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Some highlights of the physics case for running an $e^+e^-$ collider at 500 GeV and above are discussed with a particular emphasis on the experimental access to the Higgs potential via di-Higgs and (at sufficiently high energy) triple Higgs production. The information obtainable from Higgs pair production at about 500 GeV is compared with the prospects for the HL-LHC and with the indirect information that can be obtained from a Higgs factory running at lower energies.

Domain walls are a type of topological defects that can arise in the early universe after the spontaneous breaking of a discrete symmetry. This occurs in several beyond Standard Model theories with an extended Higgs sector such as the Next-to-Two-Higgs-Doublet model (N2HDM). In this talk, I will discuss the domain wall solution related to the singlet scalar of the N2HDM and demonstrate the possibility of electroweak symmetry restoration (EWSR) in the vicinity of the domain wall. Such symmetry restoration can have profound implications on the early universe cosmology as the sphaleron rate inside the domain wall would, in principle, be unsuppressed compared with the rate outside the wall.

In this work, we examine the criteria for vacuum stability in two models with extended scalar sectors namely, the N2HDM and the 2HDMS and make a detailed comparison between the two. For the purpose of demonstration, we choose a scenario which can accommodate the recently observed 95 GeV excess in both models. We further explore the impact of the measurement of the Yukawa couplings, the gauge boson couplings and most importantly the trilinear self-couplings of the scalars, in distinguishing the vacuum structure in both models. We further investigate the constraints from vacuum stability on the 2HDMS scenario that accommodates a viable dark matter candidate and compare it with the N2HDM case.

Domain walls are a type of topological defects that can arise in the early universe after the spontaneous breaking of a discrete symmetry. They can form in several beyond the Standard Model theories with an extended Higgs sector such as the Next to-Two-Higgs-Doublet model (N2HDM). In this work, we discuss the domain wall solution related to the singlet scalar of the N2HDM and demonstrate the possibility of restoring the electroweak symmetry inside and in the vicinity of the domain wall. Such symmetry restoration can have profound implications on early universe cosmology as the weak sphaleron rate inside the domain wall would, in principle, be unsuppressed compared to the rate outside the wall. We also discuss the possibility of generating CP-violating vacua localized in the vicinity of the domain wall. Our work is a first step towards the realization of electroweak baryogenesis mediated by domain walls in the N2HDM.

We study possible CP-violation effects of the 125 GeV Higgs to $Z$ boson coupling at the 250 GeV ILC with transverse and longitudinal beam polarisation via the process $e^+ e^- \rightarrow HZ \rightarrow H \mu^-\mu^+$. We explore the azimuthal angular distribution of the muon pair from the $Z$ boson decay, and constructe CP-odd observables sensitive to CP-violation effects, where we derived this observable both by analytical calculations and by $\mathtt{Whizard}$ simulations. Particularly, we can construct two CP-odd observables with the help of transversely-polarised initial beams and improve the statistical significance of CP-violation effects by combining two measurements. We defined the asymmetries between the signal regions with different signs of the CP-odd observables, and determine the CP-violation effect by comparing with the SM 95% C.L. upper bound. In this paper, we setup a scenario which assumes that the total cross-section is always fixed while CP-violation is varying, and such a scenario helps us to determine the intrinsic CP-mixing angle limit around $|\xi_{CP}|\sim 0.03$ with (90%, 40%) polarised electron-positron beams and 5 ab$^{-1}$ integrated luminosity. In addition, we determine the CP-odd coupling limit $|\widetilde{c}_{HZZ}|\sim 0.01$ as well, where we suppose that the SM tree-level cross-section is fixed and the CP-violation is the varying additional contribution. Comparing with the analysis with unpolarised beams, the sensitivity to the CP-violation effect can be improved by transverse or longitudinal polarisation.

In order to stimulate new engagement and trigger some concrete studies in areas where further work would be beneficial towards fully understanding the physics potential of an $e^+e^-$ Higgs / Top / Electroweak factory, we propose to define a set of focus topics. The general reasoning and the proposed topics are described in this document.

We study possible CP-violation effects of the Higgs to $Z$-boson coupling at a future $e^+ e^-$ collider, e.g. the International Linear Collider (ILC). We find that the azimuthal angular distribution of the muon pair, produced by $e^+ e^- \rightarrow H Z \rightarrow H \mu^+ \mu^-$, can be sensitive to such a CP-violation effect when we apply initial transversely polarized beams. Based on this angular distribution, we construct a CP sensitive asymmetry and obtain this asymmetry by \textttWhizard simulation. By comparing the SM prediction with 2$\sigma$ range of this asymmetry, we estimate the discovery limit of the CP-odd coupling in $HZZ$ interaction.

In several models of beyond Standard Model physics (BSM) discrete symmetries play an important role. For instance, in order to avoid flavor changing neutral currents (FCNC), a discrete $Z_2$ symmetry is imposed on Two-Higgs-Doublet-Models (2HDM). This can lead to the formation of domain walls (DW) as the $Z_2$ symmetry gets spontaneously broken during electroweak symmetry breaking (EWSB) in the early universe. Due to this simultaneous spontaneous breaking of both the discrete symmetry and the electroweak symmetry, the vacuum manifold has the structure of two disconnected 3-spheres and the formed domain walls can exhibit several special properties in contrast to standard domain walls. We focus on some of these properties such as CP and electric charge violating vacua localized inside the domain walls. The breaking of $U(1)_{em}$ inside the wall leads to the known phenomenon of "clash-of-symmetries" mechanism, meaning that the symmetry group inside the wall is smaller than the symmetry group far from the wall. We also discuss the scattering of top quarks off such types of domain walls and show, for example, that they can be reflected or transmitted off the wall as a bottom quark.

Many different approaches have been made to explain the nature of dark matter (DM), but it remains and unsolved mystery of our universe. In this work we examine a type II two-Higgs-doublet model extended by a complex singlet (2HDMS), where the pseudo-scalar component of the singlet acts as a natural DM candidate. The DM candidate is stabilized by a Z'2 symmetry, which is broken spontaneously by the singlet acquiring a vacuum expectation value (vev). This vev in turn causes the scalar component of the singlet to mix with the scalar components of the two doublets, which results in three scalar Higgs particles. Additionally we aim to include an excess around 95 GeV, which was observed at CSM and LEP and can be explained by one of the three scalar Higgs particles. After introducing the model, we apply experimental and theoretical constraints and find a viable benchmark point. We then look into the DM phenomenology as well as collider phenomenology.

Discrete symmetries play an important role in several extensions of the Standard Model (SM) of particle physics. For instance, in order to avoid flavor changing neutral currents, a discrete $Z_2$ symmetry is imposed on the Two-Higgs-Doublet Model (2HDM). This can lead to the formation of domain walls (DW) as the $Z_2$ symmetry gets spontaneously broken during electroweak symmetry breaking in the early universe and domain walls form between regions whose vacua are related by the discrete symmetry. Due to this simultaneous spontaneous breaking of both the discrete symmetry and the electroweak symmetry, the vacuum manifold consists of two disconnected 3-spheres. Such a non-trivial disconnected vacuum manifold leads to several choices for the vacua at two adjacent regions, in contrast to models where only the discrete symmetry gets spontaneously broken and the vacuum manifold consists of several disconnected points. Due to this, we end up with several classes of DW solutions having different properties localized inside the wall, such as charge and/or CP violating vacua. We discuss the properties of these different classes of DW solutions as well as the interaction of SM fermions with such topological defects leading to different exotic phenomena such as, for example, the top quark being transmitted or reflected off the wall as a bottom quark.

The Two-Higgs-Doublet-Standard Model-Axion-Seesaw-Higgs-Portal inflation (2hdSMASH) model consisting of two Higgs doublets, a Standard Model (SM) singlet complex scalar and three SM singlet right-handed neutrinos can embed axion dark matter, neutrino masses and address inflation. We report on an investigation of the inflationary aspects of 2hdSMASH and its subsequent impact on low energy phenomenology. In particular, we identify inflationary directions for which the parameter values required for successful inflation do not violate perturbative unitarity and boundedness-from-below conditions. By analyzing the renormalization-group flow of the parameters we identify the necessary and sufficient constraints for running all parameters perturbatively and maintaining stability from the electroweak to the PLANCK scale. We observe that stringent constraints arise on the singlet scalar self coupling from inflationary constraints, i.e, $\lambda_S\sim 10^{-10}$. Further, we find that all theoretical and experimental constraints are satisfied if the portal couplings are typically in the range $(\frac{v}{v_S})$ and $(\frac{v}{v_S})^2$ (where $v, v_S$ refer to the electroweak and singlet scalar vacuum expectation value respectively). As a consequence, inflation is realized in a variety of field space directions in the effective single field regime. Finally we provide testable benchmark scenarios at colliders.

Several designs for high-energy Lepton Colliders serving as Higgs factories but extendable to higher energies up to the TeV range are under discussion. The most mature design is the International Linear Collider (ILC), but also the Compact Linear Collider (CLIC) as well as the new concept of a Hybrid Asymmetric Linear Higgs Factory (HALHF) have a large physics potential. The first energy stage with $\sqrt{s}=250$~GeV requires high luminosity and polarized beams and imposes an effort for all positron source designs at high-energy colliders. In the baseline design of the ILC, an undulator-based source is foreseen for the positron source in order to match the physics requirements. In this contribution an overview is given about the undulator-based source, the target tests, the rotating target wheel design, as well as the pulsed solenoid and the new technology development of plasma lenses as optic matching devices.

The Two Higgs Doublet model extended with a complex scalar singlet (2HDMS) is a well-motivated Beyond Standard Model candidate addressing several open problems of nature. In this work, we focus on the dark matter (DM) phenomenology of the complex scalar singlet where the real part of the complex scalar obtains a vacuum expectation value. The model is characterized by an enlarged Higgs spectrum comprising six physical Higgs bosons and a pseudoscalar DM candidate. We address the impact of accommodating the 95 GeV excess on the 2HDMS parameter space and DM observables after including all theoretical and experimental constraints. Finally, we look into the prospects of this scenario at HL-LHC and future lepton colliders for a representative benchmark.

Heterodyne cavity experiments for gravitational wave (GW) detection experience a rising interest since recent studies showed that they allow to probe the ultra high frequency regime above $10\,\text{kHz}$. In this paper, we present a concise theoretical study of the experiment based on ideas from the former MAGO collaboration which already started experiments in turn of the millenium. It extends the former results via deriving an additional term originating from a back-action of the electromagnetic field on the cavity walls, also known as Lorentz Force Detuning. We argue that this term leads to a complex dependence of the signal power $P_{\text{sig}}$ on the coupling coefficient between the mechanical shell modes and the electromagnetic eigenmodes of the cavity. It turns out that one has to adapt the coupling over the whole parameter space since the optimal value depends on the mechanical mode $\omega_l$ and the GW frequency $\omega_g$. This result is particularly relevant for the design of future experiments.

The constituents of dark matter are still an unresolved puzzle. Several Beyond Standard Model (BSM) Physics offer suitable candidates. In this study here we consider the Two Higgs Doublet model augmented with a complex scalar singlet (2HDMS) and focus on the dark matter phenomenology of 2HDMS with the complex scalar singlet as the dark matter candidate. The parameter space allowed from existing experimental constraints from dark matter, flavour physics and collider searches has been studied. The discovery potential for such a 2HDMS at HL-LHC and at future $e^+e^-$ colliders has been worked out.

The International Linear Collider (ILC) is on the table now as a new global energy-frontier accelerator laboratory taking data in the 2030s. The ILC addresses key questions for our current understanding of particle physics. It is based on a proven accelerator technology. Its experiments will challenge the Standard Model of particle physics and will provide a new window to look beyond it. This document brings the story of the ILC up to date, emphasizing its strong physics motivation, its readiness for construction, and the opportunity it presents to the US and the global particle physics community.

The Two Higgs Doublet model augmented with a complex scalar singlet (2HDMS) is a well-motivated candidate for Beyond Standard Model (BSM) Physics. We investigate the dark matter phenomenology of the 2HDMS with the complex scalar singlet as the dark matter candidate. We perform a study of the parameter space allowed by existing experimental constraints from dark matter, flavour physics and collider searches. The distinction between real and complex scalar dark matter in the context of the 2HDMS is also performed. Furthermore, we discuss the discovery potential for the 2HDMS at the HL-LHC and at future high-energy $e^+e^-$ colliders.

We discuss a $\sim 3\,\sigma$ signal (local) in the light Higgs-boson search in the diphoton decay mode at $\sim 96$ GeV as reported by CMS, together with a $\sim 2\,\sigma$ excess (local) in the $b \bar b$ final state at LEP in the same mass range. We interpret this possible signal as a Higgs boson in the 2 Higgs Doublet Model type II with an additional Higgs singlet, which can be either complex (2HDMS) or real (N2HDM). We find that the lightest CP-even Higgs boson of the two models can equally yield a perfect fit to both excesses simultaneously, while the second lightest state is in full agreement with the Higgs-boson measurements at $125$ GeV, and the full Higgs-boson sector is in agreement with all Higgs exclusion bounds from LEP, the Tevatron and the LHC as well as other theoretical and experimental constraints. We derive bounds on the 2HDMS and N2HDM Higgs sectors from a fit to both excesses and describe how this signal can be further analyzed at future $e^+e^-$ colliders, such as the ILC. We analyze in detail the anticipated precision of the coupling measurements of the $96$ GeV Higgs boson at the ILC. We find that these Higgs-boson measurements at the LHC and the ILC cannot distinguish between the two Higgs-sector realizations.

The CMS collaboration reported a $\sim 3 \, \sigma$ (local) excess at $96\;$GeV in the search for light Higgs-boson decaying into two photons. This mass coincides with a $\sim 2 \, \sigma$ (local) excess in the $b\bar b$ final state at LEP. We show an interpretation of these possible signals as the lightest Higgs boson in the 2 Higgs Doublet Model with an additional complex Higgs singlet (2HDMS). The interpretation is in agreement with all experimental and theoretical constraints. We concentrate on the 2HDMS type II, which resembles the Higgs and Yukawa structure of the Next-to Minimal Supersymmetric Standard Model. We discuss the experimental prospects for constraining our explanation at future $e^+e^-$ colliders, with concrete analyses based on the ILC prospects.

The current challenges in High Energy Physics and Cosmology are to build coherent particle physics models to describe the phenomenology at colliders in the laboratory and the observations in the universe. From these observations, the existence of an inflationary phase in the early universe gives guidance for particle physics models. We study a supersymmetric model which incorporates successfully inflation by a non-minimal coupling to supergravity and shows a unique collider phenomenology. Motivated by experimental data, we set a special emphasis on a new singlet-like state at 97 GeV and single out possible observables for a future linear collider that permit a distinction of the model from a similar scenario without inflation. We define a benchmark scenario that is in agreement with current collider and Dark Matter constraints, and study the influence of the non-minimal coupling on the phenomenology. Measuring the singlet-like state with high precision on the percent level seems to be promising for resolving the models, even though the Standard Model-like Higgs couplings deviate only marginally. However, a hypothetical singlet-like state with couplings of about 20% compared to a Standard Model Higgs at 97 GeV encourages further studies of such footprint scenarios of inflation.

In the baseline design of the International Linear Collider (ILC) an undulator-based source is foreseen for the positron source in order to match the physics requirements. The recently chosen first energy stage with sqrt(s)=250 GeV requires high luminosity and imposes an effort for all positron source designs at high-energy colliders. In this paper we perform a simulation study and adopt the new technology of plasma lenses to capture the positrons generated by the undulator photons and to create the required high luminosity positron beam.

The Next-to-Minimal Supersymmetric Standard Model (NMSSM) can incorporate inflation, where a combination of the Higgs-doublet fields plays the role of the inflaton. At the high scale, the Higgs doublets are non-minimally coupled to supergravity; this coupling appears as an additional contribution to the $\mu$ term in the low-energy effective superpotential and potentially changes physics at the electroweak scale. In a recent publication, we investigate the extended parameter space of this model with respect to collider phenomenology at the electroweak scale, and discuss scenarios which are potentially different from the pure NMSSM. We analyse the stability of the electroweak vacuum, the masses of neutralinos/charginos and Higgs bosons as well as the mixing and decays of Higgs bosons. Some important aspects of this study are described in the following.

Little Higgs models - which can most easily be thought of as a variant of composite Higgs models - explain a light Higgs boson at 125 GeV as an pseudo-Nambu-Goldstone boson of a spontaneously broken global symmetry. The mechanism of collective symmetry breaking shifts the UV scale of these models to the 10 TeV scale and higher. T-parity is introduced as a discrete symmetry to remove tree-level constraints on the electroweak precision data. Still after run 1 of LHC, electroweak precision observables gave stronger constraints than Higgs data and direct searches. We present a full recast of all available 13 TeV searches from LHC run 2 to show that now direct searches supersede electroweak precision observables. The latest exclusion limits on the LHT model will be presented, as well as an outlook on the full high-luminosity phase of LHC.

A high-energy $e^+e^-$ Linear Collider has been considered since a long time as an important complement to the LHC. Unprecedented precision measurements as well as the exploration of so far untouched phase space for direct production of new particles will provide unique information to advance the limits of our understanding of our universe. Within this project, the physics prospects of such a collider as well as their interplay with design of the accelerator and the detectors have been investigated in a quantitative way. This kind of study required a close collaboration between theory and experiment, always taking into account results of the LHC and other relevant experiments. In this article we will summarize some of the most important developments and results, covering all core areas of the physics progamme of future $e^+e^-$ colliders.

The concept of Higgs inflation can be elegantly incorporated in the Next-to-Minimal Supersymmetric Standard Model (NMSSM). A linear combination of the two Higgs-doublet fields plays the role of the inflaton which is non-minimally coupled to gravity. This non-minimal coupling appears in the low-energy effective superpotential and changes the phenomenology at the electroweak scale. While the field content of the inflation-inspired model is the same as in the NMSSM, there is another contribution to the $\mu$ term in addition to the vacuum expectation value of the singlet. We explore this extended parameter space and point out scenarios with phenomenological differences compared to the pure NMSSM. A special focus is set on the electroweak vacuum stability and the parameter dependence of the Higgs and neutralino sectors. We highlight regions which yield a SM-like $125\,$GeV Higgs boson compatible with the experimental observations and are in accordance with the limits from searches for additional Higgs bosons. Finally, we study the impact of the non-minimal coupling to gravity on the Higgs mixing and in turn on the decays of the Higgs bosons in this model.

The particle discovered in the Higgs boson searches at the LHC with a mass of about 125 GeV is compatible within the present uncertainties with the Higgs boson predicted in the Standard Model (SM), but it could also be identified with one of the neutral Higgs bosons in a variety of Beyond the SM (BSM) theories with an extended Higgs sector. The possibility that an additional Higgs boson (or even more than one) could be lighter than the state that has been detected at 125 GeV occurs generically in many BSM models and has some support from slight excesses that were observed above the background expectations in Higgs searches at LEP and at the LHC. The couplings between additional Higgs fields and the electroweak gauge bosons in BSM theories could be probed by model-independent Higgs searches at lepton colliders. We present a generator-level extrapolation of the limits obtained at LEP to the case of a future $e^+e^-$ collider, both for the search where the light Higgs boson decays into a pair of bottom quarks and for the decay-mode-independent search utilising the recoil method. We find that at the ILC with a c.m. energy of 250 GeV, an integrated luminosity of 500 fb^-1 and polarised beams, the sensitivity to a light Higgs boson with reduced couplings to gauge bosons is improved by more than an order of magnitude compared to the LEP limits and goes much beyond the projected indirect sensitivity of the HL-LHC with 3000 fb^-1 from the rate measurements of the detected state at 125 GeV.

The public collider phenomenology computing tool CheckMATE (Check Models at Terascale Energies) was originally designed to allow theorists to quickly test their favourite BSM models against various existing LHC analyses performed by ATLAS and CMS. It offers an automatised chain of Monte Carlo event generation, detector simulation, event analysis and statistical evaluation so that it can automatically determine whether a given parameter point of a BSM model is excluded or not. Currently, it contains more than 50 individual ATLAS or CMS analyses whose several hundred signal regions target various final states as they typically appear in theories beyond the Standard Model. In this study, we extend this functionality to allow sensitivity studies for the International Linear Collider. As an example, we implemente a dark matter monophoton search and use it to analyse three benchmark scenarios with different assumptions about the interaction between dark matter and Standard Model particles. We determine the ILC sensitivity expected for a $\sqrt{s} =$ 500 GeV, $L = 500$ fb$^{-1}$and compare the results for the cases of completely unpolarised beams and for individual lepton polarisation settings.

We exploit all LHC available Run 2 data at center-of-mass energies of 8 and 13 TeV for searches for physics beyond the Standard Model. We scrutinize the allowed parameter space of Little Higgs models with the concrete symmetry of T-parity by providing comprehensive analyses of all relevant production channels of heavy vectors, top partners, heavy quarks and heavy leptons and all phenomenologically relevant decay channels. Constraints on the model will be derived from the signatures of jets and missing energy or leptons and missing energy. Besides the symmetric case, we also study the case of T-parity violation. Furthermore, we give an extrapolation to the LHC high-luminosity phase at 14 TeV as well.

The International Linear Collider is now proposed with a staged machine design, with the first stage at $\sqrt{s}=$~250 GeV and an integrated luminosity goal of 2~ab$^{-1}$. One of the questions for the machine design is the importance of positron polarization. In this report, we review the impact of positron polarization on the physics goals of the $250$ GeV stage of the ILC and demonstrate that positron polarization has distinct advantages.

We study the tunneling of virtual matter-antimatter pairs from the quantum vacuum in the presence of a spatially uniform, time-dependent electric background composed of a strong, slow field superimposed with a weak, rapid field. After analytic continuation to Euclidean spacetime, we obtain from the instanton equations two critical points. While one of them is the closing point of the instanton path, the other serves as an Euclidean mirror which reflects and squeezes the instanton. It is this reflection and shrinking which is responsible for an enormous enhancement of the vacuum pair production rate. We discuss how important features of two different mechanisms can be analysed and understood via such a rotation in the complex plane. a) Consistent with previous studies, we first discuss the standard assisted mechanism with a static strong field and certain weak fields with a distinct pole structure in order to show that the reflection takes place exactly at the poles. We also discuss the effect of possible sub-cycle structures. We extend this reflection picture then to weak fields which have no poles present and illustrate the effective reflections with explicit examples. An additional field strength dependence for the rate occurs in such cases. We analytically compute the characteristic threshold for the assisted mechanism given by the critical combined Keldysh parameter. We discuss significant differences between these two types of fields. For various backgrounds, we present the contributing instantons and perform analytical computations for the corresponding rates treating both fields nonperturbatively. b) In addition, we also study the case with a nonstatic strong field which gives rise to the assisted dynamical mechanism. For different strong field profiles we investigate the impact on the critical combined Keldysh parameter. [...]

Nonlinear phenomena of lepton-photon interactions in external backgrounds with a generalised periodic plane-wave geometry are studied. We discuss nonlinear Compton scattering in head-on lepton-photon collisions extended properly to beyond the soft-photon regime. In addition, our results are applied to stimulated lepton-antilepton pair production in photon collisions with unrestricted energies. Derivations are considered semi-classically based on unperturbed fermionic Volkov representations encoding the full interaction with the background field. Closed expressions for total probabilities considering S-matrix elements have been derived. The general formula is applied to Compton scattering by an electron propagating in an external laser-like background. We obtain additive contributions in the extended unconstrained result which turns out to be stringently required in the highly nonlinear regime. A detailed comparison of contributing harmonics is discussed for various field parameters.

We exploit offshell regions in the process $e^+e^-\rightarrow W^+W^-b\bar{b}$ to gain access to the top-quark width. Working at next-to-leading order in QCD we show that carefully selected ratios of offshell regions to onshell regions in the reconstructed top and antitop invariant mass spectra are, \emphindependently of the coupling $g_{tbW}$, sensitive to the top-quark width. We explore this approach for different centre of mass energies and initial-state beam polarisations at $e^+e^-$ colliders and briefly comment on the applicability of this method for a measurement of the top-quark width at the LHC.

A comprehensive review of physics at an e+e- Linear Collider in the energy range of sqrts=92 GeV--3 TeV is presented in view of recent and expected LHC results, experiments from low energy as well as astroparticle physics.The report focuses in particular on Higgs boson, Top quark and electroweak precision physics, but also discusses several models of beyond the Standard Model physics such as Supersymmetry, little Higgs models and extra gauge bosons. The connection to cosmology has been analyzed as well.

The importance of off-shell contributions is discussed for $H\to VV^{(*)}$ with $V\in\{Z,W\}$ for large invariant masses $m_{VV}$ involving a standard model (SM)-like Higgs boson with $m_H=125$GeV at a linear collider (LC). Both dominant production processes $e^+e^-\to ZH\to ZVV^{(*)}$ and $e^+e^-\to\nu\bar\nu H\to\nu\bar\nu VV^{(*)}$ are taken into account, and the signal processes are compared with background yielding the same final state. The relative size of the off-shell contributions is strongly dependent on the centre-of-mass energy. These contributions can have an important impact on the determination of cross sections and branching ratios. However, the combination of on- and off-shell contributions can also be utilised to lift degeneracies allowing to test higher-dimensional operators, unitarity and light and heavy Higgs interferences in extended Higgs sectors. The latter is demonstrated in the context of a 2-Higgs-Doublet model. We also discuss the impact of these aspects for the Large Hadron Collider (LHC) where they are relevant. The importance of a precise measurement of the Higgs mass for on-shell contributions in $H\to VV^{(*)}$ is emphasized. A particular focus is put on methods for extracting the Higgs width at a LC. Off-shell contributions are shown to have a negligible impact on the width determination at low $\sqrt{s}$ when applying the $Z$ recoil method to extract branching ratios in combination with an appropriate determination of a partial width. On the other hand, off-shell contributions can be exploited to constrain the Higgs width in a similar fashion as in recent analyses at the LHC. It is demonstrated that this approach, besides relying heavily on theoretical assumptions, is affected by the negative interference of Higgs and background contributions that may limit the sensitivity that is achievable with the highest foreseeable statistics at the LHC and a LC.

Non-decoupling D-term extensions of the MSSM enhance the tree-level Higgs mass compared to the MSSM, therefore relax fine-tuning and may allow lighter stops with rather low masses even without maximal mixing. We present the anatomy of various non-decoupling D-term extensions of the MSSM and explore the potential of the LHC and of the International Linear Collider (ILC) to determine their deviations in the Higgs couplings with respect to the Standard Model. Depending on the mass of the heavier Higgs $m_H$, such deviations may be constrained at the LHC and determined at the ILC. We evaluate the Higgs couplings in different models and study the prospects for a model distinction at the different stages of the ILC at $\sqrt{s}=$250, 500, 1000 GeV, including the full luminosity upgrade and compare it with the prospects at HL-LHC.

It is one of the most challenging tasks at the Large Hadron Collider and at a future Linear Collider not only to observe physics beyond the Standard Model, but to clearly identify the underlying new physics model. In this paper we concentrate on the distinction between two different supersymmetric models, the MSSM and the NMSSM, as they can lead to similar low energy spectra. The NMSSM adds a singlet superfield to the MSSM particle spectrum and simplifies embedding a SM-like Higgs candidate with the measured mass of about 125.5 GeV. In parts of the parameter space the Higgs sector itself does not provide sufficient indications for the underlying model. We show that exploring the gaugino/higgsino sectors could provide a meaningful way to distinguish the two models. Assuming that only the lightest chargino and neutralino masses and polarized cross sections $e^+e^-\to \tilde{\chi}^0_i\tilde{\chi}^0_j$, $\tilde{\chi}^+_i\tilde{\chi}^-_j$ are accessible at the linear collider, we reconstruct the fundamental MSSM parameters $M_1$, $M_2$, $\mu$, $\tan\beta$ and study whether a unique model distinction is possible based on this restricted information. Depending on the singlino admixture in the lightest neutralino states, as well as their higgsino or gaugino nature, we define several classes of scenarios and study the prospects of experimental differentiation.

One of the challenging tasks at future experiments is the clear identification of the underlying new physics model. In this study we concentrate on the distinction between different supersymmetric models, the MSSM and the NMSSM, exploring the gaugino/higgsino sector as an alternative to the Higgs sector. Under the assumption that only the light chargino and neutralino masses and polarized cross sections $e^+e^-\to \tilde{\chi}^0_i\tilde{\chi}^0_j$, $\tilde{\chi}^+_i\tilde{\chi}^-_j$ have been measured, we perform a fit of the fundamental MSSM parameters $M_1$, $M_2$, $\mu$ and $\tan\beta$ and study whether a model distinction is possible. We focus here on the challenging cases of scenarios with a relatively heavy singlino and address two classes of neutralino mixing, $\tilde{\chi}^0_1\sim$higgsino-like versus $\tilde{\chi}^0_1\sim$gaugino-like.

The future linear collider will collide dense $e^+e^-$ bunches at high energies up to 1 TeV, generating very intense electromagnetic fields at the interaction point (IP). These fields are strong enough to lead to nonlinear effects which affect all IP processes and which are described by strong field physics theory. In order to test this theory, we propose an experiment that will focus an intense laser on the LC electron beam post-IP. Similar experiments at SLAC E144 have investigated nonlinear Compton scattering, Breit-Wheeler pair production using an electron beam of 46.6 GeV. The higher beam energies available at the future LC would allow more precise studies of these phenomena. Mass-shift and spin-dependent effects could also be investigated.

This report summarizes the work of the Energy Frontier Higgs Boson working group of the 2013 Community Summer Study (Snowmass). We identify the key elements of a precision Higgs physics program and document the physics potential of future experimental facilities as elucidated during the Snowmass study. We study Higgs couplings to gauge boson and fermion pairs, double Higgs production for the Higgs self-coupling, its quantum numbers and $CP$-mixing in Higgs couplings, the Higgs mass and total width, and prospects for direct searches for additional Higgs bosons in extensions of the Standard Model. Our report includes projections of measurement capabilities from detailed studies of the Compact Linear Collider (CLIC), a Gamma-Gamma Collider, the International Linear Collider (ILC), the Large Hadron Collider High-Luminosity Upgrade (HL-LHC), Very Large Hadron Colliders up to 100 TeV (VLHC), a Muon Collider, and a Triple-Large Electron Positron Collider (TLEP).

With the discovery of the Higgs boson, the spectrum of particles in the Standard Model (SM) is complete. It is more important than ever to perform precision measurements and to test for deviations from SM predictions in the electroweak sector. In this report, we investigate two themes in the arena of precision electroweak measurements: the electroweak precision observables (EWPOs) that test the particle content and couplings in the SM and the minimal supersymmetric SM, and the measurements involving multiple gauge bosons in the final state which provide unique probes of the basic tenets of electroweak symmetry breaking. Among the important EWPOs we focus our discussion on M_W and sin^2 theta_eff^l, and on anomalous quartic gauge couplings probed by triboson production and vector boson scattering. We investigate the thresholds of precision that need to be achieved in order to be sensitive to new physics. We study the precision that can be achieved at various facilities on these observables. We discuss the calculational tools needed to predict SM rates and distributions in order to perform these measurements at the required precision. This report summarizes the work of the Energy Frontier Precision Study of Electroweak Interactions working group of the 2013 Community Summer Study (Snowmass).

This paper summarizes the physics potential of the CLIC high-energy e+e- linear collider. It provides input to the Snowmass 2013 process for the energy-frontier working groups on The Higgs Boson (HE1), Precision Study of Electroweak Interactions (HE2), Fully Understanding the Top Quark (HE3), as well as The Path Beyond the Standard Model -- New Particles, Forces, and Dimensions (HE4). It is accompanied by a paper describing the CLIC accelerator study, submitted to the Frontier Capabilities group of the Snowmass process.

In supersymmetric extensions of the Standard Model, higgsino-like charginos and neutralinos are preferred to have masses of the order of the electroweak scale by naturalness arguments. Such light $\widetilde{\chi}^0_1$, $\widetilde{\chi}^0_2$ and $\widetilde{\chi}^{\pm}_1$ states can be almost mass degenerate, and their decays are then difficult to observe at colliders. In addition to the generic naturalness argument, light higgsinos are well motivated from a top-down perspective. For instance, they arise naturally in certain models of hybrid gauge-gravity mediation. In the present analysis, we study two benchmark points which have been derived in the framework of such a model, which exhibit mass differences of O(GeV) in the higgsino sector. For chargino pair and neutralino associated production with initial-state photon radiation, we simulate the detector response and determine how accurately the small mass differences, the absolute masses and the cross sections can be measured at the International Linear Collider. Assuming that 500/fb has been collected at each of two beam-polarisations $P(e^+,e^-)=(\pm 30\%,\mp 80\%)$, we find that the mass differences can be measured to 40-300 MeV, the cross sections to 2-5%, and the absolute masses to 1.5-3.3 GeV, where the range of values correspond to the different scenarios and channels. Based on these observables we perform a parameter fit in the MSSM, from which we infer that the higgsino mass parameter $\mu$ can be measured to a precision of about $\Delta\mu=$2--7 GeV. For the electroweak gaugino mass parameters $M_1$, $M_2$, which are chosen in the multi-TeV range, a narrow region is compatible with the measurements. For both parameters independently, we can determine a lower bound.

Future linear colliders designs, ILC and CLIC, are expected to be powerful machines for the discovery of Physics Beyond the Standard Model and subsequent precision studies. However, due to the intense beams (high luminosity, high energy), strong electromagnetic fields occur in the beam-beam interaction region. In the context of precision high energy physics, the presence of such strong fields may yield sensitive corrections to the observed electron-positron processes. The Furry picture of quantum states gives a conceptually simple tool to treat physics processes in an external field. A generalization of the quasi-classical operator method (QOM) as an approximation is considered too.

Future lepton colliders will be precision machines whose physics program includes close study of the Higgs sector and searches for new physics via polarised beams. The luminosity requirements of such machines entail very intense lepton bunches at the interaction point with associated strong electromagnetic fields. These strong fields not only lead to obvious phenomena such as beamstrahlung, but also potentially affect every particle physics process via virtual exchange with the bunch fields. For precision studies, strong field effects have to be understood to the sub-percent level. Strong external field effects can be taken into account exactly via the Furry Picture or, in certain limits, via the Quasi-classical Operator method . Significant theoretical development is in progress and here we outline the current state of play.

We explore the effects of neutrino and electron mixing with exotic heavy leptons in the process e^+e^-\to W^+W^- within E_6 models. We examine the possibility of uniquely distinguishing and identifying such effects of heavy neutral lepton exchange from Z-Z' mixing within the same class of models and also from analogous ones due to competitor models with anomalous trilinear gauge couplings (AGC) that can lead to very similar experimental signatures at the e^+e^- International Linear Collider (ILC) for \sqrts=350, 500 GeV and 1 TeV. Such clear identification of the model is possible by using a certain double polarization asymmetry. The availability of both beams being polarized plays a crucial role in identifying such exotic-lepton admixture. In addition, the sensitivity of the ILC for probing exotic-lepton admixture is substantially enhanced when the polarization of the produced W^\pm bosons is considered.

At a future linear collider, very precise measurements, typically with errors of <1%, are expected to be achievable. Such an accuracy yields sensitivity to quantum corrections, which therefore must be incorporated into theoretical calculations in order to determine the underlying new physics parameters from linear collider measurements. In the context of the chargino--neutralino sector of the minimal supersymmetric standard model, this involves fitting one-loop predictions to prospective measurements of cross sections, forward-backward asymmetries and the accessible chargino and neutralino masses. Taking recent results from LHC SUSY and Higgs searches into account, we consider three phenomenological scenarios, each displaying characteristic features. Our analysis demonstrates how an accurate determination of the desired parameters is possible, and could additionally provide access to the stop masses and mixing angle.

We extend the formalism developed in Ref. [20] for the renormalisation of the chargino-neutralino sector to the most general case of the MSSM with complex parameters. We show that products of imaginary parts arising from MSSM parameters and from absorptive parts of loop integrals can already contribute to predictions for physical observables at the one-loop level, and demonstrate that the consistent treatment of such contributions gives rise to non-trivial structure, either in the field renormalisation constants or the corrections associated with the external legs of the considered diagrams. We furthermore point out that the phases of the parameters in the chargino-neutralino sector do not need to be renormalised at the one-loop level, and demonstrate that the appropriate choice for the mass parameters used as input for the on-shell conditions depends both on the process and the region of MSSM parameter space under consideration. As an application, we compute the complete one-loop results in the MSSM with complex parameters for the process h(a) to chi(i)+chi(j)- (Higgs-propagator corrections have been incorporated up to the two-loop level), which may be of interest for SUSY Higgs searches at the LHC, and for chargino pair-production at an e+e- Linear Collider, e+e- to chi(i)+chi(j)-. We investigate the dependence of the theoretical predictions on the phases of the MSSM parameters, analysing in particular the numerical relevance of the absorptive parts of loop integrals.

Very precise measurements of masses and cross sections, with errors of <1%, are expected to be achievable with a future linear collider. Such an accuracy gives sensitivity at the level of quantum corrections, which therefore must be incorporated in order to make meaningful predictions for the underlying new physics parameters. For the chargino-neutralino sector, this involves fitting one-loop predictions to expected measurements of the cross sections and forward-backward asymmetries and of the accessible chargino and neutralino masses. Our analysis shows how an accurate determination of the desired parameters is possible, providing in addition access to the mass of the lighter stop.

New heavy neutral gauge bosons Z' are predicted by many models of physics beyond the Standard Model. It is quite possible that Z's are heavy enough to lie beyond the discovery reach of the CERN Large Hadron Collider LHC, in which case only indirect signatures of Z' exchanges may emerge at future colliders, through deviations of the measured cross sections from the Standard Model predictions. We discuss in this context the foreseeable sensitivity to Z's of W^\pm-pair production cross sections at the e^+e^- International Linear Collider (ILC), especially as regards the potential of distinguishing observable effects of the Z' from analogous ones due to competitor models with anomalous trilinear gauge couplings (AGC) that can lead to the same or similar new physics experimental signatures at the ILC. The sensitivity of the ILC for probing the Z-Z' mixing and its capability to distinguish these two new physics scenarios is substantially enhanced when the polarization of the initial beams and the produced W^\pm bosons are considered. A model independent analysis of the Z' effects in the process e^+e^- \to W^+W^- allows to differentiate the full class of vector Z' models from those with anomalous trilinear gauge couplings, with one notable exception: the sequential SM (SSM)-like models can in this process not be distinguished from anomalous gauge couplings. Results of model dependent analysis of a specific Z' are expressed in terms of discovery and identification reaches on the Z-Z' mixing angle and the Z' mass.

Supersymmetric models provide many new complex phases which lead to CP violating effects in collider experiments. As an example, CP-sensitive triple product asymmetries in neutralino production and subsequent leptonic two-body decays are studied within the Minimal Supersymmetric Standard Model. A full ILD detector simulation has been performed at a center of mass energy of 500GeV, including the relevant Standard Model background processes, a realistic beam energy spectrum, beam backgrounds and a beam polarization of 80% and -60% for the electron and positron beams, respectively. Assuming an integrated luminosity of 500fb-1 collected by the experiment and the performance of the current ILD detector, a relative measurement accuracy of 10% for the CP-sensitive asymmetry can be achieved in the chosen scenario.

We study the prospects to measure the CP-sensitive triple-product asymmetries in neutralino production e+e- -> ~chi^0_i ~chi^0_1 and subsequent leptonic two-body decays ~chi^0_i -> ~l_R l, ~l_R -> ~chi^0_1 l, for l=e, mu, within the Minimal Supersymmetric Standard Model. We include a full detector simulation of the International Large Detector for the International Linear Collider. The simulation was performed at a center-of-mass energy of sqrts=500 GeV, including the relevant Standard Model background processes, a realistic beam energy spectrum, beam backgrounds and a beam polarization of 80% and -60% for the electron and positron beams, respectively. In order to effectively disentangle different signal samples and reduce SM and SUSY backgrounds we apply a method of kinematic reconstruction. Assuming an integrated luminosity of 500 fb^-1 collected by the experiment and the performance of the current ILD detector, we arrive at a relative measurement accuracy of 10% for the CP-sensitive asymmetry in our scenario. We demonstrate that our method of signal selection using kinematic reconstruction can be applied to a broad class of scenarios and it allows disentangling processes with similar kinematic properties.

If signals of new physics are discovered at the LHC it will be crucial to determine the spin structure of the new model. We discuss a method that can help to address this question with a low integrated luminosity, L=1 fb^-1, at sqrts=14 TeV. Based on the differences in angular distributions of primarily produced particles we show that a significant difference can be observed in the final state jet-pairs rapidity distance. An additional advantage of the method is that it does not rely on any particular structure of the couplings in the decay chain. We simulate samples for models with supersymmetric and UED-like spin structure and show that a distinction can be made early on.

We study the potential to observe CP-violating effects in SUSY cascade decay chains at the LHC. Asymmetries composed by triple products of momenta of the final state particles are sensitive to CP-violating effects. Due to large boosts that dilute the asymmetries, these can be difficult to observe. Extending the methods of momentum reconstruction we show that the original size of these asymmetries may be measurable. A study is done at the hadronic level with backgrounds to estimate the expected sensitivity at the LHC.

We study the potential to observe CP-violating effects in SUSY stop cascade decay chains at the LHC. Asymmetries composed by triple products of momenta of the final state particles are sensitive to CP-violating effects. Due to large boosts that dilute the asymmetries, these can be difficult to observe. If all particle masses in a cascade decay are known, it may be possible to reconstruct all momenta in the decay chains on an event-by-event basis even when we have missing momentum due to a stable LSP. After the reconstruction, the non-diluted CP-violating signal can be recovered and gets significantly enhanced so that an observation may become feasible. A fully hadronic study has been completed to define the areas of the mSUGRA parameter space that may yield a 3-sigma observation with 500 fb^(-1) at the LHC.

We discuss the potential of observing effects of CP-violation phases in squark decay chains at the LHC. As the CP-odd observable, we use the asymmetry composed by triple products of final state momenta. There are good prospects of observing these effects using the method of kinematic reconstruction for the final and intermediate state particles. We also discuss the main experimental factors and the expected sensitivity.

We discuss the potential to observe effects of CP violation in squark decay chains at the LHC. As the CP-violating observable we use the asymmetry composed by triple products of final state momenta. Extending methods for momentum reconstruction we show that there are good prospects for observation of these effects at the LHC. Finally, we include the main experimental factors and discuss the expected sensitivity.

We address the question of how to determine the stop mixing angle and its CP-violating phase at the LHC. As an observable we discuss ratios of branching ratios for different decay modes of the light stop ~t_1 to charginos and neutralinos. These observables can have a very strong dependence on the parameters of the stop sector. We discuss in detail the origin of these effects. Using various combinations of the ratios of branching ratios we argue that, depending on the scenario, the observable may be promising in exposing the light stop mass, the mixing angle and the CP phase. This will, however, require a good knowledge of the supersymmetric spectrum, which is likely to be achievable only in combination with results from a linear collider.

Discoveries at the LHC will soon set the physics agenda for future colliders. This report of a CERN Theory Institute includes the summaries of Working Groups that reviewed the physics goals and prospects of LHC running with 10 to 300/fb of integrated luminosity, of the proposed sLHC luminosity upgrade, of the ILC, of CLIC, of the LHeC and of a muon collider. The four Working Groups considered possible scenarios for the first 10/fb of data at the LHC in which (i) a state with properties that are compatible with a Higgs boson is discovered, (ii) no such state is discovered either because the Higgs properties are such that it is difficult to detect or because no Higgs boson exists, (iii) a missing-energy signal beyond the Standard Model is discovered as in some supersymmetric models, and (iv) some other exotic signature of new physics is discovered. In the contexts of these scenarios, the Working Groups reviewed the capabilities of the future colliders to study in more detail whatever new physics may be discovered by the LHC. Their reports provide the particle physics community with some tools for reviewing the scientific priorities for future colliders after the LHC produces its first harvest of new physics from multi-TeV collisions.

We study the potential to observe CP-violating effects in SUSY cascade decay chains at the LHC. We consider squark and gluino production followed by subsequent decays into neutralinos with a three-body leptonic decay in the final step. Asymmetries composed by triple products of momenta of the final state particles are sensitive to CP-violating effects. Due to large boosts these asymmetries can be difficult to observe at a hadron collider. We show that using all available kinematic information one can reconstruct the decay chains on an event-by-event basis even in the case of 3-body decays, neutrinos and LSPs in the final state. We also discuss the most important experimental effects like major backgrounds and momentum smearing due to finite detector resolution. We show that with 300 fb$^{-1}$ of collected data, CP violation may be discovered at the LHC for a wide range of the phase of the bino mass parameter $M_1$.

We argue that the allowed range of the mass of the lightest neutralino in the MNSSM is limited. We establish the theoretical upper bound on the lightest neutralino mass and obtain an approximate solution for this mass.

We study the potential observation at the LHC of CP-violating effects in stop production and subsequent cascade decays, g g -> tildet_i tildet_i, tildet_i -> t tildechi^0_j, tildechi^0_j -> tilde\chi^0_1 l^+ l^-, within the Minimal Supersymmetric Standard Model. We study T-odd asymmetries based on triple products between the different decay products. There may be a large CP asymmetry at the parton level, but there is a significant dilution at the hadronic level after integrating over the parton distribution functions. Consequently, even for scenarios where large CP intrinsic asymmetries are expected, the measurable asymmetry is rather small. High luminosity and precise measurements of masses, branching ratios and CP asymmetries may enable measurements of the CP-violating parameters in cascade decays at the LHC.

Combined analyses at the Large Hadron Collider and at the International Linear Collider are important to reveal precisely the new physics model as, for instance, supersymmetry. Examples are presented where ILC results as input for LHC analyses could be crucial for the identification of signals as well as of the underlying model. The synergy of both colliders leads also to rather accurate SUSY parameter determination and powerful mass constraints even if the scalar particles have masses in the multi-TeV range.

A short overview about the potential of polarized beams at future colliders is given. In particular the baseline design for polarized beams at the ILC is presented and the physics case for polarized $e^-$ and $e^+$ is discussed. In order to fulfil the precision requirements spin tracking from the source to the interaction point is needed. Updates concerning the theoretical calculations as well as their implementation in simulation codes are reported.

We study the neutralino sector of the Minimal Non-minimal Supersymmetric Standard Model (MNSSM) where the $\mu$ problem of the Minimal Supersymmetric Standard Model (MSSM) is solved without accompanying problems related with the appearance of domain walls. In the MNSSM as in the MSSM the lightest neutralino can be the absolutely stable lightest supersymmetric particle (LSP) providing a good candidate for the cold dark matter component of the Universe. In contrast with the MSSM the allowed range of the mass of the lightest neutralino in the MNSSM is limited. We establish the theoretical upper bound on the lightest neutralino mass in the framework of this model and obtain an approximate solution for this mass.

We consider the neutralino sector in the Minimal Non--minimal Supersymmetric Standard Model (MNSSM). We argue that there exists a theoretical upper bound on the lightest neutralino mass in the MNSSM. An approximate solution for the mass of the lightest neutralino is obtained.

Combined analyses at the Large Hadron Collider and at the International Linear Collider are important to unravel a difficult region of supersymmetry that is characterized by scalar SUSY particles with masses around 2 TeV. Precision measurements of masses, cross sections and forward-backward asymmetries allow to determine the fundamental supersymmetric parameters even if only a small part of the spectrum is accessible. Mass constraints for the heavy particles can be derived.

The physics potential of the Large Hadron Collider in combination with the planned International Linear Collider is discussed for a difficult region of supersymmetry that is characterized by scalar SUSY particles with masses around 2 TeV. Precision measurements of masses, cross sections and forward-backward asymmetries allow to determine the fundamental supersymmetric parameters even if only a small part of the spectrum is accessible. No assumptions on a specific SUSY-breaking mechanism are imposed. Mass contraints for the kinematically inaccessible particles can be derived.

We examine the allowed mass range of the lightest neutralino within the Minimal Non--minimal Supersymmetric Standard Model. Being absolutely stable if R-parity is conserved this lightest neutralino is a candidate for the dark matter of the universe. We establish the theoretical upper bound on the lightest neutralino mass and obtain an approximate solution for this mass.