LCC Physics Working Group
Keisuke Fujii$^1$
Christophe Grojean$^{2
3
4
\input{00_abstract.tex
00_abstract.tex
This paper addresses the question of whether the International Linear
Collider has the capability of discovering new particles that have not
already been discovered at the CERN Large Hadron Collider. We summarize 
the various paths to discovery offered by the ILC, and discuss them in
the context of three different scenarios: 1.~LHC does not discover any 
new particles, 2.~LHC discovers some new low mass states and 3.~LHC discovers
new heavy particles. We will show that in each case, ILC plays a critical
role in discovery of new phenomena and in pushing forward the frontiers of 
high-energy physics as well as our understanding of the universe in a manner 
which is highly complementary to that of LHC.

For the busy reader, a two-page executive summary is provided at the beginning
of the document.

01_intro.tex
%\section{Introduction}
%\label{sec:intro}


This paper addresses the question of whether the International Linear
Collider has the capability of discovering new particles that have not
already been discovered at the CERN Large Hadron Collider.   The basic
conclusions of this paper have already  been enunciated in a letter
sent in December 2015 from ICFA to the MEXT ILC Advisory Panel.   This
paper provides an appendix to that letter that summarizes the
physics studies underlying
 its conclusions.

The most important question in particle physics today is the validity
of the Standard Model of particle physics that describes the strong,
weak, and electromagnetic interactions.   This model has been tested
in precision experiments at and around the $Z$ resonance in the 1990's 
at LEP and SLC and more
recently in the measurement of complex processes at the 
Tevatron and LHC involving
the production of single and multiple $W$ and $Z$ bosons and top
quarks. So far, the Standard Model has
 passed every test.  

Still, the Standard Model is visibly inadequate as an ultimate theory
of nature.   It does not contain the dark matter and dark energy that
make up 95\% of the energy content of the universe.   It does not
explain the fact that the visible universe is made up of matter but no
antimatter. Though the Standard Model successfully predicted the
appearance and general properties of the Higgs boson, essentially
every property of this particle is determined by a parameter adjusted
by hand.   The Standard Model theory of the Higgs boson field provides
no understanding of the mass of this particle or the masses that it
 produces for the matter particles of the Standard Model, the quarks and leptons.  

Most troubling, the Standard Model does not explain the most
mysterious aspect of the Higgs field, the fact that it fills space with
a nonzero value, producing an ordered thermodynamic state like the
superconductors and superfluids of condensed matter physics.   This
state, crucial to all other properties of the Higgs field, appears
only because a particular parameter of the model is given a negative
sign. In the history of the theory of superconductivity, we passed
through a similar stage of partial understanding, in which Landau and
Ginzburg gave a quantitative description of the properties of the
superconducting state using a model with adjustable
parameters~\cite{LandauGinzburg}.  The fundamental explanation of
superconductivity came almost a decade later, from a beautiful and
subtle theory of the interactions of electrons in metals by 
 Bardeen, Cooper, and Schrieffer~\cite{BCS}.

What causes the Higgs field to behave as it does?  No interaction of
the Standard Model can account for this.  We require new interactions,
hidden at short distances or high energies.  In quantum field theory,
such new interactions also require new particles, outside those of the
Standard Model. So, particle physicists ask, where are these new
particles or forces?  
How can they be discovered?

The search for these new particles has now become the primary goal of
the experiments at the LHC.  Taking advantage of the high energy and
large reaction rates possible for proton-proton collisions, the LHC
experiments have searched for a wide variety of new particles
predicted by theories that extend the Standard Model.  So far, this
search has come up
 empty.

We will continue the search for new particle production at the LHC.
Another factor of 100 in data will be available by the end of the LHC
program in the 2030's, giving the opportunity to extend current
searches to new particle masses higher by a factor 1.3--3. Still, it is
not too early to ask whether other approaches are needed to discover
new interactions at high 
energy.

One of these approaches is the precision study of the heaviest
particles of the Standard Model --- the Higgs boson and the top quark.
New interactions that explain the behavior of the Higgs boson
necessarily alter its properties. Such new interactions can also
alter the interactions of the top quark, through its strong coupling
to the Higgs boson that is responsible for its large mass. It is
well documented that measurement of the couplings of the Higgs boson
and the top quark to the percent level can be sensitive to new interactions
at high energies, beyond the ability of the LHC to search directly.
But also,  the pattern of deviations seen in these couplings from the
predictions of the Standard Model gives information on the properties
of the new interactions that induce them~\cite{Fujii:2015jha,Kanemura:2014bqa}.
This idea was recently confirmed by an examination of the models
proposed for the 750~GeV resonance suggested by the 2015 LHC data. It
was shown that precision Higgs boson and top quark measurements would
powerfully distinguish the competing hypotheses~\cite{Fujii:2016raq}.
The ILC will give us the ability to study the heavy particles of the
Standard Model with this level of precision.  This is the principle
argument for the importance of that 
machine to the future of high energy physics.


There is still another route to the discovery of new particles beyond
those of the Standard Model.    Though the LHC experiments search for
a large range of new particles, their search is not exhaustive, even
in the range of masses that is energetically accessible.  Typical LHC
new particle searches look for reactions predicted to occur at the
rate of 1 per trillion proton-proton collisions.   These are rates at
which the Standard Model interactions are capable of producing very
complex reactions, with multiple $W$ and $Z$ bosons, top quarks, and
invisible neutrinos.    The search for new particles then depends on
the recognition of very specific predicted properties, which are used
both to select events for close analysis and to distinguish new
interactions from similar Standard Model background processes.  As a
wide variety of LHC searches have been carried out and their reach
analyzed, significant blind spots have been recognized.   The LHC
experiments have special difficulties with particles that are produced by
weak and electromagnetic rather than strong interactions, decay
primarily to $\tau$ leptons or to lighter quarks and gluons, or decay
with very small energy release.   A strength of the ILC experiments is
that they are sensitive to new particles with these hard-to-recognize
decay schemes.

Given the large number of searches carried out at the LHC, one might
be tempted to dismiss these cases as rare exceptions.   However, these
cases include some of the most important examples of new particles
that we seek to discover. The most obvious example is given by the
particle that makes up the cosmic  dark matter.  If this particle is
produced at the LHC, it will be invisible to the experimental
detectors.  The presence of dark matter particles can only be detected
by observing visible particles like e.g.\ quarks, gluons, or photons that balance the
momentum carried off by these invisible particles.  Such events are
difficult to select and are easily imitated by Standard Model
reactions with neutrino emission.  The most promising strategies for
observing dark matter involve producing a heavier particle that decays
to the dark matter particle. However, first, this particle may not
exist or be too heavy to be produced at LHC, or, second, this particle may be very close in mass to the
dark matter particle, giving too small an energy release in the decay
to be visible to the LHC experiments. Many models of dark matter
require  such close-mass partners as a part of their mechanism for
obtaining the correct observed dark 
matter density in the universe. 

New direct partners of the Higgs boson suffer from the same
difficulties.  For example, in supersymmetry, an important class of
models of new interactions,  the structure of the 
model implies that the partners of
the Higgs boson are closely spaced in mass, with the lightest one
stable and invisible to LHC detectors.   Within this model, these
particles are expected to have masses close to the mass of the
$W$, $Z$ and Higgs bosons; thus, they are expected to be found in the
energy range of the ILC.    Other models with partners of
the Higgs boson also  predict low production rates at the LHC and
decay into modes that are difficult to 
observe, making these particles targets for ILC discovery.

One of the great advantages of using electron-positron annihilation,
rather than proton-proton collisions, is that this reaction makes it
possible to search {\it comprehensively} for new reactions outside the
Standard Model.   This is the result of well-known advantages of
$e^+e^-$ reactions --- the low rate of Standard Model background processes
relative to new particle production and the high level of detail that
is visible in the final state.  In $e^+e^-$ reactions, the tight
pre-selection of events required at the LHC is not needed.  This makes
it possible not only to observe a broader range of new processes but
also to make serendipitous discoveries.   Historically, the most
unanticipated particle of the Standard Model, the $\tau$ lepton, was
discovered at an $e^+e^-$ collider.  Even the gluon, the quantum of
the strong interactions, was discovered at an $e^+e^-$ collider before
it could be recognized in proton (anti-)proton  collisions.

However it is made, whether by observation through precision measurements or
by  direct discovery of new particles, the first breakthrough to the
new particles underlying the Higgs field and the Standard Model
will be a historic event.   Not only will this be a remarkable
discovery worthy of a Nobel Prize, but also it will reveal a new stratum
of the fundamental interactions.  Its  full exploration will bring
further discoveries essential to particle physics.   Perhaps, as the
understanding of the Standard Model did, it will bring to light new possibilities
for physical laws with deep  implications throughout the physical sciences.

In this paper, we illustrate these ideas through a review of specific
studies of new particle observation that have been carried out for the
ILC. 
%Unless stated otherwise, these studies are based on the default ILC operating scenario~\cite{Barklow:2015tja}. 
Throughout the document, we refer to the ILC with a center-of-mass energy of 500\,GeV 
and its 20-year operation scenario presented in~\cite{Barklow:2015tja}.
Section~\ref{sec:BSM} introduces the theoretical models that provide the
basis for the types of new particles to be considered.   Section~\ref{sec:ILC},
included for completeness, briefly reviews the capability of the ILC
to discover new interactions indirectly through precision measurements
of the Higgs boson, the top quark, and the $W$ and $Z$ bosons.
Section~\ref{sec:ILC_NP} reviews ILC studies of new particle production in the LHC
blind spots, including the production of dark matter particles,
supersymmetric particles with only electroweak interactions, and Higgs
boson partners.  Finally, section~\ref{sec:LHC_scen} maps this information onto the
grid of possibilities provided by the ILC Advisory Panel, giving
concrete examples of the conclusions that we have
 reached for each scenario.



02_BSMscenarios.tex
% \section{Overview on BSM Scenarios}
% \label{sec:BSM}

%The theoretical structure of the Standard Model (SM) has been brilliantly 
%verified through a vast assortment of
%detailed and high precision measurements at the CERN LHC and many 
%other experiments. And yet, many physicists are confident that new structures--
%which necessitate changes in the laws of physics as we know them-- should
%appear in experiments which are currently being planned.

%Several challenges to the SM arise from cosmological theory and measurements: 
%1. How are we to explain the presence of dark matter in the universe when
%the SM contains no candidate particle(s) with the required properties:
%electrically neutral, massive, non-relativistic  particles that can 
%clump together to seed galaxy formation?
%2. How are we to understand the matter-anti-matter asymmetry in the universe?
%3. What is the nature of the scalar field(s) needed to generate inflation in the early universe?

Along with the cosmological issues mentioned earlier --- dark matter, 
dark energy, baryogenesis --- there are two troubling problems with the 
theoretical structure of the SM. The first one is closely related to the 
questions concerning the origin of the Higgs boson $h$ and its properties introduced 
above, which arise from the verified 
existence of this seemingly bonafide scalar particle. More technically speaking, 
quantum mechanical contributions to the Higgs mass diverge
quadratically with energy scale such that quantum corrections to the
Higgs mass $m_h^2$ soon exceed $m_h=125$ GeV at energy scales
$\Lambda >1$ TeV. 
While such huge mass corrections can always be cancelled off by 
adjusting some free parameters of the theory, 
such fine-tunings --- which may be as small as one part in $10^{28}$ if 
$\Lambda$ reaches as high as the grand unification scale 
$m_{GUT}\sim 2\times 10^{16}$ GeV --- 
are considered pathological and indicative of some missing ingredients 
in the theory. 
Indeed, from an historical perspective, requiring for instance not-too-large 
contributions to the $K_L-K_S$ mass difference led to the prediction of the 
charm quark mass at its measured value: 
see Ref.~\cite{Giudice:2008bi} for various other examples.

Two simple paths to addressing the Higgs mass issue include 
1. postulating a new spacetime symmetry, supersymmetry or SUSY for short, 
which guarantees cancellation of the offending quantum corrections 
to all orders in perturbation theory and 
2. understanding the Higgs field as not being fundamental but instead 
as some composite, bound state built out of new fundamental fermions.
In recent years, more intricate alternative approaches involving 
extra dimensions of spacetime or the presence of 
exotic hidden sector states 
(which may be arranged to also cancel many of the offending 
quantum corrections) have been invoked as possible solutions to the 
Higgs mass problem.

A second fine-tuning issue arises in the QCD sector in that the theory seems
to require a strongly $CP$-violating contribution to the Lagrangian whereas 
none is observed in nature. So far, the introduction of the axion field
seems the only plausible means to address the strong $CP$ problem.
As we will see, ILC may be able to shed considerable light on 
the Higgs mass problem and also some indirect light on the strong $CP$ problem.

\subsection{Dark matter}
\label{sec:DM}

There is overwhelming evidence from a large variety of observations for the existence of
dark matter (DM) in the universe.\footnote{For a recent review, 
see {\rm e.g.} \cite{Baer:2014eja}.} 
The DM content should be about five times more abundant than
baryonic matter and it must be electric (and likely color) neutral and non-relativistic in order to
seed structure formation. No particle within the SM has such properties and hence the presence of
dark matter requires new matter states to exist.

One very popular DM candidate is the weakly-interacting massive particle or WIMP, labeled as $\chi$. 
WIMPs occur in a wide variety of beyond the SM theories, including SUSY, Little Higgs
models, models with extra spacetime dimensions, and models with dark sectors among others.
A compelling feature of WIMP dark matter is that at temperatures $T\gtrsim m_\chi$, then WIMPs
are thermally produced and present in the early universe. As the universe expands and 
temperatures drop below $m_\chi$, then WIMPs can no longer be thermally produced and their
number density freezes out, locking in a relic abundance of dark matter which depends 
on the WIMP mass and its annihilation cross section. For WIMP masses in the vicinity of 
$m_{weak}\sim 100$ GeV, then roughly the measured amount of dark matter will be produced, 
a situation known as the WIMP miracle. 

% Since WIMPs are weakly interacting, then they should necessarily be produced at ILC 
% at detectable rates for $\sqrt{s}>2m_\chi$. 
% The WIMPs would escape direct detection at ILC, but could nonetheless be detected via the
% presence of initial state bremsstrahlung of a photon from the initial state of the 
% production reaction: $e^+e^-\rightarrow \chi\chi\gamma$. 
% The precision properties of ILC can even allow for WIMP mass and spin measurements in such scenarios. 
% Similar reactions such as $pp\rightarrow \chi\chi jet$ are much more difficult to study at LHC 
% due to enormous $\nu\bar{\nu}jet$ backgrounds. 
Since WIMPs are weakly interacting, they should necessarily be produced at ILC, as will be discussed in section~\ref{subsec:directNP_WIMP},
while the analoguous processes might be hidden in enormous backgrounds at hadron colliders. 
It should be noted that if WIMPs exist, they should 
ultimately also be detected by underground direct detection experiments and by indirect searches 
for WIMP-WIMP annihilation within the cosmos into gamma rays or antimatter. 
% The variety of expected data is very complementary and can be used to nail down many essential 
% WIMP properties such as mass, spin and which particle species they annihilate into.
% By detailed measurements of WIMP properties, then it is often possible to constrain or
% cross check WIMP production rates in the early universe along with WIMP scattering or annihilation
% rates and possibly the local WIMP abundance\cite{Baltz:2006fm}. Such checks could verify or falsify the simple assumptions 
% associated with thermal DM production within the WIMP miracle scenario.
Detailed measurements of WIMP properties (such as mass, spin and which particle species they annihilate into)
are required to verify or falsify the simple assumptions 
associated with thermal DM production within the WIMP miracle scenario.

It is also possible, and perhaps theoretically likely, 
that the WIMP content of the universe would be augmented by
non-thermal production of exotic particles (e.g.\ gravitinos, axinos or saxions)
followed by decay to WIMPs. It is also possible that the WIMP content might be
diminished from its thermal value for instance by decay to a lighter superWIMP
state (such as gravitino, axino or KK-graviton) 
or by significant late-time entropy injection e.g.\ by
saxion or moduli-fields, which would dilute all relics present at the 
time of decay. Late-time moduli field decay seems ubiquitous in string theory models
while early universe saxion production and decay seems required in SUSY axion models 
which address both the gauge hierarchy and strong $CP$ problems.

If indeed DM consists of super-WIMPs, then no detection of DM is expected at
either direct or indirect dark matter detection experiments. 
However, superWIMPs may leave tell-tale 
signatures at colliding beam experiments via missing energy produced as the 
end-product of cascade decays or via long-lived next-to-lightest particles 
which have suppressed decay rates into the superWIMP. 
Thus, whether or not a direct or indirect WIMP 
detection signal is found, there will remain considerable details 
regarding dark matter production in the early universe to be sorted out.
% an enterprise for which the ILC is excellently designed to carry forth.

A final mention is that DM could consist of axion particles which are required by 
the Peccei-Quinn (PQ) solution to the strong $CP$ problem. 
Axions may ultimately be detected via ongoing microwave cavity or other experiments. 
At first sight, one might not expect a connection between axion physics and ILC/LHC physics. 
However, it has been recently emphasized that there is a deep connection between
PQ symmetry and supersymmetry via the superpotential $\mu$ term which is required by
naturalness to be not too far from the weak scale\cite{Bae:2013bva}. In such a case, then
DM would consist of a WIMP-axion mixture, {\it i.e.} two dark matter species. 
% Precision measurements of SUSY particle properties at ILC could help disentangle 
% more complicated but highly motivated scenarios such as these.




\subsection{Supersymmetry}
\label{subsec:SUSY}

The minimal supersymmetrized version of the Standard Model, the MSSM or Minimal Supersymmetric Standard Model, 
contains a scalar field for each chiral fermion and a gaugino for each gauge boson, a second Higgs doublet as well as fermionic partners of the Higgs bosons, the {\em higgsinos}.  
The Lagrangian for the MSSM is derived in part from 
the {\it superpotential} which contains a mass term $\mu \hat{H}_u\hat{H}_d + {\mathrm{Yukawas}}$ 
where, very importantly, $\mu$ feeds mass to both higgsinos and Higgs bosons;
$\mu$ also feeds mass to the gauge bosons $W^\pm$ and $Z$ via the vacuum expectation values of the scalar potential.
From this, absent large unnatural cancellations in the masses of these particles, 
it may already be expected that the higgsinos should have masses of 
order the $W,\ Z$ and $h$ bosons, and thus within the discovery reach of ILC.

Since the mechanism for SUSY breaking is unknown, 
we may {\it parametrize our ignorance} via the introduction of various (independent)
soft SUSY breaking terms which include mass terms for scalars and gauginos 
along with bilinear and trilinear scalar couplings ($B$ and $A$ terms).
In more complete models, the soft terms may be calculated in terms of the
more fundamental gravitino mass $m_{3/2}$ (in the case of gravity-mediated
or anomaly-mediated SUSY breaking models) or in terms of the messenger scale
$\Lambda$ (in gauge-mediated SUSY breaking models). 
From the softly broken SUSY Lagrangian of the 
Minimal Supersymmetric Standard Model (MSSM), one may compute the various mass
eigenstates and their mixings, and consequently superparticle production and
decay rates. 

A corner-stone of SUSY is that {\it sparticles couple as particles}\cite{Dimopoulos:1981zb}. 
This is independent of the mechanism responsible for SUSY breaking. 
Thus, the couplings of particles to sparticles is completely determined in SUSY theories 
and can be {\it tested} at linear $e^+e^-$ colliders in many cases\cite{Feng:1995zd}.

The MSSM receives some impressive indirect support from experiment in that
1. the measured values of the weak scale gauge couplings from LEP 
unify under MSSM renormalization group evolution, 
2. the measured value of $m_t$ is in the range needed to trigger the 
required breakdown of electroweak symmetry and 
3. the measured value of the Higgs mass falls well within 
the window required by the MSSM, namely that $m_h<135$ GeV. 
Finally, it is perhaps under-appreciated that SUSY stabilizes the 
weak/GUT gauge hierachy in grand unified theories (GUTs). GUTs
not only unify the three forces of the SM, but also unify matter fermions
(completely for each generation in $SO(10)$) 
and explain the seemingly ad-hoc quantum numbers of the Standard Model 
fermions. 

\subsubsection{SUSY dark matter}
\label{subsubsec:SUSYDM}

Under $R$-parity conservation --- which is well-motivated by 
unified theories and the need to suppress proton decay --- 
then the lightest SUSY particle (LSP) is absolutely stable 
and may serve as a dark matter candidate. 
If the LSP is a neutralino, then it is a candidate weakly interacting 
massive particle, or WIMP. 
The SUSY WIMP may be mainly bino-, wino- or higgsino-like, 
or a comparable mixture of these states.
For the popular case of a bino-like LSP, then a low bino annihilation rate
in the early universe leads to too much dark matter. 
One special enhancement mechanism for bino annihilation is called 
co-annihilation wherein the bino
may co-exist and annihilate with states nearby in mass, such as 
the lightest stau, stop or chargino. 
Another mechanism is resonance annihilation
wherein the bino could have enhanced annihilation through some
resonance such as heavy Higgs states $A$ or $H$. 
Well-tempered neutralinos consisted of comparable bino-higgsino mixtures
and could also yield the measured abundance of dark matter; this case now seems
ruled out by direct detection experiments. 
SUSY winos as comprising all of dark matter also seem ruled out 
since they would lead to large rates for gamma ray 
production in galactic annihilations. 
Light wino dark matter as just a portion of the dark matter is also allowed
since then their galactic abundance is suppressed, leading also to
suppression of their indirect detection rates.

LSPs that are mainly higgsino-like tend to give not enough dark matter unless
their mass reaches into the (unnatural) TeV regime. 
However, low mass higgsino-like WIMPs could still exist as just a 
portion of the dark matter, wherein the remainder might be made up of axions. 
Such a scenario is appealing in models of natural supersymmetry since both
the EW and QCD (strong $CP$) naturalness problems are then solved.
 
\subsubsection{Baryogenesis in SUSY}
\label{subsubsec:SUSYbaryo}

While EW baryogenesis is not viable in the SM and barely viable in
the MSSM, SUSY provides several compelling alternative possibilities
for baryogenesis. Thermal leptogenesis (and subsequent baryogenesis
via sphaleron effects) via asymmetric decays of heavy
right-hand-neutrino states in the early universe seems to require
re-heat temperatures $T_R>\sim 10^9$ GeV. Such high re-heat brings with it
possible conflict with over production of gravitinos in the early universe.
Alternatively, Affleck-Dine leptogenesis proceeds via a lepton-number
violating condensate which forms along flat directions in the SUSY scalar
potential. Affleck-Dine leptogenesis seems viable for $T_R$ as low as $10^5$ GeV.
Other possibilities include non-thermal leptogenesis where right-handed neutrinos are 
produced via inflaton decay or oscillating sneutrino leptogenesis 
which can also occur at lower $T_R$ values. 
The capacity of ILC to perform detailed measurements on SUSY particles offers
hope to sort out amongst these promising possibilities.

%In particular, the couplings to the photon and the Z-boson are known,
%which implies that the cross-section for any \eeto ~sparticle -- anti-sparticle pair process 
%that proceeds only via the s-channel is determined by the kinematics alone, 
%i.e. by $E_{CMS}$ and the mass of the sparticle.
%Contributions from sparticle exchange in the t-channel are possible
%in $\snu_e , \sel, \XN{i}$ or $\XPM{i}$ pair-production,
%by exchange of  a $\XPM{i} , \XN{i} , \sel $ or $\snu_e$, respectively.
%Also with such diagrams contributing, all couplings are known,
%and the cross-section will still be determined by the kinematic relations,
%at the cost that one must consider both the mass of the produced and of the exchanged sparticle.
%% \begin{figure}[bh]
%%    \centering
%%       \includegraphics[scale=0.28]{plots/feyn2b}
%% \caption{A nice diagram \label{fig:1}}
%% \end{figure}

%Furthermore, by definition there is one Lightest SUSY Particle (LSP), 
%and one Next-to-Lightest SUSY Particle (NLSP). If it is stable, it is un-avoidable that
%the LSP must be neutral and weakly interacting, due to cosmological constraints. The NLSP, on
%the other hand, could be any sparticle - a slepton, a bosino, or a squark. However, there is
%only a limited number of sparticles. While an arbitrary sparticle in the spectrum of any specific
%SUSY model typically would decay through cascades of other, lighter sparticles, the NLSP only
%has one decay-mode, namely to its SM partner and to the LSP,
%which is  the topology defining a {\it simplified model}.


\subsubsection{Naturalness and upper bounds on sparticle masses}
\label{sec:BSM-SUSY-Naturalness}

An absolutely vital question --- which addresses the issue of {\it falsifiability} of
the weak scale SUSY hypothesis --- 
arises as to: how massive can the various SUSY particles be while maintaining naturalness?
In fact, this {\it naturalness} issue has led some physicists to question whether present LHC
sparticle search bounds are already putting intense pressure on the SUSY hypothesis.

Some guidance can be obtained by connecting observed particle masses to SUSY Lagrangian
parameters. For instance, the Higgs boson squared mass arises in the MSSM as
\begin{equation}
m_h^2= \mu^2+m_{H_u}^2+{\rm mixing\ terms}+{\rm radiative\ corrections},
\label{eq:mhs}
\end{equation}
where each contribution is evaluated at the weak scale and the 
latter two contributions are $\lesssim m_h^2$. 
If either $\mu^2$ or $-m_{H_u}^2$ were far larger than $m_h^2$, then the other term would have 
to be tuned to high accuracy to maintain $m_h\simeq 125$ GeV. On the contrary, to avoid
large fine-tuning in $m_h^2$, then each contribution ought to be comparable to $m_h^2$. 
This implies that $m_{H_u}^2$ should be driven to small negative values 
$\sim -(100)^2$ GeV$^2$ and also that  $\mu\sim 100-200$ GeV (the lower bound arises
from LEP2 limits on chargino production). 

Alternatively, naturalness can be expressed
in a related form in terms of $m_Z^2$ from minimizing the scalar potential:
\begin{equation}
\frac{m_Z^2}{2} =\frac{m_{H_d}^2+\Sigma_d^d-(m_{H_u}^2+\Sigma_u^u)\tan^2\beta}{\tan^2\beta -1}-\mu^2\sim -m_{H_u}^2-\Sigma_u^u-\mu^2 
\label{eq:mzs}
\end{equation} 
where the last partial equality obtains for moderate-to-large $\tan\beta$.
To avoid large unnatural cancellations on the right-hand-side, we again 
see that the weak scale magnitudes of $m_{H_u}$ and $\mu$ must be comparable to $m_Z$. 
The {\it electroweak} naturalness measure $\Delta_{EW}$\cite{Baer:2012up} 
compares the largest term on the right-hand-side of Eq.~(\ref{eq:mzs}) to $m_Z^2/2$.
A low (natural) value\footnote{The onset of fine-tuning for $\Delta_{EW}>30$ is visually displayed in Fig. 1 of Ref. \cite{Baer:2015rja}.} of $\Delta_{EW}<30$ 
implies {\it light higgsinos} $\tilde{\chi}_1^\pm$ and $\tilde{\chi}_{1,2}^0$ 
with mass $\sim 100-300$ GeV, the closer to $m_Z$ the better.
The radiative corrections $\Sigma_u^u$ and $\Sigma_d^d$ contain over 40 one-loop contributions. 
The largest of these usually arise from the top-squark sector and implies for $\Delta_{EW}<30$ 
that $m_{\tilde{t}_1}<3$ TeV as long as the stop is highly mixed 
(which helps lift $m_h$ up to $\sim 125$ GeV). 
The gluino feeds into $m_Z^2$ at two-loop order and calculations show $m_{\tilde{g}}\lesssim 4$ TeV:
well-beyond present LHC bounds and perhaps beyond the reach of HL-LHC. A more detailed discussion 
about the relations between different fine-tuning measures can be found e.g.\ in~\cite{Baer:2014ica}.

% Some previous fine-tuning evaluations indicated the need for three very light third generation 
% squarks with masses $\lesssim 500$ GeV. 
% These calculations neglected the self-contributions of the soft term $m_{H_u}^2$ to its own 
% RG running. When this is included, then large soft terms, amplified by the large top-Yukawa coupling, can radiatively drive $m_{H_u}^2$ from large, unnatural values at high mass scales to 
% natural values at the weak scale --- a phenomenon known as radiatively-driven naturalness.
% Other historical naturalness evaluations seemed to require {\it all} SUSY particles  
% not too far from the 100 GeV scale. These evaluations, using a log-derivative measure, 
% were evaluated within the context of multi-parameter effective SUSY theories that 
% didn't account for the fact that high scale soft terms are all correlated in more 
% fundamental theories. Allowing for such corrections, which lead to cancellations in the fine-tuning
% evaluations, brings both of the earlier naturalness calculations
% into accord with naturalness as expressed by Eq.~(\ref{eq:mzs})\cite{Baer:2014ica}.

\subsubsection{Overview of natural SUSY parameter space}

A grand overview of natural SUSY parameter space for the case of gravity-mediated SUSY breaking 
in the two-extra-parameter non-universal Higgs model (NUHM2) is shown in Fig.~\ref{fig:mhfvsmu} 
where we plot the unified GUT scale gaugino mass $m_{1/2}$ vs. superpotential $\mu$ parameter 
with GUT scale scalar masses $m_0=5$ TeV, $A_0=-8$ TeV, $\tan\beta =15$ and $m_A=1$ TeV\cite{Baer:2016usl}. 
The grey and blue-shaded regions were long ago excluded by chargino pair searches at LEP and LEP2. 
Current LHC13 searches require $m_{\tilde{g}}>1.9$ TeV
which excludes the region to the left of the blue line labeled LHC13.
Contours of naturalness $\Delta_{EW}$ values are indicated in red. 
Fine-tuning already sets in for $\Delta_{EW}>30$.
It can be seen that a large region of natural SUSY remains with $\Delta_{EW}<30$ 
but which is well beyond current LHC bounds. 
The future reach of HL-LHC, estimated at the generator-level with $\sqrt{s}=14$ TeV and 3000 fb$^{-1}$ of integrated luminosity is shown
by the dashed lines for the same-sign-diboson channel (SSdB) 
and the monojet plus soft dilepton channel ($\tilde{\chi}_1^0\tilde{\chi}_2^0j$). 
These reach contours can be compared to the reach of ILC with $\sqrt{s}=500$ GeV and $\sqrt{s}=1000$ GeV
(black contours). 
The location of two ILC benchmark points are denoted by green tags.
%
\begin{figure}[htbp]
\begin{center}
\hspace{3mm} \includegraphics[width=8cm]{figs/mumhfMEXT}
\end{center}
\caption{The $m_{1/2}$ vs. $\mu$ parameter space of the NUHM2 SUSY model with $m_0=5$ TeV, 
$A_0=-8$ TeV, $\tan\beta =15$ and $m_A=1$ TeV.
Naturalness contours of $\Delta_{EW}=15$, 30 50 and 75 are shown in red.
We also show the current exclusion limit of LHC13 (solid-blue) 
and future reach of HL-LHC (dashed-blue). The reach of ILC with $\sqrt{s}=500$ and 1000 GeV is also shown.
We show the locus of two ILC benchmark points which have been studied in green.
}
\label{fig:mhfvsmu}
\end{figure}

In models with gaugino mass unification such as NUHM2, then the mass hierarchy $\mu\ll M_1<M_2<M_3$ is expected
and HL-LHC should be able to identify same-sign diboson (SSdB) production signals arising from
wino pair production, or else higgsino pair production $\tilde{\chi}_1^0\tilde{\chi}_2^0 j$ followed
by $\tilde{\chi}_2^0\rightarrow \ell^+\ell^-\tilde{\chi}_1^0$ decay.

However, several compelling SUSY models which are {\it also natural} can defy this mass pattern. 
For instance, in {\it mirage mediation} SUSY models, the gauginos gain comparable mass 
contributions from moduli (gravity)-mediation and anomaly mediation. 
The GUT scale splitting of the gaugino masses, 
which is proportional to the gauge group beta functions, is compensated
by the gaugino mass RG running so that gaugino masses apparently unify at some
intermediate scale (which is determined by the relative amounts of anomaly- and moduli-mediated
mass contributions). In such mirage mediation models, the gaugino spectrum is more compressed: if the
gluino lies in the 3 TeV range, then winos and binos may also lie in the few TeV range so that
wino pair production is suppressed and the $\tilde{\chi}_2^0-\tilde{\chi}_1^0$ mass gap is
compressed to the few GeV level. In such a case, SUSY would be completely natural but largely 
inaccessible to HL-LHC searches~\cite{Baer:2016hfa}. 
However, the required light higgsinos  would still be accessible to ILC with $\sqrt{s}>2m(higgsino)$.
A concrete example is shown in Fig. \ref{fig:spect} where {\it a}) presents the mass spectrum and
{\it b}) illustrates the characteristic mirage unification of gaugino masses for a sample benchmark point.
%
\begin{figure}[tbp]
\begin{center}
\includegraphics[height=0.2\textheight]{figs/spect}
\includegraphics[height=0.2\textheight]{figs/gauginos}
\caption{{\it a}) A typical superparticle mass spectrum generated from 
natural generalized mirage mediation (nGMM)~\cite{Baer:2016hfa}.
{\it b}) Evolution of gaugino masses from the nGMM benchmark point with 
$m_{3/2}=75$ TeV, $\alpha =4$.
\label{fig:spect}}
\end{center}
\end{figure}

\subsubsection{Neutrino Physics at the ILC}

One outstanding question the Standard Model cannot explain is the smallness of neutrino masses.
Starting with the compelling data on atmospheric neutrinos in 1998, it has now been firmly established
that neutrinos are massive, and the different flavors are strongly
mixed. However, neutrino masses are much smaller than the masses of the electrically charged
fermions and flavor mixing in the leptonic sector differs considerably from that in the
quark sector. These experimental findings, along with the observed hierarchies in
the masses of the charged leptons and quarks form the so-called flavor puzzle.
Even though we can parametrize our understanding, 
we do not know the underlying principles leading to the experimentally observed patterns.

Several models have been proposed to explain the smallness of neutrino masses.
At low energies they usually lead to the so-called Weinberg operator $(LH)^2$
giving Majorana masses to light neutrinos after electroweak symmetry breaking.
The most compelling among them is the {\it seesaw mechanism} which comes in three varieties
depending on the details of how this operator is generated: type I in case of gauge singlet
fermions (usually the right-handed neutrinos), type II in case of an $SU(2)_L$ triplet
Higgs boson and type III in case of $SU(2)_L$ triplet fermions. 
The type-I see-saw easily melds with $SO(10)$ SUSY GUTs which require a right-handed neutrino
to fill out the 16-dimensional spinor representation. It should be remarked that SUSY
is highly desirable in this case to stabilize the Higgs mass from blowing
up to the see-saw scale $\sim 10^{11}-10^{13}$ GeV.

These additional particles associated with the see-saw mechanism 
are usually too heavy to be produced directly in collider experiments.
In supersymmetric models, however,  they may leave imprints on the RG evolution of
the mass parameters~\cite{Baer:2000hx} 
and give rise to additional flavor structures which are linked
to the underlying mechanism for generating neutrino masses. 
These flavor structures induce flavor violating decays of sleptons 
which can potentially be studied at the ILC\cite{Deppisch:2003wt}.

{\it See-saw scale, axions and the origin of the SUSY $\mu$-parameter:}
%\label{subsubsubsec:axion_theo}

The hidden sector SUSY breaking scale $m_{hidden}$, the Peccei-Quinn (PQ) scale $f_a$ 
and the see-saw neutrino 
scale $m_N$ are all expected to lie in the intermediate range: $\sim 10^{10}-10^{13}$ GeV.
Are these scales all related? One class of models begins with PQ symmetry which forbids the SUSY
superpotential $\mu$ term, but then both $\mu$ and the see-saw scale $m_N$ are generated due to
radiatively-broken PQ symmetry which is triggered by SUSY breaking\cite{Murayama:1992dj,Bae:2014yta}. In such models, 
the $\mu$ term is generated as $\mu=\lambda_\mu f_a^2/M_P$ leading to a Little Hierarchy 
where $\mu\ll m_{SUSY}$ arising as a consequence of $f_a<m_{hidden}$. 
A Majorana neutrino scale $m_N\sim f_a$ is also generated. In the case where an axion is detected by
microwave cavity or other experiments, where $m_a\simeq m_\pi f_\pi/f_a\cdot \sqrt{Z}/(1+Z)$ with 
$Z=m_u/m_d\simeq 0.56$, then the axion mass determines $f_a$ while
an ILC measurement of light higgsinos of mass $\sim \mu$ determines the PQ coupling $\lambda_{\mu}$.

{\it Bilinear R-parity violation}
\label{subsubsec:bRPV_theo}

Beside the usual mechanisms to generate neutrino masses, supersymmetry offers an 
additional possibility: breaking of $R$-parity in the lepton sector. The simplest
model is the one where only bilinear terms are present in the superpotential as well
as the corresponding terms in the soft SUSY sector. As $R$-parity is broken,
the lightest supersymmetric particle (LSP) is not stable anymore but decays. 
The six parameters explaining neutrino data are then also those responsible for the decay
properties of the LSP: ratios of decay branching ratios are proportional
to neutrino mixing angles, e.g.\ 
$BR(\tilde \chi^0_1 \to W \mu)/BR(\tilde \chi^0_1 \to W \tau) \simeq \tan^2\theta_\mathrm{atm}$ or
$BR(\tilde \chi^0_1 \to \nu \mu \tau)/BR(\tilde \chi^0_1 \to \nu e \tau) \simeq \tan^2\theta_\mathrm{sol}$
\cite{Porod:2000hv} where
$\theta_\mathrm{atm}$ and $\theta_\mathrm{sol}$ are the atmospheric and solar neutrino mixing angles. 
We will illustrate this possibility with a dedicated simulation study in section~\ref{subsubsec:directNP_bRPV}.
Moreover, the smallness of the neutrino masses
also implies that the lifetime of the LSP is measurable at the ILC in a large part of the parameter
space. For completeness we note that the existence of such correlations does not depend
on the nature of the LSP --- e.g.\ whether it is a neutralino or a chargino or a slepton --- but only
the concrete form of these correlations \cite{Hirsch:2003fe}. 



\subsection{Two-Higgs-Doublet Models}
\label{sec:2HDM}
%\begin{itemize}
%\item Type I vs Type II, MSSM is special case of type II
%\item 2HDM and EW Baryogenesis $\Rightarrow$ large deviation in $\lambda$ (Kanemura et al)
%\end{itemize}
% JL: from 4.3

A minimal extension of the SM Higgs sector consists of adding a second Higgs doublet to the one present in the SM.
In these Two Higgs Doublet Models (2HDMs), the scalar potential will contain mixing mass parameters and both doublets will acquire vacuum expectation values, $v_1/\sqrt{2}$ and $v_2/\sqrt{2}$, respectively,  and the gauge boson masses will keep their SM 
expressions with the Higgs vacuum expectation value $v$ replaced by $v = \sqrt{v_1^2 + v_2^2}$.  The complete 2HDM  is defined only after considering the interactions of the Higgs fields to fermions.
Yukawa couplings of the generic form
\begin{equation}
-h_{ij}^a  \bar{\Psi}^i_L H_a \Psi^j_R + h.c.
\label{eq:2HDM Yukawa}
\end{equation}
may be added to the renormalizable Lagrangian of the theory. Contrary to the SM, the two Higgs doublet structure does not ensure the alignment of the fermion mass terms  with the Yukawa couplings, and the neutral Higgses can mediate flavor changing interactions. Such flavor changing interactions should be suppressed in order to comply with flavor data constraints.  
Based on the Glashow--Weinberg criterion, it is clear that 
the simplest way of avoiding such transitions is to assume the existence of a symmetry that ensures the couplings of the fermions of each given quantum number (up-type and  down-type quarks, charged
and neutral leptons) to only one of the two Higgs doublets.  Different models may be defined depending on which of these fermion fields couple to a given Higgs boson. Models of type-I are those
in which all SM fermions couple to a single Higgs field.  In type-II models, of which the MSSM is a prime example, down-type quarks and charged leptons couple to a common Higgs field, while the up-type quarks and neutral leptons couple to
the other.  In models of type-III (lepton-specific) quarks couple to one of the Higgs bosons, while leptons couple to the other. Finally, in models of type-IV (flipped), up-type quarks and charged leptons couple to one of the
Higgs fields while down-quarks and neutral leptons couple to the other. 

In 2HDMs, the electroweak phase transition can be first order, opening the possibility for a successful baryogenesis at the weak scale. Regions of the parameter space leading to a strong first order phase transition also exhibit large deviations of the Higgs self-coupling~\cite{Kanemura:2004ch}. This is also valid in the context of various other Higgs sectors beyond the 2HDM~\cite{Noble:2007kk}.

Other extensions of the Higgs sector can include multiple copies of
SU(2)$_L$ doublets, additional Higgs singlets, triplets or more
complicated combinations of Higgs multiplets.  It is also possible to
enlarge the gauge symmetry beyond SU(2)$_L\times$U(1)$_Y$
along with the necessary Higgs structure to generate gauge boson
and fermion masses. Direct searches for additional Higgs bosons with masses below or above 125\,GeV, deviations in the 125-GeV Higgs boson couplings to fermions and gauge bosons and deviations of the its self-coupling are ways to probe these extended Higgs sectors.

\subsection{Randall-Sundrum Models}
\label{sec:RS}
%\begin{itemize}
%\item \cite{Randall:1999ee}
%\item radion (Spin-0), but no mixing with Higgs
%\item Kaluza-Klein excitations of SM particles (Spin-1)
%\end{itemize}
% JL: from 4.3
Randall-Sundrum models~\cite{Randall:1999ee} feature extra-dimensional spaces strongly curved by the gravitational effect of a bulk large vacuum energy. The 5D geometry corresponds to an Anti-de-Sitter space with two boundaries: 
the Planck brane and the TeV brane. 
The effect of the Planck brane is to render the graviton zero mode normalizable. The TeV brane provides a mass gap  into the Kaluza--Klein expansion of all types of fields.
The distance between the Planck and TeV branes is arbitrary, corresponding to the massless radion. In fact, the size of the extra dimension is unstable to small perturbations and must be stabilized~\cite{Goldberger:1999uk}, and this stabilization results in a mass for the radion. A light radion decays predominantly to gluons due to the 
trace anomaly.

 In realistic Randall-Sundrum models, the SM fermions propagate in the bulk and the support of zero-mode wavefunctions on the TeV brane, where the Higgs vev is localized, sets the size of the Yukawa interactions.  The wave functions of the third generation quarks and in particular that of the $t$ quark extends to the TeV
side of the 5th dimension, while the
wavefunctions of light fermions peak close to the Planck brane. This  constitutes an
elegant explanation of the striking mass hierarchy in the fermion
sector.   A consequence of this effect is that the couplings of the
$t$ quark to the $Z$ boson are expected to have large shifts due to
mixing with KK states, of independent size for $t_L$ and $t_R$.
Models of this effect are described, for example, in Refs.~\cite{Djouadi:2006rk,Carena:2006bn}. 


\subsection{Composite Higgs}
\label{sec:comphiggs}
%{\color{red} [Christophe] I've now significantly reduced the size, omiting some technicalities}


Within the SM, electroweak symmetry breaking is posited but has no dynamical origin. Furthermore, the Higgs boson appears to be unnaturally light. 
A scenario that remedies these two catches is to consider the Higgs boson as a bound state of new  dynamics becoming strong around the weak scale. Strongly coupled models of electroweak symmetry breaking have been recently and concisely reviewed in Ref.~\cite{Csaki:2015hcd},
that has been used as a source for the presentation below. For an exhaustive review on composite Higgs models, see Ref.~\cite{Panico:2015jxa}. 

The Higgs boson can be made significantly lighter than the other resonances of the strong sector if it appears as a pseudo-Nambu--Goldstone boson. More precisely, the  strongly coupled sector is supposed to be invariant under a global symmetry $G$ spontaneously broken to a subgroup $H$ at the  scale~$f$ and the Higgs boson belongs to the coset space $G/H$. 
To avoid conflict with EW precision measurements, it is better if the strong interactions themselves do not break the EW symmetry, hence the SM gauge symmetry itself should be contained in $H$.  A canonical and minimal example (called MCHM) is based on $SO(5)/SO(4)$.

The SM (light) fermions and gauge bosons cannot be part of the strong sector itself since LEP data have already put stringent bounds on the compositeness scale of these particles far above the TeV~scale. 
The gauge bosons couple to the strong sector by a weak gauging of a SU(2)$\times$U(1) subgroup of the global symmetry $G$. 
%Inspiration for the construction of such models comes from the AdS/CFT correspondence: the components of a gauge field along  extra warped space dimension can be interpreted as the Goldstone boson resulting from the breaking of global symmetry of the strong sector.
The couplings of the SM fermions to the strong sector could a priori take two different forms:
(i) a bilinear coupling of two SM fermions to a composite scalar operator, ${\cal O}$, of the form $ {\cal L}=y\, \bar q_L u_R {\cal O} +{\rm h.c.}$ in simple analogy with the SM Yukawa interactions. This is the way fermion masses were introduced in Technicolor theories and it generically comes with severe flavor problems and calls for extended model building gymnastics to circumvent them; 
(ii) a linear mass mixing with fermionic vector-like operators and the physical states are a linear combination of elementatry and composite fields. 
The SM fermion mass hierarchy  emerges from the dynamics controlling the mixing, $\theta_i$, between the elementary and composite sectors: the light fermions are mostly elementary states ($\sin \theta_i \ll 1$), while the third generation quarks need to have a sizable degree of compositeness.
%: ${\cal L}= \lambda_L\, \bar q_L {\cal Q}_R + \lambda_R\,  {\bar {\cal U}}_L u_R$. ${\cal Q}$ and $ {\cal U}$ are two fermionic composite operators of mass $M_Q$ and $M_U$. Being part of the composite sector, they can have a direct coupling  of generic order $Y_*$ to the Higgs boson. In analogy with the photon-$\rho$ mixing in QCD, once the linear mixings are diagonalized, the physical states are a linear combination of elementary and composite fields. Effective Yukawa couplings are generated and read for instance for the up-type quark 
%
%\begin{equation}
%y =Y_* \sin \theta_L \sin \theta_R 
%\label{eq:YukawaComposite}
%\end{equation}
%%
%where $\sin \theta_i= \lambda_i / \sqrt{M_U^2+\lambda_i^2}$, $i=L,R$, measure the amount of compositeness of the SM left- and right-handed up-type quark. If the strong sector is flavor-anarchic, i.e., if the couplings of the Higgs to the composite fermions does not exhibit any particular flavor structure, the relation~(\ref{eq:YukawaComposite}) implies that the light fermions are mostly elementary states ($\sin \theta_i \ll 1$), while the third generation quarks need to have a sizable degree of compositeness. The partial compositeness paradigm offers an appealing dynamical explanation of the hierarchies in the fermion masses. In fact, assuming the strong sector to be almost conformal above the confinement scale, the low-energy values of the mass-mixing parameters $\lambda_{L,R}$ are determined by the (constant) anomalous dimension of the composite operator they mix with. If the UV scale at which the linear mixings are generated is large, then ${\cal O}(1)$ differences in the anomalous dimensions can generate naturally large hierarchies in the fermion masses via renormalization group running. 
While the introduction of partial compositeness greatly ameliorated the flavor problem of the original composite Higgs models, nevertheless it did not solve the issue completely, at least in the case where the strong sector is assumed to be flavor-anarchic~\cite{Csaki:2008zd}. %While the partial compositeness set-up naturally emerges in models built in space-times with extra dimensions, no fully realistic microscopic realization of partial compositeness has been proposed in the literature.

Another nice aspect of the partial compositeness structure is the dynamical generation of the Higgs potential. The Higgs being a pseudo-Nambu--Goldstone boson, its mass does not receive any contribution from the strong sector itself but it is generated at the one-loop level via the couplings of the SM particles to the strong sector.% since these interactions are breaking the global symmetries under which the Higgs doublet transforms non-linearly. 
The leading contribution to the potential arises from top loops and it takes the form
\begin{eqnarray}
V(H) =&  m_\rho^4 \frac{\sin \theta_{t_L} \sin \theta_{t_R} }{16 \pi^2} \left(
\alpha \cos (H/f) +
 \beta \sin^2 (H/f)
+\gamma \sin^4 (H/f) \right),
\label{eq:CH_potential}
\end{eqnarray}
where $\alpha, \beta, \gamma$ are numbers of order 1 and $\theta_{t_L}$ and $\theta_{t_R}$ are the mixing controlling the compositeness of left-handed and right-handed top quarks respectively. 
% subject to selection rules following the transformation properties of the top quark under the global symmetries of the strong sector. The coefficient $\gamma$ also generically comes with an extra power of the top compositeness fractions, and $m_\rho \approx g_\rho f$ is the typical mass scale of the strong sector resonances. 
The gauge contribution to the potential takes the form ($g$ denotes the SU(2) gauge coupling, $g_\rho f$ is the typical coupling of the strong sector and $m_\rho \approx g_\rho f$ is the typical mass scale of the strong sector resonances)
\begin{equation}
m_\rho^4 \frac{g^ 2 /g_\rho^2}{16\pi^2} \sin^2 (H/f),
\label{eq:gauge_PC}
\end{equation}
which is parametrically suppressed with respect to the top contribution  by $g^2/(g_\rho y_t)$. The gauge term is always positive, and cannot trigger electroweak symmetry breaking by itself. When $\alpha=0$, the minimization condition of the potential simply reads
\begin{equation}
\sin^2 \frac{\langle H\rangle}{f} = - \frac{\beta}{2\gamma},
\label{eq:VEV_PC}
\end{equation}
which implies that the natural expectation is that the scale $f$ is generically of the order of the weak scale. Obtaining $v \ll f$, as required phenomenologically, requires some degree of tuning, which scales like $\xi \equiv v^2/f^2$. A mild tuning of the order of 10\% ($\xi\approx 0.1$) is typically enough to comply with electroweak precision constraints. This is an important point: in partial compositeness models, the entire Higgs potential is generated at one loop, therefore the separation between $v$ and $f$ can only be obtained at a price of a tuning. 
%This marks a difference with respect to the Little Higgs models, which realize a parametric hierarchy between the quartic and mass terms through the collective symmetry breaking mechanism. In fact in Little Higgs models, the quartic coupling is a tree-level effect, leading to a potential
%\begin{equation}
%V(H)
%\approx
%\frac{g_{\rm SM}^2}{16 \pi^2} m_\rho^2 H^2 + g_{\rm SM}^2 H^4,
%\label{eq:V_PV}
%\end{equation}
%where $g_{\rm SM}$ generically denotes the SM couplings. The minimization condition now reads $v^2/f^2 \sim g_\rho^2/(16\pi^2)$, therefore $v$ is formally loop suppressed with respect to $f$. This is the major achievement of the Little Higgs constructions, which however comes at the price of the presence of sub-TeV vectors carrying EW quantum numbers and therefore giving rise generically to large oblique corrections to the propagators of the $W$ and the~$Z$ gauge bosons. 

After minimization, the potential~(\ref{eq:CH_potential}) leads to an estimate of the Higgs mass as
\begin{equation}
m_H^2 \approx {g_\rho^3\, y_t}{2 \pi^2} v^2.
\label{mH_PC}
\end{equation}
It follows that the limit $f\to \infty$, i.e. $\xi \to 0$, is a true decoupling limit: all the resonances of the strong sector become heavy but the Higgs whose mass is protected by the symmetries of the coset $G/H$. When compared to the experimentally measured Higgs mass, this estimate puts an upper bound on the strength of the strong interactions: $g_\rho \ltsim 2$. In this limit of not so large coupling, the Higgs potential receives additional contributions. In particular, the fermionic resonances in the top sector which follow from the global symmetry structure of the new physics sector can 
help raise the Higgs mass. 
For instance in the minimal SO(5)/SO(4) model, using some dispersion relation techniques, one obtains~\cite{Matsedonskyi:2012ym}
\begin{equation}
m_H^2 \approx \frac{6}{\pi^2} \frac{m_t^2}{f^2} \frac{m_{Q_4}^2 m_{Q_1}^2}{m_{Q_1}^2-m_{Q_4}^2} \log \left( \frac{m_{Q_1}}{m_{Q_4}} \right)
\label{eq:light_partners}
\end{equation}
where $Q_4$ and $Q_1$ are fermionic color resonances transforming respectively as a weak bidoublet and a weak singlet.% transforming as a weak bi-doublet of hypercharge $Y=1/6$ and $Y=7/6$ and a weak singlet with hypercharge $Y=-1/3$. 
Therefore a 125\,GeV mass can be obtained if at least one of the fermionic resonances is lighter than $\sim 1.5\, f$. As in supersymmetric scenarios, the top sector is playing a crucial role in the dynamics of electroweak symmetry breaking and can provide the first direct signs of new physics. The direct searches for these top partners, in particular the ones with exotic electric charges 5/3 appearing in minimal models, are already exploring the natural parameter spaces of these models.

The main physics properties of a pseudo Nambu--Goldstone Higgs boson can be captured in a model-independent way by a few higher-dimensional operators, namely the ones that involve extra powers of the Higgs doublets and that are therefore generically suppressed by a factor $1/f^2$ as opposed to the operators that involve extra derivatives or gauge bosons and are suppressed by a factor $1/(g_\rho^2 f^2)$. The relevant effective Lagrangian describing a strongly interacting light Higgs is:
\begin{eqnarray}
{\cal L}_{\rm SILH} = & 
\frac{c_H}{2f^2} \left( \partial_\mu \left( \Phi^\dagger \Phi \right) \right)^2
%+ \frac{c_T}{2f^2}  \left( \Phi^\dagger  \overleftrightarrow{D}^\mu \Phi \right)^2
- \frac{c_6\lambda}{f^2}\left( \Phi^\dagger \Phi \right)^3 
%\nonumber\\&\ 
+ \left( \sum_f \frac{c_f \, y_f}{f^2}\Phi^\dagger \Phi  {\bar f}_L \Phi f_R +{\rm h.c.}\right). 
\label{eq:silh}
\end{eqnarray}
Typically, these new interactions induce deviations in the Higgs couplings that scale like ${\cal O}(v^2/f^2)$, hence the measurements of the Higgs couplings can be translated into some constraints on the compositeness scale, $4\pi f$, of the Higgs boson. The peculiarity of these composite models is that, due to the Goldstone nature of the Higgs boson, the direct couplings to photons and gluons are further suppressed and generically the coupling modifiers~\cite{LHCHiggsCrossSectionWorkingGroup:2012nn} scale like
\begin{eqnarray}
\kappa_{W, Z, f} \sim 1 + {\cal O}\left( \frac{v^2}{f^2} \right) \, , \quad
\kappa_{Z\gamma} \sim {\cal O}\left( \frac{v^2}{f^2} \right) \, , \quad
\kappa_{\gamma, g}  \sim {\cal O}\left( \frac{v^2}{f^2} \times \frac{y_t^2}{g_\rho^2} \right)\, ,
\label{eq:kappa_SILH}
\end{eqnarray}
where %$g_\rho$ denotes the typical coupling strength among the states of the strongly coupled sector and 
$y_t$ is the top Yukawa coupling that is assumed to be the largest interaction that breaks the Goldstone symmetry. 
%The $\kappa_{\Z\gamma,\gamma,g}$ coupling modifiers are not generated by the strong coupling operators of Eq.~(\ref{eq:silh}) but some subleading form-factor operator generated by loops of heavy resonances of the strong sector. The coupling modifiers also receive additional contributions from the other resonances of the strong sector, in particular the fermionic resonances of the top sector that are required to be light to generate a 125\,GeV Higgs mass. Some indirect information on the resonance spectrum could thus be inferred by a precise measurement of the Higgs coupling deviations. However, it was realized~\cite{Falkowski:2007hz} that the task is actually complicated by the fact that, in the minimal models, these top partners give a contribution to both $\kappa_t$ (resulting from a modification of the top Yukawa coupling) and $\kappa_\gamma$ and $\kappa_g$ (resulting from new heavy particles running into the loops) and the structure of interactions are such that  the net effect vanishes for inclusive quantities like $\sigma(gg\to H)$ or $\Gamma(H \to \gamma\gamma)$ as a consequence of the Higgs low energy theorem~\cite{Ellis:1975ap,Shifman:1979eb}. So one would need to rely on differential distribution, like the Higgs $p_T$ distribution, to become sensitive to the top partner effects in Higgs data, see for instance Ref.~\cite{Azatov:2016xik} and references therein. The off-shell channel $gg\to h^* \to 4\ell$~\cite{Azatov:2014jga} and the double Higgs production $gg\to hh$~\cite{Azatov:2015oxa} can also help to resolve the gluon loop and separate the top and top-partner contributions.
%
%The minimal composite Higgs models (MCHM) are concrete examples of the partial compositeness paradigm.
%The Higgs doublet is described by the coset space $SO(5)/SO(4)$ where a subgroup $SU(2)_L\times U(1)_Y$ is weakly gauged under which the four Goldstone bosons transform as a doublet of hypercharge 1. There is some freedom on how the global symmetry is acting on the SM fermions: in MCHM4~\cite{Agashe:2004rs} the quarks and leptons are embedded into spinorial representations of SO(5), while in MHCM5~\cite{Contino:2006qr} they are part of fundamental representations (it might also be interesting phenomenologically to consider larger representations like MCHM14~\cite{Pappadopulo:2013vca} with the SM fermions inside a representation of dimension 14). 
%The non-linearly realized symmetry acting on the Goldstone bosons leads to general predictions of the coupling of the Higgs boson to the EW gauge bosons.
%For instance, it can be shown that the quadratic terms in the $W$ and $Z$ bosons read
%\begin{equation}
%m^2_W(H) 
%\left( W_\mu 
%W^\mu+\frac{1}{2\cos^2\theta_W}
%Z_\mu Z^\mu\right)
%\end{equation}
%with $m_W(H)=\frac{gf}{2}\sin\frac{H}{f}$.
%Expanding around the EW vacuum, the expression of the weak scale is:
%\begin{equation}
%v= f \sin (\langle H\rangle/f),
%\label{eq:v_MCHM}
%\end{equation}
%and the values of the modified Higgs couplings to the $W$ and $Z$:
%\begin{equation}
%g_{HVV}=\frac{2m_V^2}{v} \sqrt{1-v^2/f^2}\, , 
%\quad
%g_{HHVV} = \frac{2m_V^2}{v^2} (1-2 v^2/f^2)\, .
%\label{eq:gHVV_MCHM}
%\end{equation}
%Note that the Higgs couplings to gauge bosons is always suppressed compared to the SM prediction. This is a general result~\cite{Low:2009di} that holds as long as the coset space is compact.

%{\color{red} [Christophe] I still need to add a little description on the top partners and the modification of the top quark couplings to link to section 3.2.}
  
In composite models, large corrections to the top couplings are naturally predicted and can be of two origins: (i) a strong mixing  with the composite dynamics  in scenarios with partial compositeness, (ii) the existence of top partners with exotic charges  and with a relatively low mass, as required to generate the correct Higgs mass 
(see Eq.~(\ref{eq:light_partners})). While internal left-right symmetry in the strong sector can be invoked to suppress the corrections to the $Z \bar{b}_L b_L$ vertex \cite{Agashe:2006at}, the top quark sector is not immuned and receives potentially big tree-level corrections. Two dimension-6 operators are responsible for correcting the $Z \bar{t}_L t_L$ vertex:
\begin{equation}\label{eq:operators_tL}
{\cal L} = i \frac{c_{Hq}}{f^2} (\overline q_L \gamma^\mu q_L)
\left(H^\dagger \overleftrightarrow{D_\mu} H\right)
+ i \frac{c'_{Hq}}{f^2} (\overline q_L \sigma^i \gamma^\mu q_L)
\left(H^\dagger \sigma^i \overleftrightarrow{D_\mu} H\right)\,.
\end{equation}
But the left-right symmetry of the strong sector forces $c'_{Hq} = - c_{Hq}$ and as a consequence, the leading correction to the $Z \bar{t}_L t_L$ vertex is equal to the correction to the $V_{tb}$ matrix element. The $Z \bar{t}_R b_R$
 vertex can receive tree-level corrections through the mixing, induced by the non-zero top mass, between $t_R$ and composite resonances with different quantum numbers. And these corrections are obviously enhanced if the top partners are light, see Ref.~\cite{Grojean:2013qca} for an extended discussion.

% At the same time, the values of the Higgs couplings to the $W$ and $Z$ are modified according to:
% \begin{equation}
% g_{HVV}=\frac{2m_V^2}{v} \sqrt{1-v^2/f^2}\, , 
% \quad
% g_{HHVV} = \frac{2m_V^2}{v^2} (1-2 v^2/f^2)\, .
% \label{eq:gHVV_MCHM}
% \end{equation}
% Note that the Higgs couplings to gauge bosons is always suppressed compared to the SM prediction. 

%{\color{red} [JRR] This paragraph should for sure be shortened. These
%  details are unnecessary.}
%
%The Higgs couplings to the fermions depend on the representation  which the SM fermions are embedded into.
%For the most commonly used embeddings, they take the following forms
%\begin{eqnarray}
%{\rm MCHM4:}& \ g_{Hff}= \frac{m_f}{v} \sqrt{1-v^2/f^2}\, , \nonumber\\
%{\rm MCHM5:}& \ g_{Hff}= \frac{m_f}{v} \frac{1-2 v^2/f^2}{\sqrt{1-v^2/f^2}}\, , \nonumber\\
%{\rm MCHM14: }& \ g_{Hff}= \frac{m_f}{v} \left( 1 + A(M_{1,4,9}) \frac{v^2}{f^2} + O(v^4/f^4) \right)\, ,\nonumber\\
%& {\rm with }\ A(M_{1,4,9}) = \frac{3 M_1 M_4 -11 M_1 M_9 + 8 M_4 M_9}{2 M_9 (M_1-M_4)}\, .
%\label{eq:gHff_MCHM}
%\end{eqnarray}
%While, in MHCM4 and MCHM5, the modifications of the couplings depend only on the Higgs compositeness scale, in MCHM14 the leading corrections  depend also on the mass spectrum of the resonances parametrized by $M_1,M_4$ and $M_9$~\cite{Pappadopulo:2013vca}. This is due to the fact that more than one SO(5) invariant gives rise to SM fermion masses.  The $(\kappa_V, \kappa_f)$ experimental fit of the Higgs couplings can  be used to derive a lower bound on the Higgs compositeness scale $4 \pi f \gtsim 9$\,TeV, which is less stringent than the indirect bound obtained from EW precision data, $4 \pi f \gtsim 15$\,TeV~\cite{deBlas:2016ojx} but more robust and less subject on assumptions~\cite{Grojean:2013qca}.


\subsection{Little Higgs}

Little Higgs models originally were introduced as minimal cases of
models with deconstructed
dimensions~\cite{ArkaniHamed:2001nc, ArkaniHamed:2002qx}. The main
motivation behind those models is to eliminate traces of nearby new
strong interactions in electroweak precision data, and to eliminate
the sensitivity to new physics in the Higgs sector perturbatively at
the one-loop level. Generally speaking, Little Higgs models are
composite models with an additional (global and local) symmetry
structure which allows to greatly reduce the fine tuning compared to
plain composite models. Paradigm implementations are the Littlest
Higgs~\cite{ArkaniHamed:2002qy} and the Simplest Little
Higgs~\cite{Schmaltz:2004de}. The general feature of both types of
models (with global symmetries of either simple Lie algebras or direct
sums) are heavy vector-like fermions, new heavy vector bosons and
modifications of the Higgs couplings of the order $v/f$, the ratio of
the vacuum expectation values of the electroweak symmetries and that
of the Little Higgs symmetries. In many, but not all, models an
extended Higgs sector is predicted. In general, these additional heavy
Higgs bosons are even more difficult to detect at the LHC than those
of the generic, supersymmetric or not, Two-Higgs-Doublett models. 
The mixing with the heavy vector-like fermions leads
to deviations of the electroweak and Yukawa couplings of the SM
fermions, with the effect generically being proportional to the mass
of the SM fermion. This can be seen in a benchmark model in
section~\ref{subsec:ILC_top}.

In many setups of Little Higgs models (as well as composite Higgs
models, too), there are additional $U(1)$ factors in the global
symmetry structure there are similar to the $U(1)$ that gives rise to
the $\eta$ and $\eta^\prime$ masses in the SM. These global symmetries
lead to (sometimes very) light pseudoscalar particles that decay
into the heaviest accessible fermions of the SM (mostly $b$s), two
gluons or two photons~\cite{Kilian:2004pp}.



\subsection{Twin Higgs or Neutral Naturalness models}

%{\color{red} [Christophe] This is a first attempt, again strongly inspired from the summary I wrote to the PDG Higgs chapter}

In all composite models presented above, the particles responsible for canceling the quadratic divergences in the Higgs mass are charged under the SM gauge symmetries. In particular, the top partner carries color charge, implying a reasonably large minimal production cross section at the LHC. An alternative scenario, which is experimentally quite challenging and might explain the null result in various new physics searches, is the case nowadays referred to as ``neutral naturalness"~\cite{Chacko:2005pe, Chacko:2005un,Craig:2015pha}, where the particles canceling the 1-loop quadratic divergences are neutral under the SM. The canonical example for such theories is the Twin Higgs model of Ref.~\cite{Chacko:2005pe, Chacko:2005un}. This is an example of a pseudo-Goldstone boson Higgs theory, with an approximate global $SU(4)$ symmetry broken to $SU(3)$.  The Twin Higgs model is obtained by gauging the $SU(2)_A\times SU(2)_B$ subgroup of $SU(4)$, where $SU(2)_A$ is identified with the SM $SU(2)_L$, while $SU(2)_B$ is the twin $SU(2)$ group. Gauging this subgroup breaks the $SU(4)$ symmetry explicitly, but   quadratically divergent corrections given do not involve the Higgs boson when the gauge couplings of the two SU(2) subgroups are equal, $g_A=g_B$. The $SU(4)\to SU(3)$ breaking will also result in the breaking of the twin $SU(2)_B$ group and as a result three of the seven Goldstone bosons will be eaten, leaving 4 Goldstone bosons corresponding to the SM Higgs doublet $h$.  In fact imposing the $Z_2$ symmetry on the full model will ensure the cancellation of all 1-loop quadratic divergences to the Higgs mass. Logarithmically divergent terms can however arise for example from gauge loops, leading to a Higgs mass of order $g^2 f /4\pi$, which is of the order of the physical Higgs mass for $f \sim 1$\,TeV. The quadratic divergences from the top sector can be eliminated if the $Z_2$ protecting the Higgs mass remains unbroken by the couplings that result in the top Yukawa coupling. This can be achieved by introducing top partners charged under a twin $SU(3)_c$. In this case the quadratic divergences are cancelled by top partners that are neutral under the SM gauge symmetries. 

Neutral naturalness models are the best motivated scenarios featuring exotic Higgs decays, $h \to XX \to \textrm{SM}$, with displaced vertices, the intermediate decay products being a hidden glueball or a hidden quarkonia that both decay back to the SM via an off-shell Higgs boson~\cite{Craig:2015pha}.




05_ILCandLHC.tex
%\section{ILC and LHC, Complementarity and Synergy}
% \label{sec:ILCandLHC}

%{\color{red} [content: compare the parameter space regions to be covered by the two machines in various BSM scenarios. Recap the expected discovery scenario in each of the four regions (both ILC and LHC find NPs, only LHC finds NPs, only ILC finds NPs, both find no NP). Demonstrate complementarity and synergy of the two machine in (four?) summary tables for the four regions. Emphasize that LHC is a great machine but in order to fully elucidate BSM at the EW scale and beyond, we need ILC. ] }\\

%%% moved to conclusions!
% In this paper, we have shown that in all of the three LHC discovery scenarios the ILC has significant potential to discover phenomena beyond the Standard Model. Precision measurements at the ILC have in many cases sensitivity to the new physics scale that far exceeds that of the direct searches at the LHC, ensuring the ILC's capability to probe new physics whatever happens at the LHC.
% This is due mostly to the traditional virtues of $e^+e^-$ colliders such as a well-defined initial state provided by well-controlled collisions of point-like particles, the clean environment without QCD background, and democratic production of any particles with electroweak charges via $e^+e^-$ annihilation, which lead to complementarity of and synergies between the physics programs of ILC and LHC. 

%It should be, however, emphasized that the LHC is a great machine with a high discovery potential.

The physics program of the ILC and its resulting discovery potential show a high degree of complementarity with the LHC.
But there will be also many synergies between the two machines:

The huge production rates at LHC enable, for example, a high sensitivity to rare Higgs decays with distinctive signatures such as $h \to \gamma\gamma$, $Z\gamma$, and $\mu^+\mu^-$, though measurements of their absolute branching fractions are limited by various systematic uncertainties and the lack of a possibility for model-independent total cross-section measurement at the LHC. The systematic uncertainties can be largely cancelled by taking their ratios to $h \to ZZ^*$, and together with the ILC's model-independent precision measurement of the total Higgs strahlung cross section, we can hence achieve a percent level measurements for these rare modes, which is a notable synergy of the ILC and the LHC. Independently of the direct production of new particles at the LHC or the ILC, precision measurements of all properties of the Higgs boson (c.f.\ section~\ref{subsec:ILC_Higgs}), along with those of the top quark (c.f.\ section~\ref{subsec:ILC_top}) and the $W$ and $Z$ bosons(c.f.\ section~\ref{subsec:ILC_ew}) will tell us a lot about the -- to date unknown -- energy scale of new physics, which is essential input to the physics case of energy upgrades of the ILC itself or of higher energy hadron colliders.

However, we stress that the ILC's capability is not limited to precision measurements, but that also searches for direct production
of new particles will add important information to our picture of our universe, 
%If new particles are directly accessible, precision measurements will help characterize the new particle and hence shed light on the underlying new physics. We have seen above that this is indeed the case for various models described in section~\ref{sec:BSM}.
because direct searches at the LHC and the ILC (c.f.\ section~\ref{sec:ILC_NP}) cover different parts of the parameter space: the LHC has in general a higher reach for heavier, coloured states, while ILC has higher sensitivity to subtle signals from weakly coupled new particles.
%
% Generic WIMP: 4.1 (Fig.5 left, right)
%	Lambda to 3TeV (4.5TeV) at 500 GeV (1TeV)
%	Once discovered, WIMP mass and interaction structure can be determined.
% SUSY with no loopholes: 4.2 (Fig.6a,b)
%	Discovery possible almost to the kinematical limits.
% SUSY DM: 4.3 (Fig.7a,b,c)
%	Co-annihilation preferred to be consistent with the observed cosmic DM relic density,
%	leading to a small mass difference.
%	Once discovered, possible to test if it saturates the observed relic density. 
% Light Higgsinos: 4.4 (Fig.8a,b)
%	Fundamental requirement of RNS.
%	Underlying parameter determinations and GUT scale physics 
% Additional Higgs Bosons: 4.5 (Fig.9a,b)
%	 light scalar and pseudo scalar < mZ)
% RPV SUSY: 4.6 (Fig.10a,b)
%	bino pair production (B->Wl) -> BR related to neutrino mixing.
%
A typical example of the ILC-LHC complementarity in parameter space coverage is from the generic WIMP search discussed in sections~\ref{subsec:directNP_WIMP},~\ref{subsubsec:noNP_WIMPs},~\ref{subsubsec:lightNP_WIMPs}, and~\ref{subsubsec:heavyNP_WIMPs}: The LHC is sensitive to WIMP couplings to quarks, while the ILC is sensitive to WIMP couplings to electrons.
While the LHC has a higher WIMP mass reach, the ILC has a higher sensitivity to smaller couplings, thus higher mediator mass scales.
In SUSY DM models, dark matter co-annihilation with NLSP is preferably used to reduce the DM relic density, which in turn requires a small mass difference, resulting in subtle signals. 
The ILC is capable of detecting these subtle signals as shown in sections~\ref{subsec:directNP_NoLoophole} and~\ref{subsec:directNP_SUSYDM}
Once discovered, the ILC's clean environment allows us to characterize the WIMP dark matter.

The LHC-ILC complementarity also applies to searches for other new particles including SUSY particles (c.f.\ section~\ref{subsubsec:noNP_loopholefree} and~\ref{subsubsec:noNP_NatSUSY}) or extra Higgs bosons (c.f.\ section~\ref{subsubsec:noNP_SUSYHiggs}).
While the LHC has largest sensitivity to colored SUSY particles such as gluino cascade-decaying into 
lighter uncolored SUSY particles with large energy release, the ILC is going to look directly for uncolored SUSY particles with a small mass gap and hence small energy release.
As detailed in section~\ref{subsec:SUSY}, the radiatively driven natural SUSY scenario provides a very interesting and attractive possibility, where the small mass difference is theoretically required. 
Since in this case the higgsinos reside in the blind region of the LHC in Fig.~\ref{fig:noloophole2} near the diagonal, the ILC's capability described in section~\ref{subsec:directNP_higgsinos} to cover this hole is essential to the full exploration of this scenario.
The ILC is capable not only of discovering these higgsinos but also of measuring their masses and production cross sections with high precision (c.f.\ section~\ref{subsec:directNP_higgsinos}).
These precision measurements of the higgsinos together with those of the $125$\,GeV Higgs boson will allow us to determine the model parameters such as $\mu$, $M_1$, $M_2$, etc.\ as demonstrated in section~\ref{subsubsec:noNP_NatSUSY}.

The LHC-ILC complementarity and synergy becomes most prominent in the scenario described in section~\ref{subsec:lightNP}, where the LHC experiments discover relatively light new particles. Particle masses and mixings measured at the ILC will then be important inputs to disentangle complicated cascade decays of SUSY particles at the LHC, as discussed in section~\ref{subsubsec:lightNP_SUSY}.
If a gluino is found at the LHC, we can test the gaugino unification as shown in section~\ref{subsubsec:lightNP_NatSUSY}, opening up the window to GUT scale physics, thus probing the unimaginable energy scale of $10^{16}$\,GeV.
If, however, the mass unification scale turns out to be significantly lower than the GUT scale, it would be an indication of the mirage unification scenario. If the gaugino mass unification does not happen, it would discriminate certain classes of SUSY breaking scenarios.
The predicted SUSY particle masses will again be important inputs to the physics case of energy upgrades of the ILC itself or of higher energy hadron colliders.

But also in the case that LHC discovers a relatively heavy new particle (c.f.\ section~\ref{subsec:heavyNP}), both the precision measurements and the discovery potential of the ILC are highly complemetary to the information to be expected from the LHC and remain crucial for understanding the nature of the heavy new particle -- and thus for pushing further our knowledge of the universe.

%[Scenario vs discovery mode Tables]\\


%%% moved to conclusions!
% The current LHC Run 2 data have found no other new particles beyond the Higgs boson with a mass of $125$\,GeV. 
% But a factor 100 more data will still be collected during the next 10-15 years
% which enhances the LHC discovery prospects in the strongly interacting 
% sector, but especially in the electroweak sector, where typically high luminosities are needed. Therefore,
% we cannot predict today which of the three scenarios discussed in this document will come true.
% But in any of the scenarios, even in the most pessimistic case of no further discovery, 
% the ILC potential for discovering clear signs for the existence of 
% phenomena beyond the Standard Model is highly complementary 
% to the capabilities of LHC and of great importance for a 
% deeper understanding of our universe.







06_bib-rp.tex
\bibliographystyle{apsrev}

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%%%% SH 



\bibitem{Degrassi:2012ry} 
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%\cite{Richard:2014upa}
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%%%% SH end


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%\cite{Birkedal:2004xn}
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\bibitem{Ade:2015xua}                                                                                     
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%%%microomegas:                                                                                           
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06_conclusions.tex
%\section{Conclusions} 
%\label{sec:conclusions}
In this paper, we have demonstrated that the ILC has significant potential to discover phenomena beyond the Standard Model, irrespective of the results that LHC will obtain in the coming years. 

Precision measurements at the ILC have in many cases sensitivity to the new physics scales that far exceeds that of the direct searches at the LHC.  But also, there are many scenarios in which the capabilities for the direct discovery of new particles exceed those of the LHC. Both capabilities rely on the well-appreciated properties of $e^+e^-$  colliders: the well-defined initial state, the clean environment without QCD backgrounds, and the democratic production of particles with electroweak charges. They also benefit from the ILC's extendability in energy and polarised beams. Together, these features lead to synergies between the physics programs of ILC and LHC. We have discussed explicitly how these synergies play out in three scenarios for discoveries at the LHC before the turn-on of the ILC. 

The current LHC Run 2 so far has not provided evidence for other new particles beyond the Higgs boson with a mass of $125$\,GeV. Though the LHC still has a significant window for the direct discovery of new particles, this window will narrow considerably if no evidence for new particles appears in this year's results  (with $30$-$40$\,fb$^{-1}$ at $13$\,TeV). On the other hand, the fundamental mysteries of particle physics will not have gone away. This puts additional emphasis on the Scenario 1 that we have discussed here and makes it especially important to prepare an alternative route for the discovery of new physics beyond the Standard Model. We hope very much that the LHC will break through to the discovery of new particles in the next few years. However, even in the most pessimistic case, the ILC offers distinct and very powerful strategies for the discovery of phenomena that lie beyond the Standard Model of particle physics, phenomena that will illuminate physics both at small distances and in the large-scale makeup of the universe. 



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%\cite{Baak:2013fwa}
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%\cite{Agashe:2013hma}
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%\cite{CMS:2013xfa}
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%\cite{}
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%\cite{Harnik:2013aja}
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%\cite{Godbole:2011hw}
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  %%CITATION = ARXIV:1603.04764;%%  
  
  \bibitem{Simon:2016pwp} 
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%%%%%% SH 



%\cite{Richard:2014upa}
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\bibitem{Godfrey:2005pm}
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\bibitem{Godfrey:2000hc}
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%\cite{Birkedal:2004xn}
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%\cite{Dreiner:2012xm}
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%\cite{Chae:2012bq}
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\bibitem{Bartels:2012ex}
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\bibitem{Baltz:2006fm}
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%\cite{deVries:2015hva}
\bibitem{deVries:2015hva}
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%\cite{Lehtinen:2016qis}
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                                                                           \bibitem{Bechtle:2009em}
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\bibitem{Berggren:2015qua}                                                 M.~Berggren  {\it et al.},                                                 %, A.~Cakir, D.~Kr{\"u}cker, J.~List, I.~A.~Melzer-Pellmann, B.~S.~        %  Samani, C.~Seitz, and S.~Wayand,                                        %``Non-Simplified SUSY: Stau-Coannihilation  at LHC and ILC'',
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%\cite{Baer:2016new}
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%\cite{Baer:2016new}
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%\cite{Potter:2015wsa}
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%\cite{Potter:2015wsa}
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%\cite{List:2013dga}
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\bibitem{Aad:2014vma}
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%%CITATION = ARXIV:1307.7292;%%.

\bibitem{Heister:2002mn}
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\bibitem{Baer:2013ula}                                                     
H.~Baer and J.~List, %``Post LHC8 SUSY benchmark points for ILC physics,''
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[\href{http://arxiv.org/abs/1307.0782}{{\ttfamily arXiv:1307.0782 [hep-ph]}}].                            
%%CITATION = ARXIV:1307.0782;%%.                                                                          

\bibitem{Baer:2013yha}
  H.~Baer, V.~Barger, P.~Huang, D.~Mickelson, A.~Mustafayev, W.~Sreethawong and X.~Tata,
  %``Same sign diboson signature from supersymmetry models with light higgsinos at the LHC,''
  Phys.\ Rev.\ Lett.\  {\bf 110} (2013) no.15,  151801
  doi:10.1103/PhysRevLett.110.151801
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  %42 citations counted in INSPIRE as of 22 Nov 2016

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  H.~Baer, A.~Mustafayev and X.~Tata,
  %``Monojet plus soft dilepton signal from light higgsino pair production at LHC14,''
  Phys.\ Rev.\ D {\bf 90} (2014) no.11,  115007
  doi:10.1103/PhysRevD.90.115007
  [arXiv:1409.7058 [hep-ph]].
  %%CITATION = doi:10.1103/PhysRevD.90.115007;%%
  %48 citations counted in INSPIRE as of 22 Jan 2017

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  H.~Baer, V.~Barger, J.~S.~Gainer, P.~Huang, M.~Savoy, D.~Sengupta and X.~Tata,
  %``Gluino reach and mass extraction at the LHC in radiatively-driven natural SUSY,''
  arXiv:1612.00795 [hep-ph].
  %%CITATION = ARXIV:1612.00795;%%
  %1 citations counted in INSPIRE as of 22 Jan 2017
 
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  A.~Ahmed, B.~M.~Dillon, B.~Grzadkowski, J.~F.~Gunion and Y.~Jiang,
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%\cite{Senaha:2013fva}
\bibitem{Senaha:2013fva}
  E.~Senaha,
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  %%CITATION = ARXIV:1305.1563;%%
  %3 citations counted in INSPIRE as of 21 Nov 2016

 

%\cite{Ellwanger:2016qax}
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R.~Malm, M.~Neubert, and C.~Schmell, ``{Higgs Couplings and Phenomenology in a
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\bibitem{Ellwanger:2016qax}
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  JHEP {\bf 1605} (2016) 114
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  %%CITATION = doi:10.1007/JHEP05(2016)114;%%
  %44 citations counted in INSPIRE as of 25 Nov 2016
  

\end{thebibliography}

30_ILCcap.tex
The ILC physics programme and its added value with respect to the (HL-)LHC has been
studied extensively in the past years and is documented in detailed reports~\cite{Baer:2013cma, Moortgat-Picka:2015yla, Djouadi:2007ik, Asner:2013psa, Baer:2013vqa, Baak:2013fwa, Agashe:2013hma, Vos:2016til},
including a recent update~\cite{Fujii:2015jha} according to a 20-year running strategy~\cite{Barklow:2015tja}.
In this section, we will briefly recall the most important precision measurements and highlight their
implications for physics beyond the Standard Model. We will furthermore summarize the potential to
discover new particles by direct production and present some new results in this area. 

31_ILC_Higgs.tex
%\subsection{Higgs Measurements at ILC}
%\label{sec:ILC_Higgs}
\paragraph{Higgs Couplings to Fermions and Gauge Bosons:}

With the complete ILC running scenario~\cite{Barklow:2015tja}, the couplings of the H-125 to SM fermions and gauge bosons can be
precisely determined in a model-independent way, most of them to the level of at least $1\%$ or better~\cite{Fujii:2015jha}, thus
typically gaining an order of magnitude compared to HL-LHC projections~\cite{CMS:2013xfa, bib:ATLAS_HLLHC_Higgs}. Notably
the couplings to the $Z$ and the $W$ boson are expected to be measured to better than $0.5$\% at the ILC, while invisible decay modes will be constrained to be less than $0.3$\% at $95$\%\,CL.
Beyond measuring the absolute size of the couplings, also their $CP$ properties can be probed with high precision at the ILC.
A recent full simulation study showed that 
at the ILC $\sqrt{s}=250$ GeV the $CP$ phase $\phi$ can be measured with a precision of
$3.8^\circ$ using $e^+e^-\to Zh$ and 2 $\rm{ab}^{-1}$ data \cite{Jeans:2016}, which is already a factor 2-3 better than 
expected at the HL-LHC~\cite{Harnik:2013aja}.
Better sensitivities are expected in the full H20 ILC run scenario by further including the data at $\sqrt{s}=500$ GeV. 
It is also possible to probe the Higgs $CP$ mixing in $htt$ coupling
at the ILC using the channel $e^+e^-\to t\bar{t}h$~\cite{BhupalDev:2007ftb,Godbole:2011hw,Hagiwara:2016rdv}, for which
%through the event rate is very limited and experimental reconstruction of the observable sensitive to
%$CP$ mixing angle is challenging. Nevertheless it is estimated, at the generator level, that 
a sensitivity for $\phi$ of $\sim15^\circ$ has been estimated on generator level assuming $2.5$\,ab$^{-1}$ of data at 1 TeV \cite{Asner:2013psa}.

%\label{subsubsec:Higgs_coup}
%{\color{red} [content: possible size of deviations in typical BSM scenarios, model-independency at ILC, H-20 projections, added value w.r.t.\ HL-LHC] }\\
% JL: to sec 2? 
% As a guideline, in SUSY and in composite Higgs models, one expects
% deviations of the size
% \begin{align}
% \kappa^{\rm SUSY}_b = \kappa^{\rm SUSY}_\tau \approx 1 + 1.6\% 
%            \left( \frac{1~{\rm TeV}}{M_A} \right)^2, \quad
% \kappa^{\rm comp}_V \approx \kappa_F^{\rm comp} \approx 1 - 3\%
%            \left( \frac{1~{\rm TeV}}{f} \right)^2
% \label{eqn:higgs_deviations}           
% \end{align}
% for deviations in the couplings to bottoms/taus (SUSY) and in the
% couplings to all SM gauge bosons and fermions (composite Higgs). Here
% $M_A$ denotes the general Higgs boson mass scale (SUSY) and $f$ the
% compositeness scale (composite Higgs). 



\paragraph{Higgs Self-Coupling:}
%\label{subsubsec:Higgs_self}
%{\color{red} [content: possible size of deviations in typical BSM scenarios, H-20 projections, added value
%w.r.t.\ HL-LHC] }\\
% JL: to Sec 2?
% Measuring the Higgs self-coupling allows the direct probe of the Higgs field potential, which therefore is crucial to unveil the 
% mystery of electroweak symmetry breaking. In the BSM such as electroweak baryogenesis models \cite{Senaha:2013fva}
% or the scale invariant models \cite{Hashino:2015nxa}, 
% the value of trilinear Higgs self-coupling ($\lambda$) can be largely deviated, from 20\%-100\%. 
At the ILC, current projections based on the full simulation studies suggest that $\lambda_{SM}$ 
can be measured with a precision of 26\% at $\sqrt{s}=500$ GeV for the H20 scenario~\cite{Duerig:2016dvi}, and with a 
precision of 10\% at $\sqrt{s}=1$ TeV with 8 $\rm{ab}^{-1}$ data~\cite{Tian:2013,Kurata:2013}. 
%This provides a substantial discovery potential for models of Higgs phase transition that predict a larger value. 
%It is worth noting that the projections are obtained when the SM value of $\lambda$ is assumed. 
Recent studies \cite{Tian:2015} pointed out that if $\lambda$ deviates from its SM prediction, the projections can be dramatically modified as shown in 
Fig.~\ref{fig:HHHBSM} which shows the cross sections (left) and the expected precisions of $\lambda$ (right)
 as a function of the value of $\lambda$. In the interesting case $\lambda=2\lambda_{SM}$, 
 the cross section of $e^+e^-\to Zhh$ at $\sqrt{s}=500$ GeV will be enhanced by 60\%, and the expected
 precision on $\lambda$ will be a factor of 2 better, $\delta\lambda/\lambda\sim15\%$,
 which indicates that it is possible to not only discover the trilinear Higgs self-coupling by $7\sigma$
 significance, but also to see deviation with respect to its SM expectation with $>3\sigma$ significance, at the $500$\,GeV ILC.
\begin{figure}[ht]
  \centering
    \subfigure[]{\includegraphics[width=0.44\linewidth]{figs/xsec_coupling_BSM}}
    \hspace{0.1\linewidth}
    \subfigure[]{\includegraphics[width=0.44\linewidth]{figs/precision_BSM_HHH}}   
  \caption{(a) Cross sections of the two major double Higgs production processes at the ILC normalized to the SM values
%
%  , $e^+e^-\to Zhh$ at $\sqrt{s}=500$ GeV (red) and 
%  $e^+e^-\to \nu\bar{\nu}hh$ at $\sqrt{s}=1$ TeV (blue), 
as a function of $\lambda / \lambda_{SM}$, 
  %where values of both $\lambda$ and cross sections are normalised by their SM values. 
  (b) Expected relative precisions on $\lambda$ as a function of $\lambda/ \lambda_{SM}$ for those two processes.}
  \label{fig:HHHBSM}
\end{figure}

% It has been studied that $\lambda$ contributes to the process $e^+e^-\to Zh$ through loop correction, therefore 
% $\lambda$ can be extracted indirectly by precisely 
% measuring the total cross section of $e^+e^-\to Zh$ \cite{McCullough:2013rea}. 
% It is worth pointing out that there can be many other sources from BSM contributing to the correction of $e^+e^-\to Zh$,
% therefore it is highly model dependent to extract $\lambda$ in that way. In contrast, 
% based on a general EFT analysis it is studied that 
% $\lambda$ can be determined model independently using the double Higgs production process, 
% for instance $e^+e^-\to Zhh$ \cite{Barklow:2016}, and its systematic error from uncertainties of all other parameters in the EFT
% will be below one third of its statistical error.

% JL: at the moment this doesn't occur anywhere in the discovery scenarios - thus drop for the moment. Add Instead a few lines to
% the paragraph on couplings.
% \subsubsection{Higgs $CP$}
% \label{subsubsec:Higgs_CP}
% {\color{red} [content: possible size of deviations in typical BSM scenarios,  model-independency at ILC, H-20 projections, added value
% w.r.t.\ HL-LHC] }\\
% Discovery of $h\to\gamma\gamma$ decay mode at the LHC indicates $C=+1$ and $J\neq1$. Together with 
% angular measurements in $h\to ZZ^*\to4l$, $\gamma\gamma$ and $WW^*\to l\nu l\nu$ decay modes, hypotheses in which 
% the Higgs is a pure $\rm J^{CP}$ eigenstate other than $0^{++}$ are 
% mostly ruled out with confidence level above 99.9\% \cite{Aad:2015mxa,Khachatryan:2014kca}. 
% However it remains possible that the Higgs is an admixture of $CP$-even and $CP$-odd eigenstates,
% which is of high interests in the BSM to induce additional $CP$-violating 
% source other than that from CKM to explain the baryon asymmetry in the universe.
% The Higgs $CP$ admixture can be probed by precisely measuring 
% the spin correlations between the particles which couple to Higgs.
% Experimentally, this can be explored mainly through the $hVV$ couplings, $V=Z$ or $W$,
% and the $hff$ couplings, $f=\tau$ or $t$, where the spin direction of $V$ or $f$ can be 
% inferred from its decay plane \cite{Grzadkowski:1995rx}.
% ILC offers comprehensive channels to probe the Higgs $CP$, with much higher sensitivities to the $CP$ mixing
% comparing to the capabilities at the LHC.
% 
% 
% For the $hVV$ couplings, the contribution of $CP$-odd component may mainly come from loop effect. 
% This can be studied by measuring the general Lorentz structure of $hVV$ couplings \cite{Takubo:2010tc, Ogawa:2016}
% $$\Delta{\cal L}_{hVV}=g_V(1+{a})V_\mu V^{\mu}h+\frac{b}{\Lambda}V_{\mu\nu}V^{\mu\nu}h
% +\frac{\tilde{b}}{\Lambda}V_{\mu\nu}\tilde{V}^{\mu\nu}h,$$
% where the $\tilde{b}$ term represents the contribution from $CP$-odd component with new physics scale $\Lambda$. 
% At the ILC, the Lorentz structure of $hZZ$ coupling can be mainly studied using both Higgs-strahlung and $ZZ$-fusion 
% channels where $hZZ$ coupling is probed as the production vertex. The $hWW$ Lorentz structure can be studied 
% using both the $WW$-fusion production channel and $h\to WW^*$ decay channel.
% Based on recent full simulation studies \cite{Ogawa:2016},
% $\tilde b$ in $hZZ$ coupling
% is expected to be measured with a precision of 0.005 for $\Lambda=1$ TeV in the H20 scenario.
% This sensitivity demonstrated at the ILC, in terms of the common definition of $f_{CP}$ as in paper \cite{Anderson:2013afp},
% will be two orders of magnitude higher than 
% what would be expected at the HL-LHC. As the $CP$-odd term may well be very small due to the loop effect by
% new physics, the much higher sensitivity from ILC allows detection of effect from new physics at much higher scales.
% 
% 
% In contrast to the small contribution from $CP$-odd term in $hVV$ couplings, $hff$ couplings 
% may receive large modifications from the Higgs $CP$ mixing at the tree level, 
% $$\Delta{{\cal L}_{hff}}=-\frac{\kappa_f y_f}{\sqrt{2}}h\bar{f}(\cos\phi+i\sin\phi\gamma_5)f,$$
% where the $CP$ phase parameter $\phi$ is $0^\circ$ for a pure $CP$-even state, $90^{\circ}$ for a pure $CP$-odd state, or any other 
% value for a $CP$ admixture. At the ILC, taking advantage of the known initial states and better impact parameter resolution
% for track reconstruction, momentum of neutrinos in $\tau$ hadronic decay modes can be fully reconstructed 
% \cite{Heister:2001uh,Desch:2003mw,Rouge:2005iy,Jeans:2015vaa},
% therefore the best sensitivity of $CP$ mixing can be achieved using the $h\to\tau\tau$ decay at the ILC. 
% It is studied in paper \cite{Harnik:2013aja}, though at a generator
% level, the sensitivity to $\phi$ from $e^+e^-\to Zh\to Z\tau^+\tau^-$ at the ILC $\sqrt{s}=250$ GeV 
% will be around $4.4^\circ$ using 1 $\rm{ab}^{-1}$ data, which is
% already a factor of 2-3 better than that expected at the HL-LHC. 
% A recent full simulation study further promises that 
% at the ILC $\sqrt{s}=250$ GeV $\phi$ can be measured with a precision of
% $3.8^\circ$ using $e^+e^-\to Zh$ and 2 $\rm{ab}^{-1}$ data \cite{Jeans:2016}. 
% Better sensitivities are expected in full H20 scenario
% by further including the data at $\sqrt{s}=500$ GeV. It is also possible to probe the Higgs $CP$ mixing in $htt$ coupling
% at the ILC using the channel $e^+e^-\to t\bar{t}h$ \cite{BhupalDev:2007ftb,Godbole:2011hw,Hagiwara:2016rdv},
% through the event rate is very limited and experimental reconstruction of the observable sensitive to
% $CP$ mixing angle is challenging. Nevertheless it is estimated, at the generator level, that 
% a sensitivity for $\phi$ of $\sim15^\circ$ is achievable with 2.5 $\rm{ab}^{-1}$ data at 1 TeV \cite{Asner:2013psa}.
% 

%\subsubsection{anything else?}
%\label{subsubsec:higgs_else}

32_ILC_top.tex
\newcommand{\eplus}{e^+}
\newcommand{\eminus}{e^-}
\newcommand{\epem}{\eplus\eminus}
\newcommand{\ttbar}{ \tQ \bar{\tQ}}
\def\roots{ \sqrt{s} }
\newcommand{\pem} { {\cal P}_{\e-} } 
\newcommand{\pep} {{\cal P}_{\e+} }
\definecolor{orange}{rgb}{1,0.5,0}
\definecolor{gray}{rgb}{0.5,0.5,0.5}
\definecolor{roetlich}{rgb}{1, .7, .7}
\definecolor{camel}{rgb}{0.76, 0.6, 0.42}
\definecolor{bronze}{rgb}{0.8, 0.5, 0.2}
\definecolor{britishracinggreen}{rgb}{0.0, 0.26, 0.15}


%\subsection{Top Measurements at ILC}
%\label{sec:ILC_top}
The ILC would allow to study the $t$ quark using a precisely defined leptonic initial state. Therefore individual events can be analysed in more detail. It also changes the production mechanism for $\tQ$ quark pairs from the strong to the electro-weak interactions, which are a step closer to the phenomena of electro-weak symmetry breaking. Finally, this change brings into play new experimental observables -- weak interaction polarisation and parity asymmetries -- that are very sensitive to the coupling of the $t$ quark to possible new interactions. It is very possible that, while the $t$ quark might respect Standard Model expectations at the LHC, it will break those expectations when studied with higher precision at the ILC. 
%Unless stated otherwise all results presented in the following are based on full simulation studies of the ILC detector ILD and the CLIC\_ILDdetector. 
%
%\subsubsection{Top Mass} 
\paragraph{Top Quark Mass:} 
\label{subsubsec:top_mass}
%{\color{red} [content: H-20 projections, added value w.r.t.\ HL-LHC] }\\
%\input{321_topmass.tex}
One of the unique capabilities of an $\epem$ collider is the ability
to carry out scans of particle production thresholds.  The $\ttbar$ pair production threshold around a centre-of-
mass energy $\roots \approx 2m_t$ enables a precise measurements of the $t$ quark mass
$m_t$ in theoretically well-defined mass schemes. This is in contrast to the mass measurements at the LHC, where the highest precision is obtained in measurements relying on the use of event generators, resulting in additional, currently not well understood, uncertainties when translating the experimental result to mass definitions used in theoretical frameworks. 


%as well as the $t$ quark total width $\Gamma_t$ and the QCD coupling $\alpha_s$. Because
%the top is a spin-$\frac{1}{2}$ fermion, the $\ttbar$ pair is produced in an angular S-wave state. This leads to a clearly visible rise of the cross section even when folded with e.g. the ILC luminosity spectrum as shown in Fig.~\ref{fig:topscan}.

% JL: beatiful, but no space :-(
% \begin{figure}
% \centering
% %%\includegraphics[width=0.7\columnwidth]{ttthresh_hs.pdf}
% \includegraphics[width=0.5\columnwidth]{figs/321_TopThreshold.pdf}
% \caption{
% %%Accuracy on the prediction of the top pair production 
% %%cross section at the $t\bar{t}$~threshold at the ILC as achieved by 
% %%recent calculations of QCD corrections (NNLL). For further explanations 
% %%see text. The figure has been taken from~\cite{bib:topws-ms}.
% \sl Result of a simulation study of a top threshold scan that includes the luminosity spectrum of the ILC beams. The threshold behaviour is fitted with state-of-the-art NNNLO QCD calculations taking scale uncertainties into account. The best fit as well as mass variations of 200 MeV are shown, with the corresponding scale uncertainties represented by the area between the fit template lines.} % No citation necessary in caption - Will add short explanation with refs to text (was ~\cite{Beneke:2015kwa}) 
% \label{fig:topscan}
% \end{figure}

%Figure \ref{fig:topscan} illustrates a top threshold scan at the ILC, based on a total integrated luminosity of 200$\,\ifb$ as assumed in the H-20 running scenario. 

Using the methodology described in \cite{Seidel:2013sqa} with state-of-the-art NNNLO QCD calculations of $\ttbar$ production  \cite{Beneke:2015kwa} as input, the effects of the ILC luminosity spectrum and initial state radiation as well as signal efficiencies and background contributions are taken into account. From the simulated data points, the statistical precision as well as theoretical uncertainties based on NNNLO scale uncertainties are extracted following the techniques developed in \cite{Simon:2016htt, Simon:2016pwp}, resulting in a statistical precision of $\delta m_t \approx 13 \MeV$. At present, the scale uncertainties result in a theory systematic of $\sim$40 MeV, which is comparable to the expected experimental and parametric systematics. Table \ref{tab:Top:Systematics} summarizes the current status of the estimated uncertainties of the top quark mass measurement in a threshold scan.  These studies are performed assuming unpolarized beams. In the combined uncertainties, the lower end of the given range illustrates the effect of some mild improvements assumed on $\alpha_s$ and theory uncertainties expected by the time of ILC data taking, as well as a better suppression of non-$\ttbar$ background due to the capability for high beam polarization of electrons and positrons at the ILC.


\renewcommand{\arraystretch}{1.2}
\begin{table}
\centering
\begin{tabular}{l|c|l}
\hline
\hline 
%\\[-1em]
error source & $\Delta m_t^{\mathrm{PS}}$ [MeV] & references\\
\hline
stat. error (200 fb$^{-1}$) & 13 & \cite{Seidel:2013sqa, Simon:2016pwp}\\
\hline
theory (NNNLO scale variations, PS scheme) & 40 & \cite{Simon:2016htt, Simon:2016pwp}\\
parametric ($\alpha_s$, current WA) & 35 & \cite{Simon:2016htt}\\
\hline
non-resonant contributions (such as single top) & $< 40$ & \cite{Fuster:2015jva}\\
residual background / selection efficiency & 10 -- 20 & \cite{Seidel:2013sqa}\\
luminosity spectrum uncertainty & $< 10$ & \cite{Simon:2014hna}\\
beam energy uncertainty & $< 17$ & \cite{Seidel:2013sqa}\\
\hline
\hline
combined theory \& parametric & 30 -- 50 & \\
combined experimental \& backgrounds & 25 - 50 &\\
\hline
total (stat. + syst.) & 40 -- 75 & \\
\hline
\hline
\end{tabular}
\caption{\label{tab:Top:Systematics} Summary of the estimated uncertainties of top mass measurements at threshold. The upper parts of the table reflect the current understanding, based on the references given. In the bottom part of the table, the lower end of the given ranges corresponds to moderate assumptions on improvements expected by the time of ILC data taking, see text.}
\end{table}

% JL: beatiful, but no space, and not used later
% Extending the analysis beyond the mass alone allows a simultaneous extraction of the mass together with the width $\Gamma_t$, and, if there is an external input for the strong coupling with a precision substantially beyond the current world average, the top Yukawa coupling $y_t$. The measurement of the latter becomes possible since the virtual exchange of the Standard Model Higgs boson enhances the cross section at threshold by about 9\%, an effect that is highly correlated with gluon exchange governed 

The dependence of the $t$ quark cross section shape on the $t$ quark mass and interactions is computable to high precision with full control over the renormalisation scheme dependence of the $t$ quark mass parameter. The authors of~\cite{Marquard:2015qpa} show that the PS or 1S masses as resulting from the described analysis can be translated to e.g. the $\overline{MS}$ mass, typically used in theoretical calculations to a precision of about $10 \MeV$.

\paragraph{Top Quark Electroweak Couplings:}
%\label{subsubsec:top_ewcoup}
%{\color{red} [content: H-20 projections, added value w.r.t.\ HL-LHC] }\\
%\input{322_topewcoupl.tex}
The unique feature of linear colliders to provide polarised beams allow for a largely unbiased disentangling of the individual left- and right-handed couplings of the $t$ quark to the $Z^0$ boson and the photon, $ g_{L,R}^{\gamma, Z}$ or equivalently of the form factors $F^{\gamma, Z}_{(1,2),(V,A)}$. These quantities can be measured at the sub-percent level at the ILC~\cite{Amjad:2013tlv,Amjad:2015mma}, as indicated by the red ellipse in Figure~\ref{fig:models-rp}. This is-- when referring to the results in~\cite{Baur:2004uw,Baur:2005wi}-- considerably better than will be possible at the LHC\footnote{The improving analyses of the LHC experiments, as e.g.~\cite{Khachatryan:2015sha}, will however be observed with great interest.} even with an integrated luminosity of ${\cal L} = 3000\,\ifb$. The expected precision at the ILC would allow for the verification of a great number of models for physics beyond the Standard Model, for which representative examples are given in the figure.

\begin{figure}[ht]
  \centering
\setlength{\unitlength}{1.3mm}
%\setlength{\unitlength}{\textwidth} 
\begin{picture}(150,80)
\linethickness{0.3mm}
%\begin{center}
%compose x axis
  \put(9,33){\line(0,1){2}}
  \put(5,34){\line(1,0){8}}  
  \put(15,34){...} 
  \put(21,34){\vector(1,0){60}} 
  \put(82,33){\Large{$\delta g^Z_R / g^Z_R$}}
  \multiput(31,33)(10,0){5}{\line(0,1){2}} 
  \put(6.5,31){\footnotesize{-330\%}} 
  \put(29,31){\footnotesize{-20\%}} 
  \put(39,31){\footnotesize{-10\%}}
  \put(60,31){\footnotesize{10\%}}
  \put(70,31){\footnotesize{20\%}}
%compose y-axis
  \put(51,4){\vector(0,1){60}} 
  \put(45,66){\Large{$\delta g^Z_L / g^Z_L$}}
  \multiput(50,14)(0,10){5}{\line(1,0){2}} 
  \put(44,13.5){\footnotesize{-20\%}} 
  \put(44,23.5){\footnotesize{-10\%}} 
  \put(44.5,43.5){\footnotesize{10\%}} 
  \put(44.5,53.5){\footnotesize{20\%}}
%Add models
  \put(51,34){\color{red}\circle*{2}} 
  \put(53,36){\color{red}SM}
  \put(51,24){\color{britishracinggreen}\circle*{1.5}}
  \put(53,23){\color{britishracinggreen}Light top partners~\cite{Grojean:2013qca}}
  \put(41,24){\color{britishracinggreen}\circle*{1.5}} 
  \put(15,26){\color{britishracinggreen} Light top partners}
 \put(15,23){\color{britishracinggreen} Alternative 1~\cite{bib:panico-priv}}
  \put(76,59){\color{britishracinggreen}\circle*{1.5}} 
  \put(54,61.5){\color{britishracinggreen} Light top partners Alternative 2~\cite{bib:panico-priv}}
  %\put(53,23){\color{green}Composite Higgs with SO(5)/SO(4)}%~\cite{Grojean:2013qca}}
   \put(51,19){\color{cyan}\circle*{1.5}} 
   \put(53,18){\color{cyan}Little Higgs~\cite{Berger:2005ht}}
  \put(51,14){\color{gray}\circle*{1.5}} 
  \put(53,13){\color{gray}RS with Custodial SU(2)~\cite{Carena:2006bn}}
  \put(51,9){\color{orange}\circle*{1.5}} 
  \put(53,8){\color{orange}Composite Top~\cite{Pomarol:2008bh}}
  \put(31,14){\color{magenta}\circle*{1.5}} 
  \put(20,16){\color{magenta}5D Emergent~\cite{Cui:2010ds}}
  \put(56,29){\color{camel}\circle*{1.5}} 
  \put(57,28){\color{camel}4D Composite Higgs Models~\cite{Barducci:2015aoa}}
  \put(9,34){\color{blue}\circle*{1.5}} 
  \put(1,36){\color{blue}RS with Z-Z' Mixing~\cite{Djouadi:2006rk}}
%make a box to put the ILC precision into
\multiput(56,40)(30,0){2}{\line(0,1){12}} 
\multiput(56,40)(0,12){2}{\line(1,0){30}}
\put(61,47){\large{ILC Precision}}
%embed the covariance ellipse
\color{red}
% Ellipse:  u = 71.0  v = 43.0  a = 2.45336  b = 0.631947  phi = 16.9866 Grad
\qbezier(73.3463, 43.7167)(73.2699, 43.9671)(72.5286, 43.9342)
\qbezier(72.5286, 43.9342)(71.7873, 43.9013)(70.8154, 43.6044)
\qbezier(70.8154, 43.6044)(69.8435, 43.3075)(69.2103, 42.9205)
\qbezier(69.2103, 42.9205)(68.5772, 42.5336)(68.6537, 42.2833)
\qbezier(68.6537, 42.2833)(68.7301, 42.0329)(69.4714, 42.0658)
\qbezier(69.4714, 42.0658)(70.2127, 42.0987)(71.1846, 42.3956)
\qbezier(71.1846, 42.3956)(72.1565, 42.6925)(72.7897, 43.0795)
\qbezier(72.7897, 43.0795)(73.4228, 43.4664)(73.3463, 43.7167)
\end{picture}
\caption{\sl Predictions of several Randall-Sundrum (RS) models and/or compositeness or Little Higgs models on the deviations of the left- and right-handed couplings of the $t$~quark to the $Z^0$ boson. The ellipse in the frame in the upper right corner indicates the precision that can be expected for the ILC at $\roots = 500\,\GeV$ with ${\mathcal L}=500\ifb$ of integrated luminosity shared equally between the beam polarisations $\pem,\,\pep =\pm0.8,\mp0.3$~\cite{Richard:2014upa}. 
%{\color{orange}[Figure can be replaced by the somewhat lighter version by Michael.]}
}
\label{fig:models-rp}
\end{figure}

% slightly shortened version above
% The Refs.~\cite{Amjad:2013tlv,Amjad:2015mma} report on the determination of $CP$ conserving form factors and couplings  based on a full simulation study of the reaction $\epem \rightarrow \ttbar$ at a centre-of-mass energy of $\roots = 500 \GeV$ with 80\% polarised electron beams and 30\% polarised positron beams. The unique feature of linear colliders to provide polarised beams allow for a largely unbiased disentangling of the individual {\em L}eft and {\em R}ight handed couplings of the $t$ quark to t

Beam polarisation is a critical asset for the high precision measurements of the electroweak $t$ quark couplings. Experimental and theoretical effects manifest themselves differently for different beam polarisations. It seems to be that the configuration positive electron beam polarisation is more benign in both, experimental aspects due to the suppression of migrations in the polar angle spectrum of the final state $t$ quark, see e.g.~\cite{Amjad:2013tlv,Amjad:2015mma}  and theoretical aspects due to the somewhat simpler structure of higher order electroweak corrections~\cite{Khiem:2015ofa}. 
%It is intuitively clear that the described facts would greatly support the discovery of effects due to new physics.
%Already for the standard scenario $\roots = 500\,\GeV$ after having accumulated ${\mathcal L}=500\,\ifb$ the expected precision at the ILC would allow for the verification of a great number of models for physics beyond the Standard Model. Examples for these models are extra dimensions and compositeness, see Fig.~\ref{fig:models-rp}. 
%The current results constitute therefore a perfect basis for discussions with theoretical groups. 

% JL: moved to section 4.1
%The full luminosity of the ILC increase the sensitivity to larger mass scales as shown in Fig.~\ref{fig:topcoup-lumi}. 
% \begin{figure}
% \centering
% %%\includegraphics[width=0.7\columnwidth]{ttthresh_hs.pdf}
% \includegraphics[width=0.8\columnwidth]{figs/top_H-20.pdf}
% \caption{
% %%Accuracy on the prediction of the top pair production 
% %%cross section at the $t\bar{t}$~threshold at the ILC as achieved by 
% %%recent calculations of QCD corrections (NNLL). For further explanations 
% %%see text. The figure has been taken from~\cite{bib:topws-ms}.
% \sl Extrapolation of the precision of the electroweak $t$ quark coupling $g_L$ to the full ILC luminosity of $4000\,\ifb$ expected after 25 years of running. The figure shows at the same time the improving precision (left scale, lower curves) and the mass reach expressed in terms of $M_{KK}$ (right scale upper curves) for a typical BSM scenario, here~\cite{Pomarol:2008bh}. The full lines represent the improved statistical error while other lines take into account systematic errors (theory and/or experim
% \label{fig:topcoup-lumi}
% \end{figure}


%Examples in which the LHC discovers a new particle at the TeV scale will be discussed below. 
%The exploitation of the full luminosity requires however a sustained experimental and theoretical programme to match the expected statistical precision. 

%Note at this point that the community currently discusses running scenarios for the ILC that would yield up to eight times more luminosity~\cite{Barklow:2015tja} than has been assumed so far. Moreover it can be expected that the event reconstruction will be improved by e.g. the measurement of the $b$ quark charge.  Already from the achieved precision it is mandatory that systematics are controlled to the 1\% level or better in particular for the measurement of the cross section. A study of systematic errors will therefore become very important and is addressed in ongoing studies. 

% JL: very nice, but not used later?
% Finally, the study presented in~\cite{Khiem:2015ofa} based on generated events suggests that by exploiting the polarisation of the final state $t$ quarks a simultaneous extraction of all anomalous top form factors, including the $CP$ violating $F^{\gamma, Z}_{2,A}$, to a precision below the percent level is feasible. However, $CP$ violating effects may require a precision well below the percent level. This measurement would therefore particularly benefit from the full luminosity at the ILC .
%A detailed comparison between the advantages and drawbacks of the method applied there and the method presented in~\cite{Amjad:2015mma} is left for a future study. 

% JL: not used later - save space...
% \subsubsection{Top FCNC Decays}
% \label{subsubsec:top_fcnc}
% %{\color{red} [content: possible size of deviations in typical BSM scenarios, H-20 projections, added value w.r.t.\ HL-LHC] }\\
% \input{323_topfcnc.tex}
% %{\color{blue} [proposed contributor: Filip Zarnecki] }\\

%\subsubsection{anything else?}
%\label{subsubsec:top_else}

33_ILC_ew.tex
%\subsection{Electroweak Precision Measurements at ILC}
%\label{sec:ILC_ew}
%\subsubsection{Two-fermion production}
%\label{subsubsec:ew_2f}
\paragraph{Two-fermion production:}
Precise measurements of $e^+e^- \to f \bar f$ for all types of fermions, 
making use of polarized beams, are a stringent test
on the existence of possible forces between matter particles beyond
those in the SM. Those comprise gauge interactions (generically labelled $W'$ and $Z'$), scalar resonances
like the Higgs or also tensor resonances as in gravity models or
models with new strong interactions. Also compositeness or partial
compositeness of fermions could lead to deviations of two-fermion
processes from their SM values due to mixing effects. 
In $e^+e^-$ collisions (far) below the production threshold for the extra gauge
bosons, they manifest themselves as deviations from SM predictions due to
interference between the new physics and the SM $\gamma/Z$ contributions,
see~\cite{Freitas:2013xga} for a recent review. 
This is similar to indirect observations of the presence of the $Z$ boson
in $e^+e^-\rightarrow \mu^+\mu^-$ scattering at the PETRA and TRISTAN 
colliders well before the turn-on of LEP. 

Studies for the ILC~\cite{Osland:2009dp} have shown that
already with $500$\,fb$^{-1}$ at 500\,GeV or with $1$\,ab$^{-1}$ at 1~TeV ILC 
evidence for a $Z'$ with mass exceeding $\sim 7$\,TeV and $\sim 12$\,TeV can be obtained in 
many models, respectively. The measurements will allow to distinguish between
a variety of different $Z'$ models by measuring their vector and axial-vector couplings, starting already at 
$\sqrt{s} = 500$~GeV~\cite{Godfrey:2005pm}. Similar numbers hold also for
$W'$~bosons~\cite{Godfrey:2000hc}. 

% JL: sorry, no space :(
% \begin{figure}
% \centering
% \includegraphics[width=0.9\columnwidth]{figs/Zprime.pdf}
% \caption{
%  Using the processes $e^+e^-\to\ell^+\ell^-$ (left) and $e^+e^-\to
%  b\bar b$ (right) allows determination of vector and axial-vector
%  couplings of new $Z'$ gauge bosons, i.e. new neutral weak currents 
%  (from Ref.~\cite{Baer:2013cma}). 
%  }
% \label{fig:zprime}
% \end{figure}

% \subsubsection{$W$ Mass}
% \label{subsubsec:ew_mw}
% {\color{red} [content: ILC projections, added value
% w.r.t.\ HL-LHC, impact on SM/BSM parameter space] }\\
%{\color{blue} [proposed author: Graham Wilson] }\\
%{\color{dgreen} [volunteer: JRR] }\\
\paragraph{$W$ Mass:}
At the ILC, the $W$ mass can be measured by three different methods:
(i) by a polarized threshold scan in different final states, (ii) by
measuring the $\ell\nu jj$ and partially also the $jjjj$ process at
the design energies of 250, 350 and 500\,GeV as well as (iii) from
single-$W$ production ($e^+e^- \to W \ell\nu$) favorably at the
highest possible energies. Both methods (ii) and (iii) rely on
kinematic reconstruction of the $W$ system similar to the top-quark
mass measurement at hadron colliders, while (i) uses in principle
template fits for the $WW$ threshold curve. These methods have been
shown to be able to reach precisions of $\delta M_W = 3$\,MeV~\cite{Baak:2013fwa},
which presents an improvement of a factor $5$ w.r.t.\ current precisions.

\paragraph{Weak mixing angle $\sweff$:}
When exploiting the option to operate the ILC at the $Z$ pole with polarised beams, collecting
about 1000 times more events than at LEP, the effective weak mixing angle can be measured
about a factor~10 better than today~\cite{Baak:2013fwa}. 

% pdflatex seems to frach due to next statement
%\subsubsection{$\boldmath{\sin^2{\theta_{\mathrm eff}}}$}
% \label{subsubsec:ew_sinthetaeff}
% {\color{red} [content: possible size of deviations in typical BSM scenarios, ILC projections, added value
% w.r.t.\ HL-LHC] }\\
%{\color{dgreen} [volunteer: JRR] }\\



% JL: no space, not used later, and 1 TeV rather add something about cTGCs here?
% \subsubsection{Anomalous gauge couplings}
% 
% {\color{red} [content: possible size of deviations in typical BSM scenarios, ILC projections, added value
% w.r.t.\ HL-LHC] }
% 
% Next to the measurements of trilinear electroweak self-couplings in
% $WW$ and $ZZ$, the measurements of quartic gauge couplings allows a
% stringent test of the gauge sector of electroweak interactions. Both
% (weak) vector boson scattering and triple production of heavy gauge
% \begin{figure}
% \centering
% \includegraphics[width=0.45\columnwidth]{figs/MinvW_1000GeV.pdf}
% \includegraphics[width=0.45\columnwidth]{figs/MinvZ_1000GeV.pdf}
% \caption{
%   Deviations from the SM in terms of dimension-8 operators
%  (coefficients $F_{S,0}$, $F_{S,1}$) in the processes $\nu\nu W^+W^-$
%  and $\nu\nu ZZ$, respectively.}
% \label{fig:vbs}
% \end{figure}
% bosons are among the rarest processes in the SM. At LHC, there has
% been evidence on like-sign $W^+W^+$ scattering from both experiments,
% and the measurement will heavily profit from large statistics in the
% high-luminosity phase. Due to very unfavorable signal-over-background
% ratios (particularly the large top background), opposite-sign $W+W^-$
% scattering and also the $ZZ$ final state will most likely not be
% accessible at the LHC even at highest luminosity. It is not yet clear
% whether it will be possible to separate longitudinal and transversal
% degrees of freedom. The ILC offers complementarity in those
% measurements as it allows precisely to access the channels that are
% precluded at the LHC. Fig~\ref{fig:vbs} shows deviations from the SM
% in quartic gauge couplings parameterized in terms of dimension-8
% operators with coefficients $F_{S,0}$, $F_{S,1}$, cf.~\cite{Fleper:2016frz}.


%\subsubsection{anything else?}
%\label{subsubsec:ew_else}

34_ILC_directNP.tex
% \subsection{Direct Production of New Particles at ILC}
% \label{sec:ILC_NP}
In this section, we summarize the potential of ILC for directly 
producing new particles, thus updating an earlier 
Snowmass white paper~\cite{Baer:2013vqa}. 
In particular, we highlight cases where the discovery potential of ILC 
is complementary to that of the LHC.

\subsection{Generic WIMPs}
\label{subsec:directNP_WIMP}
%{\color{red} [content: ILC discovery reach, H-20 projections, added value w.r.t.\ HL-LHC, direct/indirect detection] }\\
%{\color{dgreen} [volunteer contributor: JL] }\\
%%% The following is nice, but should go to sec 4 / 5
%The ILC has unique capabilities to detect weakly-interacting massive particles (WIMPs), which are complementary to those
%of the LHC and of direct detection experiments:
%As electron-positron collider, ILC is sensitive to DM couplings to electrons, whereas hadron colliders and direct detection experiments are sensitive to DM couplings to quarks. Moreover, while LHC can probe larger WIMP masses due to its higher center-of-mass energy, ILC can probe higher energy scales for the WIMP-electron interaction due to its higher precision.

%The primary probe at the ILC for the direct production of WIMP dark matter are photons
%emitted as initial-state radiation in association with the pair production of dark matter.
%The main backgrounds to this search are the radiative neutrino production, which is irreducible,
%and the radiative Bhabha scattering process, in which the outgoing electron and positron escape undetected in the beam pipe.
%At the ILC, WIMP pair production can be searched for with the help of an ISR photon, in analogy to mono-X searches at the LHC.
The prospects to detect WIMPs at the ILC and to determine their properties have been studied at a theoretical level~\cite{Baer:2001ia,Birkedal:2004xn,Dreiner:2012xm,Chae:2012bq} and specifically for a center-of-mass energy of $500$\,GeV in detailed detector simulation~\cite{Bartels:2012ex}. The experimental prospects have been interpreted both in the framework of effective
operators\footnote{Note that under ILC conditions the effective field theory approximation is accurate, while it is questionable
in similar analyses at hadron colliders. Reinterpretations of ILC projections in simplified models are in preparation for comparison with upcoming LHC results.} as well as in a cosmological approach, where the annihilation fraction $\kappa_e$ of WIMPs
into electron-positron pairs is a free parameter, assuming that the WIMP makes up all the dark matter in the universe.
As can be seen in the left panel of figure~\ref{fig:WIMPs}, evidence for WIMP production could be obtained over a wide range
of masses already with an initial ILC dataset of $500$\,fb$^{-1}$, even if the annihilation fraction into $e^+e^-$ pairs provides only a few percent of the total annihilation rate~\cite{bib:chaus}. The right panel shows the extrapolation of these results to a wide range of integrated luminosities and center-of-mass energies based on an effective operator approach~\cite{bib:habermehl}. For the
full $500$\,GeV-program of the ILC, scales of new physics ($\Lambda$) of up to $3$\,TeV can be probed, while the $1$\,TeV-energy-upgrade of the ILC would extend this even to $4.5$\,TeV or more, depending on the integrated luminosity.
Once a WIMP would be discovered, its properties could be determined precisely due to the known initial state of a lepton collider~\cite{Bartels:2012ex}. 
In particular, its mass could be determined with a precision of about 1\%, and the type of operator (or the angular momentum of dominant partial wave) of the WIMP pair production process can be determined. 
By such detailed measurements of WIMP properties as offered at the ILC, it is often possible to constrain WIMP production rates in the early universe along with WIMP scattering or annihilation
rates and the local WIMP abundance~\cite{Baltz:2006fm}. Such checks could verify or falsify the simple assumptions 
associated with thermal DM production within the WIMP miracle scenario, thus giving
important insights into the nature of dark matter, as explained in section~\ref{sec:DM}. 



\begin{figure}[htb]
\setlength{\unitlength}{1.0cm}
\includegraphics[width=0.55\linewidth]{figs/Sensitivity_ke_pe08pp-03}
\hspace{0.1cm}
\includegraphics[width=0.45\textwidth]{figs/V_40_scale48}
% \end{center}
  \caption{\label{fig:WIMPs} Left: Observational reach ($3\sigma$) of the ILC for a Spin-1 WIMP in terms of WIMP mass and $\kappa_e$ for three different chiralities of the WIMP-fermion couplings~\cite{bib:chaus}.
   %(Note: At WIMP masses lower than $\sim 150\,$GeV, not all assumptions in the relation between the WIMP annihilation and $e^+e^-$ production cross-sections are fully  valid anymore. Signals of low mass WIMPs could still easily be observed, but the interpretation of $\kappa_e$ becomes somewhat less straight forward, since for too low masses the WIMP production is not non-relativistic anymore). 
  Right: Expected sensitivity for a vector operator in an EFT-based interpretation as a function of integrated luminosity and center-of-mass energy~\cite{bib:habermehl}.}
\end{figure}

\subsection{SUSY with no loopholes}
\label{subsec:directNP_NoLoophole}
%\input{no_loophole_chap3}
%%-- (Assuming that this will be in chap 2:)
%%-- The corner-stone of SUSY is that {\it sparticles couple as particles} \cite{Dimopoulos:1981zb}. 
%%-- This is independent of the
%%-- mechanism responsible for SUSY breaking. In particular, the couplings to the photon and
%%-- the Z-boson are known,
%%-- which implies that the cross-section for any \eeto ~sparticle -- anti-sparticle pair process 
%%-- that proceeds only via the s-channel 
%%-- is determined by the kinematics alone, 
%%-- i.e. by $E_{CMS}$ and the mass of the sparticle.
%%-- Contributions from sparticle exchange in the t-channel are possible
%%-- in $\snu_e , \sel, \XN{i}$ or $\XPM{i}$ pair-production,
%%-- by exchange of  a $\XPM{i} , \XN{i} , \sel $ or $\snu_e$, respectively.
%%-- Also with such diagrams contributing, all couplings are known,
%%-- and the cross-section will still be determined by the kinematic relations,
%%-- at the cost that one must consider both the mass of the produced and of the exchanged sparticle.
%%-- %% \begin{figure}[bh]
%%-- %%    \centering
%%-- %%       \includegraphics[scale=0.28]{plots/feyn2b}
%%-- %% \caption{A nice diagram \label{fig:1}}
%%-- %% \end{figure}
%%-- 
%%-- Furthermore, by definition there is one Lightest SUSY Particle (LSP), 
%%-- and one Next-to-Lightest SUSY Particle (NLSP). If it is stable, it is un-avoidable that
%%-- the LSP must be neutral and weakly interacting, due to cosmological constraints. The NLSP, on
%%-- the other hand, could be any sparticle - a slepton, a bosino, or a squark. However, there is
%%-- only a limited number of sparticles. While an arbitrary sparticle in the spectrum of any specific
%%-- SUSY model typically would decay through cascades of other, lighter sparticles, the NLSP only
%%-- has one decay-mode, namely to its SM partner and to the LSP,
%%-- which is  the topology defining a {\it simplified model}.

The ILC will be able to detect new particles with electroweak interactions nearly up to the kinematic
limit of $\sqrt{s}/2$. In particular,  in SUSY 
(where, as discussed in section~\ref{subsec:SUSY}, the couplings
cannot become arbitrarily small but are given by the couplings of the corresponding SM partners and their mixings) systematic, loophole-free searches can be performed for production of pairs of NLSPs. In the $R$-parity conserving
case they have to decay into the LSP and their SM partner (either on-shell or virtual), and in the clean environment
of the ILC, these decays can be detected even for extremely small mass difference. Figure~\ref{fig:noloophole1}
shows as an example the experimentally most challenging case of a $\stone$-NLSP~\cite{Berggren:2013vna}; in other cases,
the discovery reach approaches even closer to $\sqrt{s}/2$. 

\begin{figure}[tb]
   \centering
      %\includegraphics[scale=0.25]{figs/smu_nlsp_reach}
      %\includegraphics[scale=0.25]{figs/smu_nlsp_reach_zoom}

     \subfigure[]{\includegraphics[width=0.35\linewidth]{figs/stau_nlsp_reach}}
     \hspace{0.1\linewidth}
     \subfigure[]{\includegraphics[width=0.35\linewidth]{figs/stau_nlsp_reach_zoom}}
\caption{ILC discovery reach for a $\stone$ (bottom) NLSP for $\int \mathcal{L} \, \mathrm{dt}$ = 
500 fb$^{-1}$ at $\sqrt{s}$ = 500 GeV. The mixing angle was chosen to give the lowest
possible production cross-section. (a) full scale, (b) zoom to last few GeV before the
kinematic limit~\cite{Berggren:2013vna}. \label{fig:noloophole1}}
\end{figure}

% JL: too much LHC in here - LHC will come in sec 4.
% LHC search limits for superparticle production are often presented within the context of 
% simplified models which focus on a single production subprocess and one particular sparticle decay mode.
% In contrast, from theory it is expected that, if SUSY is accessible, then a large number of subprocesses
% and a large number of decay modes (cascade decays) should be simultaneously present so that actual
% LHC SUSY events are expected to be quite complex.
% In contrast, at ILC where $\sqrt{s}$ is adjusted to a value just above kinematic production threshold, 
% then indeed a single production mode occurs and the decays occur in a single rather than multiple steps:
% at ILC, sparticle production may indeed resemble a simplified model\cite{Berggren:2013vna}. 
% This is particularly advantageous for probing reactions with nearly degenerate spectra. 
% An example comes from slepton pair production and decay (which is favored in many WIMP co-annihilation
% scenarios). Then the reaction $e^+e^-\rightarrow\tilde{\ell}^+\tilde{\ell}^-$ followed by
% $\tilde{\ell}\rightarrow \ell\tilde{\chi}_1^0$ yields a clear signal which can be extracted from
% background processes even for very tiny $\tilde{\ell}$-$\tilde{\chi}_1^0$ mass gaps. This is not the case at LHC
% where tiny mass gaps lead to soft decay products which tend to be buried beneath formidable QCD and 
% electroweak background processes. Thus, ILC searches close many of the ``degenerate mass'' loopholes
% which arise from LHC analyses.
% 
% 
% Figure \ref{fig:noloophole1} shows the expected exclusion and discovery reach
% for two different hypotheses on the nature of the NLSP: 
% an easy case of a $\smur$ NLSP and the more difficult case of a $\stone$ NLSP.
% The $\stau$ mixing angle has been chosen to give the lowest possible production cross-section
% so the analysis is conservative.
% As can be seen from these figures, at the ILC one would expect to either discover or exclude 
% slepton-NLSPs up to just a few GeV below the kinematic limit. At present, slepton mass limits from 
% the LHC are just beginning to arise.

% JL: moved to section 4.1 
% One may also compare the reach of ILC to present and projected limits from LHC 
% on electroweakino pair production.
% In figure \ref{fig:noloophole2}, the current limits in the $\MXN{1}$ - $M_{\mathrm{NLSP}}$ plane from 
% the  di- and tri-lepton searches of ATLAS \cite{Aad:2014vma} are shown, together with the
% projected discovery reach at 14 TeV with $\int \mathcal{L} \, \mathrm{dt}$ = 300 or 3000 fb$^{-1}$ \cite{ATLAS:2013hta}.
% Here it is assumed that  $\MXN{2}=\MXC{1}$, and that indeed Br($\chi\rightarrow W^{(*)}/Z^{(*)}\XN{1}$)=1
% \footnote{Note that the more difficult case $\chi\rightarrow h^{(*)}\XN{1}$ is not considered.}.
% On the same figure, the corresponding limit from LEP \cite{Heister:2002mn,Abdallah:2003xe,Abbiendi:2002vz} 
% is also shown (brown-shaded).
% For the latter, only  $\XPM{1}$ pair production is assumed, with no assumption on the decay mode.
% The expected limits for the ILC at $E_{CMS}$ = 500 or 1000 GeV are also shown with the same 
% assumptions as for the LEP plot.
% The thick diagonal line indicates where $\MXN{2}$ (or $\MXC{1}$) is twice $\MXN{1}$. Below
% this line, GUT scale unification of the gaugino mass-parameters $M_1$ and $M_2$ is not possible,
% hence no model where such unification is assumed would be allowed in that region.
% As can be seen from the (loophole) region not covered by the LHC, there is a large
% discovery potential for the ILC, even after the high luminosity LHC data has been fully exploited.
% 
% \begin{figure}[h]
%    \centering
%       \includegraphics[scale=0.4]{figs/fig_07b_w_lep_lhc14_n_hilum_ilc500_ilc1000_proc}
% \caption{Discovery or exclusion regions in the $M_{NLSP} - M_{LSP}$ plane
%   for a $\XPM{1}$ or $\XN{2}$ NLSP. Solid brown area: LEP exclusion;
%   Solid red and dashed gray lines: ATLAS exclusion (observed and expected);
%   Solid blue (green) lines: ATLAS 14 TeV discovery projections for $\int \mathcal{L} \, \mathrm{dt}$ = 300 (3000) fb$^{-1}$;
%   Dashed green (magenta) lines:  ILC discovery expectation for $E_{CMS}$ = 500 (1000) GeV;
%   Solid black line: below line, no GUT scale gaugino mass unification.
%   %% For further details, see text. 
%  \label{fig:noloophole2}}
% \end{figure}
% 
% 
% 


\subsection{SUSY Dark Matter}
\label{subsec:directNP_SUSYDM}
% {\color{red} [content: preferred mechanisms, i.e.\ co-annihilation, ILC discovery reach, H-20 projections, added value w.r.t.\ HL-LHC] }\\
%%-- (assuming the contents of these paragraphs are integrated into chep 2.)


%%-- Many observations indicate that a large fraction of the matter in the universe is
%%-- dark, i.e. is not visible in observations of electromagnetic radiation.
%%-- The data on the temperature power spectrum of the 
%%-- cosmic microwave background (CMB) and its polarisation 
%%-- obtained by the Planck mission give the most precise cosmological determination of the dark matter (DM) relic density $\Omega_{CDM}$, 
%%-- and is
%%-- $\Omega_{CDM}=0.1197 \pm 0.0022$ \cite{Ade:2015xua}.
%%-- If such an abundance of DM in the early universe is to be explained
%%-- by the existence of hitherto unknown elementary particle,
%%-- one needs a balance between early universe production and
%%-- decay of this particle.
%%-- One finds that this could be obtained by a weakly interacting, stable
%%-- massive particle (a WIMP).
%%-- Since the lightest Super-symmetric particle (the LSP) has such properties,
%%-- it is quite compelling to assume that it is the DM.
%%-- However, if the LSP is the only SUSY particle in the mass-range needed,
%%-- it would tend to be too abundant.
%%-- Fig. \ref{fig:anndiags} shows diagrams contributing to the annihilation of
%%-- the LSP. Of these, diagram (a) is effective only if it is resonant, which it could be
%%-- if one of the neutral higgses would have close to twice the LSP mass.
%%-- Diagram (b) is more effective if there are sfermions with low enough mass, however
%%-- this is challenged by the direct limits form LEP.
%%-- Diagram (c) is effective if 
%%-- some other SUSY particle was close in density
%%-- in the hot early universe, i.e. to be close to mass-degenerate with the LSP. 
%%-- Such scenarios are called a coannihilation scenarios, 
%%-- and the  coannihilation next-to-lightest SUSY particle (the NLSP) could be the $\stone$, the $\stqone$ or the $\XPM{1}$.
%%-- These scenarios are compelling for other reasons.
%%-- If the totality of current observations sensitive to SUSY are
%%-- fitted to a general SUSY model with 10 free parameters, as has been
%%-- done by the {\tt Mastercode} collaboration\cite{deVries:2015hva}, one finds that
%%-- models where the bosino and slepton-sector is quite compressed fits the
%%-- observations best, see Figure \ref{fig:mastercode_mass}.
%%-- \begin{figure}[htb]
%%-- \begin{center}
%%-- \includegraphics[scale=0.40]{figs/mastercode_mass-spectrum}
%%-- \caption{\label{fig:mastercode_mass} Best fits of SUSY masses to all available relevant data. From \cite{deVries:2015hva}.}
%%-- \end{center}
%%-- \end{figure}
%%-- 
%%-- 
%%-- \begin{figure}[htb]
%%--   \begin{center}
%%-- %%%    \subfigure[]{\includegraphics [scale=0.3]{plots/freezeout-plot-equilibrium-density.jpg}}
%%--     \subfigure[]{\includegraphics [scale=0.6]{figs/lsplsp-fermionfermion}}
%%--     \subfigure[]{\includegraphics [scale=0.6]{figs/lsplsp-tchannel-fermionfermion}}
%%--     \subfigure[]{\includegraphics [scale=0.6]{figs/lspstau-chi-taugamma}}
%%--   \end{center}
%%--   \caption{\label{fig:anndiags} \XN{1} annihilation diagrams. The coannihilation diagram is (c).}
%%-- \end{figure}
%%-- % \subsubsection*{Compressed spectra at the LHC and the ILC}
%%-- % \label{ILCandLHC}
%%-- %% \subsubsection*{{\it The ILC}}
%%-- 
%%-- 
%%-- %{\it The ILC  is not LHC}
%%-- 
%%-- 
%%-- A precise determination of the LSP and 
%%-- $\stau$ properties are essential to test whether the $\XN{1}$ is the main constituent of
%%-- DM: 
%%-- Using the {\tt micrOMEGAs} code\cite{Belanger:2006is}, it has been shown \cite{Lehtinen:2016qis}
%%-- that a 1\% variation of $\mstau$ or $\MXN{1}$ changes
%%-- the predicted relic     abundance by 5 \%, while
%%-- a similar variation of the $\stau$ mixing angle $\theta_{\stau}$ or the binoness of the LSP ($N_{11}$)
%%-- changes abundance by 1\% and 3.5 \%, respectively.
%%-- It was also found that the abundance is much less sensitive to other masses or mixings.
%%-- Therefore,  
%%-- to match the 2~\% uncertainty on the
%%-- relic abundance obtained by Planck
%%-- one needs 
%%-- per mil-level  precision measurements of the LSP and NLSP masses, 
%%-- and percent-level precision ones on the LSP and NLSP mixings.
 
%JL: attempt to shorten

Over a large region of SUSY parameter space, co-annihilation with the NLSP
is an attractive mechanism which acts to reduce the relic density of the LSP to its
cosmologically observed value~\cite{deVries:2015hva}. Co-annihilation requires
a small mass difference between the NLSP and the LSP in order to be effective,
and thus the expected value of the relic density depends strongly on the exact
masses and mixings of the involved particles, requiring measurements at the permille and
percent-level, respectively~\cite{Lehtinen:2016qis}. 
Here, we present as an example the case of $\stau$ co-annihilation, for which
the ILC prospects have been studied for different benchmark scenarios~\cite{Bechtle:2009em, Berggren:2015qua}. 
In these studies it has been shown that the relevant masses of the
NLSP and the LSP can be measured at the permille-level from kinematic edges,
but also from measuring cross sections near production thresholds. Both methods are illustrated
in Fig.~\ref{fig:sleptons}. The threshold scan in addition determines the spin of the particle
unambiguously, in this case as a scalar.
The production cross sections and
the polarisation of the $\tau$ from $\stone$ decays can be determined at the percent level~\cite{Bechtle:2009em}, 
giving access to the mixing of the $\stone$ and the bino content of the \XN{1}.

\begin{figure}[htb]
  \begin{center}
    \subfigure[]{\includegraphics[width=0.3\linewidth]{figs/STC_smuR_threhold-3curves-uni-scale}}
    \hspace{0.01\linewidth}
    \subfigure[]{\includegraphics[width=0.3\linewidth] {figs/smurr_tnphs_rl_006_selcuts_skim_whiz6_epj}}
    \hspace{0.01\linewidth}
    \subfigure[]{\includegraphics [width=0.3\linewidth]{figs/stau11-stc-epj-style_wmissch}}
  \end{center}
  \caption{\label{fig:sleptons} Property determination of SUSY Dark Matter: (a) scan of the $\eeto \smur \smur$ threshold, (b) muon and (c) $\tau$-jet energies in selected di-leptons events
after collecting
500 fb$^{-1}$ of data for beam-polarisation  $\mathcal{P}_{-80,+30}$~\cite{Berggren:2015qua}. }
\end{figure}




% As pointed out in Sec.~\ref{sec:BSM}, a precise determination of the 
% properties of $\tilde{\chi}_1^0$ and $\stau$ are essential to test whether the 
% $\XN{1}$ might constitute the bulk of dark matter under the stau co-annihilation scenario. 
% Current cosmological measurements on the relic abundance of dark matter have 
% an uncertainty of 2\%. To arrive at similar precision from measuring the direct production 
% of a SUSY WIMP candidate, uncertainties on the LSP and NLSP masses 
% must be at the per mil level while their mixing properties need to be known at the percent level. 

% Taking the well-motivated  \stau-coannhilation scenario as a reference,
% the capabilities for precision measurements in the  $\tilde{\tau}$ sector 
% have been studied with full detector simulation in Ref.~\cite{Bechtle:2009em} 
% and more recently with a fast simulation in Ref.~\cite{Berggren:2015qua}.
% The models used in these two studies differ mainly in the spectrum
% of heavy coloured sparticles: in both cases these are beyond
% kinematic reach of the ILC. The electroweakino and slepton sectors, on the other hand,
% are quite similar\footnote{The model used in Ref.~\cite{Bechtle:2009em} is, due to
% its coloured sector, excluded by recent LHC limits. The one used
% in Ref.~\cite{Berggren:2015qua} is compatible with all observations to date.}.

% \begin{figure}[htb]
%   \begin{center}
%     \subfigure[]{\includegraphics[scale=0.20] {figs/serr_tnphs_rl_006_selcuts_skim_whiz6_epj}}
%     \subfigure[]{\includegraphics[scale=0.20] {figs/smurr_tnphs_rl_006_selcuts_skim_whiz6_epj}}
%     \subfigure[]{\includegraphics [scale=0.20]{figs/stau11-stc-epj-style_wmissch}}
%   \end{center}
%   \caption{\label{fig:sleptonspect} Spectra of electron (a), muon (b) and $\tau$-jet (c) energies in selected di-leptons events
% after collecting
% 500 fb$^{-1}$ of data for beam-polarisation  $\mathcal{P}_{-80,+30}$. From Ref.~\cite{Berggren:2015qua}. }
% \end{figure}

% Figure~\ref{fig:sleptonspect} (a) and (b) show distributions from selected dileptons events
% from slepton pair production.
% The signal stands out clearly above quite small SM and SUSY backgrounds.
% By determining the position of the edges, which are kinematically related to \msle\ and \MXN{1}, 
% we obtain $\MXN{1}= 95.47  \pm 0.16 ~\GeV$ and $ M_{\selr} = 126.20 \pm 0.21 ~\GeV $ 
% from $\selr$ pair production and $\MXN{1} = 95.47 \pm 0.38 ~\GeV$
% and $M_{\smur} = 126.10 \pm 0.51 ~\GeV$ from $\smur$ pair production.
% These values are in good agreement with the output model values.
% %% the true masses in STCx are  $\MXN{1} =95.59 ~\GeV$, $M_{\selr} = 126.24 ~\GeV$ and  $M_{\smur} = 126.16 ~\GeV$.
% Combining the measurement of the LSP mass from these two  analyses
% gives an uncertainty of $\sigma_{\MXN{1}} = 147 ~\MeV$ {\it i.e.} slightly above 1 per mil.
% 
% Figure~\ref{fig:sleptonspect} (c) shows the energy-spectrum 
% % at $\Ecms = 500\GeV$ 
% of selected $\tau$-jets.
% Due both to the lower mass-difference and decay of the $\tau$-lepton, only
% the upper to endpoint is accessible to measurement. It
% could be determined to be E$_\mathrm{endpoint}=44.49^{+0.11}_{-0.09}$ \GeV,
% corresponding to an uncertainty of $M_{\stone}$ of 200 \MeV, if an uncertainty on
% the LSP mass from the $\sel$ and $\smu$ analysis of $\sim$ 100 \MeV~ is assumed.
% Hence, also $M_{\stone}$ can be determined to slightly above 1 per mil.
% 
% 
% 
% 
% In Ref.~\cite{Berggren:2015qua}, it is shown that both  \mselr\ and \msmur\
% can be determined to a similar precision from a threshold-scan
% using just a few months of ILC beam-time: see Figure \ref{fig:slept_thresh}. 
% This measurement does not depend on \MXN{1}, so it is a useful cross-check of the edge-method.
% In addition, it gives a clear test of whether the sleptons really are
% scalars, as SUSY requires.
% \begin{figure}[htb]
%   \begin{center}
% %\includegraphics[width=0.49\linewidth]{st1mass_epj_log}
% \subfigure[]{\includegraphics[width=0.49\linewidth]{figs/STC_selR_threhold-3curves}}
% \subfigure[]{\includegraphics[width=0.49\linewidth]{figs/STC_smuR_threhold-3curves-uni-scale}}
%  \end{center}
%   \caption{\label{fig:slept_thresh} Threshold scans at the $\eeto \selr \selr$ (a)  and
%       $\eeto \smur \smur$ (b) thresholds.}
%   %%   From Ref.~\cite{Brau:2012hv}}.
%   %% OR if I can find the plot:
%   %% Right: Kinematically constrained \sMuR mass-spectrum from \XI0{2} $\to$ 
%   %%\sMuR$\mu \to \mu \mu \XI0{1}.
%   %% From~\cite{...}}.
% \end{figure}

% In Ref.~\cite{Bechtle:2009em}, also the production cross-sections and
% the polarisation of the $\tau$ from $\stone$ decays were determined,
% and was found to be measurable at the percent level.
% These measurements give access to the mixing of the $\stone$ and the
% bino content of the \XN{1}.

With such precision, ILC has the capacity to cross check DM properties~\cite{Baltz:2006fm} 
such as the thermally-produced relic abundance with a precision
close to the cosmological observations on CMB from the Planck collaboration,
and hence to determine if indeed the lightest neutralino constitutes the bulk of 
dark matter in the universe --- 
or whether other non-thermal and/or non-WIMP processes play an important role in determining the
ultimate dark matter relic density, as discussed in section~\ref{subsubsec:SUSYDM}.



\subsection{Light Higgsinos}
\label{subsec:directNP_higgsinos}
% {\color{red} [content: ILC discovery reach, H-20 projections, added value w.r.t.\ HL-LHC] }\\
% {\color{dgreen} [volunteer contributor: Jacqueline Yan] }\\
% \input{light_higgsinos}
As discussed in section~\ref{sec:BSM-SUSY-Naturalness}, light higgsinos are a fundamental requirement 
of natural SUSY models, while the other SUSY particles can be more heavy.
%Supersymmetric models with radiatively-driven naturalness, as discussed in section~\ref{sec:BSM-SUSY-Naturalness}
%predict light higgsinos in the 100--300 GeV range (the lighter the better)
%as described in section~\ref{sec:BSM-SUSY-Naturalness}.
The ILC with center-of-mass energy of $\sqrt{s}=500$~GeV will be able to 
cover the most natural portion of this parameter space while the remainder
would be fully covered with an energy upgrade.
%
Mass differences within the higgsino sector are small, typically  below $20$\,GeV,
depending on the values of the other SUSY parameters, in particular the Bino and Wino
mass parameters $M_1$ and $M_2$. In the clean environment of the ILC, their soft visible
decay products can be easily detected --- without any need to rely on large-mass-gap decays
of heavier particles. The ILC capabilities 
have been studied in detector simulations performed for different benchmark points with mass 
differences ranging from $770$\,MeV~\cite{Berggren:2013vfa}
to $20$\,GeV~\cite{Baer:2016new}. 
Two examples of the striking signals and the extraction of 
kinematical endpoints are given in Fig.~\ref{fig:higgsinos}. 
The resulting precisions on masses and polarised 
cross sections reach the percent level even in the experimentally most difficult 
cases and allow to determine other SUSY parameters, as will be discussed in secttion~\ref{sec:LHC_scen}. 
They will also play an important role in unveiling the nature of dark matter: 
in this case with the result that the LSP only contributes a small fraction of the 
total abundance. Such a situation might call for additional, non-WIMP constituents of 
dark matter such as axions: c.f.\ section~\ref{subsubsec:SUSYDM}.

\begin{figure}[htb]
  \begin{center}
    \subfigure[]{\includegraphics[width=0.44\linewidth] {figs/MrecoilC_mh127}}
    \hspace{0.1\linewidth}
    \subfigure[]{\includegraphics [width=0.44\linewidth]{figs/EdleeL.pdf}}
  \end{center}
  \caption{\label{fig:higgsinos} Higgsino mass determination for (a) the charged higgsino from the recoil against an ISR photon in a scenario with a mass splitting of $770$\,MeV~\cite{Berggren:2013vfa}, (b) the neutral higgsino from the energy of its visible decay products in a scenario with a mass splitting of  $20$\,GeV~\cite{Yan:2016LCWS, Baer:2016new}. }
\end{figure}


% The signal channels are characterized by four light higgsinos,
% $\widetilde\chi^0_1$, $\widetilde\chi^0_2$, and $\widetilde\chi^\pm_1$.
% The charginos are pair produced via
% $e^+e^-\rightarrow\widetilde{\chi}^+_1\widetilde\chi^-_1$,
% while the neutralinos are dominantly produced via mixed production 
% $e^+e^-\rightarrow\widetilde{\chi}^0_1\widetilde\chi^0_2$.
% %
% The mass differences
% $\Delta M = m_{\widetilde\chi^\pm_1}-m_{\widetilde\chi^0_1}$
% and
% $\Delta M' = m_{\widetilde\chi^0_2}-m_{\widetilde\chi^0_1}$
% are typically of order $10-20$~GeV in the most natural case~\cite{Baer:2014yta}.
% The rather compressed higgsino mass spectrum leads to their visible decay products 
% being very soft, which makes them especially challenging to observe or constrain at the LHC.
% So far, the bulk of LHC sparticle searches have taken place in highly unnatural models
% such as the CMSSM or in simplified models which make no reference to naturalness.\footnote{
% The possible exception of LHC searches for light top squarks is motivated by an oversimplified
% measure of naturalness. Requiring the $\Sigma_u^u(\tilde{t}_{1,2})$ contributions of Eq. \ref{eq:mzs} 
% to be not-too-large allows for highly mixed top squarks in the several TeV range at 
% little cost to naturalness.}
% The generic features of the ILC position it as the pre-eminent machine to search for
% natural SUSY: the clean environment of ILC scattering events allows for not only the observation of--
% but also the precision measurement of-- the required light higgsinos with compressed mass spectra.
% 
% Based on a benchmark point from the NUHM2 supersymmetric model with radiatively-driven naturalness
% studied in Ref.~\cite{Baer:2014yta},
% we present new results which employ the full simulation of the ILD detector response 
% at $\sqrt{s}=$500~GeV with beam polarizations.
% The mass differences are $\Delta M=15$~GeV and $\Delta M'=21$~GeV.
% The chargino and the heavier neutralino decay only into soft SM particles plus the 
% LSP $\widetilde\chi^0_1$ which carries away missing energy.
% While the direct mass values of $\widetilde{\chi}_1^\pm$ and $\widetilde{\chi}_{1,2}^0$ are $\sim \mu$, 
% the mass {\it gaps} depend on higgsino-gaugino mixing, which in turn depends on the gaugino masses.
% We estimate the precision of the higgsino mass measurements and production cross sections at the ILC,
% which can then be used to extract the weak scale gaugino masses $M_{1,2}$ and $\mu$ and $\tan\beta$.
% 
% The chargino pair production reaction was studied in the semileptonic mode,
% $\widetilde\chi^+_1\widetilde\chi^-_1\rightarrow \widetilde\chi^0_1\widetilde\chi^0_1qq'\ell\nu$.
% The neutralino mixed production was studied in the fully leptonic mode,
% $\widetilde\chi^0_1\widetilde\chi^0_2\rightarrow \widetilde\chi^0_1\widetilde\chi^0_1\ell^+\ell^-$.
% The signal events are selected by requiring large missing energy;
% other selections are applied on the momentum, angles, and number of visible tracks,
% to reduce the SM and SUSY backgrounds.
% The selection efficiencies are approximately 25\% for the charginos
% and 50--70\% for the neutralinos.
% %In the neutralino case, a significance of about 35 and signal
% %efficiency of 50-70$\%$ is obtained. In the chargino case, a signal efficiency
% %of about 25$\%$ is obtained and almost all background are removed. 
% The final observables are the energy and invariant mass of the
% jet pair and the lepton pair, for the charginos and the neutralinos, respectively.
% The dijet and dilepton energy distributions are shown in 
% Fig.~\ref{fig:higgsino_neutralino}-\ref{fig:higgsino_chargino}.
% The kinematic edges of these distributions are used to extract the mass of the higgsinos.
% The cross sections are extracted by fitting the dijet and dilepton energy distributions.
% \begin{figure}[htbp]
%  \begin{minipage}{0.48\linewidth}
%   \begin{center}
%    \includegraphics[width=0.98\linewidth]{343_chargino_Ejj_muon}
%   \end{center}
%   \caption{Dijet energy distribution after the event selection
%   for the chargino pair production in the semileptonic mode with a muon
%   ($\widetilde\chi^+_1\widetilde\chi^-_1\rightarrow \widetilde\chi^0_1\widetilde\chi^0_1qq'\mu\nu$),
%   assuming an integrated luminosity of 500~fb$^{-1}$ and
%   beam polarization of P($e^+$, $e^-$) = (+$30\%$ ,$-80\%$).}
%   \label{fig:higgsino_chargino}
%  \end{minipage}
%  \hfill
%  \begin{minipage}{0.48\linewidth}
%   \begin{center}
%    \includegraphics[width=0.98\linewidth]{343_neutralino_Ell_muon}
%   \end{center}
%   \caption{Dilepton energy distribution after the event selection
%   for the neutralino mixed production in the leptonic mode with muons
%   ($\widetilde\chi^0_1\widetilde\chi^0_2\rightarrow \widetilde\chi^0_1\widetilde\chi^0_1\mu^+\mu^-$),
%   assuming an integrated luminosity of 500~fb$^{-1}$ and
%   beam polarization of P($e^+$, $e^-$) = (+$30\%$ ,$-80\%$).}
%   \label{fig:higgsino_neutralino}
%  \end{minipage}
% \end{figure}
% 
% Assuming an integrated luminosity of 500~fb$^{-1}$ at each of the two beam polarizations
% P($e^{+}$, $e^{-}$) = ($\pm$30$\%$ ,$\mp$80$\%$),
% we find that the higgsino masses can be measured to about 500 MeV (0.4$\%$),
% and the cross sections to 0.6-2.4$\%$,
% where the range of values corresponds to the different channels and beam polarizations.
% The precision on the higgsino masses and the cross sections are then used as input to the
% Fittino program for extraction of the underlying parameters in the supersymmetric theory:
% in this case the electroweak gaugino masses $M_{1}$, $M_{2}$, the higgsino mass parameter $\mu$,
% and the ratio of vacuum expectation values $\tan\beta$,
% by fitting the observables to the theoretical expectations.
% %The Higgs observables are also used as input.
% For the benchmark point under study, it is shown that
% $\mu$ can be measured to a precision of about 1$\%$,
% while $M_{1}$ and $M_{2}$, which are chosen to be in the range of 250--1000 GeV,
% can be measured to a precision of about 20~GeV.
% %
% In the case of the H-20 operating scenario, assuming an integrated luminosity of 1600~fb$^{-1}$ 
% has been collected at each of two beam polarizations P($e^+$, $e^-$) = ($\pm$30$\%$ ,$\mp$80$\%$),
% the higgsino masses can be measured to about 250~MeV (0.2$\%$),
% and the cross sections to 0.3-1.3$\%$.
% %Estimations for lower energies 250 and 350 GeV are made as well.
% It is estimated that $\mu$ and $\beta$ can be measured to a precision of 0.1$\%$ and 2.5$\%$, respectively,
% and $M_{1}$, $M_{2}$ to a precision of 1.5$\%$ and 1.0$\%$, respectively.

% Beginning with the weak scale gaugino masses $M_1$ and $M_2$, one may run these to high
% energy using the MSSM renormalization group equations to check the scale of unification:
% do they unify at $m_{GUT}\simeq 2\times 10^{16}$ GeV or at some intermediate scale as might occur in
% SUSY with mirage mediation?
% In the case where the gluino is discovered at LHC, and its mass measured to $\sim 10\%$ accuracy,
% then unification of all three gaugino masses can be tested as shown in
% Fig.~\ref{fig:gaugino_mass_unification}.
% This will provide an important window onto whether or not nature is
% described by a supersymmetric grand unified theory or some other framework.
% If the light higgsinos are discovered at the ILC but the gluino is not discovered at LHC,
% then downward running of $M_3$ from the $M_1$-$M_2$ unification point can
% give us a prediction on the range of the gluino mass under the assumption
% of gaugino mass unification: this information could set the target 
% energy scale required for the next hadron collider.
%
% \begin{figure}[htbp]
% \begin{center}
% \hspace{3mm} \includegraphics[width=8cm]{343_gaugino_mass_unification}
% \end{center}
% \caption{Running of the gaugino mass parameters for the benchmark point 
% from a supersymmetric theory with radiatively-driven naturalness,
% using the expected precision for the bino and wino mass parameters ($M_1$ and $M_2$) from the ILC
% and that for the gluino mass parameter $M_3$ from the LHC (10\%).}
% \label{fig:gaugino_mass_unification}
% \end{figure}

% An even more challenging analysis was carried out in Ref.~\cite{Berggren:2013vfa} where
% the mass gaps between the various higgsinos were just below and just above $1$\,GeV.
% In this case, using the tool of initial state photon radiation, 
% it was shown that the mass differences can be measured to 40--300\,MeV,
% the absolute masses to 1.5--3.3\,GeV, and the cross sections to 2--5\,$\%$.
% This scenario could allow for the extraction of the gaugino mass parameters
% even if they are in the multi-TeV range by disentangling the mixing between the 
% higgsinos and the electroweak gauginos.



\subsection{Additional Higgs Bosons}
\label{subsubsec:directNP_SUSYHiggs}
% {\color{red} [content: lighter or heavier than H125, ILC discovery reach, H-20 projections, added value w.r.t.\ HL-LHC] }\\
%{\color{blue} [proposed contributor: Chris Potter] }\\
%\input{additional_higgs}
The ILC will be able search for additional Higgs bosons of extended Higgs sectors, for instance 2HDMs (c.f.\  section~\ref{sec:2HDM}) or additional singlets as in the NMSSM. A loophole-free search for heavier Higgs bosons can probe masses up to about $\sqrt{s}/2$. Even more interesting, however, is the ILC's capability to detect
lighter Higgs bosons, even if their couplings to the $Z$ boson are strongly reduced. Figure~\ref{fig:h60_plots} shows an example from the NMSSM featuring a light scalar and a light pseudo-scalar Higgs boson with masses of $60$ and $10$\,GeV, respectively~\cite{Potter:2015wsa}.

% The NMSSM offers an interesting possibility that the 125 GeV Higgs boson may not be the lightest scalar.
% NMSSM scenarios with a light pseudoscalar are motivated by the strong $CP$ problem, naturalness, muon $g-2$, the $\eta_b$ mass spectrum, and dark matter relic density~\cite{Potter:2015wsa}.
% This possibility is illustrated in the $h_{60}$ benchmark point,
% in which the heavier scalar $h_2$ plays the role of the observed SM-like Higgs boson.
% The lighter scalar $h_1$ has $m_{h_1}\approx 60$~GeV and dominantly decays as $h_1\rightarrow a_1 a_1$ (72\%), but also decays to $h_1\rightarrow b\overline{b}$ (23\%) which is motivated by the LEP $Z\rightarrow b\overline{b}$ results.
% The pseudoscalar $a_1$ (axion) has $m_{a_1}\approx 10$~GeV and dominantly decays as $a_1\rightarrow \tau^+\tau^-$ (81\%).
% The DM candidate is the singlino LSP with $m_{\chi_1}\approx 60$~GeV, which annihilates as $\chi_1\chi_1\rightarrow b\overline{b}$.
% Due to the observed signal strengths of the $h_{125}$, the heavier scalars ($h_3$, $H^\pm$) and pseudoscalar ($a_2$) are assumed to be decoupled.
% 
% At the LHC, evidence for such signatures could be found through the cascade decay of the gluino or stop.
% The resulting $a_1$ can be reconstructed in the rare but clean $a_1\rightarrow\mu^+\mu^-$ channel,
% and is a target of the Run 2 study.

% The discovery prospect at the ILC is being evaluated using fast simulation~\cite{Potter:ECFALC2016},
% using the production of $e^+e^-\rightarrow \chi_3\chi_3$ at $\sqrt{s}=500$~GeV.
% The $\chi_3$ decays into $\chi_1 h_1$ (80\%) and $\chi_2 Z^*$ (10\%).
% The $\chi_2$ decays into $\chi_1 a_1$ (75\%).
% Figure~\ref{fig:h60_plots} shows the reconstruction of $h_1\rightarrow b\overline{b}$ (left)
% and $a_1\rightarrow \mu^+\mu^-$ (right).
% This ongoing study demonstrates that there are potentially enough signal events for reconstructing this final state.

\begin{figure}[htb]
  \begin{center}
    \subfigure[]{\includegraphics[width=0.4\linewidth] {figs/h60_Mbb}}
    \subfigure[]{\includegraphics[width=0.4\linewidth] {figs/h60_Mmumu}}
  \end{center}
  \caption{\label{fig:h60_plots} Reconstruction of $e^+e^-\rightarrow \chi_3\chi_3$ events with decays to light Higgs bosons in a NMSSM benchmark~\cite{Potter:ECFALC2016} 
%
%using Delphes fast simulation for the SiD detector:
  (a) $h_{1}\rightarrow b\overline{b}$ ,  (b) $a_1\rightarrow \mu^+\mu^-$.
   An integrated luminosity of 2000~fb$^{-1}$ at $\sqrt{s}=500$~GeV is assumed for each beam polarization configuration. Background is not yet evaluated.
%   From \cite{Potter:ECFALC2016}. 
}
\end{figure}

\subsection{$R$-Parity Violating SUSY}
\label{subsubsec:directNP_bRPV}

% {\color{red} [Jenny] For now I steal some patches from the ILC BSM Snowmass WP - 
% again needs rephrasing since literal copy from https://arxiv.org/abs/1307.5248}\\

As an example for the ILC capabilities in the case of $R$-parity violating SUSY (c.f.\ section~\ref{subsubsec:bRPV_theo}),
Fig.~\ref{fig:bRPV_reach} shows the $5\sigma$ discovery reach for selectron-mediated Bino pair production of an initial ILC run~\cite{List:2013dga}. In this scenario, the Bino-LSP decays into a $W$ boson and a lepton, where the
relative rates for the different lepton flavours are related to neutrino mixing. Fig.~\ref{fig:final_prec} compares
the resulting extraction of $\sin^2\theta_\mathrm{atm}$ from LSP branching ratios as could be measured at the ILC with the known value from neutrino oscillations.

% The ILC also offers unique opportunities in the case of $R$-parity violating SUSY. As a concrete example, 
%  we will illustrate the ILC's capabilities assuming a bilinear $R$-parity violation model introduced in section~\ref{subsubsec:bRPV_theo},
%  which can explain the smallness of neutrino masses and predicts the pattern of neutrino mixing.
%  Such a measurement, if in agreement with direct oscillation measurement, could pin down 
% the origin of neutrino masses and mixing, which would solve one of the major puzzles in our
% current understanding of nature.

% In this case, the atmospheric neutrino mixing angle can be accessed via the ratio of branching 
% ratios of the LSP: 
% $BR(\tilde \chi^0_1 \to W \mu)/BR(\tilde \chi^0_1 \to W \tau) \simeq \tan^2\theta_\mathrm{atm}$. 
% At the ILC, these branching ratios can be measured from direct LSP pair production even
% if all other SUSY particles should be out of kinematic range for direct production. The 
% cross section depends on the neutralino mixing and the selectron mass,  but can easily be 
% as large as $100$~fb, giving {\it e.g.} 25000 events per year of design ILC operation at 
% $\sqrt{s} = 500$~GeV. 
% 
\begin{figure}[htb]
  \begin{center}
\subfigure[]{\includegraphics[width=0.35\linewidth]{figs/gr_signWmuWmu00_only}\label{fig:bRPV_reach}}
\hspace{0.1\linewidth}
\subfigure[]{\includegraphics[width=0.35\linewidth]{figs/bRPV_thfinalprecision}\label{fig:final_prec}}
 \end{center}
  \caption{\label{fig:bRPV} (a) Discovery reach for an initial ILC run. (b) Derived uncertainty on the atmospheric neutrino mixing angle $\sin^2\theta_{23}$, 
  including the additional parametric uncertainty due to limited knowledge of other SUSY parameters. Both from~\cite{List:2013dga}.}
\end{figure}
% 
% %The experimental performance for the branching ratio measurements has been
% %presented in the ILC TDR~\cite{Intro1} using state-of-the-art full simulation of the ILD detector
% %concept~\cite{Intro3}. 
% The expected relative precision on 
% $BR(\tilde \chi^0_1 \to W \mu)/BR(\tilde \chi^0_1 \to W \tau)$ is shown in the left panel of 
% Fig.~\ref{fig:bRPV} as obtained from a detailed detector simulation study~\cite{List:2013dga}. Already for an integrated luminosity of $250\,$fb$^{-1}$, a relative precision of $7\%$ is 
% achieved, which decreases to $5\%$ for $500\,$fb$^{-1}$. The right panel illustrates the 
% resulting absolute precision on $\sin^2\theta_\mathrm{atm}$, assuming the current central value
% from oscillation measurements, whose precision is indicated by the dashed horizontal lines. 
% For the 
% collider measurement, the effect of the parametric uncertainty due to limited knowledge of other
% SUSY parameters which enter the relation
% between the branching ratios and the mixing angle is displayed in addition.
% 
% Such a measurement, if in agreement with direct oscillation measurement, could pin down 
% the origin of neutrino masses and mixing, which would solve one of the major puzzles in our
% current understanding of nature.



40_LHCscen.tex
In this section, we will discuss the ILC discovery potential in the light of the three LHC scenarios presented in~\cite{ICFAtoMEXT}: 
%{\color{red} [Keisuke, is there a more elegant way to refer to this document?]}:
In section~\ref{subsec:noNP}, we will address the case that the LHC experiments will not discover
any further new particles, neither in the near future nor in the long run. 
In sections~\ref{subsec:lightNP} and~\ref{subsec:heavyNP}, we will discuss
examples for the cases that the LHC experiments will discover relatively light or relatively heavy
new particles, respectively. We will show that the ILC has significant potential to 
discover phenomena beyond the Standard Model in each of the cases.

41_noNPLHC.tex
%\subsection{LHC experiments do not discover new particles}
%\label{sec:noNP}

In this section, we assume that the LHC does not discover new particles
or other significant deviations from the SM.
Concretely, we assume that even after Run~III (i.e.\ around the year
2023) with more than
${\cal O}$(300 fb$^{-1}$) and both experiments combined no evidence for new
particles will have been seen. 
The picture does not change in a relevant way if the same is assumed for the
HL-LHC, i.e.\ no evidence for new particles with ${\cal O}$(3 ab$^{-1}$).
Similarly, besides the current status
of precision measurements, we assume that neither at the LHC, nor at
another (future) experiment any significant deviation from the SM
predictions will be observed.

We will review that even in this case the physics potential of the ILC
is very rich. It will be reviewed that the ILC can discover new
physics via direct production of new particles and
via the observation of deviations in precision measurements that
cannot be done at other experiments. This ensures a strong physics
case even in the quite pessimistic scenario of no new discoveries at
the LHC.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Discovery via Precise Measurements of Standard Model Parameters}
\label{subsubsec:noNP_indirect}

% \newcommand{\MW}{\ensuremath{{M_W}}}
% \newcommand{\MH}{\ensuremath{{M_H}}}
% \newcommand{\mt}{\ensuremath{{m_t}}}
% \newcommand{\sweff}{\ensuremath{\sin^2{\theta^\ell}_{{\rm eff}}}}
% \newcommand{\als}{\ensuremath{\alpha_s}}
%\newcommand{\order}[1]{\ensuremath{{\cal O}(#1)}}

%{\color{red} [content: generic discovery potential for various types
%of NP up to high scales  via Higgs precision,  top ew couplings, 
%two-fermions etc, confirm with precise top mass and EWPO up to which scale SM i%s consistent] }\\
%{\color{blue} [proposed author: Sven Heinemeyer] }\\

In this section we concentrate on the ILC discovery potential via
precise measurements of SM quantities. This includes electroweak
precision observables (EWPO), top quark measurements, the production
of electroweak gauge bosons and Higgs-boson mass measurements. 
The corresponding Higgs-boson coupling
measurements will be discussed in the next subsection.

%\subsubsection*{Electroweak Precision Observables and the Mass of the Top Quark}
\paragraph{Electroweak Precision Observables and the Mass of the Top Quark:}
As discussed in sections~\ref{subsec:ILC_top} and~\ref{subsec:ILC_ew}, 
the ILC will improve substantially the main electroweak precision measurements,
including the masses of the
$W$~boson, \MW, and of the top quark, \mt, as
well as the effective weak mixing angle, typically
by factors $\sim 5$ to $10$ with respect
to current precisions (which will not be substantially improved at the
LHC)\footnote{For $\mt$, the improvement at the ILC in the precision of the
strong coupling constant \als by a factor of $\sim 5$ will be crucial.}.
With these results, the SM
Higgs boson mass will be predicted indirectly by
% with a precision of $\pm 6$~GeV, 
a factor of $\sim 5$ better than today~\cite{Moortgat-Picka:2015yla}, as shown
in Fig.~\ref{fig:blueband}.
% ~Using \MW, \mt\ and \sweff\ as input (as well as other, less relevant EWPO), 
Figures~\ref{fig:mwmt} 
and~\ref{fig:mwsw}~\cite{Heinemeyer:2007bw, Heinemeyer:2013dia} illustrate that 
%comparison with the measured \MH\ value can 
the high precision measurements of \MW, \mt\ and \sweff\ will have the power
to exclude the SM experimentally and reveal new, unknown physics scales
up to energy regimes far beyond the direct ILC reach.
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[trim=0 130 0 140,  clip, width=0.3\linewidth]{figs/w12lc_blueband.pdf}\label{fig:blueband}}
\hspace{0.01\linewidth}
\subfigure[]{\includegraphics[trim=0 0 50 200,  clip, width=0.275\linewidth]{figs/MWmt16LEPTevLHC-LHCfin-ILC.pdf}\label{fig:mwmt} }
\hspace{0.01\linewidth}
\subfigure[]{\includegraphics[trim=0 0 50 200,  clip, width=0.275\linewidth]{figs/MWSW16LEPTevSLD-LHCfin-ILC.pdf}\label{fig:mwsw} }\\
\caption{ILC discovery potential via electroweak precision observables: (a) The prediction of the SM Higgs boson mass
from EWPOs will improve by a factor 5 compared to today due to ILC measurements~\cite{Moortgat-Picka:2015yla}. (b),(c) With  precision offered by the ILC (red ellipses), a clear conflict with the measured value of the Higgs mass
(red lines) could be discovered over a wide range of BSM parameter space (green areas)~\cite{Heinemeyer:2007bw, Heinemeyer:2013dia}. In the case of the weak mixing angle,
ILC could resolve also the long standing discrepancy between the SLD and LEP measurements.
}
\label{fig:discEWPO}
\end{figure}


% JL: this comes in the next subsubsection anyway
% Similarly, the combination of the precise measurements of \MH\
% and \mt\ can show up to whiand/or sufficiently large mass gaps towards lighter new particles in the decay chain.
% ch high-energy scale the SM is valid (i.e.\
% at which scale the quartic Higgs coupling becomes
% zero)~\cite{Degrassi:2012ry}. According to current measurements, this is
% very likely well below the Planck scale and gives clear indications
% for physics beyond the SM.

\paragraph{Electroweak Couplings of the Top Quark: } Another guaranteed
part of the ILC physics program is measurement of the left- and right-handed couplings of the top quark to the $Z$~boson with an order of magnitude higher precision than possible at the LHC, c.f.\ section~\ref{subsec:ILC_top}. 
% This applies in particular on the
%left- and right-handed couplings to the $Z$~boson, which will allow a very
%precise test of the SM. 
Many BSM theories predict deviations from the SM in
these couplings, which cannot be resolved at the LHC. As shown in Fig.~\ref{fig:topcoup-lumi}, the excellent ILC precision
will not only enable the discovery of new particles up to energies of several tens of TeV,
far above the kinematic reach of the LHC, but
even allow for the distinction between the various BSM theories, as illustrated in Fig.~\ref{fig:models-rp}. 

\paragraph{Two-Fermion Production and New Gauge Bosons: } As discussed in section~\ref{subsec:ILC_ew}, the ILC measurements
 of $e^+e^- \to f \bar f$ for all types of fermions, 
making use of polarized beams,
will offer indirect access to a large variety of BSM models featuring
 new gauge bosons $Z'$ or $W'$. Their masses can easily be larger than the 
 kinematic reach of the LHC, i.e.\
larger than $\sim 4-5$~TeV. In this case they would escape detection 
at the LHC, while they could still be discovered at the ILC. This is
illustrated in Fig.~\ref{fig:zprime-discovery} for a variety of 
representative models~\cite{Osland:2009dp}, which then could be distinguished
by measurements of the vector and axial-vector couplings of the new gauge boson.
Thus, the discovery reach as well as the diagnostic reach of the
ILC extends far beyond the capabilities of the LHC.  

\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.6\columnwidth]{figs/top_H-20.pdf}\label{fig:topcoup-lumi}}
\hspace{0.1cm}
\subfigure[]{\includegraphics[trim=10 80 0 100, clip, width=0.325\columnwidth]{figs/zp-discovery}\label{fig:zprime-discovery}}
\caption{
 (a) Precision of the electroweak $t$ quark coupling $g_L$ (left scale, lower curves) and the mass reach expressed in terms of $M_{KK}$ (right scale upper curves) for a typical BSM scenario, here~\cite{Pomarol:2008bh} as a function of ILC running time~\cite{Barklow:2015tja}. 
%
%The full lines represent the improved statistical error while other lines take into account systematic errors (theory and/or experiment) of 0.5\%  after ${\mathcal L}=500\,\ifb$ that either scale with luminosity (dashed lines) or that are irreducible (dotted lines).
 (b) The discovery reach of ILC for new $Z'$ gauge bosons exceeds that of the LHC in various BSM models~\cite{Osland:2009dp}. 
 }
\end{figure}

% JL: moved to sec 3.3
% In $e^+e^-$ collisions (far) below the production threshold for the extra gauge
% bosons, they manifest themselves as deviations from SM predictions due to
% interference between the new physics and the SM $\gamma/Z$ contributions,
% see~\cite{Freitas:2013xga} for a recent review. 
% This is similar to indirect observations of the presence of the $Z$ boson
% in $e^+e^-\rightarrow \mu^+\mu^-$ scattering at the PETRA and TRISTAN 
% colliders well before the turn-on of LEP.
% The $e^+e^- \to f \bar f$ reactions are characterized by relatively
% clean, simple final states where $f$ could be leptons ($e$, $\mu$, $\tau$)
% or quarks ($u$, $d$, $s$, $c$, $b$, $t$), for both polarized and unpolarized
% electrons/positrons. Studies for the ILC~\cite{Osland:2009dp} have shown that
% a 500~GeV ILC with 500~fb$^{-1}$ or a 1~TeV ILC with 1~ab$^{-1}$ can see
% evidence for a $Z'$ with masses exceeding $\sim 7$~TeV and $\sim 12$~TeV for
% many models, respectively. The measurements will allow to distinguish between
% a variety of different $Z'$ models, starting already at 
% $\sqrt{s} = 500$~GeV~\cite{Godfrey:2005pm}. Similar numbers hold also for
% $W'$~bosons~\cite{Godfrey:2000hc}. 






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsubsection{Stability of the Higgs potential}
%\label{subsubsec:noNP_HiggsPot}
\paragraph{Stability of the Higgs potential:}

If no evidence for new physics is found at the LHC, the SM might be valid up
to the Planck scale. %In this case the precise measurements of $\mt$ and $\MH$
%that can be performed at the ILC play a special role. 
It was shown, see
e.g.~\cite{Degrassi:2012ry}, that the extrapolation of the Higgs self-coupling
to the Planck scale indicates at the level of $2\,\sigma$ that our vacuum state
is only meta-stable and could tunnel to the true vacuum. The life-time of our
vacuum, however, appears to be sufficiently long-lived on cosmological time
scales. 
%The precision of this determination depends crucially on the experimental
%accuracy of the top quark and the Higgs-boson mass. 
The high precision measurement of $\mt$ that can be performed 
at the ILC (c.f.\ section~\ref{subsec:ILC_top}) is indispensable to decide 
with certainty whether or not the SM vacuum is meta-stable, and to
determine its lifetime. 
Conversely, if a clearly ``too short'' lifetime is derived, 
this would be an unambigous proof
that physics beyond the SM must exist so that our vaccum could survive until today.
Or, if the SM vacuum turns out to be right on the boundary between the meta-stable
and stable regions within the ILC precisions, this would suggest the existence of
some completely new principle that leads to the apparent finetuning.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Extended Higgs Sectors}
\label{subsubsec:noNP_SUSYHiggs}

%{\color{red} [content: complementary discovery potentail in LHC wedge region, a%nd for states lighter than H125, Higgs CP, Higgs precision,
%Higgs self-coupling and EW baryogenesis] }\\
%{\color{blue} [proposed author: Sven Heinemeyer] }\\

% JL: to section 2!
%Most scenarios of BSM physics predict an extension of the SM Higgs boson
%sector. Beside a Higgs boson with SM-like couplings and a mass 
%$\sim 125$~GeV, the extended Higgs sectors imply additional Higgs bosons,
%where the masses can be above $\sim 125$~GeV, but also below this value. 
%The additional states have typically strongly suppressed couplings to SM
%gauge bosons, and thus can have escaped the LEP limits. 

\paragraph{Additional Higgs bosons heavier than \boldmath{$125$}\,GeV:}
As discussed in section~\ref{subsec:ILC_Higgs}, pair production of heavy Higgs bosons at the ILC
will be limited to masses of about half the center-of-mass energy, but it will cover this
regime in a largely model-independent way and in particular also for strongly reduced couplings to the $Z$
boson, as they are expected to be if the H(125) continues to look SM-like.
In specific MSSM scenarios (e.g.\ $m_h^{\mathrm{mod+}}$~\cite{Carena:2013ytb}), additional Higgs bosons 
with masses above $125$~GeV are already now significantly constrained by LHC searches. In more
general cases, like e.g.\ 2HDM Type I or II, this is not the case~\cite{CMS:pas-hig-16-007}.
Thus, in absence of a future discovery at the LHC, the corresponding searches at the ILC
will provide important additional information and offer complementary discovery potential. 

Furthermore, the ILC will probe Higgs masses far beyond the kinematic limit and beyond the 
reach of the LHC by its very precise and model-independent determinations of the coulings 
of H(125) to fermions and gauge bosons (c.f.\ section~\ref{subsec:ILC_Higgs}).   
Using simplified estimates,
% provided by
%equation~\ref{eqn:higgs_deviations}, {\color{red} SH: these equations have
%been deleted and they do not exist in this form in the previous sections.}
the ILC will be able to probe $m_A$ up to at least $1$\,TeV, independently of $\tan{\beta}$, 
and compositeness scales $f$ up to at least $3$\,TeV, thus investigating parameter space that
remains inaccessible at the LHC.
% JL: Roman suggests in 4.3 to cite ~\cite{Kakizaki:2015zva} for TeV scale sensitivity 
% via H-125 coupling measurements at ILC - not clear to me whether this is really added value
% compared to eqn~\ref{eqn:higgs_deviations} ?

%While their couplings to SM particles always leave the
%possibility of escaping detection at the LHC, the clean environment at
%the ILC will allow for a discovery up to the kinematic limit. In most
%cases this is given by $\sqrt{s}/2$, investigating parameter space that
%remains unaccessible at the LHC.

\paragraph{Additional Higgs bosons lighter than \boldmath{$125$}\,GeV:}
Also additional Higgs bosons below $125$~GeV can
escape searches at the LHC and have been undetectable at LEP due to their
reduced coupling to the $Z$ boson.
This scenario is a unique opportunity for the ILC. At the LHC large
backgrounds and soft decay products make the discovery of light Higgs
bosons particularly difficult. At the ILC, it will be easy to discovery
additional light Higgs bosons in a very large mass range up to 125~GeV. 
The ILC can thus considerably enlarge the discovery range of the LHC.

%The ILC also offers the opportunity to detect a BSM Higgs sector via
%precision measurements of the (SM-like) Higgs boson at $\sim 125$~GeV.
%At the LHC Higgs-boson couplings can only be determined under certain
%theory assumptions, where then precision of \order{5\%} can be reached.
%On the other hand, the ILC will enable us to perform model 
%independent Higgs-boson coupling measurements.

% JL: moved to section 3.1
% As a guideline, in SUSY and in composite Higgs models, one expects
% deviations of the size
% \begin{align}
% \kappa^{\rm SUSY}_b = \kappa^{\rm SUSY}_\tau \approx 1 + 1.6\% 
%            \left( \frac{1~{\rm TeV}}{M_A} \right)^2, \quad
% \kappa^{\rm comp}_V \approx \kappa_F^{\rm comp} \approx 1 - 3\%
%            \left( \frac{1~{\rm TeV}}{f} \right)^2
% \label{eqn:higgs_deviations}           
% \end{align}
% for deviations in the couplings to bottoms/taus (SUSY) and in the
% couplings to all SM gauge bosons and fermions (composite Higgs). Here
% $M_A$ denotes the general Higgs boson mass scale (SUSY) and $f$ the
% compositeness scale (composite Higgs). 

%In view of the current bounds on
%new physics scales, the LHC may not be able to see any deviation from
%the SM predictions. The ILC, however, with its Higgs-boson precision
%measurements in the percent (permille) range for couplings to fermions
%(massive gauge bosons), has a reach to very high scales of new physics
%in the Higgs-boson sector, far beyond the direct kinematical reach.

\paragraph{The Higgs self-coupling:} A special role is played by the Higgs-boson self-couping
$\lambda$, since it can deviate sizeably from the SM prediction even if
all other couplings of the Higgs boson are rather SM-like.
In particular values of $\lambda / \lambda_{SM} > 1.2$ are well motivated since they
would be a prerequisite for electroweak baryogenesis~\cite{Noble:2007kk}, e.g.\ as in the example of 2HDMs discussed in section~\ref{sec:2HDM}. 
According to current projections (based on fast detector simulations 
or four-vector smearing and assuming $100\%$ trigger efficiency), 
it seems highly unlikely that even the HL-LHC would 
discover double Higgs production: For $\lambda = \lambda_{SM}$, ATLAS~\cite{bib:atlas_higgsself_bbgg, bib:atlas_higgsself_bbtt} and CMS~\cite{bib:cms_higgsself} would only reach about $2\sigma$ significance for this process each, corresponding to less than $3\sigma$ in combination. For  $\lambda > \lambda_{SM}$, the cross section for
double Higgs production at the LHC diminishes further due to destructive
interference, thus further reducing the chances to observe this process.
At $95\%$\,CL, values of  $\lambda / \lambda_{SM}$ less than $-1.3$ and larger than $8.7$ could be excluded~\cite{bib:atlas_higgsself_bbgg}. 

As discussed in section~\ref{subsec:ILC_Higgs}, the situation is quite different at the ILC: Already at $\sqrt{s}=500$\,GeV, double Higgs production could be discovered with a significance of nearly $6\sigma$,
based on full detector simulation. As opposed to the situation at LHC and at $e^+e^-$ colliders with $\sqrt{s} \ge 1$\,TeV, the cross section 
{\itshape increases} for $\lambda > \lambda_{SM}$, making the discovery
even more significant in case of new physics leading to a larger values
of $\lambda$. Smaller values of $\lambda$ could be measured with at least $10\%$ precision at the $1$\,TeV upgrade of the ILC.

\paragraph{\boldmath{$CP$} odd admixtures:} Additional spectacular discovery potential is provided by the ILC's sensitivity to small $CP$-odd admixtures in the Higgs-to-fermion couplings, as discussed in section~\ref{subsec:ILC_Higgs}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Loophole-free Search for Light States }
\label{subsubsec:noNP_loopholefree}

%%-- (assuming the contents of this paragraph is integrated into chap2)
%%-- Light particles with only electroweak quantum numbers are predicted in
%%-- most models of new physics. In SUSY models with minimal field content,
%%-- there are the SUSY partners of the electroweak gauge boson, the Higgs
%%-- bosons, the leptons and neutrinos, while in non-minimal supersymmetry,
%%-- there could be e.g. additional Higgs superfield giving to additional
%%-- light scalars, pseudoscalars and neutral as well as charged
%%-- fermions. Naturalness in SUSY at least demands Higgsinos to be in the
%%-- same mass range as the Higgs boson. For extra-dimensional models,
%%-- there are correspondingly the reasonances of the electroweak SM
%%-- particles. In models with new strong
%%-- interactions like compositeness, Little Higgs etc. new chiral
%%-- symmetries that are spontaneously broken can lead to light states as
%%-- pseudo-Nambu-Goldstone bosons. As composite models treat
%%-- third-generation fermions in a special way, quite often couplings of
%%-- light states to the third-generation SM particles, particularly bottom
%%-- and top quarks are introduced. 

As pointed out in Section~\ref{sec:BSM}, many extensions to the SM predict 
light particles with only electroweak quantum numbers.
Their production rates at a hadron collider therefore is quite
small compared to the strong interaction mediated SM processes,
and are only accessible via the decay of heavier particles produced 
via the strong interaction or if they produce sizable missing transverse 
momentum or sufficiently high-energetic leptons.
Therefore, such states can quite conceivably exist even
if there is no further discovery at the LHC.
In section~\ref{subsec:directNP_NoLoophole}, it was shown that
in the case such state are SUSY partners to the SM particles,
the ILC is guaranteed to discover them nearly up to the kinematic limit, 
even in the case of very small mass splittings. 

This is illustrated by a specific example in figure~\ref{fig:noloophole2}, 
which shows the current limits in the $\MXN{1}$ - $\MXC{1}$ 
plane from ATLAS~\cite{Aad:2014vma}, together with the
projected discovery reach at 14 TeV with $\int \mathcal{L} \, \mathrm{dt}$ = 300 or 3000 fb$^{-1}$ \cite{ATLAS:2013hta}.
Here it is assumed that  $\MXN{2}=\MXC{1}$, that $\XPM{1}$ and $\XN{2}$ are pure Winos, and that Br($\chi\rightarrow W^{(*)}/Z^{(*)}\XN{1}$)=1
\footnote{Note that the more difficult case $\chi\rightarrow h^{(*)}\XN{1}$ is not considered.}.
The brown-shaded area indicates the corresponding limit from LEP \cite{Heister:2002mn,Abdallah:2003xe,Abbiendi:2002vz},
which assumes only  $\XPM{1}$ pair production, with no assumption on the decay mode.
The expected limits for the ILC at $\sqrt{s}=500$ or $1000$\,GeV are also shown with the same 
assumptions as for the LEP exclusion.
%The thick diagonal line indicates where $\MXN{2}$ (or $\MXC{1}$) is twice $\MXN{1}$. Below
%this line, GUT scale unification of the gaugino mass-parameters $M_1$ and $M_2$ is not possible,
%hence no model where such unification is assumed would be allowed in that region.
As can be seen from the (loophole) region not covered by the LHC, there is a large
discovery potential for the ILC, even after the high luminosity LHC data has been fully exploited.

\begin{figure}[h]
   \centering
      \includegraphics[width=0.5\linewidth]{figs/fig_07b_w_lep_lhc14_n_hilum_ilc500_ilc1000_proc}
\caption{Discovery or exclusion regions in the $M_{NLSP} - M_{LSP}$ plane
  for a $\XPM{1}$ or $\XN{2}$ NLSP. Solid brown area: LEP exclusion;
  Solid red and dashed gray lines: ATLAS exclusion (observed and expected);
  Solid blue (green) lines: ATLAS 14 TeV discovery projections for $\int \mathcal{L} \, \mathrm{dt}$ = 300 (3000) fb$^{-1}$;
  Dashed green (magenta) lines:  ILC discovery expectation for $E_{CMS}$ = 500 (1000) GeV;
  Solid black line: below line, no GUT scale gaugino mass unification.
  %% For further details, see text. 
 \label{fig:noloophole2}}
\end{figure}

Also in the case of other extensions to the SM,
such particles can be searched for over the whole mass range.
E.g.\ the tell-tale couplings to the third-generation SM particles
to new light states in composite models can be searched for
in the invariant mass spectrum of two $b$ jets in processes with two tagged
$W$s and four $b$ jets (c.f.\ e.g.\ \cite{Kilian:2004pp}). 
At LHC, on the other hand, such processes would lead to
anomalies in the top quark processes in fiducial regions that are
partially in background regions, so they could very likely be missed
even if they light enough to lie in the kinematic range of the ILC.
In the same way, decays into charm, tau's or photons can be detected.
Thus, by using its electroweak production modes and low backgrounds, 
the ILC can scan for all
existing particles in Nature with electromagnetic, hyper-charge or
electroweak quantum numbers and thus provides discovery potential
complementary to that of the LHC.  

%% Typcial examples for these are dark matter
%% searches such as monophotons and missing
%% energy. section~\ref{subsec:directNP_NoLoophole}.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Natural SUSY}

\label{subsubsec:noNP_NatSUSY}
% {\color{red} [content: unique discovery potential for Higgsinos, complementary to LHC, impact of Higgs precision] }\\
% {\color{dgreen} [volunteer: Howie Baer] }\\

As discussed in section~\ref{sec:BSM-SUSY-Naturalness}, a core prediction of natural SUSY models
is the existence of a triplet of light higgsinos with masses not too far above $100$\,GeV. 
In addition, top squarks may range up to $\sim 3$ TeV and gluinos up to $\sim 4$ TeV with litle cost to naturalness. 
Such heavy top squarks and gluinos may well lie beyond the reach of even HL-LHC.
Higgsinos are challenging to detect at the LHC since a) their production cross sections are significantly smaller
even than those for wino pairs (which were assumed in Fig~\ref{fig:noloophole2}), and b) because their mass splittings
are small, ranging from $20$\,GeV down to a few GeV or even less than $1$\,GeV in extreme cases.
Thus, SUSY may well remain natural even in the case where LHC ultimately discovers no new particles.
%LHC could be able to discover such scenarios in the future if either the gluinos, the stops or the Winos
%are light enough. However, from today's perspective, there is no guarantee 
%whatsoever that this will be the case.
 
The ILC on the other hand (as shown in section~\ref{subsec:directNP_higgsinos}), will be able to discover
the required light higgsinos for masses nearly up to half the center-of-mass energy even for sub-GeV mass differences --- 
{\itshape and independently of any assumption on the rest of the sparticle spectrum.} 
And proceeding beyond this unique discovery potential, 
the ILC will be able to derive information about the rest of the superparticle mass spectrum at the weak scale 
as well as on mass unification at a high scale from precision measurements of the higgsino properties.
As illustration of the power of these measurements, 
Fig.~\ref{fig:higgsino_unification} shows the RGE extrapolation of the ILC measurements 
presented in section~\ref{subsec:directNP_higgsinos} up to the GUT scale in two different
models-- the NUHM2 model with gaugino mass unification and the nGMM model with intermediate scale mirage unification
of gaugino masses-- leading to slightly different higgsino mass splittings.
As the figure shows, the two gaugino mass unification scenarios can clearly be distinguished. 
Under the assumption that all three gaugino masses unify at the same scale, 
then also the gluino mass at the weak scale can be predicted. 
Even without these model  assumptions, from the weak-scale parameter determination alone 
the masses of other sparticles (e.g.\ the stops, the heavier electroweakinos and the heavier Higgses) can be
predicted, giving important guidance for an energy upgrade of the ILC, the LHC  or for even more energetic future colliders.

\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.44\linewidth]{figs/ILC2mssm-gluino1-initial5param-M1M2-runningplot}\label{fig:runnig_ILC2_noLHC}}
\hspace{0.1\linewidth}
\subfigure[]{\includegraphics[width=0.44\linewidth]{figs/nGMM1-test1-M1M2-runningplot}\label{fig:running_nGMM1_noLHC}}
\caption{ILC probing the GUT scale: RGE extrapolation of the gaugino mass parameters $M_1$ and $M_2$ as determined from higgsinos properties measured at the ILC (c.f.\ section~\ref{subsec:directNP_higgsinos}) in two different high-scale models. (a) An NUHM2-based model with mass unification at the GUT scale (ILC2). (b) A mirage mediation model with mass unification at an intermediate mirage scale (nGMM1). Note that both figures only assume an initial ILC data set before the luminosity upgrade, with significant improvements expected after the full H-20 running scenario.
}
\label{fig:higgsino_unification}
\end{figure}

% JL: Introduce models in sec 2!
% The most direct criterion for natural SUSY derives from Eq. \ref{eq:mzs} which requires 
% the weak scale values of $\mu$ and $\sqrt{|m_{H_u}^2|}$ to be $\sim 100$ GeV. 
% Top squarks and gluino, which enter via the $\Sigma_u^u$ at the 1-loop and 2-loop level respectively, 
% can range up to $m_{\tilde{t}_1}<3$ TeV  ($m_{\tilde{g}}\lesssim 4$ TeV) whilst respecting naturalness. 
% This may be compared with the $5\sigma$ discovery reach of HL-LHC which 
% extends to $m_{\tilde{t}_1}\sim 1.1$ TeV and $m_{\tilde{g}}\sim 2.3$ TeV.
% Thus, gluinos, stops and other squarks may well lie beyond the reach of HL-LHC at little cost to naturalness.
% 
% In models with gaugino mass unification, then the mass hierarchy $\mu\ll M_1<M_2<M_3$ is expected
% and SUSY should be accessble to HL-LHC via the same-sign diboson 
% ($pp\to\tilde{\chi}_2^\pm\tilde{\chi}_4^0\to W^\pm W^\pm+$missing $E_T$ (MET) and/or 
% higgsino pair production $\tilde{\chi}_1^0\tilde{\chi}_2^0 j$ 
% (followed by $\tilde{\chi}_2^0\rightarrow \ell^+\ell^-\tilde{\chi}_1^0$ decay) 
% channels.
% 
% However, in the case of models such as natural generalized mirage mediation\cite{Baer:2016hfa}, then one
% expects a mass hierarchy $\mu\ll M_1\sim M_2\sim M_3$  where the gaugino masses may be rather
% compressed (for low mirage unification scale) and with $M_3\simeq m_{\tilde{g}}$ in the $2.5-4$ TeV range.
% In this case, natural SUSY apparently survives scrutinization by HL-LHC. Nonetheless, the required
% light higgsinos with mass $m(higgsino)\sim 100-300$ GeV (the lower the better) would still be
% accessible to ILC with $\sqrt{s}>2m(higgsino)$ although the inter-higgsino mass gaps would lie in the few GeV range, 
% leading to very soft visible decay products. Luckily, the clean environment of $e^+e^-$ colliders
% is well-suited for this very attractive manifestation of SUSY. 
% In such a scenario, ILC would be a higgsino factory. 
% Precision measurements of the higgsino mass splittings, 
% which are sensitive to the gaugino mass spectra, would allow for a test of whether gauginos arise from
% gaugino mass unification at $m_{GUT}$ or perhaps instead from mirage unification at some 
% intermediate energy scale.
% In the case of mirage unification, then ILC would be able to determine not only the mirage unification scale,
% but also the underlying fundamental {\it gravitino mass} $m_{3/2}$ which sets the scale for the gaugino masses.
% 
% Alternatively, it is possible in supergravity that the gauge kinetic function $\hat{f}_{AB}(h_m)$, 
% which transforms as the symmetric product of two adjoints, lies in a non-trivial GUT representation. 
% For $SU(5)$ SUSY GUT models, $\hat{f}_{AB}$ transforming as a $\bf 24$, $\bf 75$ or $\bf 200$
% dimensional rep has been considered\cite{Anderson:1999uia}. 
% In both the $\bf 75$ and $\bf 200$ cases, again a more  
% compressed weak scale gaugino spectrum is generated wherein natural SUSY can hide from
% HL-LHC searches. But again, the higgsinos that are required by naturalness 
% should be accessible to ILC which would also be able to determine the GUT group representation
% of the gauge kinetic function via gaugino mass measurements which influence the higgsino mass splitting.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{WIMP Dark Matter}
\label{subsubsec:noNP_WIMPs}
%{\color{red} [content: complementary discovery potential, mass determination, coupling measurement] }\\
%{\color{dgreen} [volunteer: Howie Baer] }\\

Searches for WIMP dark matter at the ILC are highly complementary to those
at hadron colliders and at direct detection experiments: as an electron-positron collider, 
ILC is sensitive to WIMP couplings to electrons, 
whereas hadron colliders and direct detection experiments are sensitive to WIMP 
couplings to quarks. Depending on the
type of particle mediating the WIMP-SM interaction, there is a priori no reason for these couplings to be of similar strength.Thus, if the LHC does not discover a deviation from the SM expectation in its ``mono-$X$'' searches, it is essential to complement the picture by probing the WIMP-lepton couplings at an electron-positron collider.

Moreover, while LHC can probe larger WIMP masses due to its higher center-of-mass energy, ILC can probe smaller couplings, thus higher energy scales for the WIMP-electron interaction due to its higher precision. As discussed in section~\ref{subsec:directNP_WIMP}, the ILC can probe new physics scales of several TeV and thus has here a unique discovery potential independent of LHC results.



% JL: this section was primarily intended for mono-photon searches. Maybe the SUSY dark matter 
% needs a separate section?
 
% As a result of recent negative search results for SUSY at the LHC, new models have been proposed
% with a very heavy spectrum for SUSY matter scalars while the gauginos remain quite light, populating the
% $100-500$ GeV range. Examples of such models include PeV SUSY\cite{Wells:2004di}, 
% split SUSY\cite{ArkaniHamed:2004fb}, spread SUSY\cite{Hall:2012zp}, 
% string models based on G2 holonomy\cite{Acharya:2008zi} and pure gravity mediation\cite{Ibe:2012hu}.
% In these models, the neutral {\it wino} is typically the lightest SUSY particle
% and is thermally underproduced as a dark matter candidate. Many authors have invoked non-thermal 
% mechanisms to boost the wino DM abundance\cite{Moroi:1999zb}, but these scenarios now seem ruled out
% because the relic winos should give rise to detectable {\it indirect} dark matter
% signals such as gamma ray line and continuum spectra\cite{Fan:2013faa,Cohen:2013ama}. 
% Alternatively, if winos along with axions comprise dark matter, then the wino-like WIMPs 
% may make up only a small fraction of dark matter and their
% galactic abundance may be suppressed so that they evade indirect DM searches\cite{Bae:2015rra}.
% In such a case, ILC could produce both charged and neutral winos using initial state photon
% radiation as a signal\cite{Berggren:2013vfa} 
% even while the bulk of the spectra is inaccessible to LHC searches.
% The quasi-stable charged winos may even leave visible tracks for a short distance before they decay
% to soft particles plus $E_{miss}$.

% More generally, WIMP pair production should be possible at ILC provided
% $\sqrt{s}>2m_{wino}$. In such a case, then initial state photon radiation may be enough to extract
% a signal for direct WIMP pair production at ILC even though the WIMPs escape as missing energy\cite{Bartels:2012ex}.

\subsubsection{$R$-Parity Violating SUSY}
\label{subsubsec:noNP_bRPV}

Another example of complementary discovery potential can be found in $R$-parity violating SUSY. Here, the LSP decays
into standard model particles and thus the characteristic feature of MET for hadron colliders 
will not be present, making it quite hard to probe such signatures at the LHC. 
For instance in the case of bRPV summarized in section~\ref{subsubsec:directNP_bRPV}, 
the relevant limits from LHC~\cite{bib:ATLAS_bRPV} are much weaker than in the $R$-parity conserving case, 
and rely even more crucially on strong production mechanisms. Thus, if the coloured SUSY particles are too heavy, LHC will most likely
not be able to detect light electroweak states which decay fully into standard model particles without significant MET.
Therefore, the example presented in section~\ref{subsubsec:directNP_bRPV}, but also other cases with similar signatures, will remain
unknown territory for the ILC to explore even if LHC does not discover further new particles. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


42_lightNPLHC.tex
%\subsection{LHC experiments discover relatively light new particles}
%\label{sec:lightNP}
In this section, we consider the scenario where LHC discovers new light 
matter states during Run II or Run III or at HL-LHC, 
and what impact such discoveries might have on the ILC physics program.

\subsubsection{Discovery via Precise Measurements of Standard Model Parameters}
If the LHC discovers relatively light new particles, these particles most likely will have
an impact on precision observables. The full ILC precision program layed out in section~\ref{subsubsec:noNP_indirect}
will not loose any of its importance, since it will provide a crucial closure test and since it will be indispensable 
in order to fully identify the kind of new physics discovered and to pin down the underlying model and its parameters.
This includes explicitly the model-independent precision determination of the 
H(125) couplings as discussed in~\ref{subsubsec:noNP_SUSYHiggs}, in particular 
in the case where the discovered particle is a candidate for a heavier Higgs boson.

\subsubsection{Loophole-free Search for Lighter States }
\label{subsubsec:lightNP_simplifiedSUSY}
%{\color{red} [content: discovery / exclusion reach for any NLSP] }\\
%\input{no_loophole_chap4_lightNPLHC}

%%-- (assuming the contents of this paragraph is integrated into chap2)
%%-- Light particles with only electroweak quantum numbers are predicted in
%%-- most models of new physics. In SUSY models with minimal field content,
%%-- there are the SUSY partners of the electroweak gauge boson, the Higgs
%%-- bosons, the leptons and neutrinos, while in non-minimal supersymmetry,
%%-- there could be e.g. additional Higgs superfield giving to additional
%%-- light scalars, pseudoscalars and neutral as well as charged
%%-- fermions. Naturalness in SUSY at least demands Higgsinos to be in the
%%-- same mass range as the Higgs boson. For extra-dimensional models,
%%-- there are correspondingly the reasonances of the electroweak SM
%%-- particles. In models with new strong
%%-- interactions like compositeness, Little Higgs etc. new chiral
%%-- symmetries that are spontaneously broken can lead to light states as
%%-- pseudo-Nambu-Goldstone bosons. As composite models treat
%%-- third-generation fermions in a special way, quite often couplings of
%%-- light states to the third-generation SM particles, particularly bottom
%%-- and top quarks are introduced. 

In case the LHC discovers new particles, these will most likely not be the lightest
states of the BSM particle spectrum, but those which either have strong interactions 
and/or sufficiently large mass gaps towards lighter new particles in the decay chain.
Therefore the complementary capabilities of the ILC to search in a loophole-free manner
for lighter new states with electroweak quantum numbers, as discussed in section~\ref{subsubsec:noNP_loopholefree}. 
This includes in particular also searches for lighter Higgs bosons as discussion in section~\ref{subsubsec:noNP_SUSYHiggs}.  
In any case, a discovery of light new particles will lead to a strong interplay of LHC and ILC measurements.
In the following we will give two explicit examples for this interplay and the joint power of the two complementary colliders.

% As pointed out in section~\ref{sec:BSM}, many extensions to the SM predict 
% light particles with only electroweak quantum numbers.
% Their production rates at a hadron collider are therefore quite
% small compared to the strong interaction mediated SM processes,
% and are only accessible via QCD
% production modes or in regions of the Drell-Yan spectrum that are not
% swamped by QCD backgrounds.
% If some such lighter states are nevertheless discovered at LHC, 
% it would still be an open question whether these are all new states,
% or if there are other, more elusive, ones.

% Therefore, further such states can quite conceivably exist even
% if they are not detectable at the LHC.
% In section \ref{subsubsec:directNP_NoLoophole}, it was shown that
% in the case such state are SUSY partners to the SM particles,
% the ILC is guaranteed to discover them, if they are within kinematic reach.
% Also, in the case of other extensions to the SM,
% such particles can be searched for over the whole mass range:
% {\it e.g.} the tell-tale couplings to the third-generation SM particles
% to new light states in composite models can be searched for
% in the
% invariant mass spectrum of two $b$ jets in processes with two tagged
% $W$s and four $b$ jets (cf. {\it e.g.}~\cite{Kilian:2004pp}). 
% At LHC, on the other hand, such processes would lead to
% anomalies in the top quark processes in fiducial regions that are
% partially in background regions, so very hard to detect. 
% In the same
% way, decays into charm, tau's or photons can be detected.
% Using its electroweak production modes, the ILC can scan for all
% existing particles in Nature with electromagnetic, hyper-charge or
% electroweak quantum numbers.  

%% Typcial examples for these are dark matter
%% searches such as monophotons and missing
%% energy. Section~\ref{subsubsec:directNP_NoLoophole}.


\subsubsection{LHC-ILC Interplay - a SUSY Dark Matter example}
\label{subsubsec:lightNP_SUSY}
%{\color{red} [content: STC study, parameter determination, relic density, impact of Higgs precision] }\\
%\input{SUSY_DM_chap4}

In this section, we take as an example a Dark Matter motivated benchmark scenario~\cite{Baer:2013ula} featuring 
$\tilde{\tau}$-coannihilation, which has been studied with respect to future LHC and ILC 
capabilities~\cite{Berggren:2015qua}, with some of
the latter also being summarized in~\ref{subsec:directNP_SUSYDM}.
In such scenarios, the LHC would discover some of the heavier SUSY particles, in particular the lighter stop and
sbottom, as well as some hints of the heavier neutralinos and charginos. However a full determination
of the mass spectrum would be extremely challenging based on LHC data alone. In particular the discovery of the 
$\tilde{\tau}_1$ and a determination of its properties, which are crucial to test whether coannihilation is indeed
the explanation for the cosmologically observed Dark Matter relic density~\cite{Lehtinen:2016qis}, will have to 
wait until the ILC turns on.

On the other hand, the detailed information from the
ILC can be used in the analysis of the LHC data in order to disentangle the contributions
of different production modes and to reconstruct quantities which are sensitive to masses of the
heavier sparticles with more complex decay chains. An example is the  case
of the \XPM{2} shown in Fig.~\ref{fig:chargino2_mass}, which is out of reach to the ILC, 
but which can --- with the knowledge of the masses of the \XPM{1}~ and the \snu\ from the ILC --- 
be isolated from the electroweakino mix at the HL-LHC. Its mass can then be reconstructed on an
event-by-event basis with a resolution of about $50$\,GeV at the HL-LHC. 
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.44\linewidth]{figs/h_3lmass_true_omc2} }
\subfigure[]{\includegraphics[width=0.44\linewidth]{figs/h_3lmass_rec_mc2_true} }\\
\caption{ \XPM{2} mass information at LHC when \MXC{1} and \msnu\ are known from ILC. 
(a) MC truth information of the invariant mass as well the reconstructed \MXC{2} for the same events. 
(b) Reconstructed \MXC{2} for all selected SM and SUSY events, where based on (a) the leftmost peak near
$400$\,GeV can be identified as \MXC{2}.
}
\label{fig:chargino2_mass}
\end{figure}

% The LHC puts strong limits on the mass of first and second generation squarks  and on the gluino.
% These states have only indirect influence on dark matter, the muon magnetic moment anomaly and
% naturalness.
% If one requires GUT-scale gaugino mass unification, then
% these limits do indirectly constrain many relevant states such as third generation squarks 
% and the EWino sector.
% 
% By relaxing the assumption of gaugino mass unification,
% \footnote{
% The {\it couplings} at the GUT scale can still be required to
% unify.
% }
% then LHC limits only remotely constrain third generation  squarks, electroweakinos and sleptons.
% These states can be important for describing SUSY contributions to precision observables and 
% the thermal WIMP abundance.
% If, in addition, the SUSY spectrum is compressed, one would expect  long cascade decay chains to
% occur  at LHC, ending up at the NLSP, which in turn decays into the LSP.
% And due to the low mass difference required in coannihilation models, one then
% expects rather soft visible particles from SUSY production processes.
% This is thus not  a  model predicting large missing E$_T$, 
% nor a simplified model where the produced heavy coloured particles exclusively decay directly
% to the LSP.
% Taken together, the current limits on  coannihilation models from LHC become
% significantly weaker than those obtained from simplified models or models yielding 
% signatures with large missing $E_T$.
% 
% In section~\ref{subsubsec:directNP_SUSYDM}, the arguments for considering SUSY 
% as the explanation for the existence of WIMP dark matter were outlined.
% The coannihilation model is a well-motivated scenario which
% presents many generic features that such a SUSY model should have. 
% The model should, in addition to containing a DM candidate,
% accommodate the current LHC limits,  the requirement from naturalness that the third generation squarks
% should be not too heavy, and that the Higgs-sector, in particular the Higgs-mass, 
% should agree with observations.
% It should also fulfil all constraints for precision measurements.
% A set of possible realisations of this kind of model (called STC4 to STC10) 
% was suggested in  \cite{Baer:2013ula}.
% The different models differ only in the magnitude of the $\stq$ mass,
% while the electroweak and slepton sectors are are quite similar.
% The coannihilating NLSP is the $\stone$, which is about 10 \GeV~ heavier than the LSP.
% The mass-spectrum and decay-modes are shown in  Fig.~\ref{fig:stc8massnbr}. 
% One can observe that the predicted decay-modes are generally quite complex
% with long chains of decays eventually ending up at a $\stone \rightarrow \tau \XN{1}$.
% Both the ILC and LHC prospects for discovering these models
% were evaluated in Ref.~\cite{Berggren:2015qua}. 
% The different models (STC4 to STC8) differ only in the magnitude of the $\stq$ mass,
% while the electroweak and slepton sectors are identical.
% A summary of the findings at LHC is that
%  \begin{itemize}\itemsep1pt \parskip0pt\parsep0pt 
%  \item The high mass first and second generation squarks and the gluino 
% will not be seen by HL-LHC since the production cross-sections are quite small.
%  \item The lower mass third generation squarks will be eventually be discovered.
%  \item The electroweakino sector will be detected, but with little possibility to 
% disentangle the different states.
%  \end{itemize}
% Hence, in this model, LHC will indeed discover relatively light new particles.
% \begin{figure}[htb]
%   \begin{center}
%     \subfigure[]{\includegraphics [scale=0.47]{figs/STC8_BR01}}
% %    \subfigure[]{\includegraphics[scale=0.20] {plots/bosino-xsect-stc}}
% %    \subfigure[]{\includegraphics[scale=0.25] {plots/sfermion-xsect-stc}}
% %.eps.gzall-stcs.png}}
% %%    \subfigure[]{\includegraphics [scale=0.6]{plots/STC4b}}
%   \end{center}
%   \caption{\label{fig:stc8massnbr} 
% %% (a): Mass-spectrum and decay modes of the model; (b) and(c): production cross-sections
% %%   at ILC for (b) bosinos, and (c) sfermions.
% Mass-spectrum and decay modes  (with BR $>$ 10 \%) of STC8. Heavier lines indicate larger
% branching ratios.
% }
% \end{figure}


%%% JL: maybe reactivate / integrate above if space?
% In section \ref{subsubsec:directNP_SUSYDM}, the ILC capabilities for this model were
% given: the masses of the lighter sleptons and the \XN{1} would be determined at the
% per mil level, and cross-sections and mixings to the percent level.
% It was pointed out that with such precisions, ILC will be capable to measure the $\tilde{\chi}_1^0$ WIMP 
% properties with a precision needed to confront the cosmological observations on 
% CMB from the Planck collaboration, and hence to determine if indeed the 
% $\tilde{\chi}_1^0$ is the main contributor to DM.

% This information from the ILC also creates new opportunities to analyse the LHC data.
% Obviously, a precise determination of the LSP mass from the ILC can be employed to turn
% edge determinations into mass measurements. Even better, the detailed information from the
% ILC can be used in the analysis of the data themselves to disentangle the contributions
% of different production modes and to reconstruct quantities which are sensitive to masses of the
% heavier sparticles with more complex decay chains. An example is the  case
% of the \XPM{2}, which is out of reach to the ILC. 
% Figure \ref{fig:chargino2_mass} shows that with the knowledge of the masses of the \XPM{1}~ 
% and the \snu\ from the ILC,
% the signal can be isolated from the electroweakino mix and its mass can be reconstructed on an
% event-by-event basis with a resolution of about 50 \GeV~ with an integrated luminosity of 
% 300 fb$^{-1}$ at 14 \TeV, with corresponding improvements at the HL-LHC. 
% \begin{figure}[!h]
% \centering
% \subfigure[]{\includegraphics[width=0.44\linewidth]{figs/h_3lmass_true_omc2} }
% \subfigure[]{\includegraphics[width=0.44\linewidth]{figs/h_3lmass_rec_mc2_true} }\\
% \caption{ \XPM{2} mass information at LHC when \MXC{1} and \msnu\ are known from ILC. 
% The left plot (a) shows the MC truth information of the invariant mass obtained with the procedure 
% described in the text as well the reconstructed \MXC{2} for those events. 
% The right plot (b) shows the reconstructed \MXC{2} for all events we would expect. 
% There the SM and the SUSY events are stacked together.  
% }
% \label{fig:chargino2_mass}
% \end{figure}


\subsubsection{LHC-ILC Interplay - a Natural SUSY example}
\label{subsubsec:lightNP_NatSUSY}
% {\color{red} [content: radiative Natural SUSY study, parameter determination with gluino (and maybe stop?) from LHC, impact of
% Higgs precision, show unification] }\\

So far LHC has gathered only $\sim 1\%$ of its projected integrated luminosity. 
Thus, there is still an excellent chance that rather light top squarks and/or gluinos could be discovered at LHC.
Also, discovery of wino pair production~\cite{Baer:2013yha} and/or higgsino pair production\cite{Baer:2014kya} 
remains as a possibility.
Any of these discoveries could provide confirmation for a SUSY spectrum compatible with naturalness
as discussed in section~\ref{sec:BSM-SUSY-Naturalness}. 
However, similar to the case explained in Sec.~\ref{subsubsec:noNP_NatSUSY}, the direct production and
observation of all three individual higgsino states as well as their precise characterisation 
(c.f.\ section~\ref{subsec:directNP_higgsinos}) will remain an important task for the ILC.
Such measurements could help determine whether the lightest higgsino comprises all of, or just a portion of, the
dark matter in the universe.

% In supersymmetric models with radiatively driven naturalness, the superpotential $\mu$ parameter is required to
% be $\mu\sim 100-300$ GeV, the closer to $m_Z$ the more natural. In addition, even if the soft terms are rather
% heavy (of order $m_{3/2}\sim$ several TeV in gravity-mediation), then the Higgs soft mass $m_{H_u}^2$ can be driven
% radiatively from large GUT scale values to natural weak scale values via the 
% large top quark Yukawa coupling which occurs in the RGEs. As mentioned previously, top squarks and gluinos may well be
% beyond the reach of LHC Run III or HL-LHC at little cost to naturalness. 
% Even so, there still exist excellent prospects for a
% SUSY discovery at LHC although it may take hundreds-to-thousands of fb$^{-1}$ to establish it. In models with gaugino mass
% unification at the GUT scale $m_{GUT}\simeq 2\times 10^{16}$ GeV, the weak scale gaugino masses occur in the 
% ratio $M_1:M_2:M_3\sim 1:2:7$ so that the mass hierarchy $\mu < M_1<M_2<M_3$ ensues. In such a case, wino pair production
% $pp\rightarrow\tilde{\chi}_2^\pm\tilde{\chi}_4^0$ followed by $\tilde{\chi}_4^0\rightarrow W^\pm\tilde{\chi}_1^\mp$ and 
% $\tilde{\chi}_2^\pm\rightarrow W^\pm\tilde{\chi}_{1,2}^0$ leads to a distinctive same-sign diboson 
% plus $E_T^{miss}$ signature with mild jet activity and very low background rates at LHC\cite{Baer:2013yha}. 
% In models with gaugino mass unification, HL-LHC should be able to explore all
% natural SUSY parameter space with $\Delta_{EW}<30$ at the $5\sigma$ level\cite{Baer:2016usl}. 
% Total rate measurements and distributions such as
% $m_{T2}$ should allow for an extraction of the wino mass $M_2$ at the 5-10\% level. 
% In addition, higgsino pair plus hard jet production $pp\rightarrow\tilde{\chi}_2^0\tilde{\chi}_1^0 j$ 
% followed by $\tilde{\chi}_2^0\rightarrow \ell^+\ell^-\tilde{\chi}_1^0$ 
% decay to soft dileptons should be seen above SM backgrounds\cite{Han:2014kaa,Baer:2014kya}. 
% The $m(\ell^+\ell^- )$ distribution is kinematically bounded by the
% $\tilde{\chi}_2^0-\tilde{\chi}_1^0$ mass difference which is expected $\sim 10-20$ GeV. 
% These signals would point to the presence of light higgsinos 
% $\tilde{\chi}_1^\pm$ and $\tilde{\chi}_{1,2}^0$ with mass $\sim 100-300$ GeV which should be accessible to ILC.
% 
% While LHC could plausibly discover the $\tilde{\chi}_1^0\tilde{\chi}_2^0j$ reaction, both the
% reactions $e^+e^-\rightarrow\tilde{\chi}_1^0\tilde{\chi}_2^0$ and $\tilde{\chi}_1^+\tilde{\chi}_1^-$ 
% should occur at large rates for ILC operating above production threshold $\sqrt{s}>2 m(higgsino)$. 
% The first reaction gives rise to a soft OS/SF dilepton
% pair recoiling against $E^{miss}$. 
% Endpoint measurements of $E(\ell^+\ell^- )$ and $m(\ell^+\ell^- )$ should allow for 
% extraction of $m_{\tilde{\chi}_1^0}$ and $m_{\tilde{\chi}_2^0}$ to sub-percent precision and an extraction of the 
% superpotential higgsino parameter $\mu$. 
% The mass splitting, measured to even better precision via the $m(\ell^+\ell^- )$
% distribution cutoff, allows indirect sensitivity to the gaugino masses $M_1$ and $M_2$. Likewise, the reaction
% $e^+e^-\rightarrow \tilde{\chi}_1^+\tilde{\chi}_1^-$ followed by $\tilde{\chi}_1^+\rightarrow\tilde{\chi}_1^0\ell\nu_{\ell}$ and
% $\tilde{\chi}_1^-\rightarrow \tilde{\chi}_1^0+(q\bar{Q})$ can allow for an $m_{\tilde{\chi}_1^\pm}$ and $m_{\tilde{\chi}_1^0}$ 
% mass extraction from the endpoints of the $E(jj)$ and $m(jj)$ distributions. 
% These extractions are directly sensitive  to $\mu$ 
% and indirectly to the wino mass $M_2$ which affects the $\tilde{\chi}_1^\pm -\tilde{\chi}_1^0$ mass splitting.
% Further measurements such as higgsino-bino $\tilde{\chi}_{1,2}^0\tilde{\chi}_3^0$ production may be available at higher energies.
% In addition, dedicated threshold measurements will be even more sensitive to the electroweak-ino mass extractions.

By just assuming the higgsino property determinations of the ILC, 
plus a $10$\% measurement of the gluino mass from LHC, most weak-scale SUSY parameters can be determined.
This enterprise will  enable predictions for possibly unobserved parts of the spectrum, 
including the heavier electroweakinos, the top squarks and the heavy Higgs bosons. 
Typical precisions for mass determinations are expected at the $10$-$20$\% level\cite{Baer:2016wkz}, 
depending on the exact scenario. 
Similar to the example given in section~\ref{subsubsec:lightNP_SUSY}, this information will provide important input 
to LHC analyses, as well as for the planning of future colliders.

Furthermore, from running the weak-scale gaugino masses via renormalization group evolution to high energy 
(as shown in Fig.~\ref{fig:higgsino_unification_withLHC}), tests can be made as to the locus of the 
gaugino mass unification scale; in this case, standard GUT-scale mass unification can clearly be 
distinguished from alternatives such as e.g.\ mirage unification. 
%
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.44\linewidth]{figs/ILC1mssm-fullrun1-5paramH20-M1M2M3-runningplot}\label{fig:runnig_ILC1_withLHC}}
\hspace{0.1\linewidth}
\subfigure[]{\includegraphics[width=0.44\linewidth]{figs/nGMM1-test1-M1M2M3-runningplot}\label{fig:running_nGMM1_withLHC}}
\caption{ILC and LHC probing the GUT scale: RGE extrapolation of the gaugino mass parameters $M_1$ and $M_2$ as determined from higgsinos properties measured at the ILC (c.f.\ section~\ref{subsec:directNP_higgsinos}) and assuming a $10\%$ measurement of the gluino mass from the LHC in two different high-scale models. (a) An NUHM2-based model with mass unification at the GUT scale (ILC1, assuming the full H-20 running scenario). (b) A mirage mediation model with mass unification at an intermediate mirage scale (nGMM1, assuming only an initial ILC dataset before the luminosity upgrade).
}
\label{fig:higgsino_unification_withLHC}
\end{figure}


\subsubsection{WIMP Dark Matter}
\label{subsubsec:lightNP_WIMPs}
%{\color{red} [content: coupling to proton vs coupling to lepton, mass determination, coupling measurement] }\\
While the LHC already has some sensitivity to WIMP masses, it continues to extend the reach
at low masses towards smaller and smaller couplings. Thus, in principle it could still find a deviation from the standard
model in the kinematic reach of the ILC. In this case, it would be important to test whether the found WIMP also couples
to leptons, and at which strength. If it couples to leptons as well and is visible at the ILC, the fact that at a lepton
collider the initial four-momentum is known and the beam helicities can be chosen will allow a full precision characterisation
of  the WIMP and the associated mediator particle, as summarized in section~\ref{subsec:directNP_WIMP}.


%\subsubsection{anything else?}
%\label{subsubsec:lightNP_else}

43_heavyNPLHC.tex
\newcommand{\lhhh}{\lambda_{hhh}}
\def\beq{\begin{equation}}
\def\eeq#1{\label{#1}\end{equation}}
\def\eeqn{\end{equation}}
\def\ee{e^+e^-}

\def\leqn#1{(\ref{#1})}
% \subsection{LHC experiments discover relatively heavy new particles}
% \label{sec:heavyNP}
%\subsubsection{Indirect Discovery Reach }
%\label{subsubsec:heavyNP_indirect}
%{\color{red} [content: identify type of NP and discriminate between models via Higgs precision, top ew couplings, two-fermions etc] }\\
% {\color{red} [Roman, TO DO list: Section heavily inspired (copied from) by 750 GeV paper, more on Spin 2 case?; connection to section 2.2 to be improved, there composite Higgs, here rather RS type models, models interlinked by duality but still; a few words, extended Higgs sector wedge region;  is it enough to cover new strong sector with composite Higgs? i.e. relation to ICFA Letter; everything shorter but how much can one reasonably integrate here? May have to make a choice]; general scrutinising}\\

In this section, we will address the case that the LHC discovers a rather heavy new particle. This is, after all, the kinematic regime
in which the LHC is most powerful, since Standard Model backgrounds by QCD start to lower. If it had turned out to be a real signal, $X750$ would have been an excellent example in this category, for which the potential of the ILC has been discussed in detail
in Ref.~\cite{Fujii:2016raq}. Here, we will summarize and generalize the points made therein and will show that also in this
case the complementary discovery potential of the ILC will add crucial information in order to complete the picture of the new physics, which in any realistic model will consist out of more than one new particle.
%It is a fair statement that the LHC is most powerful at the high end of its kinematic reach where Standard Model backgrounds by QCD start to lower. This section will review a few typical discovery scenarios of the LHC and motivates the ILC precision measurements that are mandatory in order to complete the picture of the new physics.  

\subsubsection{Extended Higgs Sectors and Higgs Coupling Measurements}
\label{subsubsec:heavyNP_SUSYHiggs}
If the discovered particle is compatible with being a spin-0 resonance, 
it could be part of an extended Higgs sector, for instance
one of the heavy $H,A$ states of a 2HDM (see section~\ref{sec:2HDM}). In this case, it would be mandatory to both search for the remaining Higgs states for which the ILC offers discovery potential complementary to the LHC. As discussed in section~\ref{subsubsec:noNP_SUSYHiggs}, the additional Higgs states could be produced directly at the ILC at some cases, while
the model-independent precision measurements of the couplings of the H(125) 
to SM particles will reliably discover further states
up to high mass scales, beyond the direct reach of the LHC.

In addition, such a new scalar could lead to very interesting enhancements in the Higgs self-coupling $\lambda$ by at least $20$\%, as they would be required for electroweak baryogenesis~\cite{Noble:2007kk}, e.g.\ as in the example of 2HDMs discussed in section~\ref{sec:2HDM}.  
Loosely speaking, the deviations on $\lambda$ scale with the mass of the new scalar. This is illustrated in Fig.~\ref{fig:delhhh} in which possible deviations of $\lambda$ (denoted as $\lhhh$) for different masses of the new scalar, denoted as $m_\Phi$, and as a function the mass parameter $M$ of the 2HDM model. As can be seen,
 the potential deviations are largely independent of the latter parameter as long as it is not too high. The contour labeled as $\phi_c/T_c$ indicates mass values above which a first order phase transition can occur. 

\begin{figure}
 \begin{center}
    \includegraphics[width=0.4\textwidth,angle=-90]{figs/Delhhh_vs_vcTc.pdf}
    \caption{\label{fig:delhhh} Deviations of the Higgs self-coupling $\lhhh$ for different masses of a heavy scalar as a function of the mass parameter $M$ of the 2HDM Model~\cite{Senaha:2013fva}}.
\end{center}
\end{figure}

In case of the new particle been a radion of an extra dimension model, the mixing with the Higgs boson may vanish as e.g.\ argued in~\cite{Ahmed:2015uqt}. In this case the H(125) couplings to fermions will be SM-like, while couplings to photons and gluons  will vary by some amount. However, as discussed in section~\ref{sec:RS}, concrete realisations of such models imply the existence of additional heavy vector (Spin-1) bosons that occur e.g.\ as so-called Kaluza-Klein (KK) excitations. These vector bosons will be discussed further below in section~\ref{subsubsec:heavyNP_indirect}.
 
But also a new Spin-1 resonance, for instance as arising in Higgs compositeness models, would introduce shifts in the couplings of the Higgs boson to heavy Standard Model particles $W$, $Z$ and $t$ that are always similar to those given in Eq.~\ref{eq:kappa_SILH}. Typical shifts for new $5$\,TeV particles are $\sim8\%$  but even a $20$\,TeV particle would produce observable effects, for both $W$ and $Z$, in the ILC precision measurements~\cite{Malm:2014gha}.
% {\color{red} [JL: I'm not quite sure about these numbers. When I insert the H-20 precisions into~\ref{eq:gHVV_MCHM}, I get
% rather 3 TeV, as written in section~\ref{subsubsec:noNP_SUSYHiggs}.]}

%\paragraph{The Top Yukawa Coupling}
In composite Higgs models as well as in RS models, effective top quark Yukawa couplings are generated. Figure~\ref{fig:ctct5RSC} shows the shifts of the Higgs boson couplings computed in \cite{Malm:2014gha} for a variety of parameters sets, as a function of the mass of the lightest KK gluon.  Note that these shifts are in general complex,
leading to $CP$ violation in the $ht\bar t$ coupling at levels observable at the ILC~\cite{Godbole:2011hw}.   
The figure shows the shifts as parameters $c_t$, $c_{t5}$ corresponding to the effective coupling
$
    \delta \L =  - {m_t\over v} \ h \  \bigl[  c_t\ \bar t t + i
    c_{t5}\  \bar t \gamma^5 t \bigr]  .
$

\begin{figure}
 \begin{center}
    \includegraphics[width=0.9\textwidth]{figs/ctct5CRS.pdf}
    \caption{\label{fig:ctct5RSC}Predictions of $CP$-even (left) 
and $CP$-odd (right) Higgs couplings to the $t$ quark in a Randall-Sundrum model with a custodial
symmetry. The parameters $c_t$, $c_{t5}$ are defined in the text.  The point clouds are a scan of the space of Yukawa couplings for three values of the free parameter $y_{\ast}$. The gray band on the right hand 
side shows the experimental bound at 90\% on the $CP$-odd predictions 
derived from the electron dipole moment (EDM)~\cite{Malm:2014gha}. 
%
%Figure and parts of the caption are taken from~\cite{Malm:2014gha}.  
}
\end{center}
\end{figure}


% JL: try out structure more similar to other sections
%\subsubsection{Bosonic resonances}
%Among the most likely discoveries in terms of heavy particles at the LHC are the appearances of bosonic resonances. It is convenient to distinguish these new bosons by their spin. Theories exists for Spin-0 (scalar), Spin-1 (vector) and Spin-2 (tensor) particles. 
%\paragraph{Spin-0:}  

% JL: attempt to be brief
%A case study for the power of the ILC in case of the discovery of a new Spin-0 particle has been carried out~\cite{Fujii:2016raq}. 
%as a reply to the sizeable fluctuation at $\roots=750\GeV$ reported by the LHC experiments ATLAS and CMS in 2015. While an early discovery at the LHC favours a comparatively light particle, the higher statistics available now and in the near future may lead to a similar signal by a particle heavier than 750\,GeV. It is therefore convenient to summarise here the arguments given in~\cite{Fujii:2016raq}.  
%For such a scalar particle one has to regard first closely a possible mixing with the Higgs Boson. Large mixings between the Spin-0 particle and the Higgs are excluded by already existing LHC data but also by electroweak precision data. However, the precision of the couplings of the Higgs to e.g.\ Standard Model fermions will be limited at the LHC to about 5\%. 
% JL: to Sec 2?
%The electroweak precision data already today put stringent bounds on a possible mixing of a new Spin-0 particle with the Higgs boson, limiting shifts in the couplings of the Higgs to SM fermions to about 6\%. Such shifts cannot be measured to a sufficient precision at the LHC while it will be well measurable at the ILC at $\roots = 500\,\GeV$. 

% JL: moved above
% In the particular case in which the new scalar will be identified as the radion of models with  such as the Randall-Sundrum (RS) Model~\cite{Randall:1999ee}, the mixing angle will vanish. In this case, the ILC will measure couplings to fermions that are consistent with the Standard Model value while couplings to photons and gluons that will vary by some amount. The realisation of a RS Model (or related models) implies the existence of heavy vector bosons that occur e.g.\ as so-called Kaluza-Klein (KK) e

% JL: discussed above
%The new scalar could be one of the heavy $H,A$ states of a 2-Higgs-doublet model (2HDM).  
% JL: to sec 2
%In this model a second doublet is added with the same quantum numbers as the standard model one giving rise to two vacuum values $v1$ and $v2$ related by $v1/v2 = {\rm tan} \beta$. The corresponding scalar potential contains a mass parameter $M$ and after symmetry breaking the Higgs sector is made of five Higgs bosons, denoted as $h, H, A, H^{\pm}$. The 2HDM entails mixing parametrised by the mixing angle $\alpha$ of the heavy Higgs bosons with the 125~GeV Higgs boson, producing shifts mainly in the $\tau\tau$ and $b\bar b$ couplings of the known Higgs boson.  

% JL: probably too detailed here compared to discussion in 4.1
% Observation of these effects of heavy $H,A$ bosons of masses at the TeV scale are expected~\cite{Kakizaki:2015zva} to be well within the 5 $\sigma$ reach of the full ILC program described in \cite{Fujii:2015jha}. An interesting phase spaceregion is the called "wedge region" that at medium ${\rm tan} \beta$ allows masses of $m_A$ down to 200\,GeV although updated LHC start to exclude values  $m_A<400$\,GeV~\cite{Aaboud:2016cre}. Even at lowest values the LHC may not have enough precsion to provide measur
% may be detectable at the ILC latest in the upgrade to 1\,TeV {\bf (please check)}. In this case also the channel $\epem \to hA$ may open up.    
%At medium ${\rm tan} \beta$ decays of the $H,A$ to heavy fermions are suppressed. In this so-called "wedge-region" only the clean environment of an $\epem$ collider would allow to study the decays of these bosons. 


%% JL: shortened version moved above
% The observation of new scalar may entail another exciting consequence. One of the main questions in today's science is the explanation of the observed Baryon Asymmetry in the Universe (BAU), i.e. the abundance of matter in form of e.g. light elements.  One plausible scenario is electroweak baryogenesis and with a precision of down to 10\% on $\lhhh$, see section~\ref{subsec:ILC_Higgs}, this can be tested right away the ILC. The Sakharov conditions require, among others, a first order phase transition f
% 
% \begin{figure}
%  \begin{center}
%     \includegraphics[width=0.5\textwidth,angle=-90]{Delhhh_vs_vcTc.pdf}
%     \caption{\label{fig:delhhh} Deviations of the Higgs self-coupling $\lhhh$ for different masses of a heavy scalar as a function of the mass parameter $M$ of the 2HDM Model. The figure is taken from~\cite{Senaha:2013fva}}.
% \end{center}
% \end{figure}

\subsubsection{Discovery via Precise Measurements of Standard Model Parameters}
\label{subsubsec:heavyNP_indirect}
Heavy resonances will impact a large variety of SM observables. As discussed already in section~\ref{subsubsec:noNP_indirect}, the
discovery reach of the ILC precision program exceeds the discovery reach by direct production at the LHC in many cases.
If a heavy resonance will be discovered at the LHC, it gives a clear proof for the existance of BSM physics. Realistic models describing the LHC observation will, however, contain more than this one new particle, but typically partners or exitations
of many SM particles. Therefore precision measurements at the ILC will not only provide essential information to characterize
the resonance found at the LHC, but they will also offer significant discovery potential for additional states.

% JL: moved above since it talks about Higgs couplings / compositeness
% \paragraph{Spin-1:} Referring to section~\ref{sec:comphiggs} Spin-1 resonances could be either resonances occurring in Higgs compositeness models or KK excitations of known particles. The LHC is in a prime position to discover heavy gluons due to the large cross-section for gluon induced processes. A new mass scale given by this or similar observations would be a key information for the different types of models that could hence then be tested at the ILC. The corrections introduced by the new vector resona


%A prominent effect is that of corrections from the tower of KK states to  the Higgs boson couplings to heavy species $W$, $Z$ and $t$. 
%For example, the models of~\cite{Malm:2014gha} predict shifts of the Higgs boson
%couplings to vector bosons $c_{V} = g(hVV)/g(hVV)|_{SM}$ of the form
%\beq
%c_W \approx c_Z \approx 1- 0.078\left( {5\ \mbox{TeV}\over  M_{g(1)} } \right)^2.  
%\eeq{Malmc}
%where $M_{g(1)}$ is the mass of the first Kaluza-Klein excitation of the gluon. For a 5\,TeV Kaluza-Klein excitation this leads to an $\sim$8\% deviation.  With the formula \leqn{Malmc}, even a 20~TeV KK gluon would produce observable effects, for both $W$ and $Z$, in the ILC precision measurements.

% JL: moved above
% \paragraph{The Top Yukawa Coupling}
% In both, Higgs composite models and RS Models, effective top quark Yukawa couplings are generated. Figure~\ref{fig:ctct5RSC} shows the shifts of the Higgs boson couplings computed in \cite{Malm:2014gha} for a variety of parameters sets, as a function of the mass of the lightest KK gluon.  Note that these shifts are in general complex,
% leading to $CP$ violation in the $ht\bar t$ coupling at levels observable at the ILC~\cite{Godbole:2011hw}.   
% The figure shows the shifts as parameters $c_t$, $c_{t5}$ corresponding to the effective coupling
% \beq
%     \delta \L =  - {m_t\over v} \ h \  \bigl[  c_t\ \bar t t + i
%     c_{t5}\  \bar t \gamma^5 t \bigr]  .
% \eeq{explainct}
% 
% \begin{figure}
%  \begin{center}
%     \includegraphics[width=0.99\textwidth]{ctct5CRS.pdf}
%     \caption{\label{fig:ctct5RSC}Predictions of $CP$-even (left) 
% and $CP$-odd (right) Higgs couplings to the $t$ quark in a Randall-Sundrum model with a custodial
% symmetry. The parameters $c_t$, $c_{t5}$ are defined in \leqn{explainct}.  The point clouds are a scan of the space of Yukawa couplings for three values of the free parameter $y_{\ast}$. The gray band on the right hand 
% side shows the experimental bound at 90\% on the $CP$-odd predictions 
% derived from the electron dipole moment (EDM)~\cite{Malm:2014gha}. 
% %
% %Figure and parts of the caption are taken from~\cite{Malm:2014gha}.  
% }
% \end{center}
% \end{figure}

\paragraph{Electroweak Couplings of the Top Quark:}
% JL: to Sec 2
% A further important consequence of the discussed models is a modification of the electroweak couplings of the
% third generation quarks. In RS models, the wave functions of these quarks and in particular that of the $t$ quark extends to the IR
% side of the 5th dimension, and so the $t$ quark shares some of the
% compositeness of the Higgs boson.  In particular, while the
% wavefunctions of light fermions peak close to the UV brane, the wavefunction of
% the $t_R$ often peaks close to the IR brane. This may constitute an
% elegant explanation of the striking mass hierarchy in the fermion
% sector.   A consequence of this effect is that the couplings of the
% $t$ quark to the $Z$ boson are expected to have large shifts due to
% mixing with KK states, of independent size for $t_L$ and $t_R$.
% Models of this effect are described, for example, in \cite{Djouadi:2006rk,Carena:2006bn}. 

% JL: where can this go?
%The $t\bar tZ$ couplings enter the expressions for the electroweak production of top quark pairs. On the pole the LHC may put a glimpse on the coupling structure of e.g. heavy $Z$ bosons to fermions using the $t$ quark as a polarimeter~\cite{Cerrito:2016qig}.  

% JL: references quoted already in 3.2.2
% At the ILC the electroweak couplings to $t_L$ and $t_R$ can be separately measured using polarised beams~\cite{Amjad:2015mma}.   In
% \cite{Fujii:2015jha}, these measurements are reviewed, and errors of order  1\%
% on the separate $Z$ couplings to $t_L$ and $t_R$ are estimated for
% the full ILC program.  As shown in Fig.~\ref{fig:models-rp}, these accuracies suffice
% not only to observe the effect but also to provide significant discrimination among models. 

As discussed in section~\ref{subsubsec:noNP_indirect}, the ILC will be able to improve the precisions on the couplings of the top quark to the $Z$ boson by about an order of magnitude, thus being sensitive to e.g.\ Kaluza-Klein resonances up to masses of several tens of TeV, as shown in Fig.~\ref{fig:topcoup-lumi}. 
In the presence of a resonance already discovered at the LHC, the capability of the ILC
to distinguish between the various BSM models as illustrated in Fig.~\ref{fig:models-rp}, is of special importance.

Note here, that the ILC will offer also a superb precision on the coupling to the $b$ quark suited to underpin observations that will be made for the $t$ quark. As the effects on the $b$ quark may be smaller than that for the $t$ quark, these measurements will benefit from the high integrated luminosity available for the H-20 scenario.

\paragraph{Two-Fermion Production and New Gauge Bosons: } 
%The new sector may contain more than one vector boson in the energy reach of the LHC.  
As discussed in section~\ref{subsubsec:noNP_indirect}, the precise measurement of polarized two-fermion production cross sections
at the ILC will allow to observe new gauge bosons, for instance $Z'$ bosons, up to very high scales, which in several models
exceed significantly the reach of the LHC. In the presence of one $Z'$-like boson already discovered at the LHC, the ILC
has complementary discovery potential for additional such states, either at higher masses, or even in the case of two
degenerate new $Z'$ bosons~\cite{Barducci:2015aoa} remaining unresolved as one state at the LHC. It is shown that beam polarisation will allow for disentangling the two $Z'$ at the ILC, underlining that its versatile design is ideally suited to decipher details of new physics scenarios. 
%The Higgs composite model presented in~\cite{Barducci:2015aoa} contains among others two degenerated new $Z'$ bosons. It is shown that beam polarisation will allow for disentangling the two $Z'$ underlining the versatile design of the ILC is ideally suited to decipher details of new physics scenarios. 
%Finally, in a more simplistic scenario as those discussed so far, the LHC may discover a new e.g. $Z'$ that can occur as a 'copy' of the Standard Model $Z$ at higher energies in generationally universal benchmark models, see e.g.~\cite{Cerrito:2016qig,Accomando:2010fz}. At the ILC a new $Z'$ would interfere with the usual photon and $Z$ exchange and lead to effects in the $\eplus \eminus \rightarrow f\bar{f}$ scattering that are well detectable at the ILC. In the contrast to the models described so far that act only on the couplings of the $t$ quark and on the $b$ quark, a pattern should repeat in the couplings to all fermions in case of more universal models. It should however remarked at this point that the introduced  gets more and more excluded by LHC with limits of up to 4\,TeV for such a $Z'$~\cite{ATLAS-CONF-2016-045}. 

%The discovery of a radion, as described above, will have interesting consequences in the vector-boson sector. The warped extra dimensions would lead to Kaluza-Klein excitations of the vector bosons. Even if these would have a mass as large as a few TeV, their effects would be observable at the ILC in particular with the H-20 scenario. The theory predicts deviations ... 750 GeV paper ...  

\subsubsection{Spin-2 Resonance} 
Clearly a major breakthrough in combining relativistic quantum field theory and gravity would be the discovery of a Spin-2 particle that may henceforth be interpreted as a KK excitation of the graviton. As before it can be expected that the KK graviton will be accompanied by KK excitations of the known Standard Model vector bosons so the observations made in the previous paragraphs may hold here again. The Ref.~\cite{Fujii:2016raq} points out that it is at least not excluded that the Spin-2 particle couples to $\ee$ pairs which could turn into an avenue for the ILC if the mass is at least within the 1 TeV upgrade scenario for the ILC. In~\cite{Fujii:2016raq} it is also reminded that a Spin-2 resonance would be a strong case for considering the option to run the ILC as a $\gamma \gamma$ collider. 


\subsubsection{Loophole-free Search for Light States }
\label{subsubsec:heavyNP_simplifiedSUSY}
If LHC discovers a relatively heavy particle without a clear indication for the existance of any lighter state,
this does not neccessarily mean that no lighter states exist, for the same reasons as given in section~\ref{subsubsec:noNP_loopholefree}, in particular for the 
there mentioned non-SUSY cases.
There might also be ambigous cases, where it is not clear whether the heavy new particle decays to SM particles
directly, or via some much lighter, exotic states whose decay products cannot be resolved in the detectors due to
the high boost, as it was also discussed as a possible explanation of the $750$\,GeV excess, see e.g.\ \cite{Ellwanger:2016qax}.
In any of these cases, the capability of the ILC to probe for all
existing particles in Nature with electromagnetic, hyper-charge or
electroweak quantum numbers still provides discovery potential complementary to that of the LHC.  


% {\color{red} [content: discovery / exclusion reach for any light electroweak state which might have been missed at LHC] }\\
% {\color{dgreen} [volunteer contributor: Mikael Berggren] }\\
% \input{no_loophole_chap4_heavyNPLHC}                                    |
%%-- (assuming the contents of this paragraph is integrated into chap2)
%%-- Light particles with only electroweak quantum numbers are predicted in
%%-- most models of new physics. In SUSY models with minimal field content,
%%-- there are the SUSY partners of the electroweak gauge boson, the Higgs
%%-- bosons, the leptons and neutrinos, while in non-minimal supersymmetry,
%%-- there could be e.g. additional Higgs superfield giving to additional
%%-- light scalars, pseudoscalars and neutral as well as charged
%%-- fermions. Naturalness in SUSY at least demands Higgsinos to be in the
%%-- same mass range as the Higgs boson. For extra-dimensional models,
%%-- there are correspondingly the reasonances of the electroweak SM
%%-- particles. In models with new strong
%%-- interactions like compositeness, Little Higgs etc. new chiral
%%-- symmetries that are spontaneously broken can lead to light states as
%%-- pseudo-Nambu-Goldstone bosons. As composite models treat
%%-- third-generation fermions in a special way, quite often couplings of
%%-- light states to the third-generation SM particles, particularly bottom
%%-- and top quarks are introduced. 
% As pointed out in section~\ref{sec:BSM}, many extensions to the SM predict 
% light particles with only electroweak quantum numbers.
% Their production rates at a hadron collider therefore is quite
% small compared to the strong interaction mediated SM processes,
% and   are only accessible via QCD
% production modes or in regions of the Drell-Yan spectrum that are not
% swamped by QCD backgrounds.
% This does not exclude that heavier states, out of ILC reach, might also exist,
% and could have strong interactions.
% Such states might well be discovered at LHC, while the lighter, un-coloured,
% states remain unseen.
% 
% Therefore, such lighter states can quite conceivably exist even
% if they are not detectable at the LHC.
% In sect \ref{subsubsec:directNP_NoLoophole}, it was shown that
% in the case such state are SUSY partners to the SM particles,
% the ILC is guaranteed to discover them, if they are in kinematic reach.
% Also in the case of other extensions to the SM,
% such particles can be searched for over the whole mass range:
% {\it e.g.} the tell-tale couplings to the third-generation SM particles
% to new light states in composite models can be searched for
% in the
% invariant mass spectrum of two $b$ jets in processes with two tagged
% $W$s and four $b$ jets (cf. e.g.~\cite{Kilian:2004pp}). 
% At LHC, on the other hand, such processes would lead to
% anomalies in the top quark processes in fiducial regions that are
% partially in background regions, so very hard to detect. 
% In the same
% way, decays into charm, taus or photons can be detected.
% Using its electroweak production modes, the ILC can scan for all
% existing particles in Nature with electromagnetic, hyper-charge or
% electroweak quantum numbers.  
%% Typcial examples for these are dark matter
%% searches such as monophotons and missing
%% energy. section~\ref{subsubsec:directNP_NoLoophole}.


%% JL: according to the definition of "heavy NP" by Michael, I think natural SUSY does not belong here
% \subsubsection{Natural SUSY}
% \label{subsubsec:heavyNP_NatSUSY}
% {\color{red} [content: if LHC discovers stop and /or gluino candidate search for Higgsinos, determine parameters, impact of Higgs precision] }\\
% {\color{dgreen} [volunteer contributor: Howie Baer] }\\
% 
% In natural SUSY defined by no large unnatural cancellations in Eq. \ref{eq:mzs}, the gluino mass
% can range up to 4 TeV for $\Delta_{EW}<30$\cite{Baer:2012cf,Baer:2015rja}. 
% This is to be compared to the current gluino mass limits from LHC13 as of Fall, 2016, 
% where in the framework of several simplified models, Atlas and CMS require $m_{\tilde{g}}\gtrsim 1.9$ TeV. 
% From this perspective, about half of natural SUSY parameter space has been explored by LHC13. 
% The naturalness upper bound on $m_{\tilde{g}}$ can also be compared to estimates of the ultimate reach
% of LHC with 300 or 3000 fb$^{-1}$. A $5\sigma$ discovery of gluinos via gluino pair production
% is possible for $m_{\tilde{g}}\sim 2$ (2.4) TeV, respectively. Thus, it is possible LHC may still find
% evidence for gluino pair production in the $m_{\tilde{g}}\sim 2-2.5$ TeV range. 
% Likewise, naturalness limits the lighter top-squark $m_{\tilde{t}_1}\lesssim 3$ TeV. 
% The current limits from LHC13 with $\sim 13$ fb$^{-1}$ 
% are $m_{\tilde{t}_1}>0.85$ TeV. The estimated $5\sigma$ discovery reach of 
% HL-LHC is to $m_{\tilde{t}_1}\sim 1.3$ TeV.
% 
% 
% In SUSY with radiatively-driven naturalness, the reaction $pp\rightarrow \tilde{g}\tilde{g}$
% followed by $\tilde{g}\rightarrow t\tilde{t}_1$ followed by $\tilde{t}_1\rightarrow b\tilde{\chi}_1^+$
% or $\tilde{t}_1\rightarrow t\tilde{\chi}_{1,2}^0$ will result in events with $\ge 4$ $b$-jets plus
% possible additional jets, isolated leptons and MET. 
% While these signatures are expected to be common for a wide variety of models with $\sim 2$ TeV gluinos, 
% a distinctive aspect\cite{Baer:2013xua} of natural SUSY is the presence of higgsino-like
% neutralinos $\tilde{\chi}_2^0$ in the cascade decay events. The $\tilde{\chi}_2^0$ should 
% decay via 3-body mode to $\tilde{\chi}_2^0\rightarrow \ell^+\ell^-\tilde{\chi}_1^0$. 
% The resultant OS/SF dilepton within 
% gluino cascade decay events would have a characteristic invariant mass bound of 
% $m(\ell^+\ell^- )<m_{\tilde{\chi}_2^0}-m_{\tilde{\chi}_1^0}\sim 10-20$ GeV. 
% A sufficiently large gluino pair event sample should allow extraction of the gluino mass (and hence $M_3$) 
% via counting to $\sim 10\%$ or even better if kinematic distributions can be used.
% 
% As described previously for natural SUSY, the ILC should be able to directly produce the higgsino pair reactions
% $e^+e^-\rightarrow \tilde{\chi}_1^0\tilde{\chi}_2^0$ and $\tilde{\chi}_1^+\tilde{\chi}_1^-$ which should allow
% for precision extraction of the various higgsino masses and to indirectly fit the electroweak gaugino 
% masses $M_1$ and $M_2$ due to their influence on the Higgsino mass splittings. In such a case, then all 
% three gaugino masses may be run to high energies to test unification at $m_{GUT}$ 
% (as shown in Fig. \ref{fig:gaugino_mass_unification}) 
% or at an intermediate mass scale, or their non-unification. 
% Input of precision electroweak measurements from ILC will allow for more detailed 
% disentangling of LHC gluino pair cascade decay events. 

\subsubsection{WIMP Dark Matter }
\label{subsubsec:heavyNP_WIMPs}
%{\color{red} [content: complementary discovery potential for invisible particles coupling to leptons, mass determination, %coupling measurement] }\\
%{\color{dgreen} [volunteer contributor: Howie Baer] }\\

In the case that a heavy new particle is discovered at the LHC, it is still important to check for lighter, invisible
particles coupling to leptons, as it can be done in a generic way via the mono-photon signature (c.f.\ section~\ref{subsec:directNP_WIMP}). Discovery and characterisation, or else a rather model-independent exclusion of such a particle as enabled by ILC data would provide important and complementary information for unveiling the nature of the
heavy particle found at the LHC.


%\subsubsection{anything else?}
%\label{subsubsec:heavyNP_else}

exsum.tex
% \section{Executive summary}
% \label{sec:exsumm}
In this Executive Summary, we give the main conclusions of this report on the potential of the ILC for the discovery of new phenomena.
Throughout this report, numerical estimates of precision or reach are based on the 20-year plan for ILC operation presented in~\cite{Barklow:2015tja}, including $4000$\,fb$^{-1}$ of luminosity at $500$\,GeV.

The ILC discovery program exploits the joint power of direct searches with the potential to produce new particles and  precision measurements  which are able to detect virtual effects of new particles at higher mass scales. The latter will shed light on the structure of physics arising from 
beyond the Standard Model even if the associated new particles are too heavy to be produced directly at LHC or ILC. 
All these measurements will rely on the clean operating environment, 
low backgrounds, and adjustable beam energy and polarization provided by the ILC. 

We will discuss the discovery of new interactions in the following programs:
\begin{enumerate}
%
\item {\bf New properties of the Higgs boson:}
The ILC will be a {\it Higgs boson factory} which offers 
absolute, model-independent measurements of the Higgs boson couplings 
to Standard Model (SM) fermions and gauge bosons, 
most of them to better than 1\% precision.
%, as well as determining their $CP$ properties. 
These measurements would probe for modifications of the Higgs interactions arising from composite structure of the Higgs or from mixings with new particles, including new, heavier Higgs bosons.
% Such a program of Higgs ``fingerprinting'' may point the way to which of 
% these routes to BSM physics is the right one.
In addition, the self-coupling of the Higgs boson can be measured to an accuracy of 27\% via reactions such as $e^+e^-\rightarrow ZHH$, improving to 10\% via $\nu\bar{\nu}HH$ at $1$\,TeV. 
This measurement is a critical test of the theory of electroweak baryogenesis, a leading contender for explaining the cosmic
matter-antimatter asymmetry.
% in the universe. 
%An observation of a deviation
%from the SM prediction in any property of the Higgs boson would be a clear discovery of new physics.


\item {\bf New properties of the Top quark:} 
The ILC will be a precision top quark factory. 
Scans of the production threshold of $e^+e^-\rightarrow t\bar{t}$  
can determine the top quark mass to a precision of 
$50$\,MeV or better, including all theoretical uncertainties. 
A precision determination of $m_t$ plays a central role in global 
electroweak fits which are indirectly 
sensitive to new particles and new interactions.
Using polarized beams, the ILC can determine separately the top-quark left- and right-handed couplings to gauge bosons to 
the sub-percent level. Such measurements offer a huge discovery potential for
a variety of composite Higgs or extra-dimensional new physics models, even if
the scale of the new physics is in the tens of TeV range.

\item {\bf New force carriers:} By measuring distributions in $e^+e^-\rightarrow f\bar{f}$
production (where $f$ stands for different SM fermions), the ILC is 
sensitive to new force particles $Z^\prime$ with masses as high as $12$\,TeV. Via fermion pair production, either directly or by virtual effects, the ILC can also explore for other new physics resonances that occur in composite Higgs or extra-dimensional models.
These measurements are not unlike the first indirect observation of the
$Z$ boson at {\sc Petra} and {\sc Tristan} via virtual effects in fermion pair production.

\item {\bf Additional Higgs bosons:} In addition to the possibility to discover relatives of the Higgs boson
via studying the properties of the $125$-GeV particle, the ILC offers unique opportunities to discover
additional lighter Higgs bosons -- or, more generally, any weakly interacting light scalar or pseudo-scalar particle -- by their direct production.

\item {\bf Supersymmetric sisters of the Higgs boson:}
If supersymmetry (SUSY) is the way nature has chosen to generate the symmetry-breaking potential of the Higgs boson, then light sister particles of the Higgs boson -- higgsinos are required. It is very possible, even theoretically preferred, that these particles have masses in the range of $\sim 100$-$300$\,GeV (the lighter the better), while all other supersymmetric particles are heavier.
Such light higgsinos are difficult, perhaps impossible, to observe at 
LHC, but their discovery would be straightforward at ILC. In that case, ILC  would  be a {\it higgsino factory}, providing quantitative tests of the hypothesis of supersymmetry and its implications for unification of forces.
% at ultra-high energies.
%Precision measurements of the higgsino properties could test gaugino mass unification at the GUT scale or at an intermediate scale. 
%We emphasize that natural SUSY is the most  plausible SUSY model since it naturally generates a weak scale in the  100 GeV range. 
%By contrast, most LHC SUSY searches have 
%assumed {\it unnatural} SUSY models which are rather implausible 
%compared to natural SUSY. Thus, t
% JL Feb 15: to keep or not to keep?
% The search for light higgsinos is much more than exploring an LHC loophole: 
% it is a search for the most natural manifestation of supersymmetry since it naturally generates a weak scale in the  100 GeV range.
%Circular $e^+e^-$ colliders may not have enough
%center-of-mass energy to pair produce the higgsinos.

\item {\bf SUSY without loop-holes:}
A central prediction of supersymmetry is that sparticles couple with the same strength 
as their SM partners. 
Thus, the rates for production of SUSY particles are well predicted as a function of mass. 
Then the clean and well-defined conditions at the ILC guarantee either discovery or exclusion. 
This is not true at the LHC, where many scenarios with light SUSY particles can evade detection.

\item {\bf Discovering dark matter particles:} It is possible that particles of dark matter are being produced 
copiously at accelerators but are invisible to their detectors. 
To search for pair-production of invisible particles, one must hunt for an associated photon or 
gluon from initial-state radiation.  
Such searches at the ILC are complementary to those at the LHC, since they probe dark matter couplings 
to leptons rather than quarks.  Because of the simplicity and calculability of background 
reactions at the ILC, the ILC mass reach for discovery of dark matter particles is 
similar to that of LHC despite the difference in center of mass energy. 

\item {\bf Identifying the nature of dark matter:} The precision capabilities of the ILC are optimal to uncover which mechanisms are responsible for generating dark matter in the early universe: 
Is it thermal or non-thermal? Does entropy-dilution play a role? 
Are there super-WIMPs? Is there a WIMP-axion admixture? 
Dark matter production in the early universe may be 
much more intricate than the simple thermal WIMP miracle scenario 
and ILC can play a key role in gaining a more complete understanding.

\item {\bf Neutrinos:} 
Though some models for neutrino mass invoke particles with masses well beyond the energy of any 
realistic accelerator, other models generate neutrino masses through new physics effects at 
energies that ILC and LHC can access.  
We give examples of models in which the ILC will discover new particles and measure properties 
that are simply related to the neutrino mixing angles.

\end{enumerate}

We conclude that the physics case for the ILC is very strong, independently of future
findings at the LHC. We illustrate this with a detailed discussion of consequences 
for ILC physics 1. if LHC discovers no new particles, 
2. if LHC discovers some new low-mass states and 
3. if LHC discovers new heavy particles. 
Under each scenario, ILC will play a critical role in the 
discovery of new phenomena. ILC will push forward our knowledge of 
high energy physics and our understanding of the universe.
%

prelude.tex
% \section{Executive summary}
% \label{sec:exsumm}
\newpage
\vspace{5cm}
\begin{center}
{\large\bf Prelude}
\end{center}
\bigskip
\begin{verse}
I saw no Way-- the heavens were stitched--\\
I felt the Columns close--\\
The Earth reversed her Hemispheres--\\
I touched the Universe--

And back it slid-- and I alone--\\
A speck upon a Ball--\\
Went out upon Circumference--\\
Beyond the Dip of Bell--

\footnotesize{Emily Dickinson, 1862}
\end{verse}

\bigskip
\bigskip

\begin{quotation}
Herein the poetess, seeing no way forward, achieves an unexpected breakthrough 
which transports her to realms beyond the limits of space and time$\cdots$
\end{quotation}
\clearpage
supportICFAtoMEXT.tex
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\def\HU{Institut f\"ur Physik, Humboldt-Universit\"at zu Berlin, 12489 Berlin, Germany}
\def\ICREA{ICREA, 08010 Barcelona, Spain and IFAE, BIST, 08193 Bellaterra, Barcelona, Spain (on leave)}
\def\kek{High Energy Accelerator Research Organization (KEK), Tsukuba,
  Ibaraki, JAPAN  }
\def\ifae{ICREA at IFAE, Univesitat Aut\'onoma de Barcelona, E-08193
  Bellaterra, SPAIN   }
\def\Tsinghua{Center for High Energy Physics, Tsinghua University, Beijing, CHINA}
\def\Toyama{Department of Physics, University of Toyama, Toyama 930-8555, JAPAN}
\def\Seoul{Department of Physics and Astronomy, Seoul National
  University, Seoul 151-747,  \\ \hskip 0.4in   KOREA}
\def\DESY{DESY, Notkestrasse 85, 22607 Hamburg, GERMANY}
\def\Cornell{Laboratory for Elementary Particle Physics, Cornell
  University, Ithaca, NY 14853, \\ \hskip 0.4in USA  }
\def\Oklahoma{Department of Physics and Astronomy, University of Oklahoma, 
Norman, OK, 73019, USA}
\def\Oregon{Center for High Energy Physics, 1274 University of Oregon, 
Eugene, Oregon 97403-1274, USA}
\def\Orsay{LAL, Centre Scientifique d'Orsay, Universit\'e Paris-Sud, 
  F-91898 Orsay CEDEX, \\ \hskip 0.4in FRANCE }
\def\OrsayTH{LPT, Universit\'e Paris-Sud, 91405 Orsay, FRANCE}
\def\MPP{Max-Planck-Institut f\"ur Physik, F\"ohringer Ring 6, 80805 Munich, GERMANY}
\def\Tokyo{ICEPP, University of Tokyo, Hongo, Bunkyo-ku, Tokyo,
  113-0033, JAPAN}
\def\UTA{Department of Physics, University of Texas, Arlington, TX 76019, USA}
\def\Madrid{Campus of International Excellence UAM+CSIC, Cantoblanco, 28049, Madrid, Spain\\
Instituto de F\'isica Te\'orica (UAM/CSIC), Universidad Aut\'onoma de Madrid, Cantoblanco, 28049, Madrid, Spain\\
Instituto de F\'isica de Cantabria (CSIC-UC), 39005, Santander, Spain}
\def\Michigan{Michigan Center for Theoretical Physics, University of
  Michigan, Ann Arbor, \\ \hskip 0.4in MI 48109, USA}
\def\Berkeley{Department of Physics, University of California,
  Berkeley, CA 94720, USA}
\def\LBL{Theoretical Physics Group, Lawrence Berkeley National
  Laboratory, Berkeley, \\  \hskip 0.4in   CA 94720, USA}
\def\IPMU{Kavli Institute for the Physics and Mathematics of the
  Universe, \\  \hskip 0.4in   University of Tokyo, Kashiwa 277-8583, JAPAN}
\def\Tohoku{Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, JAPAN}
\def\TokyoTh{Department of Physics, University of Tokyo, Tokyo
  113-0033, JAPAN}
\def\Valencia{ IFIC (UVEG/CSIC), Edificios de Investigacion,
c./ Catedratico Jose Beltran 2, E-46980 Paterna, Valencia, SPAIN}
\def\KIAS{Quantum Universe Center, KIAS, Seoul 02455, KOREA}
\def\SLAC{SLAC,
    Stanford University, Menlo Park, CA 94025, USA}
\def\Kansas{Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA}


\begin{document}
%% =================================================================
%% =====  Title Page  ==============================================
%% =================================================================
\begin{titlepage}
  \begin{flushleft}
    {\tt DESY 17-012} \\
    {\tt KEK Preprint 2016-60} \\
    {\tt SLAC-PUB-16916} \\
    {\tt LAL 17-017} \\
    {\tt MPP-2017-5} \\
    {\tt IFT-UAM/CSIC-17-008} \\
%    {\tt LC-PHSM-2017-0XX}                \\
%    {\tt January 2017}                  \\
  \end{flushleft}

\vfill
  \begin{center}
    \begin{Large}
      {\bfseries \boldmath The Potential of the ILC\\
 for Discovering New Particles} \\
 \vspace{0.5cm}
           {\normalsize Document Supporting the ICFA Response Letter to the ILC Advisory Panel} 
    \end{Large}
  \end{center}


\vfill
\Author{LCC Physics Working Group}
\bigskip
\Author{Keisuke Fujii$^1$, Christophe
Grojean$^{2,3,4}$, Michael E. Peskin$^5$(conveners); 
Tim Barklow$^5$, Yuanning Gao$^6$,
Shinya Kanemura$^7$, Hyungdo Kim$^8$, Jenny List$^2$,
Mihoko Nojiri$^{1,9}$, Maxim Perelstein$^{10}$, Roman P\"oschl$^{11}$,  
J\"urgen Reuter$^2$, Frank Simon$^{12}$, 
Tomohiko Tanabe$^{13}$,
James D. Wells$^{14}$, Jaehoon Yu$^{15}$; Howard Baer$^{16}$, Mikael Berggren$^2$, Sven Heinemeyer$^{17}$, Suvi-Leena Lehtinen$^2$,  
Junping Tian$^{13}$, Graham Wilson$^{18}$, Jacqueline Yan$^{1}$; 
Hitoshi Murayama$^{9,19,20}$, 
James Brau$^{21}$}


\vfill
  \begin{abstract}
\input{00_abstract.tex}
  \end{abstract}
  
 \vfill
 

\newpage

\begin{raggedright}
\noindent $^1$ \kek \\
$^2$  \DESY \\
$^3$  \HU \\
$^4$ \ICREA \\
$^5$ \SLAC\\
$^6$ \Tsinghua \\
$^7$  \Toyama\\
$^8$ \Seoul \\
$^9$ \IPMU \\ 
$^{10}$ \Cornell\\
$^{11}$  \Orsay\\
$^{12}$ \MPP \\
$^{13}$  \Tokyo\\
$^{14}$  \Michigan \\
$^{15}$ \UTA\\
$^{16}$  \Oklahoma \\
$^{17}$  \Madrid \\
$^{18}$  \Kansas \\
$^{19}$ \Berkeley \\
$^{20}$ \LBL  \\
$^{21}$  \Oregon\\
\end{raggedright} 
\end{titlepage}

\section*{Executive Summary}
\label{sec:exsumm}
\input{exsum.tex}

\clearpage
\input{prelude.tex}
%\begin{fmffile}{fgraphs}
% =============================================================================
\section{Introduction}
\label{sec:intro}
%
%{\color{red} [content: overall intro, aim of document, relation to ICFA letter,...] }\\
%{\color{blue} [ author: Michael Peskin] }\\
%
\input{01_intro.tex}
% =============================================================================

% =============================================================================
\section{Overview on BSM Scenarios}
\label{sec:BSM}
% 
%{\color{blue} [main authors: Christophe Grojean, Howie Baer] }\\
%
\input{02_BSMscenarios.tex}
% =============================================================================

% =============================================================================
\section{ILC Capabilities for Precision Measurements}
\label{sec:ILC}
% 
%{\color{blue} [main authors: KF/JL] }\\
\input{30_ILCcap.tex}
%
% =============================================================================
\subsection{Higgs Measurements at the ILC}
\label{subsec:ILC_Higgs}
% 
%{\color{red} [content: summary of ILC Higgs precisions and their implications for BSM] }\\
%{\color{blue} [main authors: Junping Tian,  Christophe Grojean] }\\
%
\input{31_ILC_Higgs.tex}
% 
% =============================================================================
\subsection{Top Quark Measurements at the ILC}
\label{subsec:ILC_top}
% 
%{\color{red} [content: summary of ILC top measurements and their implications for BSM] }\\
%{\color{blue} [main authors: Roman P\"oschl, Frank Simon] }\\
%
\input{32_ILC_top.tex}
% 
% % =============================================================================
\subsection{Electroweak Precision Measurements at the ILC}
\label{subsec:ILC_ew}
% 
%{\color{red} [content: summary of ILC EWP measurements and their implications for BSM] }\\
%{\color{blue} [main authors: Graham Wilson, J\"urgen Reuter ] }\\
%
\input{33_ILC_ew.tex}
% 
% =============================================================================
\section{Direct Production of New Particles at the ILC}
\label{sec:ILC_NP}
% 
% {\color{red} [content: summary of ILC direct discovery potential] }\\
% {\color{blue} [main authors: Tomohiko Tanabe, Mikael Berggren] }\\
%
\input{34_ILC_directNP.tex}
% 
% =============================================================================


\section{LHC Discovery Scenarios}
\label{sec:LHC_scen}
%
%{\color{blue} [main authors: KF/JL ] }\\
\input{40_LHCscen.tex}
%
% =============================================================================
\subsection{Scenario 1: LHC Does Not Discover New Particles}
\label{subsec:noNP}
% 
%{\color{red} [content: complementary discovery potential and probing high scales through precision observables] }\\
%{\color{blue} [main authors: Sven Heinemeyer, Howie Baer] }\\
%
\input{41_noNPLHC.tex}
% 
% =============================================================================
\subsection{Scenario 2: LHC Discovers Relatively Light New Particles}
\label{subsec:lightNP}
% 
% {\color{red} [content: precision spectroscopy, additional discovery potential and complementary information from precision observables, in particular Higgs and electroweak] }\\
% {\color{blue} [main authors: Mikael Berggren, Howie Baer] }\\
%
\input{42_lightNPLHC.tex}
% 
% =============================================================================
\subsection{Scenario 3: LHC Discovers Relatively Heavy New Particles}
\label{subsec:heavyNP}
% 
% {\color{red} [content: model discrimination via precision observables, in particular top and electroweak, complementary discovery potential] }\\
% {\color{blue} [main authors: Roman P\"oschl, NN "theory"] }\\
%
\input{43_heavyNPLHC.tex}
% 
% =============================================================================

% =============================================================================
%\section{Conclusion on Complementarity and Synergy of LHC and ILC}
\section{Complementarity and Synergy of LHC and ILC}
\label{sec:ILCandLHC}
% 
%{\color{red} [content: compare the parameter space regions to be covered by the two machines in various BSM scenarios. Recap the expected discovery scenario in each of the four regions (both ILC and LHC find NPs, only LHC finds NPs, only ILC finds NPs, both find no NP). Demonstrate complementarity and synergy of the two machine in (four?) summary tables for the four regions. Emphasize that LHC is a great machine but in order to fully elucidate BSM at the EW scale and beyond, we need ILC.] }\\
%{\color{blue} [main authors: JL/KF] }\\
%
\input{05_ILCandLHC.tex}
% 
% =============================================================================


% =============================================================================
\section{Conclusions} 
%
\label{sec:conclusions}
\input{06_conclusions.tex}

% =============================================================================



%\end{fmffile}

% \begin{acknowledgement}
\section*{Acknowledgements}
The authors thank the ILD and SiD detector concepts groups for providing material for this document.
We thank M.~Savoy and H.~Serce for help with some figures.
M.~Berggren, C.~Grojean, S.~Lehtinen, J.~List and J.~Reuter thankfully acknowledge the support of the SFB676 
of the Germany Science Foundation. F.~Simon acknowledges the support of the DFG cluster of excellence `Origin and Structure of the Universe'. C.~Grojean is supported by the European Commission through the Marie Curie Career Integration Grant 631962 and by the Helmholtz Association.
The work of S.~Heinemeyer is supported in part by CICYT (Grant FPA 2013-40715-P) and by the Spanish MICINN's Consolider-Ingenio 2010 Program under Grant MultiDark CSD2009-00064. The work of  T.~Barklow and M.~Peskin is supported by the U.S. Department of Energy, contract
DE-AC02-76SF00515.
Some material is based upon work supported by the U.S. Department of Energy, Office of Science, 
Office of High Energy Physics under Award Number DE-SC-0009956.
This work is also supported by the Grants-in-Aid for Science Research No. 16H02173 and 16H02176 of the Japan Society for Promotion of Science (JSPS).

% \end{acknowledgement}

\appendix
\begin{footnotesize}
\input{07_bib.tex}
\end{footnotesize}

\end{document}


1702.05333.extracted_bib
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1702.05333.extracted_bib
{Wells:2004di J.~D.~Wells, %``PeV-scale supersymmetry,'' Phys.\ Rev.\ D {71 (2005) 015013 doi:10.1103/PhysRevD.71.015013 [hep-ph/0411041]. %%CITATION = doi:10.1103/PhysRevD.71.015013;%% %222 citations counted in INSPIRE as of 20 Sep 2016
1702.05333.extracted_bib
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{Baer:2013ula H.~Baer and J.~List, %``Post LHC8 SUSY benchmark points for ILC physics,'' \href{http://dx.doi.org/10.1103/PhysRevD.88.055004{ Phys. Rev. {series D88 (2013) 055004, [\href{http://arxiv.org/abs/1307.0782{{\ttfamily arXiv:1307.0782 [hep-ph]]. %%CITATION = ARXIV:1307.0782;%%.
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{Baer:2013yha H.~Baer, V.~Barger, P.~Huang, D.~Mickelson, A.~Mustafayev, W.~Sreethawong and X.~Tata, %``Same sign diboson signature from supersymmetry models with light higgsinos at the LHC,'' Phys.\ Rev.\ Lett.\ {110 (2013) no.15, 151801 doi:10.1103/PhysRevLett.110.151801 [arXiv:1302.5816 [hep-ph]]. %%CITATION = doi:10.1103/PhysRevLett.110.151801;%% %42 citations counted in INSPIRE as of 22 Nov 2016
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{Baer:2014kya H.~Baer, A.~Mustafayev and X.~Tata, %``Monojet plus soft dilepton signal from light higgsino pair production at LHC14,'' Phys.\ Rev.\ D {90 (2014) no.11, 115007 doi:10.1103/PhysRevD.90.115007 [arXiv:1409.7058 [hep-ph]]. %%CITATION = doi:10.1103/PhysRevD.90.115007;%% %48 citations counted in INSPIRE as of 22 Jan 2017
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{Baer:2016wkz H.~Baer, V.~Barger, J.~S.~Gainer, P.~Huang, M.~Savoy, D.~Sengupta and X.~Tata, %``Gluino reach and mass extraction at the LHC in radiatively-driven natural SUSY,'' arXiv:1612.00795 [hep-ph]. %%CITATION = ARXIV:1612.00795;%% %1 citations counted in INSPIRE as of 22 Jan 2017
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{Ahmed:2015uqt A.~Ahmed, B.~M.~Dillon, B.~Grzadkowski, J.~F.~Gunion and Y.~Jiang, %``Higgs-radion interpretation of 750 GeV di-photon excess at the LHC,'' arXiv:1512.05771 [hep-ph]. %\cite{Senaha:2013fva
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{Senaha:2013fva E.~Senaha, %``Overview of Electroweak Baryogenesis,'' arXiv:1305.1563 [hep-ph]. %%CITATION = ARXIV:1305.1563;%% %3 citations counted in INSPIRE as of 21 Nov 2016 %\cite{Ellwanger:2016qax
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{Malm:2014gha R.~Malm, M.~Neubert, and C.~Schmell, ``{Higgs Couplings and Phenomenology in a Warped Extra Dimension'' \href{http://dx.doi.org/10.1007/JHEP02(2015)008{{\em JHEP {02 (2015) 008, \href{http://arxiv.org/abs/1408.4456{{\tt arXiv:1408.4456 [hep-ph]. %%CITATION = ARXIV:1408.4456;%%.
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{Ellwanger:2016qax U.~Ellwanger and C.~Hugonie, %``A 750 GeV Diphoton Signal from a Very Light Pseudoscalar in the NMSSM,'' JHEP {1605 (2016) 114 doi:10.1007/JHEP05(2016)114 [arXiv:1602.03344 [hep-ph]]. %%CITATION = doi:10.1007/JHEP05(2016)114;%% %44 citations counted in INSPIRE as of 25 Nov 2016