-
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
\gamma\gamma and\gamma Z modes, which are severely constrained by current indirect detection experiments [37]. Thus, we start from DM-diboson operators:\begin{array}{*{20}{l}} {\cal O}_B^\phi = \phi^*\phi B^{\mu\nu}B_{\mu\nu}, \qquad {\cal O}_W^\phi = \phi^*\phi W^{a,\mu\nu}W^a_{\mu\nu}, \end{array}
(1) \begin{array}{*{20}{l}} {\cal O}_B^\chi = \overline{\chi}\chi B^{\mu\nu}B_{\mu\nu}, \qquad {\cal O}_W^\chi = \overline{\chi}\chi W^{a,\mu\nu}W^a_{\mu\nu}. \end{array}
(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
\mu_{\rm EW} . Consequently, two extra dimension-6 (7) operators\begin{array}{*{20}{l}} {\cal O}_y^\phi = y_q\phi^*\phi \bar q H q, \qquad {\cal O}_H^\phi = \phi^*\phi(H^\dagger H)^2, \end{array}
(3) \begin{array}{*{20}{l}} {\cal O}_y^\chi = y_q\bar \chi \chi \bar q H q, \qquad {\cal O}_H^\phi = \bar \chi \chi (H^\dagger H)^2, \end{array}
(4) must be considered. In the above equations,
y_q=\sqrt{2}m_q/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 (\mu_{\rm EW} < \mu < \Lambda ) is given by\mathcal{L}_{\rm eff}^{\phi,\chi} = \sum\limits_{k=B,W,y,H} \frac{C^{\phi,\chi}_k(\mu)}{\Lambda^2} {\cal O}_k\,.
(5) At the leading logarithmic (LL) order, the Wilson coefficients of the effective operators
{\cal O}_y and{\cal O}_H at the scale\mu_{\rm EW} are [38]C_y(\mu_{\rm EW}) \simeq \frac{6Y_{qL}Y_{qR}\alpha_1}{\pi}\ln\left(\frac{\mu_{\rm EW}}{\Lambda}\right),
(6) C_H(\mu_{\rm EW}) \simeq -9 \alpha^2_2\ln\left(\frac{\mu_{\rm EW}}{\Lambda}\right) C_W(\Lambda),
(7) where
Y_{qL}\,(Y_{qR}) denotes the hypercharges of the left-handed (right-handed) quarks withY_{uL}=Y_{dL}= 1/6 ,Y_{uR}=2/3 , andY_{dR}=1/3 , and\alpha_1 and\alpha_2 are the gauge coupling constants of theU(1)_Y andS U(2)_Y gauge groups, respectively. In our calculation, we set\mu_{\rm EW}\equiv m_Z with\alpha_1\simeq 1/98 and\alpha_2\simeq 1/29 . After electroweak symmetry breaking (EWSB), the gauge eigenstate fieldsB_\mu andW_\mu^3 mix into the physical massless photon fieldA_\mu and massive gauge fieldZ_\mu . Subsequently, the effective DM-diboson and DM-quark operators are recast as\begin{aligned}[b]& {\cal O}_{FF}^\phi = \phi^*\phi F^{\mu\nu}F_{\mu\nu}, \quad {\cal O}_{FF}^\chi = \overline{\chi}\chi F^{\mu\nu}F_{\mu\nu}, \\& {\cal O}_{q}^\phi = m_q \phi^*\phi\bar{q}q, \quad {\cal O}_{q}^\chi = m_q \overline{\chi}\chi\bar{q} q, \end{aligned}
(8) where
FF=AA,\;AZ,\;ZZ,\;WW . The relevant matching conditions are\begin{aligned}[b] {C_{AA}(\mu_{\rm EW})}=&C_B(\mu_{\rm EW}) \cos ^{2} \theta_{W}+C_W(\mu_{\rm EW}) \sin ^{2} \theta_{W}, \\ {C_{Z Z}(\mu_{\rm EW})} =&C_W(\mu_{\rm EW}) \cos ^{2} \theta_{W}+C_B(\mu_{\rm EW}) \sin ^{2} \theta_{W}, \\ {C_{A Z}(\mu_{\rm EW})} =&\left(C_W(\mu_{\rm EW})-C_B(\mu_{\rm EW})\right) \sin 2 \theta_{W}, \\ {C_{W W}(\mu_{\rm EW})}=&C_W(\mu_{\rm EW}), \\ {C_{q}(\mu_{\rm EW})}=&C_y(\mu_{\rm EW})-\frac{v^2}{m_h^2}C_H(\mu_{\rm EW}), \end{aligned}
(9) with
\theta_{W} is the Weinberg angle.Integrating out the heavy top quarks, the effective DM-gluon operators
\begin{array}{*{20}{l}} {\cal O}_{q}^\phi = \alpha_s \phi^*\phi G^{a,\mu\nu}G_{\mu\nu}^a, \quad {\cal O}_{q}^\chi = \alpha_s \overline{\chi}\chi G^{a,\mu\nu}G_{\mu\nu}^a \end{array}
(10) can be generated, where
G_{\mu\nu}^a denotes the gluon field strength. The matching condition at leading order is given by [39]C_{G}(m_t)=-\frac{1}{12\pi}C_t(m_t).
(11) When RGEs are evolved from the EW scale down to the hadronic scale
\mu_{\rm had}\simeq 1 GeV,{\cal O}_{AA} will contribute to{\cal O}_{q} through the exchanging of virtual photons. For scale μ withm_q<\mu<\mu_{\rm EW} , we obtain [40]{C_{q}(\mu)}\simeq C_{q}(\mu_{\rm EW})+ \frac{6 Q_q^2\alpha_{\rm em}}{\pi} \ln\left(\frac{\mu}{\mu_{\rm EW}}\right) C_{AA}(\mu_{\rm EW})
(12) at the LL order, where
Q_q is the electric charge of a quark, and the electromagnetic coupling constant\alpha_{\rm em} \simeq 1/137 .At the
m_b andm_c threshold, we must integrate out the corresponding heavy bottom and charm quarks by considering Eq. (11) again. At the scale\mu_{\rm had}\simeq 1 GeV, we obtain\begin{aligned}[b] C_{q}\left(\mu_{\rm had}\right) \simeq& \left(\frac{6 Y_{q_{L}} Y_{q_{R}} \alpha_{1}}{\pi} C_{B}(\Lambda)+ 9 \alpha_2^2 \frac{v^{2}}{m_{h}^{2}} C_{W}(\Lambda)\right) \ln \left(\frac{\mu_{\rm EW}}{\Lambda}\right) \\ &+\frac{6 Q_{q}^{2} \alpha_{\rm em}}{\pi} C_{AA}(\mu_{\rm EW}) \ln \left(\frac{\mu_{\rm had}}{\mu_{\rm EW}}\right), \end{aligned}
(13) \begin{aligned}[b] C_{G}\left(\mu_{\rm had}\right) \simeq& -\frac{1}{12 \pi}\Bigg\{\left(\frac{\alpha_{1}}{\pi} C_{B}(\Lambda)+ 27 \alpha_2^2 \frac{v^{2}}{m_{h}^{2}} C_{W}(\Lambda)\right) \ln \left(\frac{\mu_{\rm EW}}{\Lambda}\right) \\ &+\frac{2 \alpha_{\rm em}}{3 \pi} C_{AA}(\mu_{\rm EW})\left[\ln \left(\frac{m_b}{\mu_{\rm EW}}\right)+4 \ln \left(\frac{m_c}{\mu_{\rm EW}}\right)\right]\Bigg\}, \end{aligned}
(14) where the Wilson coefficient
C_{AA}(\mu_{\rm EW}) 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
\gamma/Z modes of VBF processes. In CC production, DM can be generated through the process ofe^- p \to \nu_e j \chi\bar\chi(\phi\phi^*) viaWW 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 processe^- p \to e^-j \chi\bar\chi\; (\phi\phi^*) , as depicted in Fig. 1. Thus, the corresponding signature is characterized as the{\not E}_T+e^-j channel.For SM background estimation, we consider both the reducible background (RB) and irreducible background (IB). The IB is primarily due to the processes
e^-p\to e^-j\nu_\ell\bar\nu_\ell\; (\ell=e,\; \mu,\; \tau) . Among them, bothW/Z -boson bremsstrahlung processese^-p\to e^-jZ/\nu_ejW^-\; (Z\to\nu_e\bar\nu_e/ W^-\to e^-\bar\nu_e) contribute to\nu_e final state, whereas for\nu_{\mu,\tau} , only the bremsstrahlung processe^-p\to e^-jZ\; (Z\to\nu_{\mu,\tau}\bar\nu_{\mu,\tau}) is relevant. For the RB, a major contribution is frome^- p \to e^- j W^\pm with a W leptonic decayW^\pm\to\ell^\pm \nu_\ell . 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:
e^-p\to e^-j\nu_\ell\bar\nu_\ell ,● RB:
e^- p \to e^- j \ell^\pm \nu_\ell\; (\bar\nu_\ell) ,where
\ell=e,\mu,\tau . -
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:
\begin{aligned}[b] &p^{j,\ell}_T > 5\,{\rm GeV}, \quad \ |\eta_{j,\ell}|\ < 5, \\ &\Delta R_{j\ell} > 0.4, \quad \Delta R_{\ell\ell} > 0.4, \end{aligned}
(15) where
p_T and η are the transverse momentum and pseudorapidity of the corresponding particles, respectively, and\Delta R =\sqrt{\Delta \phi^2+\Delta y^2} is the separation in the azimuthal angle-rapidity (\phi-y ) 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
{\not E}_T ,● events containing any extra jets with
p_T^j>3 GeV and|\eta_j|<2 or leptons withp_T^{e,\mu}>5 GeV or tagged τ jets are vetoed.Fig. 2 and Fig. 3depict the normalized distributions of the missing transverse energy
{\not E}_T , invariant mass of tagged electron and jet systemM(e^-,j) , and inelasticity variable y for both signal and backgrounds after the basic cuts, at the 140 GeV\otimes 7 TeV LHeC and 60 GeV\otimes 50 TeV FCC-ep, respectively. For the signal, we assume benchmark values ofm_{\chi,\; \phi}=1 GeV,C_{B,\; W}=1 , and\Lambda= 500 GeV. For the inelasticity variable y, we follow the definition in Ref. [41], which is given asFigure 2. (color online) Normalized distributions of the missing transverse energy
{\not E}_T (left), invariant mass of electron-jet systemM(e^-,j) (middle), and inelasticity variable y (right) withm_\chi=m_\phi=1 GeV,\Lambda=500 GeV, andC_B=C_W=1 at the 140 GeV\otimes 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. (color online) Same as Fig. 2 but at the 60 GeV
\otimes 50 TeV FCC-ep.y\equiv\frac{k_P\cdot(k_e-p_e)}{k_P\cdot k_e},
(16) where
k_{P,\; e} are the 4-momenta of the initial proton and electron, andp_e is the 4-momenta of the final electron. We can observe the following characteristics:● for
{\not E}_T distribution, the SM RB tends to have a smaller value,{\not E}_T <100 GeV, 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:
\begin{aligned}[b] &{\not E}_T > 100\quad(220)\,{\rm GeV}, \\ &M(e^-,j) > 100 \quad (200)\,{\rm GeV}, \\ &0.3 < y < 0.8 \quad (0.1 < y < 0.9). \end{aligned}
(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
{\cal S}=N_S/\sqrt{N_B} and signal-to-background ratio{\cal R}= N_S/N_B with the total integrated luminosityL = 2\; {\rm ab}^{-1} are also shown, whereN_{S,\; B}=\sigma_{S,\; B}\cdot L are the surviving event numbers of signals and backgrounds, respectively. Both{\cal S} and{\cal 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{\not E}_T cut, which only survives about1.26 % (0.07%) at the LHeC (FCC-ep).Process Signal (ϕ) Signal (χ) RB IB {\cal S}_\phi {\cal S}_\chi {\cal R}_\phi {\cal R}_\chi 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 {\not E}_T > 100 GeV1.71 2.62 18.28 38.33 10.16 15.58 0.030 0.046 M(e^-,j)> 100 GeV1.56 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 Table 1. Remaining cross-section (in units of fb) of the signal and background after corresponding cuts with
m_{\chi, \phi}=1 GeV,C_{B, W}=1 , and\Lambda=500 GeV at the 140 GeV\otimes 7 TeV LHeC. The corresponding significance{\cal S}=N_S/\sqrt{N_B} and signal-to-background ratio{\cal R}=N_S/N_B were obtained forL=2 ~{\rm ab}^{-1} , whereN_S andN_B are the numbers of signal and background events, respectively.Process Signal (ϕ) Signal (χ) RB IB {\cal S}_\phi {\cal S}_\chi {\cal R}_\phi {\cal R}_\chi 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 {\not E}_T > 220 GeV3.10 13.44 0.86 4.32 60.88 264.17 0.060 2.60 M(e^-,j)> 200 GeV2.59 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 Table 2. Same as Table 1 but at the 60 GeV
\otimes 50 TeV FCC-ep.At the LHeC, after all cuts, the fermion (scalar) DM signal can survive
13.2 % (14.9 %), whereas the RB (IB) only survives0.37 % (3.5 %). At the FCC-ep, the signal for fermion (scalar) DM survived13.3 % (14.6 %), and the RB (IB) ultimately survives0.047 % (0.65 %). Note that the remaining cross-section of the background at the FCC-ep is only14 % of ones that at the LHeC, whereas the signal for fermion (scalar) DM production at the FCC-ep can reach6 (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 processe^+ e^- \to V^{\prime*} \to \phi\phi^*(\bar \chi \chi) + V\ (V=\gamma,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\sqrt{s}=240\ \mathrm{GeV} with a total luminosity of\sim 5.6\ \mathrm{ab}^{-1} , the Z factory mode for a two year run at\sqrt{s}=91.2\ \mathrm{GeV} with a total luminosity of\sim 16\ \mathrm{ab}^{-1} , and theW^+W^- threshold scan mode for a one year run at\sqrt{s} \sim 158-172\ \mathrm{GeV} with a total luminosity of\sim 2.6\ \mathrm{ab}^{-1} .1 Figure 4. Feynman diagrams for the process
e^+ e^- \to \phi\phi^*(\bar \chi \chi) + Z ore^+ e^- \to \phi\phi^*(\bar \chi \chi) + \gamma .For the monophoton signature at the CEPC, an irreducible background results from the neutrino production
e^+e^-\to\nu_l\bar\nu_l\gamma processes, where\nu_l=\nu_e,\nu_\mu,\nu_\tau . 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 atE_\gamma^Z=\frac{s-M_Z^2}{2\sqrt{s}}
(18) with a full-width-at-half-maximum of
\Gamma_\gamma^Z=M_Z\Gamma_Z/\sqrt{s} . To suppress the irreducible background contribution, we veto the events within5\Gamma_\gamma^Z at the Z resonance in the monophoton energy spectrum [27]. The vetoing cut can be expressed as\begin{array}{*{20}{l}} |E_\gamma-E_\gamma^Z|<5\Gamma_\gamma^Z. \end{array}
(19) The main reducible SM backgrounds result from the
e^+e^-\to\gamma+{\not X} processes, where only one photon is visible in the final state, and{\not X} denotes the other undetectable particle(s) owing to the limited detection capability of the detectors. In our analysis, the parameters for the EMC coverage,|\cos\theta| <0.99 andE>0.1 GeV, are adopted based on CEPC CDR [50]. The processese^+e^-\to f\bar f \gamma ande^+e^-\to\gamma\gamma\gamma provide dominant contributions to the reducible background when the finalf\bar f and\gamma\gamma are emitted with|\cos\theta| >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]E_\gamma^m(\theta_\gamma) = \sqrt{s}\left(\frac{\sin\theta_\gamma}{\sin\theta_b}\right)^{-1},
(20) where the polar angle
\theta_b corresponds to the boundary of the EMC with|\cos\theta| <0.99 . To suppress the monophoton events resulting from the reducible background, we adopt the detector cutE_\gamma>E_\gamma^m(\theta_\gamma) on the final state photon in our analysis. -
Through the DM-diboson operators, a WIMP can be produced via the processes
q q^\prime \to V^{\prime*} \to \phi\phi^*(\bar \chi \chi) + V (V=\gamma,Z,W) , which can exhibit a mono-photon ({{\not E}_T+\gamma)} or mono-jet ({{\not E}_T+W/Z\ (\to {\rm hardons})} ) signature, respectively. The representative Feynman diagrams are shown in Fig. 5. Moreover, the loop-induced effective DM-quarks ({\cal O}_q ) and DM-gluons ({\cal O}_G ) operators can result in the mono-jet signature via theq \bar q \to \bar\chi \chi +g ,gg \to \bar\chi \chi +g , andq g \to \bar\chi \chi +q processes. The representative Feynman diagrams are shown in Fig. 6.Figure 5. Representative Feynman diagrams for mono-photon (left) and mono-jet (right) signatures resulting from DM-diboson operators.
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 fb
^{-1} at a center-of-mass energy of 13 TeV.For the
{{\not E}_T+\gamma} search channel, events in our analysis are required to have a leading γ withE_{{T}}^{\gamma}>150 \mathrm{GeV}, \ |\eta^\gamma|<1.37 \text { or } 1.52<|\eta^\gamma|<2.37,\ \Delta \phi\left(\gamma,\ \boldsymbol{E}_{{T}}^{\text {miss }}\right)>0.4, 0 or 1 jet withp_{{T}}>30 \mathrm{GeV},\ |\eta|<4.5 \text { and } \Delta \phi\left(\mathrm{jet},\ \boldsymbol{E}_{{T}}^{\mathrm{miss}}\right)>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{E}_{{T}}^{\mathrm{miss}} . We observe that the strongest bounds on the parameter space in our case are given with the most severe cut of{E}_{{T}}^{\mathrm{miss}}> 375 GeV. The corresponding 95% C.L. limit on the fiducial cross-section is\sigma_{\rm fid}< 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
p_T>150 GeV and|\eta|<2.4 , and up to three additional jets withp_T>30 GeV and|\eta|<2.8 . Separation in the azimuthal angle between the missing transverse momentum direction and each selected jet\Delta \phi\left(\mathrm{jet},\ \boldsymbol{E}_{{T}}^{\mathrm{miss}}\right)>0.4 (0.6) is required for events with{E}_{\mathrm{T}}^{\mathrm{miss}}>250 GeV (200\; \mathrm{GeV} < {E}_{{T}}^{\mathrm{miss}} < 250 GeV). We observe that the stronger restriction of{E}_{{T}}^{\mathrm{miss}} > 1200 GeV provides the best limits. At 95% C.L., the bound on the corresponding fiducial cross section is given by\sigma_{\rm fid}< 0.3 fb.Table 3 presents the ratio of the number of events for mono-jet signature at LHC from the process
{pp\to \chi\bar\chi+W/Z\ (\to {\rm hardons})} via dimension-7 DM-diboson operators with the Feynman diagrams shown in Fig. 5 to one from the process{pp\to \chi\bar\chi+ 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.,C_{B}=C_{W}=500 ,C_{B}=0, \; C_{W}=500 , andC_{B}=500,\; C_{W}=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 forC_{B}=1, C_{W}=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.m_\chi 1 300 800 2000 3000 C_B=500,C_W=0 638.22 699.35 752.11 1435.32 2733.95 C_B=0,C_W=500 95.00 99.55 104.29 181.55 302.49 C_B=500,C_W=500 79.20 80.33 91.64 158.94 271.41 Table 3. Ratio of the number of events for the mono-jet signature at the LHC from the process
{pp\to \chi\bar\chi+W/Z\ (\to {\rm hardons})} via DM-diboson operators with the Feynman diagrams shown in Fig. 5 to{pp\to \chi\bar\chi+ q/g} via loop-induced DM-quarks and DM-gluons operators with the Feynman diagrams shown in Fig. 6 at Λ=1000 GeV. -
In this section, we discuss the bounds on Λ from current direct and indirect search experiments. The effective operators
{\cal O}_B and{\cal O}_W lead to the non-relativistic cross-sections of DM annihilation\chi\bar\chi,\ \phi\phi^*\to\gamma\gamma/\gamma Z/ZZ/WW as follows:\begin{array}{*{20}{l}} \langle\sigma_k v\rangle = a_k + b_k v^2+ {\cal O}(v^4) \end{array}
(21) with
k=B,W annihilation channels. The coefficients of a and b are [37]\begin{array}{*{20}{l}} \chi\left\{\begin{aligned} a_B =&a_W=0,\\ b_B =&\frac{\left|C_{B}(\Lambda)\right|^{2}}{\pi} \frac{m_{\chi}^{4}}{\Lambda^{6}}\Big[c_{w}^{4}+\frac{c_{w}^{2} s_{w}^{2}}{8} \beta_{\gamma Z}^{2}\left(x_{Z}-4\right)^{2}\Theta(M_Z<2m_\chi) +\frac{s_{w}^{4}}{8} \beta_{Z Z}\left(3 x_{Z}^{2}-8 x_{Z}+8\right)\Theta(M_Z<m_\chi)\Big], \\ b_W =&\frac{\left|C_{W}(\Lambda)\right|^{2}}{\pi} \frac{m_{\chi}^{4}}{\Lambda^{6}}\Big[s_{w}^{4}+\frac{c_{w}^{2} s_{w}^{2}}{8} \beta_{\gamma Z}^{2}\left(x_{Z}-4\right)^{2}\Theta(M_Z<2m_\chi) +\frac{c_{w}^{4}}{8} \beta_{Z Z}\left(3 x_{Z}^{2}-8 x_{Z}+8\right)\Theta(M_Z<m_\chi)\\ &+\frac{1}{4} \beta_{W W}\left(3 x_{W}^{2}-8 x_{W}+8\right)\Theta(M_W<m_\chi)\Big], \end{aligned}\right. \end{array}
(22) \begin{array}{*{20}{l}} \phi\left\{\begin{aligned} a_B =& \frac{\left|C_{B}(\Lambda)\right|^{2}}{\pi} \frac{2m_{\phi}^{2}}{\Lambda^{4}}\Big[c_{w}^{4} +\frac{c_{w}^{2} s_{w}^{2}}{8} \beta_{\gamma Z}^{2}\left(x_{Z}-4\right)^{2}\Theta(M_Z<2m_\phi)+\frac{s_{w}^{4}}{8} \beta_{Z Z}\left(3 x_{Z}^{2}-8 x_{Z}+8\right)\Theta(M_Z<m_\phi)\Big],\\ a_W =& \frac{\left|C_{W}(\Lambda)\right|^{2}}{\pi} \frac{2m_{\phi}^{2}}{\Lambda^{4}}\Big[s_{w}^{4} +\frac{c_{w}^{2} s_{w}^{2}}{8} \beta_{\gamma Z}^{2}\left(x_{Z}-4\right)^{2}\Theta(M_Z<2m_\phi)+\frac{c_{w}^{4}}{8} \beta_{Z Z}\left(3 x_{Z}^{2}-8 x_{Z}+8\right)\Theta(M_Z<m_\phi) \\ &+\frac{1}{4} \beta_{W W}\left(3 x_{W}^{2}-8 x_{W}+8\right)\Theta(M_W<m_\phi)\Big], \\ b_{B} =&\frac{\left|C_{B}(\Lambda)\right|^{2}}{\pi} \frac{2m_{\phi}^{2}}{\Lambda^{4}}\Big[c_{w}^{4}/2 +\frac{c_{w}^{2} s_{w}^{2}}{4} \beta_{\gamma Z}^{2}\left(x_{Z}-4\right)\Theta(M_Z<2m_\phi)+s_{w}^{4} \beta_{Z Z}\left(1/2-x_Z\right)\Theta(M_Z<m_\phi)\Big], \\ b_{W} =&\frac{\left|C_{W}(\Lambda)\right|^{2}}{\pi} \frac{2m_{\phi}^{2}}{\Lambda^{4}}\Big[s_{w}^{4}/2 +\frac{c_{w}^{2} s_{w}^{2}}{4} \beta_{\gamma Z}^{2}\left(x_{Z}-4\right)\Theta(M_Z<2m_\phi)+c_{w}^{4} \beta_{Z Z}\left(1/2-x_Z\right)\Theta(M_Z<m_\phi)\\ &+\beta_{W W}\left(1-2x_Z\right)\Theta(M_W<m_\phi)\Big]. \end{aligned}\right. \end{array}
(23) In above equations, we have defined the phase space factor
\beta_{i j}=\sqrt{1-\left(m_{i}+m_{j}\right)^2/\left(4 m_{\rm DM}^2\right)} andx_i=m_i^2/m_{\rm DM}^2 with{\rm DM}=\chi,\phi . Using the results fora_{B,W} andb_{B,W} , the relic density can then be calculated using\Omega_{\rm DM} h^{2}=\frac{1.09 \times 10^{9} x_{F} \mathrm{GeV}^{-1}}{M_{P l} \sqrt{g_{*}\left(x_{F}\right)}\left(a+3 b / x_{F}\right)},
(24) where
a=a_B+a_W andb=b_B+b_W .x_F=m_{\rm DM}/T_F is the ratio of the mass of DMm_{\rm DM} and the early-universe freeze-out temperatureT_F , which can be obtained by solvingx_F=\ln\left[c(c+2)\sqrt{\frac{45}{8}}\frac{m_{\rm DM}M_{Pl}(a+6b/x_F)}{2\pi^3\sqrt{g_*(x_F)}}\right].
(25) The loop-induced effective DM-quarks (
{\cal O}_q ) and DM-gluons ({\cal O}_G ) 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\begin{aligned}[b] \sigma_{\rm SI}^{\mathrm{\phi}} \simeq& \frac{\mu_N^{2} m_{N}^{2}}{\pi m_\phi^2 \Lambda^{4}}\Bigg|\frac{\alpha Z^{2}}{A} f_{A}^{N} C_{AA}\left(\mu_{\rm had}\right)\\&+\frac{Z}{A} f_{p}\left(\mu_{\rm had}\right)+\frac{A-Z}{A} f_{n}\left(\mu_{\rm had}\right)\Bigg|^{2}\,, \end{aligned}
(26) \begin{aligned}[b] \sigma_{\rm SI}^{\mathrm{\chi}} \simeq& \frac{\mu_N^{2} m_{N}^{2}}{\pi \Lambda^{6}}\Bigg|\frac{\alpha Z^{2}}{A} f_{A}^{N} C_{AA}\left(\mu_{\rm had}\right)\\&+\frac{Z}{A} f_{p}\left(\mu_{\rm had}\right)+\frac{A-Z}{A} f_{n}\left(\mu_{\rm had}\right)\Bigg|^{2}\,. \end{aligned}
(27) In above equations,
m_N\simeq 0.939 GeV is the average nucleon mass, and\mu_N=m_{\phi,\chi}m_N/(m_{\phi,\chi}+m_N) is the reduced mass of the DM-nucleon system. The form factorf_{A}^{N} describes the Rayleigh scattering of two photons on the entire nucleus. At a zero-momentum transfer limit,f_{A}^{N}\simeq0.08 for a xenon target. The form factorf_{N=p,n} describing the couplings between the DM and nucleon is expressed as [40, 53]f_{N}\left(\mu_{\rm had}\right)=\sum\limits_{q=u, d, s} f_{q}^{N} C_{q}\left(\mu_{\rm had}\right)-\frac{8 \pi}{9} f_{G}^{N} C_{G}\left(\mu_{\rm had}\right),
(28) where the form factors
f_{q}^{N} describing the scalar couplings between quarks and nucleon are given by [54]\begin{aligned}[b]& f_{d}^{p}=0.0191, \;\;\ f_{u}^{p}=0.0153, \;\;\ f_{d}^{n}=0.0273,\\& f_{u}^{n}=0.0110,\ \;\; f_{s}^{p}=f_{s}^{n}=0.0447, \end{aligned}
(29) and
f_{G}^{N}=1-\sum\limits_{q=u, d, s} f_{q}^{N}.
(30)
Exploring dark matter-gauge boson effective interactions at current and future colliders
- Received Date: 2023-09-07
- Available Online: 2024-06-15
Abstract: In this study, we systematically investigate collider constraints on effective interactions between Dark Matter (DM) particles and electroweak gauge bosons. We consider the simplified models in which scalar or Dirac fermion DM candidates couple only to electroweak gauge bosons through high dimensional effective operators. Considering the induced DM-quarks and DM-gluons operators from the Renormalization Group Evolution (RGE) running effect, we present comprehensive constraints on the effective energy scale Λ and Wilson coefficients