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ISOSPIN CIASSIFICATION OF HADRONS BY FINITE GROUP

  • Starting from the straton model, we discuss the possibility of slassifying the isos-pin of both mesons and baryons by one and the same finite group. We find that thefinite group Td can be used for the purpose.
  • 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 as all of such observations are based on its gravitational effects. Weakly Interacting Massive Particles (WIMPs) are canonical DM candidates. Such a scenario can easily fulfill the relic abundance through the thermal freeze-out mechanism (the “WIMP miracle”) [2] and can be naturally embedded in many popular theoretical frameworks [3, 4]. Moreover, owing to their weak interaction with SM particles, WIMPs 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 exploring the complete model landscape is signifcantly challenging. Instead of exhausting all possible theoretical model parameter spaces, the Effective Field Theory (EFT) framework is extensively used, which enables 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 [534].

    Among them, the parameter space of DM-quark interaction that fulfills the correct relic density has a conflict with the exclusion limits from direct detection, indirect detection, and collider searches. In this work, we consider dimension-6 (7) DM-diboson effective operators for scalar (Dirac fermion) DM and examine their sensitivity at the current LHC and future electron and electron-proton colliders. The current limits of DM-diboson operators at the LHC are primarily constrained by mono-j and mono-γ signal channels, and the mono-γ signature is a sensitive channel at the CEPC. The two most currently promising proposals for electron-proton colliders are the LHeC and FCC-ep. The LHeC is based on an economic LHC upgrade, and a 60−140 GeV electron beam is planned for collision with a 7 TeV proton beam in the LHC ring during the High Luminosity LHC (HL-LHC) run. The 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 a T+ej signal.

    The remainder of this paper is organized as follows. In Sec.II, we briefly describe the DM-diboson effective operators. Subsequently, the relevant collider phenomenology of the T+ej signal channel at the LHeC and FCC-ep is discussed in Sec. III, including event simulations, kinematical distributions of final states, and cut selections related to signals and backgrounds. In Sec. IV, we present the sensitivity at the CEPC with a 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 discussed in Sec. VI. The results are presented in Sec. VII. Finally, we provide a summary and conclusion in Sec. VIII.

    We consider the DM particle as a Dirac fermion (χ) or scalar (ϕ) that connects with a SM sector through high-dimensional DM-gauge boson effective operators. At the lowest order, such operators enable DM to couple with a single gauge boson, which induces electromagnetic dipole and anapole interactions [35, 36]. However, these interactions lead to monochromatic gamma-rays when the unsuppressed cross-section of DM annihilates into the γγ and γZ modes, which are severely constrained by current indirect detection experiments [37]. Thus, we start from DM-diboson operators:

    OϕB=ϕϕBμνBμν,OϕW=ϕϕWa,μνWaμν,

    (1)

    OχB=¯χχBμνBμν,OχW=¯χχWa,μνWaμν.

    (2)

    These operators are assumed to be generated at the energy scale Λ, with other high energy resonances much heavier than Λ being all decoupled. Note that in this scenario, the coupling between DM and the SM quarks can also be induced through the RGE from the Λ scale down to the electroweak scale μEW. Consequently, two extra dimension-6 (7) operators

    Oϕy=yqϕϕˉqHq,OϕH=ϕϕ(HH)2,

    (3)

    Oχy=yqˉχχˉqHq,OϕH=ˉχχ(HH)2,

    (4)

    must be considered. In the above equations, yq=2mq/v denotes the Yukawa couplings of SM quarks, and v is the vacuum expectation value (VEV) of the SM Higgs doublet H. Subsequently, the total effective Lagrangian at scale μ (μEW<μ<Λ) is given by

    Lϕ,χeff=k=B,W,y,HCϕ,χk(μ)Λ2Ok.

    (5)

    At the leading logarithmic (LL) order, the Wilson coefficients of the effective operators Oy and OH at the scale μEW are [38]

    Cy(μEW)6YqLYqRα1πln(μEWΛ),

    (6)

    CH(μEW)9α22ln(μEWΛ)CW(Λ),

    (7)

    where YqL(YqR) denotes the hypercharges of the left-handed (right-handed) quarks with YuL=YdL=1/6, YuR=2/3, and YdR=1/3, and α1 and α2 are the gauge coupling constants of the U(1)Y and SU(2)Y gauge groups, respectively. In our calculation, we set μEWmZ with α11/98 and α21/29. After electroweak symmetry breaking (EWSB), the gauge eigenstate fields Bμ and W3μ mix into the physical massless photon field Aμ and massive gauge field Zμ. Subsequently, the effective DM-diboson and DM-quark operators are recast as

    OϕFF=ϕϕFμνFμν,OχFF=¯χχFμνFμν,Oϕq=mqϕϕˉqq,Oχq=mq¯χχˉqq,

    (8)

    where FF=AA,AZ,ZZ,WW. The relevant matching conditions are

    CAA(μEW)=CB(μEW)cos2θW+CW(μEW)sin2θW,CZZ(μEW)=CW(μEW)cos2θW+CB(μEW)sin2θW,CAZ(μEW)=(CW(μEW)CB(μEW))sin2θW,CWW(μEW)=CW(μEW),Cq(μEW)=Cy(μEW)v2m2hCH(μEW),

    (9)

    with θW is the Weinberg angle.

    Integrating out the heavy top quarks, the effective DM-gluon operators

    Oϕq=αsϕϕGa,μνGaμν,Oχq=αs¯χχGa,μνGaμν

    (10)

    can be generated, where Gaμν denotes the gluon field strength. The matching condition at leading order is given by [39]

    CG(mt)=112πCt(mt).

    (11)

    When RGEs are evolved from the EW scale down to the hadronic scale μhad1 GeV, OAA will contribute to Oq through the exchanging of virtual photons. For scale μ with mq<μ<μEW, we obtain [40]

    Cq(μ)Cq(μEW)+6Q2qαemπln(μμEW)CAA(μEW)

    (12)

    at the LL order, where Qq is the electric charge of a quark, and the electromagnetic coupling constant αem1/137.

    At the mb and mc threshold, we must integrate out the corresponding heavy bottom and charm quarks by considering Eq. (11) again. At the scale μhad1 GeV, we obtain

    Cq(μhad)(6YqLYqRα1πCB(Λ)+9α22v2m2hCW(Λ))ln(μEWΛ)+6Q2qαemπCAA(μEW)ln(μhadμEW),

    (13)

    CG(μhad)112π{(α1πCB(Λ)+27α22v2m2hCW(Λ))ln(μEWΛ)+2αem3πCAA(μEW)[ln(mbμEW)+4ln(mcμEW)]},

    (14)

    where the Wilson coefficient CAA(μEW) is determined by the first matching condition in Eq. (9).

    In this section, we investigate the sensitivity of the DM-gauge boson effective operators at future electron-proton colliders.

    From the effective interactions in Eqs. (1) and (2), DM pairs can be produced at the electron-proton collider through both the Charged Current (CC) and Neutral Current (NC), which respectively correspond to the W boson and γ/Z modes of VBF processes. In CC production, DM can be generated through the process of epνejχˉχ(ϕϕ) via WW fusion, resulting in a mono-jet plus missing energy signature. Note that the mono-jet signature coincidentally aligns with the background of CC deeply inelastic scattering. Owing 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 production – specifically the process epejχˉχ(ϕϕ), as depicted in Fig. 1. Thus, the corresponding signature is characterized as the T+ej channel.

    Figure 1

    Figure 1.  Feynman diagram for the process epejχˉχ(ϕϕ).

    For SM background estimation, we consider both the reducible background (RB) and irreducible background (IB). The IB is primarily due to the processes epejνˉν(=e,μ,τ). Among them, both W/Z-boson bremsstrahlung processes epejZ/νejW(Zνeˉνe/Weˉνe) contribute to νe final state, whereas for νμ,τ, only the bremsstrahlung process epejZ(Zνμ,τˉνμ,τ) is relevant. For the RB, a major contribution is from epejW± with a W leptonic decay W±±ν. Therefore, two possibilities must be considered. One is that electron and muon final states exceed the detector acceptance, and the τ final state mimics a hard jet in the detector. Another possibility is that the products of hadronic τ decays are too soft to be tagged [41, 42]. Consequently, the SM background can be summarized as follows:

    ● IB: epejνˉν,

    ● RB: epej±ν(ˉν),

    where =e,μ,τ.

    For an event simulation, we generate a Universal FeynRules Output (UFO) [43] model file using the FeynRules package [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:

    pj,T>5GeV, |ηj,| <5,ΔRj>0.4,ΔR>0.4,

    (15)

    where pT and η are the transverse momentum and pseudorapidity of the corresponding particles, respectively, and ΔR=Δϕ2+Δy2 is the separation in the azimuthal angle-rapidity (ϕy) plane. Note that all of the cuts in Eq. (15) are defined in the laboratory frame. We then use Pythia6.4 [46] to implement parton shower and hadronization and Delphes3.4 [47] for rapid detector simulation according to the LHeC designed parameters [48].

    To suppress the RB, we apply the following veto criteria [42] as basic cuts:

    ● events must contain exactly one hard electron, one hard jet, and T,

    ● events containing any extra jets with pjT>3 GeV and |ηj|<2 or leptons with pe,μT>5 GeV or tagged τ jets are vetoed.

    Fig. 2 and Fig. 3depict the normalized distributions of the missing transverse energy T, 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, respectively. For the signal, we assume benchmark values of mχ,ϕ=1 GeV, CB,W=1, and Λ= 500 GeV. For the inelasticity variable y, we follow the definition in Ref. [41], which is given as

    Figure 2

    Figure 2.  (color online) Normalized distributions of the missing transverse energy T(left), invariant mass of electron-jet system M(e,j) (middle), and inelasticity variable y (right) with mχ=mϕ=1 GeV, Λ=500 GeV, and CB=CW=1 at the 140 GeV 7 TeV LHeC. The red and black curves correspond to scalar and fermion DM signals, respectively, and the green and blue lines show the SM RB and IB, respectively.

    Figure 3

    Figure 3.  (color online) Same as Fig. 2 but at the 60 GeV 50 TeV FCC-ep.

    ykP(kepe)kPke,

    (16)

    where kP,e are the 4-momenta of the initial proton and electron, and pe is the 4-momenta of the final electron. We can observe the following characteristics:

    ● for T distribution, the SM RB tends to have a smaller value, T<100GeV, compared with signals, whereas the SM IB has values similar to those of signals,

    ● for y distribution, the SM RB and IB tend to distribute at small and large y values, respectively, whereas signals exhibit a relatively flat distribution.

    Such distinct behaviors of signals and SM backgrounds indicate that we can impose the following kinematical cuts for the LHeC (FCC-ep) to further extract signals:

    T>100(220)GeV,M(e,j)>100(200)GeV,0.3<y<0.8(0.1<y<0.9).

    (17)

    Table 1 and Table 2 show the cut-flows for the fermion and scalar DM signals and backgrounds at the LHeC and FCC-ep, respectively. The corresponding significance S=NS/NB and signal-to-background ratio R=NS/NB with the total integrated luminosity L=2ab1 are also shown, where NS,B=σS,BL are the surviving event numbers of signals and backgrounds, respectively. Both S and R increase when each cut is imposed. After the pre-selection cuts, RB contribution is still large, whereas the basic cuts suppress the RB by a factor of 4.5 (5.0) at the LHeC (FCC-ep). Moreover, the RB is dramatically reduced by the T cut, which only survives about 1.26% (0.07%) at the LHeC (FCC-ep).

    Table 1

    Table 1.  Remaining cross-section (in units of fb) of the signal and background after corresponding cuts with mχ,ϕ=1 GeV, CB,W=1, and Λ=500 GeV at the 140 GeV 7 TeV LHeC. The corresponding significance S=NS/NB and signal-to-background ratio R=NS/NB were obtained for L=2 ab1, where NS and NB are the numbers of signal and background events, respectively.
    Process Signal (ϕ) Signal (χ) RB IB Sϕ Sχ Rϕ Rχ
    pre-selection cuts 7.57 10.28 1421.87 260.39 8.26 11.20 0.0045 0.0061
    basic cuts 4.91 6.84 316.83 155.21 10.11 14.08 0.010 0.014
    T>100 GeV 1.71 2.62 18.28 38.33 10.16 15.58 0.030 0.046
    M(e,j)>100 GeV 1.56 2.33 15.46 29.28 10.46 15.56 0.035 0.052
    0.3<y<0.8 1.01 1.54 5.40 9.16 11.80 18.05 0.070 0.106
    DownLoad: CSV
    Show Table

    Table 2

    Table 2.  Same as Table 1 but at the 60 GeV 50 TeV FCC-ep.
    Process Signal (ϕ) Signal (χ) RB IB Sϕ Sχ Rϕ Rχ
    pre-selection cuts 18.17 64.85 1232.28 222.80 21.31 76.03 0.012 0.045
    basic cuts 11.56 42.58 244.95 125.09 26.87 98.98 0.031 0.12
    T>220 GeV 3.10 13.44 0.86 4.32 60.88 264.17 0.060 2.60
    M(e,j)>200 GeV 2.59 10.71 0.61 2.13 70.21 289.95 0.95 3.92
    0.1<y<0.9 2.41 9.48 0.57 1.44 75.94 298.87 1.19 4.71
    DownLoad: CSV
    Show Table

    At the LHeC, after all cuts, the fermion (scalar) DM signal can survive 13.2% (14.9%), whereas the RB (IB) only survives 0.37% (3.5%). At the FCC-ep, the signal for fermion (scalar) DM survived 13.3% (14.6%), and the RB (IB) ultimately survives 0.047% (0.65%). Note that the remaining cross-section of the background at the FCC-ep is only 14% of ones that at the LHeC, whereas the signal for fermion (scalar) DM production at the FCC-ep can reach 6 (1.5) times that at the LHeC, which implies that the FCC-ep exhibits a considerable improvement in significance compared with the LHeC.

    Owing to the effective DM-diboson operators in Eqs. (1) and (2), a WIMP can be produced at e+e colliders via the process e+eVϕϕ(ˉχχ)+V (V=γ,Z), which results in mono-photon or mono-Z signatures. The corresponding Feynman diagrams are shown in Fig. 4. In this section, we focus on the mono-photon signature at the future projected CEPC [49, 50] with three different running modes: the Higgs factory mode for a seven year run at s=240 GeV with a total luminosity of 5.6 ab1, the Z factory mode for a two year run at s=91.2 GeV with a total luminosity of 16 ab1, and the W+W threshold scan mode for a one year run at s158172 GeV with a total luminosity of 2.6 ab1. 1

    Figure 4

    Figure 4.  Feynman diagrams for the process e+eϕϕ(ˉχχ)+Z or e+eϕϕ(ˉχχ)+γ.

    For the monophoton signature at the CEPC, an irreducible background results from the neutrino production e+eνlˉνlγ processes, where νl=νe,νμ,ντ. Owing to the Breit-Wigner distribution of the Z boson in the irreducible BG, a resonance occurs in the final photon energy spectrum, which is located at

    EZγ=sM2Z2s

    (18)

    with a full-width-at-half-maximum of ΓZγ=MZΓZ/s. To suppress the irreducible background contribution, we veto the events within 5ΓZγ at the Z resonance in the monophoton energy spectrum [27]. The vetoing cut can be expressed as

    |EγEZγ|<5ΓZγ.

    (19)

    The main reducible SM backgrounds result from the e+eγ+ processes, where only one photon is visible in the final state, and denotes the other undetectable particle(s) owing to the limited detection capability of the detectors. In our analysis, the parameters for the EMC coverage, |cosθ|<0.99 and E>0.1 GeV, are adopted based on CEPC CDR [50]. The processes e+efˉfγ and e+eγγγ provide dominant contributions to the reducible background when the final fˉf and γγ are emitted with |cosθ|>0.99. Owing 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]

    Emγ(θγ)=s(sinθγsinθb)1,

    (20)

    where the polar angle θb corresponds to the boundary of the EMC with |cosθ|<0.99. To suppress the monophoton events resulting from the reducible background, we adopt the detector cut Eγ>Emγ(θγ) on the final state photon in our analysis.

    Through the DM-diboson operators, a WIMP can be produced via the processes qqVϕϕ(ˉχχ)+V(V=γ,Z,W), which can exhibit a mono-photon (T+γ) or mono-jet (T+W/Z (hardons)) signature, respectively. The representative Feynman diagrams are shown in Fig. 5. Moreover, the loop-induced effective DM-quarks (Oq) and DM-gluons (OG) operators can result in the mono-jet signature via the qˉqˉχχ+g, ggˉχχ+g, and qgˉχχ+q processes. The representative Feynman diagrams are shown in Fig. 6.

    Figure 5

    Figure 5.  Representative Feynman diagrams for mono-photon (left) and mono-jet (right) signatures resulting from DM-diboson operators.

    Figure 6

    Figure 6.  Representative Feynman diagrams for a mono-jet signature resulting from loop-induced effective DM-quarks and DM-gluons operators.

    In this section, we consider the sensitivities from LHC mono-j and mono-γ searches. To simulate the signal from dark matter, we genrate the parton-level events using MadGraph@NLO [45], and the subsequent parton shower and hadronization are generated using Pythia6.4 [46]. We use Delphes3.4 [47] to conduct a rapid detector simulation for the ATLAS or CMS detector with the corresponding parameter setup. We follow the procedure of the latest mono-photon [51] and mono-jet [52] analysis by the ATLAS Collaboration with an integrated luminosity of 139 fb1 at a center-of-mass energy of 13 TeV.

    For the T+γ search channel, events in our analysis are required to have a leading γ with EγT>150GeV, |ηγ|<1.37 or 1.52<|ηγ|<2.37, Δϕ(γ, Emiss T)>0.4, 0 or 1 jet with pT>30GeV, |η|<4.5 and Δϕ(jet, EmissT)>0.4, and no leptons. Moreover, the ATLAS Collaboration performs the measurement in seven different signal regions with a varying cut on the missing transverse momentum EmissT. We observe that the strongest bounds on the parameter space in our case are given with the most severe cut of EmissT>375 GeV. The corresponding 95% C.L. limit on the fiducial cross-section is σfid< 0.53 fb.

    For mono-jet search, events with identified muons, electrons, photons, or τ-leptons in the final state are vetoed. Selected events have a leading jet with pT>150 GeV and |η|<2.4, and up to three additional jets with pT>30 GeV and |η|<2.8. Separation in the azimuthal angle between the missing transverse momentum direction and each selected jet Δϕ(jet, EmissT)>0.4(0.6) is required for events with EmissT>250 GeV (200GeV<EmissT<250 GeV). We observe that the stronger restriction of EmissT>1200 GeV provides the best limits. At 95% C.L., the bound on the corresponding fiducial cross section is given by σfid< 0.3 fb.

    Table 3 presents the ratio of the number of events for mono-jet signature at LHC from the process ppχˉχ+W/Z (hardons) via dimension-7 DM-diboson operators with the Feynman diagrams shown in Fig. 5 to one from the process ppχˉχ+q/g via loop-induced DM-quarks and DM-gluons operators with the Feynman diagrams shown in Fig. 6 with Λ=1000 GeV. Here, we consider three cases, i.e., CB=CW=500, CB=0,CW=500, and CB=500,CW=0. We observe that all the ratios in the three cases increase with an increment in the DM mass. Furthermore, the contributions from tree-level DM-diboson operators to the mono-jet signature at the LHC are about two orders of magnitude above those from loop-induced DM-quarks and DM-gluons operators, especially for CB=1,CW=0, with the contributions being about three orders higher; thus, we ignore the contributions from DM-quarks and DM-gluons operators in the following analysis.

    Table 3

    Table 3.  Ratio of the number of events for the mono-jet signature at the LHC from the process ppχˉχ+W/Z (hardons) via DM-diboson operators with the Feynman diagrams shown in Fig. 5 to ppχˉχ+q/g via loop-induced DM-quarks and DM-gluons operators with the Feynman diagrams shown in Fig. 6 at Λ=1000 GeV.
    mχ 1 300 800 2000 3000
    CB=500,CW=0 638.22 699.35 752.11 1435.32 2733.95
    CB=0,CW=500 95.00 99.55 104.29 181.55 302.49
    CB=500,CW=500 79.20 80.33 91.64 158.94 271.41
    DownLoad: CSV
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    In this section, we discuss the bounds on Λ from current direct and indirect search experiments. The effective operators OB and OW lead to the non-relativistic cross-sections of DM annihilation χˉχ, ϕϕγγ/γZ/ZZ/WW as follows:

    σkv=ak+bkv2+O(v4)

    (21)

    with k=B,W annihilation channels. The coefficients of a and b are [37]

    χ{aB=aW=0,bB=|CB(Λ)|2πm4χΛ6[c4w+c2ws2w8β2γZ(xZ4)2Θ(MZ<2mχ)+s4w8βZZ(3x2Z8xZ+8)Θ(MZ<mχ)],bW=|CW(Λ)|2πm4χΛ6[s4w+c2ws2w8β2γZ(xZ4)2Θ(MZ<2mχ)+c4w8βZZ(3x2Z8xZ+8)Θ(MZ<mχ)+14βWW(3x2W8xW+8)Θ(MW<mχ)],

    (22)

    ϕ{aB=|CB(Λ)|2π2m2ϕΛ4[c4w+c2ws2w8β2γZ(xZ4)2Θ(MZ<2mϕ)+s4w8βZZ(3x2Z8xZ+8)Θ(MZ<mϕ)],aW=|CW(Λ)|2π2m2ϕΛ4[s4w+c2ws2w8β2γZ(xZ4)2Θ(MZ<2mϕ)+c4w8βZZ(3x2Z8xZ+8)Θ(MZ<mϕ)+14βWW(3x2W8xW+8)Θ(MW<mϕ)],bB=|CB(Λ)|2π2m2ϕΛ4[c4w/2+c2ws2w4β2γZ(xZ4)Θ(MZ<2mϕ)+s4wβZZ(1/2xZ)Θ(MZ<mϕ)],bW=|CW(Λ)|2π2m2ϕΛ4[s4w/2+c2ws2w4β2γZ(xZ4)Θ(MZ<2mϕ)+c4wβZZ(1/2xZ)Θ(MZ<mϕ)+βWW(12xZ)Θ(MW<mϕ)].

    (23)

    In above equations, we have defined the phase space factor βij=1(mi+mj)2/(4m2DM) and xi=m2i/m2DM with DM=χ,ϕ. Using the results for aB,W and bB,W, the relic density can then be calculated using

    ΩDMh2=1.09×109xFGeV1MPlg(xF)(a+3b/xF),

    (24)

    where a=aB+aW and b=bB+bW. xF=mDM/TF is the ratio of the mass of DM mDM and the early-universe freeze-out temperature TF, which can be obtained by solving

    xF=ln[c(c+2)458mDMMPl(a+6b/xF)2π3g(xF)].

    (25)

    The loop-induced effective DM-quarks (Oq) and DM-gluons (OG) operators can result in the interaction between DM and nucleons. The spin-independent (SI) cross section for elastic scalar and Dirac WIMP scattering on a nucleon has the form

    σϕSIμ2Nm2Nπm2ϕΛ4|αZ2AfNACAA(μhad)+ZAfp(μhad)+AZAfn(μhad)|2,

    (26)

    σχSIμ2Nm2NπΛ6|αZ2AfNACAA(μhad)+ZAfp(μhad)+AZAfn(μhad)|2.

    (27)

    In above equations, mN0.939 GeV is the average nucleon mass, and μN=mϕ,χmN/(mϕ,χ+mN) is the reduced mass of the DM-nucleon system. The form factor fNA describes the Rayleigh scattering of two photons on the entire nucleus. At a zero-momentum transfer limit, fNA0.08 for a xenon target. The form factor fN=p,n describing the couplings between the DM and nucleon is expressed as [40, 53]

    fN(μhad)=q=u,d,sfNqCq(μhad)8π9fNGCG(μhad),

    (28)

    where the form factors fNq describing the scalar couplings between quarks and nucleon are given by [54]

    fpd=0.0191, fpu=0.0153, fnd=0.0273,fnu=0.0110, fps=fns=0.0447,

    (29)

    and

    fNG=1q=u,d,sfNq.

    (30)

    We apply simple χ2 analysis to derive the lower bounds on Λ for effective operators in Eq. (5) at the future LHeC, FCC-ep, and CEPC at 95% C.L. by requiring χ2=N2S/NB=3.84 [55] with the specified luminosities. For LHeC and FCC-ep, the total integrated luminosities are assumed as L=2ab1. The limits of Λ as a function of the mass of Dirac fermion and scalar DM are shown 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 ΩDMh2=0.1186±0.0020 . The neutrino floor is also shown, which represents the WIMP-discovery limit obtained using an assumed exposure of 1000 8B neutrinos detected on a xenon target [60]. For the scalar DM, we also show the constraints from the DM indirect searches through the gamma-ray and WW observations by the Fermi-LAT Collaboration [61]. Here, we consider three typical cases, i.e., CB=0, CW=500, CB=500,CW=0, and CB=CW=500.

    Figure 7

    Figure 7.  (color online) Constraints in the mχΛ plane for the fermion DM with C6(7)B=0,C6(7)W=500 (left panel), C6(7)B=500,C6(7)W=0 (middle panel), and C6(7)B=500,C6(7)W=500 (right panel). The 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, respectively. 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 ΩDMh2=0.1186 are also plotted as gray dash-dotted curves. The neutrino floor is shown as 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 dash-dotted lines for W+W threshold scan mode, red dashed lines for Higgs factory mode) are shown.

    Figure 8

    Figure 8.  (color online) Same as Fig. 7 but in the mϕΛ plane for the fermion DM. The constraints from the indirect search of DM through the cosmic gamma-ray (black solid line) and WW (black dashed line) observations by the Fermi-LAT collaboration [61] are also shown.

    For fermion DM, the bounds from the CEPC with all the three running modes can touch the neutrino floor with the DM mass less than a few GeV, whereas the e+e collider CEPC is not competitive with other ep and pp colliders. At the LHC, the sensitivity from mono-jet search is better with CB=0,CW=500, and CB=CW=500 but is slightly worse with CB=500,CW=0 than the mono-photon search. Moreover, the LHC mono-jet search can provide the best constraint with CB=0,CW=500 compared with other collider experiments. The figures further validate that the FCC-ep has significant advantages to constrain DM-gauge boson effective interactions compared with the LHeC, as mentioned earlier. For CB=500,CW=0, and CB=CW=500, the FCC-ep can incorporate the region unconstrained by other experiments when the DM mass in less than several hundred GeV and extend the limits up to about 8000 and 9000 GeV, respectively.

    For scalar DM, in the case of CB=0,CW=500, all the collider searches cannot escape the neutrino floor, whereas the indirect searches of DM by the Fermi-LAT Collaboration provide the leading constraints and can include the neutrino floor in the mass regions of 5 GeVmϕ7 GeV and mϕ130 GeV with gamma-ray observation and mϕ280 GeV with WW observation. For CB=500,CW=500, the FCC-ep can reach the neutrino floor in the mass region of 5 GeVmϕ6 GeV, whereas other collider searches cannot escape the neutrino floor. Similar to the case of CB=0,CW=500, DM searches by the Fermi-LAT Collaboration can reach the neutrino floor for 3 GeVmϕ7 GeV and mϕ80 GeV with cosmic gamma-ray observation and mϕ>250 GeV with WW observation. ForCB=500,CW=0, the cosmic gamma-ray observation by the Fermi-LAT Collaboration almost lies above the neutrino floor in the plotted region, and WW observation cannot provide any constraint because ϕˉϕW+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 mϕ less than a few hundred GeV. Moreover, for CB=500,CW=0, the FCC-ep and CEPC in the Z factory mode can probe a light DM (mϕ7 GeV) reaching the neutrino floor.

    In this study, we focus on the constraints on dimension-6 (7) effective operators between scalar (Dirac fermion) DM and SM gauge bosons at future ep colliders LHeC and FCC-ep via the T+ej signature for the first time. We observe that SM irreducible and reducible backgrounds can be effectively suppressed by imposing appropriate kinematic cuts. We also consider the sensitivity at the future e+e collider CEPC under three different modes with a mono-photon signature and update the limits from the LHC with current mono-j and mono-γ searches. In addition to the contribution from T+W/Z(hardons) channel due to the DM-diboson operators for mono-j at LHC, we investigate the contribution from the loop induced effective DM-quarks and DM-glouns operators. We observe that the contribution from DM-quarks and DM-gluons operators can be ignored. We present the constraints on the effective energy scale Λ as a function of DM mass with three typical cases of Wilson coefficients: C6(7)B=C6(7)W=500, C6(7)B=0,C6(7)W=500, and C6(7)B=500,C6(7)W=0. The FCC-ep exhibits better sensitivity than the LHeC in all cases for scalar and Dirac fermion DM. We observe that the collider searches can probe light DM reaching the neutrino floor.

    1We take \begin{document}$ \sqrt s= 160 $\end{document} GeV for the \begin{document}$ W^+W^- $\end{document} threshold scan mode throughout our analysis.

    [1] K.M. Case, R. Karplus and C. N. Yang, Phys. Rev. 101 (1956), 874.[2] K.Yamada, Phys. Rev., D18 (1978), 935.[3] 北京大学理论物理研究室,中国科学院数学研究所理论物理研究室,北大学报,2(1966), 103;中国科学院原子能研究所基本粒于理论组,原子能,3(1966), 137,[4] G. F. Koster et al., Properties of the Thirty-Two Point Groups, (1963).[5] A. B. Cokoлoв, B. П. шиpoкoвeкий, уcnexu Фuзuuecrcкuǔ Hayrc, 60 (1956), 617-688.
  • [1] K.M. Case, R. Karplus and C. N. Yang, Phys. Rev. 101 (1956), 874.[2] K.Yamada, Phys. Rev., D18 (1978), 935.[3] 北京大学理论物理研究室,中国科学院数学研究所理论物理研究室,北大学报,2(1966), 103;中国科学院原子能研究所基本粒于理论组,原子能,3(1966), 137,[4] G. F. Koster et al., Properties of the Thirty-Two Point Groups, (1963).[5] A. B. Cokoлoв, B. П. шиpoкoвeкий, уcnexu Фuзuuecrcкuǔ Hayrc, 60 (1956), 617-688.
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Xu Jiao-lin, Wu Yong-shi, Chen Shi and Guo Han-ying. ISOSPIN CIASSIFICATION OF HADRONS BY FINITE GROUP[J]. Chinese Physics C, 1980, 4(2): 251-255.
Xu Jiao-lin, Wu Yong-shi, Chen Shi and Guo Han-ying. ISOSPIN CIASSIFICATION OF HADRONS BY FINITE GROUP[J]. Chinese Physics C, 1980, 4(2): 251-255. shu
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Revised: 1900-01-01
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ISOSPIN CIASSIFICATION OF HADRONS BY FINITE GROUP

Abstract: Starting from the straton model, we discuss the possibility of slassifying the isos-pin of both mesons and baryons by one and the same finite group. We find that thefinite group Td can be used for the purpose.

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