Future Electron-Positron Colliders and the 4-Dimensional Composite Higgs Model

In this note we analyse the prospects of a future electron-positron collider in testing a particular realisation of a composite Higgs model encompassing partial compositeness, namely, the 4-Dimensional Composite Higgs Model. We study the main Higgs production channels for three possible energy stages and different luminosity options of such a machine and confront our results to the expected experimental accuracies in the various Higgs decay channels accessible herein and, for comparison, also at the Large Hadron Collider.


I. INTRODUCTION
The long time effort of the physics community to understand the origin of mass has now produced an outstanding result, as the discovery at the Large Hadron Collider (LHC) of a Higgs boson with a mass of roughly 125 GeV has finally been confirmed by both the ATLAS [1] and CMS [2] collaborations. While this is an extraordinary outcome of the CERN machine operations, one of the primary questions that we need to ask now is whether this particle corresponds to the Higgs boson embedded in the Standard Model (SM) or else belongs to some Beyond the SM (BSM) scenario.
In fact the SM suffers from a key theoretical drawback, the so-called "hierarchy" problem, pointing out that the it could be a low energy effective theory valid only up to some cut-off energy Λ. The latter can well be at the TeV scale, hence in an energy range accessible at the LHC, so that new physics could be discovered at the CERN machine in the coming years.
For this reason many BSM scenarios, which could be revealed at the TeV scale, were proposed in the last decades, the most popular being Supersymmetry (SUSY) that solves the instability of the Higgs mass under radiative corrections by predicting new particle states in the spectrum, differing from the SM ones by half unit of spin, i.e., by postulating a symmetry between fermions and bosons [3,4]. The behaviour of the new SUSY particles with respect to the SM ones is such that the Higgs mass becomes stable to all orders in perturbation theory (see, e.g., [5] and references therein).
Another intriguing possibility is that the Higgs particle may be a composite state arising from some strongly interacting dynamics at a high scale instead of being an elementary state.
This will solve radically the hierarchy problem owing to compositeness form factors taming the otherwise divergent growth of the Higgs mass upon quantum effects. Furthermore, the now measured light mass of this potentially composite Higgs state is consistent with the fact that the latter could arise as a Pseudo Nambu-Goldstone Boson (PNGB) from a particular coset of a global symmetry breaking [6][7][8][9].
Models with a composite Higgs state arising as a PNGB generally predict modifications of its couplings to both bosons and fermions of the SM [10], hence the measurement of these quantities represents a powerful way to test the possible non-fundamental nature of the newly discovered state. Furthermore, the presence of additional particles in the spectrum of such composite Higgs models leads to both mixing effects with the SM states as well as new Feynman diagram topologies both of which would represent a further source of deviations from the SM expectations.
The advantages of an electron-positron collider with respect to a hadron collider, e.g., the cleanliness of the environment, the precision of the measurements and the large number of Higgs bosons produced, make an analysis of the prospect of these types of machine in disentangling the fundamental from the composite nature of the Higgs boson of primary importance in view of a future decision of the physics community regarding the new generation of accelerators.
For this reason, in this note we will be analysing the generic potential of the proposed e + e − colliders in testing a particular realisation of a composite Higgs model, the so-called 4-Dimensional Composite Higgs Model (4DCHM) of Ref. [11]. We will therefore borrow energy and luminosity configurations from machines prototypes such as the International Linear Collider (ILC), the Compact Linear Collider (CLIC) and the Triple Large Electron-Positron (TLEP) collider.
The plan of the paper is at follows. In the next section we describe the features of the model we have chosen. Sect. III illustrates our analysis and presents our results. Sect. IV summarises our findings.

II. MODEL
In this section we briefly introduce the salient features of the 4DCHM, that is, the specific realisation of the composite Higgs model with partial compositeness [12] that we have chosen for our analysis.
The 4DCHM is an effective low-energy Lagrangian approximation that represents an extremely deconstructed version of the Minimal Composite Higgs Model (MCHM) described in [13]. It is based on the SO(5)/SO(4) coset that gives four PNGBs, one of which being the physical Higgs state. The explicit breaking of the SO(4) global symmetry via linear mixing between the SM and new fields give rise to a radiative Higgs potentialà la Coleman-Weinberg [14] and hence to a mechanism of dynamical Electro-Weak Symmetry Breaking (EWSB). The particular choice of the fermionic sector of the 4DCHM makes the Higgs potential Ultra-Violet (UV) finite at one loop and, remarkably, for a natural choice of the model parameters, it has the correct form to provide EWSB [11] and a Higgs boson mass compatible with the experimental value of the new state discovered at the LHC.
The extra SO(5) ⊗ U (1) X gauge group present in the model implies 11 new vectorial degrees of freedom among which 5 are neutral and 3 + 3 are charged gauge bosons, collectively called Z and W , respectively. The choice of the four fundamental representations of SO(5) ⊗ U (1) X (called Ψ T,B andΨ T,B ) that build up the fermionic sector gives 20 new fermions, 16 with standard electric charge, +2/3 and −1/3, denoted collectively as t and b , respectively, and 4 with exotic charges, +5/3 and −4/3, denoted collectively ast andb , respectively. In our notation, the new particles present in the 4DCHM in addition to the SM ones are 1 then the following: Together with the SM matter and gauge fields and neglecting the SU (3) C part that is left untouched with respect to the SM (besides the fact that also the new spin 1/2 resonances carries colour charge and so the corresponding QCD terms ought to be added) the 4DCHM is described by the following Lagrangian 1 An increasing number labelling the 4DCHM bosonic and fermionic states corresponds to their increasing masses.
In L gauge the covariant derivatives with respect to the SMW fields and the extraÃ fields, with coupling g 0 and g ρ , are given by The link fields Ω n in the unitary gauge are given by where Π = √ 2hâTâ is the PNGB matrix and the Tâ's are the SO(5)/SO(4) broken generators, withâ = 1, 2, 3, 4.
The field Φ is a vector of SO(5) that describes the spontaneous symmetry breaking of The f i 's are the link coupling constants and f the strong sector scale (or 'compositeness' scale), which are related through and the choice f 1 = f 2 = √ 2 f has then been made.
The fermionic Lagrangian L f erm contains kinetic terms, mass (m * ) and interaction (Y T,B , m Y T,B ) parameters between the extra fermions and also mixing parameters between the extra matter and the SM content. L el f erm describes the kinetic terms of the SM fermions.

III. RESULTS
Being our goal the study of this particular composite Higgs framework in its completeness and its detailed analysis with its full particle spectrum included, we have chosen to implement the 4DCHM into numerical tools that allow one to perform dedicated analyses up to event generation. Our simulations have been mainly performed with the help of the CalcHEP package [15] in which the model under discussion had been previously implemented via the LanHEP tool [16], as already illustrated in some detail in Refs. [17,18]. Moreover, since CalcHEP only allows by default the analysis of only tree-level processes, we have also added by hand the one-loop Hgg, Hγγ and HγZ vertices (again computed with the full model particle spectrum inside them).
A feature specific to future e + e − colliders is the presence of Initial State Radiation (ISR) and Beamstrahlung. For the former, CalcHEP implements the Jadach, Skrzypek and Ward expressions of Refs. [19,20]. Regarding the latter, we adopted the parametrisation specified for the ILC project in [21], that is 2 , • beam size (x + y): 645.7 nm, • bunch length: 300 µm, • bunch population: 2 · 10 10 .
We will be considering throughout three values for the Centre-of-Mass (CM) energy, which are standard benchmark energies for a future e + e − collider: 250 GeV, 500 GeV and 1 TeV. Within the 4DCHM, we have studied the phenomenology of a Higgs boson obtained via the standard production mechanisms of lepton colliders: i.e., Higgs-Strahlung (HS) from neutral vector bosons and Vector Boson Fusion (VBF) (the latter primarily onset by an offshell charged vector boson pairs), as per Feynman diagram shown in Fig. 1. Furthermore, we have focused the analysis on two other Higgs processes, which are the specialities of future electron-positron machines: i.e., Higgs production in association with top-antitop quark pairs (see Fig. 2) and double Higgs production, considering both the case of a second Higgs boson emitted in the HS channel, e + e − → ZHH, and the one via VBF, e + e − → ν eνe HH (see Fig. 3). When combining production cross sections and decay Branching Ratios (BRs), our simulated data have been always related to the experimental accuracies presented in Refs. [24][25][26]. There and here, we will indicate the production cross section with σ(ZH) for HS, σ(W W ) for VBF, σ(ttH) for Higgs production in association with tt pairs, σ(ZHH) for double HS and σ(ννHH) for double Higgs production via VBF. In keeping with the aforementioned references, we have always assumed a luminosity of 250/500/1000 fb −1 in correspondence to an energy of 250/500/1000 GeV. 2 While fine details of the emerging electron and positron spectra may be different in other parametrisations (see [22] for CLIC and [23] for TLEP), we can confirm that the gross features of the ensuing 4DCHM phenomenology are generically captured by the present one. In the following (sub)sections we will present several results concerning the studies of the aforementioned Higgs production processes organised as follows. Initially we will confirm the results that in the so-called "decoupling limit" (whereby, essentially, the additional gauge bosons and fermions present in the 4DCHM particle spectrum are made sufficiently heavy) sizable deviations from the SM couplings of the discovered Higgs object are possible and may indicate indirect evidence of a composite Higgs sector. However, we will eventually also argue that such decoupling does not really onset yet for values of the new particle masses pertaining to the 4DCHM which are still compatible with current experimental data: i.e., from Tevatron and LHC direct searches for new gauge bosons and fermions. That is, we will show that genuine 4DCHM effects cannot be relegated to a simple rescaling of the relevant Higgs couplings, as the presence of W , Z and/or t propagator effects in all the production processes analysed cannot generally be neglected. This will in fact be shown to be a relevant effect for HS Higgs production and production in association with tt pairs, albeit less so for single Higgs production via VBF and both double Higgs production channels, via HS and via VBF, the latter two being also limited in a phenomenological analysis by the poor experimental accuracies which are expected.
In essence, to generalise our findings, quantitative studies of Higgs boson phenomenology in composite Higgs models at future electron-positron colliders cannot ignore effects from realistic mass spectra. Conversely, studies in the decoupling scenario remain valid so long that such mass spectra are made significantly heavy.

A. Decoupling limit in the 4DCHM
In recent years, a popular approach introduced by [29] (for a more recent review, see also [30,31]) in order to parametrise effects of generic New Physics (NP), has been adapted to the study of Higgs couplings (see, e.g., [32][33][34]) in generic decoupling scenarios. The rationale is as follows. It is commonly accepted that any effects arising from NP (at some scale Λ NP Λ EW ) can be embedded in the SM via 6-dimensional effective operators (thereby assuming that the new physical fields are integrated away), realising the aforementioned decoupling limit, so long that the probing energy is around the EW scale. Then, after a specific theory is chosen, the coefficients of such effective operators can be expressed in terms of coupling modifications of the SM interactions.
Now, focusing onto the 4DCHM as an illustrative example of composite Higgs scenarios with additionally heavy matter and gauge fields, the modifications of the Higgs couplings arise via three effects: 1. the non-linear realisation of the Goldstone symmetry; 2. the mixing between SM and extra particles due to the partial compositeness mechanism; 3. the possibility that, in both tree-level and loop-induced processes, exchange of new particles can occur.
In the aforementioned effective approach, the first two effects are normally captured while the latter is not whenever the masses of the additional states are close to Λ NP , as it is naturally the case. In fact, sometimes, in current literature, even the second effect is not accounted for, so that one achieves a true decoupling of all NP, i.e., all the non-SM particle effects in the model spectrum are neglected, and the analysis consists in the trivial rescaling of the Higgs couplings due to its Goldstone nature. In particular, defining ξ = v 2 /f 2 and assuming the case of the 4DCHM (i.e., a SO(5)/SO(4) coset with extra fermions in fundamental representations of SO(5)), this rescaling values, for vector bosons V = W/Z and SM fermions f , are [10] g SM In such a regime of decoupling we can also compute the values of the so-called Higgs signal strengths, defined as where X represents anything produced in association with the Higgs boson (e.g., Z in the case of HS and ν eνe or e + e − in the case of VBF) and i simply labels a possible final state of the Higgs boson decay (for example bb, γγ, W W , ZZ), and relate a measurement of the various µ i 's to the model parameter f .
An example of this is found in Fig. 4, limited for illustration purposes to the case of the HS process, which indeed shows that there exists sensitivity at future e + e − machines to the compositeness scale f . Further exercises can be performed along similar lines and various such statements can be finally made for a variety of production modes and decay channels as recently done, e.g., in Ref. [28].
However, in the light of our previous discussion, a legitimate question arises: does the decoupling limit represent a good approximation for interpreting the phenomenological implication of the 4DCHM? The answer is "generally not". In the following, we will set out to prove this by studying the impact of the 4DCHM particle spectrum onto the main Higgs production and decay channels for customary values of the NP scale.
We have started by performing a parameter scan 4 for various benchmark points of our model. We have found that the largest deviations from the SM expectations arise, naturally, for the lowest possible values of M Z , M W ≈ f g ρ and, correspondingly, for the lowest choices of the compositeness scale f . We have considered f in the TeV range and fixed the g ρ parameter in order to have M Z , M W ≈ 2 TeV, which indeed represents an allowed configuration from both EWPTs (see, e.g., Fig. 1 of [28]) and direct Z , W searches (see Ref. [36]). Furthermore, notice that keeping the mass of the extra neutral vector resonance fixed renders clearer the dependence of the observable onto the compositeness scale f .
In order to separate (factorisable) rescaling effects, due to both the non-linear realisation of the Goldstone symmetry and the mixing between SM and extra particles, from the ones due to the additional propagators, we have introduced the R and ∆ parameters for the inclusive HS production cross section as follows: Then, by numerical computation, we have proven that, if the new class of neutral gauge bosons are completely stripped off the calculations, R tends to κ 2 HZZ (equivalently, ∆ tends to 0) with a negligible deviation ∼ 0.01% due to a slight shift in the C V and C A couplings of the SM-like Z to the fermions (herein, e + e − ), due to the aforementioned mixing. Since HS is generically the most useful process from which deviations of the Higgs couplings from the SM values can be extracted, we are essentially making the generic statement that, even when the CM energy of the collider is below the scale of NP, f in this case, where the additional boson and fermion masses of the 4DCHM naturally tend to cluster, the HS cross section is always affected by propagator effects (see also [37] and [38] for studies in the context of the Littlest Higgs and Minimal Composite Higgs models, respectively). This is well illustrated by propagator effects. In fact, the trend of R (or equivalently ∆) is almost constant but, from some threshold on, it decreases with Γ Z 3 (somewhat linearly) reflecting the nature of the interference contribution that is proportional to 1/Γ Z 3 when the CM energy is smaller than the Z 3 masses involved (modulus some dilution induced by the Z 2 state, which is however 5 We explicitly verified that all the features of the Z 2 and Z 3 contributions are the same, with the exception of the couplings involved: the Z 3 one to leptons is about twice the Z 2 one to the same particles. Z 1 and Z 4 do not couple to leptons and are thus irrelevant in this process. Z 5 is always much heavier. responsible for the interference in the region where Γ Z 3 is small). The deviations from the SM limit span from ∼ 2% when √ s = 250 GeV up to ∼ 25% when √ s = 1 TeV. We have also verified that the effect is completely due to the constructive interference term arising from the SM-like Z resonance and the Z 2 + Z 3 contributions, with Z 3 being dominant among the two extra vectors, as already mentioned. A feature of such interference effect is that the R values are always above the expected "reduction" onsetting in the decoupling limit, to the point that, at √ s = 1 TeV for the three benchmarks and even at √ s = 500 GeV for f = 1000 GeV and f = 1200 GeV, the R value is above 1, which is never expected if the new resonances are totally decoupled.
We have then verified that choosing a bigger M Z , M W will render the effects of the extra gauge bosons smaller until one reaches the limit in which they are completely decoupled from the theory, thereby recovering the results of [28] although, for example, even for M Z = 3 TeV, the effects due to the aforementioned interference are still present and lead to deviations up to ∼ 10% with respect to the scaling of the couplings, when √ s = 1 TeV.
As a conclusion then, we believe that a complete study of composite Higgs models via the HS process should also take into account the possibility of non-decoupled spectra for the extra resonances (in the context of composite Higgs scenarios the effects of resonances, although in different processes, has already been analysed in [39]).
For completeness, we have also analysed the case of VBF, where we have established that such interference effect is taking place but with inverted sign and with a contribution that is one order of magnitude lower than in the HS case. The largest effect is again seen at √ s = 1 TeV, yielding a destructive interference of ∼ 2 − 3%.
B. Away from the decoupling limit in the 4DCHM

Higgs couplings analysis at the LHC
As a first step we consider the LHC capability to disentangle the 4DCHM from the SM by means of the measurement of the signal strengths µ i 's by comparing our predictions against the LHC experimental accuracies that can be found in Refs. [24,26]. In particular, we plot in Fig. 6   produced via gg fusion (left frame) and in associated production with tt (right frame) as given in Refs. [24,26] at the LHC with 14 TeV and 300 fb −1 of luminosity.

Higgs couplings analysis at e + e − colliders in HS and VBF
We have shown in Sect. III A that the exchange of sufficiently light new particles in the tree-level diagrams can affect Higgs production cross sections due to interference effects between SM and 4DCHM diagrams, so that the low energy approach cannot be sufficiently accurate in the processes that concern our analysis.    [24,26] at a future e + e − collider for various energy and luminosity stages, as detailed in the text.

The top Yukawa coupling from e + e − → ttH
At the running energy stages of 500 and 1000 GeV one of the most important process to be analysed at a future e + e − collider is e + e − → ttH 7 . In this channel, beside the effects already seen in the HS channel due to the exchange of s-channel extra gauge bosons, mainly Z 2 and Z 3 , there could also occur the exchange of t s that, if light enough and/or produced resonantly, could affect this production process significantly. In fact, in general composite Higgs models these particles are predicted to be relatively light, around the TeV scale (so as to avoid a large fine tuning: see, e.g., [40] for a review on this) which is precisely the region that is starting to be tested by the LHC experiments. As already mentioned, in order to naively take into account the direct search limits of extra coloured quarks coming from the LHC, we have restricted our parameter scans by only allowing for masses of these states greater than 600 GeV. This does not intend to be a precise and accurate exclusion, which is beyond the scope of this work, rather an indication of what a realistic mass bound on these states (which depends non-trivially on different BRs, interference effects and other subtleties) could be, an approach that is indeed sufficient for the purposes our analysis.
Following again the guidance provided by Refs. [24,26], we quote in Tab. III the expected accuracies, in correspondence to the chosen energies of a future e + e − collider, of the bb signal 7 Note that the contribution of the Higgs-strahlung diagrams is negligible at the energies considered with respect to those depicted in Fig. 2, see [26].  strength (the most easily accessible one) and we show the results of our scans in Fig. 9. We present the results for the two benchmark points already used in the previous subsection in one case with the inclusion of t fermions while in the other we do so by excluding these  extra states from the Higgs production process. From the comparison of the two panels of Fig. 9 it is clear that the enhancement of the signal strength up to a value of 2.5 (or more) relative to the decoupling limit result at √ s = 1 TeV is due to the exchange of t states with a mass smaller than √ s − m t that can indeed be resonant in the subsequent production of a Higgs-top pair in the final state. In contrast, at 500 GeV, the more moderate (yet still detectable) departure of the µ bb value from the the decoupling limit is due to non-resonant Z and t effects (from off-shell propagators and/or mixing), as at this energy (owing to the aforementioned 600 GeV mass limit) no t mass can at the same time be larger than m t +m H and smaller than √ s − m t 8 . Once again, the potential of a future e + e − machine in accessing the 4DCHM and also assessing its finite mass effects is very significant also via this process.  boson produced via associated production with tt as given in Refs. [24,26] at a future e + e − collider for two energy and luminosity stages, as detailed in the text. Only the bb decay mode is considered.

The triple Higgs self-coupling
As mentioned in Sect. II, in composite Higgs models the Higgs potential is radiatively generatedà la Coleman-Weinberg [14] and with the 4DCHM choice of the fermionic sector this turns out to be UV finite. From the effective potential and its derivatives at the minimum, one can extract the Higgs vacuum expectation value v, its mass m H and its triple self-coupling λ. At the leading order in the contribution of the gauge and fermionic loops we get in agreement with [28].
This modified coupling intervenes in one of the two series of three Feynman diagrams of Fig. 3, which are, as mentioned, the ones concerning the HS and VBF double Higgs production cross sections at a future e + e − collider. Our scatter plot in Fig. 10 illustrates 8 Notice that, for √ s = 500 or 1000 GeV, t t production for m t ≥ 600 GeV is not possible, so that single t production followed by t → tH could be a discovery process of the heavy fermions or else an independent confirmation of a possible LHC signal of the latter. that sizable deviations in λ are possible in the 4DCHM with respect to the SM, reflecting in a variation of production and/or decay rates at the level of several tens of percent. However, accuracies in the corresponding measurements at a future electron-positron collider ought to be reduced by a factor of two or so, in order to disentangle composite Higgs model effects.
This is in line with the pursuit of the so-called ILC(LumUP) high luminosity scenario [25], in the hope to decrease the expected error on such an observable down to a level comparable to the expected departures from the SM value.

IV. CONCLUSIONS
In summary, we have used the 4DCHM to illustrate the potential of future e + e − machines operating at energies between 250 and 1000 GeV, with appropriate luminosity options, in testing the salient features of composite Higgs models. We have in particular shown that ZHH @ 500 GeV ννHH @ 1000 GeV bb 0.64 0.38 boson produced in pair via HS as given in Ref. [26] at a future e + e − collider for two energy and luminosity stages, as detailed in the text. Only the bb decay mode of the Higgs boson is considered. the approach of reducing the problem to a simple rescaling of SM production and decay rates and consequent extrapolations could be inaccurate in certain regions of the parameter space, as it fails to account for significant effects that can arise from mixing between SM and 4DCHM particles or interference effects between the two, whenever the mass spectra of extra gauge bosons or fermions present in these scenarios are allowed to exist around the compositeness scale, as indeed natural in these scenarios. In fact, propagator effects of the additional heavy states present in composite Higgs models are often the most visible signals.
Such an approach instead naively integrates out all heavy particles. Clearly, the ensuing inaccuracies are the larger the smaller(higher) the new particle masses(collider energies) are.
In short, for mass spectra in either or both the gauge and fermionic sectors of the 4DCHM consistent with all current experimental and theoretical bounds, the effects studied here are tangible in most hallmark observables, such as Higgs production in single mode via HS, VBF, in association with tt pairs as well as in (potentially) double Higgs production, both via HS and VBF. Hence, our results indicate that this kind of studies must be based on a complete implementation of the relevant composite Higgs models, one example of which we make available for public use via the High Energy Physics Model Data-Base (HEPMDB) [41] at https://hepmdb.soton.ac.uk/ under the "4DCHM" name.