Lepton flavor of four-fermion operator and fermion portal dark matter

While the Standard Model (SM) of particle physics has achieved striking success, its lack of explanation of fundamental phenomena, such as the origins of neutrino masses and fermion flavor structure, and providing particle candidates for dark matter (DM) underscores the need for physics beyond the SM (BSM). In light of null results in searches for new resonances at the Large Hadron Collider (LHC), the effective field theory (EFT) framework offers a powerful way to explore new physics Isidori:2023pyp .

In the EFT approach, BSM interactions are systematically parameterized as a series of higher-dimensional operators with the corresponding Wilson coefficients suppressed with inverse powers of the new physics scale $\Lambda$. At the lowest order with mass dimension 5, only one operator is present that gives masses to neutrinos Weinberg:1979sa , while exploding numbers of effective operators emerge at higher orders Buchmuller:1985jz ; Grzadkowski:2010es ; Henning:2015alf .

There has been growing interest in dimension-6 four-fermion operators, as being utilized to fit precision data and address anomalies observed at the LHC and in low-energy experiments Falkowski:2017pss ; Cirigliano:2023nol ; Fernandez-Martinez:2024bxg ; Karmakar:2024gla , and intriguing connections with DM models Cepedello:2023yao . Focusing on the semileptonic operators with two leptons and two quarks, the one-by-one constraints on the lepton-flavor-conserving (LFC) are rather stringent, and global fits are essential to mitigate or even eliminate the tensions in different observables Cirigliano:2023nol . On the other hand, for the lepton-flavor-changing (LFV) operators, possible tensions are milder since less experimental data is available Fernandez-Martinez:2024bxg .

Consider the dimension-6 four-fermion operator Grzadkowski:2010es

where $L=(\nu_{L},e_{L})^{T}$ and $Q=(u_{L},d_{L})^{T}$ denote the left-handed $SU(2)_{L}$ doublets of leptons and quarks, respectively. $j$ is the isospin index, and $\alpha$, $\beta$, $s$, $t$ are flavor indices. This operator is forbidden under the $U(3)^{5}$ flavor symmetry, and highly suppressed within minimal flavor violation hypothesis DAmbrosio:2002vsn ; Isidori:2023pyp , rendering it extraordinarily sensitive probe of BSM physics. We will consider the first-generation quarks, $s=t=1$, and the lepton flavor indices $\alpha,\beta=1,2$.

In general, the stringent constraints on the Wilson coefficients of four-fermion operators challenge direct searches for ultraviolet (UV) physics. However, one-loop UV completions of the four-fermion operators with the participation of DM particles in the box diagram can notably alleviate the restrictions on the mass scale $\Lambda$. In the class of DM models Cepedello:2023yao , the $Z_{2}$ symmetry that stabilizes the DM prohibits tree-level contributions to the Wilson coefficients of four-fermion operators. Consequently, upon integrating out heavy new particles, the Wilson coefficients are suppressed by a factor of $1/(16\pi^{2})$. Moreover, the Wilson coefficients are proportional to $f_{\rm NP}^{4}$, where $f_{\rm NP}$ denotes new physics couplings. This dependency has the potential to further lower the new physics scale for small values of $f_{\rm NP}$. Previous studies of connections between cLFV observables and DM were conducted in Refs. Herrero-Garcia:2018koq ; Toma:2013zsa ; Vicente:2014wga .

In this work, we will investigate the complementarities of indirect constraints on the Wilson coefficients of the four-fermion operator $O_{ledq}^{\alpha\beta 11}$ in both LFC and LFV scenarios from the measurements of neutrino NSI and cLFV searches for $\mu-e$ conversion, and DM relic density, direct detection, and collider searches in the context of fermion portal DM model Bai:2013iqa12 ; DiFranzo:2013vra ; An:2013xka ; Bai:2014osa with two mediators and Majorana fermionic DM. We highlight that

• •

  In the LFC scenario, the anticipated future sensitivity to the neutrino NSI in next-generation neutrino oscillation experiments could play a nontrivial role in investigating new physics, while ensuring that tensions among various observables are eliminated.

• •

  In the LFV scenario, the scattering of Majorana DM with nuclei can appear at tree level, and $\mu-e$ conversion arises solely from the contribution of the box diagram at the one-loop level, with no other cLFV processes occurring at this order.

• •

  In both scenarios, the model-independent constraints on the Wilson coefficients of the four-fermion operator offer a distinctive probe of the fermion portal DM model, particularly in the parameter space where the masses of the DM and scalar are either close or large, depending on the requirement of DM relic density.

The remainder of the paper is organized as follows. In Sec. 2, the fermion portal DM model is studied in detail. In Sec. 3, the Wilson coefficients of the four-fermion operator $O_{ledq}^{\alpha\beta 11}$ are calculated. In subsequent sections, the sensitivities in DM relic density, direct detection, and collider searches are investigated. In Sec. 7, the combined results are discussed in benchmark scenarios of masses and coupling with the relic density of $\chi$ satisfying $\Omega_{\chi}h^{2}\leq 0.1199$ or $\Omega_{\chi}h^{2}=0.1199$. We conclude in Sec. 8. More details are provided in Appendix A and Appendix B about the anomalous magnetic moment for leptons and DM direct detection with one-loop exchange of photon, respectively.

We consider a UV completion model that induces the four-fermion operator $O_{ledq}^{\alpha\beta 11}$ , incorporating Majorana fermionic DM $\chi$, which remains a more viable DM candidate compared to Dirac fermionic DM. In addition to the DM particle $\chi$, we introduce a fermion doublet $F=(F^{+},F^{0})^{T}$, scalar $S$, and colored mediator $\phi_{d}$, all of which are $Z_{2}$ odd. The quantum numbers of these fields under $\rm SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}\times Z_{2}$ are presented in Table 1, and are documented in the systematic classification of DM models for four-fermion operators Cepedello:2023yao .

The Lagrangian for the UV model of the four-fermion operator is given by

where $f_{LS}$, $f_{\chi S}$, $f_{FQ}$, and $f_{d\chi}$ are coupling constants. The flavor indices are omitted, and the couplings are assumed to be real and positive for simplicity.

In the presence of the dark mediators $S$ and $\phi_{d}$ in this fermion portal DM model Bai:2013iqa12 ; An:2013xka ; DiFranzo:2013vra ; Bai:2014osa , the DM candidate $\chi$ interacts with SM particles, leading to distinct observables in direct detection, indirect detection, and collider experiments. In the subsequent sections, we will analyze each type of signature separately.

Given the interactions in Eq. (2), the one-loop diagram in Fig. 1 can be generated. We refer to this type of box diagram as “dark loop”, which was proposed to explain the flavor anomalies in $B$ physics Huang:2020ris ; Capucha:2022kwo ; Cepedello:2022xgb .

The Wilson coefficient of the effective operator $O_{ledq}^{\alpha\beta 11}$ is calculated using Package-X Patel:2015tea ; Patel:2016fam , which is expressed as follows

where the effective coupling $f_{\rm NP}$ and the loop function $I(x,y,z,w)$ are defined as

This result agrees with Ref. Bischer:2018zbd by taking $m_{\chi}=m_{F}=0$.

The Wilson coefficient is constrained by searches in low-energy experiments and at colliers depending on the lepton flavor of the four-fermion operator. According to the global analysis in Ref Cirigliano:2023nol , the allowed value of the Wilson coefficient for the LFC operator $O_{ledq}^{2211}$ is $C_{ledq}^{2211}/\Lambda^{2}=\left(0.017\pm 0.039\right)\leavevmode\nobreak\ \text{TeV}^{-2}$, which is consistent with zero within $1\sigma$. Refs. Du:2020dwr ; Du:2021rdg have investigated the prospects of next-generation neutrino oscillation experiments in the search for neutrino non-standard interactions (NSIs) Proceedings:2019qno , which are described by certain SMEFT operators at the electroweak scale. These studies indicate the highest sensitivity for the operator $O_{ledq}^{2211}$, $\mid C_{ledq}^{2211}\mid/\Lambda^{2}<\left(12.3\leavevmode\nobreak\ \text{TeV}\right)^{-2}$ Du:2021rdg . This enhanced sensitivity is attributed to the significant increase in neutrino production from pion decay, particularly in the presence of charged-current (CC) neutrino NSIs induced by $O_{ledq}^{2211}$ at lower energies. Note that this operator does not interfere with the CC neutrino interactions in the SM, so the experimental measurements are blind to the sign of the Wilson coefficient. Henceforth, we will consider the magnitude of the Wilson coefficient $C_{ledq}^{2211}/\Lambda^{2}$ with the absolute value symbol being omitted.

We illustrate the constraints on $f_{\mathrm{NP}}$ and $m_{\chi}$ from future LFC searches in the left panel of Fig. 2 with two benchmark values of the scalar mass $m_{S}=200\leavevmode\nobreak\ \text{GeV}$ (red region) and $500\leavevmode\nobreak\ \text{GeV}$ (blue region) for $m_{\phi}=m_{F}=2\leavevmode\nobreak\ \text{TeV}$ (solid curve), or $m_{\phi}=2.5\leavevmode\nobreak\ \text{TeV}$, $m_{F}=3\leavevmode\nobreak\ \text{TeV}$ (dashed curves). We find that this constraint does not strongly depend on $m_{\chi}$, but it is highly sensitive to $f_{\mathrm{NP}}$, which is expected to be in the range of approximately 1.7 to 2.1. It is important to note that there are currently no constraints from existing LFC searches, as the results are consistent with the SM predictions.

In the case of the LFV operator, the current constraint on the Wilson coefficient $C_{ledq}^{1211}/\Lambda^{2}<\left(2.2\times 10^{3}\leavevmode\nobreak\ \text{TeV}\right)^{-2}$ Fernandez-Martinez:2024bxg is derived from charged-lepton-flavor-violation (cLFV) searches for $\mu-e$ conversion in nuclei Badertscher:1981ay ; SINDRUMII:1993gxf ; SINDRUMII:1996fti ; SINDRUMII:2006dvw . The projected limit $C_{ledq}^{1211}/\Lambda^{2}<\left(2.9\times 10^{4}\leavevmode\nobreak\ \text{TeV}\right)^{-2}$ Haxton:2024lyc in the upcoming Mu2e Mu2e:2014fns ; Bernstein:2019fyh and COMET COMET:2018auw ; COMET:2018wbw experiments, representing an improvement by approximately two orders of magnitude. The cLFV searches are also insensitive to the sign of the Wilson coefficient of the operator $O_{ledq}^{1211}$, and we omit the absolute value symbol for convenience.

In the right panel of Fig. 2, we present the contours of LFV limits as functions of $f_{\rm NP}$ and $m_{\chi}$, by choosing the benchmark values of $m_{S}$, $m_{F}$, and $m_{\phi}$ as the left panel of Fig. 2. Unlike the LFC scenario, constraints from both the current and future experimental efforts are depicted. The current limit is shown on the lower axis, while the anticipated future limit is displayed on the upper axis. The current limit for $f_{\rm NP}$ will be around 0.13 to 0.15, with expectations that the future limit could improve by an order of magnitude, potentially reaching as low as 0.04. It is important to note that despite the stringent constraint on the LFV interaction, the BSM particles could potentially exist at the TeV scale or even lower, owing to the suppression of loop factor $\sim 1/(16\pi^{2})$ and the coupling dependence $\sim f_{\rm NP}^{4}$.

Besides the effective interaction expressed as the four-fermion operator $\mathcal{O}_{ledq}^{\alpha\beta 11}$, the dark particles can also contribute to the anomalous magnetic momentum of leptons. To address the discrepancies between the experimental measurements and the SM predictions, improved SM calculations or other new physics contributions are needed; see Appendix A for details.

Given the Lagrangian in Eq. (2), the Majorana DM pair $\chi$ can annihilate to muon pair and $d$ quark pair, mediated by $S$ and $\phi_{d}$, respectively. In our setup, where $m_{\phi}\gg m_{S}$ and the ratio $f_{d\chi}/f_{\chi S}\sim\mathcal{O}(1)$, the contribution from the process mediated by $\phi_{d}$, is negligible. Tree-level Feynman diagram for the annihilation $\chi\bar{\chi}\to\mu^{+}\mu^{-}$ is depicted in Fig. 3. The corresponding thermal-averaged cross section is Liu:2021mhn

where $x\equiv m_{\chi}^{2}/m_{S}^{2}$, and $v$ is the relative velocity of two DM particles, which is typically around $0.3c$ at the freeze-out temperature. Since $m_{\mu}^{2}/m_{S}^{2}\ll v^{2}$, the $p$-wave contribution to the thermal-averaged annihilation cross section is dominant Bai:2013iqa12 ; Bai:2014osa ; Liu:2021mhn .

In Fig. 4, we present the DM relic density contours for $m_{S}$ and $m_{\chi}$ with different choices of $f_{\chi S}$, considering the relic density $\Omega_{\chi}h^{2}=0.1199\pm 0.0022$ Planck:2018vyg . Regions below the contour curves indicate overabundance.

The Majorana DM $\chi$ can interact with the SM quark through the $f_{d\chi}$ term in Eq. (2), giving rise to the tree-level scattering process depicted in Fig. 5.

By using the Fierz identity Bai:2013iqa

and the relation for Majorana fermion

we can obtain the following effective Lagrangian Jungman:1995df ; Hisano:2010ct ; Mohan:2019zrk ; Arcadi:2023imv :

where the effective operators are

Here, the notation of the operator $\mathcal{O}_{D}$ follows Refs. Hisano:2010ct ; Hill:2014yka ,

The Wilson coefficients are given by

Based on whether the DM-nucleus interactions depend on the spin of nucleus or not, the DM-nucleus scattering is classified as spin-dependent (SD) and spin-independent (SI) scatterings, respectively. The latter is typically proportional to $A^{2}$, where $A$ is the mass number of nucleus. Table 2 summarizes the suppression factors of DM-nucleon cross sections for the above operators. Here, $\vec{v}$ is the DM-nucleon relative velocity, $\vec{q}$ denotes the momentum transfer, and the transverse relative velocity is defined as $\vec{v}^{\perp}\equiv\vec{v}-\vec{q}/(2\mu_{N})$, where $\mu_{N}\equiv m_{\chi}m_{N}/(m_{\chi}+m_{N})$ denotes the DM-nucleon reduced mass.

The DM-nucleon SI and SD cross sections for the operators $\mathcal{O}_{V}$ and $\mathcal{O}_{A}$ depend on the kinematic suppression factors, which agree with the earlier studies Bai:2013iqa12 ; DiFranzo:2013vra ; An:2013xka ; Mohan:2019zrk . On the contrary, the SI cross sections for the operators $\mathcal{O}_{S}$ and $\mathcal{O}_{D}$ are not kinematically suppressed, but suppressed by $m_{\chi}^{2}m_{N}^{2}/m_{\phi}^{4}$ Jungman:1995df ; Hisano:2010ct ; Mohan:2019zrk ; Arcadi:2023imv . This is because $\mathcal{O}_{S}$ and $\mathcal{O}_{D}$ are obtained by expanding the propagator of $\phi_{d}$ at next-to-leading order. Their contributions to the SD cross section are not displayed as they are further suppressed by the DM velocity and momentum transfer Kumar:2013iva .

We first consider the SI contributions from these operators. From Eq. (5), the Wilson coefficients of the operators $\mathcal{O}_{V}$ and $\mathcal{O}_{A}$ are the same, so we readily obtain that the contribution of $\mathcal{O}_{V}$, which is suppressed by $v^{\perp 2}\sim 10^{-6}$, is larger that that of $\mathcal{O}_{A}$. To derive the exclusion limit for the operator $\mathcal{O}_{V}$, we evaluate the non-relativistic events generated in the XENON1T experiment XENON:2018voc with an exposure of $w=1.0\ \text{ton-year}$ using DirectDM Bishara:2017nnn and DMFormFactor Anand:2013yka . The 90% confidence level (C.L.) constraint is determined under the assumption of 7 signal events after accounting for SM backgrounds Kang:2018rad ; Liang:2024tef . Our result agrees with the EFT analysis by the XENON Collaboration XENON:2022avm . The exclusion region for $m_{\chi}$ and $m_{\phi}$ is depicted in orange color in Fig. 6.

The suppression factor for the contributions to the SI cross section from the operators $\mathcal{O}_{S}$ and $\mathcal{O}_{D}$ is $m^{2}_{\chi}m^{2}_{N}/m_{\phi}^{4}\sim 10^{-6}\times(m_{\chi}/m_{\phi})^{2}$. Following Refs. Jungman:1995df ; Hisano:2010ct ; Mohan:2019zrk ; Arcadi:2023imv ; Hill:2014yka , we express the contributions to the SI cross section as a combination of the Wilson coefficients of $\mathcal{O}_{S}$ and $\mathcal{O}_{D}$. In Fig. 6, we present the exclusion of $m_{\chi}$ and $m_{\phi}$ using the upper limit on the DM-nucleon SI cross sections by the XENON1T experiment XENON:2018voc with a 1 ton-year exposure. We find that the contribution from $\mathcal{O}_{V}$ dominates over that from the combination of $\mathcal{O}_{S}$ and $\mathcal{O}_{D}$ within our parameter space.

On the other hand, the leading contribution to the SD cross section comes from the operaotr $\mathcal{O}_{A}$, as clearly shown in Table 2, where the contributions from $\mathcal{O}_{S}$ and $\mathcal{O}_{D}$ are highly suppressed and not presented. The SD DM-nucleon cross section for $\mathcal{O}_{A}$ is expressed as Bai:2013iqa12 ; DiFranzo:2013vra ; An:2013xka ; Mohan:2019zrk

where $\Delta_{d}^{n}=0.842\pm 0.012$ Belanger:2008sj and $\mu_{n}$ denotes the DM-neutron reduced mass. Here, we only consider the limit on the DM-neutron cross section, which is much stronger than that on the DM-proton cross section for the XENON1T XENON:2019rxp , PandaX-4T PandaX:2022xas , and LZ LZ:2022lsv experiments. Using Eq. (13) and the upper limit on the DM-nucleon SD cross section given by the XENON1T experiment XENON:2019rxp with a 1 ton-year exposure, we derive the exclusion of $m_{\chi}$ and $m_{\phi}$ for the operator $\mathcal{O}_{A}$, which is shown as the red region in Fig. 6. It is evident that in our case, the exclusion limit from DM direct detection is dominated by the SD interactions associated with the operator $\mathcal{O}_{A}$.

In Fig. 7, we show the exclusion limits on the DM mass $m_{\chi}$ and colored mediator mass $m_{\phi}$ from the constraints on the SD cross sections measured in the LUX-ZEPLIN (LZ) experiment LZ:2022lsv and the PandaX-4T experiment PandaX:2022xas assuming coupling values $f_{d\chi}=1$ or 2. Our result for $f_{d\chi}=1$ using the limit by the LZ experiment agrees with Ref. Arcadi:2023imv . It is important to mention that we assume the relic density of $\chi$ as the observed total DM relic density. If the DM relic density depends on a given $f_{\chi S}$, the limits could be notably weaker, as we will see in Sec. 7.

It is noted that direct detection can also occur by exchanging photons at the one-loop level, which leads to much weaker direct detection signals compared to the contribution from exchanging $\phi_{d}$ at the tree level; see Appendix B for details.

The DM $\chi$ can interact with SM particles via $F$, $S$ and $\phi_{d}$, resulting in detectable DM signatures at collider experiments. We focus on DM searches at the LHC, where the presence of missing energy is a typical signature indicating the possible existence of DM particles. Assuming the mass hierarchy $m_{F}\geq m_{\phi}>m_{S}>m_{\chi}$, and the decay branching ratios of $S$ and $\phi_{d}$ are determined as

The particle $F$ can decay to leptons or jet via the processes $F\to\phi_{d}j(S^{\pm}\ell^{\mp})$ or $F^{\pm}\to\phi_{d}j(S^{\pm}\nu)$, where $j$ represents any first-generation quark. The most relevant collider search for DM involves these channels: (1) leptons$+\not{E}_{T}$ and (2) jet(s)$+\not{E}_{T}$.

We have explored the phenomenological implications of the four-fermion operator $O_{ledq}^{\alpha\beta 11}$ in our UV model, including DM relic density, direct detection, and collider searches, as well as the model-independent LFC and LFV constraints on the Wilson coefficients. To analyze their complementarities, we examine four benchmark (BM) scenarios detailed in Table 3.

Based on the collider searches and DM direct detection discussed, we choose the values:

• •

  BM (a) or (b): $f_{d\chi}=2$, $m_{\phi}=2.5{\leavevmode\nobreak\ \rm TeV},m_{F}=3{\leavevmode\nobreak\ \rm TeV}$;

• •

  BM (c) or (d): $f_{d\chi}=1$, $m_{\phi}=m_{F}=2{\leavevmode\nobreak\ \rm TeV}$.

As discussed in Sec. 4, the DM relic density can be determined by the coupling $f_{\chi S}$ and the mass $m_{\chi}$, $m_{S}$. In BM (a) and BM (b), the coupling $f_{\chi S}$ is taken as an input parameter. Given the constraints on the LFC and LFV Wilson coefficients $C_{ledq}^{\alpha\beta 11}/\Lambda^{2}$, as illustrated in Fig. 2, the other couplings are set to be equal, such that their products yield $f_{\rm NP}=2.05$ and $0.1314$ (cf. Eq. (4)), respectively. In BM (b) and BM (d), $f_{\chi S}$ is fixed by the observed DM relic density for comparison.

In Figs. 16 and 17, we combine the sensitivities to the DM mass $m_{\chi}$ and scalar mass $m_{S}$ from DM relic density, direct detection, collider searches, and model-independent indirect constraints on the Wilson coefficients of the four-fermion operator $O_{ledq}^{\alpha\beta 11}$. For all benchmark scenarios in Table 3, a large portion of parameter space with $m_{\chi}\lesssim 200\leavevmode\nobreak\ \text{GeV}$ and $m_{S}\lesssim 450\leavevmode\nobreak\ \text{GeV}$ is excluded by the existing searches at the LEP and LHC.

For the LFC operator, we consider the projected future sensitivity to Wilson coefficient, $C_{ledq}^{2211}/\Lambda^{2}<(12.3\leavevmode\nobreak\ \text{TeV})^{-2}$ Du:2020dwr ; Du:2021rdg from the neutrino NSI measurements, given that the currently allowed value is consistent with zero as mentioned in Sec. 3. For the LFV operator, we utilize the current constraint, $C_{ledq}^{1211}/\Lambda^{2}<\left(2.2\times 10^{3}\leavevmode\nobreak\ \text{TeV}\right)^{-2}$ Fernandez-Martinez:2024bxg derived from the cLFV searches for $\mu-e$ conversion in nuclei, since our aim is to show that the new physics scale can be notably alleviated within the dark loop paradigm.

In the left panel of Fig. 16, we consider $f_{\chi S}=2$, and require the DM relic density $\Omega_{\chi}h^{2}\leq 0.1199$ to avoid an overabundance of DM, shown as the gray shaded region. Given the values of $f_{d\chi}$ and $m_{\phi}$, DM direct detection in the LZ experiment LZ:2022lsv rules out a narrow range of $m_{S}$ and $m_{\chi}$, as depicted in purple slash shading area. The sensitivity of DM direct detection in this case is rather weak, which is suppressed by the relic density of $\chi$ for a relatively large $f_{\chi S}$. The exclusion by the LFC constraint on $C_{ledq}^{2211}/\Lambda^{2}$ from the neutrino NSI measurements is shown in red color. It is evident that this model-independent constraint exhibits better sensitivity, probing parameter regions beyond the reach of collider searches, and DM direct detection, especially in scenarios where $m_{\chi}$ and $m_{S}$ are close.

In the right panel of Fig. 16, the coupling $f_{\chi S}$ is not independent but determined by the DM relic density $\Omega_{\chi}h^{2}=0.1199$. The coupling $f_{\chi S}$ decreases as $m_{S}$ becomes smaller. The purple slash shading band for $15\leavevmode\nobreak\ \text{GeV}<m_{\chi}<100\leavevmode\nobreak\ \text{GeV}$ is ruled out by DM direct detection in the LZ experiment. The region excluded by the model-independent LFC constraint is depicted in red, with the boundary corresponding to $f_{\chi S}\simeq 2$. One can see that the Yukawa coupling $f_{\chi S}\sim\mathcal{O}(1)$ can yield the correct relic density for the electroweak scale $m_{\chi}$ and $m_{S}$ Liu:2021mhn . In this scenario, the most effective constraints are provided by LHC searches and DM direct detection. For larger values of $m_{S}$ and $m_{\chi}$, however, the neutrino NSI measurements are more significant.

In Fig. 17, the exclusions are derived with the same experimental results as Fig. 16, except for the current model-independent constraint on the Wilson coefficient of the LFV four-fermion operator $C_{ledq}^{1211}/\Lambda^{2}<(2.2\times 10^{3}\leavevmode\nobreak\ \text{TeV})^{-2}$. As highlighted in Sec. 2, the model-independent constraint on the Wilson coefficient of LFV operator is more stringent and can probe smaller new physics couplings, compared to that on the Wilson coefficient of the LFC operator. For illustration, we consider $f_{d\chi}=1$, for which the colored mediator mass $m_{\phi}\gtrsim 1.76\leavevmode\nobreak\ \text{TeV}$ is allowed by the current LHC jet(s)$+\not{E}_{T}$ searches. Taking $m_{\phi}=2\leavevmode\nobreak\ \text{TeV}$, the DM direct detection in the LZ experiment LZ:2022lsv puts no constraints as shown in Fig. 7. In the case of BM (c) with $f_{\chi S}=1.5$, the prevailing cLFV constraint rules out most of the parameter space where $m_{\chi}\lesssim 300\leavevmode\nobreak\ \text{GeV}$ and $m_{S}\lesssim 450\leavevmode\nobreak\ \text{GeV}$, exceeding the sensitivity of the LHC searches, especially for the region where $m_{\chi}$ and $m_{S}$ are close. On the other hand, for BM (d) with the relic density $\Omega_{\chi}h^{2}=0.1199$ being fixed, the Yukawa coupling $f_{\chi S}\sim\mathcal{O}(1)$ can also yield the correct relic density for the electroweak scale $m_{\chi}$ and $m_{S}$. The LHC searches remain the most sensitive probes of the fermion portal DM model for the scalar mass $m_{S}\lesssim 420\leavevmode\nobreak\ \text{GeV}$. As $m_{S}$ increases or DM mass $m_{\chi}\gtrsim 200\leavevmode\nobreak\ \text{GeV}$, the cLFV searches exhibit better sensitivity than the LHC searches.

In this work, we have studied the UV completion of the four-fermion operator $O_{ledq}^{\alpha\beta 11}$ in both lepton-flavor-conserving (LFC) and lepton-flavor-violating (LFV) scenarios incorporating the Majorana dark matter (DM). Due to the $Z_{2}$ symmetry that stabilizes the DM, the four-fermion operator $O_{ledq}^{\alpha\beta 11}$ is firstly generated at one-loop level via box diagram, which could effectively mitigate the lower bounds on the new physics scale.

We investigated the interplay between the model-independent constraints on the Wilson coefficients of the four-fermion operator $O_{ledq}^{\alpha\beta 11}$ from the measurements of neutrino NSI in the next-generation neutrino oscillation experiments, charged-lepton-flavor-violation (cLFV) searches in $\mu-e$ conversion, as well as DM relic density, direct detection, and collider searches in the context of fermion portal DM model with two mediators. In order to illustrate the complementarities, we consider four benchmark scenarios as outlined in Table 3.

In the cases of BM (a) and BM (c), where the Yukawa coupling $f_{\chi S}$ is considered as an independent parameter, the model-independent constraints on the Wilson coefficients of $O_{ledq}^{2211}$ and $O_{ledq}^{1211}$ provide a complementary investigation of the parameter space of the DM mass $m_{\chi}$ and scalar mass $m_{S}$, which extends the reach of collider searches and DM direct detection, especially for the region where $m_{\chi}$ and $m_{S}$ are close.

In the cases of BM (b) and BM (d), the coupling $f_{\chi S}$ is determined by the requirement of DM relic density, $\Omega_{\chi}h^{2}=0.1199$. For electroweak scale DM and scalar, the LHC searches demonstrate greater sensitivity than the indirect constraints on the four-fermion operator, while the latter are more significant for larger values of $m_{\chi}$ and $m_{S}$.