Interaction operators for fermionic WIMP of 6-dimensional Effective Field Theory in tree level, which contains four-fermion interactions and vector, scalar, fermion interactions, are implemented.

For 6-dimensional EFT, fermionic DM is a SM gauge singlet and odd under global Z2 symmetry. The tree level interactions between this fermionic DM and SM particles can be chosen as the Lagrangian form of the operators given [1,2,3]

\begin{itemize} \item 4-fermion vectoral interactions: \begin{eqnarray}

\mathcal{L}_{(uR,dR,eR)\chi}=\frac{g^{u_{R}}{2 \Lambda}2}(\bar{u}\gamma^{{\mu}u)(\bar{\chi}\gamma_{\mu}\chi) +\frac{g}d_{R}}{2 \Lambda^{2}(\bar{d}\gamma}{\mu}d)(\bar{\chi}\gamma_{\mu}\chi) + \frac{g^{e_{R}}{2\Lambda}2}(\bar{e}\gamma^{{\mu}e)(\bar{\chi}\gamma_{\mu}\chi) \label{eq-lag-4ferm-vekt} \end{eqnarray}}

\item 4-fermion scalar interactions: \begin{eqnarray}

\mathcal{L}_{(\ell,q)\chi}=\frac{g^{{\ell}_L}{\Lambda}2}(\bar{\ell}\chi)(\bar{\chi}\ell) + \frac{g^q_L}{\Lambda2} (\bar{q}\chi)(\bar{\chi}q) \label{eq-lag-4ferm-sca} \end{eqnarray}

\item Fermion-vector-scalar interactions: \begin{eqnarray}

\mathcal{L}_{\phi\chi}=\frac{i \alpha_{\phi \chi}}{\Lambda^2}({\phi}{\dagger}D^{{\mu}\phi)(\bar{\chi}\gamma_{\mu}\chi)+h.c. \label{eq-lag-ferm-vec-sca} \end{eqnarray} \end{itemize}}

where, $\chi$ is fermionic DM field, $u,d,e$'s are right-handed fermions, $\gamma$'s are gamma matrices, $q,\ell$ denotes left-handed quarks and leptons, $\phi$ is Higgs field $\Lambda$ is cut-off scale of new physics, $g_{R}^{{u(d,e)}$ and $g_L}{\ell(q)}$'s are the coupling parameters related to dark operators $\alpha$'s. The apparent relation between $g$'s and $\alpha$'s are given as: \begin{eqnarray} g^{{u}_{L}=-\frac{1}{2}\alpha_{q\chi}, \qquad g}{u}_{R}=\frac{1}{2}\alpha_{u\chi} \nonumber \end{eqnarray} \begin{eqnarray} g^{{d}_{L}=-\frac{1}{2}\alpha_{q \chi}, \qquad g}{d}_{R}=\frac{1}{2}\alpha_{d\chi} \nonumber \end{eqnarray} \begin{eqnarray} g^{{e}_{L}=-\frac{1}{2}\alpha_{\ell \chi},\qquad g}{e}_{R}=\frac{1}{2}\alpha_{e\chi} \nonumber \end{eqnarray} \begin{eqnarray} g^{{\nu}_{L}=-\frac{1}{2}\alpha_{\ell \chi}, \qquad g}{\nu}_{R}=0 \nonumber \end{eqnarray}

[1] Ayşe Elçiboğa KUDAY, Erdinç Ulaş SAKA, Ferhat ÖZOK. June 2022. Analysis of Direct and Indirect Detection of Fermionic Dark Matter of 6-Dimensional Effective Field Theory. International Journal of Geometric Methods in Modern Physics, Vol. 19, No. 13 (2022) 2250202,​2305.02302 [hep_ph. [2] Ayşe Elçiboğa KUDAY, Erdinç Ulaş SAKA, Ferhat ÖZOK. 2020. Probing Dark Matter via Effective Field Theory Approach. International Journal of Geometric Methods in Modern Physics, Vol. 17, No. 2 (2020) 2050028, ​2305.02592 [hep_ph]. [3] Zhang, H., Cao, QH., Chen, CR. et al. Effective dark matter model: relic density, CDMS II, Fermi LAT and LHC. J. High Energ. Phys. (2011) 2011: 18 ​0912.4511v2 [hep_ph]