A fermionic portal to a non-abelian dark sector Alexander Belyaev1,2, Aldo Deandrea3,4, Stefano Moretti1,2,5*, Luca Panizzi1,5, Douglas A. Ross1 and Nakorn Thongyoi1 1School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom, 2Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom, 3Université de Lyon, Université Claude Bernard Lyon 1, Villeurbanne, France, 4Department of Physics, University of Johannesburg, Johannesburg, South Africa, 5Department 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. KEYWORDS dark matter, large Hadron collider, vector-like fermions, dark gauge group, relic density, direct dark matter detection 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 allowed1 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 VD μ � (V0 D+μ V0 D0μ V0 D−μ), where, here and in the following, the electric charge is specified in the field superscripts, while the isospin under SU(2)D (D-isospin) is specified in the field subscripts. The covariant derivative associated with SU(2)D is OPEN ACCESS EDITED BY Roman Pasechnik, Lund University, Sweden REVIEWED BY Urjit Yajnik, Indian Institute of Technology Bombay, India Giorgio Arcadi, University of Messina, Italy *CORRESPONDENCE Stefano Moretti, s.moretti@soton.ac.uk RECEIVED 17 November 2023 ACCEPTED 25 March 2024 PUBLISHED 13 May 2024 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 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. 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. Frontiers in Physics frontiersin.org01 TYPE Brief Research Report PUBLISHED 13 May 2024 DOI 10.3389/fphy.2024.1339886 https://www.frontiersin.org/articles/10.3389/fphy.2024.1339886/full https://www.frontiersin.org/articles/10.3389/fphy.2024.1339886/full https://crossmark.crossref.org/dialog/?doi=10.3389/fphy.2024.1339886&domain=pdf&date_stamp=2024-05-13 mailto:s.moretti@soton.ac.uk mailto:s.moretti@soton.ac.uk https://doi.org/10.3389/fphy.2024.1339886 https://creativecommons.org/licenses/by/4.0/ https://www.frontiersin.org/journals/physics https://www.frontiersin.org https://www.frontiersin.org/journals/physics https://www.frontiersin.org/journals/physics#editorial-board https://www.frontiersin.org/journals/physics#editorial-board https://doi.org/10.3389/fphy.2024.1339886 Dμ � ∂μ − i gD� 2 √ V0 D±μT ± D + igDV 0 D0μT3D( ), (1) where gD is the SU(2)D coupling constant and T3D is the D-isospin. The fields responsible for breaking the gauge symmetries are two scalar doublets: ΦH � ϕ+ ϕ0( )T ⇝ 〈ΦH〉 � 1� 2 √ 0 v( )T, ΦD � φ0 D+1 2 φ0 D−1 2 ( )T ⇝ 〈ΦD〉 � 1� 2 √ 0 vD( )T, (2) 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 vD. The scalar potential for ΦH and ΦD reads V ΦH,ΦD( ) � −μ2Φ† HΦH − μ2DΦ† DΦD + λ Φ† HΦH( )2 +λD Φ† DΦD( )2 + λΦHΦDΦ† HΦHΦ† DΦD, (3) 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 Φ† HΦHΦ† DΦD is not protected by any symmetry and cannot thus be removed from the potential, coupling λΦHΦD can have any value. If it is small enough, the dark sector would be effectively decoupled from the SM and only observable through gravitational effects. Moreover, the Higgs portal induces scalar mixing, which modifies Higgs–SM couplings and generates Higgs–DM interactions, all of which are strongly constrained [28]. Here, we suggest a different mechanism of communication between the dark and visible sectors via a fermion doublet Ψ = (ψD 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 fermions2. The mass and Yukawa interaction terms of Ψ read −Lf � MΨ �ΨΨ + y′�ΨLΦDf SM R + h.c( ), (4) wherefSM R generically denotes an SM right-handed singlet and y′ is a new Yukawa coupling connecting the SM fermion with ΨL � (1−γ5) 2 Ψ through the ΦD doublet. At this point, it would be possible to write an additional Yukawa term y′′ �ΨLΦc Df SM R , which would violate the stability of DM due to the simultaneous mixing of ψD and ψ with SM fermions and would induce a direct coupling V0 D± �f SM fSM. The appearance of such y′′ term and the respective stability of DM can be avoided by imposing an unbroken continuous global symmetry U(1)D ≡ eiΛYD , unrelated to local SU(2)D. Without this symmetry, such a term would be compulsory since the scalar doublet, ΦD, is in the pseudo-real representation. The symmetry-breaking pattern is SU(2)D × U(1)D → U(1)dD. With U(1)D phase assignments YD � 1 2 for dark doublets and YD = 0 for triplets, there is still an invariance under the subgroup Z2 ≡ (−1)QD , where QD � T3 D + YD. The new particles are summarized in Table 1. The lightest Z2-odd particle is stable and could be either V0 D± or ψD, with very different consequences from a cosmological point of view [27]. We consider the case where the lightest Z2-odd particle is V0 D±, which we label fermion portal vector dark matter (FPVDM). The theory contains six massive gauge bosons (Z, W±, V0 D0, and V0 D±), and therefore, six Goldstone bosons correspond to their longitudinal components. The remaining two degrees of freedom correspond to physical scalars, which include the SM Higgs boson and another CP-even scalar. By denoting the neutral scalars in terms of their components in the unitary gauge as ϕ0 � 1� 2 √ (v + h1) and φ0 D−1/2 � 1� 2 √ (vD + φ1), the mass terms of the scalar Lagrangian reads LS m � h1 φ1( ) λv2 λΦHΦD 2 v vD λΦHΦD 2 v vD λDv2D ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ h1 φ1 ( ). (5) Upon diagonalization, the mass eigenvalues read m2 h,H � λv2 + λDv 2 D ∓ ���������������������� λDv2D − λv2( )2 + λ2ΦHΦD v2v2D √ , (6) with the mixing angle sin θS � ������������ 2 m2 Hv 2λ−m2 h v2DλD m4 H−m4 h √ . In the fermion sector, the component with T3D = +1/2 gets only a VL mass; therefore, mψD � MΨ. However, the other fermion masses are generated after both scalars acquire a VEV. The fermionic mass Lagrangian reads Lf m � �f SM L �ψL( ) y v� 2 √ 0 y′ vD� 2 √ MΨ ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ fSM R ψR ( ). (7) The mass eigenvalues are m2 f,F � 1 4 Δ ∓ ������������ Δ2 − 8y2v2M2 Ψ √[ ], (8) where Δ � y2v2 + y′2v2D + 2M2 Ψ, f identifies the SM fermion, and F identifies its heavier partner. The mass hierarchy is mf 0, affects the relic density and ID signals. The complementarity of various constraints is especially evident for small values of gD in the low mVD region. The region excluded by ID corresponds to small values of mVD for gD = 0.05, FIGURE 2 Excluded and allowed region of the parameter space of the model from the full five-dimensional scan (sin θS = 0) of the parameter space projected into a (gD ,mVD) plane. White areas represent non- perturbative regions. FIGURE 3 Combination of constraints from LHC, relic density, DD, and ID for the benchmark points in the {mtD ,mVD} (left and right panels) and {mtD , 1 − mVD/mtD} (center panel) planes. The colored regions are excluded. For relic density, the under-abundant region is considered allowed and the borders of the excluded region correspond to the measured relic density value. The non-perturbative region corresponds to one-loop corrections to gauge boson masses larger than 50%. An estimate of the region of large kineticmixing is shown as a hatched area. Contours corresponding to different tD 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. Frontiers in Physics frontiersin.org04 Belyaev et al. 10.3389/fphy.2024.1339886 https://www.frontiersin.org/journals/physics https://www.frontiersin.org https://doi.org/10.3389/fphy.2024.1339886 largely overlapping with the region excluded by relic density, and rapidly vanishes as gD 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 mVD values (i.e., below few hundreds GeV). The LHC bound is almost independent of the mass of tD until its mass difference with the DM reaches the top-quark threshold: in that region, Emiss T decreases and the sensitivity of the CMS search reduces, allowing a small mass-gap region. As the process is QCD- initiated, the bound is also almost independent of other parameters of the model. Processes of 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 gD in the small DM- tD mass-gap region where the correct relic density is reproduced. In this region, tD is long-lived (its lifetime in the small mass-gap region is shown in the central panel, Figure 33) 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). 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