Gauge Singlet Vector-like Fermion Dark Matter, LHC Diphoton Rate and Direct Detection

We study a gauge-singlet vector-like fermion hidden-sector dark matter model, in which the communication between the dark matter and the visible standard model sector is via the Higgs-portal scalar-Higgs mixing, and also via a hidden-sector scalar with loop-level couplings to two gluons and also to two hypercharge gauge bosons induced by a vector-like quark. We find that the Higgs-portal possibility is stringently constrained to be small by the recent LHC di-Higgs search limits, and the loop-induced couplings are important to include. In the model parameter space, we present the dark matter relic-density, the dark-matter-nucleon direct-detection scattering cross-section, the LHC diphoton rate from gluon-gluon fusion, and the theoretical upper-bounds on the fermion-scalar couplings from perturbative unitarity.


Introduction
The nature of dark matter is yet to be established and many particle physics candidates in theories beyond the standard model (BSM) are being considered. In this work, we add to the standard model (SM) a gauge-singlet "hidden sector" containing a vector-like fermion (VLF) dark matter ψ and a scalar φ. We also add an SU (3) c (color) triplet, SU (2) L singlet, U (1) Y hypercharge 2/3, vector-like quark (VLQ) U . These states could be the lower energy remnants of a more complete theory which we do not need to specify here. The hidden sector is coupled to the standard model (SM) via the "Higgs-portal" mechanism due to the φ mixing with the SM Higgs boson h, and also via the loop-level couplings of the φ to two gluons and the φ to two hypercharge gauge bosons induced by the U .
We analyze the large hadron collider (LHC) constraints on the model and find that the LHC di-Higgs channel imposes a tight constraint on the scalar-Higgs mixing. If the parameters are such that the Higgs-portal mixing is tiny, it becomes important to include the loop-induced couplings of the φ to the SM induced by the VLQ that offers another mechanism of communication between the hidden sector and the SM and generates the required size of the self-annihilation cross section that sets the dark matter relic density. Thus the presence of the VLQ is crucial for obtaining an acceptable phenomenology in the small Higgs-portal mixing limit. We present a scenario for which the scalar-Higgs mixing is tiny, and only the loop induced couplings communicate between the hidden sector and the SM. We show that the required values of the scalar-fermion Yukawa couplings are consistent with perturbative unitarity constraints by considering the ψψ → ψψ process.
The presence of the VLQ in the model affords a way to probe the model, and ongoing direct searches at the LHC are important. We briefly make contact with the extensive literature on searches for a VLQ at the LHC and the present constraints. Another signature of the model is a diphoton resonance signal due to φ production at the LHC via its digluon coupling and subsequent decay into photons via its diphoton coupling, both of these induced by the VLQ at loop-level. We study the diphoton signature of the model in detail. In our model, we obtain expressions for the one-loop scalar-gluon-gluon (φgg) and scalar-photon-photon (φγγ) effective couplings, and explore the phenomenology including these couplings. We obtain expressions for some relevant decay modes of the φ, present direct LHC constraints, the LHC gluon-gluon-fusion rate for φ production, and the LHC diphoton rate from φ decay for various total widths of the φ. For processes involving the dark matter, in addition to the Higgs boson contributions, the new contributions here are the s-channel φ contribution to the ψψ → SM self-annihilation process that sets the dark matter relic-density in the early universe, and the t-channel φ contribution to the interaction of the dark matter with a nucleon that leads to a direct-detection signal. We compute the dark matter relic-density and the dark-matter-nucleon direct-detection cross section for this model. Indirect detection of the dark matter via cosmic ray observables is another potential probe of the model, which we do not purse in this study but leave for future work.
We summarize next other studies in the literature that have some overlap with our work. Ref. [1] studies loop induced couplings of a singlet scalar to electroweak gauge bosons. Precision electroweak observables, and scalar and Higgs phenomenology at the LHC with a singlet scalar and VLFs present are analyzed in Ref. [2]. Singlet scalar decays to electroweak gauge bosons and to di-Higgs is studied in Ref. [3]. An analysis of a gauge-singlet fermionic dark matter in the Higgs portal scenario with significant φ ↔ h mixing is carried out for example in Refs. [4,5,6,7,8,9]. The phenomenology of a singlet scalar coupled to VLFs in the context of the earlier 750 GeV diphoton excess [10,11] which also discuss the dark matter implications of the neutral VLFs present in those models is studied in Refs. [12,13,14,15,16,17,18]. With more data accumulated at the LHC, it appears that the earlier diphoton excess at 750 GeV was a statistical fluctuation and is no longer significant at both ATLAS and CMS [19,20].
In this work, we study the prospects of a singlet VLF to be dark matter for various dark matter masses, taking a benchmark value of the φ mass of 1 TeV. We also present the constraint on the h − φ mixing angle (θ h ) from the LHC hh channel results [21], which is not analyzed in the references mentioned above. Usually in the literature, only the h mediated processes are included in the dark matter direct-detection cross section calculations. However, for small θ h (or when there is no mixing), the h mediated processes are suppressed, and the φ mediated process due to the φgg and φγγ effective coupling induced by the VLQ that we include here are important. Ref. [16] does include this contribution, although in the context of scalar dark matter and when the dominant contribution is the Higgs-scalar mixing contribution.
The rest of the paper is organized as follows. In Sec. 2 we present a model with a gaugesinglet vector-like fermion dark matter, that also contains a singlet scalar and a vector-like quark. We present a scenario that leads to a tiny singlet-Higgs mixing, in which case the loop induced couplings we include in this work become significant. We present the formulas for the SM fermion (SMF) and VLF contributions to the scalar-gluon-gluon (φgg) and scalar-photon-photon (φγγ) loop-level couplings. We compute the dominant φ decay modes. We infer the perturbative unitarity constraints on the φ couplings to the VLFs. In Sec. 3 we compute expressions for φ production in gluon-gluon fusion, discuss the direct LHC constraints on the model, including from the di-Higgs channel, and present the LHC diphoton rate. In Sec. 4 we present the preferred regions of parameter-space of the model that give the correct dark matter relic-density and are consistent with direct detection constraints, also showing the future prospects. In Sec. 5 we offer our conclusions. In App. A we show the range of possible diphoton rates by saturating the upper bound from the perturbative unitarity constraint, and also present diphoton rates in terms of the φgg and φγγ effective couplings.
2 Vector-like fermion dark matter model A VLF is composed of two different Weyl fermions as its L and R chiralities that belong to conjugate representations of the gauge group. In contrast to this, a chiral fermion contains a Weyl fermion without its conjugate representation partner. A gauge-singlet vector-like fermion again contains two different singlet Weyl fermions in contrast to a Majorana fermion which contains one. For a VLF, due to the presence of both chiralities, a mass term can be written in a gauge-invariant way without involving a Higgs field. This allows us to add TeV-scale mass terms for the VLFs. For VLFs, fermion number is a conserved quantity.
For us, the hidden-sector is any sector that is not charged under the SM guage symmetry, and we remain agnostic to the possibility that there are new symmetries in this sector that may even be gauged. For example, in theories with the factor group structure, G SM ⊗ G BSM , where G SM is the SM gauge group SU (3)⊗SU (2)⊗U (1), and G BSM is any new-physics group, the states charged only under G BSM and singlets under G SM will look like a hidden-sector to us. To include the possibility of the hidden sector scalar Φ to be in a nontrivial representation of G BSM , we take Φ to be complex, with the real component denoted as φ/ √ 2. For example, Ref. [7] discusses a model in which G BSM is a U (1) gauge symmetry.
Here, we present a model with a SM gauge-singlet hidden-sector containing a vector-like fermion dark matter candidate ψ with mass M ψ , and a CP-even scalar φ with mass M φ , that couples to the visible SM sector via loop-induced couplings due to an SU(2)-singlet color triplet VLQ U having hypercharge 2/3 and mass M U . (The color-triplet is the fundamental representation of the gauged SU (3) c of the SM.) This representation of the VLQ is just one choice out of many possible, and we take this for definiteness and to explore the phenomenology.
For a thermal dark matter candidate, the hidden-sector dark matter must couple to the visible SM sector by some operators. Some possibilities already considered in the literature include communication via: (a) an abelian gauge boson in the hidden sector mixing with the SM hypercharge gauge boson (see for example Ref [22] and references therein); (b) mixing between the φ and the SM Higgs (h), commonly called the 'Higgs-portal' scenario (see for example Ref. [7] and references therein). Here, we add another possibility (c) in which the communication between the hidden sector and the visible sector is mediated by a hidden sector scalar φ with loop induced couplings to the SM. The φ directly couples to the dark matter at tree-level, and at loop-level to the SM, in particular to two gluons and two hypercharge gauge bosons, induced by a vector-like quark (VLQ) U . The loop-level coupling of the φ to two hypercharge gauge bosons imply φγγ, φZγ and φZZ couplings. The U is the only new state that is charged under both G SM and G BSM and serves to connect the two sectors when the scalar-Higgs mixing is small, leading to an acceptable dark matter phenomenology. In this work, we do not explore option (a), and present a model in which (b) and (c) are both present. We show that in this model, the recent large hadron collider (LHC) di-Higgs channel constraints limits the Higgs-singlet mixing in possibility (b) to be small, and therefore including the loop-induced couplings of the φ to the SM, as in (c), will be important. Interestingly, in this model, the visible and hidden sectors do not decouple in the limit of the Higgs-portal mixing going to zero since the loop-level couplings induced by the VLQ remain as couplings between the two sectors. We explore this limit also.
The Lagrangian of the model is 1 where we show only the relevant terms in L SM and do not repeat all the SM terms, and the "·" represents the anti-symmetric product in SU (2) space. We have also not shown possible Φ and Φ 3 operators since they do not affect the phenomenology being studied here. The q L , u R , d R , , e R , ν R are the 3-generation SMFs, and we suppress the generation index on these fields. Here we have included right-handed neutrinos (ν R ) also for completeness; whether this is present in nature is still being probed in experiments. The VLQ U is in the fundamental of SU (3) c and has EM charge +2/3, and thus has gauge interactions with the gluons (g µ ) and hypercharge gauge bosons (B µ ) exactly as the SM up-type right-chiral quarks, and are not shown explicitly. The ψ, being an SM gauge singlet, has no SM gauge interactions. We have demanded that L respect a Z 2 symmetry under which only ψ is odd (i.e. ψ → −ψ) and all other fields are even. This leads to an absolutely stable ψ which we identify as our dark matter candidate. This Z 2 symmetry forbids theψ L · H operator (where L is the SM lepton doublet) which would have otherwise been allowed and caused the ψ to decay. 2 To not have a cosmologically stable colored relic, the decay of the VLQ U can be ensured by allowing the mixed operators where q L is a left-chiral SM quark doublet and u R a right-chiral SM quark singlet. We chose the hypercharge of U to be 2/3 to be able to write these operators in Eq. (2) that singly couple the U to the SM, allowing it to decay into SM final states, thus preventing a stable colored relic. (The same objective can be achieved by taking a hypercharge assignment of −1/3 instead, which then allows us to couple the VLQ to a down-type right-chiral SM quark.) One can ensure that experimental constraints are not violated by takingỹ U 1 andm M U , and allowing mixings with third generation quarks only (for details see for example Ref. [24]).
In Fig. 1 we show schematically the two contributions to the coupling between the φ and the SM. On the left we show the Higgs-portal contribution due to h-φ mixing, while on the right we show the loop-induced couplings to two gluons and to two hypercharge gauge bosons (B µ ) due to the VLQ U . The latter coupling implies the φγγ coupling that leads to the diphoton signature explored in Sec. 3.
We can contemplate other hypercharge assignments for the U , even a hypercharge neutral assignment with U being an electroweak singlet and only charged under SU (3) c . In this case, the U cannot be singly coupled to the SM since the operators in Eq. (2) cannot be written down, and therefore the theory will have to be extended to allow the U to decay. We do not develop this  possibility any further, other than to state that this assignment will remove the diphoton signature in Sec. 3, but the dark matter phenomenology of Sec. 4 will remain unchanged since that only relies on the φgg effective coupling.
Next, we study the φ ↔ h mixing that leads to a communication between the hidden sector and the visible SM sector, the Higgs-portal scenario. We point out a scenario in which this mixing is suppressed. Following this, we work out the 1-loop φgg and φγγ couplings induced by the VLQ U .

Higgs-scalar mixing
If the scalar potential is such that nonzero vacuum expectation values (VEVs) are generated, namely Φ = ξ/ √ 2 and H = (0, v/ √ 2) T , and the fluctuations around these are denoted asφ/ √ 2 and (0,ĥ/ √ 2) T respectively, theφ andĥ mix due to spontaneous symmetry breaking. Theφĥĥ interaction strength in Eq. (1) is given by (µ + κξ)/2. Diagonalizing theφ ↔ĥ mixing terms, we go from the (ĥ,φ) basis to the mass basis (h, φ), and define the mass eigenstates to be h = c hĥ − s hφ and φ = s hĥ + c hφ with mass eigenvalues M h , M φ respectively. The mixing angle sin θ h ≡ s h is given by In Fig. (2) we show the regions of parameter space that result in a small s h . We show s h = 0.001, 0.01, 0.1 contours in the (µ + κξ) -µ φ plane. In our numerical analysis below, we treat s h as an input parameter, and one can always relate it to the L parameters if needed using Eq. (3). The phenomenology due to the κ operator is discussed in detail for example in Ref. [7]. In the (h, φ) mass basis we have where we have defined the dimensionless coupling . We identify the mass eigenstate h as the 125 GeV Higgs boson discovered at the LHC.
We identify here a scenario in which s h 1, implying a suppressed Higgs-portal coupling. Consider the situation when µ is either very small or zero, and ξ = 0. The former is the case when Φ has non-zero charge in G BSM (see for example Ref. [7]), and the latter when there is no spontaneous symmetry breaking in the Φ sector, due to taking a positive mass-squared term for Φ in the potential rather than the negative mass-squared term shown in Eq. (1). In such a case, although it is broken, it is useful to consider another discrete symmetry, which we call Z 2 , under which the Φ is odd and all other fields are even. The full discrete symmetry under consideration then is Z 2 × Z 2 , where the former Z 2 being exact is what is keeping the dark matter absolutely stable. The consequence of the Z 2 symmetry is that if µ is zero at some scale, then s h = 0 at the tree-level at that scale, as can be seen from Eq. (3). The Z 2 however is broken explicitly by the y U and y ψ operators, and will generate the µ term at loop-level (in fact at 3-loop). This will result in a tiny µ ∼ 10 −6 M φ , and s h also correspondingly small. This serves as an example of a scenario in which s h 1. When s h is suppressed, the loop induced couplings of the φ to the SM due to the VLQ U becomes important to include. We discuss these loop-induced couplings next.

The κ φγγ and κ φgg loop-level effective couplings
When s h is small (of the order of 0.01), the loop induced couplings of theφ to the SM induced by exchange of the VLQ U will become important. Theφψψ tree-level coupling and these loop induced κ φgg and κ φγγ couplings will then couple the dark matter VLF ψ to the SM. The κ φgg and κ φγγ effective couplings induced by the VLF are detailed in Ref. [23]. Here we summarize these contributions for easy reference.
The effective Lagrangian defining the effective couplings κ φgg and κ φγγ can be written for the CP-even φ following the general definitions in Ref. [23] as where F µν , G µν are the photon and gluon field-strengths respectively, M is an arbitrary mass scale which we introduce to make the κ φγγ and κ φgg effective couplings dimensionless, and we show the numerical results of these effective couplings for M = 1 TeV. This choice is motivated by the presence of new physics at around the TeV scale in our model. The observables do not depend on M since it cancels out of expressions for all observables, as can be verified easily. We compute these effective couplings for the model Lagrangian defined in Eq. (1) at 1-loop. Defining r f = m 2 f /P 2 , with P 2 the invariant-mass-squared of the scalar, f running over all colored fermion species (includes SMFs and VLFs) with mass m f and Yukawa couplings y f , and with the electric charge of the fermion (f ) denoted by Q f , the κ φgg and κ φγγ at 1-loop are (for details see Ref. [23]) with F The expressions for F 1/2 in Eq. (6) reduce to the closed form expressions given in Ref. [25]. The color-factor in κ ab φgg is .., 8} are the adjoint color indices. Computing a decay rate or cross-section by summing over a, b gives a,b |C ab | 2 = 8(1/2) 2 = 2 resulting in a color-factor of 2. In the numerical results below, we include this color factor in the κ φgg and suppress the color indices. Analogous expressions hold for the hγγ and hgg effective couplings, and in our numerical analysis we include the contribution of the VLQ U in addition to the usual SMF contributions. In Fig. 3 we show the numerical values of the 1-loop effective couplings κ φgg and κ φγγ generated by the VLQ U for M = 1000 GeV, M φ = 1000 GeV,

φ decay
In our analysis, we include the decay modes φ → ψψ, hh, gg, γγ, tt, ττ , where ψ is the vector-like dark matter, while the rest are SM final states. The other SM decay final states are not important for our analysis. We write the φ total width Γ φ in terms of κ Γ , which we define as The contribution of each decay mode to κ 2 Γ includes the couplings and phase-space factors relevant to that decay. Expression for the Γ(φ → XX) can be found for example in Refs. [23,25]. For instance, for the decay φ → ψψ, via a Yukawa coupling y ψ / √ 2, we have a contribution κ 2 For the decay φ → QQ into a quark-pair, the same formula holds but is multiplied by the color factor N c . The κ φhh coupling identified in Eq. (4) leads to the φ → hh decay, which contributes to κ 2 Γ an amount κ 2 For the loop-level decays φ → gg and φ → γγ, as detailed in Ref. [23], we have κ 2

Perturbative unitarity constraint
If the φf f Yukawa coupling y f for any fermion f becomes very large, certain processes will violate perturbative unitarity. Thus, demanding perturbative unitarity will imply an upper bound on y f . We assume that λ φ , κ and µ/µ φ of Eq. (1) are all small enough that there is no constraint on these.
Here, we take f = {ψ, U } and obtain upper-bounds on y ψ and y U , the φψψ and φŪ U Yukawa couplings defined in Eq. (1), from perturbative unitarity of the ff → ff process at tree-level for s M 2 φ , m 2 f , where s is one of the Mandelstam variables as usual. The l th partial wave a l of the elastic scattering amplitude is bounded by perturbative unitarity to be |a l | ≤ 1 [26,27]. For the process ff → ff , the helicity amplitude in the limit of s M 2 φ , m 2 f is given by M(++ → ++) ≈ y 2 f /2 [28,29], where the "+" denote the helicities of the fermions. The 0 th partial wave amplitude is then readily written down as a 0 ≈ y 2 f /(32π). There is no t-channel contribution to this helicity configuration, and other helicity configurations that are non-zero have similar sized amplitudes [29] and therefore should result in similar bounds.
Compared to considering the ff → ff channel for a single f , a stronger bound could result from scattering channels with different initial and final state fermions, i.e. from the "coupled channels" To find this, we consider in the basis (ψψ, U αŪ α ) (no sum on α) with α = {r, g, b} the color index, the 4 × 4 coupled channel a 0 matrix a 0 = 1 32π The largest eigenvalue of this coupled channel matrix is a max 0 = (y 2 ψ + 3y 2 U )/32π. Applying the perturbative unitarity bound |a max 0 | ≤ 1 on the coupled channel corresponding to this maximum eigenvalue thus implies (y 2 ψ + 3y 2 U ) ≤ 32π .
We ensure that this bound is satisfied in the numerical analysis of the following sections.

LHC Phenomenology
The dark matter ψ, when produced at the LHC, will exit the detector as missing energy. Searches are underway at the LHC to look for missing energy events above the SM background, in which the dark matter recoils against one or more visible leptons, photons or jets. In addition to such missing energy signatures, one can search for the other BSM particles in the model defined in Sec. 2. These include the singlet scalar φ and the VLQ U , and in this section we discuss the LHC signatures of these particles. We work in the narrow width approximation (NWA) in which we can write In this work, we only focus on the φ → γγ signature at the LHC, since in comparison, the BR(φ → Zγ, ZZ) in our model are typically smaller by a factor that ranges from about 4 to 10 depending on M φ .

φ production at the LHC
We consider here φ production via the gluon-gluon fusion channel at the LHC. Rather than compute σ(gg → φ) ourselves, we relate it to the SM-like Higgs production c.s. at this mass and make use of the vast literature on this by writing where h denotes a scalar with SM-Higgs-like couplings to other SM states with the mass varied.
We take from Ref. [30] the 14 TeV LHC σ(gg → h ) for M h = 1000 GeV and multiply by 0.9 to get the √ s = 13 TeV values [31]. As can be inferred from Eq. (10) and detailed in Ref. [23], a quark Q coupled to φ via a Yukawa coupling y Q / √ 2 as in Eq. (1), gives a contribution to σ(gg → φ) given by where the sum over Q includes all quarks, including the top-quark and VLQ contributions, y t = √ 2m t /v is the top htt Yukawa coupling (we ignore the effect of running this to the scale µ = M φ ), F 1/2 is defined in Eq. (6) whose argument is r Q ≡ m 2 Q /P 2 with P 2 = M 2 φ for obtaining the on-shell φ resonant cross-section. We include contributions from Q = {t, U }, and in the small scalar-Higgs mixing region (s h 0.1), the U contribution dominates.

LHC constraints
We discuss here the LHC constraints on the model from the tt, τ τ , dijet and di-Higgs channels. In Fig. 2 of Ref. [23], constraints on the κ φgg from the 8 TeV LHC exclusion limits are shown. For an SM-like Higgs h with mass 1000 GeV, we have κ h gg = 7 with σ(pp → h ) ≈ 30 fb at the 8 TeV LHC due mainly to the top contribution. From the κ φgg expression in Eq. (6), with r f = m 2 f /M 2 φ here, since P 2 = M 2 φ for on-shell φ production, and with BR i = κ 2 i /κ 2 Γ , we derive the bound Q y Q y t where the sum over Q is as explained below Eq. (11), κ 2 t = N c y 2 φtt (1 − 4r t ) 3/2 , and κ 2 τ = y 2 φτ τ (1 − 4r τ ) 3/2 . The index (i) runs over various channels {tt, ττ , hh, gg, ...} i.e. (i) = {t, τ, h, g}, and we have κ max φgg (t) = 20, κ max φgg (τ ) = 4 (corresponding to BR i = 1) as derived in Ref. [23]. The limits on our model due to φ → tt, τ τ will be very weak for small mixings s h ∼ 0.07. The LHC upper limit on the dijet channel at a mass of 1000 GeV is about 30 pb [32], and for the sizes of cross-section and dijet BR in this model, this will be a loose constraint. The 95 % CL limit on σ(gg → φ) × BR(φ → hh) at a resonance mass of M φ = 1000 GeV is about 10 fb as can be read-out from the experimental exclusion plot in Fig. 6 of Ref. [21] from the ATLAS collaboration. (The H of Ref. [21] in our case is the φ decaying into hh.) This translates into κ max φgg (h) = 4 in Eq. (12). For the parameter ranges in our study, we find the di-Higgs limit is stringent and limits s h 0.07 for κ 2 Γ ≈ 0.1 (cf Fig. 6 for the limits for a range of s h ).
Generically, in new physics models including the one under consideration here, there are shifts in the h couplings to SM states, which are constrained by the LHC data (see for example Ref. [33]). Once the above constraint s h 0.07 is enforced, the constraints from the Higgs coupling measurements are satisfied.
The precise direct limit on the mass of the VLQ U depends on the BRs. The lower limit on the U mass is presently in the 920 − 1000 GeV range [34,35,36,37,38], For a long-lived VLQ with life-times in the range 10 −7 −10 5 s, the bound is looser with M U 525 GeV being allowed [39,40]. 3 3 It may be possible to weaken the VLQ mass bound somewhat by allowing the decay U → tφ where φ is an SU(2) singlet and will lead to missing energy at the LHC. This for example can be achieved by introducing the operators U φ t c where U is the VLQ and t c is the SM right-chiral top quark. Due to the new decay mode, the usual assumption that the BRs into the SM final states (bW, tZ, th) sum to one fails, and the limits have to be reanalyzed. The BRs into the SM final states are decreased and since the new mode has substantially larger SM irreducible SM tt + / ET backgrounds, the VLQ lower limits should be weaker. A detailed investigation of the implications of this proposal is beyond the scope of this work. For instance, the model discussed in Ref. [41] has this possibility.

LHC diphoton rate
From Eqs. (7) and (10), the LHC σ φ × BR(φ → γγ) in terms of the effective couplings can be written as where, as already explained below Eq. (5), M is a reference mass-scale which we take to be 1 TeV. The σ φ × BR γγ can be obtained from Eq. (13), and the expressions for κ φgg , κ φγγ are given in Eq. (6) with SMF and VLQ contributions included. For κ h gg in Eq. (6) only the SMF is included. As a representative benchmark point, we present results in this section for M φ = 1000 GeV.
In Fig. 4 we show contours of σ φ × BR γγ (in fb), and various κ 2 Γ as colored regions (darker to lighter shades correspond to smaller to larger κ 2 Γ ), with the parameters not varied along the axes fixed at s h = 0.01, M ψ = 475 GeV, M U = 1200 GeV, M φ = 1000 GeV, y ψ = 2, y U = 2.5. These parameter choices are motivated by obtaining the observed dark matter relic density and direct-detection constraints (cf Sec. 4). For these central values of the parameters we find that σ φ 0.25 pb, and the partial widths Γ {hh,tt,gg} are 0.015, 0.004, 0.065 GeV respectively. The current sensitivity of the LHC searches is about 1 fb. For the parameters in the figure, the diphoton rate range is 0.001-0.25 fb, which the LHC will probe in the future. The entire parameter region shown in the plot satisfies the unitarity constraint in Eq. (9). For very small y ψ or M ψ > M φ /2, Γ(φ → ψψ) 0 and Γ φ is dominated by Γ {hh,tt,gg} ; in this limit BR γγ 2.4 × 10 −3 and σ φ × BR γγ 0.4 fb for the set of parameters chosen with s h = 0.01. For M ψ < M φ /2 and y ψ large, Γ φ is large, being dominated by φ → ψψ decay, resulting in a very small σ φ × BR γγ . In the region where M ψ is within about 5 MeV of M φ /2 and if Γ ψ < 0.1 MeV, a large threshold enhancement is possible [42], which we do not include in our analysis.
In App. A, we present model-independently the range of diphoton rates as a function of the effective couplings, valid more generally than for the specific model considered here. We overlay on the plots there the diphoton rate for the model considered here. We also present the range of diphoton rate for the model considered here by varying y U and y ψ from very small values all the way up to saturating the unitarity constraint of Eq. (9). While helping in probing the model considered here, these results also help more generally in probing other such models through the diphoton channel.

Dark Matter Phenomenology
In this section we identify the region of parameter space of the model of Sec. 2 where the VLF ψ is a viable dark matter candidate. We also discuss constraints from dark matter direct detection experiments and prospects for the future. Another way to probe this scenario is through indirect detection via cosmic ray observables, which we do not take up in this study and leave for future work. Figure 4: The contours of σ φ × BR γγ (in fb), and regions (darker to lighter shades) of κ 2

Dark matter relic density
The dark matter relic density is set by the self-annihilation processes ψψ → SM mediated by schannel h, φ exchange. The relic density can be computed as detailed, for example, in App. A of Ref. [24]. We have for our case [7,24] the self-annihilation thermally averaged cross-section given by where x f ≡ M ψ /T f ≈ 25 with T f the freeze-out temperature, the sum is over all self-annihilation processes ψψ → f i f i for final states f i kinematically allowed, the |B i | 2 is the coefficient of v 2 rel in the amplitude squared for each process, v rel being the relative velocity of the two initial state ψ; thê Π i P S ≡ (1 − 4m 2 i /s) 1/2 is a phase-space factor with m i the mass of the final-state particle, and s is the Mandelstam variable, which for a cold-dark matter candidate during freeze-out is s ≈ 4M 2 ψ . In our analysis we include the two-body final states bb, W W, ZZ, hh, tt, gg, whichever are kinematically allowed for that given M ψ . The loop-level γγ, Zγ final states are insignificant compared to gg, and therefore we do not include them. The |B i | 2 for each of these final states are extracted from Ref. [7] to which we add |B gg | 2 here. These are given by where s ≈ 4M 2 ψ ,Ŝ hφ BW is a Breit-Wigner resonance factor including the s-channel {h, φ} contributions, ff = {bb, tt}, the M ZZ is identical to M W W except for an additional factor of 1/(2c 2 W ) and M W → M Z , and in |M hh | we do not include the t-channel (and u-channel) contributions as it is suppressed by an extra factor of s h and can be ignored for s h 1. M is a mass scale which we set to 1 TeV for numerical evaluations as explained below Eq. (5). We evaluate κ φgg and κ hgg using Eq. (6) taking r f = m 2 f /(4M 2 ψ ), since P 2 = s ≈ 4M 2 ψ here. The mixing angle θ h enters in κ φgg and κ hgg through φU U, φtt and htt couplings. Although τ τ and γγ final states are also possible, we neglect them in our analysis since these contributions are small owing to a small y τ for the former and a small EM coupling for the latter, compared to the larger QCD coupling and the presence of a color factor in the gg case. For small s h 0.07, the gg contribution becomes comparable or even larger than the tree-level contributions.

Dark matter direct detection
The dark-matter direct-detection elastic scattering cross-section on a nucleon is mediated by h, φ exchange. If s h 0.07, the φ contribution is also important even though it is much heavier than h. The h exchange contribution is given for example in Ref. [7], which we generalize here to include φ contribution also since we consider s h 0.07. The scalar-nucleon-nucleon coupling is generated due to the scalar coupling to the quark content of the nucleon, and also due the scalar coupling to the gluon content of the nucleon via the ggh, ggφ effective couplings. We define an effective Lagrangian for the scalar-nucleon-nucleon interaction as where N denotes the nucleon, and in the second line we write in the mass basis. We derive λ hN N and λ φN N using the formalism of Ref. [43] updated in Ref. [44] (for a review, see App. C of Ref. [45]) in which the scalar-nucleon coupling is denoted as f p,n . Identifying λ hN N ≡ f p,n , we have [44] with [44] f ≈ 0.85. We then find numerically λ hN N = 2 × 10 −3 , which we take for our numerical analysis. The λ φN N coupling is induced via the φ couplings to the gluon content of the nucleon, i.e. via the φgg effective coupling induced by the VLQ U . Following the same procedure as above, we derive this coupling from the second term in Eq. (17) as λ φN N = (2/27) f (p,n) T G y U m (p,n) /M U ≈ 0.063 y U m N /M U , which we use for our numerical studies. We content ourselves with this simple estimate of the coupling. Reliably computing the effective coupling is of critical importance, and our direct-detection rates can be scaled quite straightforwardly for a more accurately computed coupling. Ref. [46] examines recent developments and argues for a smaller value of the coupling λ hN N ≈ 1.1×10 −3 . Other sophisticated analyses can be found for example in Refs. [14,47].
We can now write the spin-averaged ψ elastic scattering cross section on a nucleon for q 2 m 2 where p ψ ≈ M ψ v ψ with v ψ ∼ 10 −3 [45], m N ≈ 1 GeV is the nucleon mass, ∆ h = (λ φN N /λ hN N )(s h /c h ), and ∆ φ = (λ hN N /λ φN N )(s h /c h ). This is the generalization of the direct-detection elastic cross section Eq. (13) of Ref. [7] which included only the h contribution, to now include the φ contribution also that becomes In addition to uncertainties in the dark matter nucleon effective coupling mentioned above, there is uncertainty in the local dark matter halo density and its velocity distribution. Given these uncertainties (see for example Refs. [48]), the direct-detection exclusion limits should be taken to be accurate only up to unknown O(1) factors.

Dark matter preferred regions of parameter space
Here we show regions of parameter space of the model of Sec. 2 for which we obtain the observed relic-density and are consistent with the dark matter direct detection limits. We also present  the prospects in upcoming direct detection experiments. In order to get the correct relic density of Ω dm = 0.26 ± 0.015 [49], we need the thermally averaged self-annihilation cross-section to be σv ≈ 2.3 × 10 −9 GeV −2 .
In Figs. 5 and 6 we plot contours of Ω dm = 0.1, 0.25, 0.3 with both the loop-induced couplings due to the the U and the Higgs-portal couplings present for M φ = 1000 GeV and M U = 1200 GeV, and show the regions with σ DD > 5 × 10 −45 cm 2 , 10 −45 cm 2 < σ DD < 5 × 10 −45 cm 2 , 10 −46 cm 2 < σ DD < 10 −45 cm 2 , 10 −47 cm 2 < σ DD < 10 −46 cm 2 , 10 −48 cm 2 < σ DD < 10 −47 cm 2 , 10 −49 cm 2 < σ DD < 10 −48 cm 2 , σ DD < 10 −49 cm 2 with parameters not varied along the axes fixed at s h = 0.01, M ψ = 475 GeV, y U = 2.5. The entire parameter region shown in the plots satisfies the unitarity constraint in Eq. (9). We see that for the choice of parameters we make, the direct-detection cross section is less than the current experimental limit, which is σ DD ≤ (0.1 − 1) × 10 −45 cm 2 [50] for dark matter mass in the 10−1000 GeV range. The correct self-annihilation cross-section is obtained only with an enhancement of the cross-section at the φ, h pole with M ψ ∼ M φ,h /2. Being close to the φ pole suppresses the φ → ψψ decay rate due to the limited phase-space available, leading to a small κ 2 Γ 0.1 as can be seen from Fig. 4. We first explore the s h = 0 limit, i.e. when the dark matter couples to the SM entirely via the φgg and φγγ effective couplings induced by the VLQ at the loop-level, with no contribution from the Higgs portal. This limit can be straightforwardly taken in Eqs. (14)- (18). The correct relic density can be achieved in this limit as shown in Fig. 5 (left), and we obtain σ DD < 10 −49 cm 2 . The required relic-density, for example, can be obtained for y ψ = 2, y U = 2.5, M U = 1200 GeV, M φ = 1000 GeV, M ψ = 467 GeV, for which σ DD = 2.4 × 10 −51 cm 2 . Thus the s h = 0 limit provides an example scenario in which the relic density is satisfied but direct-detection is very challenging.
In Fig. 5 (right) we show the situation for s h = 0.05, i.e. when the Higgs-portal is also turned on. For y U 1 the loop induced couplings due to the U are significant, while for smaller y U the Higgs-portal contribution dominates. Thus, for y U 0.5 the relic density contour starts losing dependence on y U , and for y U = 0.1 the loop-induced couplings are completely negligible and the dark matter phenomenology is that of the Higgs-portal scenario.
Since we are required to have s h 1 in which case the gg contribution dominates, the dark matter relic-density scales as ∼ (y ψ y U ) −2 to a very good approximation as can be inferred from Eqs. (14) and (15). Similarly, the dark matter direct-detection rate also scales the same way in this limit, as evident from Eq. (18). Thus, for M φ = 1000 GeV and for a given value of M ψ , other values of (y ψ , y U ) that give the correct relic-density and direct-detection rates can be obtained from those in Fig. 6, by scaling y ψ → (2.5/y U )y ψ . Thus, for s h 1, since the couplings of the dark matter with SM states is via loop-level effective couplings, we find for M φ = 1000 GeV and M ψ = 475 GeV, moderately large values y ψ y U ≈ 5 are required in order for the dark matter self-annihilation crosssection to be of sufficient size to give the correct relic-density. Taking smaller values of y ψ y U will require tuning M ψ closer to M φ /2 (or to M h /2). The regions we identify are safe from present direct-detection constraints, and will be probed in upcoming experiments.

Conclusions
In this work, we study a BSM model with a hidden sector containing a stable gauge-singlet vectorlike fermion dark matter ψ, and a gauge-singlet scalar φ. The φ couples to the SM via its mixing to the Higgs (the Higgs-portal scenario), and via loop-level couplings to two gluons and also to two hypercharge gauge bosons induced by an SU(2) singlet vector-like quark U carrying hypercharge 2/3. We point out a scenario in which the Higgs-portal mixing is suppressed, due to which the loop-level couplings are the dominant communication mechanism between the hidden sector and the SM. We study the LHC and dark matter phenomenology of this model.
We highlight the LHC direct constraints relevant to the model. We show that the LHC di-Higgs channel constrains the Higgs-singlet mixing to be very small (sin θ h 0.07), and therefore the loop-induced couplings are important to include. We present the rate for LHC scalar production via gluon-gluon fusion and its decay into the diphoton channel. We identify viable regions of parameter-space where the observed dark matter relic density is obtained and that are consistent with dark matter direct detection constraints.
When the mixing is tiny, and the dark matter is coupled to the SM via loop-induced operators, we show that moderately large φ Yukawa couplings to the vector-like fermions y ψ y U ≈ 5 are required in order to get a large enough dark matter self-annihilation cross section to obtain the correct relic density. Furthermore, (M ψ , M φ ) needs to be in the pole enhanced region, i.e. M ψ should be within a few tens of GeV of M φ /2 (or a few tenths of GeV of M h /2). We show that these large couplings are within the bounds of perturbative unitarity, by computing the upper bounds on these couplings from the ff → ff coupled channel scattering process for f = {ψ, U }.
The diphoton rate when the scalar-fermion couplings are varied is shown in Fig. 4. These diphoton rates are accessible at the LHC. We find regions of parameter space that are compatible with dark matter direct-detection bounds, and the rate we find is accessible in current and upcoming direct-detection experiments. We show these in Fig. 6, with the region consistent with the direct Figure 7: For the model of Sec. 2, the σ φ ×BR γγ (in fb) vs. κ 2 Γ , for M ψ = 475 GeV, M U = 1200 (left) and 1500 GeV (right), M φ = 1000 GeV, s h = 0.01 and y U , y ψ scanned over the range 0 < y U < y max U , 0 < y ψ < y max ψ subject to the unitarity constraint in Eq. (9).
LHC hh bound. In addition to the direct production signals of the vector-like quark at the LHC, another promising signal is the φ → hh mode which already imposes very tight constraints on the parameter-space. For the benchmark values of the parameters we study, the regions that yield the correct dark matter relic density have direct detection cross-sections that range between the current limits from experiments to about 10 −51 cm 2 . The lower value, very challenging to experimentally detect, is obtained when the Higgs-portal mechanism is shut-off with the dark matter coupled to the SM only via the loop-level couplings.

A Range of diphoton rate
In Sec. 2.4 we derived an upper bound on y ψ and y U from perturbative unitarity. Here, we show the range of diphoton rate at the LHC by saturating this upper bound. In Fig. 7 we show σ φ × BR γγ vs. κ 2 Γ in the model of Sec. 2, for M ψ = 475 GeV, M U = 1200 and 1500 GeV, M φ = 1000 GeV, s h = 0.01 and scanning over y U , y ψ in the range 0 < y U < y max U , 0 < y ψ < y max ψ , subject to the unitarity constraint in Eq. (9). For example, for s h = 0.01, we can get σ φ × BR γγ 2.9 fb.
In Fig. 8 we show contours of various κ 2 Γ in the κ φgg -κ φγγ plane that give σ φ × BR γγ = 0.1 fb for M φ = 1000 GeV. This cross-section is presently allowed with the 95% CL exclusion limit being about 1 fb [19,20]. We show in Fig. 8 (right) a band of diphoton rate 0.01 ≤ σ φ × BR γγ ≤ 0.5 fb for two representative total width values κ 2 Γ = 0.01 and 3, a wide range with the former being 0.1% of M φ , and the latter 5%. The latter width is rather large, and for M U > M φ /2, it is obtained for M ψ < M φ /2 for large y ψ as we discuss below. For such large couplings, there is a danger of tree-level unitarity being violated, and our analysis of Sec. 2.4 becomes relevant. Fig. 8 shows the situation model independently in any model with a φ as here, in which the κ φgg and κ φγγ effective couplings can be calculated. In the same figure, we overlay a "×" to depict the situation for the particular model of Sec.