W = 500, C B = 0, C W = 500, and C B = 500, C W = 0.It can be found that the ratios in the three cases all increase with the increment of the DM mass.
Besides, to the mono-jet signature at LHC the contributions from tree-level DM-diboson operators are about two orders of magnitudes over from loop-induced DM-quarks and DM-gluons operators, especially about three orders in the case of C B = 1, C W = 0, thus we ignore the contribution from DM-quarks and DM-gluons operators in our following analysis.III.The ratio of the number of events for the mono-jet signature at LHC from the process pp → χ χ + W/Z (→ hardons) via DM-diboson operators with the Feynman diagrams shown in Fig. 5 to pp → χ χ + q/g via loop-induced DM-quarks and DM-gluons operators with the Feynman diagrams shown in Fig. 6 with Λ=1000 GeV.

VI. RELIC ABUNDANCE, DIRECT AND INDIRECT CONSTRAINTS
In this section, we discuss the bounds on Λ from current direct and indirect search experiments.
The effective operators O B and O W lead to the non-relativistic cross-sections of DM annihilation χ χ, ϕϕ * → γγ/γZ/ZZ/W W as with k = B, W annihilation channels.The coefficients of a and b are [37] χ In above equations, we have defined the phase space factor β ij = 1 − (m i + m j ) 2 / (4m 2 DM ) and x i = m 2 i /m 2 DM with DM = χ, ϕ.Using the results for a B,W and b B,W , the relic density can then be calculated with where a = a B + a W and b = b B + b W . x F = m DM /T F , being the ratio of the mass of DM m DM and the early-universe freeze-out temperature T F , can be obtained by solving The loop-induced effective DM-quarks (O q ) and DM-gluons (O G ) operators, could result in the interaction between DM and nucleon.The spin-independent (SI) cross section for elastic scalar and Dirac WIMP scattering on a nucleon has the form In above equation, m N ≃ 0.939 GeV is the average nucleon mass, and µ N = m ϕ,χ m N /(m ϕ,χ + m N ) is the reduced mass of the DM-nucleon system.The form factor f N A describes the Rayleigh scattering of two photons on the entire nucleus.At zero-momentum transfer limit, f N A ≃ 0.08 for xenon target.The form factor f N =p,n describing the couplings between the DM and nucleon reads [40,53] f N (µ had ) = q=u,d,s where from factors f N q describing the scalar couplings between quarks and nucleon are given by [54] and

VII. RESULTS
We apply simple χ 2 analysis to derive the lower bounds on Λ for effective operators in Eq. ( 5) at future LHeC, FCC-ep, and CEPC at 95% C.L., by requiring χ 2 = N 2 S /N B = 3.84 [55] with the specified luminosities.For LHeC and FCC-ep, the total integrated luminosities are assumed as L = 2 ab −1 .The limits of Λ as a function of the mass of Dirac fermion and scalar DM are displayed in Figs.7 and 8, respectively.We also present the exclusion limits derived from the mono-photon [51] search and mono-jet [52] search at the LHC and from the direct DM experiments XENON1T [56,57], XENONnT [58] and PandaX-4T [59].For illustration, we plot the contours of the relic abundance Ω DM h 2 = 0.1186 ± 0.0020 [60].The neutrino floor is also shown, which represents the WIMP-discovery limit obtained with an assuming exposure of 1000 8 B neutrinos detected on a Xenon target [61].For the scalar DM, we also show the constraints from the DM indirect searches through the gamma-ray and W W observations by Fermi-LAT collaboration [62].Here we consider three typical cases, i.e., C B = 0, C W = 500, C B = 500, C W = 0, and C B = C W = 500.
For fermion DM, the bounds from CEPC with all the three running modes can touch the neutrino floor with the DM mass less than a few GeV, while the e + e − collider CEPC is not competitive with other ep and pp colliders.At LHC, the sensitivity from mono-jet search is better with

VIII. SUMMARY AND CONCLUSION
In this work, we focus on the constraints on the dimension-6 (7) effective operators between scalar (Dirac fermion) DM and SM gauge bosons at future ep colliders LHeC and FCC-ep via / E T + e − j signature for the first time.We find that SM irreducible background and reducible background can be effectively suppressed by imposing appropriate kinematic cuts.We also consider the sensitivity at the future e + e − collider CEPC in three different modes with mono-photon signature and update the limits from the LHC with current mono-j and mono-γ searches.Besides the contribution from / E T + W/Z (→ hardons) channel due to the DM-diboson operators for mono- = 500 (right panel).purple and orange solid lines denote the exclusion limits from the mono-photon [51] and mono-jet [52] searches at the 95% C.L. at the 13 TeV LHC.For the direct searches, recent bounds from XENON1T [56,57], XENONnT [58] and PandaX-4T [59] are shown as cyan, dark cyan and steel blue solid lines, respectively.For illustration, the contours of the relic abundance Ω DM h 2 = 0.1186 are also plotted with gray dash-dotted curves.The

2 ≃ 1 / 29 .
qL (Y qR ) denotes the hypercharges of the left-handed (right-handed) quarks with Y uL = Y dL = 1/6, Y uR = 2/3 and Y dR = 1/3, α 1 and α 2 are the gauge coupling constants of the U (1) Y and SU (2) Y gauge group, respectively.In our calculation, we set µ EW ≡ m Z with α 1 ≃ 1/98 and α After Electroweak symmetry breaking (EWSB), the gauge eigenstate fields B µ and W 3 µ mix into the physical massless photon field A µ and massive gauge field Z µ .Then the effective DM-diboson and DM-quark operators are recast as

⊗ 7 ⊗ 7
FIG. 2. The normalized distribution for the missing transverse energy / E T (left), the invariant mass of electron-jet system M (e − , j) (middel) and the inelasticity variable y (right) with m χ = m ϕ = 1 GeV, Λ = 500 GeV and C B = C W = 1 at the 140 GeV ⊗ 7 TeV LHeC.The red and black curves respectively correspond to scalar and fermion DM signals, and SM RB and IB are shown by green and blue lines.
37, ∆ϕ γ, E miss T > 0.4, 0 or 1 jet with p T > 30GeV, |η| < 4.5 and ∆ϕ jet, E miss T > 0.4, and none leptons.Besides, the ATLAS Collaboration performs the measurement in seven different signal regions with a varying cut on the missing transverse momentum E miss T .We find that the strongest bounds on the parameter space in our case are given with the most severe cut of E miss T > 375 GeV.The corresponding 95% C.L. limit on the fiducial cross-section reads σ fid < 0.53 fb.For mono-jet search, events having identified muons, electrons, photons or τ -leptons in the final state are vetoed.Selected events have a leading jet with p T > 150 GeV and |η| < 2.4, and up to three additional jets with p T > 30 GeV and |η| < 2.8.Separation in the azimuthal angle between the missing transverse momentum direction and each selected jet ∆ϕ jet, E miss T > 0.4(0.6) is required for events with E miss T > 250 GeV (200GeV < E miss T < 250 GeV).It turns out that the stronger restriction of E miss T

C
B = 0, C W = 500, and C B = C W = 500, but is slightly worse with C B = 500, C W = 0 than mono-photon search.Besides, the LHC mono-jet search can give the best constraint with C B = 0, C W = 500 compared with other collider experiments.From the figures, it is further validated that FCC-ep has great advantages to constrain DM-gauge boson effective interactions compared with LHeC as mentioned before.In the cases of C B = 500, C W = 0 and C B = C W = 500, FCC-ep can touch the region unconstrained by other experiments when the DM mass in less than several hundred GeV, and push the limits up to about 8000 GeV and 9000 GeV respectively.For scalar DM, in the case of C B = 0, C W = 500 all the collider searches cannot escape the neutrino floor, while the indirect searches of DM by the Fermi-LAT collaboration provide the leading constraints and can touch the neutrino floor in the mass region of 5 GeV ≲ m ϕ ≲ 7 GeV and m ϕ ≳ 130 GeV with gamma-ray observation, and m ϕ ≳ 280 GeV with W W observation.In the case of C B = 500, C W = 500, except FCC-ep can reach the neutrino floor in the mass region of 5 GeV ≲ m ϕ ≲ 6 GeV, other collider searches cannot escape the neutrino floor.Similar to the case of C B = 0, C W = 500, DM searches by the Fermi-LAT collaboration can touch the neutrino floor for 3 GeV ≲ m ϕ ≲ 7 GeV and m ϕ ≳ 80 GeV with cosmic gamma-ray observation, and m ϕ > 250 GeV with W W observation.In the case of C B = 500, C W = 0, the cosmic gammaray observation by the Fermi-LAT collaboration almost lies above the neutrino floor in the plotted region, and W W observation cannot provide any constraint since ϕ φW + W − coupling vanishes.For the collider searches, the future FCC-ep provides the most stringent restrictions relative to the other colliders for all the considered three cases with m ϕ less than a few hundred GeV.Besides, in the case of C B = 500, C W = 0, FCC-ep and CEPC in the Z factory mode can probe a light DM ( m ϕ ≲ 7 GeV ) touching the neutrino floor.

FIG. 7 .
FIG. 7. Constraints in the m χ − Λ plane for the fermion DM with C 6(7) B = 0, C 6(7) W = 500 (left panel) neutrino floor is shown in green-shaded regions.The expected 95% C.L. bounds at the future LHeC (blue dotted lines), FCC-ep (dark blue dashed lines), and CEPC (red dotted lines for Z factory mode, red dashdotted lines for W + W − threshold scan mode, red dashed lines for Higgs factory mode) are shown.

FIG. 8 .
FIG.8.Same as Fig.7, but in the m ϕ − Λ plane for the fermion DM.The constraints from the indirect search of DM through the cosmic gamma-ray (black solid line) and W W (black dashed line) observations by the Fermi-LAT collaboration[62] are also shown.

TABLE I .
The remaining cross-section (in a unit of fb) of signal and background after corresponding cuts with m χ, ϕ = 1 GeV, C B, W = 1 and Λ = 500 GeV at 140 GeV ⊗ 7 TeV LHeC.The corresponding significance S = N S / √ N B and the signal-to-background ratio R = N S /N B are obtained for L = 2 ab −1 ,where N S and N B are the number of signal and background events.

TABLE II .
Same as Table I, but at the 60 GeV ⊗ 50 TeV FCC-ep.