Dark matter in the fully flipped 3-3-1-1 model

We present the features of the fully flipped 3-3-1-1 model and show that this model leads to dark matter candidates naturally. We study two dark matter scenarios corresponding to the triplet fermion and singlet scalar candidates, and we determine the viable parameter regimes constrained from the observed relic density and direct detection experiments.


I. INTRODUCTION
The extension of the Standard Model (SM) gauge symmetry based upon the higher weakisospin symmetry SU (3) L is best known for solving the number of observed fermion generations, It is noteworthy that the electric charge Q and the baryon-minus-lepton charge B − L neither commute nor close algebraically with SU (3) L [10]. To close the symmetries on which we can base our theory, the isospin symmetry SU (3) L must be enlarged (i.e. flipped) to SU (3) L ⊗U (1) X ⊗U (1) N by symmetry principles, called 3-3-1-1 when including the color group [12,13]. Here the new abelian charges X and N are related to Q and B −L through the Cartan generators of SU (3) L , respectively.
Hence, the fully flipping in gauge sector yields a complete gauge symmetry and it may of course contain the above flipped fermion content. In this work, we discuss a model that is based on this complete gauge symmetry with the flipped fermion content, called fully flipped 3-3-1-1 model. We show that the model supplies the novel schemes of single-component dark matter (DM).
The rest of this paper is organized as follows. In Sec. II, we provide the features of the model. The mass spectra for the scalar and gauge boson sectors are considered in Sec. III. In Sec. IV we compute relevant interactions. Sec. V is devoted to the DM observables. Finally, we conclude this work in Sec. VI.
Here α = 2, 3 and a = 1, 2, 3 are generation indices, the component fields ξ's compose a real triplet of SU (2) L , and the fields E, U have the same electric charge as e, u, respectively. The above fermion content is free from all the anomalies, including the gravitational anomaly.
To break the gauge symmetry and generate appropriate masses for the particles, the scalar content is introduced as φ ∼ (1, 1, 0, 2), (12) which have the corresponding vacuum expectation values (VEVs) to be where we assume κ u, v w, Λ ∆. Notice that the other neutral scalars are W P -odd, hence possessing vanished VEV due to the W P conservation, as shown below.
The gauge symmetry is broken via three steps, Here W P is a residual gauge symmetry of U (1) N which is continuously broken along with SU (3) L defining a final residual symmetry W P = (−1) 3(B−L) with B − L = (2/ √ 3)T 8 + N [12,13]. When including the spin symmetry (−1) 2s , we have This symmetry divides the model particles into two classes, as displayed in Table I, where the new fermions and some other particles are odd under W P , while all the remaining particles, including the SM ones, are even under this symmetry. Since W P is conserved, the odd particles are only coupled in pairs in interactions and the lightest odd particle is stabilized and thus can be a DM candidate if it is electrically neutral and colorless.
The second term of (16) contains Yukawa interactions, which were previously presented in [10]. There, all the fermion masses were properly produced, so we do not refer to them hereafter.

III. SCALAR AND GAUGE SECTORS
Because of the condition ∆ w, V , the field φ can be integrated out from the low-energy effective potential of η, ρ, χ, and S. As a result, the scalar potential below ∆ has a form similar to V (η, ρ, χ, S) where its couplings become effective due to the modification of heavy particles.
Indeed, let us expand φ = 1 √ 2 (∆ + H C + iG C ), where H C and G C are the new Higgs and Goldstone bosons associate to U (1) N , respectively. The masses of the new Higgs H C and the new gauge C are proportional to ∆, m H C √ 2λ 19 ∆ and m C 2g N ∆, which are decoupled from the low energy particle spectra. Hence, we will neglect the U (1) N sector. Now, we consider the scalar potential (24). To obtain the potential minimum and physical scalar spectrum, we expand the neutral scalar fields around the VEVs as where the fields ρ 0 3 , χ 0 2 , and S 0 23 are odd under W P , cannot develop VEVs, as mentioned. Additionally, since κ constrained by the ρ-parameter is tiny, its contribution would be neglected.
The mass part V mass consists of the quadratic terms in fields, grouped into V mass = V S mass +V A mass + V charged mass +V S mass +V A mass . Here the charged term includes charged scalars, while the remaining terms describe CP -even and CP -odd scalar fields, and notice that the primed fields are decoupled from the normal fields due to the W P conservation. That said, after integrating φ out, the scalar potential (24) gives a mass spectrum of the scalar sector including 22 massive Higgs fields, summarized as The mentioned Higgs and Goldstone bosons are related to those in the gauge basis, such as where we define s α 1 ≡ sin α 1 , c α 1 ≡ cos α 1 , tan α 1 ≡ t α 1 = v/u, and The masses of the mentioned Higgs fields (the masses of the mentioned Goldstone bosons vanish, For the gauge boson sector, recall that the U (1) N gauge boson is heavy, which is integrated out as φ is. Hence, the spectrum of the remaining gauge bosons is identical to those obtained in [10].
That said, we have two new non-Hermitian gauge bosons X, Y with the masses at the w, Λ scales, besides the W boson of the SM, as follows The mass of W implies u 2 + v 2 (246 GeV) 2 . For the neutral gauge sector, the physical fields are related to the gauge fields as defines the Z-Z mixing angle, ϕ. Here A is the massless photon field, while Z is the SM neutral weak boson with mass m Z m W /c W . Z is the new neutral gauge boson, obtaining a mass at w, Λ scale, such as The lower bound of w, Λ is given by the rho parameter constraint to be √ w 2 + 4Λ 2 ∼ 5−7 TeV dependent on u, v relation, as well as the LHC search for Z through dilepton products makes a constraint on Z mass, m Z > 2.25-2.8 TeV, which translates to in agreement to the rho parameter [10]. We made a study deduced from the LEPII bound for the process e + e − → µ + µ − via Z exchange [14] which reveals m Z > 2.7 TeV, implying a similar bound for w, Λ.

A. Fermion and gauge boson interactions
The gauge interactions of fermions arise from, Using the result in (46), we get the charged current interactions of W , X, and Y to be where the currents are given by Using the result in (48), we get the neutral current interactions for the neutral gauge bosons, where e = gs W and f refers to every fermion of the model, except for the right-handed neutrinos.
The vector and axial-vector couplings of Z are collected in Table II as put in Appendix A. Notice that all the interactions between Z boson and ordinary fermions are consistently recovered in the limit ϕ → 0.
The couplings of Z with fermions can be obtained from those for Z, by replacing which need not necessarily to be determined, but pointed out in the Table II caption.
Notice that ν R 's have only gauge interaction with the U (1) N gauge boson and are obviously integrated out as C is, since they possess a large mass proportional to ∆ via the coupling ν R ν R φ [10].

B. Scalar and gauge boson interactions
The gauge interactions of the scalar fields arise from Substituting the physical fields from (33), (46), and (48) to this Lagrangian, we get the interactions between a gauge boson and two scalars, two gauge bosons and a scalar, and two gauge bosons and two scalars in the model, given in III, IV, V, VI, VII, VIII, IX, X, XI, and XII, which are gathered in Appendix A. At the leading order, we have verified that all the interactions between the SMlike Higgs boson and the gauge bosons are consistently recovered. However, only the interactions relevant to the following diagrams are needed, yielding the cross-sections as given.

V. SEARCH FOR DM
In this model, the two scalars H 0 1,2 , the fermion ξ 0 , and the gauge boson Y 0 , which all have masses at w, Λ scale, are W P -odd particles and electrically neutral and colorless. Hence, they may be responsible for the DM candidate. As indicated in [12] and updated in [15], the relic density contribution of the gauge boson Y 0 is small compared to the observed dark matter density, so we do not interpret the vector field as DM. Additionally, if the scalar H 0 2 that transforms as a SU (2) L doublet has a correct relic density to be DM, it should be ruled out by the direct detection experiments due to a large scattering cross-section induced by Z [16,17]. Therefore, in the following we study the DM phenomenology associated with the candidates, singlet scalar H 0 1 and triplet fermion ξ 0 . The presence of the triplet candidate, a result of the fully flipped, would make dark matter phenomena of the model completely different from the other theories of this type, such as the ordinary 3-3-1 [18,19], 3-3-1-1 [20][21][22], 3-2-3-1 [23], and flipped trinification [24].

A. DM as a singlet scalar
We now consider the DM scenario where H 0 velocity is approximately given by σv where the annihilation channels into the SM-like Higgs bosons (H 1 ) involving the H 1 , H 1 propagators as well as the last annihilation channel exchanged by Z are infinitesimal, as neglected, because of the small coupling or non-relativistic dark matter momentum suppression. Indeed, although the channels exchanged by H 1,2 are the same, the H 1 H 1 H * 1 coupling as proportional to u, v is radically smaller than the H 1 H 1 H * 2 coupling as proportional to w, f 2 . So the channels exchanged by H 1 are suppressed as compared to those by H 2 , for which only the H 2 contribution appears in Eq. (60). Note that the H 2 H 1 H * 1 coupling is also proportional to u, v. Hence, for the same reason, the s-channel contributions of H 1,2 to the process H 1 H 1 → H 1 H 1 are much smaller than those by H 3,4 . The SM Higgs (H 1 ) contribution is straightforwardly prevented, while the H 2 one may become significant, as kept, due to a resonance in the relic density when the DM mass is close to its mass. The amplitude of the Z -exchanged diagram is proportional to the DM To study the direct detection for the H 1 via the spin-independent (SI) scattering on nuclei, we write the effective Lagrangian describing DM-nucleon interaction in the limit of zero-momentum transfer through the exchange of the Higgs boson H 1 as follows The SI cross-section for the scattering of H 1 on a target nucleus is given by [25] σ SI where N = p, n and m H 1 N = m H 1 m N /(m H 1 + m N ) m N is the DM-nucleon reduced mass. The nucleus factor, C N , is where Z and A correspond to the nucleus charge and the total number of nucleons in the nucleus, respectively, and [26] f For numerical computation in this subsection and the next one for DM fermion, we take the following values of known parameters as [27] Note that λ 1 is related to λ 2 via the SM Higgs mass. Although λ 2 can be changed, we will choose only three values of it that characterize its viable range, simultaneously making the DM phenomenology suitable, as supplied in the following figures.
In Figure 2 m H 1 = m H 4 /2, where the relic density is largely decreased. The variation of λ 2 slightly separates the relic density and this is also valid for the whole viable range of λ 2 ∼ 0-0.7, since λ 1 > 0.
When w, Λ are large as in the right panel, the DM mass upper bound is increased but appearing an excluded intermediate region according to Ωh 2 > 0.12.
Even new scalar fields we depict the SI scattering cross-section of H 1 on the nucleon according to the above choices of λ 2 as well as show the experimental bounds from XENON1T [28], where the black line is the SI WIMP-nucleon cross-section limit at 90% confidence level, while the green and yellow bands are the 1σ and 2σ sensitivities, respectively. We see that the scalar DM mass below around 500 GeV is excluded by the direct detection experiment for λ 2 = 0.12. Hence, in this case the viable scalar DM mass regime is around 500 GeV to a few TeV (cf. Figure 2). However, the SI DM cross-section is quite separated for the variation of λ 2 . In the latter cases for λ 2 = 0.05 and 0.01, the respective lower bounds of H 1 mass are above 1 TeV. If λ 2 is bigger, i.e. λ 2 > 0.12, the DM mass is close to the weak scale, and in this case the DM would be subject to the electroweak precision test and collider bounds, which is not studied in this work. Additionally, when λ 2 is smaller than 0.01, the viable DM mass region is narrow as constrained by the relic density and the unstable regime.

Varying w, Λ while fixing relevant couplings
Since the new physics scales w and Λ govern all the DM annihilation channels, it is interesting to recast the above investigation in which w and Λ are varied instead of f 2 . For this aim, we choose f 2 = −4 TeV, without loss of generality. As mentioned, w, Λ commonly set the strength of the SU (3) L breaking, it is suitably to impose w ∼ Λ, while both these parameters are simultaneously run from a lower bound as supplied in Eq. (51). We also vary λ 2 while fixing the values of the other parameters as in the previous case. The results of the relic density and the mass variations are shown in Fig. 5 and Fig. 6, respectively.
As we see from That said, the viable scalar DM mass regime is now either from 5.25 TeV to 6.85 TeV according to Λ = 1.1w or from 3.65 TeV to 4.2 TeV and from 5.4 TeV to 5.67 TeV according to Λ = 1.5w, as bounded by the unstable region and the relic density Ωh 2 < 0.12. Such ranges are appropriate to the SI DM cross-section bounds.

B. DM as a triplet fermion
The ξ triplet components have a degenerate mass, at the tree level, as induced by the VEV of the scalar sextet. But, the loop effects of gauge bosons can make ξ ± mass larger than ξ 0 mass by an amount, as shown in [16]. Hence, ξ 0 is first regarded as the lightest of the triplet components. We further assume that ξ 0 is the lightest particle among all of the other W P -odd particles and thus ξ 0 is responsible for the DM candidate. This scenario was briefly discussed in [10], in which the field ξ 0 yielded the correct abundance and satisfied the direct detection bounds, provided that it had a mass m ξ 0 2.86 TeV. Here in the present work, we will explore the full viable mass region of ξ 0 and investigate physical density resonances which are crucial for the experimental detections.
It is obvious that the DM candidates mainly annihilate to the SM particles. The dominant channels for the fermion DM pair annihilation ξ 0 to the SM particles are presented in Figure 7.
The thermal average annihilation cross-section times the relative velocity is approximated as . 7. Dominant contributions to the fermion DM pair annihilation into the SM particles.
follows which are not significant in the scalar DM case. For this aim, the H 3,4 contributions to the total annihilation cross-section can directly be suppressed, as omitted, by assuming thatλ is radically smaller than the gauge coupling constant, g.
Note that ξ 0 does not interact with Z at the effective limit u, v w, Λ. To investigate the direct detection of the field ξ 0 via the SI scattering on nuclei, we write the effective Lagrangian that describes the interactions of ξ 0 with fundamental level quarks, induced through the t-channel exchange of the field Z , such as Thus, we obtain the SI cross-section for the scattering of the ξ 0 on a target nucleus [25], where m ξ 0 N m N .
For numerical computation in this subsection, we take m Z 91.187 GeV and relevant parameters given in (69). In Figure 8, we plot the relic density of the DM as a function of its mass according to the several choices of w and Λ. We see that each density curve always contain a quite narrow resonance at m ξ 0 = m Z /2, at which the relic density is substantially reduced, tending to zero. Whereas, outside the resonance region, the relic density slowly increases. Since the gauge couplings are fixed and that the Z-Z mixing angle is small, the relic density only depends on m ξ 0 and w, Λ through m X,Y,Z . It is proportional to m 4 X,Y,Z /m 2 ξ 0 for m ξ 0 close to the weak scale, while it is proportional to m 2 ξ 0 for m ξ 0 much beyond m X,Y,Z . Because the range of the DM mass considered in Fig. 8 is narrow, the corresponding relic density outside the resonance region is weakly changed as scaled by the resonance mass m Z . To see a significant change, we make an estimation, Ωh 2 = 2.8, 0.7, and 3.75 for m ξ 0 = 1, 20, and 50 TeV, respectively, according to the right panel. Whereas, Ωh 2 = 0.6 and 3.6 correspond to m ξ 0 = 20 and 50 TeV, respectively, according to the left panel. In Figure 9, we plot the SI scattering cross-section limit as a function of the DM mass according to the above choices. The limits are in good agreement the constraint from XENON1T [28]. Combining the results in Fig. 8 and Fig. 9, the viable DM mass regime is as follows: m ξ 0 = 1.3-3.2 TeV for w = 5 TeV, Λ = 6 TeV; m ξ 0 = 3.5-4.4 TeV for w = 8 TeV, Λ = 9 TeV; and m ξ 0 = 4.9-5.5 TeV for w = 11 TeV, Λ = 12 TeV. Here the middle one is set by the relic density with a plot similar to the right panel in Fig. 8, which was not displayed. Last, but not least, the scalar sextet (S) and triplet (χ) are decomposed as singlets respectively, hence they can develop the VEVs, S 33 = Λ/ √ 2 and χ 3 = w/ √ 2, for breaking SU (3) L down to SU (2) L as conserved. At this stage, w, Λ contribute to all the new particle masses, as seen in the new gauge and Higgs bosons above. Additionally, w gives exotic quark mass, while Λ provides ξ mass, and both w, Λ supply E mass. All that implies a similar role between w, Λ as mutually contributing to the new physics and interacting of their fields with the SM. Although we have chosen Λ > w in interpreting the results, we reexamined that an opposite choice, w > Λ, or including both cases, but always ensuring w ∼ Λ, lead to the same physics.

VI. CONCLUSION
In this work, the fully flipped 3-3-1-1 model has been interpreted. The scalar sector has explicitly been diagonalized, yielding the appropriate particle spectrum. We have shown that the model   c ϕ → s ϕ and s ϕ → −c ϕ .

Appendix A: Couplings of fermions and scalars with gauge bosons
This appendix is devoted to determine all the couplings of fermions and scalars with gauge bosons, given throughout Tables II, III, IV, V, VI, VII, VIII, IX