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

Gopalakrishna, Shrihari; Mukherjee, Tuhin Subhra

doi:https://doi.org/10.1155/2017/8689270

Advances in High Energy Physics

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Research Article | Open Access

Volume 2017 | Article ID 8689270 | https://doi.org/10.1155/2017/8689270

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

Shrihari Gopalakrishna¹and Tuhin Subhra Mukherjee¹

Academic Editor: Anna Cimmino

Received15 Sept 2017

Accepted31 Oct 2017

Published17 Dec 2017

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.

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 matter and a scalar . We also add an (color) triplet, singlet, hypercharge , and vector-like quark (VLQ). 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 mixing with the SM Higgs boson and also via the loop-level couplings of to two gluons and to two hypercharge gauge bosons induced by .

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 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 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 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 () and scalar-photon-photon () effective couplings and explore the phenomenology including these couplings. We obtain expressions for some relevant decay modes of , present direct LHC constraints, LHC gluon-gluon-fusion rate for production, and LHC diphoton rate from decay for various total widths of . For processes involving the dark matter, in addition to the Higgs boson contributions, the new contributions here are the -channel contribution to the self-annihilation process that sets the dark matter relic density in the early universe and the -channel 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 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 the mass of 1 TeV. We also present the constraint on the mixing angle () from the LHC channel results [21], which is not analyzed in the references mentioned above. Usually in the literature, only the mediated processes are included in the dark matter direct detection cross section calculations. However, for small (or when there is no mixing), the mediated processes are suppressed, and the mediated processes due to and 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 Section 2 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 () and scalar-photon-photon () loop-level couplings. We compute the dominant decay modes. We infer the perturbative unitarity constraints on the couplings to the VLFs. In Section 3 we compute expressions for 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 and effective couplings.

2. Vector-Like Fermion Dark Matter Model

A VLF is composed of two different Weyl fermions as its and 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, , where is the SM gauge group , and is any new physics group, the states charged only under and singlets under will look like a hidden sector to us. To include the possibility of the hidden sector scalar to be in a nontrivial representation of , we take to be complex, with the real component denoted as . For example, [7] discusses a model in which is gauge symmetry.

Here, we present a model with a SM gauge-singlet hidden sector containing a vector-like fermion dark matter candidate with mass and a CP-even scalar with mass that couples to the visible SM sector via loop induced couplings due to an -singlet color triplet VLQ having hypercharge and mass . (The color triplet is the fundamental representation of the gauged 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, e.g., [22] and references therein) and (b) mixing between and the SM Higgs (), 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 with loop induced couplings to the SM. 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). The loop-level coupling of to two hypercharge gauge bosons imply , , and couplings. is the only new state that is charged under both and 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 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 model)where we show only the relevant terms in and do not repeat all the SM terms, and “” represents the antisymmetric product in space. We have also not shown possible and operators since they do not affect the phenomenology being studied here. The are the 3-generation SMFs, and we suppress the generation index on these fields. Here we have included right-handed neutrinos () also for completeness; whether this is present in nature is still being probed in experiments. The VLQ is in the fundamental of and has EM charge and thus has gauge interactions with the gluons () and hypercharge gauge bosons () exactly as the SM up-type right-chiral quarks, which are not shown explicitly. , being an SM gauge singlet, has no SM gauge interactions. We have demanded that respect a symmetry under which only is odd (i.e., ) and all other fields are even. This leads to an absolutely stable which we identify as our dark matter candidate. This symmetry forbids the operator (where is the SM lepton doublet) which would have otherwise been allowed and caused to decay. (If the symmetry is not imposed, this operator would be allowed and is the neutrino Yukawa operator, and can then be identified as the right-handed neutrino . 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 VLQ can be ensured by allowing the mixed operators where is a left-chiral SM quark doublet and a right-chiral SM quark singlet. We chose the hypercharge of to be 2/3 to be able to write these operators in (2) that singly couple 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 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 and and allowing mixings with third-generation quarks only (for details see, e.g., [24]).

In Figure 1 we show schematically the two contributions to the coupling between and the SM. On the left we show the Higgs-portal contribution due to - mixing, while on the right we show the loop induced couplings to two gluons and to two hypercharge gauge bosons () due to the VLQ. The latter coupling implies the coupling that leads to the diphoton signature explored in Section 3.

(a)

(b)

We can contemplate other hypercharge assignments for , even a hypercharge neutral assignment with being an electroweak singlet and only charged under . In this case, the 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 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 effective coupling.

Next, we study the 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 and couplings induced by the VLQ.

2.1. Higgs-Scalar Mixing

If the scalar potential is such that nonzero vacuum expectation values (VEVs) are generated, namely, and , and the fluctuations around these are denoted as and , respectively, and mix due to spontaneous symmetry breaking. The interaction strength in (1) is given by . Diagonalizing the mixing terms, we go from basis to the mass basis and define the mass eigenstates to be and with mass eigenvalues , respectively. The mixing angle is given byIn Figure 2 we show the regions of parameter space that result in a small . We show contours in the plane. In our numerical analysis below, we treat as an input parameter, and one can always relate it to the parameters if needed using (3). The phenomenology due to the operator is discussed in detail, for example, in [7]. In the mass basis we have where we have defined the dimensionless coupling . We identify the mass eigenstate as the 125 GeV Higgs boson discovered at the LHC.

We identify here a scenario in which , implying a suppressed Higgs-portal coupling. Consider the situation when is either very small or zero, and . The former is the case when has nonzero charge in (see, e.g., [7]), and the latter is when there is no spontaneous symmetry breaking in the sector, due to taking a positive mass-squared term for 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 , under which the is odd and all other fields are even. The full discrete symmetry under consideration then is , where the former being exact is what is keeping the dark matter absolutely stable. The consequence of the symmetry is that if is zero at some scale, then at the tree-level at that scale, as can be seen from (3). however is broken explicitly by the and operators and will generate the term at loop level (in fact at 3-loop level). This will result in a tiny and also correspondingly small. This serves as an example of a scenario in which . When is suppressed, the loop induced coupling of to the SM due to the VLQ becomes important to include. We discuss these loop induced couplings next.

2.2. The and Loop-Level Effective Couplings

When is small (of the order of ), the loop induced couplings of the to the SM induced by exchange of the VLQ will become important. The tree-level coupling and these loop induced and couplings will then couple the dark matter VLF to the SM. The and effective couplings induced by the VLF are detailed in [23]. Here we summarize these contributions for easy reference.

The effective Lagrangian defining the effective couplings and can be written for the CP-even following the general definitions in [23] aswhere are the photon and gluon field-strengths, respectively; is an arbitrary mass scale which we introduce to make the and effective couplings dimensionless, and we show the numerical results of these effective couplings for 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 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 , with , the invariant-mass-square of the scalar, running over all colored fermion species (includes SMFs and VLFs) with mass and Yukawa couplings , and the electric charge of the fermion () denoted by , and at 1-loop level are (for details see [23])The expressions for in (6) reduce to the closed form expressions given in [25]. The color factor in is , where are the adjoint color indices. Computing a decay rate or cross section by summing over gives resulting in a color factor of 2. In the numerical results below, we include this color factor in and suppress the color indices. Analogous expressions hold for the and effective couplings, and in our numerical analysis we include the contribution of the VLQ in addition to the usual SMF contributions. In Figure 3 we show the numerical values of the 1-loop effective couplings and generated by the VLQ for GeV, GeV, and , in the – plane.

(a)

(b)

2.3. Decay

In our analysis, we include the decay modes , where 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 total width in terms of , which we define as The contribution of each decay mode to includes the couplings and phase-space factors relevant to that decay. Expression for the can be found, for example, in [23, 25]. For instance, for the decay , via a Yukawa coupling , we have a contribution . For the decay into a quark-pair, the same formula holds but is multiplied by the color factor . The coupling identified in (4) leads to the decay, which contributes to , an amount . For the loop-level decay and decay , as detailed in [23], we have and .

2.4. Perturbative Unitarity Constraint

If the Yukawa coupling for any fermion becomes very large, certain processes will violate perturbative unitarity. Thus, demanding perturbative unitarity will imply an upper bound on . We assume that , , and of (1) are all small enough so that there is no constraint on these. Here, we take and obtain upper bounds on and , the and Yukawa couplings defined in (1), from perturbative unitarity of the process at tree-level for , where is one of the Mandelstam variables as usual.

The partial wave of the elastic scattering amplitude is bounded by perturbative unitarity to be [26, 27]. For the process , the helicity amplitude in the limit of is given by [28, 29], where “+” denote the helicities of the fermions. The partial wave amplitude is then readily written down as . There is no -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 the channel for a single , a stronger bound could result from scattering channels with different initial and final state fermions, that is, from the “coupled channels” with . To find this, we consider in the basis (no sum on ) with , the color index, the coupled channel matrix: The largest eigenvalue of this coupled channel matrix is . Applying the perturbative unitarity bound on the coupled channel corresponding to this maximum eigenvalue thus implies We ensure that this bound is satisfied in the numerical analysis of the following sections.

3. LHC Phenomenology

The dark matter , 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 and the VLQ, 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 , where . In this work, we only focus on the signature at the LHC, since in comparison in our model are typically smaller by a factor that ranges from about 4 to 10 depending on .

3.1. Production at the LHC

We consider here production via the gluon-gluon fusion channel at the LHC. Rather than computing 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 denotes a scalar with SM-Higgs-like couplings to other SM states with the mass varied. We take from [30] the 14 TeV LHC for GeV and multiply by to get the TeV values [31]. As can be inferred from (10) and detailed in [23], a quark coupled to via a Yukawa coupling as in (1) gives a contribution to given by where the sum over includes all quarks, including the top quark and VLQ contributions, is the top Yukawa coupling (we ignore the effect of running this to the scale ), and is defined in (6) whose argument is with for obtaining the on-shell resonant cross section. We include contributions from , and in the small scalar-Higgs mixing region (), the contribution dominates.

3.2. LHC Constraints

We discuss here the LHC constraints on the model from the , dijet, and di-Higgs channels. In Figure of [23], constraints on from the 8 TeV LHC exclusion limits are shown. For an SM-like Higgs with mass GeV, we have with fb at the 8 TeV LHC due mainly to the top contribution. From the expression in (6), with here, since for on-shell production, and with , we derive the bound where the sum over is as explained below (11), , and . The index runs over various channels , that is, , and we have , (corresponding to ) as derived in [23]. The limits on our model due to will be very weak for small mixings . The LHC upper limit on the dijet channel at a mass of GeV is about 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 at a resonance mass of GeV is about fb as can be read out from the experimental exclusion plot in Figure of [21] from the ATLAS collaboration. ( of [21] in our case is decaying into .) This translates into in (12). For the parameter ranges in our study, we find the di-Higgs limit is stringent and limits for (cf. Figure 6 for the limits for a range of ).

Generically, in new physics models including the one under consideration here, there are shifts in the couplings to SM states, which are constrained by the LHC data (see, e.g., [33]). Once the above constraint is enforced, the constraints from the Higgs coupling measurements are satisfied.

The precise direct limit on the mass of the VLQ depends on the BRs. The lower limit on the mass is presently in the – GeV range [34–38]. For a long-lived VLQ with life-times in the range – s, the bound is looser with GeV being allowed [39, 40]. (It may be possible to weaken the VLQ mass bound somewhat by allowing the decay where is an singlet and will lead to missing energy at the LHC. This, e.g., can be achieved by introducing the operators where is the VLQ and is the SM right-chiral top quark. Due to the new decay mode, the usual assumption that the BRs into the SM final states () 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 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 in terms of the effective couplings can be written aswhere, as already explained below (5), is a reference mass scale which we take to be TeV. can be obtained from (13), and the expressions for , are given in (6) with SMF and VLQ contributions included. For in (6) only the SMF is included. As a representative benchmark point, we present results in this section for GeV.

In Figure 4 we show contours of (in fb) and various as colored regions (darker to lighter shades correspond to smaller to larger ), with the parameters not varied along the axes fixed at , GeV, GeV, GeV, , and . 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 pb, and the partial widths are GeV, respectively. The current sensitivity of the LHC searches is about fb. For the parameters in the figure, the diphoton rate range is – 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 or , and is dominated by ; in this limit and fb for the set of parameters chosen with . For and large , is large, being dominated by decay, resulting in a very small . In the region where is within about 5 MeV of and if MeV, a large threshold enhancement is possible [42], which we do not include in our analysis.

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 varying and 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 TeV channel constraints discussed in Section 3.2 constrain . For example, this constraint leads to the bound for and . For , the is dominant and largely controls . For , can reach only about fb for . For very small , the total width (i.e., ) is small and dominated by top and loops and the tree-level decay. For , , both and come from loops and scales as and , respectively; increases with up to around .

4. Dark Matter Phenomenology

In this section we identify the region of parameter space of the model of Section 2 where the VLF 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 processes mediated by s-channel exchange. The relic density can be computed as detailed, for example, in Appendix of [24]. We have for our case [7, 24] the self-annihilation thermally averaged cross section given by where with the freeze-out temperature; the sum is over all self-annihilation processes for final states kinematically allowed; is the coefficient of in the amplitude squared for each process, being the relative velocity of the two initial states ; is a phase-space factor with being the mass of the final state particle, and is the Mandelstam variable, which for a cold-dark matter candidate during freeze-out is . In our analysis we include the two-body final states , whichever are kinematically allowed for that given . The loop-level final states are insignificant compared to , and therefore we do not include them. for each of these final states are extracted from [7] to which we add here. These are given by where , is a Breit-Wigner resonance factor including the -channel contributions, , is identical to except for an additional factor of and , and in we do not include the -channel (and -channel) contribution as it is suppressed by an extra factor of and can be ignored for . is a mass scale which we set to TeV for numerical evaluations as explained below (5). We evaluate and using (6) taking , since here. The mixing angle enters in and through , and couplings. Although and final states are also possible, we neglect them in our analysis since these contributions are small owing to a small 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 case. For small , the 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 by exchange. If , the contribution is also important even though it is much heavier than . The exchange contribution is given, for example, in [7], which we generalize here to include contribution also since we consider . 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 effective couplings. We define an effective Lagrangian for the scalar-nucleon-nucleon interaction as where denotes the nucleon, and in the second line we write in the mass basis. We derive and using the formalism of [43] updated in [44] (for a review, see Appendix of [45]) in which the scalar-nucleon coupling is denoted as . Identifying , we have [44]with [44] , , , , , , and . We then find numerically , which we take for our numerical analysis. coupling is induced via the couplings to the gluon content of the nucleon, that is, via the effective coupling induced by the VLQ. Following the same procedure as above, we derive this coupling from the second term in (17) as , 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 . Other sophisticated analyses can be found, for example, in [14, 47–49].

We can now write the spin-averaged elastic scattering cross section on a nucleon for as where with [45], GeV is the nucleon mass, , and . This is the generalization of direct detection elastic cross section Eq. of [7] which included only the contribution, to now include the contribution also that becomes important for very small . For instance, for , GeV, the extra factor in (18); namely, , with which can be dropped. Including the contribution thus decreases the elastic cross section by about % 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 factors.

4.3. Dark Matter Preferred Regions of Parameter Space

Here we show regions of parameter space of the model of Section 2 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 [53], we need the thermally averaged self-annihilation cross section to be .

In Figures 5 and 6 we plot contours of with both the loop induced couplings due to and the Higgs-portal couplings present for GeV and GeV and show the regions with cm², cm², cm², cm², cm², cm², and cm² with parameters not varied along the axes fixed at , GeV, and . 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 cm² [54–56] for dark matter mass in the – GeV range. The correct self-annihilation cross section is obtained only with an enhancement of the cross section at the pole with . Being close to the pole suppresses the decay rate due to the limited phase-space available, leading to a small as can be seen from Figure 4.

(a)

(b)

We first explore the limit, that is, when the dark matter couples to the SM entirely via the and 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 . The required relic density, for example, can be obtained for , , GeV, GeV, and GeV, for which . Thus the limit provides an example scenario in which the relic density is satisfied but direct detection is very challenging.

In Figure 5(b) we show the situation for , that is, when the Higgs-portal is also turned on. For the loop induced couplings due to are significant, while for smaller the Higgs-portal contribution dominates. Thus, for the relic density contour starts losing dependence on , and for 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 in which case the contribution dominates, the dark matter relic density scales as 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 GeV and for a given value of , other values of that give the correct relic density and direct detection rates can be obtained from those in Figure 6, by scaling . Thus, for , since the coupling of the dark matter with SM states is via loop-level effective couplings, we find, for GeV and GeV, moderately large values 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 will require tuning closer to (or to ). 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 matter and a gauge-singlet scalar . 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 singlet vector-like quark carrying hypercharge . 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 (), 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 Yukawa couplings to the vector-like fermions are required in order to get a large enough dark matter self-annihilation cross section to obtain the correct relic density. Furthermore, needs to be in the pole enhanced region, that is, should be within a few tens of GeV of (or a few tenths of GeV of ). We show that these large couplings are within the bounds of perturbative unitarity, by computing the upper bounds on these couplings from the coupled channel scattering process for .

The diphoton rate when the scalar-fermion couplings are varied is shown in Figure 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 Figure 6, with the region consistent with the direct LHC bound. In addition to the direct production signals of the vector-like quark at the LHC, another promising signal is the 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 . 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.

Appendix

Range of Diphoton Rate

In Section 2.4 we derived an upper bound on and from perturbative unitarity. Here, we show the range of diphoton rate at the LHC by saturating this upper bound. In Figure 7 we show versus in the model of Section 2, for GeV, and GeV, GeV, and and scanning over in the range , , subject to the unitarity constraint in (9). For example, for , we can get fb.

(a)

(b)

In Figure 8 we show contours of various in the plane that give fb for GeV. This cross section is presently allowed with the CL exclusion limit being about 1 fb [19, 20]. We show in Figure 8(b) a band of diphoton rate fb for two representative total width values and 3, a wide range with the former being % of and the latter . The latter width is rather large, and for , it is obtained for for large as we discuss below. For such large couplings, there is a danger of tree-level unitarity being violated, and our analysis of Section 2.4 becomes relevant. Figure 8 shows the situation model-independently in any model with as here, in which the and effective couplings can be calculated. In the same figure, we overlay “” to depict the situation for the particular model of Section 2, with the choice of GeV, GeV, and for and for .

(a)

(b)

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2017 Shrihari Gopalakrishna and Tuhin Subhra Mukherjee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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