Spin-$0^\pm$ portal induced Dark Matter

Standard model (SM) spin-zero singlets are constrained through their di-Bosonic decay channels via an effective coupling induced by a vector-like quark (VLQ) loop at the LHC for $\sqrt{s}$ = 13 TeV. These spin-zero resonances are then considered as portals for scalar, vector or fermionic dark matter particle interactions with SM gauge bosons. We find that the model is validated with respect to the observations from LHC data and from cosmology, indirect and direct detection experiments for an appreciable range of scalar, vector and fermionic DM masses greater than 300 GeV and VLQ masses $\ge$ 400 GeV, corresponding to the three choice of portal masses 270 GeV, 500 GeV and 750 GeV respectively.


Introduction
ATLAS and CMS [1][2][3][4] have been assiduously searching for di-Boson production in the semileptonic, fully leptonic and di-Bosonic channels at √ s = 13 TeV with the integrated luminosity of 3.2 f b −1 . They have looked for a scalar spin zero resonance of mass > 200 GeV and a spin 2 Randall-Sundrum graviton state as benchmark model of mass > 500 GeV. Assuming a scaling of cross-section for an s-channel resonance produced by gluon fusion (light quark-antiquark annihilations) the consistency between the 13 TeV data and the data collected at the 8 TeV is found at the level of 1.2 (2.1) standard deviation. An excess of di-photon events at a mass of 750 GeV reported by the LHC's ATLAS and CMS experiments in 2015 had led to a flurry of activity resulting in more than 500 papers in a short span of time (see for example [5] and references therein). The LHC phenomenology of the 750 GeV di-photon resonance was also studied in the framework of the effective field theory (EFT) and extended to include this new found resonance induced interactions of the standard model (SM) singlet fermionic and/ or scalar DM with the SM gauge Bosons of the visible world [6][7][8][9][10]. The excess reported in 2015 however did not show up in 2016 data. ATLAS and CMS results of run 1 at LHC also saw the enhanced production of SM Higgs Boson in association with a top quark. A possible explanation put forward in [11] was to interpret the data due to the existence of another heavier scalar with the best fit mass of 272 +12 −9 GeV. This particle unlike the SM Higgs was supposed to interact with dark matter. The most promising mechanism for the production of di-photon resonance discussed in the literature is through gluon-gluon fusion and its subsequent decay into SM di-Bosons viz gg → φ 0 /A 0 → γγ. We will examine the possibility of this resonance to act as a portal between the dark matter particle (DM) of any spin (0, 1/2, 1) with the SM particles and examine the constraints on the model parameters from the observed relic density, direct and indirect observations while satisfying the constraints from ATLAS and CMS results [1][2][3][4].
In section 2 we describe a simple renormalizable model by augmenting the particle content of the SM to include an SU (2) L singlet scalar/pseudo-scalar particle and vectorlike SM colour-triplet fermions of exotic charge Q ψ . The DM particles in this model are neutral SM SU (2) L × U (1) Y singlets which are odd under a discrete Z 2 symmetry and can be scalars, fermions or vectors. These particles interact with the SM gauge Bosons through the scalar/pseudo-scalar portal. We compute the partial decay-widths of the scalar and pseudo-scalar in the subsection 2.1. In the subsection 2.2 we analyse the di-Boson production cross-section observed by the ATLAS and the CMS experiments [1] in p − p collision in the context of the model discussed here and obtain constraints on the coupling of the di-Boson resonance with vector-like fermions. With these constrained couplings of the portal scalar and pseudo-scalar, we compute the relic density contribution of the viable DM candidates through their interactions with the visible world in the subsection 3.1. The indirect detection of the DM candidates through the emission of monochromatic γ-rays by Fermi-LAT [12,13], a satelite based γ ray observatory and the ground based Cherenkov telescope H.E.S.S. [14,15] is discussed in the subsection 3.2. Further, we investigate the possibility of direct detection of such DM particles in the elastic DM -nucleon scattering experiments in Dark-Side50 (2016) [16], LUX [17,18], XENON [19,20] and PANDA [21] collaborations in the subsection 3.3. Section 4 summaries our analysis and results through composite figures, where all experimental constraints are used to look out for allowed region of the model.

The Model
We consider a portal induced dark matter model in which the di-Boson resonance is either a CP even scalar (Φ) or a CP odd (P) scalar. The di-Boson coupling is introduced through vector-like SU (3) C triplet fermion with exotic charge Q ψ = +5/3. In addition we propose the dark matter to be a real scalar, a real vector or a spin 1/2 Dirac fermion. In order to avoid mixing of the VLQ's with SM quarks and to stabilise the DM particles we invoke an Abelian U (1) d gauge symmetry. The U (1) d sector gauge Lagrangian contains where ϕ is a charged scalar, V 0 µν is the U (1) d field strength tensor of the gauge field V 0 µ and V (ϕ) is the scalar potential. The charged scalar field in the dark sector allows the spontaneous breaking of U (1) d gauge symmetry to a Z 2 subgroup after ϕ develops a non-zero VEV v ϕ . The imaginary part of ϕ is eaten up by V 0 µ to give it a mass m V 0 = λ ϕ v ϕ /2, where λ ϕ is the gauge coupling [22,23]. The usual charge conjugation Z 2 symmetry V 0 µ → − V 0 µ makes this massive gauge field a viable stable DM candidate.
The Lagrangian of the model is given as follows: Here H is the SM Higgs SU (2) L doublet and µ 2 , µ 2 Φ < 0 and µ 2 P > 0. After spontaneous symmetry breaking, CP even scalar Φ picks up a VEV and can be written as The Lagrangian after the electroweak and U (1) d symmetry breaking is re-written as The Quantum number assignments of new scalars, pseudo-scalar, vector-Boson and fermions of the dark U (1) d sector under the gauge symmetry group SU Table 1.
In general Φ and P will mix with SM Higgs and consequently the DM scalar η will interact with SM particles through the Higgs portal unless λ HΦ (λ HP ) is zero. Even in the absence of λ HΦ (λ HΦ ) term, a mixing term h 0 − φ 0 (h 0 − A 0 ) will be generated radiatively through multi-loop diagrams as shown in Figure 1 and will be highly suppressed. Recently, the authors of reference [24] have reviewed the scalar induced DM models and considered such ∼ 10 % Higgs mixing with the scalar portal, to generate the required DM relic density in the universe. However, we will assume this term is absent and the radiatively induced portal φ 0 / A 0 -Higgs mixing is negligible such that the DM interacts with SM particles albeit gauge-boson pairs only through the di-Boson portal. The decay of (3, 1, 5/3) VLQ is protected by Z 2 symmetry and the lifetime of such singlet VLQs is naturally large, as there is no renormalizable, gauge invariant operator to mediate their decays into SM particles. However, VLQs form a bound state which can decay through non-trivial mechanism. This subject has been analysed in detail by the authors of reference [25], where they have also considered the possibility of introducing an additional Z 2 odd VLQ doublet ψ in the (3, 2, 7/6) representation facilitating the decay of the singlet VLQ via an off-shell ψ with the constraint m Ψ > m Ψ such that the extra VLQ does not contribute appreciably to the di-photon spectrum. On the other hand, the authors of reference [26] while exploring the various co-annihilation scenarios where the colored VLQ's are slightly heavier than some new dark matter state so that they lie in the compressed spectra also found that VLQ's are likely to hadronize before they can decay. This bound state formation of VLQ opens up the frontier to look for the new resonances at the ongoing and proposed particle accelerators. In LHC, one can expect to observe the second peak at ∼ 2 m ψ in the di-Boson invariant mass distribution in pp → γγ, γZ, and ZZ channels, following the primary peak due to the scalar/ pseudo-scalar portal at m φ 0 / A 0 and the preliminary theoretical exercise has been performed in reference [25].

Partial decay-widths to Gauge Bosons
Interactions described in equations (2.5)-(2.9) indicate that there are no tree level couplings of the singlet scalar and pseudo-scalar with the SM neutral gauge Bosons. However, the non-vanishing couplings are generated at the level of one loop which are induced by fermions.

Particle
Spin       Figure 3: Figure 3a, depicts the variation of the partial decay-widths of the pseudo-scalar with the VLQ mass m ψ for the three values of the decaying pseudo-scalar masses 270, 500, 750 GeV respectively and Figure 3b shows the partial decay-widths of the 270 GeV pseudo-scalar to pair of gg, γγ, γZ and ZZ gauge Bosons respectively.
The dominant one loop contribution comes from the VLQ which is the heaviest fermion available in our model. They are evaluated in the appendix A. These one loop amplitudes can then be translated in the language of the effective field theory as the effective couplings of the three interacting fields. They become the coefficients of the four distinct effective interacting three point vertices, each for scalar and pseudo-scalar. The resulting effective Lagrangians can then be written as even and odd scalars are calculated in terms of the loop integrals in Appendix A. We studied the variation of the partial decay-widths Γ φ 0 →γ γ with the VLQ mass m ψ varying between 400 GeV -1.6 TeV for the three choice of the scalar portal masses namely m φ 0 = 270, 500 and 750 GeV in Figure 2a. We also compare the partial decaywidth of the scalar to the di-photon channel with the dominant channel φ 0 → g g and the suppressed channels of φ 0 → γZ and φ 0 → Z Z in Figure 2b.
Similarly the pseudo-scalar partial decay-width variations are shown in Figures 3a and 3b.

Constraints on the model from LHC
We study the production cross-section of these exotic scalar and/ or pseudo-scalar at the LHC (pp → φ 0 / A 0 → V 1 V 2 ). The di-photon production cross-section σ(p p → φ 0 / A 0 → γ γ) is mainly through gluon fusion. As shown in the previous section that the partial decay-widths of scalar / pseudo-scalar being negligibly small in comparison to their masses, we can calculate the production of SM di-Bosons V 1 and V 2 (V 1, 2 ≡ γ, Z) in the narrow width approximation of the scalar (or pseudo-scalar). The cross-section is given as (2.11) Using equation (2.11), we evaluate the cross-sections for the processes pp → γγ, gg, γZ, ZZ and compare with the observations for a spin 0 ± resonance mass in the narrow width approximation from the CMS and ATLAS run II collaboration at √ s = 13 TeV and an integrated luminosity of 3.2 fb −1 [1][2][3]. We have tabulated these results only for the γγ and γZ channels for the benchmark resonance masses m φ 0 / m A 0 = 270 GeV, 500 GeV and 750 GeV in Table 2.
Figures 4a and 4b show the 2-sigma limits on the couplings allowed by the LHC data at √ s = 13 TeV (as given in Table 2) in the plane defined by the VLQ mass m ψ and its coupling with the y φ 0 / y A 0 for the scalar and pseudo-scalar portal respectively. Each plot depicts the three different bands of allowed region corresponding to the three choices of scalar or pseudo-scalar masses 270, 500 and 750 GeV respectively.

Portal induced Dark matter Scenarios
DM which are popularly known as weakly interacting massive particle (WIMP) do not have either electromagnetic or strong interaction. One of the most challenging tasks today is to identify the nature of the DM particle [27].  Since the investigation of the nature of the DM particles needs an understanding of the underlying physics of the model and vice-versa, we would like to begin our analysis by considering the spin of DM particle to be either 0, 1 and/ or 1/2. Before, proceeding with the analysis, we consider the existing cosmological constraints on such a DM candidate from the the WMAP [28,29] and Planck data [30].

DM pair-Annihilation and Relic Density
In the model described above in section 2, the proposed scalar η, vector V 0 and fermion χ DM candidates can interact with the SM gauge-bosons through CP even φ 0 and odd A 0 scalars respectively. In the early universe SM particles remained in thermal equilibrium as long as their reaction rate was faster than expansion rate of the universe. As the universe cooled, the reaction rate fell below the expansion rate and DM particles de-coupled from the thermal bath and contributed to the relic density observed today. The equilibrium in the early universe was maintained via the leading DM pair annihilation processes viz into pair of SM particles. The vector-like quark -antiquark pair and a pair of portal scalar/ pseudo-scalar can also be produced as a result of the annihilation of DM particles provided mass of the DM particle is higher than those of ψ and/ or φ 0 /A 0 .
Therefore as a next logical step we compute the thermal averaged DM pair-annihilation cross section. The DM pair annihilation is facilitated through the portal mediated s channel processes assuming the momentum transfer in the scattering to be much less than the portal mass. These annihilation processes lead to the following visible final states: g g, γγ, γZ and ZZ. Therefore, the s channel processes at such low energy appears to be an effective point interaction among the SM vector bosons and pair of DM candidates, suppressed by the portal mass squared. As to the couplings of Dark Matter with the portal, we consider two different cases, namely (a) Scalar portal couplings to the scalar, vector and Fermion DM pair which are defined by κ ηφ 0 , κ V 0 φ 0 , κ χφ 0 respectively.
(b) Pseudo-scalar portal couplings to the fermions which is defined as κ χA 0 .
In the Appendix B.1, we calculate the thermal averaged cross-section for the scalar DM pair annihilation via scalar portal to the above visible states and are given in equations (B.1)-(B.4). In addition, to these the pair annihilation also lead to a production of the vector like quark-antiquark pair via the scalar portal and its thermal averaged cross-section is given in equation (B.5), which is kinematically possible, only for DM mass greater than VLQ mass.
The corresponding thermal averaged cross-section for the annihilation of vector DM to SM gauge Bosons can be obtained directly by substituting κ ηφ 0 by For the VLQ pair production we multiply a factor 1/6 to the contribution given by scalar DM in (B.5).
The thermalized fermionic DM pair annihilation cross-section for the corresponding final states are given in equations (B.7)-(B.11). These cross-sections for every annihilation channel are p-wave suppressed unlike the scalar DM case and thus result in low thermally averaged cross-sections. Therefore, the sensitivity of the fermionic DM -scalar mediator coupling is an order of magnitude less sensitive than those of the corresponding couplings of the scalar and vector DM candidates with the portal.
In addition, the t-channel annihilation diagrams also contribute to the relic density, where DM pair annihilation to a pair of portal scalars can become kinematically feasible for DM mass ≥ m φ 0 . The thermal averaged cross-sections for these processes are given in equations (B.6) and (B.12) for pair annihilation of scalar and fermionic DM respectively. The contribution of the vector DM pair annihilation to the pair production of such scalars via t channel is found to be identical to that of the scalar as given in (B.6).
Unlike the scalar portal case, pseudo-scalar cannot decay to two scalar or vector DM pairs and leaving no choice but to consider relevant spin 1/2 DM candidate. We are now well equipped to calculate the present day relic abundance of DM by solving the Boltzmann equation: where H =ȧ a = 8πρ 3M ρ l , σ|v| is the thermally averaged cross-section and n EQ DM = g where the number of degrees of freedom g are 1, 2 and 3 for scalar, fermionic and vector DM respectively. The freeze-out occurred when DM is nonrelativistic with v c and then σ|v| can be written as The Boltzman equation is solved numerically following the reference [31] to give the thermal relic density where g (x F ) is the total number of effective degrees of freedom at the freeze-out temper-ature T F and x F = m DM /T F is obtained by solving where C is of order 1. For the Dirac fermionic DM, the additional contribution from the anti-particle will make the Ω DM h 2 twice of that is given in equation (3.2). We compute the relic density numerically using MadDM [32,33], which require the pair annihilation cross-sections to be calculated by the event generator MadEvent [34,35]. We have generated the input model files containing all the Feynman rules from the Lagrangian given in equations (2.5)-(2.9), (2.10a) and (2.10b) for the MadEvent using FeynRules [36,37].
To analyse and study the model we consider a single DM candidate with a specific intrinsic spin quantum number associated with a given portal at a time. In other words, all DM-portal couplings bar the one under discussion shall be switched off to zero. To keep our calculation in compliance with the LHC data, we use the central values of the portal-VLQ coupling y φ 0 (y A 0 ) for a given m ψ and the portal mass m φ 0 (m A 0 ) as computed from the experimental cross-section curves for the gauge-boson production at ATLAS [1][2][3] and given in the Figures 4a and 4b respectively.
We have verified analytically the relic DM abundance by taking g (x F ) = 92 and C = 1/2 in equations (3.2) and (3.3) and found them to be in agreement with the numerical calculations done by MadDM.
In Figure 5a, we depict the contours (for the scalar portal mass of 270 GeV and VLQ masses 0.4, 0.8, 1.2 and 1.6 TeV) in the plane defined by the varying scalar DM mass between 0.02 -4.0 TeV and the scalar portal -scalar DM coupling κ ηφ 0 which generate the correct amount of present day energy density for the scalar DM. In Figure 5b, we show the variation of relic density ∼ 0.11 curves corresponding to the three scalar portal masses 270, 500 and 750 GeV for a fixed VLQ mass of 400 GeV. In Figures 5c and 5d we plot the relic density ∼ 0.11 contours in the plane defined by the varying vector DM mass between 0.02 -4.0 TeV and the scalar portal -vector DM coupling κ V 0 φ 0 for the values of portal and VLQ masses. These contours are evaluated corresponding v Φ = 1 TeV. In Figures 6a and  6c we plot the constant relic density contours for the fermionic DM in the plane defined by the DM mass and DM-portal coupling w.r.t. corresponding to the scalar and pseudo-scalar portals respectively. Three contours in Figures 5b, 5d, 6b and 6d correspond to the three choices of the portal masses 270, 500 and 750 GeV respectively. We observe that • the unshaded region below the curve is disallowed as it would over-close the universe with the DM. The perturbativity requirement that the coupling should be less than √ 4 π further shrinks the allowed parameter region.
• the pair annihilation cross-section on varying with DM mass maximizes at the DM mass m φ 0 /2 for which the constant relic density contours drop sharply w.r.t. the coupling. The sharp fall in the constant relic density contours are again observed at the two different values of DM masses a) first at DM mass ≈ m ψ where the portal mediated s channel pair annihilation process opens up for the pair production of VLQ's and b) then at DM mass ≈ m φ 0 (m A 0 ) where DM mediated t channel annihilation process opens up for pair production of portal scalars (pseudo-scalars) respectively.
• the relic density curve corresponding to the lowest VLQ mass spans the minimal allowed region as depicted via blue shaded region in Figures 5 and 6. Increasing the mass of VLQ requires its coupling with the portal to be large so that it is consistent with the gauge Boson pair production at LHC, which in turn pull down the DMportal coupling to a much lower value such that enough relic density is generated. Therefore, the contribution of the DM greater than 1 TeV to the relic density can be made favourable with the choice of high VLQ masses > 400 GeV.
• the absence of the portal VEV dependence in the interaction Lagrangian of the fermionic DM renders its coupling with the portal to be more sensitive. The portal coupling with the DM are found to be an order of magnitude higher than the scalar and vector DM for an appreciable range of DM mass to generate the same relic density. Consequently, the allowed parameter region for which the perturbativity is satisfied becomes highly constrained.
We have performed the rest of our analysis with the conservative choice for VLQ mass of 400 GeV, corresponding to the three choices of portal masses 270, 500 and 750 GeV.

Indirect Detection : Monochromatic Gamma Rays
DM annihilation to SM photons (high energy gamma rays) in galactic halos can be generated from various astrophysical targets for example the Dwarf Spheroidal galaxies, Galactic centre and Galaxy clusters [12,13,38]. These gamma rays can travel galactic distances and their flux can be observed by the satellite based γ ray observatory Fermi-LAT [12,13] and the ground-based Cherenkov telescope H.E.S.S. [14,15]. The annihilation rates are roughly velocity independent in the non-relativistic region. Here in particular we compare the bounds from the indirect detection experiments in the γγ and γZ channels.
In our model, the production of the monochromatic photons are realised in the DM pair annihilation to γγ and γZ two body final states. These processes are studied in the context of the scalar portal induced interactions for the scalar, vector and fermionic DM candidates while the pseudo-scalar induced interactions are only allowed for spin 1/2 DM candidates. We can directly use the thermal averaged cross sections which are expressed in terms of the local velocity of the DM particle in the Appendix B. We analyse the variation of the thermal averaged DM annihilation cross-sections to γγ and γZ w.r.t. the DM mass in detail for all the cases and are depicted in the Figures 7 and 8. To calculate the thermal averaged cross-section for the indirect detection we shall use the conservative lower bound on the DM-portal effective coupling obtained from the relic density criterion for a given DM mass. Therefore, the region below the curves defined in the DM mass and DM-portal coupling plane become cosmologically disfavoured. We plot and compare our results in the   and 7d show the thermal averaged cross-sections for vector DM pair annihilation V 0 V 0 → φ 0 → γγ and V 0 V 0 → φ 0 → γZ processes via scalar portal respectively. All the cross-sections are drawn for the upper limit on DM couplings allowed by the relic density constraints for a given DM mass. We have also exhibited the Fermi-LAT 1σ and 2σ limits for DM mass range < 500 GeV [13] and H.E.S.S. 2013 upper limit on the thermal averaged cross-section for DM mass range > 500 GeV [15] corresponding to γγ and γZ channels. Shaded regions appearing in blue, golden yellow and red correspond to the relic density forbidden regions for the portal mass of 270, 500 and 750 GeV respectively.
γγ mode with the limits obtained from Fermi-LAT [13], for the restricted DM mass range (< 500 GeV).
We find that the model calculated thermal averaged DM pair annihilation cross-section in the γγ channel lies below the limit obtained from the experimental results for an appreciable range of DM mass. The region trapped between the experimental curve and our results from the top and below respectively depicts the allowed region of the thermal averaged cross-section, which can further be translated in terms of the allowed model parameter space w.r.t. relic density and the indirect detection. Therefore the indirect experimental results naturally provides the upper bound on the coupling, for a given DM mass. We note   and 8d show the thermal averaged cross-sections for fermionic DM pair annihilation χχ → A 0 → γγ and χχ → A 0 → γZ processes via pseudo-scalar portal respectively. All the cross-sections are drawn for the upper limit on DM couplings allowed by the relic density constraints for a given DM mass. We have also exhibited the Fermi-LAT 1σ and 2σ limits for DM mass range < 500 GeV [13] and H.E.S.S. 2013 upper limit on the thermal averaged cross-section for DM mass range > 500 GeV [15] corresponding to γγ and γZ channels. Shaded regions appearing in blue, golden yellow and red correspond to the relic density forbidden regions for the portal masses of 270, 500 and 750 GeV respectively.
that the σ χχ → φ 0 → γγ v is p-wave suppressed and therefore lies much below than that of the null result obtained from the FermiLAT.

Direct Detection
Direct detection of the DM identifies the nature of the low-energy effective DM-nucleon scattering interaction. Therefore, the direct detection experiments aim to establish a first confirmed detection of DM particles. Direct detection data analysis focuses on formulation and computation of the momentum (energy) and velocity dependent cross-section scenarios, where DM couples to target nuclei by spin-independent or spin-dependent interactions. Spin-independent cross-section scales coherently with the nucleon number, so nuclei of larger atomic mass are always more effective in the direct detection searches. The spindependent scattering cross-section on the other hand is most effectively probed by a nucleus with larger spin.
Direct detection experiments Dark-Side50 [16], LUX [17,18,21], XENON1T [19] etc. are set to observe the recoil energy transferred to target nucleus in an elastic collision with the DM particles. The current experimental null results [19], constrain the maximum value of the elastic nucleon-DM cross-sections. The constraints on the upper limit of the elastic cross-sections will be considerably lowered in the future projected sensitivities of the super CDMS experiments [39].
The elastic scattering of DM particles η, V 0 and χ from a heavy nucleus can be illustrated in terms of the effective DM scattering off the gluons, where the DM is attached to the triangle loop of charged virtual VLQ via the portal which in turn interacts with the nucleons via two gluons exchange. Since, this occurs at very low energy and momentum O(1) GeV, all the propagators are approximated by their respective masses. The nucleus recoil which is of the order of few MeV is then measured in the detector. DM particles scatter-off nucleons through the t channel exchange of portal scalar and the scattering of gluons via triangle VLQ loop, gives the dominant contribution to the DM-nucleon scattering cross-section. The scalar gluon-gluon coupling is described by the effective Lagrangian given in (2.9a) and (2.9b) as where the loop integral I gg is given in Appendix A. The effective DM-gluon interaction can be described by the effective Lagrangian through the Feynman diagram given in Figure 9. This interaction contributes to the spinindependent part of the DM-nucleon scattering cross-section to give is the nucleon-DM reduced mass and f N T G ≡ − 9 αs(µ) m N 8 π N |O g | N is the gluon contribution to the zero-momentum hadronic matrix element. f N T G can be extracted in terms of light quark contribution to the zero-momentum hadronic matrix element as Here the hadronic matrix element refers to a definite spin state of the nucleon. The f N T G is calculated to be 0.92 using the values for f N T q quoted in the literature [27,[40][41][42].
We estimate the direct detection cross-section using the central value of y φ 0 obtained from the LHC for a given portal scalar mass and m ψ = 400 GeV along with the cosmologically allowed lower bound on the DM -scalar portal coupling for a given DM mass. The variation of the scalar DM -Nucleon scattering cross-section via scalar portal is depicted in the Figure 10a corresponding to the three scalar portal masses 270, 500 and 750 GeV respectively.
On the similar note, the spin-independent cross-sections for the vector and fermionic dark matter interaction via the scalar mediator are given by The variation of the vector and fermionic DM -Nucleon scattering cross-sections via scalar portal are depicted in the Figures 10b and 10c respectively. Each figure depicts the scattering cross-section for three scalar portal masses 270, 500 and 750 GeV respectively.
On comparison we find that for the most of the DM mass range our direct detection cross-section curves in Figures 10a, 10b and 10c lie much below the null result of the experiment [19] which in turn gives an upper bound on the DM -portal coupling for a given DM mass and thus constrains the portal induced DM parameter space. We note that the scalar and vector DM scattering cross-section is an order of magnitude higher than that of the fermionic DM contribution. Thus the DM scattering experimental constraints do not shrink the allowed parameter space for the fermionic DM below the mass region 300 GeV which is in contrary to the scalar and vector DM contributions. In each of the figure three graphs are exhibited for the fixed VLQ mass 400 GeV and scalar portal masses 270, 500, 750 GeV respectively. All the cross-sections are drawn for the upper limit on the respective DM couplings allowed by the relic density constraints for a given DM mass. We have also exhibited the experimental cross-section from LUX (2017) [17,18], XENON (2017) [19] and Dark-Side50 (2016) [16]. Shaded regions appearing in blue, golden yellow and red correspond to the relic density forbidden regions for the portal masses of 270, 500 and 750 GeV respectively.
For the direct detection cross-section induced by the CP odd pseudo-scalar mediator the corresponding A 0 g g coupling and the effective DM-gluon interaction is given by The effective Lagrangian (3.11) generates a spin-dependent DM-nucleon cross-section which is however suppressed by the square of the momentum exchanged and is therefore sub-dominant. The available experimental results on the spin-dependent cross-sections are comparatively less tightly constrained [18,21,43], hence not shown graphically.   [15] (red) and the perturbativity condition (golden yellow) respectively in the plane defined by m η − κ ηφ 0 , Figures 11a, 11b, 11c, and 11d respectively. Constraint from the direct detection experiments for the pseudo-scalar portal are not shown (see text).

Summary and Conclusions
Our analysis provide a conservative complementary scenario to that appeared in the review [24], where we have neglected the portal mixing with the SM Higgs and instead we have generated the required relic density of DM by considering an effective loop induced interaction of the VLQ with the portal.
In this paper we considered a U (1) d extension of the standard model with a dark sector and a singlet scalar or a pseudo-scalar di-Boson resonance which interacts with the SM gauge Bosons through a vector-like SM colour triplet fermion of exotic charge Q = 5/3. The dark matter particle considered here is a neutral SM singlet real scalar, a real vector or a spin-1/2 fermion interacting with the standard model gauge Bosons through a scalar/pseudo-scalar di-Boson resonance. As a first step we obtained the constraints on the coupling of di-Boson resonance with the vector-like fermions from the ATLAS and   [15] (red) and the perturbativity condition (golden yellow) respectively in the plane defined by m η − κ ηφ 0 , Figures 12a, 12b, 12c, and 12d respectively. Constraints from the direct detection experiments for the pseudo-scalar portal are not shown (see text).
CMS experimental searches through spin-zero di-photon production cross-section σ (pp → φ 0 /A 0 → γγ). The constrained parameter region in (y φ 0 /A 0 − m ψ ) plane is depicted in Fig 4 at the 2σ level. With these constraints in place, we obtained relic density contours in the dark matter mass-coupling parameter space assuming that the dark matter particles considered here saturate the observed relic density Ω c h 2 0.1138 ± .0045. We then study that the prediction of the LHC and relic density constrained model for a) the air-borne indirect detection of DM pair annihilation into a pair of γγ/ γZ in the galactic halo and b) the ground based DM -nucleon scattering direct detection experiments.
On comparing the thermal averaged annihilation cross-sections for the indirect detection with the limits on the cross-section obtained from the Fermi-LAT and H.E.S.S. (2013) in the mass range less and greater than 500 GeV respectively through the observation of monochromatic gamma-rays, we found that the model predicted annihilation cross-section   [15] (red) and the perturbativity condition (golden yellow) respectively in the plane defined by m η − κ ηφ 0 , Figures 13a, 13b, 13c, and 13d respectively. Constraints from the direct detection experiments for the pseudo-scalar portal are not shown (see text).
with the allowed lower bound of scalar/ vector -scalar portal and fermionic DM -pseudoscalar portal couplings are being favoured by the limits obtained from these experiments for the DM masses 400 GeV and above as shown in Figures 7, 8c and 8d. However, the model prediction of the annihilation cross-section for fermionic DM induced by the scalar portal lies far below the experimental upper limit and therefore favours the large model parameter space spanning the mass range upto 4 TeV as shown in the Figures 8a and 8b. Comparing the constrained model prediction for spin-independent direct detection DMnucleon elastic scattering experiments with the recent experimental results from LUX (2016) [17], XENON1T (2017) [19], PANDA (2017) [21] and Dark-Side50 (2016) [16] collaborations as shown in Figures 10a, 10b and 10c respectively, we find that almost the entire parameter space allowed by the LHC searches and the observed relic density is also allowed by the direct detection experiments for the fermionic dark matter. The case of real scalar and vector dark matter particles is different, where the parameter region is favoured only for the mass range greater than 300 GeV.
Finally, it is important to comment upon the model restrictions, if it has to satisfy each and every experimental results. This helps in narrowing down the search for such DM candidates in the ongoing and upcoming collider based experiments. To this end we translate the experimental results in terms of restriction on the DM mass and DM -portal coupling for the mass range varying between 10 GeV and 4 TeV. Therefore, we summarise our analysis with the help of the three composite Figures 11, 12 and 13 corresponding to the three choice of the scalar portal masses 270, 500 and 750 GeV respectively, where we have put all the constraints from the relic density, two indirect detection experiments and two direct detection experiments on the model for a specific DM candidate in a plane spanned by its mass and its coupling to the scalar and/ or pseudo-scalar portal. These Figures spell out the implications of the experimental results on the model parameter space and depict • the contours drawn with the coordinates of the lower limit on the DM -portal coupling for a given DM mass derived from the constant relic density [28,30] • the contours drawn with the coordinates of the upper limit on the respective DMportal coupling for a given DM mass by demanding such DM candidates to satisfy the observed null results of the thermal averaged pair production cross-section for γγ final state due to a DM pair annihilation from Fermi-LAT [13] (H.E.S.S. 2013 [15]) for DM mass range less (greater) than 500 GeV.
• the contours drawn with the coordinates of the upper limit on the respective DMportal coupling for a given DM mass by demanding such DM candidates to satisfy the observed null results of the sensitive DM -Nucleon scattering cross-section from the recent experiments XENON1T (2017) [19] and PANDA (2017) [21]. However, since we do not have a stringent experimental constraints on the spin dependant fermionic DM -nucleon cross-sections, we do not show any direct detection contours for the pseudo-scalar induced fermionic DM.
It is remarkable to note that we have now shrunk the allowed parameter space which appear as white unshaded region in each of these figures. However, area of the allowed unshaded white region in Figures 11-13 for the DM masses greater than 1 TeV can be enhanced by increasing the VLQ mass above 400 GeV. Thus this tightly constrained spin 0 ± induced DM model has become challenging enough to be probed in the colliders. computed and are given as The expression for Γ(φ 0 → Z Z) is correct to ∼ 0.1%.
Similarly, the effective couplingsκ V V of the pseudo-scalar to the pair of g g, γ γ, γ Z and Z Z are obtained by replacing y φ 0 → y A 0 , m φ 0 → m A 0 and I V V →Ĩ V V in equations (A.1a)-(A.1d) respectively. The corresponding loop integralsĨ V V are obtained by substituting m φ 0 → m A 0 and f (x, y) by 1 in equations (A.2a)-(A.2c) respectively. Accordingly, the partial decay-widths of the pseudo-scalar to a pair of SM neutral gauge Bosons can be obtained by replacing y φ 0 by y A 0 , m φ 0 by m A 0 and I V V byĨ V V in equations (A.3a)-(A.3d).