A Singlet Dark Matter in the SU(6)/SO(6) Composite Higgs Model

Singlet scalar Dark Matter can naturally arise in composite Higgs models as an additional stable pseudo-Nambu-Goldstone boson. We study the properties of such a candidate in a model based on $SU(6)/SO(6)$, with the light quark masses generated by 4-fermion interactions. The presence of non-linearities in the couplings allows to saturate the relic density for masses $400<m_{\rm DM}<1000$ GeV, and survive the bound from Direct Detection. The viable parameter regions are in reach of the sensitivities of future upgrades, like XENONnT and LZ.


INTRODUCTION
The standard cosmology model, "ΛCDM" based on a flat prior, can well describe an expanding universe from the early to late times. The combination of radiation, matter and dark energy determines the Hubble expansion H(t) =ȧ(t)/a(t) as governed by the Friedmann equations. According to astrophysical measurements, the present Universe consists of roughly 30% matter and 70% dark energy after the dark age and large scale structure emergence [1]. However, the density of baryonic matter today is Ω b h 2 ∼ 0.02, comprising only a small portion of the total matter. Thus, the remaining 85% of the total matter is made of a Dark Component, expected to be distributed as spherical halos around Galaxies. Despite the convincing evidences for Dark Matter (DM) from various sources, such as the galaxy rotation curves, gravitational lensing and observations of cosmic microwave background (CMB), the particle identity of DM has not been identified yet. In the Standard Model of Particle Physics (SM), the only candidates, neutrinos, are too light and have too small a relic density to account for the observation. Therefore, many theories Beyond the Standard Model (BSM) have been proposed, with the most popular ones advocating weakly interacting massive particles (WIMPs) stabilised by a discrete symmetry: this scenario can naturally provide a suitable DM candidate thanks to the thermal decoupling.
One of the possibilities is the Higgs mediated singlet Scalar DM model. Without consideration for the naturalness of the light scalar mass, the DM physics is simply described by two free parameters: the mass m S and Higgs coupling to the singlet scalar λ hSS [2][3][4]. Because of the small parameter space, this model has high predictive power. The dominant DM annihilation channels, which determine the thermal relic density, are as follows: for m S < m W , a DM pair mainly annihilates intobb, while in the high mass region its annihilation cross section into W W , ZZ and hh turns to be very effective. However, the viable parameter space for the model is tightly squeezed [5,6] since the λ hSS coupling is subject to strong constraints from direct detection experiments, e.g. Xenon1T [7], PandaX [8] and LUX [9], as well as by the bounds on the Higgs invisible decay width and the upper bounds on events with large missing energy at the LHC experiments. In this work we plan to investigate this scenario in the context of a Composite Higgs Model (CHM) that enjoys an underlying gauge-fermion description and can be UV completed. The scalar DM candidate emerges as a pseudo-Nambu-Goldstone boson (pNGB) together with the Higgs boson itself. In our scenario, the composite nature of both DM and the Higgs boson can substantially modify the DM couplings to the SM states and alter the relative importance of various annihilation channels. This is mainly due to the presence of higher order couplings, generated by non linearities in the pNGB couplings, which can enhance the annihilation cross sections while the coupling to the Higgs (constrained by Direct Detection) is small. This kind of scenarios was first proposed in Ref. [10] in the context of the minimal coset SU (4)/Sp(4), however the scalar candidate is allowed to decay if a topological anomaly is present, like it is always the case for models with a microscopic gaugefermion description [11]. Also, in Ref. [12] it has been pointed out that the DM has dominant derivative couplings to the Higgs, which again ensures the suppression of Direct Detection rates [13]. However, this is a basis dependent statement, as one can always choose a basis for the pNGBs where the derivative coupling is absent [14]. In this paper we will work in this basis.
The CHM we studied is based on the coset SU (6)/SO (6), which enjoys a microscopic description with underlying fermions transforming in a real representation of the confining gauge group [15]. Top partial compositeness can also be implemented along the lines of [16]. The pNGB sector is similar to the SU (5)/SO(5) model [17], which is the minimal realistic coset in the SU (N )/SO(N ) family: the pNGBs include a bi-triplet of the custodial SU (2) L × SU (2) R global symmetry, like the Georgi-Machacek (GM) model [18]. However, unlike the GM model, the direction inside the bi-triplet that does not violate the custodial symmetry is CP-odd, thus it usually cannot develop a vacuum expectation value in a CP conserving theory. On the other hand, the interactions of fermions to the composite sector typically induce a tadpole for the custodial triplet component, thus generating unbearable contributions to the ρ parameter. For the top, coupling to composite fermions in the adjoint representation of SU (5) allows to avoid this issue [19]. In our model, the adjoint of SU (6) serves the same purpose, while the masses of the light fermions can be generated by other mechanisms. This results in a violation of custodial symmetry of the order of m 2 b /m 2 h , thus being small enough to evade precision bounds, as we demonstrate in this paper. The extension of the model to SU (6) also contains a second Higgs doublet and a singlet, which can be protected by a Z 2 symmetry for suitable couplings of the top quark [15]. For other examples of CHMs with DM, see Refs [20][21][22][23][24][25][26][27].
In this work, we investigate the properties of the singlet, Z 2 -odd, pNGB as candidate for Dark Matter. We find that, notwithstanding the presence of additional couplings, the model is tightly constrained, especially by direct detection. The small parameter space still available will be tested by the next generation direct detection experiments, with DM masses in the 400 to 1000 GeV range.

THE MODEL
The main properties of the low energy Lagrangian associated to this model haven been studied in detail in [15], where we refer the reader for more details. In this section, we will briefly recall the main properties of the pNGBs, and discuss in detail how custodial violation is generated via the masses of the light SM fermions. The latter point was not discussed in the previous work. Following Refs [15,19], we will embed the SM top fields in the adjoint representation of the global SU (6): this is the only choice that allows for vanishing triplet VEV, thus preserving custodial symmetry. For the light fermions, we will add direct four-fermion interactions, to generate effective Yukawa couplings to the composite Higgs sector.
For a start, we recall the structure of the 20 pNGBs generated in this model. To do so, it is convenient to define them around a vacuum that preserves the EW symmetry, incarnated in a 6 × 6 symmetric matrix Σ EW (for the explicit form, see [15]). The pNGBs can thus be classified in terms of the EW gauge symmetry SU (2) L × U (1) Y , and of the global custodial symmetry envelope SU (2) L × SU (2) R , which needs to be present in order to preserve the SM relation between the Z and W masses [28]. These quantum numbers are given in Table I. A non-linearly transforming pNGB matrix can be defined as (1, 0) (1, 1) + where Π(x) contains the pNGB fields [15]. We can now define a Z 2 transformation that is from a broken global U (1): where H 2 → −H 2 and η 2 → −η 2 in Σ (x). Thus, they are the Z 2 -odd states, while all the other pNGBs are even, as indicated in the last column in Table I. This parity commutes with the EW and custodial symmetries, and with a suitable choice of the top couplings in the adjoint spurion [15], thus it can remain an exact symmetry of this model. The DM candidates, therefore, can be either the singlet pseudo-scalar η 2 or the component field A 0 in the second doublet H 2 , which are both CP-odd.
To study the properties of the DM candidate, we need to introduce the effects due to the breaking of the EW symmetry. The latter is due to some pNGBs acquiring a VEV: this effects can be introduced as a rotation of the vacuum by a suitable number of angles. In our case, as we want to preserve the Dark Z 2 , we will assume that only H 1 will acquire a VEV, and check the consistency of this choice a posteriori when studying the potential for the pNGBs. For generality, we also introduce a VEV for the custodial triplet, corresponding to . The rotation allows to define a new pNGB matrix where This rotation can be interpreted as follows: the exponential containing α is generated by a VEV for H 1 , aligned with the broken generator X 10 ; then, we operate a rotation generated by the unbroken generator S 8 that misaligns the VEV along the direction of λ 0 by an angle γ. As the effect of the VEVs is an SU (6) rotation of the pNGB matrix, the couplings among pNGBs are unaffected as the chiral Lagrangian is invariant under such type of rotation: thus, no derivative couplings between one Higgs boson with two DM states is generated in this basis.
Here we normalise f such that v = 246 GeV = f sin α for γ = 0. The precise value of the two angles is determined by the total potential, generated by couplings that explicitly break the global symmetry SU (6). After turning on the γ, we can evaluate the potential in a new basisΠ(x), which is equivalent to a pNGB field redefinition, with the full mapping of other pNGBs provided in the Appendix. We proved that, as expected, only tadpoles forh (i.e. the Higgs) andλ 0 are generated. Furthermore, the tadpole terms from all contributions observe the same structure: ∂γλ 0 , like the Taylor expansions. Thus, the vanishing of the tadpoles is guaranteed at the minimum of the potential. This is a main result of this paper and validates our choice for the vacuum misalignment in Eq. (4).
First, we study the potential V 0 coming from the top, gauge and underlying fermion mass, at γ = 0 as studied in [15]. The V 0 depends on 4 independent parameters: Q A and R S in the top sector, the underlying mass Bm (where B is a dimension-less form factor), and a form factor for the gauge loops C g . The Q A represents the coupling of the quark doublet to top partner, and the R S is the coupling of the quark singlet (R S = rR S is irrelevant to V 0 ). Although C g and B can be computed on the lattice once an underlying dynamics is fixed, we will treat them as free parameters. The Q A and R S can be traded in terms of the misalignment angle α and the Higgs boson mass m h : where m h = 125 GeV is the measured Higgs mass. We now introduce the masses for the light fermions via direct couplings. This means, in practice, that we introduce effective Yukawa couplings: e.g., for the bottom quark where Y b is the Yukawa coupling. Note that this kind of operators may also derive from partial compositeness upon integrating out the heavy partners of the light quarks. At one loop, this will generate a contribution to the potential for α and γ of the form which generates a tadpole forλ 0 that does not vanish for γ = 0. More details on all potential contributions can be found in the Appendix. Expanding for small α and small γ, the cancellation of the tadpole ∂V0(α,γ) ∂γ = 0 yields the following result As a consequence of a non-vanishing γ, the model suffers from a tree-level correction to the ρ parameter: To study the impact of this correction, we determined the constraints from the EW precision tests, in the form of the oblique parameters S and T . Besides the tree level correction, which impact directly T , we also included loops deriving from the modification of the Higgs couplings to gauge bosons, and a generic contribution to S from the strong sector. We thus define [29]: where N D counts the number of EW doublets in the underlying theory. Numerically, we plot the bounds in Fig. 1 for two realistic models [16]: only two models are relevant, both based on a confining G HC = SO(N c ), with N c = 7, 9, and underlying fermions in the spinorial representation. Following the nomenclature of [30,31], we show M3 (N D = 16) in solid and M4 (N D = 32) in dashed. The blue shaded wedge is the region or parameters spanned in our model, where we vary 0 < C g < 0.1. The plot shows that the value of γ is always very small, and that the bound on the parameter space is always dominated by the contribution of α to the S parameter. We should note that the generic contribution of the strong dynamics to the S parameter can be reduced in various way: replacing it by loops of the heavy states [32], considering a cancellation between vector and axial resonances [33], or including the effect of a light-ish 0 ++ state [34]. For our purposes, the main point is to show that the effect of custodial breaking via γ is under control. We will not consider the bound on α from EW precision in the following because it can be reduced in a model-dependent way.

DARK MATTER PHENOMENOLOGY
Since the Z 2 -odd states in our model have sizeable couplings to the SM, the relic abundance can be produced by the thermal freeze-out mechanism. The DM candidate(s) remain in thermal equilibrium with the SM, until the DM annihilation rate drops below the Hubble expansion rate. After the decoupling from the thermal bath (freeze-out), the DM density remains constant in a co-moving volume. Defining the yield Y = n(x)/s(x), where s(x) is the entropy density, the DM density evolution is described by the Boltzmann equation [35]: where with x = m/T . The effective thermal averaged crosssection can normally be expanded as σ eff v = a (0) + 3 2 a (1) x −1 + · · · , unless the particle masses are near a threshold or a resonant regime [36]. Using an analytic approach, the relic density is calculated to be: Assuming s-wave dominance for the annihilation crosssections, the above equations yield an approximate solution σ eff v 2.0 × 10 −26 cm 3 /s in order to saturate the relic density Ωh 2 = 0.12 ± 0.001 observed in the present universe [1]. In the following we will use these results to calculate the favourable region of parameter space in our model.
Due to the DM parity described in Eq (2), all odd particles η 2 and φ i = (A 0 , H 0 , H ± ) ∈ H 2 participate in the thermal equilibrium before the freeze out. There exist a vertex of η 2 -A 0 -λ 0 , thus via the Yukawa operators for the light quarks, the heavier component will quickly decay into the lightest mass eigenstate after freeze-out. In principal, the effective averaged cross-section need to take into account all co-annihilation processes [36,37].
However, the mass hierarchy in the spectrum is of paramount importance for co-annihilation. In our model, the inert Higgs doublet observes a mass hierarchy m H0 > m H ± m A0 [15]. In this work, we only investigate a simplified scenario m A0 > m η2 with a large mass gap of ∆ = (m A0 −m η2 )/m η2 > 0.2, so that the actual DM is the singlet. At a typical freeze out temperature x f ∼ 25, the ratio of number densities, n eq φi /n eq η2 , is highly suppressed by a Boltzmann factor (1 + ∆) 3/2 exp(−∆ x f ) ∼ 1/200. Since the direct annihilation σ η2η2 v is not subdominant, the co-annihilation effect can be safely neglected and one can only consider the direct annihilation of the singlet η 2 . The dominant channels are: where η p/m are linear combinations of the Z 2 -even η 1,3 , which we should include in because they are typically much lighter than other pNGBs and will eventually decay into SM particles [15]. The invariant cross-section and the Møller velocity in the lab frame are: and The amplitudes |M XY | 2 are given in the Appendix , as function of all relevant couplings in the model. The key ingredient for relic density is the thermal averaged crosssection, which can be evaluated by an integral (without velocity expansion of σv lab ) [38]: with K 1,2 (x) being the modified Bessel functions of the second kind. In the region far from the resonance (Higgs mass), like the singlet DM with a mass of hundreds of GeV, the integral approach precisely matches with the s, p and d-wave limits [39]. We now discuss the impact of Direct Detection experiments, which are sensitive to the recoil energy deposited by the scattering of DM to nucleus. In our model, the relevant interactions at the microscopic level are: Besides the Higgs portal, we also have direct couplings to quarks from non-linearities in the pNGB couplings (For light quarks, the last term comes from an effective Yukawa coupling). Thus, the spin-independent DMnucleon cross section (factoring out to σ SI Nucleus ) can be parameterised as [10,40]: where f p,n characterise the interaction between η 2 and nucleons: with q summing over all quark flavours. Note that setting C q = 0 will exactly match the result in [41]. For the form factors f [42], we will use the values from [43] for the light quarks, u, d and s, while for heavy flavours the form factor can be computed via an effective coupling to gluons at one loop, giving For this model, the η 2 annihilation will mainly proceed in the following channels η 2 η 2 → W + W − , ZZ, hh,tt,bb, and η p,m η p,m . The interactions and masses of pNGBs are determined by four parameters: (α, C g , Bm, r), where we have neglected γ and set δ = 0. The latter indicates a universal mass for all underlying fermions, and the δ only affects the mass splitting between η p and η m with minor impact on this analysis. Note that the r will influence the mass spectrum of DM parity odd particles η 2 , A 0 , H 0,± but not others [15]. These parameters are preliminarily constrained by the non-tachyon conditions, i.e. m 2 π > 0 for all pNGBs.
In Figure 2, we show the prospect for η 2 to play the role of DM, as opposed to the current bounds from direct detection. There is no constraint from the Higgs decay width since all singlet masses are heavier than m h . For the two panels, we highlight the regions in red that satisfy the relic density bound Ωh 2 < 0.123, as well as non-tachyon conditions and m A0 > 1.2 m η2 . While the light blue regions are permitted by the direct detection since we impose σ SI < σ exp SI . Only the overlaps between the red and blue are viable, where the relic density almost saturates Ω η2 h 2 ∼ 0.12, thus it is not necessary to rescale σ SI by a factor of Ω η2 /Ω DM . The green regions indicate near the edge 2m η2 = m h , the Higgs resonance is active, but they are far from the viable region, thus validate our analytical computation of the relic density. The region allowed by all the bounds, therefore, is fairly limited. We get two branches centered around (α, r) ≈ (0.08±0.01, 0.6±0.15) or (0.02±0.01, 0.35±0.05), which implies f ≈ v/α ∼ 3.0 or 12.0 TeV. We want to stress that the DM mass can be computed for each point of (α, C g , Bm, r), so it varies over the slices of parameter space in the figure.
To better investigate the DM physics, we selected two illustrative benchmark points in Figure 3: we show the thermal averaged cross-section σv for thett channel and for the combined W + W − , ZZ, hh channels at r = 0.4 and r = 0.6 respectively, with C g = 0.05 and Bm = 20GeV fixed. Note that the intersection with the dashed green line indicates points saturating the observed relic density. We do not add the direct detection bound for Figure 3. For r = 0.4, the lower limit of m η2 derives from the non-tachyon condition, m 2 ηm > 0, while the upper limit is bounded by the requirement of m A0 > 1.2 m η2 . At this benchmark, thett cross-section is dominant, reaching above 80% percent of the total contribution. The dot-dashed red line intersects with the reference crosssection line for a DM mass of m η2 ∼ 500 GeV. Instead, for r = 0.6, the averaged cross-section mainly comes from the combined di-bosons and di-Higgs channels, with the blue solid line intersecting the reference value for a DM mass around m η2 ∼ 700 GeV. As inferred from Figure 2, a smaller mass with a larger cross section is excluded by the direct detection. What this plot reveals is that due to the model-specific couplings, the DM cross section is dominated by thett channel for lower masses, and by the di-boson one for larger masses. For a further illustration, in Figure 4, we show the results of a random scan of (α, C g , Bm, r), where we retain only the points passing all the constraints, i.e. relic density, direct detection and no-tachyon conditions. As expected, we find two distinct branches of viable points: for 400 < m η2 < 600 GeV, thē tt channel dominates, while for 600 < m η2 < 1000 GeV, it is the di-boson W + W − , ZZ, hh that dominates the annihilation cross section. Note that this latter section is similar to the traditional Higgs portal singlet model. One new ingredient in our model is the annihilation into the lighter even pNGBs η p and η m : we have checked that this channel is always subleading, with a contribution of less that 3% for all points. Also, since the quartic couplings are relatively small, the cubic coupling λ hη 2 2 is always close to the upper limit of the direct detection bound in order to put the relic density into the correct ballpark. This implies that the singlet DM candidate in our model is on the verge of being excluded (or discovered) by future direct detection experiments, like XENONnT [7] and LUX-ZEPLIN [44], which will improve the reach of the current XENON1T experiment by at least one order of magnitude.

CONCLUSION
In this work, we analysed the properties of a pseudoscalar singlet Dark Matter candidate that emerges as a pseudo-Nambu-Goldstone boson from a composite Higgs model, based on the coset SU (6)/SO (6). Since the light quarks, in particular the bottom, have to obtain their masses via an effective Yukawa operator originating from four fermion interaction, the custodial symmetry is unavoidably broken. We have proved that all the tadpole terms in a generic vacuum with (α, γ) angles will vanish after imposing the minimum conditions. More importantly, the γ value is suppressed by an order of m 2 b /m 2 h , making the custodial symmetry breaking well under control. Our result should also apply to the CHM model in SU (5)/SO (5), which shares the same bi-triplet structure but without DM candidate.
Like most Higgs portal singlet models, the parameter space in this model is tightly constrained, mainly by Direct Detection experiments. Yet, due to the non-linear nature of the DM couplings to the SM particles, we find that a small region of the parameter space is still allowed, with masses 400 < m DM < 1000 GeV. For masses below ∼ 600 GeV, the annihilation cross section is dominated bytt, unlike the traditional models, while larger masses go back to dominant di-boson channels. This pattern comes from the complicated pNGB couplings. Furthermore, requiring the correct relic density leaves the allowed regions within reach of future experiments, XENONnT and LUX-ZEPLIN, which would be able to discover or exclude the η 2 as a DM.
Finally, the SU (6)/SO(6) model has another DM candidate in an inert second Higgs doublet, if it is lighter than the singlet. The phenomenology of this state is similar to the traditional inert Higgs doublet model [45,46], which is also tightly constrained by observations. We leave an exploration of this limit for future work.

Tadpole terms in a general vacuum
To ensure the DM stability, only two CP-even neutral pNGBs in this model, i.e. h and λ 0 = − i √ 2 (Λ 0 − Λ * 0 ) can develop VEVs. This means that we set β = 0 so that the second Higgs doublet remains inert. The pNGB matrix is parametrized as: where the Σ EW is rotated away to be misaligned with the direction of SU (2) L × U (1) Y by U (α, γ). In analogy to the β rotation, the SU (6) transformation can be decomposed into: with U γ = e iγ √ 2S8 defined in term of an unbroken generator. The inner U † γ operation on Σ(x) can be fully absorbed by the pNGB field redefinition. Thus it is the outer U γ that determines the γ dependence. And Eq. (25) can be re-written as: withΣ(x) = U † γ · Σ(x) · U * γ . Thus we can find out an exact mapping for Π(x) →Π(x) in terms of pNGB fields. The field redefinition can be split into several blocks and leaving one pion field H + unchanged. For h and λ 0 , they transform in a SO(2) rotation defined by γ:h This also applies to the DM candidates A 0 and η 2 : The charged Goldstone G + eaten by W + can mix with the Λ + and φ + in the bi-triplet.