# Little composite dark matter

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## Abstract

We examine the dark matter phenomenology of a composite electroweak singlet state. This singlet belongs to the Goldstone sector of a well-motivated extension of the Littlest Higgs with *T*-parity. A viable parameter space, consistent with the observed dark matter relic abundance as well as with the various collider, electroweak precision and dark matter direct detection experimental constraints is found for this scenario. *T*-parity implies a rich LHC phenomenology, which forms an interesting interplay between conventional natural SUSY type of signals involving third generation quarks and missing energy, from stop-like particle production and decay, and composite Higgs type of signals involving third generation quarks associated with Higgs and electroweak gauge boson, from vector-like top-partners production and decay. The composite features of the dark matter phenomenology allows the composite singlet to produce the correct relic abundance while interacting weakly with the Higgs via the usual Higgs portal coupling \(\lambda _{\text {DM}}\sim O(1\%)\), thus evading direct detection.

## 1 Introduction

The Hierarchy problem of the Standard Model (SM) could be solved by assuming that the Higgs is a pseudo Nambu-Goldstone Boson (pNGB) of a spontaneously broken global symmetry [1, 2, 3, 4]. In this scenario, the Higgs is not an elementary particle but rather a composite state whose constituents are held together by some new strong force. In this respect, the Composite Higgs resembles other scalars found in nature, the QCD pions. An extended composite sector could also explain the origin of dark matter (DM) [5, 6, 7, 8, 9, 10, 11, 12]. The same strong dynamics responsible for the Higgs may produce a stable neutral scalar bound state, a composite DM candidate. This could be considered in analogy to the Proton, another QCD bound state, which is an abundant particle in our universe, whose stability is insured by an (accidental) global symmetry. The composite DM candidate is a pNGB and it could be naturally as light as the weak scale, which fits in the weakly interacting massive particle (WIMP) paradigm.

One realization of the composite Higgs scenario is the Littlest Higgs [13, 14, 15, 16, 17, 18, 19]. The original model is strongly constrained by electroweak precision tests (EWPT) due to tree level contributions to electroweak observables [20, 21, 22, 23, 24, 25, 26, 27]. These constraints required the symmetry breaking scale *f* to be a few TeV, thus reintroducing considerable fine-tuning.

*T*-Parity has been proposed in order to prevent tree-level exchanges of heavy states [28, 29, 30, 31]. The new heavy states are odd under a discrete *T*-parity, therefore contributions to electroweak observables are possible only at the 1-loop level. This allows the symmetry breaking scale *f* to be *O*(1) TeV. As an added benefit, *T*-Parity can be used as a stabilizing symmetry for a DM candidate, as the lightest *T*-odd particle is guaranteed to be stable.

In this work, we consider the phenomenology of a Littlest Higgs model with *T*-parity (LHT) with a consistent implementation of *T*-parity in the fermionic sector [32]. Compared to the simplest LHT model, one enlarges the symmetry breaking pattern and also the unbroken symmetry group *H* which allows a complete composite representation containing just one fermion doublet. In particular, we analyze the DM phenomenology of a composite singlet scalar. In Sect. 2 we present the model and motivate the extension leading to the larger Goldstone sector. In Sect. 3 we briefly review the scalar potential structure. A detailed discussion of the scalar potential can be found in Appendix 1. In Sects. 4 and 5 we derive constraints on the model parameters from recent LHC searches and EWPT. In Sect. 6 we discuss the DM phenomenology of the composite singlet DM. We finally summarize our results and conclude in Sect. 7.

## 2 Model

*SU*(5):

*SU*(5) to

*SO*(5). The 10 unbroken

*SO*(5) generators denoted by \(T_i\) satisfy

*SU*(5), denoted by \([SU(2)~\times ~ U(1)]_i\) with \(i=1,2\). The gauged generators are

*SU*(5) generators decompose under the SM gauge group to the following representations

*H*which we identify as the SM Higgs field, and a heavy charged triplet \(\Phi \). The gauge spectrum contains the SM gauge fields and additional heavy gauge fields with masses \(m_{W_H}\sim gf, m_{B_H}\sim g' f,\) with \(g,g'\) the SM gauge couplings. The heavy gauge states contribute at tree level to the electroweak oblique parameters. These contributions lead to stringent constraints from electroweak precision tests (EWPT), pushing the symmetry breaking scale of the original LH model \(f \sim \) a few TeV (e.g Ref. [20]). The corrections to electroweak observables from the heavy gauge states are made smaller by introducing a discrete symmetry which forbids tree level exchanges of heavy states. The addition of a discrete symmetry stabilizes the lightest odd particle, making it a viable DM candidate. This discrete symmetry, usually referred to as

*T*-parity, is defined as [28]

*SU*(5) generators. This definition determines the

*T*-parity of all the fields associated with the

*SU*(5) generators, namely the Goldstone and gauge fields. The \(\Omega \) rotation is introduced to make the Higgs even under

*T*-parity, while keeping the rest of the Goldstone fields odd. For the gauge fields, the

*T*-parity transformation can be interpreted as an exchange symmetry between the gauge groups \(1 \leftrightarrow 2\). Hence the diagonal combination is even, and the broken combination is odd.

*SU*(5) transform under

*T*-parity. One can use Eq. (9) to show that each transformation \(g = e^{i \alpha _j X_j + i \beta _i T_i}\in SU(5)\) is mapped under

*T*-parity to

*SU*(5) are mapped to each other

*SU*(5) indices, so under

*T*-parity

### 2.1 A UV doubling problem, making the *T*-odd doublet massive

*T*-parity is in tension with the SM matter content [32, 33]. The low energy theory must contain a

*T*-even massless

*SU*(2) doublet, the left-handed quark doublet of the SM. Since

*T*-parity can be understood as an exchange symmetry between the two gauged

*SU*(2) subgroups of

*SU*(5) (we omit the

*U*(1) factors for the following discussion), one must therefore introduce two doublets \(\psi _i\), each transforming under a different \(SU(2)_i\) with \(i=1,2\). Under

*T*-parity the two doublets are mapped into each other

*T*-odd combination \(\psi _- \equiv (\psi _1-\psi _2)\) that respects the SM gauge group. Let us introduce a right-handed field \(\psi ^c\) transforming as a doublet under the SM gauge group \([SU(2)]_{1+2}\)

Let us examine now the VEV’s which we can use to write this term in a gauge-invariant way. In Sects. 2.1.1 and 2.1.2 we briefly examine two different constructions presented in the literature that generate the mass term of Eq. (14). We mention possible shortcomings of these constructions, which motivate the construction used in this work, presented in Sect. 2.1.3. Readers interested only in the details of the model used in this work, may skip directly to Sect. 2.1.3.

#### 2.1.1 Non-linear formulation of a massive odd doublet

One construction commonly presented in the literature uses the CCWZ formalism [34, 35]. The main advantage of this approach is that no new sources of spontaneous symmetry breaking are needed.

*SU*(5) [29]

*T*-parity transformation

*T*-odd combination is constructed using a non-linearly transforming field

*T*-parity symmetric form

*T*-parity

*T*-parity is

*SU*(5), e.g \({\mathbf {5}}\) and \({\bar{\mathbf {5}}}\)

*SO*(5) representation, otherwise the kinetic term for \(\tilde{\Psi }^c\) would explicitly break the global symmetry protecting the Higgs mass [29]. The field \(\psi _1^c\) is still massless at this point. One could formally introduce an additional doublet \(\eta \) and write a mass term

#### 2.1.2 Adding a third \(SU(2)~\times ~ U(1)\)

We conclude that the model requires additional structure in order to give mass to the *T*-odd combination without explicit breaking of the global symmetry.

*T*-even gauge fields. The new

*T*-even gauge fields can be made heavy by making the coupling constant of the third \(SU(2)~\times ~U(1)\) gauge group large, effectively decoupling them from the theory without spoiling the naturalness of the model. One has the choice of how to enlarge the global symmetry to incorporate this additional gauge group. The most naive extension is

*U*(1) charges), namely

*T*-parity

*T*-odd doublet gets a massafter \(\Phi _1\) and \(\Phi _2\) acquire VEV’s given by

*SO*(5). These 10 additional states decompose under the SM gauge group as

*T*-even gauge fields. This naive approach unavoidably introduces additional physical NGB’s in the form of a

*T*-odd doublet \(\mathbf {2}_{1/2}\) and a

*T*-even complex scalar \(\mathbf {1}_{1/2}\). These states must be made massive without spoiling the symmetry protection of the SM Higgs. Additional NGB’s are a generic result of the enlarged global symmetry structure, even more so when the additional

*SU*(2) is a gauged subgroup of a larger global symmetry [33].

#### 2.1.3 Mirroring the \(1 \leftrightarrow 2\) exchange symmetry

*X*, which transforms linearly under \([SU(2)\times U(1)]_L \times [SU(2)\times U(1)]_R\)

*X*acquire VEV’s, \(\langle \Sigma \rangle = \Sigma _0\) and \(\langle X \rangle = \mathbb {1}_2\), the symmetry is spontaneously broken to

*X*using the non-linearly transforming Goldstone fields associated with this symmetry breaking,

*f*, the symmetry breaking scale of the original coset defined in Eq. (7).

*T*-parity in the additional coset is realized as an \(L~\leftrightarrow ~R\) exchange, mirroring the \(1 \leftrightarrow 2\) exchange symmetry of the original coset. Under

*T*-parity,

*SU*(2) doublet to the spectrum, \(\psi ^c\), and write a mass term for the

*T*-odd doublet, without any explicit breaking of the global symmetry. In additional to the 14 original NGB’s of Eq. (6), our spectrum includes now an additional NGB’s, a real singlet \(\mathbf {1}_0 \) and a real triplet \(\mathbf {3}_0 \).

### 2.2 Gauge sector

### 2.3 Goldstone sector

*H*and the charged triplet \(\Phi \), the Goldstone sector includes additional physical states: a real singlet

*s*and a real triplet \(\varphi \equiv \frac{1}{2} \varphi _a \sigma ^a\), defined as the following linear combinations

### 2.4 Matter sector

*SU*(5) multiplets

*T*-parity,

*T*-parity,

*i*,

*j*,

*k*are summed over 1, 2, 3. We define the

*T*-parity eigenstates

*T*-parity eigenstates are defined as

*T*-even fields, and in particular \(\tilde{t}_L,\tilde{t}_R\), are not the mass eigenstates (hence the tilde). After the Higgs field acquires its VEV, \(\langle H \rangle = \frac{1}{\sqrt{2}}( 0,v)^T\), we find the following mass matrix for the

*T*-even fermions

*T*-odd sector we must introduce a mass term for the doublet similar to the term in Eq. (45). We introduce a RH doublet \(\psi _R^-\) transforming non-linearly under \([SU(2)~\times ~ U(1)]_L ~\times ~ [SU(2)~\times ~ U(1)]_R\) according to the CCWZ formalism. \(\psi _R^-\) is odd under

*T*-parity

*T*-odd singlet \(T^-\) and a

*T*-odd doublet \(\psi ^-\) with the following masses

## 3 Scalar potential

*O*(1) numbers originating from unknown UV contributions to these operators. \( \Lambda \sim 4 \pi f \) is the cutoff scale of the theory. Expanding the scalar potential \(V = V_{f} +V_{\text {V}} \) in the NGB fields, we find that

*T*-parity, integrating out \(\Phi \) at tree-level does not influence any of the couplings explicitly written in the scalar potential of Eq. (73). Like \(m_{\Phi }^2\), the Higgs quartic \(\lambda \) is generated by 1-loop quadratically divergent diagrams.

The rest of the operators in Eq. (73), including the Higgs mass term \(\mu ^2\), are generated through logarithmically divergent loops, and as such they exhibit a mild dependence on the UV cutoff scale. The explicit calculations, found in Appendix 1, give us an order of magnitude estimation for the IR contribution to these operators at 1-loop. However quadratically divergent 2-loop diagrams as well as UV contributions can have comparable effects on these operators. Therefore we do not presume to be able to predict these couplings accurately in terms of the fundamental parameters of this model. In this work we treat the couplings in Eq. (73) as free parameters, except \(\mu ^2\) and \(\lambda \) which are already fixed by experiment. Our goal is to allow the free parameters to take values that are reasonable in light of the approximation given by the 1-loop IR contribution, and state explicitly when this is not the case.

In addition to \( {m}_\varphi ^2,\lambda _{\text {DM}} ,\lambda _{\varphi }\), we must introduce a mass term for the singlet *s*. The singlet remains massless at 1-loop, and a mass for *s* is generated at the 2-loop level. We take the pre-EWSB mass term of the singlet, denoted as \(\tilde{m}_s^2\), as a free parameter as well. The sizes and ranges of \( {m}_\varphi ^2,\lambda _{\varphi },\tilde{m}_s^2,\lambda _{\text {DM}}\) are dictated by the DM phenomenology and are discussed in Sect. 6.

## 4 LHC phenomenology

### 4.1 *T*-even singlet \(T^+\)

*T*-even singlet is responsible for cancelling the quadratically divergent top loop contribution to the Higgs mass, hence it is the standard top partner predicted by composite Higgs models. It can be doubly produced at the LHC via QCD processes, as well as singly produced with an associated third generation quark through the following EW interactionsIn this model,

#### 4.1.1 Decay modes

#### 4.1.2 LHC searches

*Single production* \(T_+\) can be singly produced at the LHC in association with a third generation quark. A recent search from CMS [38] looked for \((T^+ \rightarrow Z\;t) bq\) with a fully leptonic *Z* decay. The search places a lower bound on the mass of the singlet LH Top partner at 1.2 TeV, assuming negligible width and BR\([T^+\rightarrow Z t]=0.25\). The bound strongly relies on a model-dependent production cross-section, which in term depends on the coefficients of Eq. (76). In the CMS search the coupling is fixed at \(C_{bW}=0.5\). Conservatively we consider the \(m_{T^+}>1.2\) TeV bound at face value, although we expect a smaller value for \(C_{bW}\), as can be seen in Eq. (76). \(C_{bW}\) is further suppressed for \(\lambda _2>1\), which is the region in parameters space that, as we later show, is consistent with the LHC constraints on the *T*-odd top partners masses. The mass of the *T*-odd singlet is bound from below to be \(m_{T^+} > \sqrt{2} f\). The lower bound of 1.2 TeV can be trivially satisfied by taking \(f>850\) GeV.

*Double production*\(T_+\) can also be doubly-produced via QCD processes. A recent search from ATLAS [39] looked for a pair produced top partners in a range of final states, assuming that at least one of the top partner decays to

*th*. The quoted nominal bound of the singlet top partner is

### 4.2 *T*-odd singlet \(T^-\)

The phenomenology of the *T*-odd singlet resembles that of a stop squark with conserved R-parity. It can be doubly produced at the LHC via QCD processes, and consequently decay to tops and missing energy.

#### 4.2.1 Decay modes

#### 4.2.2 LHC searches

We performed a simple recast of recent stop bounds by accounting for the enhanced production cross section of the fermionic \(T^-\) relative to the scalar stop squark case. We would like to account for the presence of the *T*-odd doublet, which contributes to the same final states as \(T^-\). We postpone the derivation of these bounds to Sect. 4.3.

### 4.3 *T*-odd doublet \(\psi ^-\)

The phenomenology of the *T*-odd doublet resembles that of a mass-degenerate stop and sbottom squarks with conserved R-parity. The upper (lower) component up \(\psi ^-\) can be doubly produced at the LHC via QCD processes, and consequently decay to tops (bottoms) and missing energy.

#### 4.3.1 Decay modes

#### 4.3.2 LHC searches

*T*-odd sector contains two top-like and one bottom-like fermions. We perform a recast of recent bounds on stop and sbottom masses by accounting for the enhanced production cross section of a fermionic colored top partner, along the lines of [40] and [41]. The quoted bounds in Ref. [42] for the stop and sbottom masses are

*s*coloured particle with mass

*M*as \(\sigma ^s_{\text {pair}}(M)\). We require that

*T*-odd doublet and

*T*-odd singlet top partners respectively. We use \(\sigma ^0_{\text {pair}}(M)\) reported by the CMS collaboration [43] and \(\sigma ^{1/2}_{\text {pair}}(M)\) calculated using HATHOR [44]. The combination of Eqs. (90) and (90) in the \((m_{\psi ^-},m_{T^-})\) plane is plotted in the left panel of Fig. 3. We conservatively assume all the branching ratios to be \(100\%\). We thus obtain the following lower bounds on the

*T*-odd fermion masses

*f*,

## 5 Electroweak precision tests

*T*-even singlet \(T^+\) [31]

*T*-even top partner mass of Eq. (80), we expect \(x_t \le 0.03 \ll 1\). We therefore expand Eq. (93) to leading order in \(x_t\):

*T*-odd heavy gauge bosons. The correction is proportional to the mass splitting after EWSB,

*T*-odd gauge loops generate the following correction to the T parameter [31]

*S*and

*U*parameters also receive corrections due to the mixing the LH fermion sector. As noted in Ref. [31], the size of these corrections are an order of magnitude smaller than the correction to the

*T*parameter and are therefore sub-leading. Additionally, the \(Z \bar{b}_L b_L\) vertex receives corrections due to \(T_+\) loops [31]

*T*-odd gauge loops to the T-parameter is the dominant constraint in the allowed region where \(\lambda _2\) is large. We find the following lower bound on

*f*from Eq. (98) at \(3\sigma \) after taking \(\Lambda = 4 \pi f\)

## 6 Dark matter phenomenology

### 6.1 Spectrum

*T*-odd particle (LTP) in the spectrum is stable and therefore a natural DM candidate. One possible LTP is the gauge field \(B_H\). This possibility has been considered in the past in the context of the original LHT model [46]. In this work we explore the possibility of DM being part of the composite scalar sector, in particular the singlet

*s*. The singlet mass \(m_s\) is a free parameter in our model. The mass \(m_{B_H}\), given in Eq. (48), is of order

*O*(200) GeV. The region in which

*s*is the LTP corresponds to \(r \sim 2-3\) and thus would be the focus of our study. In this region we may safely neglect co-annihilation effects of

*s*with \(B_H\). Since larger values of

*r*correspond to heavier

*T*-odd gauge bosons, there is a small increase in the fine-tuning of the Higgs mass from the gauge sector. We can easily see by comparing the logarithmically divergent contributions to the Higgs mass from the two sectors

### 6.2 Singlet-triplet mixing

*s*and the neutral component of the triplet \(\varphi _3\) after EWSB. The effects of singlet-triplet mixing on the DM phenomenology have been considered in Ref. [47]. We focus on the composite nature of the singlet DM. For simplicity, we limit ourselves to the region in parameter space where we may neglect the mixing effects. The mixing angle is given by

^{1}We note that the assumption \( \lambda _\varphi \sim 1\) as well as the lower bound on \(m_\varphi \) are consistent with the IR contribution of Eq. (73) to these operators. We find that the operator corresponding to \(\lambda _\varphi \) enjoys an accidental factor \(\sim 5\) enhancement to its coefficient in the CW potential. The IR contributions can be found in Appendix 1 in Eqs. (B.47) and (B.48). We conclude that a moderate mass separation is sufficient in order to neglect the singlet-triplet mixing effects.

### 6.3 Annihilation cross section

*T*is given by [51]

#### 6.3.1 Higgs portal

*s*-channel annihilation to SM gauge fields and fermions. The annihilation channel \(s s\rightarrow h h\) is also possible via the

*s*,

*t*and

*u*channels as well as directly via the dimension 4 operator \(s^2 h^2\). We assume that freeze-out occurs after the EW phase transition. In unitary gauge, we can rewrite Eq. (109) as

#### 6.3.2 Goldstone derivative interaction

*f*. They affect all the annihilation channels of the Higgs portal couplings, typically resulting in destructive interferences [5]. We discuss this effect in detail in Sect. 6.4. As

*r*increases, \(s_0\) increases and approaches unity. This is equivalent to decreasing the effective scale of this operator \(\tilde{f} = f / s_0\), thus making these interactions stronger for lower DM masses.

#### 6.3.3 Contact term

*r*increases, the effective scale of this operator \(\tilde{f} = f / \sqrt{s_0}\) decreases, thus making this interaction stronger for lower DM masses.

### 6.4 Relic abundance

*x*. Furthermore, \(\sigma _0,f_1,f_2\) depend in general on \(f,r,\lambda _2\). \(f_1(x)\) parametrizes the destructive effects of the dimension 6 operator of Eq. (112), hence we expect \(f_1 \sim x^2\) . \(f_2(x)\) accounts for the dimension 5 operator of Eq. (113), which allows the singlet to annihilate into two tops independently of the Higgs interactions, therefore we expect \(f_2 \sim x\).

#### 6.4.1 Portal coupling dominance

*f*, this region is characterized by a large portal couplings or small DM masses. The mass region \(m_s < m_h/2\) is severely constrained by the LHC due to the Higgs invisible width to singlets. For \(m_s \approx m_h/2\), the Higgs mediator is resonantly produced and \(\lambda _{\text {DM}}\) must be extremely suppressed in order to produce the correct relic abundance, making this finely tuned region hard to probe experimentally. We shall focus on DM masses above \(m_h/2\) the avoid the above-mentioned issues.

This region can be seen in the left panel of Fig. 5 where \(m_s<150~\)GeV. In this region the total annihilation cross section for a fixed portal coupling decreases with \(m_s\), as expected in the standard singlet DM scenario for \(m_s > m_h/2\). In the right panel of Fig. 5, the portal coupling dominance region is to the right of the minima of the curves. In this region, for a fixed value of the mass, the total annihilation cross section increases with \(\lambda _{\text {DM}}\).

#### 6.4.2 Contact term dominance

The relevant parameter space in the left panel of Fig. 5 corresponds to the region where \(m_s \sim 220\) GeV, close to the minimal value of the cross section. The annihilation to the Higgs and gauge bosons is effectively suppressed by the destructive interference between the portal coupling and the derivative interactions. As this suppression occurs where \(m_s>m_t\), the remaining annihilation cross section is exclusively to tops. In the right panel of Fig. 5, the minima of the different curves are precisely mapped to this area of maximal interference. For the fixed mass \(m_s=150\) GeV, the singlet is not allowed kinematically to decay into tops and the annihilation cross section vanishes. Conversely, for \(m_s=200\) GeV the decay into tops is allowed and the annihilation cross section is dominated by the contact term. Lastly, the minimum of the curve corresponding to \(m_s=250\) GeV is approximately 1 pb, meaning that for this particular point in the \((\lambda _2,r,f)\) parameter space, \(x_{\text {max}} \approx 250/1000 = 1/4\).

#### 6.4.3 Derivative interaction dominance

### 6.5 Direct detection

The model was implemented using FeynRules [57] and exported to micrOMEGAs [58]. The strongest direct detection bounds are due to XENON1T [59] after 34.2 live days. Scan results for this model can be seen in Fig. 6. The two branches appearing in each panel represent the two possible solutions for \(\lambda _{\text {DM}}\) for each mass value which produce the observed relic abundance. The branches meet at some maximal DM mass, above which the singlet is always under-abundant. The upper branch is ruled out by direct detection. Some of the lower branch is still consistent with experimental bounds. In the region where \(m_s \approx \sqrt{x_{\text {min}}}f\), \(\lambda _{\text {DM}}\) can be arbitrarily small, thus avoiding direct detection. In this regions, the theory gives a sharp prediction for the DM mass. At mentioned previously, the naive IR contribution to \(\lambda _{\text {DM}}\) is too big and of \(O(10\%)\). We therefore assume that additional contributions from UV physics and higher loops generate mild cancellations, allowing this coupling to take the allowed \(O(1\%)\) values.

*f*can be seen in Fig. 7. The largest effect is seen for increasing

*r*, which in turn raises the importance of the non-renormalizable interactions at lower DM masses. A smaller effect due to the increase of \(\lambda _2\) can be seen in the meeting point of the two branches. Larger values of \(\lambda _2\) decrease the contact term, pushing \(m_s^{\text {max}} = \sqrt{x_{\text {max}}}f\) to higher values.

## 7 Conclusions

In this work we have presented a viable composite dark matter (DM) candidate within the Littlest Higgs with *T*-parity framework. We started by motivating a minimal extension of the original coset which allows the *T*-odd doublet to acquire a mass without introducing additional sources of explicit symmetry breaking. The extended coset contains a *T*-odd electroweak singlet. This singlet is naturally light and therefore it is reasonable to assume it is the lightest *T*-odd particle, which insures its stability.

The top sector is implemented using a collective breaking mechanism, insuring the absence of quadratically divergent contribution to the Higgs mass at 1 loop. *T*-parity implies a rich LHC phenomenology: in addition to the usual (*T*-even) top partners, the top sector contains *T*-odd top partners. This *T*-odd top partners can be doubly produced via QCD in the LHC and decay to standard model (SM) particles and missing energy. We have derived lower bounds on the masses of the *T*-even and *T*-odd top partners from various LHC searches. When combined with electroweak-precision-test (EWPT) bounds, we derived a set of constraints on the parameter space of the model.

We examined the DM phenomenology of the composite singlet DM within the allowed parameter space. The usual “elementary” singlet DM scenario is heavily constrained by direct detection experiments. In the composite singlet DM scenario, the composite nature of the DM allows it to escape detection in areas with \(O(1\%)\) portal coupling, while still producing the observed relic abundance via its derivative interactions with the Higgs. The “elementary” singlet can only hide in the finely tuned “resonance” valley where \(m_s \approx m_h/2\). Conversely, the composite singlet can exist in a broader region, corresponding to different values of *f* and *r*, in which it can evade detection. In these regions the correct relic abundance can be produced only due to the derivative interactions. The small portal coupling needed in these regions would in general require some mild amount of fine tuning, unless one can find a way to suppress it e.g using symmetries or additional dynamics.

## Footnotes

- 1.
The LHC phenomenology of \(\varphi \) resembles that of the Wino, which implies that the charged components can be doubly produced via electroweak processes and decay to \(W^\pm \) and missing energy. However, the relevant SUSY searches, e.g [48, 49], do not pose strong constraints on \(m_{\varphi }\), especially in light of the reduced production cross section of the scalar triplet in comparison to the fermionic Wino.

## Notes

### Acknowledgements

We thank D. Pappadopulo, M. Ruhdorfer, E. Salvioni, and A. Vicchi for useful discussions. The work of RB is supported by the Minerva foundation. The work of GP is supported by grants from the BSF, ERC, ISF, Minerva, and the Weizmann-UK Making Connections Programme. RB and AW have been partially supported by the DFG cluster of excellence EXC 153 “Origin and Structure of the Universe”, by the Collaborative Research Center SFB1258, the COST Action CA15108, and the European Union’s Horizon 2020 research and innovation programme under the Marie Curie grant agreement, contract No. 675440.

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