The left-right symmetric model (LRSM) [1–7] is a well-known extension of the Standard Model (SM) that has been developed over several decades, primarily to account for parity violation in weak interactions and to provide an explanation for neutrino masses through the seesaw mechanism [8,9]. Despite its theoretical appeal, the LRSM does not automatically provide a viable dark matter (DM) candidate. It is therefore natural to investigate whether the framework can be extended to address the dark matter problem.

Many attempts have been made to incorporate a DM candidate within the LRSM framework. One possibility consists in lowering the mass of a right-handed neutrino (RHN) to the keV scale, thereby rendering it a long-lived warm DM candidate [10]. Alternatively, by imposing a discrete Z2 symmetry on the scalar triplets –under which the left triplet ΔL is odd and the right triplet ΔR is even– and setting the vacuum expectation value (VEV) of ΔL to zero, the neutral component ΔL0 becomes a viable DM candidate [11]. Unfortunately, the first approach is rather unnatural within the LRSM framework, while in the second approach ΔL0 cannot reproduce the observed relic density due to its small annihilation cross section [12].

These limitations have motivated various extensions of the LRSM to address the DM problem. For example, the LRSM can be extended by adding a scalar singlet [13] or a fermionic singlet [14] that plays the role of DM. Other possibilities include left-right fermion triplets and quintuplets forming a viable two-component DM scenario [11]. Additional proposals involve stable fermionic or scalar multiplets whose stability is ensured purely by the gauge structure, without the need for an ad hoc stabilizing symmetry [15], or models in which the gauge group is extended to SU(3)C×SU(2)L×U(1)L×SU(2)R×U(1)R [16] containing a scalar bidoublet and a singlet fermionic DM candidate. Models in which DM arises as a mixture of two or more multiplets have also been studied [17], as well as scenarios where the fermion sector is extended by an additional heavy right-handed copy, with the lightest heavy neutrino acting as DM [18]. Another possibility, which we explore in this work, is to extend the LRSM by introducing vectorlike leptons (VLLs) [19].

In this paper, we extend the gauge structure of the LRSM by introducing an additional SU(2) gauge symmetry, under which one generation of VLF doublets are charged. The model, first introduced in Ref. [20], aims to address several open questions, including the origin of parity violation, neutrino mass generation, and the nature of dark matter. Here, we are primary interested in the possibility that the vectorlike neutrino (N) could serve as a DM candidate. To ensure cosmological stability of N, we assume the existence of a new symmetry in the dark sector. However, imposing a discrete Z2 symmetry—commonly invoked in the literature to stabilize DM—is not viable in this scenario due to the structure of the Yukawa interactions between the VLFs, the chiral fermions, and the scalar sector. To overcome this issue, we introduce a new parity symmetry that forbids mixing between the VLLs and the chiral leptons [19]. As a consequence, the dark sector particles interact exclusively through the vector bosons portal or through the VLLs portal. The dark sector consists of the VLLs, namely the DM candidate N and its charged partner E±. For simplicity, we assume that the scalar sector is very heavy and effectively decoupled from the energy scale of interest. Moreover, we neglect the presence of vectorlike quarks (VLQs), as their inclusion does not significantly affect the constraints on the extra gauge bosons derived in Ref. [20]. The most stringent bounds arise from gauge boson decays into the second generation of heavy neutrinos (HNs), which are unaffected by the presence of VLQs.

As in other models where RHNs couple to new gauge bosons [21], the DM relic density falls within the narrow range measured by Planck [22] when the total mass of the initial particles in the annihilation (or coannihilation) processes is close to the mass of the mediator, namely one of the neutral or charged gauge bosons of the model. Our goal in this paper is to assess the viability of this LRSM extension in light of current dark matter constraints. After imposing collider limits on the new gauge bosons [20], we require that the predicted relic density does not exceed the upper bound determined by Planck observations. Additional constraints from Large Electron Positron Collider (LEP) and LHC searches for new fermions, together with recent limits from direct detection (DD) experiments such as LUX-ZEPLIN (LZ) [23], push the DM mass above the TeV scale. We also consider limits from dark matter indirect detection (ID) searches for γ rays from dwarf spheroidal galaxies by the Fermi Large Area Telescope (Fermi-LAT) [24,25]. However, these mainly constrain the low-mass region, which is already excluded by collider and cosmological bounds.

Finally, we evaluate the prospects for probing this model in future experiments. In particular, we consider projected sensitivities from next-generation liquid xenon direct detection experiments such as Xenon-Lux-Zeplin-Darwin (XLZD)/Dark Matter Wimp Search with Liquid Xenon (DARWIN) [26], and from indirect detection experiments such as Cherenkov Telescope Array (CTA) [27], which are especially relevant for TeV-scale dark matter.

The paper is organized as follows: the model is briefly reviewed in Sec. II. Section III focuses on properties of the dark sector. Section IV includes constraints on the gauge bosons masses as well as other collider constraints. Section V includes a discussion of the dark matter observables for a few benchmarks, while the results of a general scan of the model defining the currently allowed parameter space as well as future probes is performed in Sec. VI. Section VII contains our conclusions.

We consider an extension of the LRSM which includes one generation of vectorlike leptons. The VLLs belong to doublets of an extra gauge symmetry SU(2)V. The vectorlike nature of the VLLs requires that both their chiral components belong to the same representation (the fundamental in this case). Consequently, their masses can be included directly in the Lagrangian without spoiling gauge invariance. The left and right chiral components of SM fermions belong to doublets under the fundamental representations of SU(2)L and SU(2)R, respectively, as invoked in the LRSM. Under the SU(2)V×SU(2)L×SU(2)R×U(1)B-L gauge group, the VLLs transform as LL,R=(NE)L,R∼(2,1,1)-1,(1)where the subscript “−1” denotes the B-L quantum number.

The gauge bosons fields and the gauge couplings of the model are denoted by SU(2)V:  WVi,gVSU(2)L:  WLi,gLSU(2)R:  WRi,gR  U(1)B-L:B,g′,(2)where i=1, 2, 3 is the index of the gauge bosons components.¹

In the mass basis, the gauge bosons include those of the standard model γ,W±,Z, as well as four heavier gauge bosons W′±,Z′,W′′±, and Z′′.

The scalar sector of the model contains a bidoublet field Φ and left (right) triplets ΔL (ΔR), which are introduced to spontaneously break the left/right symmetry of the model. These fields generate the chiral fermions interaction in the Yukawa sector and provide Majorana mass term for the neutrinos. To break the SU(2)V symmetry and allow the VLLs to mix with the chiral leptons, we need to introduce the two following self-dual bidoublet scalar fields: ΦVL=(ϕ10-ϕ1+ϕ1-ϕ10*)∼(2,2,1)0,ΦVR=(ϕ20-ϕ2+ϕ2-ϕ20*)∼(2,1,2)0.(5)

We recall that the VEVs of ΔR and ΦVR can be taken to be zero as shown in Ref. [20]. The pattern of the symmetry breaking of the model is schematized as follows: SU(2)V×SU(2)L×SU(2)R×U(1)B-L⟶⟨ΦVL⟩=0⟨ΦVR⟩=uRSU(2)L×SU(2)R×U(1)B-L⟶⟨ΔL⟩=0⟨ΔR⟩=vRSU(2)L×U(1)Y⟶⟨Φ⟩=diag(k1,k2)U(1)EM,(6)where the VEVs uR and vR are both at the TeV scale, and k12+k22=vEW2, where vEW is the SM VEV. The bidoublet Φ VEVs can be parametrized as follows: k1=vEWcβk2=vEWsβ.(7)We assume k2≪k1, following Ref. [28].²

To ensure the known mass hierarchy for the ordinary top and bottom quarks, we must take k2≪k1; see Ref. [28] for more details.

The mixing of the VLLs with the chiral leptons is achieved through the self-dual bidoublet scalar fields. Thus, the Yukawa Lagrangian describing this mixing is given by Lint=-L¯Rλl†ΦVLlL-L¯Lλl′ΦVRlR+H.c.,(12)where λl and λl′ are 3×1 (1×3) Yukawa coupling matrices.

The scenario where N plays the role of DM requires that it must be stable on cosmological times scales. This stability strongly suggests the existence of a new symmetry in the dark sector. There are many possibilities for symmetries that stabilize DM; for example, the discrete Z2 symmetry which is commonly used [29–39]. Obviously, this possibility cannot be used in our case because this symmetry imposes that both the VLLs and the self-dual bidoublets must be odd fields [i.e., LL,R→-LL,R, ΦVL(R)→-ΦVL(R)] under Z2, while all other fields should be even. ΦVL could be an odd field when the VEV is zero, ⟨ΦVL⟩=0. In contrast, the VEV of ΦVR should be at TeV scale; hence, it could not be an odd field under Z2. To circumvent this problem, one can include a new parity symmetry which disallows mixing between the VLLs and the ordinary leptons [19]; under this symmetry, only the VLLs are odd fields while all other fields are even, such that λl=λl′=0. The latter symmetry implies independent free masses for the vectorlike neutrino N and the charged partner E±. As will be seen in the following sections, their masses will be constrained by phenomenological requirements, mainly by imposing the DM relic density constraint as well as the constraints from the DD and ID approaches. In this scenario, neutrinos get their masses through the seesaw mechanism as invoked, usually within the LRSM, since there is no mixing with the VLL [4,5,8,9].

In the following sections, we will examine various experimental constraints on the relevant parameter space for the DM study; for this, it is more convenient to use the physical parameters as free parameters. The free parameters include the coupling gV, the VEV of ⟨ΦVR⟩ (uR), as well as the mass of one extra gauge boson, mW′. In the following, we fix uR=vR+100  GeV, thus the masses of new gauge bosons can all be rewritten in terms of gV and mW′. This choice is based on the pattern of symmetry breaking followed in Eq. (6), which ensures that both VEVs are at the TeV scale and that uR is larger than vR, but not excessively so. This choice is convenient because it guarantees that the first-generation extra gauge bosons W′ and Z′ are lighter than the second-generation ones W′′ and Z′′, while the latter remain accessible at colliders and contribute to dark matter physics. In addition, we fix the mass of the HNs to be mW′/2 since this relation is used to obtain the collider limits on the heavy gauge bosons. Moreover, with this assumption, the mass of the heavy neutrinos should be at the TeV scale as expected in the seesaw mechanism for neutrino masses. We keep the same feature for the rest of the analysis. For the sake of simplicity, all the masses of the extra scalar bosons are considered to decouple from the scale of interest. Finally, in the dark sector, the two parameters relevant for computing DM observables are the mass of the DM candidate mN and its mass splitting with the charged partner E± which is denoted by Δm.

In the following, we discuss the interactions among the dark sector particles and their relevance for DM annihilation and coannihilation processes. There are two types of channels for which the annihilation and the coannihilation of dark particles can proceed: first, the s-channel processes mediated exclusively by the vector bosons (see the upper diagrams of Fig. 1), and second, the t-channel annihilation processes into vector bosons mediated by the VLLs of the dark sector (see the lower diagrams of Fig. 1). Technically, any combination of particles that interact with one of the gauge bosons (i.e., the fermions,³

Including the SM fermions (f) and the heavy neutrinos.

Although any combinations of the gauge bosons that satisfy the conservation of electric charge can be produced in the final state for the coannihilation processes, in our model, the Z′′/W′′± are kinematically forbidden. Moreover, one should mention that there is no interaction between the DM particle and the photon (i.e., in middle lower diagram of Fig. 1). All possible final states for the s-channel diagrams (i.e., the upper diagrams of Fig. 1) are listed here: NN¯(E+E-)→V0→{qq¯,li+li-νiνi,νiNi,NiNiW+W-,W±W′∓Zh,Z′hN(N¯)E±→V±→{qq¯′νili±,Nili±γW±ZW±,ZW′±,Z′W±W±h,W′±h{NN¯(E+E-)→Z/Z′→W′±W′∓N(N¯)E±→W±/W′±→γW′±,Z′W′±E+E-→γ→{qq¯,li+li-W+W-,W′±W′∓W±W′∓,(13)where q represent the quarks and i the generation index for the leptons [charged leptons (li±), Majorana neutrinos (νi), and HNs (Ni)], and V0(V±) stands for the neutral (charged) gauge bosons.

The couplings for the interaction of the VLLs with the gauge bosons are summarized in Table I. They depend on the angles used for the diagonalization of the mass matrices of the gauge sector in Ref. [20], namely: c1=gVg2+gV2c2=ggVg′2(g2+gV2)+g2gV2.(14)

Clearly, the interactions via Z and W± are suppressed since they are proportional to the suppressed parameters ε1,2 and sβ defined in Eq. (8). Moreover, the NN¯Z coupling is very strongly suppressed because of the multiple powers of small mixing angles entering its definition. This interaction do not contribute significantly to the dark matter formation, as we will see. Thus, the efficient annihilation can be realized exclusively via Z′ and Z′′. On the other hand, efficient coannihilation can proceed via the three charged gauge bosons (V±), although the interaction with W± is somewhat suppressed in comparison with the others.

In this section, we calculate some DM observables for fixed gauge boson masses. We recall that the feynrules package [48] is used to generate the model in CalcHEP [49] format. Throughout this article, we use micromegas [47] to calculate DM observables.

The four free parameters for our study have been discussed in Sec. II. The gauge coupling gV is fixed to four benchmarks values; see Table III. The remaining three parameters are varied randomly, taking into consideration the collider constraints on the gauge bosons masses from Table II. The range considered is represented in Table III.

In this work, we have shown that an extension of the LRSM with an extra SU(2)V factor and one generation of VLLs can provide a good DM candidate, the neutral component of the VLL doublet. For stabilizing the DM particle, a discrete parity symmetry that forbids the mixing of VLLs-leptons has been imposed, thus the dark sector particles interact only with vector bosons. DM annihilation and coannihilation into fermions proceed mainly through exchange of the vector bosons portal in s-channel, otherwise annihilation into final state bosons proceed through VLLs exchange in t-channel.

After imposing an upper bound on the relic density, we found that DM could be either at the electroweak scale or at the (multi) TeV scale. For the low mass region, a small mass splitting between the DM particle and its charged partner is required. However, searches for charged leptons at LEP and LHC almost completely exclude this possibility. The only remaining points above 104 GeV are found to be incompatible with direct detection results of LZ. For DM above the TeV scale, all viable points escape the current limits from dwarf spheroidal galaxies by Fermi-LAT; however, the model is partly constrained by direct detection results of LZ and will be further probed by future large detectors. Moreover, we have highlighted a complementarity between DD and ID. In particular, the future CTA telescope will be able to probe multi-TeV DM that escape multiton scale direct detectors such as XLZD, and even to probe some points that fall below the neutrino floor.

In this work, we have made simplifying assumptions; for one, we only considered the case where the mass of the HNs is half that of mW′ in order to adapt simply the existing LHC limits on new gauge bosons coming from the W′±→N2μ± channel. We found that the contribution of final states with pair of HNs give a non-negligible contribution to DM annihilation, although the annihilation into quarks and leptons remain dominant. We checked that removing the HNs from the final state did not affect significantly the photon spectra. We conclude that allowing a larger mass range for the HNs is more likely to impact the DM observables only if it impacts the allowed masses for the new gauge bosons.

Finally, in our analysis all the masses of extra scalars were fixed at a high scale, an open question that we leave for future work is whether taking the scalar sector particles into consideration—where neutral, charged, and doubly charged scalars could appear in the final states of annihilation and coannihilation processes—would impact DM observables.