Higgs Physics at the CLIC Electron-Positron Linear Collider

The CLIC accelerator design is based on a two-beam acceleration scheme. It uses a high-intensity drive beam to efficiently generate radio frequency (RF) power at 12 GHz. The RF power is used to accelerate the main particle beam that runs parallel to the drive beam. CLIC uses normal-conducting accelerator structures, operated at room temperature. These structures permit high accelerating gradients, while the short pulse duration discussed below limits ohmic losses to tolerable levels. The initial drive beams and the main electron/positron beams are generated in the central complex and are then injected at the ends of the two linac arms. The feasibility of the CLIC accelerator has been demonstrated through prototyping, simulations and large-scale tests, as described in the Conceptual Design Report CLICCDR_vol1 . In particular, the two-beam acceleration at gradients exceeding 100 MV/m has been demonstrated in the CLIC test facility, CTF3. High luminosities are achievable by very small beam emittances, which are generated in the injector complex and maintained during transport to the interaction point.

CLIC will be operated with a bunch train repetition rate of 50 Hz. Each bunch train consists of 312 individual bunches, with 0.5 ns between bunch crossings at the interaction point. The average number of hard $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$ interactions in a single bunch train crossing is much less than one. However, for CLIC operation at $\sqrt{s}>1\,\text{TeV}$, the highly-focussed intense beams lead to significant beamstrahlung (radiation of photons from electrons/positrons in the electric field of the other beam). Beamstrahlung results in high rates of incoherent electron–positron pairs and low-$Q^{2}$ $t$-channel multi-peripheral $\mathup{}\mathup{}\to\text{hadron}$ events, where $Q^{2}$ is the negative of the four-momentum squared of the virtual space-like photon. In addition, the energy loss through beamstrahlung generates a long lower-energy tail to the luminosity spectrum that extends well below the nominal centre-of-mass energy, as shown in Figure 1. Both the CLIC detector design and the event reconstruction techniques employed are optimised to mitigate the influence of these backgrounds, which are most severe at the higher CLIC energies; this is discussed further in Section 4.2.

The baseline machine design allows for up to $\pm$80 % longitudinal electron spin-polarisation by using GaAs-type cathodes CLICCDR_vol1 ; and provisions have been made to allow positron polarisation as an upgrade option. Most studies presented in this paper are performed for zero beam polarisation and are subsequently scaled to account for the increased cross sections with left-handed polarisation for the electron beam.

The detector concepts used for the CLIC physics studies, described here and elsewhere CLIC_PhysDet_CDR , are based on the SiD Aihara:2009ad ; ilctdrvol4:2013 and ILD ildloi:2009 ; ilctdrvol4:2013 detector concepts for the International Linear Collider (ILC). They were initially adapted for the CLIC $3\,\text{TeV}$ operation, which constitutes the most challenging environment for the detectors in view of the high beam-induced background levels. For most sub-detector systems, the $3\,\text{TeV}$ detector design is suitable at all energy stages, the only exception being the inner tracking detectors and the vertex detector, where the lower backgrounds at $\sqrt{s}=350\,\text{GeV}$ enable detectors to be deployed with a smaller inner radius.

The key performance parameters of the CLIC detector concepts with respect to the Higgs programme are:

• •

  excellent track-momentum resolution of $\sigma_{p_{T}}/p_{T}^{2}\lesssim 2\cdot 10^{-5}$ $\text{GeV}^{-1}$, required for a precise reconstruction of leptonic $\mathup{{{Z}}}$ decays in $\mathup{{{Z}}}\mathup{{{H}}}$ events;

• •

  precise impact parameter resolution, defined by $a\lesssim 5\,\upmu\text{m}$ and $b\lesssim 15\,\upmu\text{m}\,\text{GeV}$ in $\sigma_{d_{0}}^{2}=a^{2}+b^{2}/(p^{2}\sin^{3}\theta)$ to provide accurate vertex reconstruction, enabling flavour tagging with clean $\mathup{{{b}}}$-, $\mathup{{{c}}}$- and light-quark jet separation;

• •

  jet-energy resolution $\sigma_{E}/E\lesssim 3.5\,\%$ for light-quark jet energies in the range $100\,\text{GeV}$ to $1\,\text{TeV}$, required for the reconstruction of hadronic $\mathup{{{Z}}}$ decays in $\mathup{{{Z}}}\mathup{{{H}}}$ events and the separation of $\mathup{{{Z}}}\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ and $\mathup{{{H}}}\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ based on the reconstructed di-jet invariant mass;

• •

  detector coverage for electrons extending to very low angles with respect to the beam axis, to maximise background rejection for $\mathup{{{W}}}\mathup{{{W}}}$-fusion events.

The main design driver for the CLIC (and ILC) detector concepts is the required jet-energy resolution. As a result, the CLIC detector concepts CLIC_PhysDet_CDR , CLIC_SiD and CLIC_ILD, are based on fine-grained electromagnetic and hadronic calorimeters (ECAL and HCAL), optimised for particle-flow reconstruction techniques. In the particle-flow approach, the aim is to reconstruct the individual final-state particles within a jet using information from the tracking detectors combined with that from the highly granular calorimeters thomson:pandora ; Marshall2013153 ; ALEPH:pflow ; CMS:pflow . In addition, particle-flow event reconstruction provides a powerful tool for the rejection of beam-induced backgrounds CLIC_PhysDet_CDR . The CLIC detector concepts employ strong central solenoid magnets, located outside the HCAL, providing an axial magnetic field of 5 T in CLIC_SiD and 4 T in CLIC_ILD. The CLIC_SiD concept employs central silicon-strip tracking detectors, whereas CLIC_ILD assumes a large central gaseous Time Projection Chamber. In both concepts, the central tracking system is augmented with silicon-based inner tracking detectors. The two detector concepts are shown schematically in Figure 2 and are described in detail in CLIC_PhysDet_CDR .

The studies presented in this paper are based on a scenario in which CLIC runs at three energy stages. The first stage is at $\sqrt{s}=350\,\text{GeV}$, around the top-pair production threshold. The second stage is at $\sqrt{s}=1.4\,\text{TeV}$; this energy is chosen because it can be reached with a single CLIC drive-beam complex. The third stage is at $\sqrt{s}=3\,\text{TeV}$; the ultimate energy of CLIC. At each stage, four to five years of running with a fully commissioned accelerator is foreseen, providing integrated luminosities of $500\,\text{fb}^{-1}$, $1.5\,\text{ab}^{-1}$ and $2\,\text{ab}^{-1}$ at $350\,\text{GeV}$, $1.4\,\text{TeV}$ and $3\,\text{TeV}$, respectivelyⁱⁱiAs a result of this paper and other studies, a slightly different staging scenario for CLIC, with a first stage at $\sqrt{s}=380\,\text{GeV}$ to include precise measurements of top quark properties as a probe for BSM physics, and the next stage at 1.5 TeV, has recently been adopted and will be used for future studies staging_baseline_yellow_report .. Cross sections and integrated luminosities for the three stages are summarised in Table 1.

The choice of the CLIC energy stages is motivated by the desire to pursue a programme of precision Higgs physics and to operate the machine above $1\,\text{TeV}$ at the earliest possible time; no CLIC operation is foreseen below the top-pair production threshold.

From the Higgs physics perspective, operation at energies much below $1\,\text{TeV}$ is motivated by the direct and model-independent measurement of the coupling of the Higgs boson to the $\mathup{{{Z}}}$, which can be obtained from the recoil mass distribution in $\mathup{{{Z}}}\mathup{{{H}}}\to\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\mathup{{{H}}}$, $\mathup{{{Z}}}\mathup{{{H}}}\to\mathup{{}^{\scriptstyle{+}}}\mathup{{}^{\scriptstyle{-}}}\mathup{{{H}}}$ and $\mathup{{{Z}}}\mathup{{{H}}}\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}\mathup{{{H}}}$ production (see Section 5.1.1 and Section 5.1.2). These measurements play a central role in the determination of the Higgs couplings at an $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$ collider.

However, from a Higgs physics perspective, there is no advantage to running CLIC at around $\sqrt{s}=250\,\text{GeV}$ where the $\mathup{{{Z}}}\mathup{{{H}}}$ production cross section is larger, compared to running at $\sqrt{s}=350\,\text{GeV}$. Firstly, the reduction in cross section at $\sqrt{s}=350\,\text{GeV}$ is compensated, in part, by the increased instantaneous luminosity achievable at a higher centre-of-mass energy. The instantaneous luminosity scales approximately linearly with the centre-of-mass energy, ${\cal{L}}\propto\gamma_{\mathup{{{e}}}}$, where $\gamma_{\mathup{{{e}}}}$ is the Lorentz factor for the beam electrons/positrons. For this reason, the precision on the coupling $g_{\mathup{{{H}}}\mathup{{{Z}}}\mathup{{{Z}}}}$ at $350\,\text{GeV}$ is comparable to that achievable at $250\,\text{GeV}$ for the same period of operation. Secondly, the additional boost of the $\mathup{{{Z}}}$ and $\mathup{{{H}}}$ at $\sqrt{s}=350\,\text{GeV}$ provides greater separation between the final-state jets from $\mathup{{{Z}}}$ and $\mathup{{{H}}}$ decays. Consequently, the measurements of $\sigma(\mathup{{{Z}}}\mathup{{{H}}})\times BR(\mathup{{{H}}}\to X)$ are more precise at $\sqrt{s}=350\,\text{GeV}$. Thirdly, and most importantly, operation of CLIC at $\sqrt{s}\approx 350\,\text{GeV}$ provides access to the $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{H}}}\mathup{{{\mathup{}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}}\mathup{{\overline{{\mathup{}}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}}$ fusion process; this improves the precision with which the total decay width $\Gamma_{\mathup{{{H}}}}$ can be determined at CLIC. For the above reasons, the preferred option for the first stage of CLIC operation is $\sqrt{s}\approx 350\,\text{GeV}$.

Another advantage of $\sqrt{s}\approx 350\,\text{GeV}$ is that detailed studies of the top-pair production process can be performed in the initial stage of CLIC operation. Finally, the Higgs boson mass can be measured at $\sqrt{s}=350\,\text{GeV}$ with similar precision as at $\sqrt{s}=250\,\text{GeV}$.

The majority of CLIC Higgs physics studies presented in this paper are performed assuming unpolarised $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}$ and $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$ beams. However, in the baseline CLIC design, the electron beam can be polarised up to $\pm 80\,\%$. There is also the possibility of positron polarisation at a lower level, although positron polarisation is not part of the baseline CLIC design. For an electron polarisation of $P_{-}$ and positron polarisation of $P_{+}$, the relative fractions of collisions in the different helicity states are:

By selecting different beam polarisations it is possible to enhance/suppress different physical processes. The chiral nature of the weak coupling to fermions results in significant possible enhancements in $\mathup{{{W}}}\mathup{{{W}}}$-fusion Higgs production, as indicated in Table 3. The potential gains for the $s$-channel Higgsstrahlung process, $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{Z}}}\mathup{{{H}}}$, are less significant, and the dependence of the $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{H}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$ cross section on beam polarisation is even smaller. In practice, the balance between operation with different beam polarisations will depend on the CLIC physics programme taken as a whole, including the searches for and potential measurements of BSM particle production.

The Higgsstrahlung process, $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{Z}}}\mathup{{{H}}}$, provides an opportunity to study the couplings of the Higgs boson in an essentially model-independent manner. Such a model-independent measurement is unique to a lepton collider. Higgsstrahlung events can be selected based solely on the measurement of the four-momentum of the $\mathup{{{Z}}}$ boson through its decay products, while the invariant mass of the system recoiling against the $\mathup{{{Z}}}$ boson peaks at $m_{\mathup{{{H}}}}$. The most distinct event topologies occur for $\mathup{{{Z}}}\to\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$ and $\mathup{{{Z}}}\to\mathup{{}^{\scriptstyle{+}}}\mathup{{}^{\scriptstyle{-}}}$ decays, which can be identified by requiring that the di-lepton invariant mass is consistent with $m_{\mathup{{{Z}}}}$ (see Section 5.1.1). SM background cross sections are relatively low. A slightly less clean, but more precise, measurement is obtained from the recoil mass analysis for $\mathup{{{Z}}}\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ decays (see Section 5.1.2).

Recoil-mass studies provide an absolute measurement of the total $\mathup{{{Z}}}\mathup{{{H}}}$ production cross section and a model-independent measurement of the coupling of the Higgs to the $\mathup{{{Z}}}$ boson, $g_{\mathup{{{H}}}\mathup{{{Z}}}\mathup{{{Z}}}}$. The combination of the leptonic and hadronic decay channels allows $g_{\mathup{{{H}}}\mathup{{{Z}}}\mathup{{{Z}}}}$ to be determined with a precision of $0.8\,\%$. In addition, the recoil mass from $\mathup{{{Z}}}\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ decays provides a direct search for possible Higgs decays to invisible final states, and can be used to constrain the invisible decay width of the Higgs, $\Gamma_{\text{invis}}$.

By identifying the individual final states for different Higgs decay modes, precise measurements of the Higgs boson branching fractions can be made. Because of the high flavour tagging efficiencies CLIC_PhysDet_CDR achievable at CLIC, the $\mathup{{{H}}}\to\mathup{{{b}}}\mathup{{\overline{{\mathup{{{b}}}}}}}$ and $\mathup{{{H}}}\to\mathup{{{c}}}\mathup{{\overline{{\mathup{{{c}}}}}}}$ decays can be cleanly separated. Neglecting the Higgs decays into light quarks, the branching ratio of $\mathup{{{H}}}\to\mathup{{{g}}}\mathup{{{g}}}$ can also be inferred and $\mathup{{{H}}}\to\mathup{{}^{\scriptstyle{+}}}\mathup{{}^{\scriptstyle{-}}}$ decays can be identified.

Although the cross section is lower, the $t$-channel $\mathup{{{W}}}\mathup{{{W}}}$-fusion process $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{H}}}\mathup{{{\mathup{}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}}\mathup{{\overline{{\mathup{}}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}}$ is an important part of the CLIC Higgs physics programme at $\sqrt{s}\approx 350\,\text{GeV}$. Because the visible final state consists of the Higgs boson decay products alone, the direct reconstruction of the invariant mass of the Higgs boson or its decay products plays a central role in the event selection. The combination of Higgs production and decay data from Higgsstrahlung and $\mathup{{{W}}}\mathup{{{W}}}$-fusion processes provides a model-independent extraction of Higgs couplings.

For CLIC operation above $1\,\text{TeV}$, the large number of Higgs bosons produced in the $\mathup{{{W}}}\mathup{{{W}}}$-fusion process allow relative couplings of the Higgs boson to the $\mathup{{{W}}}$ and $\mathup{{{Z}}}$ bosons to be determined at the ${\cal{O}}(1\,\%)$ level. These measurements provide a strong test of the SM prediction for:

where $\theta_{\mathrm{W}}$ is the weak-mixing angle. Furthermore, the exclusive Higgs decay modes can be studied with significantly higher precision than at $\sqrt{s}=350\,\text{GeV}$. For example, CLIC operating at $3\,\text{TeV}$ yields a statistical precision of $2\,\%$ on the ratio $g_{\mathup{{{H}}}\mathup{{{c}}}\mathup{{{c}}}}/g_{\mathup{{{H}}}\mathup{{{b}}}\mathup{{{b}}}}$, providing a direct comparison of the SM coupling predictions for up-type and down-type quarks. In the context of the model-independent measurements of the Higgs branching ratios, the measurement of $\sigma(\mathup{{{H}}}\mathup{{{\mathup{}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}}\mathup{{\overline{{\mathup{}}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}})\times BR(\mathup{{{H}}}\to\mathup{{{W}}}\mathup{{{W}}}^{*})$ is particularly important. For CLIC operation at $\sqrt{s}\approx 1.4\,\text{TeV}$, the large number of events allows this cross section to be determined with a precision of $1\,\%$ (see Section 6.3). When combined with the measurements at $\sqrt{s}\approx 350\,\text{GeV}$, this places a strong constraint on $\Gamma_{\mathup{{{H}}}}$.

Although the $\mathup{{{W}}}\mathup{{{W}}}$-fusion process has the largest cross section for Higgs production above $1\,\text{TeV}$, other processes are also important. For example, measurements of the $\mathup{{{Z}}}\mathup{{{Z}}}$-fusion process provide further constraints on the $g_{\mathup{{{H}}}\mathup{{{Z}}}\mathup{{{Z}}}}$ coupling. Moreover, CLIC operation at $\sqrt{s}=1.4\,\text{TeV}$ enables a determination of the top Yukawa coupling from the process $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}\mathup{{{H}}}\to\mathup{{{b}}}\mathup{{{W}}}^{+}\mathup{{\overline{{\mathup{{{b}}}}}}}\mathup{{{W}}}^{-}\mathup{{{H}}}$ with a precision of $4.2\,\%$ (see Section 8). Finally, the self-coupling of the Higgs boson at the $\mathup{{{H}}}\mathup{{{H}}}\mathup{{{H}}}$ vertex is measurable in $1.4\,\text{TeV}$ and $3\,\text{TeV}$ operation.

In the SM, the Higgs boson originates from a doublet of complex scalar fields $\phi$ described by the potential:

where $\mu$ and $\lambda$ are the parameters of the Higgs potential, with $\mu^{2}<0$ and $\lambda>0$. The measurement of the strength of the Higgs self-coupling provides direct access to the coupling $\lambda$ assumed in the Higgs mechanism. For $m_{\mathup{{{H}}}}$ of around $126\,\text{GeV}$, the measurement of the Higgs boson self-coupling at the LHC will be extremely challenging, even with $3000\,\text{fb}^{-1}$ of data (see for example Dawson:2013bba ). At a linear collider, the trilinear Higgs self-coupling can be measured through the $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{Z}}}\mathup{{{H}}}\mathup{{{H}}}$ and $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{H}}}\mathup{{{H}}}\mathup{{{\mathup{}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}}\mathup{{\overline{{\mathup{}}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}}$ processes. The $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{Z}}}\mathup{{{H}}}\mathup{{{H}}}$ process at $\sqrt{s}=500\,\text{GeV}$ has been studied in the context of the ILC, where the results show that a very large integrated luminosity is required ILCPhysicsDBD . However for $\sqrt{s}\geq 1\,\text{TeV}$, the sensitivity for the process $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{H}}}\mathup{{{H}}}\mathup{{{\mathup{}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}}\mathup{{\overline{{\mathup{}}}}{}_{\scriptstyle{\!\mathup{{{e}}}}}}$ increases with increasing centre-of-mass energy and the measurement of the Higgs boson self-coupling (see Section 9) forms a central part of the CLIC Higgs physics programme. Ultimately a precision of approximately $20\,\%$ on $\lambda$ can be achieved.

Because of the presence of beamstrahlung photons in the colliding electron and positron beams, it is necessary to generate MC event samples for $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$, $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{}$, $\mathup{}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$, and interactions. The main physics backgrounds, with up to six particles in the final state, are generated using the Whizard 1.95 Kilian:2007gr program. In all cases the expected energy spectra for the CLIC beams, including the effects from beamstrahlung and the intrinsic machine energy spread, are used for the initial-state electrons, positrons and beamstrahlung photons. In addition, low-$Q^{2}$ processes with quasi-real photons are described using the Weizsäcker-Williams approximation as implemented in Whizard. The process of fragmentation and hadronisation is simulated using Pythia 6.4 Sjostrand2006 with a parameter set tuned to OPAL $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$ data recorded at LEP Alexander:1995bk (see CLIC_PhysDet_CDR for details). The decays of leptons are simulated using Tauola tauola . The mass of the Higgs boson is taken to be $126\,\text{GeV}$ⁱⁱⁱⁱiiA Higgs boson of $125\,\text{GeV}$ is used in the process $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}\mathup{{{H}}}$. and the decays of the Higgs boson are simulated using Pythia with the branching fractions listed in Dittmaier:2012vm . The events from the different Higgs production channels are simulated separately. The background samples do not include Higgs processes. MC samples for the measurement of the top Yukawa coupling measurement (see Section 8) with eight final-state fermions are obtained using the PhysSim gen:physsim package; again Pythia is used for fragmentation, hadronisation and the Higgs boson decays.

The Geant4 detector simulation toolkits Mokka Mokka and Slic Graf:2006ei are used to simulate the detector response to the generated events in the CLIC_ILD and CLIC_SiD concepts, respectively. The QGSP_BERT physics list is used to model the hadronic interactions of particles in the detectors. The digitisation, namely the translation of the raw simulated energy deposits into detector signals, and the event reconstruction are performed using the Marlin MarlinLCCD and org.lcsim Graf:2011zzc software packages. Particle flow reconstruction is performed using PandoraPFA thomson:pandora ; Marshall2013153 ; Marshall:2015rfa .

Vertex reconstruction and heavy flavour tagging are performed using the LcfiPlus program Suehara:2015ura . This consists of a topological vertex finder that reconstructs secondary interactions, and a multivariate classifier that combines several jet-related variables such as track impact parameter significance, decay length, number of tracks in vertices, and vertex masses, to tag bottom, charm, and light-quark jets. The detailed training of the multivariate classifiers for the flavour tagging is performed separately for each centre-of-mass energy and each final state of interest.

Because of the 0.5 ns bunch spacing in the CLIC beams, the pile-up of beam-induced backgrounds can affect the event reconstruction and needs to be accounted for. Realistic levels of pile-up from the most important beam-induced background, the $\mathup{}\mathup{}\to\text{hadrons}$ process, are included in all the simulated event samples to ensure that the impact on the event reconstruction is correctly modelled. The $\mathup{}\mathup{}\to\text{hadrons}$ events are simulated separately and a randomly chosen subset, corresponding to 60 bunch crossings, is superimposed on the physics event before the digitisation step LCD:overlay . 60 bunch crossings is equivalent to 30 ns, which is much longer than the assumed offline event reconstruction window of 10 ns around the hard physics event, so this is a good approximation CLIC_PhysDet_CDR . For the $\sqrt{s}=350\,\text{GeV}$ samples, where the background rates are lower, 300 bunch crossings are overlaid on the physics event. The impact of the background is small at $\sqrt{s}=350\,\text{GeV}$, and is most significant at $\sqrt{s}=3\,\text{TeV}$, where approximately $1.2\,\text{TeV}$ of energy is deposited in the calorimeters in a time window of 10 ns. A dedicated reconstruction algorithm identifies and removes approximately $90\,\%$ of these out-of-time background particles using criteria based on the reconstructed transverse momentum $p_{\mathrm{T}}$ of the particles and the calorimeter cluster time. A more detailed description can be found in CLIC_PhysDet_CDR .

Jet finding is performed on the objects reconstructed by particle flow, using the FastJet Fastjet package. Because of the presence of pile-up from $\mathup{}\mathup{}\to\text{hadrons}$, it was found that the Durham Catani:1991hj algorithm employed at LEP is not optimal for CLIC studies. Instead, the hadron-collider inspired $k_{t}$ algorithm Catani:1993hr ; Ellis:1993tq , with the distance parameter $R$ based on $\Delta\eta$ and $\Delta\phi$, is found to give better performance since it increases distances in the forward region, thus reducing the clustering of the (predominantly low transverse momentum) background particles together with those from the hard $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$ interaction. Instead, particles that are found by the $k_{t}$ algorithm to be closer to the beam axis than to any other particles, and that are thus likely to have originated from beam-beam backgrounds, are removed from the event. As a result of using the $R$-based $k_{t}$ algorithm, the impact of the pile-up from $\mathup{}\mathup{}\to\text{hadrons}$ is largely mitigated, even without the timing and momentum cuts described above. Further details are given in CLIC_PhysDet_CDR . The choice of $R$ is optimised separately for different analyses. In many of the following studies, events are forced into a particular $N$-jet topology. The variable $y_{ij}$ is the smallest $k_{t}$ distance when combining $j$ jets to $i=(j-1)$ jets. These resolution parameters are widely used in a number of event selections, allowing events to be categorised into topologically different final states. In several studies it is found to be advantageous first to apply the $k_{t}$ algorithm to reduce the beam-beam backgrounds, and then to use only the remaining objects as input to the Durham algorithm.

The event simulation and reconstruction of the large data samples used in this study was performed using the iLCDirac Grefe:2014sca ; Tsaregorodtsev:2008zz grid production tools.

In the process $\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{Z}}}\mathup{{{H}}}$, it is possible to identify efficiently $\mathup{{{Z}}}\to\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$ and $\mathup{{{Z}}}\to\mathup{{}^{\scriptstyle{+}}}\mathup{{}^{\scriptstyle{-}}}$ decays with a selection efficiency that is essentially independent of the $\mathup{{{H}}}$ decay mode. The four-momentum of the (Higgs boson) system recoiling against the $\mathup{{{Z}}}$ can be obtained from $E_{rec}=\sqrt{s}-E_{\mathup{{{Z}}}}$ and $\vec{p}_{rec}=-\vec{p}_{\mathup{{{Z}}}}$, and the recoil mass, $m_{\mathrm{rec}}$, peaks sharply around $m_{\mathup{{{H}}}}$. The recoil mass analysis for leptonic decays of the $\mathup{{{Z}}}$ is described in Section 5.1.1. While these measurements provide a clean model-independent probe of $\mathup{{{Z}}}\mathup{{{H}}}$ production, they are limited by the relatively small leptonic branching ratios of the $\mathup{{{Z}}}$. Studies of $\mathup{{{Z}}}\mathup{{{H}}}$ production with $\mathup{{{Z}}}\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ are inherently less clean, but are statistically more powerful. Despite the challenges related to the reconstruction of hadronic $\mathup{{{Z}}}$ decays in the presence of various Higgs decay modes, a precise and nearly model-independent probe of $\mathup{{{Z}}}\mathup{{{H}}}$ production can be obtained by analysing the recoil mass in hadronic $\mathup{{{Z}}}$ decays, as detailed in Section 5.1.2. When all these measurements are taken together, a model-independent measurement of the $g_{\mathup{{{H}}}\mathup{{{Z}}}\mathup{{{Z}}}}$ coupling constant with a precision of $<1\,\%$ can be inferred Thomson:2015jda .