Searches for vector-like quarks at future colliders and implications for composite Higgs models with dark matter

Searches for vector-like quarks at future
colliders and implications for composite Higgs models
with dark matter

Mikael Chala,    Ramona Gröber    and Michael Spannowsky mikael.chala@durham.ac.uk ramona.groeber@durham.ac.uk michael.spannowsky@durham.ac.uk
Abstract

Many composite Higgs models predict the existence of vector-like quarks with masses outside the reach of the LHC, e.g. TeV, in particular if these models contain a dark matter candidate. In such models the mass of the new resonances is bounded from above to satisfy the constraint from the observed relic density. We therefore develop new strategies to search for vector-like quarks at a future TeV collider and evaluate what masses and interactions can be probed. We find that masses as large as () TeV can be tested if the fermionic resonances decay into Standard Model (dark matter) particles. We also discuss the complementarity of dark matter searches, showing that most of the parameter space can be closed. On balance, this study motivates further the consideration of a higher-energy hadron collider for a next generation of facilities.

institutetext: Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham, DH1 3LE, UK\preprint

IPPP/18/6

1 Introduction

Composite Higgs Models (CHMs) Dimopoulos:1981xc (); Kaplan:1983fs (); Kaplan:1983sm () are among the most compelling solutions to the hierarchy problem. They involve a new strongly-interacting sector (approximately) symmetric under a global group that is spontaneously broken to at a new physics scale . The Higgs boson is assumed to be a pseudo Nambu-Goldstone Boson (pNGB) of this symmetry breaking pattern. The smoking-gun signature of this setup is the presence of new resonances, in particular Vector-Like Quarks (VLQs), whose masses scale as , with being the coupling of the strong sector. The Higgs mass in these models is also expected to scale as , hence requiring some tuning to make it light ( GeV) for large .

Current experimental bounds on range from to GeV for top-like VLQs111We emphasize that the corresponding searches are motivated by the minimal CHM Agashe:2004rs (). Thus, the results are obtained under the assumption that VLQs decay only into SM particles. For non-minimal symmetry breaking, the bounds can be significantly altered Chala:2017xgc (). Araque:2016jrb (); Aaboud:2017zfn (); Aaboud:2017qpr (). These numbers are still compatible with expectations from naturalness arguments: e.g. Refs. Contino:2006qr (); Matsedonskyi:2012ym (); Redi:2012ha (); Marzocca:2012zn (); Pomarol:2012qf (); Panico:2012uw () highlighted values for in the range TeV. Moreover, latter references (e.g. Ref. Panico:2012uw ()) have also shown that masses as large as TeV are compatible with a tuning on the Higgs mass of the order of in some classes of CHMs. Such masses are most probably beyond LHC reach Azatov:2013hya (); Ortiz:2014iza (); Matsedonskyi:2015dns () (at least in the model-independent pair-production mode). Therefore, it is conceivable that VLQs might not be discovered at the LHC and new facilities will be required to probe such models.

This conclusion can be further strengthened by exploring non-minimal CHMs containing extra stable pNGBs that can play the role of Dark Matter (DM) particles. These models are well motivated by two main reasons. (i) One single mechanism explains why the electroweak (EW) and the DM scales are of the same order, as suggested by the WIMP paradigm. (ii) The Higgs boson can have naturally small portal couplings to the pNGB DM, which evades the strong constraints from low-energy direct detection experiments. At the same time, the observed relic density can be produced by effective derivative couplings ( at momenta at which DM annihilation occurs). Thus, we will show that scenarios of this kind require few TeV to accommodate the correct DM abundance. Consequently, we study the reach of a future TeV collider to the VLQs, including decays into SM particles, which are also relevant for the minimal CHM, as well as into DM particles.

We extend previous works on the interplay between collider and DM searches in CHMs Frigerio:2012uc (); Marzocca:2014msa (); Fonseca:2015gva () in several ways: (i) Instead of focusing on a particular model, we adopt a generic parametrization that captures the main features of cosets like  Gripaios:2009pe (),  Chala:2016ykx (); Balkin:2017aep (),  Chala:2012af (); Ballesteros:2017xeg (),  Gripaios:2016mmi (), etc. (ii) We match to representations not previously considered in the literature (e.g the in ) as well as symmetry breaking patterns not yet studied (e.g. or ). (iii) In what concerns LHC constraints, we consider the latest experimental data, including LHC searches for heavy pair-produced resonances at TeV. (iv) We quantify the effect of having all resonances of a multiplet at once, instead of considering constraints on each separately.

This article is organized as follows. In Section 2, we introduce briefly the effective parametrization describing the different models of interest. In Section 3 we discuss the DM phenomenology in light of the parameters discussed previously. In Section 4 we detail the current LHC constraints on new fermionic resonances and discuss new analysis strategies for future colliders. We devote Section 4.1 to searches for VLQs with SM decays and Section 4.2 to searches for VLQs decaying into DM particles. In Section 5 we match the coefficients of the aforementioned parametrization to concrete non-minimal CHMs. We discuss the interplay between collider and DM searches in Section 6 and highlight the characteristics of the most viable models.

2 Parametrization

We will denote by the SM Higgs doublet with hypercharge . Likewise, we assume the presence of a single scalar DM field , singlet under the SM gauge group, whereas odd under a symmetry . The relevant Lagrangian for our study is parameterized by and , namely the typical mass of the fermionic resonances and the typical coupling of the strong sector, as well as a number of dimensionless coefficients. We can write the Lagrangian explicitly as

(2.1)

where and are , , is the number of colours and is the top Yukawa coupling. Note that not all these parameters are physical: for instance, a scaling of could be reabsorbed in the dimensionless coefficients. Likewise, only a particular combination of these coefficients enter into physical observables, e.g. the DM annihilation cross section (see below). This parametrization is simple, predictive, yet flexible enough, and it can comprise very different CHMs. Moreover, it reflects the expected power counting in these setups Giudice:2007fh (); Chala:2017sjk (). Finally, we emphasize that, being a strong coupling, is expected to be , while perturbative unitarity implies222The reduction on this estimation in comparison with the naive has been pointed out e.g. in Refs. Lee:1977eg (); Panico:2015jxa (). Often, the upper value is also used in the literature. . We restrict ourselves to this range henceforth.

One can easily link the phenomenology of pair-produced VLQs with that of . As a matter of fact, the former depends only on the mass of the VLQs, which is just given by . The number of such resonances and their charges depend crucially on the coset structure. In all our cases of interest, however, there is always a fourplet of VLQs transforming as under the custodial group and/or a VLQ decaying 100% into , with the physical Higgs boson and the SM top quark. For concreteness we assume that the decay rates of the different components in the fourplet are Serra:2015xfa ()

(2.2)

3 Dark matter phenomenology

Contrary to the derivative interactions in Eq. 2, the effective coupling driven by is not enhanced by the DM mass, , at the annihilation scale. Additionally, it is suppressed by an additional factor with respect to the Higgs portal coupling in the potential (proportional to ) in the low-energy DM-nucleon interactions. For these reasons, we take it to be zero hereafter for simplicity, but we will comment on the implications of switching it on when relevant.

Thus, the annihilation cross section is driven by the interaction, receiving contributions from both the sigma model Lagrangian and the potential. In particular, the Feynman rule associated to the quartic coupling between two DM particles and two Higgses (or two longitudinal gauge bosons) reads

(3.1)

where and are the four-momenta of the Goldstone bosons while and stand for those of the DM particles. All momenta are assumed to be incoming. Due to momentum conservation we have the relation . The last equality in Eq. (3) holds in the limit in which . We have neglected terms in , where GeV denotes the EW vacuum expectation value. In CHMs where both and come in the same multiplet of a larger group , one expects . In such a case, the derivative interactions drive the DM annihilation provided . Given that all these couplings are expected to be , derivative interactions are expected to be highly relevant in this class of models. Similar results have been previously pointed out in Refs. Frigerio:2012uc (); Fonseca:2015gva (); Chala:2016ykx (); Bruggisser:2016ixa (); Ballesteros:2017xeg (); Balkin:2017aep (); Balkin:2017yns (). For , in Eq. 3 is very small, and therefore can dominate the DM annihilation rate; see Ref. Balkin:2017yns () for an explicit example.

Regarding and , the relic density scales as . As a consequence, very large values of , as well as very small values of , are excluded by the requirement , where stands for the measured value of the total relic abundance Ade:2013zuv (). Concerning direct detection constraints, derivative interactions are irrelevant as they are velocity suppressed. We consider current LUX limits Akerib:2016vxi () and future LZ goals Szydagis:2016few () on spin-independent cross sections. In our notation, the theoretical prediction for the spin-independent cross section reads

(3.2)

with  Alarcon:2011zs (); Alarcon:2012nr () and GeV. Bounds from direct searches are therefore complementary to those set by the upper limit on . Furthermore, both are parametrically complementary to the quantity tested by collider searches for VLQs. Finally, indirect searches are of little relevance for scalar singlet models like the ones considered here Ballesteros:2017xeg (); Duerr:2015aka ().

4 Searches for new resonances

The masses of the top partners in Eq. 2 scale like . Thus, collider searches for VLQs are also complementary to the bounds set by direct-detection experiments and the measurement of the relic abundance. Moreover, contrary to direct detection tests, they are independent of the value of .

In order to compute the reach of current LHC data to the heavy resonances, we use VLQlimits Chala:2017xgc (). This code includes the information of several experimental searches, at both 8 TeV Aad:2015kqa () and 13 TeV ATLAS-CONF-2016-102 (); ATLAS-CONF-2016-104 (); ATLAS-CONF-2017-015 () with the largest luminosity, as well as SUSY searches sensitive to  CMS:2016hxa (). The code takes into account the simultaneous presence of all vector-like fermions in Eq. 2. The limits obtained in this way set a robust constraint on

(4.1)

For a high-luminosity run of the LHC with ab, we estimate the corresponding bound by rescaling the signal and background events with the luminosity. We obtain

(4.2)

4.1 Search for vector-like quarks with Standard Model decays at 100 TeV

In order to estimate the prospects for the searches of VLQs at 100 TeV, we consider a 3-lepton final state. In case of charged VLQs such a final state arises in decays where one boson and one top quark decays leptonically. For charged or charged VLQs this final state arises from decays . While at the LHC searches for VLQs are mainly performed in 1-or 2-lepton final states (with the exception of the search Aad:2014efa ()), due to the larger cross section, the 3-lepton final state has the advantage of smaller backgrounds. Since the cross sections are in general larger at the 100 TeV collider than at the LHC, the 3-lepton final state is an optimal choice for an estimate of the prospects for searches of VLQs.

We generate the signal with Madgraph5_aMC@NLO Alwall:2014hca () and use HERWIG Bellm:2015jjp (); Bellm:2017bvx () for the parton shower and the hadronization. The model file for the signal was generated with FeynRules Alloul:2013bka (). The dominant background processes are , and , with . For the latter we generate samples with an exclusive jet and we merge an additional jet using Sherpa 2.2.2 Gleisberg:2008ta (); Gleisberg:2008fv (). The background events for the processes and are generated with Madgraph5_aMC@NLO and showered and hadronized with HERWIG. Jets are clustered with the anti- algorithm as implemented in FastJet Cacciari:2011ma (), with a radius parameter . For the analysis we use Rivet Buckley:2010ar ().

We apply the following basic selection cuts:

  • exactly three leptons with and , and

  • at least four jets with and

  • a cut on the transverse momentum of the leading jet

  • an angular separation between the jets and the leptons of

  • We consider leptons to be isolated if they satisfy the so-called “mini-isolation” criterion Rehermann:2010vq (), as for very boosted objects the angle between the leptons and other decay products of the boosted object decreases. Hence we require for our analysis that

    (4.3)

    if

    (4.4)

    and

    (4.5)

    where the sum runs over all tracks (except the respective lepton track) with .

Furthermore, we apply the following cuts in the order

  1. we require that exactly two of the jets are tagged as  –jets, . The angular separation between  –jets and light jets is required to be . The tagging efficiency of the  –jets is set to 70% and the mistagging efficiency to 2%.

  2. we set a cut on where .

  3. if we discuss the searches for 2/3 charged fermions we also request that the one boson mass is reconstructed from either a pair or a pair in the window .

The cutflow for the background processes and the signal for under the assumption that is shown in Table 1. We give also the cutflow for pair production for under the assumption that . Note that this is equivalent to production. As it can be inferred from the table, the cut on gives a very good handle on the signal over the background. We exemplify this also further in Fig. 1 where the distribution for the different background processes and the signal for the masses (red) and (blue) is shown. For simplicity of the figure, we unify the background processes , and and the processes and into one, since they have a very similar shape (with the tending to slightly larger than and ) . As can be inferred from the figure, the variable can be used to distinguish very well between signal and background processes. With increasing mass of the top partner , the distribution of the signal peak at higher .

cut flow [fb] one
3.71 0.706
0.306
0.133
1.38 0.111
2.45 0.111 0
86.8 2.93
Table 1: Cut flow for the different background processes and the signal for with and with .
Figure 1: distribution for the signal process with (red) and (blue) and the background processes (violet), (orange) and (green).

We implement a simple counting approach and hence compute the significance by

(4.6)

where is the number of signal events and is the number of background events.

We then find that at a 100 TeV collider with () masses of the top partner of up to () can be excluded at , assuming . The discovery reach is for and for . For a or charged VLQ we find that masses up to () for or can be excluded. Note that the sensitivity is a bit less stringent than for the top partners, since we could not exploit the reconstruction of the leptonically decaying boson. However, this only results in a small effect on the significance.

In Fig. 2 we show the BRs that can be excluded at the level as a function of the mass, for a top partner (left plot) and a bottom partner (right plot). Note that for low masses lower BRs can be potentially excluded if a smaller cut is applied. Our cut is optimised for large masses of the VLQ. For large BRs into other final states, e.g. , other searches are needed. Under the assumption that the BRs add up to one, this will allow to exclude also lower BRs, as shown in Fig. 2. A closer assessment is however beyond the scope of this paper.

Figure 2: BRs that can be excluded at the 2 level for (left) and (right) for (light blue) and (dark blue) at a 100 TeV collider in the 3-lepton final state.

Finally, we show exclusion limits in the presence of several VLQ transforming in a under assuming their masses are approximately given by and their BRs are as given in Eq. (2). In such a case we can add up their cross sections. We then obtain that masses up to () can be probed for ().

4.2 Search for at 100 TeV

Figure 3: Left) Normalized distribution for in pair-produced stops (thin lines) and VLQs (thick lines) for TeV (orange) and TeV (green). Right) Pair-production cross section at TeV for stops (green dashed line) and for VLQs (solid orange line). In all cases, stands for either the mass of the stop or the mass of the VLQ.

Prospects for can be obtained from those for pair-produced stops Cohen:2014hxa (). Although scalars and fermions present a priori different kinematics due to the different structure of their interactions with SM particles, the kinematic differences are small. To show that, we depict in the left panel of Fig. 3 the boost factor () distribution of pair-produced stops (thin line) and pair-produced VLQs (thick lines) for masses TeV (orange) and TeV (green). Consequently, the reach of the projected analysis in Ref. Cohen:2014hxa () for VLQs can be obtained by rescaling by the larger VLQ pair-production cross section. The latter is shown in the right panel of Fig. 3 (orange solid line). The corresponding cross section for stops is also drawn (green dashed line). As it can be seen, there is almost an order of magnitude of difference between the two.

The concrete bound on depends also on . We obtain the excluded regions in the plane in Fig. 4. In the left panel the integrated luminosity at a future 100 TeV collider is assumed to be fb; in the right panel, fb. The regions below the solid orange line and the dashed green, red and blue lines are excluded at the 95 % C.L. assuming and and , respectively. Note that for as large as , . For smaller values of , is even smaller. Thus, with fb, resonances of mass below TeV can be excluded.

Figure 4: Left) The area below the orange solid, green dashed, red dashed and blue dashed lines can be excluded at the 95 % C.L. by recasting the analysis of Ref. Cohen:2014hxa () with fb assuming and , respectively. Right) Same as Left) but for fb.

5 Matching to concrete models

In this section we consider different coset structures containing at least a Higgs doublet and an additional scalar singlet. The latter is supposed to be stabilized by an external symmetry compatible with the strong dynamics. By studying different representations in which the SM fermions (mainly the third generation and ) can be embedded, we will see that definite coefficients in Eq. 2 are predicted.

In order to fix the notation, let us call () the unbroken (broken) generators of any symmetry breaking pattern . The spectator is typically required to reproduce the fermion hypercharges. Hereafter we will omit both this and the colour group. Let us also define with running over the pNGBs. The sigma-model Lagrangian at the leading order in derivatives reads

(5.1)

where is the projection of the Maurer-Cartan one-form of the broken generators. It is explicitly defined by the equality

(5.2)
Table 2: Summary of the values that the coefficients in Eq. 2 can take in different CHMs. See the text in Secs. (5.1-5.5) for details on the possible assumptions made in each case.

5.1 and related models

We start considering the symmetry breaking pattern  Gripaios:2009pe (). The generators can be chosen to be

(5.3)
(5.4)

expand the coset space of the Higgs doublet. The broken generator associated to is provided by . Expanding in powers of we get

where the ellipsis stands for terms with higher powers of . This Lagrangian already fixes the values of in Eq. 2. The exact values of the other parameters depend on the pairs of representations that embed the third generation and . Different choices can generate a scalar potential and the SM-like Yukawa Lagrangian without breaking the external symmetry stabilizing the scalar . Among others, can transform in: , , , or the . Note that we are implicitly assuming that the third generation and mix with only one composite operator. This implies that lighter generations might be embedded in different representations. For example, if the choice is made for the top quark, cannot be embedded in the same representation, because it should carry a different charge under than the one associated to (the charge matches the one of ). Under this assumption, the hierarchical Yukawa matrices can originate simply from the renormalization group evolution of non-hierarchical couplings in the UV Panico:2015jxa ().

Among the highlighted choices of representations, the first one, , is problematic because does not break the global symmetry. It must be instead broken by , which decomposes as under . Therefore, the Higgs mass would depend on only one free parameter to leading order333It is well known Panico:2015jxa () that the number of free parameters (encoding the details of the strong dynamics) in the leading-order term in the one-loop induced potential (namely that containing the smallest number of symmetry breaking insertions) is one unit smaller than the number of invariants that can be constructed out of the irreducible representations into which a particular representation of decomposes. In the case under study, two such invariants exist: the singlet resulting from the product and the one given by . Its smallness, required to explain why the Higgs mass is much smaller than the scale TeV, can only be explained at the expense of unexpected cancellations in the strong sector.

In the second case, , however, does break the global symmetry and partially cancel the contribution of to the Higgs mass. Nevertheless, the leading-order Higgs potential has only a minimum at , so it does not drive EWSB. Next-to-leading order terms must be considered, which not only need to be tuned to be of similar size to those generated at leading order, but they also increase the number of free parameters. Definite predictions for and cannot be made. The same holds true for the next two cases, and . However, the embedding of in the can respect the shift symmetry of and hence make it light. Where this the case, and would be predicted to be . We will come to this point again in Section 6.

Finally, the choice can lead to definite predictions. Despite the fact that does not break the global symmetry, , which decomposes as , generates both a Higgs mass term and a quartic coupling that depend on only two unknowns; see footnote 3. Expanding the one-loop induced potential in powers of and restricting to the renormalizable terms, we find

(5.5)

We stress that the small differences with the results obtained, for example in Eq. 4.11 of Ref. Chala:2017sjk () (in their stability-preserving limit ), are due to the fact that we are using a shift-symmetry preserving basis that does not resum higher-orders of . The two unknowns, and , can then be traded for the Higgs mass and its quartic coupling:

(5.6)

Putting all together, the relevant Lagrangian reads

(5.7)

where denotes the usual Higgs quartic coupling. For a mild value of , this corresponds to

(5.8)

Departures from this relation might appear if the gauge contribution to the scalar potential were sizable, too.

Similar results to the ones discussed so far apply also to other models based on the coset . They all develop a Higgs doublet and singlets. Thus, for example, representations such as the , or in require the inclusion of both the leading and the next-to-leading terms in the one-loop induced potential to achieve EWSB. The choice , instead, gives rise to a Lagrangian very similar to the one in the equation above:

(5.9)

The phenomenology of the DM particle is thus identical to the case of , provided that the extra pNGB, , is heavier.

A mass splitting between the two singlets cannot come from gauging (which induces a potential only for the Higgs doublet). Instead, it has to arise due to global-symmetry breaking induced through the fermionic sector. For example, a small increase of the mass of the second pNGB singlet arises if is embedded in the appropriate irreducible representation of a of . Note that this representation reduces to

(5.10)

under and , respectively. If has components in the first singlet (which is a total singlet of ) and the second one (which is also a singlet of ), then the mass of the non-DM Goldstone can be increased. In the base analogous to that of Eq. 5.3, this embedding can be achieved by

(5.11)

This gives a mass splitting of order

(5.12)

Note that small differences in the coefficients of the potential would be expected if and mix and the physical DM were a linear combination of both and . The parameters and in the derivative interactions, instead, would remain the same, given that the derivative Lagrangian is invariant under an arbitrary transformation of and .

Finally, the coset provides another very similar scenario. The pNGB spectrum consists of the Higgs doublet, a neutral scalar and singly-charged singlet . Among the smallest representations that can embed the third generation and we find and . The second leads again to the predictions of Eq. 5.8. The DM phenomenology is, therefore, similar to that of with , provided, again, that is heavier than and hence cannot be produced in DM annihilations. Likewise in the case of , this splitting can be triggered by if it is embedded, for example, in the within the of . A larger splitting arises however from the hypercharge interactions (note that is charged while is not). It can be estimated to be

(5.13)

5.2

This coset is special in the sense that the pNGBs transform in a reducible representation of the unbroken group , namely . In the basis of Eq. 5.3, these are expanded by , by and by , respectively. The sigma-model Lagrangian can then be written to leading order in derivatives as

(5.14)

Contrary to the previous cases, it depends on several free parameters, . The different symbols correspond to

(5.15)

They do not mix under a generic transformation.

The coefficients and in the derivative interactions in Eq. 2 can then get different values. As a particular scenario, let us discuss the case . This can be interpreted as two step symmetry breaking: . The first takes place at a scale at which the second doublet can get a mass of similar order. A phenomenological study of a scenario similar to this interpretation has been given in Ref. Sanz:2015sua (). Provided the singlet remains light, which can occur if its associated shift symmetry is only slightly broken, the phenomenology at the scale is that described by the parametrization of Eq. 2. The relevant sigma Lagrangian reads

(5.16)

With and depending on the fermion representation that we did not specify explicitly for this case, we can conclude that

(5.17)

5.3

This coset has been previously considered in Ref. Gripaios:2016mmi (). The product structure of the global group makes the pNGBs transform also in a reducible representation: . As a consequence, the sigma model Lagrangian is written as

(5.18)

where the ellipsis stand for terms with higher powers of and which, however, do not make and interact. Note also that in the equation above, is constructed out of the four broken generators of . This model provides then, a DM phenomenology very similar to that of the elementary Higgs portal. It has been also pointed out that, within this scenario, the scalar potential for vanishes unless the top quark mixes with several composite operators transforming in different representations of the global group. As we have argued before, renormalization group evolution of anarchical couplings in the UV can not explain the absence of flavour-violating effects by itself. Although this problem might be circumvent by advocating extra symmetries (see for example Ref. Chala:2017sjk ()), one might still expect the top quark to mix mostly with one composite resonance. In that case, we could obtain

(5.19)

5.4

In the same vein as , this model develops three multiplets of the unbroken group, transforming in a reducible representation: . Following the discussion (and notation) of Section 5.2, again in the limit , one might expect the sigma-model Lagrangian to read

(5.20)

where stands for the complete singlet of . As in the case of , and could mix, for example, if they were both protected by a symmetry. The difference is that, in this case, the sigma-model is not invariant under a general transformation rotating into .

Assuming that the physical DM particle is mostly , one obtains

(5.21)

5.5 Complex dark matter in

As we have mentioned before, the sigma-model Lagrangian in respects an additional symmetry under which rotates into . This symmetry is of course not broken by the gauge interactions (unless that is also gauged giving rise to an extended model with a boson). If it were also not broken by the mixings between the elementary and the composite fermions, and would be degenerated in mass and would form a complex DM candidate. (Note that, in such a case, there is no need to assume that the strong sector is compatible with the symmetry stabilizing the DM.) A thorough study of this model in the representation has been carried out in Ref. Balkin:2017aep ().

Another possibility is considering, for example, . is explicitly embedded in

(5.22)

An arbitrary rotation of angle mixing and can be implemented by

(5.23)

which, when acting on , gives . Being an eigenstate of means that the elementary-composite mixing does not break the symmetry. The relevant Lagrangian reads in this case

(5.24)

For a mild value of , this can be mimicked by a real scalar scenario444Note that actually there are two solutions for and . We remark also that the reason that we can do such a matching to the real scalar case is that both derivative and potential couplings are present. Instead for a elementary singlet it is impossible because the relic abundance fixes the coupling which can then not be adjusted anymore for the direct detection cross section. with: