^{1}School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom^{2}Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom^{3}Université de Lyon, Université Claude Bernard Lyon 1, Villeurbanne, France^{4}Department of Physics, University of Johannesburg, Johannesburg, South Africa^{5}Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden

We introduce a new class of renormalizable models for dark matter with a minimal particle content, consisting of a dark *SU*(2)_{D} gauge sector connected to the standard model through a vector-like fermion mediator, not requiring a Higgs portal, in which a massive vector boson is the dark matter candidate. These models are labeled fermion portal vector dark matter (FPVDM). Multiple realizations are possible, depending on the properties of the vector-like partner and scalar potential. One example is discussed in detail. Fermion portal vector dark matter models have a large number of applications in collider and non-collider experiments, with their phenomenology depending on the mediator sector.

The nature of DM, whose existence has been established beyond any reasonable doubt by several independent cosmological observations, is one of the greatest puzzles of contemporary particle physics. Models with a vector DM, especially in the non-abelian case, are the least explored but well-motivated, as the gauge principle offers guidance and constraints limiting the possible theoretical constructions (see, e.g., [1–26], for a discussion of non-abelian DM in different setups, in particular using non-renormalizable kinetic mixing terms or Higgs portal scenarios). In this article, we develop a new minimal framework that extends the gauge sector of the standard model (SM) by a new non-abelian gauge group for which no renormalizable kinetic mixing terms are allowed^{1} and under which all SM particles are singlets. The full model structure, Lagrangian, and particle content are presented in the following sections, along with the main results and immediate prospects for experimental testing, while more technical details can be found in [27].

The simplest non-abelian group is *SU*(2), which in the following will be labeled *SU*(2)_{D} as it connects the SM to the dark sector. The gauge bosons associated with *SU*(2)_{D} are labeled as *SU*(2)_{D} (D-isospin) is specified in the field subscripts. The covariant derivative associated with *SU*(2)_{D} is

where *g*_{D} is the *SU*(2)_{D} coupling constant and *T*_{3D} is the D-isospin.

The fields responsible for breaking the gauge symmetries are two scalar doublets:

where the first is breaking *SU*(2)_{L} × *U*(1)_{Y}, while the second is breaking *SU*(2)_{D} via their respective vacuum expectation values (VEVs) *v* and *v*_{D}.

The scalar potential for Φ_{H} and Φ_{D} reads

which was introduced in [2] and ensures that the gauge bosons of *SU*(2)_{D} are degenerate and stable because of the custodial symmetry of the scalar Lagrangian. Although the operator *ψ*_{D} *ψ*), vector-like (VL) under *SU*(2)_{D}, and both elements of which are singlets under *SU*(2)_{L}, sharing the same hypercharge as one of the SM right-handed fermions^{2}. The mass and Yukawa interaction terms of Ψ read

where *y*′ is a new Yukawa coupling connecting the SM fermion with _{D} doublet. At this point, it would be possible to write an additional Yukawa term *ψ*_{D} and *ψ* with SM fermions and would induce a direct coupling *y*′′ term and the respective stability of DM can be avoided by imposing an unbroken continuous global symmetry *SU*(2)_{D}.

Without this symmetry, such a term would be compulsory since the scalar doublet, Φ_{D}, is in the pseudo-real representation. The symmetry-breaking pattern is *U*(1)_{D} phase assignments *Y*_{D} = 0 for triplets, there is still an invariance under the subgroup

**Table 1**. Quantum numbers of the new particles under the electro-weak (EW) and *SU*(2)_{D} gauge groups.

The lightest *ψ*_{D}, with very different consequences from a cosmological point of view [27]. We consider the case where the lightest *Z*, *W*^{±},

Upon diagonalization, the mass eigenvalues read

with the mixing angle

In the fermion sector, the component with *T*_{3D} = +1/2 gets only a VL mass; therefore,

The mass eigenvalues are

where *f* identifies the SM fermion, and *F* identifies its heavier partner. The mass hierarchy is

The Yukawa couplings and mixings can be expressed in terms of the masses of the physical fermions. The new fermion sector is completely decoupled in the limit *y* = *y*_{SM} and *y*′ = 0. When the full flavor structure of the SM is taken into consideration, different possibilities can be considered. A VL fermion can interact with one or more SM flavors, and there can be multiple VL fermions. The Cabibbo–Kobayashi–Maskawa (CKM) matrix of the SM might also receive contributions from new physics induced by the mixing of SM and VL quarks.

The masses of the SM gauge bosons are not altered by the presence of Φ_{D}. The gauge bosons of *SU*(2)_{D} are all degenerate in mass at the tree level: *SU*(2)_{D} gauge bosons. Such different contributions might also affect loop corrections to *Z* and *W* masses, addressing the CDF anomaly [32]. In the following, to simplify notation, we will label *m*_{V′}. The leading contribution to the radiative mass split of *V*′ and *V*_{D} bosons, *F* and *ψ*_{D} loops and reads

In the following, we assume that new VL fermions interact only with one flavor of the SM. Six independent input parameters are thus necessary to describe the new physics sector of the model, namely, *g*_{D}, *m*_{H}, sin *θ*_{S}, *m*_{F}, and

Let us now discuss a specific realization of the model, assuming only one VL partner interacting exclusively with the SM top quark and no mixing between *h* and *H*, i.e., *θ*_{S} = 0. This choice significantly simplifies the Lagrangian: the Higgs sector of the SM is not affected by the new physics at the tree level, and the potential of Φ_{D} has the very same structure as the Higgs potential. A mixing between *h* and *H* is induced only by fermionic loops and will be neglected in the following. Therefore, in this case, the model is described by the following five parameters: *g*_{D}, *m*_{H}, *m*_{F} = *m*_{T}, and

The hierarchy between the masses in the fermion sector is *H* can have any mass allowed by experimental bounds, even being lighter than the SM Higgs boson.

In our study, we tested this realization of the model against multiple observables from cosmology, DM direct, and indirect detection (DD and ID) experiments and LHC searches. For this purpose, the Lagrangian has been implemented in LanHEP [33] and FeynRules [34], while model files have been generated in CalcHEP [35], UFO [36], as well as FeynArts [37] formats and are available on the HEPMDB [38]. This implementation has been used in micrOMEGAs v5.2.7 [39] for the evaluation of various DM observables and for extracting the respective limits. The model implementation in UFO format has been used in MG5_aMC [40] for the determination of the LHC constraints. Collider simulations have been performed at LO using the NNPDF3.0 LO set [41] through the LHAPDF6 library [42] (LHA index 262400). A simplified version of the model has been implemented to calculate cross-sections at one loop in MG5_aMC and FormCalc9.8 [43].

The amount of relic density is determined by the interplay of annihilation and co-annihilation processes, a subset of which is shown in Figure 1A. ID constraints are associated with DM annihilation rates at CMB time, excluding regions of parameter space where the injection into SM-plasma in the early universe is too large to be consistent with CMB data. Both the relic density and ID processes are tested against PLANCK data [44]. DD processes arise from diagrams such as those shown in Figure 1B and are tested against limits from XENON 1T [45].

**Figure 1**. Representative diagrams for **(A)** *t*-channel and resonant contributions to DM annihilation and DM-mediator co-annihilation processes; **(B)** DD processes; **(C)** production processes at the LHC: *hV*′, and *V*′*V*′.

The LHC bound has been obtained via testing of *t*_{D} pair production with subsequent decay into *V*_{D} and top quarks against CMS searches for top squark pair production decaying into DM [46]. The relevant limit from *m*_{T} [47, 48]. Single *T* production is less constrained, as it is driven by the small *T* − *t* mixing.

We also estimated the relevance of *V*′ pair production and associate production of *V*′ with the Higgs boson, occurring at LO via fermion loops. Representative topologies for the tested processes are shown in Figure 1C.

The complementarity of cosmological and collider constraints has been studied by performing a comprehensive scan over the parameter space (excluding the fixed parameter sin *θ*_{S} = 0) projected onto the *H*) and DM-*t*_{D} co-annihilation regions, respectively, which satisfy the relic density constraint from PLANCK within 5%. The allowed regions of the parameter space are superimposed on top of the forbidden ones to provide their best visualization in this projection of the five-dimensional scan into the two-dimensional plane.

**Figure 2**. Excluded and allowed region of the parameter space of the model from the full five-dimensional scan (sin *θ*_{S} = 0) of the parameter space projected into a

The generic DM annihilation determines a lower limit on *g*_{D} as a function of *H*-resonant region allows for the reduction of *g*_{D} values by up to two orders of magnitude, while the strong DM-*t*_{D} co-annihilation channel allows for even lower values of *g*_{D} for not-so-heavy DM. For *H*-resonant annihilation requires larger *g*_{D} coupling for higher DM mass to provide the right amount of relic density. Therefore, the region with *g*_{D} couplings, corresponding to the bulk *V*_{D}*V*_{D} → *V*′*V*′ annihilation and *H*-resonant annihilation presented in Figure 2 by green and light-blue colors, respectively. Notice also that the regions with

To assess the relative role of the different constraints, we identify representative benchmarks characterized by different gauge couplings, *g*_{D} = 0.05 and *g*_{D} = 0.5, and fixed values for the masses, *m*_{T} = 1,600 GeV and *m*_{H} = 1,000 GeV. For these points, gauge coupling is small enough to allow a perturbative treatment in a region of parameter space, which can be tested by both collider and cosmological observables.

We show in Figure 3 the exclusion regions in the *V*_{D} and mediator *t*_{D} are left as free parameters.

**Figure 3**. Combination of constraints from LHC, relic density, DD, and ID for the benchmark points in the *t*_{D} lifetimes are shown for the small mass splitting region. Cross-sections for collider processes that are at one-loop at LO are shown as orange and blue contours.

The predicted relic density is consistent with PLANCK results only in specific regions: for *g*_{D} = 0.05 (left and central panels of Figure 3), most of the parameter space predicts an over-abundant relic density, except for an area where the mass difference between *t*_{D} and DM is less than *t*_{D}-*t*_{D} and DM-*t*_{D} co-annihilation processes dominate), a small area around *H*), and *g*_{D} (right panel of Figure 3), annihilation processes become more effective, reducing the size of the excluded area in the lower *V*_{D}*V*_{D} → *V*′*V*′ process, due to Δ*m*_{V} > 0, affects the relic density and ID signals. The complementarity of various constraints is especially evident for small values of *g*_{D} in the low *g*_{D} = 0.05, largely overlapping with the region excluded by relic density, and rapidly vanishes as *g*_{D} increases. The large region excluded by DD is mainly determined by processes (see Figure 1B) with sizable kinetic mixing or DM multipole moment contributions, taking place in the regions with low

The LHC bound is almost independent of the mass of *t*_{D} until its mass difference with the DM reaches the top-quark threshold: in that region, *V*′ pair production and associated production of *V*′ with the Higgs boson would only be potentially testable in a region already excluded by DD constraints (see orange and blue contours in the right panel of Figure 3). The model has an important feature, especially for small values of *g*_{D} in the small DM-*t*_{D} mass-gap region where the correct relic density is reproduced. In this region, *t*_{D} is long-lived (its lifetime in the small mass-gap region is shown in the central panel, Figure 3^{3}) and can be probed by dedicated searches at the LHC or future colliders. Different *T* or *H* masses would not modify this qualitative picture.

The FPVDM scenario introduced in this paper connects a vectorial DM candidate from a non-abelian *SU*(2)_{D} gauge group to the fermionic sector of the SM without the necessity of a Higgs portal at the tree level, and the mechanism is realized in the most economical way, with a minimal set of new parameters and new particles. Even the simplest realization of FPVDM, involving interactions of the dark sector with only one SM fermion, has great potential to explain DM phenomena and has several important implications for collider and non-collider DM searches. Minimal FPVDM realizations involving other SM fermions can be used to explain outstandingly observed anomalies. For example, if the VL fermion interacts with the leptonic sector of the SM, new contributions might explain (*g* − 2)_{μ} [49] and, at the same time, provide novel physics cases for future *e*^{+}*e*^{−} colliders [50–53]. Non-minimal realizations, including mixing in the scalar sector, further VL partners, or additional interactions of the same VL representation, would open up a vast range of possibilities for future studies, both phenomenological and experimental, and would allow one to explore the complementarity between collider and non-collider observables in multiple scenarios.

## Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

## Author contributions

AB: writing–original draft and investigation. LP: writing–original draft and investigation. AD: writing–original draft and investigation. SM: writing–original draft and investigation. DR: writing–original draft and investigation. NT: investigation and writing–original draft.

## Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. AB and SM acknowledge support from the STFC Consolidated Grant ST/L000296/1 and are partially financed through the NExT Institute. AB also acknowledges support from Soton-FAPESP and Leverhulme Trust RPG-2022-057 grants. LP’s work is supported by the Knut and Alice Wallenberg Foundation under the SHIFT project (grant KAW 2017.0100). AD is grateful to the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) for its financial support within the program “Investissements d’Avenir”. AD acknowledges partial support from the National Research Foundation in South Africa. NT is supported by the scholarship from the Development and Promotion of Science and Technology Talents Project (DPST).

## Acknowledgments

All authors acknowledge the use of the IRIDIS High-Performance Computing Facility and associated support services at the University of Southampton in completing this work.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Footnotes

^{1}Contributions to gauge kinetic mixing may arise at the loop level, depending on the structure of the Higgs sector, but they correspond to suppressed higher operator terms.

^{2}VL portals have also been explored in [29, 30], but for scalar DM candidates, and in [26, 31] for vector dark matter, but with either the simplifying assumptions of setting the new Yukawa coupling to zero [31] or with a much larger, hence non-minimal, particle content [26].

^{3}*t*_{D} decay can only occur through the *t*_{D} → *V*_{D}*t*^{(}*^{)} process, as it is *V*′ and *H* can be into different final states if kinematically allowed:

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Keywords: dark matter, large Hadron collider, vector-like fermions, dark gauge group, relic density, direct dark matter detection

Citation: Belyaev A, Deandrea A, Moretti S, Panizzi L, Ross DA and Thongyoi N (2024) A fermionic portal to a non-abelian dark sector. *Front. Phys.* 12:1339886. doi: 10.3389/fphy.2024.1339886

Received: 17 November 2023; Accepted: 25 March 2024;

Published: 13 May 2024.

Edited by:

Roman Pasechnik, Lund University, SwedenReviewed by:

Urjit Yajnik, Indian Institute of Technology Bombay, IndiaGiorgio Arcadi, University of Messina, Italy

Copyright © 2024 Belyaev, Deandrea, Moretti, Panizzi, Ross and Thongyoi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Stefano Moretti, s.moretti@soton.ac.uk