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

Authors: Shrihari Gopalakrishna; Tuhin Subhra Mukherjee
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

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.


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## Body

## 1. 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 matterψ and a scalar ϕ. We also add an SU3c (color) triplet, SU2L singlet, U1Y hypercharge 2/3, and 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 ϕ to two gluons and ϕ to two hypercharge gauge bosons induced by 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 ofϕ 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 toϕ 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 ϕ, present direct LHC constraints, LHC gluon-gluon-fusion rate for ϕ production, and LHC diphoton rate from ϕ decay for various total widths of 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 other next studies in the literature that have some overlap with our work. Reference [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 [2]. Singlet scalar decaying to electroweak gauge bosons and to di-Higgs is studied in [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 [4–9]. The phenomenology of a singlet scalar coupled to VLFs in the context of the earlier 750 GeV diphoton excess [10, 11] also discussing the dark matter implications of the neutral VLFs present in those models is studied in [12–18]. 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 theϕ mass of 1 TeV. We also present the constraint on the h-ϕ mixing angle (θh) 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 θh (or when there is no mixing), the h mediated processes are suppressed, and the ϕ mediated processes due to ϕgg and ϕγγ effective coupling induced by the VLQ that we include here are important. Reference [16] does include this contribution, although it is 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 Section2 we present a model with a gauge-singlet 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 Section 3 we compute expressions for ϕ production in gluon-gluon fusion, discuss the direct LHC constraints on the model, including those from the di-Higgs channel, and present the LHC diphoton rate. In Section 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 Section 5 we offer our conclusions. In the Appendix 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 itsL 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 gauge 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,GSM⊗GBSM, where GSM is the SM gauge group SU(3)⊗SU(2)⊗U(1), and GBSM is any new physics group, the states charged only under GBSM and singlets under GSM 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 GBSM, we take Φ to be complex, with the real component denoted as ϕ/2. For example, [7] discusses a model in which GBSM is U(1) gauge symmetry.Here, we present a model with a SM gauge-singlet hidden sector containing a vector-like fermion dark matter candidateψ with mass Mψ and a CP-even scalar ϕ with mass Mϕ that couples to the visible SM sector via loop induced couplings due to an SU(2)-singlet color triplet VLQU having hypercharge 2/3 and mass MU. (The color triplet is the fundamental representation of the gauged SU3c 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 athermal 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, e.g., [22] and references therein) and (b) mixing between ϕ and the SM Higgs (h), commonly called the “Higgs-portal” scenario (see, e.g., [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. 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 ϕ to two hypercharge gauge bosons imply ϕγγ, ϕZγ, and ϕZZ couplings. U is the only new state that is charged under both GSM and GBSM 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 constraint limits the Higgs-singlet mixing in possibility (b) to be small, and therefore including the loop induced couplings of 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 (this model and the couplings to the VLF parallel the SVU model of [23], and in the notation of that paper, this model may be termed as the SVUψ model)(1)LSM⊃μh2H†H-λHH†H2-yuq¯L·H∗uR+ydq¯LHdR+yνl¯L·H∗νR+yel¯LHeR+h.c.,LBSM⊃μϕ2Φ†Φ-λϕΦ†Φ2-κΦ†ΦH†H-μΦH†H+h.c.-Mψψ¯ψ-MUU¯U-yψΦψ¯ψ+yUΦU¯U+h.c.,where we show only the relevant terms in LSM and do not repeat all the SM terms, and “·” represents the antisymmetric product in SU(2) space. We have also not shown possible Φ and Φ3 operators since they do not affect the phenomenology being studied here. The qL,uR,dR,l,eR,νR are the 3-generation SMFs, and we suppress the generation index on these fields. Here we have included right-handed neutrinos (νR) also for completeness; whether this is present in nature is still being probed in experiments. The VLQU is in the fundamental of SU(3)c and has EM charge +2/3 and thus has gauge interactions with the gluons (gμ) and hypercharge gauge bosons (Bμ) exactly as the SM up-type right-chiral quarks, which are not shown explicitly. ψ, being an SM gauge singlet, has no SM gauge interactions. We have demanded that L respect a Z2 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 Z2 symmetry forbids the ψ¯lL·H operator (where lL is the SM lepton doublet) which would have otherwise been allowed and caused ψ to decay. (If the Z2 symmetry is not imposed, this operator would be allowed and is the neutrino Yukawa operator, and ψR can then be identified as the right-handed neutrino νR. This possibility is not considered here since we are motivated by having a stable dark matter candidate but is extensively studied in the literature in the context of neutrino mass models.)To not have a cosmologically stable colored relic, the decay of the VLQU can be ensured by allowing the mixed operators (2)LU-SM⊃-y~UU¯RqL·H-m~u¯RUL+h.c.,where qL is a left-chiral SM quark doublet and uR a right-chiral SM quark singlet. We chose the hypercharge of U to be 2/3 to be able to write these operators in (2) that singly couple 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 taking y~U≪1 and m~≪MU and allowing mixings with third-generation quarks only (for details see, e.g., [24]).In Figure1 we show schematically the two contributions to the coupling between 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 (Bμ) due to the VLQU. The latter coupling implies the ϕγγ coupling that leads to the diphoton signature explored in Section 3.Figure 1
ϕ couples to the SM via the Higgs-portal mixing contribution (a) and via the loop induced couplings to two gluons and two hypercharge gauge bosons due to the VLQU (b).
(a)
(b)We can contemplate other hypercharge assignments forU, 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 (2) cannot be written down, and therefore the theory will have to be extended to allow U to decay. We do not develop this possibility any further, other than to state that this assignment will remove the diphoton signature in Section 3, but the dark matter phenomenology of Section 4 will remain unchanged since that only relies on the ϕgg effective coupling.Next, we study theϕ↔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 VLQU.


### 2.1. 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 as ϕ^/2 and (0,h^/2)T, respectively, ϕ^ and h^ mix due to spontaneous symmetry breaking. The ϕ^h^h^ interaction strength in (1) is given by (μ+κξ)/2. Diagonalizing the ϕ^↔h^ mixing terms, we go from (h^,ϕ^) basis to the mass basis (h,ϕ) and define the mass eigenstates to be h=chh^-shϕ^ and ϕ=shh^+chϕ^ with mass eigenvalues Mh,Mϕ, respectively. The mixing angle sin⁡θh≡sh is given by(3)tan⁡2θh=2μ+κξvμϕ2-μh2.In Figure 2 we show the regions of parameter space that result in a small sh. We show sh=0.001,0.01,0.1 contours in the (μ+κξ)–μϕ plane. In our numerical analysis below, we treat sh as an input parameter, and one can always relate it to the L parameters if needed using (3). The phenomenology due to the κ operator is discussed in detail, for example, in [7]. In the (h,ϕ) mass basis we have (4)Lϕhh=-κϕhhMϕ22ϕhh,where we have defined the dimensionless coupling κϕhh≡tan⁡2θh(ch3-2chsh2)(Mϕ2-Mh2)/(2vMϕ). We identify the mass eigenstate h as the 125 GeV Higgs boson discovered at the LHC.Figure 2
The scalar-Higgs mixing parametersh=0.001,0.01,0.1 contours as a function of the Lagrangian parameters of the model.We identify here a scenario in whichsh≪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 nonzero charge in GBSM (see, e.g., [7]), and the latter is 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 (1). In such a case, although it is broken, it is useful to consider another discrete symmetry, which we call Z2′, under which the Φ is odd and all other fields are even. The full discrete symmetry under consideration then is Z2×Z2′, where the former Z2 being exact is what is keeping the dark matter absolutely stable. The consequence of the Z2′ symmetry is that if μ is zero at some scale, then sh=0 at the tree-level at that scale, as can be seen from (3). Z2′ however is broken explicitly by the yU and yψ operators and will generate the μ term at loop level (in fact at 3-loop level). This will result in a tiny μ~10-6Mϕ and sh also correspondingly small. This serves as an example of a scenario in which sh≪1. When sh is suppressed, the loop induced coupling of ϕ to the SM due to the VLQU becomes important to include. We discuss these loop induced couplings next.


### 2.2. Theκϕγγ and κϕgg Loop-Level Effective Couplings

Whensh is small (of the order of 0.01), the loop induced couplings of the ϕ^ to the SM induced by exchange of the VLQU will become important. The ϕ^ψ¯ψ tree-level coupling and these loop induced κϕgg and κϕγγ couplings will then couple the dark matter VLF ψ to the SM. The κϕgg and κϕγγ effective couplings induced by the VLF are detailed in [23]. Here we summarize these contributions for easy reference.The effective Lagrangian defining the effective couplingsκϕgg and κϕγγ can be written for the CP-even ϕ following the general definitions in [23] as(5)Leff=-κϕγγ64π2MϕFμνFμν-κϕgg64π2MϕGμνGμν,where Fμν,Gμν are the photon and gluon field-strengths, respectively; M is an arbitrary mass scale which we introduce to make the κϕγγ and κϕgg 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 expressions for all observables, as can be verified easily. We compute these effective couplings for the model Lagrangian defined in (1) at 1-loop level. Defining rf=mf2/P2, with P2, the invariant-mass-square of the scalar, f running over all colored fermion species (includes SMFs and VLFs) with mass mf and Yukawa couplings yf, and the electric charge of the fermion (f) denoted by Qf, κϕgg and κϕγγ at 1-loop level are (for details see [23])(6)κϕγγ=2e2∑fNcfQf2yf2MmfF1/21rf,κϕgg=gs2∑fyf2MmfF1/21rf,withF1/21rf=4rf∫01dy∫01-ydx1-4xyrf-xy.The expressions for F1/2(1) in (6) reduce to the closed form expressions given in [25]. The color factor in κϕggab is Cab=(1/2)δab, where a,b={1,…,8} are the adjoint color indices. Computing a decay rate or cross section by summing over a,b gives ∑a,bCab2=8(1/2)2=2 resulting in a color factor of 2. In the numerical results below, we include this color factor in κϕgg 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 VLQU in addition to the usual SMF contributions. In Figure 3 we show the numerical values of the 1-loop effective couplings κϕgg and κϕγγ generated by the VLQU for M=1000 GeV, Mϕ=1000 GeV, and P2=Mϕ2, in the yU–MU plane.Figure 3
κ
ϕ
g
g (a) and κϕγγ (b) for Mϕ=1000 GeV with M=1000 GeV.
(a)
(b)


### 2.3.ϕ Decay

In our analysis, we include the decay modesϕ→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 Γϕ in terms of κΓ, which we define as (7)Γϕ≡κΓ216πMϕ.The contribution of each decay mode to κΓ2 includes the couplings and phase-space factors relevant to that decay. Expression for the Γ(ϕ→XX) can be found, for example, in [23, 25]. For instance, for the decay ϕ→ψψ, via a Yukawa coupling yψ/2, we have a contribution κΓ2(ψψ)=yψ2(1-4Mψ2/Mϕ2)3/2. For the decay ϕ→QQ into a quark-pair, the same formula holds but is multiplied by the color factor Nc. The κϕhh coupling identified in (4) leads to the ϕ→hh decay, which contributes to κΓ2, an amount κΓ2(hh)=(κϕhh2/4)(1-4mh2/Mϕ2)1/2. For the loop-level decay ϕ→gg and decay ϕ→γγ, as detailed in [23], we have κΓ2(gg)=2κϕgg2/(16π2)2(Mϕ2/M2) and κΓ2(γγ)=κϕγγ2/(32π2)2(Mϕ2/M2).


### 2.4. Perturbative Unitarity Constraint

If theϕf¯f Yukawa coupling yf for any fermion f becomes very large, certain processes will violate perturbative unitarity. Thus, demanding perturbative unitarity will imply an upper bound on yf. We assume that λϕ, κ, and μ/μϕ of (1) are all small enough so that there is no constraint on these. Here, we take f={ψ,U} and obtain upper bounds on yψ and yU, the ϕψ¯ψ and ϕU¯U Yukawa couplings defined in (1), from perturbative unitarity of the ff¯→ff¯ process at tree-level for s≫Mϕ2,mf2, where s is one of the Mandelstam variables as usual.Thelth partial wave al of the elastic scattering amplitude is bounded by perturbative unitarity to be al≤1 [26, 27]. For the process ff¯→ff¯, the helicity amplitude in the limit of s≫Mϕ2,mf2 is given by M(++→++)≈yf2/2 [28, 29], where “+” denote the helicities of the fermions. The 0th partial wave amplitude is then readily written down as a0≈yf2/(32π). There is no t-channel contribution to this helicity configuration, and other helicity configurations that are nonzero have similar sized amplitudes [29] and therefore should result in similar bounds.Compared to considering theff¯→ff¯ channel for a single f, a stronger bound could result from scattering channels with different initial and final state fermions, that is, from the “coupled channels” f1f¯1→f2f¯2 with f1,f2={ψ,U}. To find this, we consider in the basis (ψψ¯,UαU¯α) (no sum on α) with α={r,g,b}, the color index, the 4×4 coupled channel a0 matrix: (8)a0=132πyψ2yψyUyψyUyU213×3.The largest eigenvalue of this coupled channel matrix is a0max=(yψ2+3yU2)/32π. Applying the perturbative unitarity bound a0max≤1 on the coupled channel corresponding to this maximum eigenvalue thus implies (9)yψ2+3yU2≤32π.We ensure that this bound is satisfied in the numerical analysis of the following sections.


## 2.1. 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 as ϕ^/2 and (0,h^/2)T, respectively, ϕ^ and h^ mix due to spontaneous symmetry breaking. The ϕ^h^h^ interaction strength in (1) is given by (μ+κξ)/2. Diagonalizing the ϕ^↔h^ mixing terms, we go from (h^,ϕ^) basis to the mass basis (h,ϕ) and define the mass eigenstates to be h=chh^-shϕ^ and ϕ=shh^+chϕ^ with mass eigenvalues Mh,Mϕ, respectively. The mixing angle sin⁡θh≡sh is given by(3)tan⁡2θh=2μ+κξvμϕ2-μh2.In Figure 2 we show the regions of parameter space that result in a small sh. We show sh=0.001,0.01,0.1 contours in the (μ+κξ)–μϕ plane. In our numerical analysis below, we treat sh as an input parameter, and one can always relate it to the L parameters if needed using (3). The phenomenology due to the κ operator is discussed in detail, for example, in [7]. In the (h,ϕ) mass basis we have (4)Lϕhh=-κϕhhMϕ22ϕhh,where we have defined the dimensionless coupling κϕhh≡tan⁡2θh(ch3-2chsh2)(Mϕ2-Mh2)/(2vMϕ). We identify the mass eigenstate h as the 125 GeV Higgs boson discovered at the LHC.Figure 2
The scalar-Higgs mixing parametersh=0.001,0.01,0.1 contours as a function of the Lagrangian parameters of the model.We identify here a scenario in whichsh≪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 nonzero charge in GBSM (see, e.g., [7]), and the latter is 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 (1). In such a case, although it is broken, it is useful to consider another discrete symmetry, which we call Z2′, under which the Φ is odd and all other fields are even. The full discrete symmetry under consideration then is Z2×Z2′, where the former Z2 being exact is what is keeping the dark matter absolutely stable. The consequence of the Z2′ symmetry is that if μ is zero at some scale, then sh=0 at the tree-level at that scale, as can be seen from (3). Z2′ however is broken explicitly by the yU and yψ operators and will generate the μ term at loop level (in fact at 3-loop level). This will result in a tiny μ~10-6Mϕ and sh also correspondingly small. This serves as an example of a scenario in which sh≪1. When sh is suppressed, the loop induced coupling of ϕ to the SM due to the VLQU becomes important to include. We discuss these loop induced couplings next.


## 2.2. Theκϕγγ and κϕgg Loop-Level Effective Couplings

Whensh is small (of the order of 0.01), the loop induced couplings of the ϕ^ to the SM induced by exchange of the VLQU will become important. The ϕ^ψ¯ψ tree-level coupling and these loop induced κϕgg and κϕγγ couplings will then couple the dark matter VLF ψ to the SM. The κϕgg and κϕγγ effective couplings induced by the VLF are detailed in [23]. Here we summarize these contributions for easy reference.The effective Lagrangian defining the effective couplingsκϕgg and κϕγγ can be written for the CP-even ϕ following the general definitions in [23] as(5)Leff=-κϕγγ64π2MϕFμνFμν-κϕgg64π2MϕGμνGμν,where Fμν,Gμν are the photon and gluon field-strengths, respectively; M is an arbitrary mass scale which we introduce to make the κϕγγ and κϕgg 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 expressions for all observables, as can be verified easily. We compute these effective couplings for the model Lagrangian defined in (1) at 1-loop level. Defining rf=mf2/P2, with P2, the invariant-mass-square of the scalar, f running over all colored fermion species (includes SMFs and VLFs) with mass mf and Yukawa couplings yf, and the electric charge of the fermion (f) denoted by Qf, κϕgg and κϕγγ at 1-loop level are (for details see [23])(6)κϕγγ=2e2∑fNcfQf2yf2MmfF1/21rf,κϕgg=gs2∑fyf2MmfF1/21rf,withF1/21rf=4rf∫01dy∫01-ydx1-4xyrf-xy.The expressions for F1/2(1) in (6) reduce to the closed form expressions given in [25]. The color factor in κϕggab is Cab=(1/2)δab, where a,b={1,…,8} are the adjoint color indices. Computing a decay rate or cross section by summing over a,b gives ∑a,bCab2=8(1/2)2=2 resulting in a color factor of 2. In the numerical results below, we include this color factor in κϕgg 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 VLQU in addition to the usual SMF contributions. In Figure 3 we show the numerical values of the 1-loop effective couplings κϕgg and κϕγγ generated by the VLQU for M=1000 GeV, Mϕ=1000 GeV, and P2=Mϕ2, in the yU–MU plane.Figure 3
κ
ϕ
g
g (a) and κϕγγ (b) for Mϕ=1000 GeV with M=1000 GeV.
(a)
(b)


## 2.3.ϕ Decay

In our analysis, we include the decay modesϕ→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 Γϕ in terms of κΓ, which we define as (7)Γϕ≡κΓ216πMϕ.The contribution of each decay mode to κΓ2 includes the couplings and phase-space factors relevant to that decay. Expression for the Γ(ϕ→XX) can be found, for example, in [23, 25]. For instance, for the decay ϕ→ψψ, via a Yukawa coupling yψ/2, we have a contribution κΓ2(ψψ)=yψ2(1-4Mψ2/Mϕ2)3/2. For the decay ϕ→QQ into a quark-pair, the same formula holds but is multiplied by the color factor Nc. The κϕhh coupling identified in (4) leads to the ϕ→hh decay, which contributes to κΓ2, an amount κΓ2(hh)=(κϕhh2/4)(1-4mh2/Mϕ2)1/2. For the loop-level decay ϕ→gg and decay ϕ→γγ, as detailed in [23], we have κΓ2(gg)=2κϕgg2/(16π2)2(Mϕ2/M2) and κΓ2(γγ)=κϕγγ2/(32π2)2(Mϕ2/M2).


## 2.4. Perturbative Unitarity Constraint

If theϕf¯f Yukawa coupling yf for any fermion f becomes very large, certain processes will violate perturbative unitarity. Thus, demanding perturbative unitarity will imply an upper bound on yf. We assume that λϕ, κ, and μ/μϕ of (1) are all small enough so that there is no constraint on these. Here, we take f={ψ,U} and obtain upper bounds on yψ and yU, the ϕψ¯ψ and ϕU¯U Yukawa couplings defined in (1), from perturbative unitarity of the ff¯→ff¯ process at tree-level for s≫Mϕ2,mf2, where s is one of the Mandelstam variables as usual.Thelth partial wave al of the elastic scattering amplitude is bounded by perturbative unitarity to be al≤1 [26, 27]. For the process ff¯→ff¯, the helicity amplitude in the limit of s≫Mϕ2,mf2 is given by M(++→++)≈yf2/2 [28, 29], where “+” denote the helicities of the fermions. The 0th partial wave amplitude is then readily written down as a0≈yf2/(32π). There is no t-channel contribution to this helicity configuration, and other helicity configurations that are nonzero have similar sized amplitudes [29] and therefore should result in similar bounds.Compared to considering theff¯→ff¯ channel for a single f, a stronger bound could result from scattering channels with different initial and final state fermions, that is, from the “coupled channels” f1f¯1→f2f¯2 with f1,f2={ψ,U}. To find this, we consider in the basis (ψψ¯,UαU¯α) (no sum on α) with α={r,g,b}, the color index, the 4×4 coupled channel a0 matrix: (8)a0=132πyψ2yψyUyψyUyU213×3.The largest eigenvalue of this coupled channel matrix is a0max=(yψ2+3yU2)/32π. Applying the perturbative unitarity bound a0max≤1 on the coupled channel corresponding to this maximum eigenvalue thus implies (9)yψ2+3yU2≤32π.We ensure that this bound is satisfied in the numerical analysis of the following sections.


## 3. LHC Phenomenology

The dark matterψ, 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 Section 2. These include the singlet scalar ϕ and the VLQU, 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 σ(pp→ϕ→XX)≈σ(pp→ϕ)×BR(ϕ→XX)≡σϕ×BRXX, where BR(ϕ→XX)≡Γ(ϕ→XX)/Γϕ. In this work, we only focus on the ϕ→γγ signature at the LHC, since in comparison BR(ϕ→Zγ,ZZ) in our model are typically smaller by a factor that ranges from about 4 to 10 depending on Mϕ.


### 3.1.ϕ Production at the LHC

We consider hereϕ production via the gluon-gluon fusion channel at the LHC. Rather than computing 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 (10)σgg→ϕ=σgg→h′Γϕ→ggΓh′→gg,where h′ denotes a scalar with SM-Higgs-like couplings to other SM states with the mass varied. We take from [30] the 14 TeV LHC σ(gg→h′) for Mh′=1000 GeV and multiply by 0.9 to get the s=13 TeV values [31]. As can be inferred from (10) and detailed in [23], a quark Q coupled to ϕ via a Yukawa coupling yQ/2 as in (1) gives a contribution to σ(gg→ϕ) given by (11)σgg→ϕ=σgg→h′∑QyQytF1/21rQF1/21rtmtMQ2,where the sum over Q includes all quarks, including the top quark and VLQ contributions, yt=2mt/v is the top htt Yukawa coupling (we ignore the effect of running this to the scale μ=Mϕ), and F1/2(1) is defined in (6) whose argument is rQ≡mQ2/P2 with P2=Mϕ2 for obtaining the on-shell ϕ resonant cross section. We include contributions from Q={t,U}, and in the small scalar-Higgs mixing region (sh≲0.1), the U contribution dominates.


### 3.2. LHC Constraints

We discuss here the LHC constraints on the model from thett,ττ, dijet, and di-Higgs channels. In Figure 2 of [23], constraints on κϕgg from the 8 TeV LHC exclusion limits are shown. For an SM-like Higgs h′ with mass 1000 GeV, we have κh′gg=7 with σ(pp→h′)≈30 fb at the 8 TeV LHC due mainly to the top contribution. From the κϕgg expression in (6), with rf=mf2/Mϕ2 here, since P2=Mϕ2 for on-shell ϕ production, and with BRi=κi2/κΓ2, we derive the bound (12)∑QyQytF1/21rQF1/21rtmtMQ2κi2κΓ2<κϕggimaxκh′gg2,where the sum over Q is as explained below (11), κt2=Ncyϕtt21-4rt3/2, and κτ2=yϕττ21-4rτ3/2. The index (i) runs over various channels {tt¯,ττ¯,hh,gg,…}, that is, (i)={t,τ,h,g}, and we have κϕgg(t)max=20, κϕgg(τ)max=4 (corresponding to BRi=1) as derived in [23]. The limits on our model due to ϕ→tt,ττ will be very weak for small mixings sh~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 Mϕ=1000 GeV is about 10 fb as can be read out from the experimental exclusion plot in Figure 6 of [21] from the ATLAS collaboration. (H of [21] in our case is ϕ decaying into hh.) This translates into κϕgg(h)max=4 in (12). For the parameter ranges in our study, we find the di-Higgs limit is stringent and limits sh≲0.07 for κΓ2≈0.1 (cf. Figure 6 for the limits for a range of sh).Generically, in new physics models including the one under consideration here, there are shifts in theh couplings to SM states, which are constrained by the LHC data (see, e.g., [33]). Once the above constraint sh≲0.07 is enforced, the constraints from the Higgs coupling measurements are satisfied.The precise direct limit on the mass of the VLQU depends on the BRs. The lower limit on the U mass is presently in the 920–1000 GeV range [34–38]. For a long-lived VLQ with life-times in the range 10-7–105 s, the bound is looser with MU≳525 GeV being allowed [39, 40]. (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, e.g., can be achieved by introducing the operators Uϕ′tc where U is the VLQ and tc 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 [41] has this possibility.)


### 3.3. LHC Diphoton Rate

From (7) and (10), the LHC σϕ×BR(ϕ→γγ) in terms of the effective couplings can be written as(13)σϕ×BRγγ=σgg→h′κϕgg2κh′gg214κϕγγ16π2M2Mϕ2κΓ2,where, as already explained below (5), M is a reference mass scale which we take to be 1 TeV. σϕ×BRγγ can be obtained from (13), and the expressions for κϕgg, κϕγγ are given in (6) with SMF and VLQ contributions included. For κh′gg in (6) only the SMF is included. As a representative benchmark point, we present results in this section for Mϕ=1000 GeV.In Figure4 we show contours of σϕ×BRγγ (in fb) and various κΓ2 as colored regions (darker to lighter shades correspond to smaller to larger κΓ2), with the parameters not varied along the axes fixed at sh=0.01, Mψ=475 GeV, MU=1200 GeV, Mϕ=1000 GeV, yψ=2, and yU=2.5. These parameter choices are motivated by obtaining the observed dark matter relic density and direct detection constraints (cf. Section 4). For these central values of the parameters we find that σϕ≃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 (9). For very small yψ or Mψ>Mϕ/2, Γ(ϕ→ψψ)≃0 and Γϕ is dominated by Γ{hh,tt,gg}; in this limit BRγγ≃2.4×10-3 and σϕ×BRγγ≃0.4 fb for the set of parameters chosen with sh=0.01. For Mψ<Mϕ/2 and large yψ, Γϕ is large, being dominated by ϕ→ψψ decay, resulting in a very small σϕ×BRγγ. In the region where Mψ is within about 5 MeV of Mϕ/2 and if Γψ<0.1 MeV, a large threshold enhancement is possible [42], which we do not include in our analysis.Figure 4
The contours ofσϕ×BRγγ (in fb) and regions (darker to lighter shades) of κΓ2<0.1 (red), 0.1<κΓ2<0.5 (blue), 0.5<κΓ2<1 (pink), 1<κΓ2<2 (gray), 2<κΓ2<3 (orange), and κΓ2>3 (light green). Parameters not varied along the axes are fixed at sh=0.01, Mψ=475 GeV, MU=1200 GeV, Mϕ=1000 GeV, yψ=2, and yU=2.5. These parameter choices are motivated by obtaining the observed dark matter relic density and direct detection constraints.In the Appendix, 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 varyingyU and yψ from very small values all the way up to saturating the unitarity constraint of (9). While helping in probing the model considered here, these results also help more generally in probing other such models through the diphoton channel.The 8 TeVhh channel constraints discussed in Section 3.2 constrain κϕhh≪1. For example, this constraint leads to the bound sh≲0.17 for yU=2.5 and κΓ2=0.25. For yψ≳0.1, the BR(ϕ→ψψ) is dominant and yψ largely controls κΓ2. For κΓ2=1, σϕ×BRγγ can reach only about 0.03 fb for sh=0.01. For very small yψ≲0.1, the total width (i.e., κΓ2) is small and dominated by top and U loops and the tree-level ϕ→hh,tt decay. For yψ→0, sh→0, both σϕ×BRγγ and κΓ2 come from U loops and scales as yU4 and yU2, respectively; σϕ×BRγγ increases with κΓ2 up to around κΓ2≃0.03.


## 3.1.ϕ Production at the LHC

We consider hereϕ production via the gluon-gluon fusion channel at the LHC. Rather than computing 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 (10)σgg→ϕ=σgg→h′Γϕ→ggΓh′→gg,where h′ denotes a scalar with SM-Higgs-like couplings to other SM states with the mass varied. We take from [30] the 14 TeV LHC σ(gg→h′) for Mh′=1000 GeV and multiply by 0.9 to get the s=13 TeV values [31]. As can be inferred from (10) and detailed in [23], a quark Q coupled to ϕ via a Yukawa coupling yQ/2 as in (1) gives a contribution to σ(gg→ϕ) given by (11)σgg→ϕ=σgg→h′∑QyQytF1/21rQF1/21rtmtMQ2,where the sum over Q includes all quarks, including the top quark and VLQ contributions, yt=2mt/v is the top htt Yukawa coupling (we ignore the effect of running this to the scale μ=Mϕ), and F1/2(1) is defined in (6) whose argument is rQ≡mQ2/P2 with P2=Mϕ2 for obtaining the on-shell ϕ resonant cross section. We include contributions from Q={t,U}, and in the small scalar-Higgs mixing region (sh≲0.1), the U contribution dominates.


## 3.2. LHC Constraints

We discuss here the LHC constraints on the model from thett,ττ, dijet, and di-Higgs channels. In Figure 2 of [23], constraints on κϕgg from the 8 TeV LHC exclusion limits are shown. For an SM-like Higgs h′ with mass 1000 GeV, we have κh′gg=7 with σ(pp→h′)≈30 fb at the 8 TeV LHC due mainly to the top contribution. From the κϕgg expression in (6), with rf=mf2/Mϕ2 here, since P2=Mϕ2 for on-shell ϕ production, and with BRi=κi2/κΓ2, we derive the bound (12)∑QyQytF1/21rQF1/21rtmtMQ2κi2κΓ2<κϕggimaxκh′gg2,where the sum over Q is as explained below (11), κt2=Ncyϕtt21-4rt3/2, and κτ2=yϕττ21-4rτ3/2. The index (i) runs over various channels {tt¯,ττ¯,hh,gg,…}, that is, (i)={t,τ,h,g}, and we have κϕgg(t)max=20, κϕgg(τ)max=4 (corresponding to BRi=1) as derived in [23]. The limits on our model due to ϕ→tt,ττ will be very weak for small mixings sh~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 Mϕ=1000 GeV is about 10 fb as can be read out from the experimental exclusion plot in Figure 6 of [21] from the ATLAS collaboration. (H of [21] in our case is ϕ decaying into hh.) This translates into κϕgg(h)max=4 in (12). For the parameter ranges in our study, we find the di-Higgs limit is stringent and limits sh≲0.07 for κΓ2≈0.1 (cf. Figure 6 for the limits for a range of sh).Generically, in new physics models including the one under consideration here, there are shifts in theh couplings to SM states, which are constrained by the LHC data (see, e.g., [33]). Once the above constraint sh≲0.07 is enforced, the constraints from the Higgs coupling measurements are satisfied.The precise direct limit on the mass of the VLQU depends on the BRs. The lower limit on the U mass is presently in the 920–1000 GeV range [34–38]. For a long-lived VLQ with life-times in the range 10-7–105 s, the bound is looser with MU≳525 GeV being allowed [39, 40]. (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, e.g., can be achieved by introducing the operators Uϕ′tc where U is the VLQ and tc 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 [41] has this possibility.)


## 3.3. LHC Diphoton Rate

From (7) and (10), the LHC σϕ×BR(ϕ→γγ) in terms of the effective couplings can be written as(13)σϕ×BRγγ=σgg→h′κϕgg2κh′gg214κϕγγ16π2M2Mϕ2κΓ2,where, as already explained below (5), M is a reference mass scale which we take to be 1 TeV. σϕ×BRγγ can be obtained from (13), and the expressions for κϕgg, κϕγγ are given in (6) with SMF and VLQ contributions included. For κh′gg in (6) only the SMF is included. As a representative benchmark point, we present results in this section for Mϕ=1000 GeV.In Figure4 we show contours of σϕ×BRγγ (in fb) and various κΓ2 as colored regions (darker to lighter shades correspond to smaller to larger κΓ2), with the parameters not varied along the axes fixed at sh=0.01, Mψ=475 GeV, MU=1200 GeV, Mϕ=1000 GeV, yψ=2, and yU=2.5. These parameter choices are motivated by obtaining the observed dark matter relic density and direct detection constraints (cf. Section 4). For these central values of the parameters we find that σϕ≃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 (9). For very small yψ or Mψ>Mϕ/2, Γ(ϕ→ψψ)≃0 and Γϕ is dominated by Γ{hh,tt,gg}; in this limit BRγγ≃2.4×10-3 and σϕ×BRγγ≃0.4 fb for the set of parameters chosen with sh=0.01. For Mψ<Mϕ/2 and large yψ, Γϕ is large, being dominated by ϕ→ψψ decay, resulting in a very small σϕ×BRγγ. In the region where Mψ is within about 5 MeV of Mϕ/2 and if Γψ<0.1 MeV, a large threshold enhancement is possible [42], which we do not include in our analysis.Figure 4
The contours ofσϕ×BRγγ (in fb) and regions (darker to lighter shades) of κΓ2<0.1 (red), 0.1<κΓ2<0.5 (blue), 0.5<κΓ2<1 (pink), 1<κΓ2<2 (gray), 2<κΓ2<3 (orange), and κΓ2>3 (light green). Parameters not varied along the axes are fixed at sh=0.01, Mψ=475 GeV, MU=1200 GeV, Mϕ=1000 GeV, yψ=2, and yU=2.5. These parameter choices are motivated by obtaining the observed dark matter relic density and direct detection constraints.In the Appendix, 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 varyingyU and yψ from very small values all the way up to saturating the unitarity constraint of (9). While helping in probing the model considered here, these results also help more generally in probing other such models through the diphoton channel.The 8 TeVhh channel constraints discussed in Section 3.2 constrain κϕhh≪1. For example, this constraint leads to the bound sh≲0.17 for yU=2.5 and κΓ2=0.25. For yψ≳0.1, the BR(ϕ→ψψ) is dominant and yψ largely controls κΓ2. For κΓ2=1, σϕ×BRγγ can reach only about 0.03 fb for sh=0.01. For very small yψ≲0.1, the total width (i.e., κΓ2) is small and dominated by top and U loops and the tree-level ϕ→hh,tt decay. For yψ→0, sh→0, both σϕ×BRγγ and κΓ2 come from U loops and scales as yU4 and yU2, respectively; σϕ×BRγγ increases with κΓ2 up to around κΓ2≃0.03.


## 4. Dark Matter Phenomenology

In this section we identify the region of parameter space of the model of Section2 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.


### 4.1. Dark Matter Relic Density

The dark matter relic density is set by the self-annihilation processesψψ→SM mediated by s-channel h,ϕ exchange. The relic density can be computed as detailed, for example, in Appendix A of [24]. We have for our case [7, 24] the self-annihilation thermally averaged cross section given by (14)σv=6xf18πs∑iBi2Π^PSi,where xf≡Mψ/Tf≈25 with Tf the freeze-out temperature; the sum is over all self-annihilation processes ψψ→fifi for final states fi kinematically allowed; Bi2 is the coefficient of vrel2 in the amplitude squared for each process, vrel being the relative velocity of the two initial states ψ; Π^PSi≡(1-4mi2/s)1/2 is a phase-space factor with mi being 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≈4Mψ2. In our analysis we include the two-body final states bb¯,WW,ZZ,hh,tt¯,gg, whichever are kinematically allowed for that given Mψ. The loop-level γγ,Zγ final states are insignificant compared to gg, and therefore we do not include them. Bi2 for each of these final states are extracted from [7] to which we add Bgg2 here. These are given by (15)Bff¯2=Ncfyf2yψ2sh2ch21-4mi2sMψ4S^BWhϕ;S^BWhϕ=Mϕ2-Mh22s-Mh22+Mh2Γh2s-Mϕ22+Mϕ2Γϕ2,BWW2=yψ2g4v2sh2ch2Mψ2412+s/2-MW224MW4S^BWhϕ,Bhh2=Mψ2yψ264sh2ch6κ3h2v2s-Mh22+Mh2Γh2+ch8κϕhh2Mϕ2s-Mϕ22+Mϕ2Γϕ2-2shch7κ3hvκϕhhMϕs-Mh2s-Mϕ2+MhMϕΓhΓϕ,Bgg2=16yψ2Mψ616π2M2ch2κϕgg2s-Mϕ22+Mϕ2Γϕ2+sh2κhgg2s-Mh22+Mh2Γh2-2chshκϕggκhggs-Mh2s-Mϕ2+MhMϕΓhΓϕ,where s≈4Mψ2, S^BWhϕ is a Breit-Wigner resonance factor including the s-channel {h,ϕ} contributions, ff¯={bb¯,tt¯}, MZZ is identical to MWW except for an additional factor of 1/(2cW2) and MW→MZ, and in Mhh we do not include the t-channel (and u-channel) contribution as it is suppressed by an extra factor of sh and can be ignored for sh≪1. M is a mass scale which we set to 1 TeV for numerical evaluations as explained below (5). We evaluate κϕgg and κhgg using (6) taking rf=mf2/(4Mψ2), since P2=s≈4Mψ2 here. The mixing angle θh enters in κϕgg and κhgg through ϕUU,ϕtt, and htt couplings. Although ττ and γγ final states are also possible, we neglect them in our analysis since these contributions are small owing to a small yτ 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 sh≲0.07, the gg contribution becomes comparable or even larger than the tree-level contributions.


### 4.2. Dark Matter Direct Detection

The dark matter direct detection elastic scattering cross section on a nucleon is mediated byh,ϕ exchange. If sh≲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 [7], which we generalize here to include ϕ contribution also since we consider sh≲0.07. The scalar-nucleon-nucleon coupling is generated due to the scalar coupling to the quark content of the nucleon and also due to 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 (16)L⊃λhNN2h^N¯N+λϕNN2ϕ^N¯N,L⊃chλhNN-shλϕNN2hN¯N+chλϕNN+shλhNN2ϕN¯N,where N denotes the nucleon, and in the second line we write in the mass basis. We derive λhNN and λϕNN using the formalism of [43] updated in [44] (for a review, see Appendix C of [45]) in which the scalar-nucleon coupling is denoted as fp,n. Identifying λhNN≡fp,n, we have [44](17)λhNN=∑q=u,d,sfTqp,nyqmp,nmq+fTGp,n227∑q=c,b,tyqmp,nmq,with [44] fTu(p)=0.02±0.004, fTd(p)=0.026±0.005, fTs(p)=0.118±0.062, fTu(n)=0.014±0.003, fTd(n)=0.036±0.008, fTs(n)=0.118±0.062, and fTG(p,n)=1-∑q=u,d,sfTq(p,n)≈0.85. We then find numerically λhNN=2×10-3, which we take for our numerical analysis. λϕNN coupling is induced via the ϕ couplings to the gluon content of the nucleon, that is, via the ϕgg effective coupling induced by the VLQU. Following the same procedure as above, we derive this coupling from the second term in (17) as λϕNN=(2/27)fTG(p,n)yUm(p,n)/MU≈0.063yUmN/MU, 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. Reference [46] examines recent developments and argues for a smaller value of the coupling λhNN≈1.1×10-3. Other sophisticated analyses can be found, for example, in [14, 47–49].We can now write the spin-averagedψ elastic scattering cross section on a nucleon for q2≪mN2 as (18)σψN→ψN=yψ28πshchλhNN-shλϕNNMh2-chchλϕNN+shλhNNMϕ22pψ2+mN2,σψN→ψN=yψ2sh2ch2λhNN28πpψ2+mN2Mh41-λϕNNλhNNchsh1+Δϕ1-ΔhMh2Mϕ22,where pψ≈Mψvψ with vψ~10-3 [45], mN≈1 GeV is the nucleon mass, Δh=(λϕNN/λhNN)(sh/ch), and Δϕ=(λhNN/λϕNN)(sh/ch). This is the generalization of direct detection elastic cross section Eq. (13) of [7] which included only the h contribution, to now include the ϕ contribution also that becomes important for very small sh. For instance, for sh=0.01, Mϕ=1000 GeV, the extra factor in (18); namely, ⋯2≈1-0.125(1200GeV/MU)(yU/2.5)2, with Δϕ,h≪1 which can be dropped. Including the ϕ contribution thusdecreases the elastic cross section by about 25 % for the central values we choose.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, e.g., [50–52]), the direct detection exclusion limits should be taken to be accurate only up to unknown O(1) factors.


### 4.3. Dark Matter Preferred Regions of Parameter Space

Here we show regions of parameter space of the model of Section2 for which we obtain the observed relic density which 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 Ωdm=0.26±0.015 [53], we need the thermally averaged self-annihilation cross section to be σv≈2.3×10-9GeV-2.In Figures5 and 6 we plot contours of Ωdm=0.1,0.25,0.3 with both the loop induced couplings due to U and the Higgs-portal couplings present for Mϕ=1000 GeV and MU=1200 GeV and show the regions with σDD>5×10-45 cm2, 10-45cm2<σDD<5×10-45cm2, 10-46cm2<σDD<10-45 cm2, 10-47cm2<σDD<10-46 cm2, 10-48cm2<σDD<10-47 cm2, 10-49cm2<σDD<10-48 cm2, and σDD<10-49 cm2 with parameters not varied along the axes fixed at sh=0.01, Mψ=475 GeV, and yU=2.5. The entire parameter region shown in the plots satisfies the unitarity constraint in (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 σDD≤(0.1–1)×10-45 cm2 [54–56] 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 Mψ~Mϕ,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 κΓ2≲0.1 as can be seen from Figure 4.Figure 5
Contours ofΩdm=0.1,0.25,0.3, for yψ=2, Mϕ=1000 GeV, and MU=1200 GeV, for sh=0 (a) and sh=0.05 (b) with the colored bands showing σDD as marked.
(a)
(b)Figure 6
Same as in Figure5 but with the parameters not varied along the axes fixed at yU=2.5, sh=0.01, and Mψ=475 GeV. The thick red line shows the 8 TeV LHC hh channel constraint.We first explore thesh=0 limit, that is, 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 (14)–(18). The correct relic density can be achieved in this limit as shown in Figure 5(a), and we obtain σDD<10-49cm2. The required relic density, for example, can be obtained for yψ=2, yU=2.5, MU=1200 GeV, Mϕ=1000 GeV, and Mψ=467 GeV, for which σDD=2.4×10-51cm2. Thus the sh=0 limit provides an example scenario in which the relic density is satisfied but direct detection is very challenging.In Figure5(b) we show the situation for sh=0.05, that is, when the Higgs-portal is also turned on. For yU≳1 the loop induced couplings due to U are significant, while for smaller yU the Higgs-portal contribution dominates. Thus, for yU≲0.5 the relic density contour starts losing dependence on yU, and for yU=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 havesh≪1 in which case the gg contribution dominates, the dark matter relic density scales as ~yψyU-2 to a very good approximation as can be inferred from (14) and (15). Similarly, the dark matter direct detection rate also scales the same way in this limit, as evident from (18). Thus, for Mϕ=1000 GeV and for a given value of Mψ, other values of (yψ,yU) that give the correct relic density and direct detection rates can be obtained from those in Figure 6, by scaling yψ→(2.5/yU)yψ. Thus, for sh≪1, since the coupling of the dark matter with SM states is via loop-level effective couplings, we find, for Mϕ=1000 GeV and Mψ=475 GeV, moderately large values yψyU≈5 are required in order for the dark matter self-annihilation cross section to be of sufficient size to give the correct relic density. Taking smaller values of yψyU will require tuning Mψ closer to Mϕ/2 (or to Mh/2). The regions we identify are safe from present direct detection constraints and will be probed in upcoming experiments.


## 4.1. Dark Matter Relic Density

The dark matter relic density is set by the self-annihilation processesψψ→SM mediated by s-channel h,ϕ exchange. The relic density can be computed as detailed, for example, in Appendix A of [24]. We have for our case [7, 24] the self-annihilation thermally averaged cross section given by (14)σv=6xf18πs∑iBi2Π^PSi,where xf≡Mψ/Tf≈25 with Tf the freeze-out temperature; the sum is over all self-annihilation processes ψψ→fifi for final states fi kinematically allowed; Bi2 is the coefficient of vrel2 in the amplitude squared for each process, vrel being the relative velocity of the two initial states ψ; Π^PSi≡(1-4mi2/s)1/2 is a phase-space factor with mi being 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≈4Mψ2. In our analysis we include the two-body final states bb¯,WW,ZZ,hh,tt¯,gg, whichever are kinematically allowed for that given Mψ. The loop-level γγ,Zγ final states are insignificant compared to gg, and therefore we do not include them. Bi2 for each of these final states are extracted from [7] to which we add Bgg2 here. These are given by (15)Bff¯2=Ncfyf2yψ2sh2ch21-4mi2sMψ4S^BWhϕ;S^BWhϕ=Mϕ2-Mh22s-Mh22+Mh2Γh2s-Mϕ22+Mϕ2Γϕ2,BWW2=yψ2g4v2sh2ch2Mψ2412+s/2-MW224MW4S^BWhϕ,Bhh2=Mψ2yψ264sh2ch6κ3h2v2s-Mh22+Mh2Γh2+ch8κϕhh2Mϕ2s-Mϕ22+Mϕ2Γϕ2-2shch7κ3hvκϕhhMϕs-Mh2s-Mϕ2+MhMϕΓhΓϕ,Bgg2=16yψ2Mψ616π2M2ch2κϕgg2s-Mϕ22+Mϕ2Γϕ2+sh2κhgg2s-Mh22+Mh2Γh2-2chshκϕggκhggs-Mh2s-Mϕ2+MhMϕΓhΓϕ,where s≈4Mψ2, S^BWhϕ is a Breit-Wigner resonance factor including the s-channel {h,ϕ} contributions, ff¯={bb¯,tt¯}, MZZ is identical to MWW except for an additional factor of 1/(2cW2) and MW→MZ, and in Mhh we do not include the t-channel (and u-channel) contribution as it is suppressed by an extra factor of sh and can be ignored for sh≪1. M is a mass scale which we set to 1 TeV for numerical evaluations as explained below (5). We evaluate κϕgg and κhgg using (6) taking rf=mf2/(4Mψ2), since P2=s≈4Mψ2 here. The mixing angle θh enters in κϕgg and κhgg through ϕUU,ϕtt, and htt couplings. Although ττ and γγ final states are also possible, we neglect them in our analysis since these contributions are small owing to a small yτ 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 sh≲0.07, the gg contribution becomes comparable or even larger than the tree-level contributions.


## 4.2. Dark Matter Direct Detection

The dark matter direct detection elastic scattering cross section on a nucleon is mediated byh,ϕ exchange. If sh≲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 [7], which we generalize here to include ϕ contribution also since we consider sh≲0.07. The scalar-nucleon-nucleon coupling is generated due to the scalar coupling to the quark content of the nucleon and also due to 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 (16)L⊃λhNN2h^N¯N+λϕNN2ϕ^N¯N,L⊃chλhNN-shλϕNN2hN¯N+chλϕNN+shλhNN2ϕN¯N,where N denotes the nucleon, and in the second line we write in the mass basis. We derive λhNN and λϕNN using the formalism of [43] updated in [44] (for a review, see Appendix C of [45]) in which the scalar-nucleon coupling is denoted as fp,n. Identifying λhNN≡fp,n, we have [44](17)λhNN=∑q=u,d,sfTqp,nyqmp,nmq+fTGp,n227∑q=c,b,tyqmp,nmq,with [44] fTu(p)=0.02±0.004, fTd(p)=0.026±0.005, fTs(p)=0.118±0.062, fTu(n)=0.014±0.003, fTd(n)=0.036±0.008, fTs(n)=0.118±0.062, and fTG(p,n)=1-∑q=u,d,sfTq(p,n)≈0.85. We then find numerically λhNN=2×10-3, which we take for our numerical analysis. λϕNN coupling is induced via the ϕ couplings to the gluon content of the nucleon, that is, via the ϕgg effective coupling induced by the VLQU. Following the same procedure as above, we derive this coupling from the second term in (17) as λϕNN=(2/27)fTG(p,n)yUm(p,n)/MU≈0.063yUmN/MU, 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. Reference [46] examines recent developments and argues for a smaller value of the coupling λhNN≈1.1×10-3. Other sophisticated analyses can be found, for example, in [14, 47–49].We can now write the spin-averagedψ elastic scattering cross section on a nucleon for q2≪mN2 as (18)σψN→ψN=yψ28πshchλhNN-shλϕNNMh2-chchλϕNN+shλhNNMϕ22pψ2+mN2,σψN→ψN=yψ2sh2ch2λhNN28πpψ2+mN2Mh41-λϕNNλhNNchsh1+Δϕ1-ΔhMh2Mϕ22,where pψ≈Mψvψ with vψ~10-3 [45], mN≈1 GeV is the nucleon mass, Δh=(λϕNN/λhNN)(sh/ch), and Δϕ=(λhNN/λϕNN)(sh/ch). This is the generalization of direct detection elastic cross section Eq. (13) of [7] which included only the h contribution, to now include the ϕ contribution also that becomes important for very small sh. For instance, for sh=0.01, Mϕ=1000 GeV, the extra factor in (18); namely, ⋯2≈1-0.125(1200GeV/MU)(yU/2.5)2, with Δϕ,h≪1 which can be dropped. Including the ϕ contribution thusdecreases the elastic cross section by about 25 % for the central values we choose.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, e.g., [50–52]), the direct detection exclusion limits should be taken to be accurate only up to unknown O(1) factors.


## 4.3. Dark Matter Preferred Regions of Parameter Space

Here we show regions of parameter space of the model of Section2 for which we obtain the observed relic density which 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 Ωdm=0.26±0.015 [53], we need the thermally averaged self-annihilation cross section to be σv≈2.3×10-9GeV-2.In Figures5 and 6 we plot contours of Ωdm=0.1,0.25,0.3 with both the loop induced couplings due to U and the Higgs-portal couplings present for Mϕ=1000 GeV and MU=1200 GeV and show the regions with σDD>5×10-45 cm2, 10-45cm2<σDD<5×10-45cm2, 10-46cm2<σDD<10-45 cm2, 10-47cm2<σDD<10-46 cm2, 10-48cm2<σDD<10-47 cm2, 10-49cm2<σDD<10-48 cm2, and σDD<10-49 cm2 with parameters not varied along the axes fixed at sh=0.01, Mψ=475 GeV, and yU=2.5. The entire parameter region shown in the plots satisfies the unitarity constraint in (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 σDD≤(0.1–1)×10-45 cm2 [54–56] 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 Mψ~Mϕ,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 κΓ2≲0.1 as can be seen from Figure 4.Figure 5
Contours ofΩdm=0.1,0.25,0.3, for yψ=2, Mϕ=1000 GeV, and MU=1200 GeV, for sh=0 (a) and sh=0.05 (b) with the colored bands showing σDD as marked.
(a)
(b)Figure 6
Same as in Figure5 but with the parameters not varied along the axes fixed at yU=2.5, sh=0.01, and Mψ=475 GeV. The thick red line shows the 8 TeV LHC hh channel constraint.We first explore thesh=0 limit, that is, 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 (14)–(18). The correct relic density can be achieved in this limit as shown in Figure 5(a), and we obtain σDD<10-49cm2. The required relic density, for example, can be obtained for yψ=2, yU=2.5, MU=1200 GeV, Mϕ=1000 GeV, and Mψ=467 GeV, for which σDD=2.4×10-51cm2. Thus the sh=0 limit provides an example scenario in which the relic density is satisfied but direct detection is very challenging.In Figure5(b) we show the situation for sh=0.05, that is, when the Higgs-portal is also turned on. For yU≳1 the loop induced couplings due to U are significant, while for smaller yU the Higgs-portal contribution dominates. Thus, for yU≲0.5 the relic density contour starts losing dependence on yU, and for yU=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 havesh≪1 in which case the gg contribution dominates, the dark matter relic density scales as ~yψyU-2 to a very good approximation as can be inferred from (14) and (15). Similarly, the dark matter direct detection rate also scales the same way in this limit, as evident from (18). Thus, for Mϕ=1000 GeV and for a given value of Mψ, other values of (yψ,yU) that give the correct relic density and direct detection rates can be obtained from those in Figure 6, by scaling yψ→(2.5/yU)yψ. Thus, for sh≪1, since the coupling of the dark matter with SM states is via loop-level effective couplings, we find, for Mϕ=1000 GeV and Mψ=475 GeV, moderately large values yψyU≈5 are required in order for the dark matter self-annihilation cross section to be of sufficient size to give the correct relic density. Taking smaller values of yψyU will require tuning Mψ closer to Mϕ/2 (or to Mh/2). The regions we identify are safe from present direct detection constraints and will be probed in upcoming experiments.


## 5. Conclusions

In this work, we study a BSM model with a hidden sector containing a stable gauge-singlet vector-like fermion dark matterψ and a gauge-singlet scalar ϕ. 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⁡θh≲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 largeϕ Yukawa couplings to the vector-like fermions yψyU≈5 are required in order to get a large enough dark matter self-annihilation cross section to obtain the correct relic density. Furthermore, (Mψ,Mϕ) needs to be in the pole enhanced region, that is, Mψ should be within a few tens of GeV of Mϕ/2 (or a few tenths of GeV of Mh/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 Figure4. 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 Figure 6, with the region consistent with the direct 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 10-51cm2. 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.



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*Source: 8689270-2017-12-17.xml*