Light Scalars in Composite Higgs Models

A composite Higgs boson is likely to be accompanied by additional light states generated by the same dynamics. This expectation is substantiated when realising the composite Higgs mechanism by an underlying gauge theory. We review the dynamics of such objects, which may well be the first sign of compositeness at colliders. We also update our previous analysis of the bounds from LHC searches to the latest results, and discuss the projected reach of the High-Luminosity run.


INTRODUCTION
Models of composite Higgs are a valid option for describing new physics beyond the Standard Model (SM). In this approach, the Higgs sector is replaced by confining dynamics, with the merit of solving the problem of hierarchy, as the only mass scale in the sector is dynamically generated, like in quantum chromo-dynamics (QCD). Furthermore, the breaking of the electroweak (EW) symmetry also arises dynamically, in contrast to the SM where it is merely described by a wrong-sign mass term.
The idea of dynamical EW symmetry breaking is as old as the SM itself [1], however in its first form lacked the presence of a light scalar degree of freedom, the Higgs boson. Later, it was proposed that the Higgs may arise as a pseudo-Nambu Goldstone boson (pNGB) of a global symmetry breaking [2]. This latter class of models saw a revival in the 2000's, following the development of holography in warped extra dimensions [3]. A minimal model of composite pNGB Higgs was thus proposed in Agashe et al. [4], and it has since been extensively studied in the literature (see [5][6][7], and references therein). The Higgs thus arises as a pNGB from the symmetry breaking pattern SO(5)/SO(4), together with the three Goldstones eaten by the W and Z bosons.
A key ingredient is the concept of partial compositeness [8] for the SM fermions, as a means to generate their masses and the SM flavor structures. The generation of a sizeable top-quark mass is particularly challenging and partial compositeness provides a possible solution by mixing the elementary fermions with a composite operator that has a large scaling dimension. This feature, again, follows from the constructions in warped space [9,10], where the SM fermions mix with bulk ones. We want to stress here that the main motivation behind the introduction of partial compositeness was to address the mass and flavor problems while avoiding the generic appearance of large flavor changing neutral currents among SM fermions. Only later, inspired by the holographic principle [11], did the role of the composite top partners extended to the role of regulators of the loop divergences of the Higgs mass, by assuming the finiteness of the full one loop expression via sum rules [11,12]. This, in turn, implies the necessity for light and weakly coupled spin-1/2 resonances [12,13]. Nevertheless, alternatives to regulate the top loops exist, and the potential generated by such loops can be stabilized by, for instance, the introduction of masses for the underlying fermions [14,15].
Another approach to composite dynamics, closer in spirit to the origin of the dynamical EW symmetry breaking of Technicolor, consists in defining an underlying theory in terms of gauge and fermion degrees of freedom, that confine at low energies [15]. In this approach, it is not possible to naturally obtain the minimal coset. 1 In turn, once the underlying dynamics is specified, only three kinds of patterns are allowed [19,20]: SU(N)/Sp(N), SU(N)/SO(N) and SU(N) × SU(N)/SU(N). The minimal model is thus based on SU(4)/Sp (4), which can be obtained with an underlying SU(2) gauge theory [14,21] and features only 5 pNGBs: the Higgs doublet plus a CP-odd singlet [14,15]. Other minimal cosets are SU(5)/SO(5) [22] and SU(4) × SU(4)/SU(4) [23].
The inclusion of partial compositeness poses additional constraints in building the model, in primis the fact that many additional underlying fermions are needed, therefore loss of asymptotic freedom follows. In Ferretti and Karateev [24], a systematic construction of underling models, with partial compositeness for the top, was done. The main new ingredient was the sequestering of QCD color charges, which need to be carried by the underlying fermions in order to give color to the spin-1/2 resonances, to a new species of fermions, χ, that transforms under a different representation of the confining group than that of the fermions, ψ, giving rise to the composite Higgs. Thus, no dangerous mixing between the EW symmetry breaking and potential color breaking arises. The spin-1/2 bound states, therefore, arise as "chimera baryons" [25] made of ψψχ or ψχχ, depending on the model. There are few other cases where partial compositeness can be achieved with a single species of fermions: a confining SU(3) gauge symmetry with fermions in the fundamental, à la QCD, as proposed in Vecchi [26]; SU(6) with fermions in the two-index anti-symmetric representation and E 6 with the 27. The QCD colored fermions, in the latter cases, act as "heavy flavors", in order to avoid light QCD colored pNGBs.
Phenomenologically, the most interesting feature of this class of underlying theories is the fact that global symmetries in the effective low-energy model are determined. In particular, one realizes that a symmetry comprising of QCD is unavoidable. Furthermore, there is always a non-anomalous U(1) charge, acting on both species of fermions, which is broken by (at least) the chiral condensate in the EW (Higgs) sector of the theory. This results in one light pNGB singlet under all the SM gauge symmetries. This state may be the lightest of the pNGB spectra, as it typically does not receive any mass contribution from top and gauge loops [27]. The properties of this state have been studied in Cai et al. [28], Belyaev et al. [29], Belyaev et al. [27], DeGrand et al. [30], and Cacciapaglia et al. [31]. At the LHC, it can be copiously produced via gluon fusion with the coupling to gluons being generated by the Wess-Zumino-Witten anomaly term [32,33] via the presence of the χ-fermions in the pNGB wave function. Couplings to other pNGBs and tops can also be predicted, once the underlying theory is specified. Furthermore, they can be produced via the decays of the top partner resonances [34]. The fact that the properties of this state can be predicted in terms of the underlying theory, and their potential lightness, is the most attractive feature. As a historical note, they were perfect candidates to explain the WW/WZ resonance at 2 TeV Cai et al. [28] and the γ γ resonance at 750 GeV [29] hinted at by the LHC data, which later appeared to have been statistical fluctuations. Other light states comprise of additional EW-charged pNGBs arising from the Higgs sector, and QCD-colored states coming from the condensation of the χ's.
In this work, we will mainly focus on the singlet pNGB associated to the global U(1) symmetry. If both fermion species condense, it is accompanied by a second pseudo-scalar singlet associated to the anomalous U(1) charges. The latter will receive a mass term from the anomaly, in a similar fashion to the η ′ in QCD. Nevertheless, it may be relatively light, as expected at large-N c for instance. We will therefore consider the phenomenology at the LHC to come from the presence of both states. This work follows Belyaev et al. [27] closely, and our main new contribution is the update of the bounds to the latest LHC searches, and the addition of projections at the High-Luminosity-LHC (HL-LHC) run. We will see that the bounds on the compositeness scale derived from the non-discovery of such a state can be much stronger than the typical bounds from electroweak-precision tests. The latter are usually considered the main constraints on models of Composite Higgs. Conversely, they appear to have the best prospects for being discovered at the LHC. The HL-LHC run will be crucial in this case, due to the lightness of such states and the paucity of current searches focusing on the low mass region between 14 and 65 GeV.
Before presenting our results, we should stress that these theories are not full Ultra-Violet (UV) completions of composite Higgs models with partial compositeness. One point is that the number of fermions we can introduce before loosing confinement (asymptotic freedom) is limited, thus one can only have enough to give mass to the top quark in this way. Furthermore, the theory needs to lie outside the conformal Infra-Red (IR) window [35]. It was shown that only 12 models are consistent with these requirements, while having the minimal Higgs cosets [36]. The second point is that the origin of the fourfermion interactions giving rise to the mixing between the SM tops and the composite fermions is not explained. Finally, the consistency of flavor bounds usually requires the theory to enjoy an IR conformal phase right above the condensation scale. This allows to split the scale where the masses of light quarks and leptons are generated from the confinement scale [37,38], which should not be far from the TeV. In the underlying theory under study, this can be achieved by adding a few additional fermions at a mass close to the condensation scale, such that the theory above this scale is right inside the conformal window. Being just above the lower edge of the conformal window is crucial if one needs the composite fermions to have large anomalous dimensions, as the theory is expected to be strongly interacting around the IR fixed point near the lower edge of the conformal window. A first step toward the construction of truly UV complete theories can be found in Cacciapaglia et al. [39], based on the potential presence of a UV safe fixed point, due to large multiplicities of fermions. As a final introductory word, we should also mention one main benefit of this approach: once an underlying theory is defined, it can be studied on the lattice. Thus, spectra and various properties of the theory in the strong sector can, in principle, be computed. This includes low-energy constants, which are crucial for the generation of the Higgs misalignment potential and the Higgs boson mass [40]. So far, theories based on confining SU(4) [25,41] and Sp(4) [42][43][44][45] are being studied. For SU(4), preliminary results on the spectra [25] show that the chimera baryons tend to be heavy and beyond the reach of the LHC, while first calculations of the relevant form factors Ayyar et al. (Unpublished) show a suppressed mixing to the top. This would disqualify them as "light" top partners that regulate the Higgs mass loop [12,13], however they would still play a role in generating the top mass and help with the flavor issue. It should be mentioned however that current lattice results do not yet include a realistic multiplicity of fermions, which may be crucial as the realistic models are close to the conformal window. Finally, we mention the possibility that spin-1/2 resonances may arise as a bound state between a fermion and a scalar, both carrying underlying color charges [46] (see also [17]). The price to pay, in this case, is the presence of fundamental scalars in the theory (unless the underlying scalars arise themselves as bound states of a theory that confines at higher energies or are protected by supersymmetry at high scales).
The paper is organized as follows: in section 2 we recap the main properties of the 12 underlying models. In section 3 we summarize the main properties of the pseudo-scalars associated with the two spontaneously broken U(1) global symmetries and present the updated bounds on the singlet pNGBs in section 4. We offer our conclusions in section 5.

UNDERLYING MODELS FOR A COMPOSITE HIGGS WITH TOP PARTIAL COMPOSITENESS
In this work we are interested in the underlying models for composite Higgs with top partial compositeness defined in Ferretti and Karateev [24]. These models characterize the underlying dynamics below the condensation scale ≈ 4πf , f being the decay constant of the pNGBs. As such, the need to be outside of the conformal window leaves only 12 models [36], listed in Table 1. They are defined in terms of a confining gauge interaction, that we call hypercolor (HC), and two species of fermions in two different irreducible representations of the HC. The two species of fermions play different roles: the EW charged ψ's generate the Higgs and the EW symmetry breaks upon condensation, and their multiplicity is chosen to match the minimal cosets; the QCD charged χ's consist of a triplet and an anti-triplet of QCD color, thus always amounting to 6 Weyl spinors. We will also assume that both fermions condense and thus the chiral symmetry in each sector is broken. In principle, the χ's may not condense and the 't Hooft anomaly matching condition may lead to the presence of light composite fermions, that may play the role of top partners [47]. However, assuming the persistent mass condition, it is possible to show that chiral symmetry breaking must occur in both cosets [36]: the argument goes that by giving a common mass to one class of fermions at a time, the chimera baryons that saturate the global 't Hooft anomaly would become massive and thus ineffective. The final answer can only be found on the lattice. The phenomenology of two of the models have been studied in detail, M8 in Barnard et al. [48] and M6 in Ferretti [49]. Lattice studies for the two models are also underway based on SU(4) HC [25] (which also applies to M11), and Sp(4) HC [44,45] (which also applies to M5). Note that a study based on a Nambu Jona-Lasinio effective model of M8 can be found in Bizot et al. [50]. As shown in Table 1, the baryons that enter partial compositeness for the top arise either as ψψχ or ψχχ bound states, depending on the representations under HC.
It is expected that the lightest states in these models are the pNGBs, that arise from the breaking of the chiral symmetries in the two sectors, while the fermionic and spin-1 resonances are expected to be heavier. The quantum numbers of the pNGBs in the 12 models are listed in Table 2. They can be organized in three classes: A) The ones arising from the EW coset, i.e., the chiral symmetry breaking in the ψ sector, only carry EW quantum numbers. All cosets contain at least one singlet, thus being nonminimal compared to the holographic SO(5)/SO(4) model. The production rate of these states at the LHC is typically very small, as it is due to EW interactions, and thus are very difficult to observe at the LHC. The neutral components may also couple to two gluons via loops of tops, however still give rise to small production rates. The case of the singlet in the SU(4)/Sp(4) coset has been studied in detail in Galloway et al. [14] and Arbey et al. [51], note however that the same considerations apply to singlets in the other cosets. The SU(5)/SO(5) case can be found in Ferretti [49,52]. Finally, the SU(4) 2 /SU(4) case is special compared to the other two, as it allows for a stable pNGB that may play the role of Dark Matter [53]. B) The ones arising from the chiral breaking in the χ sector, i.e., QCD coset, always carry QCD charges. A ubiquitous member of this class is a neutral color octet [27,54]. For all those pNGBs, pair production via QCD interactions can be substantial at the LHC [55] for masses below or around 1 TeV. The phenomenology of the color sextet in the context of model M8 has been studied in Cacciapaglia et al. [54]. After Run-I at the LHC, the bound on their masses can be estimated around the 1 TeV scale. This bound is still compatible with the fact that one-loop self-energy diagrams, involving a gluon, put their masses roughly in that range. C) The U(1) singlets are ubiquitous to all models. Their phenomenology has been studied in detail in Belyaev et al. [27]. They will be the main focus of this work. While they are singlets under the gauge symmetries of the SM, couplings arise via the topological WZW anomalies, which include coupling to gluons. In this, they differ from the EW The first column shows the EW and QCD color cosets, respectively, followed by the representations under the confining hypercolor (HC) gauge group of the EW sector fermions ψ and the QCD colored ones χ. The −qχ /qψ column indicates the ratio of charges of the fermions under the non-anomalous U(1) combination, while "Baryon" indicate the typical top partner structure. The column "Name" contains the model nomenclature from Belyaev et al. [27], while the last column marks the models that are currently being considered on the lattice. Note that Sp indicates the spinorial representation of SO(N), while F and A 2 stand for the fundamental and two-index anti-symmetric representations.
For the EW coset (ψψ condensate), we list the SU(2) L multiplets with their hypercharge, for the QCD coset (χχ condensate) the QCD representation and hypercharge. We remark that the only ubiquitous ones are the color octet and the two U(1) singlets, plus one singlet in the EW coset.
cosets, where couplings to gluons can only arise via top loops. We can therefore expect larger production rates for them.
All models in M1-M12 preserve custodial symmetry. Indeed, this requirement is central in their construction and determines the minimum amount of fermionic matter present in each model. For custodial symmetry to be preserved one needs to be able to embed a SU(2) L × SU(2) R group into the unbroken group H of the electroweak cosets G/H. This requirement is satisfied by However, the further requirement that there be a Higgs field in the bi-fundamental of SU(2) L × SU(2) R , requires to take N o ≥ 5. Thus, ρ = 1 at tree level in these constructions, as long as the triplet pNGBs (when present), do not acquire a vacuum expectation value.

LIGHT U(1) PSEUDO-SCALARS
In this section we summarize the main properties of the two U(1) pseudo-scalars, one of which associated with non-anomalous global symmetry. Most of the results shown in this section can be found in Belyaev et al. [27], where a more detailed analysis can be found. We refer to other results in the literature when necessary. This section can be considered a handbook for anybody who is interested in studying the phenomenology of such states, as we will provide all the relevant couplings and formulas necessary to compute cross-sections and branching ratios. Following the notation in Belyaev et al. [27], we call two mass eigenstates {a, η ′ }, with a being the lighter one, which is also closer to the anomaly-free U(1) boson. The masses, which also determine the mixing angle between the two states, receive three contributions: two from the masses of the underlying fermions ψ and χ, and one from the anomalous U(1) combination. Assuming that m χ ≫ m ψ , and neglecting the latter, the mixing angle can be determined in terms of the mass eigenvalues. We define the mixing angle α between the mass eigenstates and the pseudo-scalars associated to the U(1) ψ and U(1) χ charges. Thus, in the decoupling limit M η ′ ≫ M a , the mixing angle is given by where q ψ and q χ are the charges of the anomaly-free U(1) (see Table 1), f ψ,χ are the decay constants in the two sectors, and N ψ,χ their multiplicity. Note that only the ratio f ψ /f χ is not fixed but depends on the strong dynamics (thus calculable on the lattice [25]). However, we can fix it by applying the Maximal Attractive Channel (MAC) hypothesis [56], see Table 3 in Appendix A. Once this is fixed, all the couplings of the pseudoscalars to SM states are fixed in terms of the properties of the underlying dynamics, as we will show below. The relevant effective Lagrangian for both pseudo-scalars, i.e., φ = {a, η ′ }, can be generically parameterized as withF µν = 1 2 ǫ µνρσ F ρσ for F = {G a , W i , B}. Note that we have normalized the couplings with the decay constant in the Higgs sector, f ψ , which is directly related to the tuning in the misalignment potential as v = f ψ sin θ [27]. We could also have defined a U(1)-singlet decay constant as in Cacciapaglia et al. [31]. The relation between the two decay constants is given in Table 3.
The Lagrangian in Equation (2) matches with a generic Axion-Like Particle (ALP) Lagrangian [57][58][59], except that the various coefficients can be computed. The couplings in the last two lines are generated by loops of tops and gauge bosons (dominantly), but differ from the results from a generic ALP Lagrangian [59,60] due to non-linear couplings of the pNGBs in the composite models [31]. In the following, we shall review how each of the terms in the effective Lagrangian can be calculated. All the numerical coefficients, in the decoupling limit and in the minimal mass splitting limit, are given in Tables 3, 4 in Appendix A. The numbers we provide here assume the MAC relation between the decay constants, as used in Cacciapaglia et al. [31], while the values in Belyaev et al. [27] assume f ψ = f χ .
The computability of all the coefficients is one of the main appeals of these models, having an underlying gauge theory construction. For each model that has a fixed gauge group and representation for the underlying fermions, once a discrete choice of the representation of the top partners under the global symmetry is done, the phenomenology of the pseudo-scalars is determined in terms of only three independent continuous parameters (the masses m φ with φ = a , η ′ and a common decay constant f ψ ). All the couplings and ratios of the decay constants for the various cosets can be computed as shown in Tables 3, 4 in Appendix A. The only assumption we make is that the tops couple dominantly to only one composite operator.

Couplings to Gauge Bosons
The general couplings of the singlet pseudo-scalars to gauge bosons are almost entirely dictated by the quantum numbers of the underlying dynamics, i.e., with K η ′ V obtained from the above expression with the replacement α → α + π/2. In the above expression, c 5 = √ 2 for models with SU(5)/SO(5) breaking and 1 otherwise, C ψ,χ V are the anomaly coefficients of the singlets associated with U(1) χ ,ψ groups which are fully determined by the SM charges of the underlying fermions 2 . Thus, the only dependence on the mixing angle α remains, which is determined by the masses of the two states. In the Tables in Appendix A we give values in the two limiting cases of minimal mass splitting and decoupling.
One can rewrite the WZW interactions in terms of the physical gauge bosons, i.e., The couplings of a and η ′ to gauge bosons are thus determined purely from the underlying dynamics with one assumption, i.e., the validity of the MAC hypothesis. The only external dependence arises from the masses via the mixing angle α. Table 3 in Appendix A shows the resulting couplings of a and η ′ for all 12 underlying models. Typically, in a generic mixing angle, the couplings vary between the two limits shown. The couplings to two-gauge bosons also receive contributions at loop-level, in particular from top-loops, which are particularly relevant at low masses and can affect the production rate via gluon fusion and the decays. These contributions were fully computed in Belyaev et al. [27], and their effect expressed in terms of the Branching Ratio formulas: m 2 t C i (p 2 1 , (p 1 + p 2 ) 2 , p 2 2 ; m f , m t , m t ) the Passarino-Veltman functions with the normalization given in Package-X [61]. We have used the short-hand notation C 1+2 ≡ C 1 + C 2 and analytical expression for some of the simplest loop function can be found in Belyaev et al. [27]. C φ t is the coupling to tops, which is discussed in the following subsection.

Coupling to Tops, Light Quarks, and Leptons
The coupling to tops only depends on the charges under the two U(1)'s of the composite operators that mix to the left-handed and right-handed tops. If we assume that the two top chiralities mix dominantly to one operator, there are only six possible charges that enter the coupling to tops via the top mass operator: (n ψ , n χ ) = (±4, 2) , (0, ±2) , (±2, 0) , for ψψχ , (n ψ , n χ ) = (2, ±4) , (0, ±2) , (±2, 0) , for ψχχ , (9) where n ψ and n χ are the net numbers of ψ and χ fields, respectively in the two operators coupling to the two top chiralities (see Belyaev et al. [27] for more details). Thus, the C a t coefficient reads Like above, C η ′ t is given by α → α + π/2. For the light quarks and leptons, we will assume, for simplicity, that their mass is coming from a direct coupling to a bilinear of ψ's, i.e., via an effective Yukawa coupling. This corresponds to the top case, but with fixed {n ψ , n χ } = {2, 0}.
The coupling to tops above has been computed by writing the effective operators generating the top mass, as in Golterman and Shamir [40] and Golterman and Shamir [62]. However, in Bizot et al. [34] it was noted that computing the coupling of the pseudo-scalars starting from the mixing to the top partners would lead to a different expression, differing by the presence of the mixing angles in the partial compositeness. For the top this has a minor impact on the numerical results, and we therefore chose to remain using the operator case.

Loop-induced Couplings to the Higgs and to Zh
Models with a pseudo-scalar state generically contain a coupling to Zh [60], which is generated at loop level. In our models, the leading contributions to the effective coupling between the singlet pseudo-scalars, Z and Higgs bosons are given by the diagrams in Figure 1 [31]. Explicit calculation for the coupling K φ eff hZ defined in Equation (2) gives: with B 0 (τ p/t ) ≡ B 0 (p 2 ; m t , m t ), see Belyaev et al. [27] for the analytic expression. In the formula, the κ t and κ Z are the corrections to the Higgs coupling to tops and Z, respectively, normalized by the SM value. The loop function B 0 is UVdivergent and we have parameterized it in terms of a cutoff, i.e., 1/ǫ → −1 + ln(16π 2 f 2 ψ /µ 2 ). Note that the UV-sensitivity is only present in the term proportional to (κ t − κ Z ), which reflects the  non-linearities in the Higgs couplings, a common feature in all composite Higgs models. The partial width for the pseudo-scalar decay gives with λ(x, y, z) the Källén function. For very light pseudo-scalars the decay h → φZ is allowed, with a partial width given by the formula above, with the replacement of m φ ↔ m h . At loop level, a coupling hφ 2 is also generated. This is relevant for m φ < m h /2, for which Higgs decays into two pseudo-scalars are open. Explicit calculation of the leading diagrams, shown in Figure 2, gives The Higgs decay to two pseudo-scalars is then given by 3

LHC BOUNDS AND HIGH-LUMINOSITY PROJECTIONS
The presence of the light composite pseudo-scalars can be tested at the LHC via the single production via gluon fusion, which is the dominant production mode, and further decays into a resonant pair of SM states. In this work we include both the effect from 3 There is also an additional contribution coming from the diagrams in Figure 2 that is proportional to p 2 h . This signals the presence of an effective term of the form φ 2 h, however, such contribution is always negligible. the WZW direct coupling to gluons, and the contribution of top and bottom loops. The cross-section calculation is performed at NLO in QCD by use of the HIGLU [63] code. For the tops, as shown above, we have six possible top partner assignment choices: following Belyaev et al. [27] and Cacciapaglia et al. [31], in the numerical results we chose the case {n ψ , n χ } = {2, 0}. A discussion of the effect of other choices can be found in Appendix B.
The strategy for applying bounds follows Belyaev et al. [27]. We collected all available searches, looking for resonant final states that may come from the pseudo-scalars, and extract a bound from the production cross section times branching ratio, assuming that the efficiencies of the experimental searches are the same on our model. This is a reasonable assumption, as the searches are mainly sensitive to the resonant nature of the signal, and much less on the possible kinematical differences in the production. Furthermore, we do not attempt to do a statistical combination of various searches, as we cannot take into account correlations of the systematic uncertainties in the experiments. Thus, we simply consider the most constraining search or signal region to extract a bound from for each final state. The final result is shown in Figure 3 for two representative models, M8 and M9. What connects the two is the fact that the global symmetries are the same, thus they can be characterized by the same low energy effective action based on the minimal SU(4)/Sp(4) EW coset and SU(6)/SO(6) QCD coset. However, as shown in the plot, the properties of the two pseudo-scalars are very different, hence leading to very different bounds. Note that we have re-expressed the bound on the cross sections into a bound on the decay constant of the Higgs. This is possible because all the coefficients of the couplings are calculable, as detailed in the previous section.
Before commenting on the numerical results, we will list here all the searches we implemented.
i) The tt final state is only relevant for large masses and indicated in orange (Run-II at 13 TeV) and green (Run-I at 8 TeV) on the side-bands of the plots. We implemented a  , shown by the dot-dashed blue line. In our specific models, this is stronger than the direct searches, mainly because at the final states the searches focused on have small branching ratios. x) Finally, we checked that constraints coming from associated production of the pseudo-scalars with bb [102,103] and tt [104] are not competitive with production via Z decays [105] (Z → aγ ).
The plots on the left column of Figure 3 show the limit on the Higgs decay constant f ψ in the plane of the two pseudo-scalar masses and for models M8 and M9. For each point in the m am η ′ plane we compute independently the bounds on f ψ coming from the a and η ′ resonances and then show the most stringent one. In the two side-bands we show the strongest bound coming from a (top band) and η ′ (right band), split into the various channel we considered. One important observation is that the limit often passes the 1 TeV mark. This is significant as typical electroweak precision bounds on this class of models give a lower limit on f around this scale [106][107][108]. Cases where the limit can be relaxed have been discussed in Contino and Salvarezza [109], Ghosh et al. [110], and Buarque Franzosi et al. (Unpublished). We note, therefore, that searches for these light pseudo-scalars can be the most constraining probe for these class of models. Also note the presence of a poorly constrained region for the 14 < m a < 65 GeV window of the lightest pseudo-scalar (most evident for M9). This is mainly due to the paucity of direct searches that are significant in this low mass window, the strongest bound being on the new physics Higgs decay rate. Note that the latter will not significantly improve at the end of the HL-LHC [111]. It is therefore crucial to close this gap with searches dedicated to this region, which is present for all models. Note also that the constraints on M8 are always rather mild: this is due to the coupling to gluons, which is particularly low in this specific model. The plots, therefore, show how the constraints are particularly sensitive to the details of the underlying models, as the twin models M8 and M9 dramatically show. For comparison, in Figure 4 we show the bounds for another model, M7, based on the SU(5)/SO (5)  A new result we show in this paper is the inclusion of projections for the HL-LHC run. First, we would like to attack the low mass window, which is left open after the Run-II searches, as shown in all plots. In this window, the main decay channels are in two jets (either gluons or b quarks), followed by taus. Di-photon final states are also present, however current searches [83, 86, 87] cannot reach this low mass region due to trigger limitations.
In Cacciapaglia et al. [31] we proposed a new search based on the di-tau final state. To be able to pass the trigger requirements, we proposed to aim at the production of a single a that recoils against a high-p T jet. This also allows to reduce the background level, while the reduction in the cross section still leaves a large signal rate. We analyzed in detail the case of leptonic decays of the two taus into different flavor leptons. Due to the high boost, the angular separation between the two leptons is typically very small. Thus, imposing an upper cut on the angular separation, R eµ < 1, allows to efficiently reduce the main background, coming from tt and Drell-Yan di-tau production. Fakes in this channel should have a limited impact, thus allowing us to derive reliable estimates for the reach. A key ingredient to improve the reach in the case of a small mass below 30 − 40 GeV is the reduction of the lower cut on the separation angle between the two leptons. The current minimal separation used at the LHC, see Khachatryan et al. [112] for instance, is R eµ > 0.1 ÷ 0.2, as such it would lead to a degradation of the sensitivity for low invariant masses where the boost produces very low angles [31]. It would be necessary, therefore, to relax the isolation criteria and remove the minimal separation in order to optimize the reach. Furthermore, due to the low statistics, it is crucial to reduce at the maximum the systematic errors on the lepton reconstructions. For this reason, we focused on the fully leptonic case. The main systematics in boosted di-tau searches [113] come from hadronic tau decays and from the invariant mass reconstruction, which are not required in our study. We optimistically assume, therefore, that systematic uncertainties below the % level can be achieved. In the right plots of Figures 3, 4 and Figures 7-9 in Appendix C, we show the projected reach of this proposed search in black. The plots show that in most models it can effectively cover the low mass open window, with enhanced sensitivity to the low mass end. Note also that we only use the opposite-flavor fully leptonic channel. Nevertheless, semi-leptonic decays may also be used, by implementing advanced techniques, like the "mini-isolation" proposed in Rehermann and Tweedie [114], while tests of fully-hadronic di-tau tagging can be found in Katz et al. [115] and Conte et al. [116].
Another method that would allow to cover the low mass window is extracting indirect bounds from the diphoton differential cross section measurements, as proposed in Mariotti et al. [117]. We added a projection of this bound at High-Luminosity in red. Figure 3 effectively shows the complementarity between the two searches: for M8, the di-tau search gives stronger bounds in the full mass range, while for M9 the di-photon bound is more stringent while di-tau can only compete at the low mass end of the window. In Figure 4 we show another case, M7, where the complementarity between the two methods at the low and high ends of the open mass window is more evident. To complete the High-Luminosity projections, we also include projections for tt [118][119][120] (in blue), di-jet [119,121,122] (in green), Zh [123] (in orange), WW [124] (in cyan), τ τ [125] (in violet), and bb [119,122] (in red).
The plots on the right side of the Figures 3, 4 and Figures 7-9 in Appendix C show that the High-Luminosity run of the LHC will allow to effectively cover the full parameter space of the pseudo-scalar masses for nearly all models, provided that the searches addressing the low mass window are implemented. This is a last chance situation, as the sensitivity of high-energy future colliders to such low masses will be much lower.
Before concluding the section, we would like to comment on another search that can be potentially useful to cover the low-mass open window, i.e., the LHCb search for dark photons in the di-muon final state [126]. The main strength of this search relies on the cancellation of all systematic uncertainties. A recast of this search in the context of a two Higgs doublet model can be found in Haisch et al. [127]. While the systematics associated to the detector effects are reasonably similar between the pseudo-scalar resonance and the dark photon, the production channel (gluon fusion vs. Drell-Yann) remains different, thus a more detailed determination of the acceptances and systematics is needed for a recast in our case. The results of ongoing work will be presented in a separate publication.

CONCLUSIONS AND OUTLOOK
We have updated the bounds from various experimental searches on two potentially light pseudo-scalar mesons, which arise in models of composite Higgs with top partial compositeness, with an underlying gauge-fermion description. We have provided a handbook containing all the relevant information necessary to study the phenomenology in any of the variations of the 12 possible basic models. In each model, the couplings of the two states can be computed in terms of the properties of the underlying gauge theory and of the two decay constants in the two sectors, one related to the EW symmetry breaking and the other to QCD carrying states.
We found that, in most models, scanning for masses up to 10 TeV, the non-observation of a resonance allows to set a bound on the compositeness scale, that surpasses the typical bound from electroweak precision tests. This result shows how the observation of these states can be a smoking gun for these class of theories, while also carrying precious information on the details of the underlying models. In all cases, there is a poorly constrained region for masses between 10 and 65 GeV, where the "standard" channels relying on Higgs decays or di-muon searches, give very weak bounds in these models.
We thus reviewed two proposals to cover this window: one based on the search for boosted di-tau systems, and the other on indirect bounds from the di-photon differential cross section measurements. At the High-Luminosity LHC, these two strategies would allow to close the gap. In fact, they are complementary in two aspects: the di-tau is more sensitive to small masses while the photon is more sensitive to larger masses; in models where the photon coupling is suppressed, the tau Frontiers in Physics | www.frontiersin.org channel is most constraining, and vice versa. Finally, we included the projected sensitivity of Zh, WW, γ γ , tt, bb, τ τ and di-jet searches at High-Luminosity, to push the bounds higher. Our results also show the necessity to keep looking for tt resonances down to the mass threshold, as this is the most sensitive channel, in these models, above 350 GeV.

AUTHOR CONTRIBUTIONS
All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

ACKNOWLEDGMENTS
We would like to thank Xiabier Cid Vidal, Mike Williams and Martino Borsato for useful feedback on the LHCb dark photon search, and Uli Haisch for discussion on the recast. TF was supported by IBS under the project code IBS-R018-D1. GF is supported in part by a grant from the Wallenberg foundation no. KAW-2017.0100. GC acknowledges partial support from the Institut Franco-Suedois (project Tör), the Labex Lyon Institute of the Origins -LIO (under grant ANR-10-LABX-66 and FRAMA FR3127 Fédération de Recherche André Marie Ampère) and the LIA FKPPL. HS has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 668679).