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%	Author:			Daniel S. Mickelson
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%	Description:	Dissertation for OU Homer L. Dodge Department of
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%                              Radiative Natural Supersymmetry
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	\titlebg UNIVERSITY OF OKLAHOMA \\
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	GRADUATE COLLEGE \\
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	RADIATIVELY-DRIVEN NATURAL SUPERSYMMETRY \\
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	\titlebg A DISSERTATION \\
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	SUBMITTED TO THE GRADUATE FACULTY \\
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	in partial fulfillment of the requirements for the  \\
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	\normalsize Degree of \\
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	DOCTOR OF PHILOSOPHY \\
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	By \\
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	\singlespacing DANIEL S. MICKELSON \\
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	Norman, Oklahoma \\
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	2014
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	RADIATIVELY-DRIVEN NATURAL SUPERSYMMETRY \\
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	A DISSERTATION APPROVED FOR THE\\
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	HOMER L. DODGE DEPARTMENT OF PHYSICS AND ASTRONOMY\\
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	BY \\
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	\hfill Dr. Howard Baer, Chair \\
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	\hfill Dr. Braden Abbott \\
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	\hfill Dr. Bruce Mason \\
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	\hfill Dr. Kimball Milton \\
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	\hfill Dr. Nikola Petrov
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	\copyright Copyright by DANIEL S. MICKELSON 2014 \\
	All Rights Reserved.
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\textit{To Kristina, without whom this would not have been possible.}
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\section*{Abstract}
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Within the framework of supersymmetric theories, a question arises:
how can the $W, Z$, and $h$ masses be so low ($\sim 100$~GeV) when the
superpartner masses are so high ($m_{SUSY}> 1-2$~TeV)? This is the little hierarchy problem, and can be quantified by
studying the \textit{fine-tuning} of a particular model. Quantifying 
a model in such a way provides a unique opportunity to give upper bounds on the supersymmetric particle
masses. Introduced in this dissertation is the model called \textit{Radiatively-driven Natural
  Supersymmetry}, wherein low fine-tuning is achieved while
maintaining a light Higgs scalar $\simeq 125$ GeV. In addition, RNS offers a
particle spectrum that evades searches at all current collider
experiments, and satisfies cosmological constraints. It is shown that
RNS could be discovered with high luminosity at LHC14 in multiple
channels, having a soft trilepton + MET signature, a unique same-sign 
diboson signature accompanied by jets, and gluino cascade decays in
the trilepton+jets channel. An International Linear Collider operating 
at $\sqrt{s}=600$ GeV would either discover RNS or rule it out as a
feasible model. Dark matter direct and indirect detection experiments
also offer a means of discovery, with a 1-ton noble gas detector
effectively probing the entirety of RNS parameter space.
%
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\section{Introduction}
\pagenumbering{arabic}
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%%%%
%The recent successful runs of LHC7 and LHC8 at CERN have
%provided no conclusive evidence of physics beyond the Standard
%Model (SM). The lack of new physics puts stress on theories with
%{\it natural supersymmetry} (NS) appearing at the weak scale.  With the upcoming
%operation of the LHC at its design energy of 14 TeV along with talks
%of a new linear collider in the works, looking ahead at how new physics may be discovered is of utmost importance. 
The Standard Model (SM) has demonstrated huge success in the current understanding of particle physics and, as predicted, the LHC has discovered the long
sought after Higgs boson. Despite the success
of the Higgs discovery with $m_h\simeq 125$~GeV, the Higgs mass in the SM suffers from quadratic divergences. These quadratic divergences are removed when one considers
supersymmetry (SUSY). However, the introduction of SUSY also introduces
supersymmetric partner particles. No sign of
supersymmetry has yet emerged at the LHC, leading to mass
limits $m_{\tg}\agt 1.8$ TeV (for $m_{\tg}\simeq m_{\tq}$) and
$m_{\tg}\agt 1.3$ TeV (for $m_{\tg}\ll
m_{\tq}$)\cite{atlas_susy,cms_susy}. These limits are obtained in the
context of popular models such as mSUGRA/CMSSM\cite{msugra,sugraphen}, but are
qualitatively also valid in other frameworks as long as one
understands that
the squark mass limit refers to first generation squarks.  These
squark and gluino mass limits have caused 
concern for some physicists, arguing that in
order to maintain naturalness in SUSY models, sparticles ought to be
well below the TeV scale\cite{ellis,bg,ac1,dg,ac2,CCN,ellis2,king,ross,nonuni}. This
leads to the little hierarchy problem (LHP): How do SUSY parameters at
or above the TeV scale lead to $W$, $Z$, and $h$ masses of just $\sim 100$ GeV? Models of {\it natural supersymmetry} (NS) \cite{kn} address the LHP by
positing a spectrum of light higgsinos $\alt 200$~GeV and light top- and
bottom-squarks with $m_{\tst_{1,2},\tb_1}\alt 600$~GeV along with very
heavy first/second generation squarks and~TeV-scale
gluinos\cite{ah,others,delhs,bbht}. The absence of any hint of
deviations from the SM in the LHC8 data have led some to question
whether SUSY could be the solution to the naturalness problem of the
SM. 

This dissertation describes a model within the framework of
the MSSM that satisfies all of the constraints experimental data
requires, while allowing for desirable theoretical
considerations. The model, Radiatively-driven Natural Supersymmetry
(RNS), has low electroweak fine-tuning, a cold dark matter (CDM)
candidate, and satisfies the experimental constraints set by the LHC and
LEP2.

This chapter introduces the Standard Model, outlines the notation, and
sets the framework for the remaining chapters. This will also be
useful information concerning the later discussions of physics at
hadron and linear colliders. The deficiancies of the SM are
pointed out, along with what one may expect from a theory beyond the SM
(BSM). Supersymmetry (SUSY) and SUSY theories in general are then introduced, before moving
onto the simplest example, the Minimal Supersymmetric Standard Model
(MSSM). In Chapter~\ref{sec:fine}, `Naturalness' is defined along with
various fine-tuning measures and how these affect the search for
acceptable SUSY theories. Chapter~\ref{sec:rns} introduces the
framework of RNS and points out the advantages
of RNS over other theories. Chapters~\ref{sec:lhc} and ~\ref{sec:ilc} investigate prospects for
discovery at the LHC and a future linear collider, respectively.
In Chapter~\ref{sec:dark}, the lightest neutralino as a
WIMP along with prospects for both direct and indirect detection of
dark matter is discussed.
%%
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%\begin{center}
\subsection{The Standard Model}
%--------------------------------------------------------------------
%%
%--------------------------------------------------------------------
\subsubsection{The Electroweak Theory and Quantum Chromodynamics}
%--------------------------------------------------------------------
The Standard Model is the most successful theory of elementary
particle physics constructed and was finalized in the
1970's. It describes the electromagnetic, weak and strong
interactions successfully across a large range of energies. The SM is a
quantum field theory built up of two parts: the $SU(3)_C$ color theory
of strong interactions, Quantum Chromodynamics (QCD), and the $SU(2)_L \times U(1)_Y$
electroweak theory (EW). The complete theory, based on $SU(3)_C
\times SU(2)_L \times U(1)_Y$, combines them.

The electroweak (EW) theory describes the electromagnetic and weak
interactions in the SM. The EW symmetry is spontaneously broken from
$SU(2)_L\times U(1)_Y \rightarrow U(1)_{QED}$ via the Higgs mechanism, giving rise to
massive $W$ and $Z$ bosons, quarks, leptons, and a physical
Higgs scalar. The EW Lagrangian is given by
\be
\l_{EW}=\l_{gauge}+\l_{matter}+\l_{Higgs}+\l_{Yukawa}
\label{eq:ewl}
\ee
where
\be
\l_{gauge}=-\frac{1}{4}W_{A\mu\nu}W_A^{\mu\nu} - B_{\mu\nu}B^{\mu\nu},
\ee
Here, $W_A$ and $B$ represent the $SU(2)$ and $U(1)$ gauge
fields, respectively. 
\be
\l_{matter}=i\pbar \slashed{D} \psi
\ee
$D$ is the electroweak gauge covariant derivative, 
\be
D_{\mu}=\partial_\mu + igW_{\mu}\cdot T + \frac{1}{2}i g'B_{\mu}Y
\ee
where $g$ and $g'$ are the $SU(2)$ and $U(1)$ coupling
constants, $W$ is a gauge isotriplet for $SU(2)$, $B$ is a gauge
singlet for $U(1)$, $T$ is the weak isospin operator,
and $Y$ is the weak hypercharge operator.
\be
\l_{Higgs}=(D\phi)^\dagger(D\phi)+\mu^2\phi^\dagger\phi-\lambda(\phi^\dagger\phi)^2,
\ee
where $\mu$ is the Standard Model Higgs mass parameter (not to be confused
with the supersymmetric bilinear Higgs/higgsino mass term which will have the same notation
introduced in Sec.~\ref{sssec:mssm}). After the symmetry is broken,
the Yukawa coupling of the Higgs field to the fermions is given by
\be
\l_{Yukawa}=-\displaystyle\sum\limits_i \frac{\lambda_{f_i}}{\sqrt{2}}\bar{f}_if_ih
\ee
where the sum is over all of the quarks and leptons in the theory.

The QCD Lagrangian is given by
\be
\l_{QCD}=-\frac{1}{4}G_{A\mu\nu}G_A^{\mu\nu}+ \bar{q}_i(i\slashed{D}-m_i)q_i
\label{eq:qcdl}
\ee
where $i$ is the quark index and is summed over, the $G$'s are the $SU(3)$ gauge fields of QCD:
\be
G_{A\mu\nu}=\partial_\mu G_{A\nu}-\partial_\nu G_{A\mu}-g_S f_{ABC}G_{B\mu}G_{C\nu},
\ee
and $D$ is the gauge covariant derivative:
\be
D_\mu=\partial_\mu+ig_S\frac{\lambda_A}{2}G_{A\mu}.
\ee
Here, $g_S$ is the strong coupling constant, and $\lambda_A$ are the
eight Gell-Mann matrices, a representation of the generators of the theory.
The color indices on the quarks take on the values $i=1,2,3$, usually
called red, blue, and green. The corresponding anti-colors are
anti-red, anti-blue, and anti-green. For the gluons, $A,B,C=1,...,8$,
where the eight colors are combinations of red, blue, green, and their
anti-colors. All QCD interactions must conserve color charge, in the
same way electric charge must be conserved. 
%%
%--------------------------------------------------------------------
\subsubsection{The SM Higgs Mechanism}
\label{sssec:higgs}
%--------------------------------------------------------------------
%%
The Higgs mechanism\cite{higgs,englert} is the process
which breaks the $SU(2)_L\times U(1)_Y$ electroweak symmetry and gives
mass to particles in the standard model. This process also introduces
a $CP$-even scalar particle into the theory, called the Higgs scalar. The Higgs
sector of the SM consists of an $SU(2)_L$ doublet of spin zero fields, 
\be
\Phi=\left(\begin{array}{c}\phi^+\\ \phi^0\end{array}\right). 
\ee
The corresponding Higgs Lagrangian is given by
\be
\l_{Higgs}=(D\phi)^\dagger(D\phi)-V(\lvert\phi\rvert^2)
\ee
where the Higgs potential is
\be
V(\lvert\phi\rvert^2)=-\mu^2\phi^\dagger\phi+\lambda(\phi^\dagger\phi)^2
\ee
and $\phi^0=(\phi^0_R+i\phi^0_I)/\sqrt{2}$. Minimization of this
potential leads to a non-zero vacuum expectation value ({\it vev})
$v=\left<\phi_R\right>/\sqrt{2}=\sqrt{-\mu^2/2\lambda}$. The presence
of a {\it vev} signifies the spontaneous breakdown of EW symmetry and gives mass to the vector
bosons and fermions.

Within the SM, the Higgs mass is a free parameter that must be
measured. In contrast, the MSSM puts an upper
limit on the SM-like Higgs mass as $m_h \alt 130$ GeV. 
The recent highly successful runs at the Large Hadron Collider at CERN
in Switzerland led to the discovery of a Standard Model (SM) Higgs-like
resonance by the ATLAS and CMS collaborations
\cite{atlas_h,cms_h}. More analysis needs to be done to confirm that
it is in fact the SM Higgs; however it appears that $m_h \simeq 125.5$
GeV.
%%
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\subsubsection{Motivations for Physics Beyond the Standard Model}
%--------------------------------------------------------------------
%%
Despite the success of the SM there are several questions - both
experimental and theoretical - that it does not or cannot address; thus
it appears incomplete. On the experimental side, these include:
\bi
\item the lack of an explanation for gravity.
\item a wealth
of cosmological data suggests that the universe is dominated by
dark matter and dark energy.
\item neutrino oscillation data imply that neutrinos have some (small)
mass. This is contradictory to the SM, where neutrinos are treated as
massless (there are simple extensions to the SM which do include neutrino mass). 
\ei
On the theoretical side:
\bi
\item the big gauge hierarchy problem: Why is there
such a discrepancy between the weak scale, and the scale which the
forces become unified? 
\item the strong CP fine-tuning problem,
which seemingly requires the Peccei-Quinn mechanism and introduction
of new matter, axions, to be solved. 
\item gauge coupling unification is expected at a grand unified theory (GUT) energy, however
this does not occur in the SM.
\ei

To address these issues, one must move beyond the SM. One might consider
supersymmetry, introducing new matter, or a combination of both. As
more experiments search for answers, one may even discover unexpected
new physics.

%%%%%%%%%%%%%%%%%
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\subsection{Supersymmetry}
\label{ssec:susy}
%--------------------------------------------------------------------
%%
Supersymmetry is a
symmetry between bosons and fermions, i.e. for each boson (fermion)
there exists a partner fermion (boson) that requires an extension of
spacetime to include anti-commuting dimensions. The representations of
supersymmetry contain both the fermion and boson superpartners and are
called supermultiplets. Each supermultiplet has the same number of
bosonic and fermionic degrees of freedom, so that $n_F=n_B$. A simple
supermultiplet is a {\it chiral supermultiplet} containing a Majorana
spinor field (spin 1/2) and a complex scalar field (spin zero). The
standard model fermions are contained in chiral supermultiplets along
with their superpartners, the spin-0 sfermions. Another supermultiplet is the
{\it gauge} or {\it curl supermultiplet}. The SM gauge bosons and
their supersymmetric partner spin-1/2 gauginos are contained within gauge
supermultiplets. In the following, all supersymmetric formalism will follow the notation
in \cite{wss}, where four component spinor notation is used. 
%---------------------------------------------------------------------
\subsubsection{Supermultiplets and the SUSY Lagrangian}
%---------------------------------------------------------------------
A left chiral supermultiplet consists of a Majorana spinor field $\psi_L$, a
complex scalar field $\s$, and an {\it auxiliary} field
$\f$. The simplest model using this supermultiplet with massless and
non-interacting particles is the Wess-Zumino model, with the
Lagrangian:
\be
\l_{WZ}=\dmu\s^{\dagger}\partial^{\mu}\s +
\frac{i}{2}\bar{\psi}\slashed{\partial}\psi + \f^{\dagger}\f.
\label{eq:lwz}
\ee
The $\f$-field in the Lagrangian does not have a kinetic term, and
therefore does not propogate. Instead, it satisfies an {\it algebraic}
equation of motion. Such a term is required to be present to allow the
Lagrangian to be invariant under supersymmetric transformations even
when off-shell particles are present. The supersymmetric
transformations for these fields are given by
\bea
\delta\s&=&-i\sqrt{2}\bar{\alpha}\psi_L,\nonumber\\
\delta\psi_L&=&-\sqrt{2}\f\alpha_L +
\sqrt{2}\slashed\partial\s\alpha_R\\
\delta\f&=&i\sqrt{2}\bar{\alpha}\slashed\partial\psi_L,\nonumber
\eea
where $\alpha$ is a spacetime independent anti-commuting Majorana
spinor parameter. The three fields $\psi_L$, $\s$, and $\f$ can be combined into a {\it
  superfield} 
\be
\hat{\s}(\hat{x})=\s(\hat{x})+i\sqrt{2}\bar{\theta}\psi_L(\hat{x})+i\theta\bar{\theta}\f(\hat{x})
\ee
where
$\hat{x}_{\mu}=x_{\mu}+\frac{i}{2}\bar{\theta}\gamma_5\gamma_{\mu}\theta$
and the $\theta$ are anti-commuting Grassmann numbers which, along with
$x$, make up an extension to spacetime called {\it superspace}. Interactions among
such superfields may be included by introducing the superpotential,
$\hat{f}$ which can be chosen by the model builder. Combining this with
\ref{eq:lwz}, the guaranteed supersymmetric Lagrangian then becomes
\bea
\l_{chiral}&=&\dmu\s^{\dagger}\partial^{\mu}\s +
\frac{i}{2}\bar{\psi}\slashed{\partial}\psi + \f^{\dagger}\f\\
&-&i\displaystyle\sum_i\frac{\partial
  \hat{f}}{\partial\hat{\s}}\bigg|_{\hat{\s}=\s}\f_i-\frac{1}{2}\displaystyle\sum_{i,j}\frac{\partial^2\hat{f}}{\partial\hat{\s}_i\partial\hat{\s}_j}\bigg|_{\hat{\s}=\s}\bar{\psi}_iP_L\psi_j\nonumber\\
&+&i\displaystyle\sum_i\left(\frac{\partial
    \hat{f}}{\partial\hat{\s}_i}\right)^{\dagger}\bigg|_{\hat{\s}=\s}\f^{\dagger}_i-\frac{1}{2}\displaystyle\sum_{i,j}\left(\frac{\partial^2\hat{f}}{\partial\hat{\s}_i\partial\hat{\s}_j}\right)\bigg|_{\hat{\s}=\s}\bar{\psi}_iP_R\psi_j
\label{eq:chiral}
\eea

The curl supermultiplet, sometimes called the gauge (when applied in a
gauge theory as will be the case in this dissertation) or vector
supermultiplet, contains the gauge fields of the theory. It contains a
field strength tensor $F_{\mu\nu}$ containing the vector field
$V_{\mu}$, a Majorana field $\lambda$, and an auxiliary field
$\mathcal{D}$. The Majorana field $\lambda$ is called a {\it gaugino},
and will be the fermionic partner to the SM gauge bosons. The spin
zero bosonic field $\mathcal{D}$ will play the same role as $\f$ in the chiral Lagrangian. A general gauge Lagrangian will take the form
\be
\l_{gauge}=-\frac{1}{4}F_{A\mu\nu}F_A^{\mu\nu}+\frac{i}{2}\bar{\lambda}_A\slashed{D}_{AC}\lambda_C+\frac{1}{2}\mathcal{D}_A\mathcal{D}_A,
\ee
where $D$ is the gauge covariant derivative, given by
\be
(\slashed{D}\lambda)_A=\slashed{\partial}\lambda_A + igf_{ABC}\slashed{V}_B\lambda_C.
\ee
The SUSY transformations of these fields
are
\bea
\delta
F^{\mu\nu}&=&-i\bar{\alpha}[\gamma^{\nu}\partial^{\mu}-\gamma^{\mu}\partial^{\nu}],\nonumber\\
\delta \lambda &=&
-i\gamma_5\alpha\mathcal{D}+\frac{1}{4}[\gamma_{\nu},\gamma_{\mu}]F^{\mu\nu}\alpha,\\
\delta \mathcal{D}&=&\bar{\alpha}\slashed{\partial}\gamma_5\lambda.
\eea
The three fields in the curl supermultiplet can be combined into a
{\it curl superfield} in a similar way as the chiral superfield. The
curl superfield $\hat{W}_A(\hat{x})$ in the Wess-Zumino gauge is given by
\be
\hat{W}_A(\hat{x})=\lambda_{LA}(\hat{x})+\frac{1}{2}\gamma^{\mu}\gamma^{\nu}F_{\mu\nu
  A}(\hat{x})\theta_L-i\bar{\theta}\theta_L(\slashed{D}\lambda_R)_A-i\mathcal{D}_A(\hat{x})\theta_L.
\ee

The action for supersymmetric gauge theories can be written as an
integral over superspace. One finds
\bea
S=&-&\frac{1}{4}\int
d^4xd^4\theta\left[\hat{\s}^{\dagger}e^{-2gt_A\hat{\Phi}_A}\hat{\s}\right]\\
&-&\frac{1}{2}\left[\int d^4xd^2\theta_L\hat{f}(\hat{\s}+h.c.)\right]-
\frac{1}{4}\int d^4xd^2\theta_L\overline{\hat{W}_A^c}\hat{W}_A,
\eea
where $\hat{\Phi}_A$ is a set of gauge potential superfields in which
the vector potential resides and the $t_A$ are matrix representations of the generators of the
gauge group that satisfy the Lie algebra $[t_A,
t_B]=if_{ABC}t_C$. Also allowed, but not shown, are Fayet-Iliopoulos
terms which appear for each $U(1)$ factor of the gauge group. 

The complete SUSY Lagrangian for a renormalizable gauge theory will take the form 
\bea
\l&=&\l_{chiral}+\l_{gauge}+\l_{int}\nonumber\\
&=&(D_{\mu}\s)^{\dagger}(D^{\mu}\s) +
\frac{i}{2}\bar{\psi}\slashed{D}\psi + \f^{\dagger}\f
-\frac{1}{4}F_{A\mu\nu}F_A^{\mu\nu}+\frac{i}{2}\bar{\lambda}_A\slashed{D}_{AC}\lambda_C+\frac{1}{2}\mathcal{D}_A\mathcal{D}_A\nonumber\\
&+&\displaystyle\sum_i\left[-\sqrt{2}\left(\s_i^{\dagger}gt_{A}\bar{\lambda}_A\frac{1-\gamma_5}{2}\psi_i+h.c.\right)+\s_i^{\dagger}gt_A\s_i \right]\\
&-&i\displaystyle\sum_i\frac{\partial
  \hat{f}}{\partial\hat{\s}}\bigg|_{\hat{\s}=\s}\f_i-\frac{1}{2}\displaystyle\sum_{i,j}\frac{\partial^2\hat{f}}{\partial\hat{\s}_i\partial\hat{\s}_j}\bigg|_{\hat{\s}=\s}\bar{\psi}_iP_L\psi_i
\nonumber\\
&+&i\displaystyle\sum_i\left(\frac{\partial
    \hat{f}}{\partial\hat{\s}_i}\right)^{\dagger}\bigg|_{\hat{\s}=\s}\f^{\dagger}_i-\frac{1}{2}\displaystyle\sum_{i,j}\left(\frac{\partial^2\hat{f}}{\partial\hat{\s}_i\partial\hat{\s}_j}\right)\bigg|_{\hat{\s}=\s}\bar{\psi}_iP_R\psi_j.\nonumber
\label{eq:totlag}
\eea
Here $i, j$ denote the matter field types and $A$ is the gauge group
index. Once again, the Fayet-Iliopoulos terms are not included. 

Symmetries other than SUSY must also be considered when constructing a supersymmetric
Lagrangian, such as lepton and baryon number conservation. These can be
broken by non-perturbative effects in general SUSY theories, but should be conserved at this
level to prevent rapid decay rates of the proton. A symmetry that will
remove B and L violating terms from the SUSY Lagrangian is matter parity,
which is the same as R-parity, defined by
\be
R\equiv(-1)^{3(B-L)+2s},
\ee
where $s$ is the spin of the field. Here, squark, slepton, gaugino,
and higgsino fields
are odd, whereas the matter, gauge, and Higgs fields are even. For the
remainder of this dissertation, R-parity is assumed to be conserved in
nature. This implies that the lightest supersymmetric (LSP) in R-parity
conserving models is stable. When constructing soft breaking terms,
these symmetries are necessarily conserved.

In addition to the protection of symmetries, there are other
constraints on SUSY that must be taken into consideration when
considering the phenomenological features of the theory. Collider experiments
put limits on decays such as $b\rightarrow s \gamma$ and
$B_S\rightarrow\mu^+\mu^-$ give high precision flavor limits by which
model builders must abide. LEP2 constraints put a limit on lightest
chargino mass $m_{\tw_1}> 103.5$ GeV \cite{lep2ino} in addition to the
squark and gluon constraints mentioned earlier (these particles will
be introduced in Sec~\ref{sssec:mssm}).  Cosmological
observations may also put limits
on SUSY through dark matter direct and indirect detection.

%--------------------------------------------------------------------
\subsubsection{SUSY Breaking}
\label{sssec:ssb}
%--------------------------------------------------------------------
%%
If supersymmetry were unbroken, the known particles would each have a
superpartner of the same mass. Experimentally this is not the case,
and supersymmetry must be a broken symmetry, giving
larger masses to the superpartners. Broken SUSY arises spontaneously
if the $F$ or $D$ terms in the Lagrangian acquire $vev$s, however,
such breaking cannot be realized in accord with phenomenology for
global SUSY models. One must move to local SUSY or supergravity (SUGRA)
models. Then SUSY could be broken in some {\it hidden sector} which couples to the {\it visible
  sector}. The hidden sector would be essentially decoupled from the
our world, and only communicated to the visible sector through
messenger interactions. There are many models which employ this method
of SUSY breaking, with the most studied being {\it gravity mediation}.

In gravity mediated scenarios, gravitational interactions act as the
mediator between the two sectors. The gravity force carrier is
the spin 2 graviton, and its supersymmetric partner is the spin 3/2
gravitino. This is the case in supergravity models, such as
mSUGRA/CMSSM along with its extensions, such as the Non-Universal
Higgs Mass (NUHM) model which will be discussed further in
Sec.~\ref{sssec:ext}. In these models, for a suitably chosen
superpotential, SUGRA can be broken at a mass scale $m\sim 10^{11}$
GeV and gives rise to supersymmetric
Goldstone bosons called goldstinos. The goldstino degrees of freedom
are absorbed by the gravitino, giving it a mass $m_{3/2}$. The visible
sector soft terms are then of order $m^2/M_{\rm Pl}$, where the Planck
mass $M_{\rm Pl}\simeq 1.2 \times 10^{19}$ GeV.

Without knowledge of the actual supersymmetry breaking mechanism, we
can parameterize the effects of SUSY breaking by adding to the
Lagrangian {\it soft SUSY breaking} (SSB) terms. These terms must include
all possible SUSY breaking terms, however they are required to respect
the desired symmetries (such as lepton and baryon number) at the SUSY
breaking scale. It is also required that they do not re-introduce
quadratic divergences into the theory. In this way, SUSY is
explicitly broken, however particular models of SUSY breaking may still be
communicated into the soft terms. For example, in gravity mediated
theories, the soft term masses are proportional to the
gravitino mass $m_{3/2}$ which is assumed to lie around the TeV
scale. 

%--------------------------------------------------------------------
\subsubsection{The Minimal Supersymmetric Standard Model}
\label{sssec:mssm}
%--------------------------------------------------------------------
%%
The Minimal Supersymmetric Standard Model (MSSM) is the direct
supersymmetric extension of the Standard Model, introducing the
minimum number of extra particles and interactions to the
theory. Supersymmetry is broken by adding to the theory all allowed soft SUSY
breaking terms. The gauge symmetry of the MSSM is chosen to be $SU(3)_C
\times SU(2)_L \times U(1)_Y$, which is also the Standard Model gauge symmetry. Table~\ref{tab:mssmmatter} lists the fields in the MSSM, along
with the SM particles and their supersymmetric partners. For a full
treatment of the MSSM see, for example, \cite{wss}.

The MSSM requires the introduction of two Higgs
doublets, the reason being twofold: 1. in SUSY, a doublet can only give mass to
either the up-type quarks or the down-type quarks and leptons, a
single doublet cannot do both, and 2. the doublets each carry
higgsinos which can circulate in triangle anomalies. This upsets the
delicate anomaly cancellation that occurs in the SM model, therefore
one set of higgsinos with $Y=1/2$ must be paired by a second with $Y=-1/2$. These appear in a chiral supermultiplet, consisting of both
scalar/pseudo-scalar Higgs bosons and higgsinos. Before symmetry
breaking the two Higgs doublets are specified as being up-type with mass
$m_{H_u}$ or down-type with mass $m_{H_d}$. Electroweak symmetry is
broken when $m_{H_u}^2$ is driven to negative values via
renormalization group equation (RGE)
running. At this point the neutral scalar fields each acquire a {\it vev},
$\left<h^0_u\right>\equiv v_u$ and $\left<h^0_d\right>\equiv v_d$. Thus we
define a new parameter,
\be
\tan\beta\equiv \frac{v_u}{v_d}
\ee
After the electroweak symmetry is
broken five physical Higgs states remain, the light Higgs scalar
$h$ (typically associated with the SM Higgs), the heavy Higgs scalar,
two charged Higgs $H^{\pm}$, and a pseudo-scalar Higgs $A$.
The fermionic partners of the Higgs are the {\it higgsinos}. The
supersymmetric Higgs/higgsino bilinear term $\mu$ gives mass to both
SM particles (the gauge and Higgs bosons) and the SUSY partner
higgsinos. This arises from the superpotential term
\be
\hat{f}\ni \mu(\hat{h}^0_u\hat{h}^0_d+\hat{h}^+_u\hat{h}^-_d).
\ee


The gauge sector of the MSSM is built up of three curl superfields,
$\hat{B}_{\mu}, \hat{W}^a_{\mu},$ and $\hat{g}^A_{\mu}$. These
correspond to the $U(1)_Y, SU(2)_L,$ and $SU(3)_C$ symmetries,
respectively. The Lagrangian contribution from the
soft SUSY breaking gaugino masses is 
\be
\l_{mass}=-\frac{1}{2}[M_1\bar{\lambda}_0\lambda_0
+M_2(\bar{\lambda}_3\lambda_3+2\bar{\lambda}\lambda)
+M_3\bar{\tilde{g}}\tilde{g}].
\label{eq:gagmass}
\ee
Here, $M_1$, $M_2$, and $M_3$ are the bino, wino, and gluino mass
parameters. The gluino is the only color octet fermion; since
$SU(3)_C$ is not broken, it cannot mix with any other fermion and is
therefore a mass eigenstate with $m_{\tg}=|M_3|$.

The bino, winos, and higgsinos mix to give physical mass eigenstates, the four
neutralinos ($\tz_{1,2,3,4}$) and four charginos
($\tw^{\pm}_{1,2}$). The charginos and neutralinos are labeled
according to increasing mass, with $\tz_1 (\tw_1)$ the lightest and
$\tz_4 (\tw_2)$ the heaviest. These masses depend on the complex
mixing patterns of $M_1, M_2,$ $\mu$, and $\tan\beta$, however
they can be described as gaugino-like, higgsino-like, or some mixture. If
$\lvert M_{1,2} \rvert \gg\lvert \mu \rvert, m_W,$ then the lighter chargino
and two lighter neutralinos are higgsino-like, and the heavier chargino
and two heavier neutralinos are gaugino-like. If $\lvert \mu \rvert
\gg \lvert M_{1,2} \rvert, m_W,$ then the situation is reversed. 

Partnered to the leptons are the sleptons, such as the scalar electron
(selectron), scalar electron neutrino (sneutrino), etc. The partners
to the quarks are called {\it squarks}, including the stop, sbottom,
etc. As a whole, these are sometimes referred to as {\it
  sfermions}. Just as in the SM, the sfermions appear in three generations.


%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\centering
\begin{tabular}{| c | c | c | c | c | c |}
\hline 
Field & spin 0 & spin 1/2 & $SU(3)_C$ & $SU(2)_L$ & $U(1)_Y$ \\
\hline 
$\hat{L}= \left(\ba{c} \hat{\nu}_{eL} \\ \hat{e}_{L}\ea\right)$ &
  $(\tnu ~ \te _{L})$ & $(\nu ~ e _{L})$ & {\bf 1} & {\bf 2} & -1  \\
\hline 
 $\hat{E}^c$ & $\tilde{e}_R^*$ & $e_R^{\dagger}$& {\bf 1} & {\bf 1} &
 2\\
\hline
$\hat{Q}= \left(\ba{c} \hat{u}_{L} \\ \hat{d}_{L}\ea\right)$ &
  $(\tu _{L} ~ \td _{L})$ & $(u _{L} ~ d _{L})$ & {\bf 3} & {\bf 2} & 1/3  \\
\hline
$\hat{U}^c$ &  $\tilde{u}_R^*$ & $u_R^{\dagger}$ & {\bf 3$^*$} & {\bf 1} & -4/3\\
\hline
$\hat{D}^c$ &  $\tilde{d}_R^*$ & $d_R^{\dagger}$ & {\bf 3$^*$} & {\bf 1} & 2/3\\
\hline
$\hat{H}_u= \left(\ba{c} \hat{h}^+_{u} \\ \hat{h}^0_{u}\ea\right)$ &
  $(\tilde{h}^+ _{u} ~ \tilde{h}^0 _{u})$ & $(h^+_{u} ~ h^0
  _{u})$ & {\bf 1} & {\bf 2} & 1  \\
\hline 
$\hat{H}_d= \left(\ba{c} \hat{h}^-_{d} \\ \hat{h}^0_{d}\ea\right)$ &
  $(\tilde{h}^- _{d} ~ \tilde{h}^0 _{d})$& $(h^- _{d} ~ h^0
  _{d})$ & {\bf 1} & {\bf 2$^*$} & -1  \\
\hline
\end{tabular}
\caption{The matter and Higgs chiral supermultiplet content
  in the MSSM for one generation.}
\label{tab:mssmmatter}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\centering
\begin{tabular}{| c | c | c | c | c | c |}
\hline
Field & spin 1/2 & spin 1 & $SU(3)_C$ & $SU(2)_L$ & $U(1)_Y$  \\
\hline
$\hat{g}_A$ A=1,...,8 & $\tilde{g}$ & $g$ & {\bf 8} & {\bf 1} & $0$ \\
\hline
$\hat{W}_a$, a=1,2,3 & $\tilde{W}^{\pm}~\tilde{W}^0$ &$W^{\pm}~ W^0$ &
{\bf 1} &{\bf 3} & $0$\\
\hline
$\hat{B}$ & $\tilde{B}$ & $B$ & {\bf 1} & {\bf 1} & $0$ \\
\hline
\end{tabular}
\caption [The gauge supermultiplets in the MSSM.]{The gauge
  supermultiplets in the MSSM.}
\label{tab:inos}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Within the MSSM there are 124 free parameters.
One of the successes of the MSSM is gauge
coupling unification at the grand unified theory (GUT) scale $m_{\rm GUT}\simeq 2\times
10^{16}$ GeV. One may then assume that the MSSM is the correct
effective field theory describing nature from the electroweak scale $Q=1$ TeV to
the high scale (HS) $Q=m_{\rm GUT}$. Motivated by the universality of gravity and by
the phenomenological need to suppress FCNC and CP violating
processes, many physicists adopt the universality hypothesis, defining at the GUT scale
\begin{align*}
g&=g'=g_S \equiv g_{\mbox{GUT}}\\
m^2_{Q_i}&=m^2_{U_i}=m^2_{D_i}=m^2_{L_i}=m^2_{E_i}=m^2_{H_u}=m^2_{H_d}\equiv
m^2_0 \\
M_1&=M_2=M_3\equiv m_{1/2}\\
A_t&=A_b=A_\tau\equiv A_0.
\end{align*}
This assumption greatly simplifies the parameter space. Now it is possible to specify just five
parameters, $m_0,m_{1/2},A_0, sign(\mu), \tan\beta$, to fully
determine the weak scale spectra of the theory. This is called mSUGRA
(minimal SUperGRAvity), or CMSSM (Constrained MSSM) model. Through
renormalization group equation (RGE) running, the term $m_{H_u}^2$
evolves from large $m_0^2$ through zero to negative values, breaking
electroweak symmetry. The RGE running is heavily dependent on the
radiative corrections, therefore this mechanism is referred to as {\it
  radiative electroweak symmetry breaking} (REWSB) \cite{rewsb}.

The MSSM may be embedded within a larger framework, such as a GUT or
string model. Since the MSSM is free of quadratic divergences, masses
at the weak scale remain stable under radiative corrections and allow
the predictions of the MSSM to be extended to the HS. However,
logarithmic divergences will still remain, and for energies $Q\sim
m_{\rm GUT}$ calculations will contain terms proportional to
$\frac{\alpha_i}{4\pi}\log(m_{\rm GUT}/m_Z)$. These large logarithms
will become a part of the discussion of fine-tuning in SUSY in Chapter~\ref{sec:fine}.

%%
%--------------------------------------------------------------------
\subsubsection{Extensions and Alternatives to mSUGRA}
\label{sssec:ext}
%--------------------------------------------------------------------
%%
Another model, the Non-Universal Higgs Mass (NUHM) model is included
in the supergravity class of models. Within the NUHM framework, one may make the choice of
additional free parameters such that the GUT scale Higgs masses are not equal to the common scalar mass $m_0$. NUHM models may have one, two, or three extra parameters where:
\bi
\item NUHM1 \cite{bmpbt} has $m^2_{H_u}=m^2_{H_d}\neq
  m^2_0$ at the GUT scale.
\item NUHM2\cite{nuhm2} has two free parameters leading to non-universal Higgs
  masses, $m^2_{H_u}\neq m^2_{H_d}\neq m^2_0$. This leads to a
  parameter set consisting of 
\be
m_0, m_{1/2}, A_0, \tan\beta, m^2_{H_u}, m^2_{H_d}
\label{eq:nuhm2}
\ee
\item NUHM3 is the same as NUHM2, but with different values $m^2_0$
  for the third generation from the first and second,
  i.e. $m^2_0(3) \neq m^2_0(1,2) $.
\ei
The NUHM models are the focus of Chapter~\ref{sec:rns}.

A popular model that includes solving the Strong CP Problem by introducing
axions ($a$) is the Peccei-Quinn MSSM (PQMSSM). The partners to the
axion are the saxion ($\tilde{s}$) and axino ($\tilde{a}$) \cite{pq1,pq2,weincp,wilccp}. This model
will be discussed further in Chapter~\ref{sec:dark}.
%--------------------------------------------------------------------
\newpage
\section{Naturalness in SUSY \cite{comp,demise}}
\label{sec:fine}
%--------------------------------------------------------------------
%%
The ultra-violet properties of softly broken SUSY
theories, with SUSY broken near the weak scale, ensures that the low
energy theory is at most logarithmically sensitive to high scale (HS)
physics. The realization \cite{hier} that weak scale SUSY solves the
big gauge hierarchy problem endemic to the SM
\cite{gild}, embedded into a high scale framework,
provided much impetus for its study over the last three
decades. Thus, the question arises: are SUSY models now unnatural, and if so,
how unnatural are they?  Or, do there exist portions of parameter
space where SUSY remains natural? If so, a credible goal of
collider\cite{lhcltr,gurrola} and dark matter\cite{bbm} search experiments is
to leave no stone unturned in the search for natural SUSY.  Although
some authors maintain that naturalness is inherently subjective, it is
shown here that this is not so by examining several different proposed
measures. One finds that -- if applied properly -- all measures
agree with one another and allow for much heavier top squarks than
previously considered.

%-------------------------------------------------------------------
\subsection{Standard Model Fine-Tuning}
\label{ssec:smfine}
%-------------------------------------------------------------------
In the SM, the mass of the Higgs boson can 
be calculated as the sum of the tree level contributions and the 
radiative corrections:
\be
m^2_h=m^2_{h~tree}+\delta m^2_{h~rad}.
\ee
where the quadratically divergent radiative corrections
\be
\delta m^2_{h~rad}\simeq \frac{3}{4\pi^2}\left(-\lambda^2_t
  +\frac{g^2}{4} + \frac{g^2}{8\cos^2\theta_W} +\lambda\right)\Lambda^2
\ee
are independent of the tree level Higgs mass (here, $\lambda_t$ is the
SM top Yukawa coupling, $g$ is the $SU(2)_L$ gauge coupling, $\lambda$ is
the SM Higgs quartic coupling and $\Lambda$ is the effective theory
energy cutoff scale).
A SM fine-tuning measure can be defined as
\be
\Delta_{SM}\equiv \delta m^2_{h~rad}/(m^2_h/2). 
\ee
One may
require a fine-tuning with $\Delta_{SM}\approx 1$, thus for large
$\Lambda$ the large radiative corrections must be balanced by the fine
tuning of $m^2_{h~tree}$. Alternatively, $\delta m^2_{h~rad}\sim
m^2_{h~tree}$, requiring
$\Lambda \sim 1$ TeV and the SM
should only be valid up to the TeV scale.  

%The recent discovery of a SM-like Higgs
%boson with mass $m_h=125-126$~GeV \cite{atlas_h,cms_h} at the LHC
%seemingly provides credence to the simplest supersymmetric (SUSY)
%models of particle physics \cite{wss} which had predicted $m_h\sim
%115-135$~GeV \cite{mhiggs}.  
%--------------------------------------------------------------------
\subsection{Electroweak Fine-Tuning $\delew$}
%--------------------------------------------------------------------
%%
The value of $m_Z$ that is obtained from the minimization of the
one-loop-corrected Higgs boson potential
%
\be 
\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,
\label{eq:mZsSig}
\ee 
%
is the starting point for most discussions of fine-tuning
\cite{ellis,bg,CCN} in the MSSM and its extensions.  This expression
is obtained using the weak scale MSSM Higgs potential and all
parameters in Eq.~(\ref{eq:mZsSig}) are evaluated at the scale $Q=M_{SUSY}$.
The $\Sigma$s in Eq.~(\ref{eq:mZsSig}) arise from one loop corrections
to the Higgs potential. Explicit forms for the $\Sigma_u^u$ and
$\Sigma_d^d$ are given in the Appendix of Ref. \cite{rns}.  Requiring
that the observed value of $m_Z$ is obtained without large
cancellations between the various terms on the right-hand-side of
(\ref{eq:mZsSig}) leads to the suggestion of 
%
\be 
\Delta_{\rm EW} \equiv max_i \left|C_i\right|/(m_Z^2/2)\;, 
\ee 
%
as a measure of fine-tuning. Here, $C_{H_d}=m_{H_d}^2/(\tan^2\beta -1)$,
$C_{H_u}=-m_{H_u}^2\tan^2\beta /(\tan^2\beta -1)$ and $C_\mu =-\mu^2$.
Also, $C_{\Sigma_u^u(k)} =-\Sigma_u^u(k)\tan^2\beta /(\tan^2\beta -1)$
and $C_{\Sigma_d^d(k)}=\Sigma_d^d(k)/(\tan^2\beta -1)$, where $k$ labels
the various loop contributions included in Eq.~(\ref{eq:mZsSig}). By
studying (\ref{eq:mZsSig}) one sees that for low $\delew$
we must have low $\mu^2\sim m_Z^2$ so that higgsino masses are
necessarily bounded from above.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp!]
  \begin{center}
\includegraphics[width=\textwidth]{mass_hierarchy_xerxes_not.eps}
  \end{center}
  \caption[Typical mass spectrum from low $\Delta_{\rm EW}$
  models.]{Typical mass spectrum from low $\Delta_{\rm EW}$ models.}
\label{fig:spectra}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The fine-tuning measure $\delew$ has several attractive features that merit consideration. 
%
\bi
\item {\it Model independent} (within the context of models which reduce 
to the MSSM at the weak scale): 
$\delew$ is essentially determined by the sparticle
spectrum\cite{sugra}, and -- unlike other
measures of fine-tuning -- does not depend on the mechanism by which
sparticles acquire masses.
Since $\delew$ is determined only from weak scale Lagrangian parameters, the
phenomenological consequences which may be derived by requiring low
$\Delta_{EW}$ will apply not only for the NUHM2 model considered here,
but also for other possibly more complete (or less complete, such as pMSSM) 
models which lead to look-alike spectra at the weak scale.
\item {\it Conservative}: $\delew$ captures the minimal fine-tuning 
that is necessary for any given sparticle spectrum, 
and so leads to the most conservative
conclusions regarding fine-tuning considerations.
\item {\it Measureable}: $\delew$ is in principle measurable, in that it 
can be evaluated   if the underlying weak scale parameters 
can be extracted from data.
\item {\it Unambiguous}: Fine-tuning measures which depend 
on high scale parameter choices are
highly sensitive to exactly which set of model input parameters 
one adopts. 
There is no such ambiguity in the fine-tuning sensitivity as
measured by both $\delew$.
\item {\it Predictive}: While $\delew$ is less restrictive than than another
  measure of fine-tuning $\Delta_{HS}$ (introduced in Sec.~\ref{ssec:delhs}), 
it still remains highly restrictive. The requirement of low $\Delta_{EW}$ 
highly disfavors models such as mSUGRA/CMSSM\cite{sugra}, 
while allowing for very distinct predictions from more general 
models such as NUHM2.
\item {\it Falsifiable}: The most important prediction from 
requiring low $\Delta_{EW}$ is that $|\mu |$ cannot be too far removed from 
$m_Z$. This implies the existence of light higgsinos $\sim 100-300$ GeV 
which are hard to see at hadron colliders, but which are easily detected at 
a linear $e^+e^-$ collider with $\sqrt{s}\agt 2|\mu |$.
\item {\it Simple to calculate}: 
$\delew$ is extremely simple to encode in sparticle mass spectrum
programs, even if one adopts models with very large numbers of 
input parameters.
\ei 
Fig.~\ref{fig:spectra} shows see the typical SUSY spectra that
occurs in models with low $\delew$. As expected by the condition
$\mu^2\sim m_Z^2$, the higgsinos are roughly of the same order as the
SM gauge bosons and the Higgs. The stops, sbottoms and gluinos lie
$\sim 1-5$ TeV, safely above current collider limits. 
%

Before proceeding further, note that $\delew$ as defined here entails
only weak scale parameters, and therefore has no information about the
$\log\Lambda$ terms mentioned in Sec.~\ref{sssec:mssm} that cause weak
scale physics to exhibit logarithmic sensitivity to HS physics. Since
many broad features of the phenomenology are determined by the
spectrum, much of the phenomenology of the (unknown) underlying theory
is the same as those of the NUHM2 model with the same spectrum.

%%
%--------------------------------------------------------------------
\subsection{Fine-Tuning of the Higgs mass}
\label{ssec:delhs}
%--------------------------------------------------------------------
%%

In the MSSM, it is found that,
%
\be m_h^2 \simeq \mu^2 +m_{H_u}^2|_{tree}+\delta m_{H_u}^2|_{rad}, 
\ee 
where now $\mu^2$ is the supersymmetric Higgs/higgsino bilinear term
mentioned earlier. If one assumes the MSSM is valid up to the GUT scale,
then the value of $\delta m^2_{H_u}$ can be found by integrating the
renormalization group equation \cite{barger94}
\be
\frac{dm^2_{H_u}}{dt}=\frac{1}{8\pi}\left(-\frac{3}{5}g^2_1M^2_1-3g^2_2M^2_2+\frac{3}{10}g^2_1S+3f^2_tX_t\right)
\label{eq:rgediff}
\ee
where $t=\sqrt{Q^2/Q_0^2}$, $S=m^2_{H_u}-m^2_{H_d}+Tr[{\bf m}^2_Q -{\bf m}^2_L -2{\bf m}^2_U +{\bf m}^2_D +{\bf m}^2_E]$, and $X_t=m^2_{Q_3}+m^2_{U_3}+m^2_{H_u}+A^2_t$. Neglecting the gauge
terms, the $S$ term, the $m^2_{H_u}$ contribution to $X_t$, and the
fact that $f_t$ and the soft terms evolve under $Q^2$ variation,
equation~\ref{eq:rgediff} can be integrated to give 
\be \delta m_{H_u}^2|_{rad}\simeq
-\frac{3f_t^2}{8\pi^2}(m_{Q_3}^2+m_{U_3}^2+A_t^2)\ln\left(\Lambda^2/M_{\rm
  SUSY}^2 \right). 
\label{eq:intrun}
\ee 
Here, $\Lambda$ is again the cut-off scale which
may be taken as high as $m_{\rm GUT}$ or even the reduced Planck mass
$M_P\simeq 2.4\times 10^{18}$~GeV in models with a high scale origin, or close to 
$M_{SUSY}^2 \simeq m_{\tst_1}m_{\tst_2}$. By requiring the
measure $\delhs$ \cite{kn,delhs}$\sim \delta m^2_{H_u}/(m^2_h/2)\alt10$, one expects
$m_{\tst_{1,2}\tb_1}\alt600$ GeV. Using $\delhs$ along with $m_h\simeq
125$ GeV, some popular SUSY models are fine-tuned to 0.1\%.

This case is different from the SM case because $m^2_{H_u}$ and $\delta
m^2_{H_u~rad}$ are {\it not independent}. Therefore, using $\delhs$ could lead to the
erroneous conclusion that the model is fine-tuned. If, instead, these two terms
are grouped together, then
\be
(m^2_{H_u}+\delta m^2_{H_u~rad})=m_{H_u}^2.
\ee
Such a regrouping of terms leads back to the $\delew$ measure.

%--------------------------------------------------------------------
\subsection{Barbieri-Giudice Fine-Tuning $\Delta_{\rm BG}$}
%--------------------------------------------------------------------
%%
The traditional fine-tuning measure $\Delta_{BG}$,
introduced over 25 years ago \cite{ellis,bg,dg}, uses
fractional change in the output value of $m_Z^2$ 
given by (\ref{eq:mZsSig}) relative
to the corresponding change in the input parameters, and is defined by,
%
\be \Delta_{\rm BG}\equiv max_i|B_i|=\left|\frac{\partial\ln
  m_Z^2}{\partial\ln c_i}\right|
=max_i\left|\frac{c_i}{m_Z^2}\frac{\partial m_Z^2}{\partial
  c_i}\right|\;.
\label{eq:DBG}
\ee
%
Here, the $c_i$ constitute the fundamental parameters of the
theory. These would be the weak scale parameter set in the case of the 
pMSSM, but the HS parameter set for models such as mSUGRA. 
If the dependence of $m_Z^2$ on $a_i$ is linear as in
(\ref{eq:mZsSig}) with weak scale MSSM parameters as inputs,
$\Delta_{\rm BG}$ coincides with $\delew$ at the tree-level. An
advantage for $\delbg$ over $\delhs$ is that it maintatins the
correlation between $m^2_{H_u}$ and $\delta
m^2_{H_u~rad}$ by its expression in terms of the high scale
parameters. 

To evaluate $\Delta_{\rm BG}$ in HS models, one needs to know the
explicit dependence of the weak scale parameters on the fundamental
parameters. At tree-level, for moderate to large values of $\tan\beta$,
the $m_{H_d}^2$ term in the expression for $m_Z^2$ is unimportant, and
so, to a good approximation, one needs only the $m_{H_u}^2$ and $\mu^2$ in
terms of the HS parameters.  Expressions for these can be gained by
semi-analytic solutions to the one-loop renormalization group equations, as found for instance in Refs \cite{munoz}.  For instance, for
$\tan\beta =10$, it is found in Ref. \cite{abe,martin,feng} that
%
\bea 
-2\mu^2(m_{\rm weak})
&=& -2.18\mu^2 \\ 
-2m_{H_u}^2(m_{\rm weak})&=& 3.84 M_3^2+0.32M_3M_2+0.047
M_1M_3-0.42 M_2^2 \nonumber \\ & & +0.011 M_2M_1-0.012M_1^2-0.65
M_3A_t-0.15 M_2A_t\nonumber \\ & &-0.025M_1 A_t+0.22A_t^2+0.004
M_3A_b\nonumber \\ & &-1.27 m_{H_u}^2 -0.053 m_{H_d}^2\nonumber \\ &
&+0.73 m_{Q_3}^2+0.57 m_{U_3}^2+0.049 m_{D_3}^2-0.052 m_{L_3}^2+0.053
m_{E_3}^2 \\ & &+0.051 m_{Q_2}^2-0.11 m_{U_2}^2+0.051
m_{D_2}^2-0.052 m_{L_2}^2+0.053 m_{E_2}^2\nonumber \\ & &+0.051
m_{Q_1}^2-0.11 m_l{U_1}^2+0.051 m_{D_1}^2-0.052 m_{L_1}^2+0.053
m_{E_1}^2 , \nonumber
\label{eq:mHu}
\eea 
%
where the parameters on the right-hand-side are evaluated at the
GUT scale.  For different values of $\tan\beta$, the functional form
on the right-hand-side is the same except for somewhat different
values of the coefficients.  Substituting this into the tree level
expression for $m_Z^2$, it is straightforward to evaluate the
derivatives in Eq.~(\ref{eq:DBG}), and $\Delta_{BG}$ can be calculated
using any sparticle mass spectrum code which includes solving the MSSM
coupled RGEs starting from the GUT scale. For the mSUGRA
model where the GUT scale scalar mass parameters are universal, the
scalar mass squared terms coincidentally sum to just $0.013m_0^2$. 

Note that the parameters
in Eq.~\ref{eq:mHu} may even be further correlated. For any fully
specified hidden sector in supergravity theories, one expects each SSB
parameter to be some multiple of $m_{3/2}$: {\it e.g.}
\bea
m^2_{H_u}&=&a_{H_u}\cdot m^2_{3/2}\\
m^2_{Q_3}&=&a_{Q_3}\cdot m^2_{3/2}\\
m^2_{A_t}&=&a_{A_t}\cdot m_{3/2}\\
M_i&=&a_i\cdot m_{3/2}\\
& &\cdots .
\eea
Here, the coefficients $a_i$ parameterize ignorance of the exact
model for SUSY breaking. By using several adjustable parameters, a
wide net is cast which encompasses a large range of hidden sector SUSY
breaking possibilites. Using these terms,
Eq.~\ref{eq:mHu} can be rewritten as 
\be
m_Z^2=-2.18\mu^2+a\cdot m^2_{3/2}
\ee
where now $a$ is just some number which is the sum of all of the
coefficients of the terms $\propto m^2_{3/2}$, assuming $\mu$ is
independent of $m_{3/2}$. Now, naturalness simply requires a small
value of $a$ such that 
\bi
\item$m_z^2\sim a\cdot m^2{3/2}$ 
\ei
and
\bi
\item $m_z^2\sim \mu^2$.
\ei
The first of these implies $m^2_{3/2}\sim m^2_Z$ or $a$ quite
small. The former is unlikely due to the lack of SUSY signal at LHC,
while in the latter case the SUSY soft terms conspire such that there
are large cancellations among the various coefficients of
$m^2_{3/2}$. This is what occurs in radiatively-driven natural SUSY,
which will be introduced in the next chapter. One may equate the value of $m^2_Z$ in terms of weak scale parameters
with the value of $m^2_Z$ in terms of GUT scale parameters:
\be
m^2_Z\simeq-\mu^2(weak)-m^2_{H_u}(weak)\simeq-2.18\mu^2(GUT)+a\cdot m^2_{3/2}.
\ee
Since $\mu$ hardly evolves under RG running, then the BG
condition for low fine-tuning is
\be
-m^2_{H_u}(weak)\sim a \cdot m^2_{3/2}\sim m^2_Z,
\ee
{\it i.e.} that the value of $m^2_{H_u}$ must be driven to small
negative values $\sim -m^2_Z$ at the weak scale. These are exactly the
conditions required by the $\delew$ measure. In other words,
\be
\lim_{n_{SSB}\to 1}\delbg\rightarrow\delew
\ee
where $n_{SSB}$ is the the number of independent soft SUSY breaking
terms. Therefore, if applied properly, $\delbg\to\delew$ and $\delhs\to\delew$.
%--------------------------------------------------------------------
\newpage
\section{Radiative Natural Supersymmetry \cite{ltr,rns}}
\label{sec:rns}
%--------------------------------------------------------------------

Spectra from low $\Delta_{EW}$ models are characterized by (as seen in
Fig~\ref{fig:spectra}):
\bi
\item four light higgsinos $\tw_1^\pm$, $\tz_1$ and $\tz_2$ with mass $\sim \mu\sim 100-300$~GeV,
\item well-mixed top and bottom squarks in the few TeV regime,
\item $m_{\tg}\alt 2-4$~TeV and
\item first/second generation squarks and sleptons in the $5-30$~TeV
regime, consistent with at least a partial decoupling solution to the SUSY
flavor, $CP$, gravitino and $p$-decay problems\cite{dine,am}.
\ei
%
Models with such spectra have been described as radiatively-driven
natural supersymmetry (RNS) models since the value of $m_{H_u}^2$ is
radiatively driven to values close to $m_Z^2$.  RNS spectra can be
realized in the 2-extra-parameter non-universal Higgs models (NUHM2),
but not in more constrained models such as mSUGRA/CMSSM.  For the case
of mSUGRA, while $\mu$ can become low in the HB/FP region, the rather
heavy top squarks $m_{\tst_{1,2}}\sim 7-15$~TeV produce large
$\Sigma_u^u(\tst_{1,2})$ leading again to substantial
fine-tuning\cite{sugra}. An example may be seen in Fig.~\ref{fig:barplot}.
Plotted here are the
contributions to $m_Z^2$ from the mSUGRA/CMSSM model and RNS model
scaled to $m_Z^2/2$ for $m_0=7025$~GeV, $m_{1/2}=568.3$~GeV,
$A_0=-11426.6$~GeV, and $\tan\beta=8.55$. Red bars denote negative
contributions, while blue bars denote positive contributions. In frame
{\it a}), the situation is shown for the mSUGRA model (parameters as
above with $m_{H_u}=m_{H_d}=m_0$). The value of $\mu^2$ must be
fine-tuned so that a large, unnatural cancellation between $m^2_{H_u}$
and $\mu^2$ is needed to obtain a $Z$ mass of just 91.2 GeV. In frame
{\it b}), the case is shown for RNS with the same parameters as mSUGRA
but with $\mu=150$ GeV and where now $m_{H_u}(\Lambda)\neq
m_{H_d}(\Lambda)\neq m_0$. All contributions are now roughly
comparable, so that it is easy to understand why $m_Z$ and $m_h$ both
naturally occur around $\sim 100$~GeV.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{ft_bar_plot.eps}
\end{center}
\caption{Plot of contributions to $m_Z^2/2$ from the mSUGRA/CMSSM
  model with parameters as listed, and also for the RNS2 benchmark
  point with the same $m_0, m_{1/2}, A_0,$ and $\tan\beta$ values but
  with $\mu=150$ GeV. Red bars denote negative contributions, while
  blue bars denote positive contributions.}
\label{fig:barplot}
\end{figure}

The RNS model automatically maintains the SUSY
success stories of gauge coupling unification and radiative breaking of
electroweak symmetry due to a large top quark mass.  These features
require the MSSM as the effective field theory from $M_{weak}$ up to a scale $\Lambda$,
which is taken to be $M_{GUT}\simeq 2\times 10^{16}$~GeV.
The low value of $m_{H_u}^2({\rm weak})$ 
that is required to obtain small $\delew$ can always be realized 
via RGE running, once the GUT-scale value of $m_{H_u}^2$ is decoupled 
from matter scalar masses.

To illustrate how a low value of $m_{H_u}^2(\msusy)$ is obtained, shown in
Fig.~\ref{fig:evol} is the running of various SUSY parameters
versus the renormalization scale $Q$ for the RNS2 benchmark point from
Ref.~\cite{ltr}.  The RNS2 point has parameters $m_0=7025$~GeV,
$m_{1/2}=568.3$~GeV, $A_0=-11426.6$~GeV, $\tan\beta =8.55$ with $\mu
=150$~GeV and $m_A=1000$~GeV.  The gaugino and matter scalar mass
parameters evolve from $m_{1/2}$ and $m_0$ to their weak scale values,
resulting in a pattern of masses very similar to that in mSUGRA. The
parameter $\mu$ hardly evolves, and for such a low value of $\tan\beta$,
$m_{H_d}^2$ also suffers little evolution. Of most interest here
is the RG evolution of $m_{H_u}^2$. As is well known, the SUSY breaking
parameters $m_{Q_3}^2$, $m_{U_3}^2$ and $m_{H_u}^2$ of the scalar fields
that couple via the large top quark Yukawa coupling are driven down with
reducing values of the scale $Q$. The reduction is the greatest for
$m_{H_u}^2$ which, in fact, is driven negative, triggering the radiative
breakdown of electroweak symmetry \cite{rewsb}. It is seen from the figure
that the weak scale value of $-m_{H_u}^2$ has a magnitude $\sim m_Z^2$,
and is much smaller than the weak scale value of other mass parameters.
This is not an accident because the NUHM2 model provides the
flexibility to adjust the GUT scale value of $m_{H_u}^2$ so that it
barely runs to negative values at the weak scale.  
%Since $m_{H_u}^2$ is
%driven radiatively to $\sim -m_Z^2$ at the weak scale, this scenario has
%been dubbed {\it Radiative Natural SUSY}, or RNS for short.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{mass_plots.eps}
\end{center}
\caption[Evolution of SSB parameters from $M_{\rm GUT}$ to $M_{weak}$ for
the RNS2 benchmark point taken from in Ref.~\cite{ltr} whose parameters
are given in the text.]{Evolution of SSB parameters from $M_{\rm GUT}$ to $M_{weak}$ for
the RNS2 benchmark point taken from in Ref.~\cite{ltr} whose parameters
are given in the text. The graph extends to values below
$Q^2=m_{\tst_1}m_{\tst_2}$ where the Higgs mass parameters are extracted. }
\label{fig:evol}
\end{figure}
%

This chapter will explore what parameter choices
lead to low values of $\Delta_{\rm EW}$.  While $\Delta_{EW}$ seems
bounded from below by about 100 in mSUGRA/CMSSM\cite{sugra}, one finds
that $\Delta_{EW}$ as low as $\sim 10$ can be obtained in NUHM2. The goal is to provide a rather complete characterization
of RNS.  This should provide a
comprehensive picture as to where in model parameter space one can find
1)~$m_h\sim 125$~GeV along with 2)~low EWFT $\delew\alt 30$
while at the same time 3)~respecting LHC constraints on sparticle
masses. In Sec.~\ref{sssec:nuhm2},
parameter space regions leading to low $\delew$ from scans over
the NUHM2 model which allow for
RNS are shown.  In Sec.~\ref{sssec:nuhm3}, the
results are extended
to include the split generation non-universal Higgs model NUHM3, wherein
high scale third generation scalar masses $m_0(3)$ need not equal
first/second generation scalar masses $m_0(1,2)$.  While the former
implementation allows for fewer parameters, the additional freedom in
the NUHM3 model allows for a more robust decoupling solution to the SUSY
flavor and $CP$ problems because heavier multi-TeV first/second
generation sfermion masses are then possible. Section~\ref{ssec:b} shows
that constraints from $B$-physics -- especially $BF(b\to s\gamma )$ are
much more easily respected in RNS than in generic natural SUSY models.

Motivated by the possibility of cancellations occurring in
$m_{H_u}^2(\msusy)$, the EWSB minimization condition
(\ref{eq:mZsSig}) is now returned to, and the radiative corrections embodied in $\Sigma_u^u$ and
$\Sigma_d^d$ that have not yet been discussed will be examined. These affect the
minimization condition in an important way when 
$m_{H_u}^2(m_{\rm SUSY})$ and $\mu^2$ are much smaller 
than the scale of other weak scale SUSY breaking parameters. 
At the one-loop level, $\Sigma_u^u$
contains the contributions\cite{an,deboer} $\Sigma_u^u(\tst_{1,2})$,
$\Sigma_u^u(\tb_{1,2})$, $\Sigma_u^u(\ttau_{1,2})$,
$\Sigma_u^u(\tw_{1,2})$, $\Sigma_u^u(\tz_{1-4})$, $\Sigma_u^u(h,H)$,
$\Sigma_u^u(H^\pm)$, $\Sigma_u^u(W^\pm)$, $\Sigma_u^u(Z)$, and
$\Sigma_u^u(t)$. $\Sigma_d^d$ contains similar terms along with
$\Sigma_d^d(b)$ and $\Sigma_d^d(\tau)$ while $\Sigma_d^d(t)=0$. There are additional contributions from first/second
generation sparticles from their $D$-term couplings to Higgs scalars.
If these squarks, and separately sleptons, are degenerate then these
contributions cancel within each generation because the sum of weak
isospins/hypercharges of squarks/sleptons total to zero\cite{sugra}.
In the parameter space region where RNS is realized,
{\it i.e.} where $-m_{H_u}^2(\msusy)\sim \mu^2\sim m_Z^2$, the radiative correction
terms from $\Sigma_u^u$ may give the largest contributions to
$\delew$.

The largest of the $\Sigma_u^u$ terms almost always come from top
squarks for which it can be found
%
\bea
\Sigma_u^u(\tst_{1,2} )&=&\frac{3}{16\pi^2}F(m_{\tst_{1,2}}^2)\times
\left[ f_t^2-g_Z^2\mp \frac{f_t^2 A_t^2-8g_Z^2(\frac{1}{4}-\frac{2}{3}x_W)\Delta_t}{m_{\tst_2}^2-m_{\tst_1}^2}
\right]
\label{eq:Siguu}
\eea
%
where $\Delta_t=(m_{\tst_L}^2-m_{\tst_R}^2)/2+m_Z^2\cos
2\beta(\frac{1}{4}-\frac{2}{3}x_W)$, $g_Z^2=(g^2+g^{\prime 2})/8$,
$x_W\equiv \sin^2\theta_W$ and $F(m^2)=m^2\left(\log
(m^2/Q^2)-1\right)$, with $Q^2=m_{\tst_1}m_{\tst_2}$.  In
Ref.~\cite{ltr}, it is shown that for the case of the $\tst_1$
contribution, as $|A_t|$ gets large there is a suppression of
$\Sigma_u^u(\tst_1)$ due to a cancellation between terms in the square
brackets of Eq.~(\ref{eq:Siguu}).  The $\tst_2$ contribution is
suppressed
if there is a
sizeable splitting between $m_{\tst_2}$ and $m_{\tst_1}$ due to a large
cancellation within $F(m_{\tst_2}^2)$ because
$\log(m_{\tst_2}^2/Q^2) = \log (m_{\tst_2}/m_{\tst_1})\simeq 1$.
The large $|A_t|$ values suppress both top squark
contributions to $\Sigma_u^u$, and at the same time lift up the
value of $m_h$, which is near maximal for large negative $A_t$.
Combining all effects, one sees that the same mechanism responsible for
boosting the value of $m_h$ into accord with LHC measurements can also
suppress the $\Sigma_u^u$ contributions to EWFT, leading to a model with
low EWFT.

To display the quality of EWFT explicitly,
Fig.~\ref{fig:mzcont}{\it a} shows the various {\it signed} contributions to
$m_Z^2/2$ that enter Eq.~(\ref{eq:mZsSig}) for the RNS2 point from
Fig.~\ref{fig:evol} and Ref.~\cite{ltr}.  In this figure,
the signed contributions are labeled by $C_i$ where $i=H_u, H_d, \mu, \Sigma_u^u,
\Sigma_d^d$.  The largest contributions come from $C_{\Sigma_u^u}\sim
0.04$~TeV$^2$ and $C_{H_u}\sim -0.03$~TeV$^2$.  Frame {\it b})
shows these same quantities for the mSUGRA model (where $\mu$ and $m_A$
are outputs instead of input parameters).  Here, the maximal
contributions $C_{H_u}\sim 15$~TeV$^2$ and $C_\mu\sim -15$~TeV$^2$.
Frame {\it c}) compares results from the two models using a common
scale.  Here, it is clearly seen that the mSUGRA model is enormously
fine-tuned compared to the RNS2 benchmark point.

%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{shift_plot.eps}
\end{center}
\caption[Signed contributions to $m_Z^2/2$ from terms in the EWSB
minimization condition Eq.~\ref{eq:mZsSig}.]{Signed contributions to $m_Z^2/2$ from terms in the EWSB
minimization condition Eq.~\ref{eq:mZsSig} from {\it a}) the RNS2
benchmark point defined in the text and {\it b}) the corresponding
mSUGRA model as RNS2 with $\mu$ and $m_A$ as outputs rather than inputs.
In frame {\it c}), the results for both models are plotted on a common
scale.  }
\label{fig:mzcont}
\end{figure}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Radiative natural SUSY from the NUHM models}
\label{ssec:nuhm2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Since the mass scale of the
MSSM is stable to radiative corrections even when the MSSM is embedded
into a high scale framework, it is tempting to speculate that the MSSM
arises as the low energy limit of an underlying SUSY grand unified
theory with a unification scale $M_{\rm GUT}\simeq 2\times 10^{16}$~GeV.
Indeed, the MSSM receives some indirect support
from experiment in that 1)~the measured weak scale gauge couplings
nearly unify at $M_{\rm GUT}$ under MSSM RG evolution,
2)~radiative corrections due to the large top quark Yukawa
coupling -- consistent with $m_t\sim 173$~GeV -- dynamically breaks electroweak
symmetry, and 3)~a light SM-like Higgs boson has been discovered
to be lying squarely within the narrow mass window predicted
by the MSSM.

Motivated by these successes, the interesting question arises as to
whether a natural SUSY sparticle mass spectrum, {\it i.e.} one with a
modest value of $\delew$, can be consistently generated from a model
with parameters defined at the high scale $Q=M_{\rm GUT}$.


\subsubsection{RNS from the NUHM2 model}
\label{sssec:nuhm2}
Naturalness requires $|\mu| \sim m_Z\sqrt{\delew/2}$, while the
recently measured\cite{lhcb} value of the branching fraction $BF(B_s
\to\mu^+\mu^- )$ qualitatively agrees with the predicted SM value,
which in turn requires the $CP$ odd boson $A$ to be relatively
heavy. One is then led to adopt the 2-parameter non-universal Higgs
model (NUHM2)\cite{nuhm2}, wherein weak scale values of $\mu$ and
$m_A$ may be used as inputs in lieu of GUT scale values of $m_{H_u}^2$
and $m_{H_d}^2$.
%\footnote{Since the Higgs fields belong to different
 % multiplets from matter fields, it is easy to envisage models with
  %independent SUSY breaking mass parameters for Higgs and matter
  %scalars.}. 
Also, in this section a common GUT scale mass
parameter $m_0$ is taken for {\it all} the matter scalars. Thus the parameter
set is given by:
\begin{equation}
m_0,\ m_{1/2},\ A_0,\ \tan\beta,\ \mu,\ m_A .
\end{equation}
%
For these calculations, $m_t=173.2$~GeV and the Isajet 7.83~\cite{isajet} SUSY
spectrum generator Isasugra\cite{isasugra} are used.

Radiative Natural SUSY solutions are searched for by
first performing a random scan over the following NUHM2 parameter ranges:
%
\bea
m_0 &:& \ 0-20\ {\rm TeV}, \nonumber\\
m_{1/2} &:& \  0.3-2\ {\rm TeV},\nonumber\\
-3 &<& A_0/m_0 \ <3,\nonumber\\
\mu &:& \ 0.1-1.5\ {\rm TeV}, \label{eq:param1}\\
m_A &:& \ 0.15-1.5\ {\rm TeV},\nonumber\\
\tan\beta &:& 3-60 . \nonumber
\eea
%
For the solutions found, it is required that:
\bi
 \item electroweak symmetry be radiatively broken (REWSB),
 \item the neutralino $\tz_1$ is the lightest MSSM particle,
 \item the light chargino mass obeys the model
independent LEP2 limit, $m_{\tw_1}>103.5$~GeV\cite{lep2ino},
\item The parameters $m_0$ and $m_{1/2}$ respect the
recent LHC search bounds on $m_{\tg}$ and $m_{\tq}$ obtained within
the mSUGRA model~\cite{atlas_susy,cms_susy} are respected,
\item $m_h=125\pm 2$~GeV, allowing for estimated uncertainty in the
  calculation of $m_h$.
\ei

To begin the investigation of NUHM2 model parameters which lead to low
$\delew$, Fig.~\ref{fig:nuhm2} shows each scan point as a red ``+''
in frames of $\delew$ versus {\it a})~$m_0$, {\it b})~$m_{1/2}$, {\it
  c})~$A_0/m_0$, {\it d})~$\tan\beta$, {\it e})~$\mu$, and {\it
  f})~$m_A$.
Since low $\delew$ solutions are only
possible for low values of $\mu$, a separate narrow scan was performed,
but this time with $\mu$ restricted between 100--300~GeV. The results of this
second scan are shown by the blue crosses in the figure.

One sees from the plots that $\delew$ varies from as low as $\sim
10$ ($\Delta_{EW}^{-1}=10\%$ EWFT) to over 1000.  While the bulk of points shown are
fine-tuned with large $\delew\agt 100$, there do exist many
solutions with $\delew\alt 30$, corresponding to better than
3\% EWFT. The RNS solutions with $\delew\alt 30$ are obtained
for values of $m_0\sim 1-8$~TeV.  In the cases where $m_0$ is as high as
5-10~TeV, the top squark masses are driven to much lower values via
1)~the large top-quark Yukawa coupling $f_t$ which suppresses top-squark
soft masses during RG evolution, 2)~large mixing effects which can
suppress $m_{\tst_1}$ and yield a large $m_{\tst_1}-m_{\tst_2}$
splitting, and 3)~two-loop RGE suppression of diagonal top squark mass
terms arising from large first/second generation sfermion
masses\cite{am,graesser,imh}.  If $m_0$ is too large -- in this case
above $\sim 10$~TeV -- then these suppression mechanisms are insufficient
to drive $m_{\tst_{1,2}}$ to low enough values to allow for low EWFT.
Thus, the span of points shown in frame {\it a}) trends upward in
$\delew$ as $m_0$ increases past about 8~TeV.  One also sees that
for the red pluses in frame {\it a}) $\delew$ has an upper
bound close to about 500 if $m_0\alt 10$~TeV. For still larger values of
$m_0$ then $\delew$ increases with $m_0$. This is because while
$\mu^2$ (or equivalently $-m_{H_u}^2$) is the largest of the quantities
in (\ref{eq:mZsSig}) for the lower range of $m_0$, for very large
values of $m_0$ then $\Sigma_u^u$ begins to dominate.  The blue crosses
from the narrow scan with small $\mu$ have a different shape from the
red broad scan since the upper edge is mostly determined by
$\Sigma_u^u$, and so increases with $m_0$.
%
\begin{figure}[htbp!] 
\begin{center}
  \includegraphics[width=7.2cm,clip]{delew_m0_nuhm2.eps}
  \includegraphics[width=7.2cm,clip]{delew_mhf_nuhm2.eps}
  \includegraphics[width=7.2cm,clip]{delew_a0_nuhm2.eps}
  \includegraphics[width=7.2cm,clip]{delew_tanb_nuhm2.eps}
  \includegraphics[width=7.2cm,clip]{delew_mu_nuhm2.eps}
  \includegraphics[width=7.2cm,clip]{delew_ma_nuhm2.eps}
\end{center}
\caption[The dependence of $\delew$ on various NUHM2 parameters
  from a scan (\ref{eq:param1}) over parameter space.]{The dependence of $\delew$ on various NUHM2 parameters
  from a scan (\ref{eq:param1}) over parameter space (red pluses) and
  for the dedicated scan with 100~GeV$<\mu<$300~GeV (blue
  crosses). The line at $\delew=30$ is to guide the eye.  }
\label{fig:nuhm2}
\end{figure}


Frame {\it b}) of Fig.~\ref{fig:nuhm2} shows
$\delew\ vs.\ m_{1/2}$. Here, the low values of $\delew$
span a wide range of $m_{1/2}$ values from $0.3-1.5$~TeV. Since
$m_{\tg}\sim (2.5-3)m_{1/2}$, one expects $\delew\alt 30$ for
$m_{\tg}$ values up to about 4~TeV.  For the entire parameter space
(red pluses) $\delew$ is roughly evenly distributed with
respect to the gaugino mass parameter.
Frame {\it c}) shows $\delew\ vs.\ A_0/m_0$. A
clear trend is evident for low values of EWFT when $|A_0/m_0|\sim  1.5-2$.
The reason is that the hole at low magnitudes of $A_0/m_0$ and
small values of $\delew$
occurs because of the Higgs mass constraint.
Large magnitudes of GUT scale $A_0$ lead to
correspondingly large weak scale $A_t$ parameters,
which, in turn, provide large mixing in the top-squark sector.  This leads to
low EWFT and also heightened values of $m_h\sim 125$~GeV.  Frame {\it
  d}) shows $\delew\ vs.\ \tan\beta$. There is a slight preference
for low $\tan\beta \sim 10-20$ but otherwise no structure to speak of.
Frame {\it e}) shows $\delew$ versus the weak scale value of
$\mu$. The parabolic lower edge of the span of points reflects the upper
bound on $\mu$ necessary for low EWFT.  From the plot, bounds on $\mu$
can be conveniently read off: for instance, requiring $\delew\alt
30$ then requires $\mu \alt 350$~GeV. Models with low
$\mu\sim 100$~GeV but multi-TeV top squarks can still be very
fine-tuned if the dominant contributions to $\delew$ 
arise from $\Sigma_u^u(\tst_i )$. In frame {\it f})
$\delew\ vs.\ m_A$ is shown.  Low $\delew$ can be found
over the entire range of $m_A\sim 0.15-1.5$~TeV, so this parameter is
not so relevant towards achieving low EWFT.

Next, to gain a sense of the sparticle mass ranges expected from RNS, $\delew$ is plotted versus selected sparticle masses.  First, since
$m_0\sim 2-8$~TeV for $\delew\alt 30$, the first and
second generation squark and slepton masses are expected to also be within this range
(which is for the most part inaccessible LHC SUSY searches).  Next, in
frame {\it a}) of Fig.~\ref{fig:mass1}, $\delew\ vs.\ m_{\tg}$ is shown.
Requiring $\delew \alt 30$ requires $m_{\tg}\sim
1-4$~TeV. The lower portion of this range should be accessible to LHC14
searches, while the upper part lies beyond any LHC luminosity
upgrade\cite{lhchl}.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=7.2cm,clip]{delew_glu_nuhm2.eps}
\includegraphics[width=7.2cm,clip]{delew_t1_nuhm2.eps}
\includegraphics[width=7.2cm,clip]{delew_t2_nuhm2.eps}
\includegraphics[width=7.2cm,clip]{delew_b1_nuhm2.eps}
\end{center}
\caption[The value of $\delew$ versus gluino and third generation
  squark masses from a scan over NUHM2 parameter space.]{The value of $\delew$ versus gluino and third generation
  squark masses from a scan over NUHM2 parameter space. As in
  Fig.~\ref{fig:nuhm2}, the red pluses denote the distributions from
  the complete scan, whereas the blue crosses depict the results for
  the dedicated low $\mu$ scan. The line at $\delew=30$ is to
  guide the eye. }
\label{fig:mass1}
\end{figure}

Frame {\it b}) shows $\delew$ versus the lighter top squark mass
$m_{\tst_1}$. Here, $\delew\alt 30$ allows $m_{\tst_1}\sim
0.5-2.5$~TeV range. This is well above the range expected in generic
NS models\cite{ah,others,delhs}, where $m_{\tst_{1,2}}$ has
been advocated to lie below about 600~GeV.
Frame {\it c}) shows $\delew\ vs.\ m_{\tst_2}$. Here, $m_{\tst_2}$ can range up to $\sim 6$~TeV for $\delew\alt 30$.
%\footnote{We note that the absolute upper limit of about 8~TeV on
%  $m_{\tst_2}$ is an artifact of the rather low upper limit on
%  $m_{1/2}$ in our scan. For LHC phenomenology, this is completely
%  irrelevant.}  
Such high values of $m_{\tst_2}$ are helpful to
increase radiative corrections to the light Higgs mass $m_h$ into the
125~GeV range.  However, such heavy top squarks lie far beyond any
conceivable LHC reach.  In frame {\it d}), $\delew\ vs.\
m_{\tb_1}$ is shown. Here, $m_{\tb_1}\sim 0.8-6$~TeV, which again allows for far heavier bottom squarks than previous NS models, 
where $m_{\tst_{1,2}}$ and $m_{\tb_1}$ all were suggested to be $\alt 600$~GeV.

Figure~\ref{fig:mass2}{\it a}) shows $\delew\ vs.\ m_{\tw_1}$.  For
RNS models, $m_{\tw_1}\simeq m_{\tz_{1,2}}\sim |\mu |$, {\it i.e.} its
mass is roughly equal to that of the two lighter neutralinos.  Since
$\tw_1$ is mainly higgsino-like near the lower edge of the envelope of
points, the distribution follows a similar pattern as for the $\delew\
vs.\ \mu$ plot in Fig.~\ref{fig:nuhm2}. For $\delew\alt 20$,
$m_{\tw_1}\alt 250$~GeV; thus, a linear collider operating with
$\sqrt{s}>2m_{\tw_1}$ will directly probe the lowest values of $\delew$ if the relatively soft visible
daughters of the chargino can be distinguished over two-photon
backgrounds\cite{krupov}. In this sense, it has been emphasized that for
models of natural SUSY, a linear $e^+e^-$ collider would be a higgsino
factory in addition to a Higgs factory\cite{bbh,bbht,ltr}. This will
be discussed in Chapter~\ref{sec:ilc}.  In frame
{\it b}), $\delew\ vs.\ m_{\tw_2}$ is shown.  In the RNS model,
the $\tw_2$ is nearly pure wino-like and its mass can range between
$\sim 0.3-1.2$~TeV for $\delew\alt 30$.  Since RNS as presented
here includes gaugino mass unification, then typically $\tz_{1,2}$ are
higgsino-like, $\tz_3$ is bino-like and $\tz_4$ is wino-like. Since the
$SU(2)$ gauge coupling $g$ is rather large, one expects significant
rates for $\tw_2^\pm \tz_4$ production at LHC, at least for the lower
portion of the range of $m_{\tw_2}$.  Frame {\it c}) shows the
$m_{\tz_2}-m_{\tz_1}$ mass difference in RNS versus $\delew$.  For most
points with $\delew\alt 30$, it is seen that $m_{\tz_2}-m_{\tz_1}\alt
10-20$~GeV. Some points with $\delew\sim 30-40$ have a mass
difference as large as 100~GeV; these points arise from sampling the
lower portion of the $m_{1/2}$ range, which gives rise to gaugino masses
comparable in magnitude to $|\mu |$ so that the lighter electroweakinos
are actually gaugino-higgsino mixtures.  For the small mass
gap case, the lighter neutralinos are dominantly higgsino-like and decay
via $\tz_2 \to \tz_1f\bar{f}$ (here $f$ denotes SM-fermions) through the
virtual $Z$.  Then decays into opposite-sign same-flavor (OS/SF)
isolated dileptons should occur at $\sim 3\%$ for each charged lepton
species.  The presence of low invariant mass OS/SF isolated dileptons
from boosted $\tz_2$ produced in gluino or gaugino cascade decay events
could then be a distinctive signature of RNS at the LHC as will be
discussed in Chapter~\ref{sec:lhc}.  For NUHM2
models with larger values of $\delew$, falling outside the RNS low EWFT
requirement, $m_{\tz_2}$ can be greater than $m_{\tz_1}+m_Z$ or
$m_{\tz_1}+m_h$ so that two body decays of $\tz_2$ are then allowed.
Finally, in frame {\it d}), $\delew\ vs.\ m_h$ is shown. Here, the
lower $m_h\sim 123-124$~GeV values are just slightly preferred by EWFT over
the higher range, although values of $m_h$ as high as $\sim 126.5$~GeV
occur for $\delew =30$.
%

\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=7.2cm,clip]{delew_w1_nuhm2.eps}
\includegraphics[width=7.2cm,clip]{delew_w2_nuhm2.eps}
\includegraphics[width=7.2cm,clip]{delew_zd_nuhm2.eps}
\includegraphics[width=7.2cm,clip]{delew_higgs_nuhm2.eps}
\end{center}
\caption[The value of $\delew$ versus electroweak -ino
   and Higgs boson masses from a scan over NUHM2 parameter space.]{The value of $\delew$ versus electroweak -ino
   and Higgs
  boson masses from a scan over NUHM2 parameter space. As in
  Fig.~\ref{fig:nuhm2}, the red pluses denote the distributions from
  the complete scan, whereas the blue crosses depict the results for
  the dedicated low $\mu$ scan. The line at $\delew=30$ is to
  guide the eye. }
\label{fig:mass2}
\end{figure}

Finally, in Fig.~\ref{fig:mHuGUT} a
scatter plot of $\delew$ versus the GUT scale ratio $m_{H_u}^2/m_0$
from the scan over NUHM2 models is shown. Here is is seen that for points with
$\delew\alt 30$,
then $m_{H_u}(M_{\rm GUT})\sim (1-2) m_0$.
%

\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{delew_hum0_nuhm2.eps}
\end{center}
\caption[The value of $\delew$ versus $m_{H_u}/m_0 (M_{\rm GUT})$
from the scan over the NUHM2 parameter space.]{The value of $\delew$ versus $m_{H_u}/m_0 (M_{\rm GUT})$
from the scan over the NUHM2 parameter space. As before, the red pluses
are for the scan over the entire range of $\mu$ while the blue crosses
are for the dedicated scan with $\mu$ limited to the 100-300~GeV range.
The line at $\delew=30$ is to guide the eye.}
\label{fig:mHuGUT}
\end{figure}


\subsubsection{RNS from the NUHM1 model}
The NUHM1 model\cite{bmpbt} is inspired by $SO(10)$ SUSY GUT models
where the Higgs doublets live in the 10-dimensional fundamental
representation while the matter scalars inhabit the 16- dimensional
spinor representation. In this case, the parameter set is expanded by
one and now the scan is over
\bi
\item $m_0 : 0-15$ TeV,
\item $m_{H_u} = m_{H_d} \equiv m_H : 0-15$ TeV,
\item $m_{1/2} : 0-2$ TeV,
\item $−2.5 < A_0/m_0 < 2.5$,
\item $\tan\beta : 3-60$.
\ei
By increasing $m_H \gg m_0$, then $m^2_{H_u}$ is only driven to small instead of
large negative values, while if $m^2_{H_u}$ is increased too much, then $m^2_{H_u}$
is never driven negative and electroweak symmetry is
not broken. If $m_H$ is taken smaller than $m_0$, even with $m^2_H < 0$ as a
possibility, then $m_{H_d} \sim m_A$ can be decreased while $m^2_{H_u}$
is driven to very large negative values. In the former case, where
$m^2_{H_u}$ is driven to small negative values, then $\mu$ also
decreases since its value is set to yield the measured $Z$ mass. In such cases, one expects reduced values of $\delew$.
In the scan results shown in Fig.~\ref{fig:nuhm1}, this is
true, as the minimal value of $\delew$ reaches as
low as $\sim$ 30. Values of
$\delew$ in the $30-50$ range which obey B-decay constraints and
$m_h\sim$125 GeV can be found for $m_0\sim$ $3-10$ TeV. With such large
$m_0$ values, then the top squarks also tend to be in the $3-10$ TeV regime
and the top squark radiative corrections prevent $\delew$ from
reaching below $\sim 30$.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth]{nuhm1_m0.png}
\end{center}
\caption{Plot of $\delew$ vs. $m_0$ from a scan over NUHM1 parameter
  space while maintaining $m_h=125.5\pm2.5$~GeV.}
\label{fig:nuhm1}
\end{figure}


\begin{comment}
Up to this point the focus has been on RNS from the NUHM2 model. However,
it is of interest to see if low EWFT is also possible within the NUHM1
\cite{nuhm2} framework in which $H_u$ and $H_d$ have equal GUT scale
mass parameters; {\it i.e.}  $m_{H_u}^2(M_{\rm GUT}) = m_{H_d}^2(M_{\rm
GUT}) \equiv m_\phi^2$. The common GUT scale Higgs mass parameter
$m_\phi^2$ is then adjusted so that $m_{H_u}^2(M_{\rm SUSY}) \sim m_Z^2$. The present
investigation is confined to model lines where $m_0$, $m_{1/2}$ and
$A_0$ are fixed to
be the same as for the RNS2 model point, but $m_\phi$ and
$\tan\beta$ are varied. Figure~\ref{fig:nuhm1} shows the value of {\it a})~$\mu$, {\it
b})~$m_A$, and {\it c})~$\delew$ versus $m_{\phi}$ for $\tan\beta =
8.85$ (the RNS2 value), 25, 40 and 50. The various curves in
frame {\it a}) are quite close (except at very large values where they
dive down). This is essentially because the top Yukawa coupling that
dominantly affects $m_{H_u}^2$ %(remember that $m_{H_u}^2(M_{\rm SUSY})$
%determines $\mu$) 
hardly varies with $\tan\beta$; the small differences
arise from the (subdominant) effects of bottom Yukawa couplings.  In
contrast, the $m_A$ values in frame {\it b}) reduce considerably as
$\tan\beta$ increases. This is because the
bottom Yukawa coupling-- which increases with $\tan\beta$-- drives
$m_{H_d}^2$ to low values thus reducing $m_A^2\simeq m_{H_d}^2 +
\mu^2$ for larger $\tan\beta$ values. Turning to frame {\it
c}) one sees that for this model line with $\tan\beta=8.55$ (uppermost
curve), $\delew$ reduces with increasing $m_\phi$ as in the $\mu$ curve
in frame {\it a}) until the kink at which it starts increasing. The kink occurs when $\mu^2$ becomes so low that the
$m_{H_d}^2$ term becomes larger than all other terms in 
(\ref{eq:mZsSig}). For larger values of $\tan\beta$, the $m_{H_d}^2$
contribution is suppressed, resulting in smaller values of $\delew$.
However, in none of the cases shown does $\delew$ drop below
$\sim 80$.  It may be a useful exercise to scan the NUHM1 parameter
space to see just how small the EWFT can be when all LHC constraints are satisfied.
%

\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=7.2cm,clip]{mu_mphi.eps}
\includegraphics[width=7.2cm,clip]{ma_mphi.eps}
\includegraphics[width=10cm,clip]{delew_mphi.eps}
\end{center}
\caption[Plot of {\it a}) $\mu$, {\it b}) $m_A$ and {\it c}) $\delew$
in the 1-parameter NUHM1 model versus $m_{\phi}$ for
RNS2 model parameters.]{Plot of {\it a}) $\mu$, {\it b}) $m_A$ and {\it c}) $\delew$
in the 1-parameter NUHM1 model versus $m_{\phi}$ for
RNS2 model parameters $m_0=7025$~GeV, $m_{1/2}=568.3$~GeV, $A_0=-11426.6$~GeV
and for several values of $\tan\beta$. In all the frames the order of
the lines is that of increasing $\tan\beta$, with $\tan\beta=8.55$ on
the top and $\tan\beta=50$ on the bottom.
}
\label{fig:nuhm1}
\end{figure}
\end{comment}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{RNS from the NUHM3 (split generation) model}
\label{sssec:nuhm3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This section explores the
possibility of split first/second versus third generation matter
scalars to see if any advantage can be gained for RNS.
Here, the third generation GUT scale mass parameter $m_0(3)$ is allowed to
differ from $m_0(1,2)$ for the first/second generation scalars.
Universality within each generation is well-motivated by $SO(10)$ GUT
symmetry, since all matter multiplets of a single generation belong to a
16-dimensional spinor rep of $SO(10)$. One can also envisage some degree
of non-universality between $m_0(1)$ and $m_0(2)$ as long as both lie in
the tens of~TeV regime: such a scenario invokes a partial
decoupling-partial degeneracy solution to the SUSY flavor and $CP$
problems (for constraints from FCNC processes \cite{silvestrini}, see
{\it e.g.}  Ref.~\cite{nmh}).  For convenience,
$m_0(1)$ and $m_0(2)$ will be set equal, and the parameter set
%
\begin{equation}
m_0(1,2),\ m_0(3),\ m_{1/2},\ A_0,\ \tan\beta,\ \mu,\ m_A \qquad {\rm (NUHM3)}
\end{equation}
%
will be adopted. Here, $m_0(3)$ is the GUT scale third generation soft SUSY breaking mass
parameter and $m_0(1,2)$ is the corresponding (common) parameter for the
first/second generation.

RNS solutions from the split
generation parameter space are searched for by implementing a random scan over the
parameters:
%
\bea
m_0(3) &:&\ 0-20\ {\rm TeV},\nonumber \\
m_0(1,2) &:&\ m_0(3)-30\ {\rm TeV},\nonumber\\
m_{1/2} &:&\  0.3-2\ {\rm TeV},\nonumber \\
-3 & < & A_0/m_0\ <3, \label{eq:SGparams} \\
\mu &:&\ 0.1-1.5\ {\rm TeV}, \nonumber \\
m_A &:&\ 0.15-1.5\ {\rm TeV},\nonumber \\
\tan\beta &:&\ 3-60 .\nonumber
\eea
%
The same LHC sparticle mass and $m_h=125\pm 2$~GeV
constraints are implemented as before.

Figure~\ref{fig:nuhm3} shows $\delew$ versus $m_0(3)$ and also
versus $m_0(1,2)$. The results for $\delew$ versus other
parameters are very similar to Fig.~\ref{fig:nuhm2} and are not
repeated here.  From Fig.~\ref{fig:nuhm3}{\it a}), one sees that RNS
solutions with $\delew\alt 30$ can be found for $m_0(3)$ values
ranging between 1-8~TeV, similar to results found in
Fig.~\ref{fig:nuhm2} for the NUHM2 model. It is interesting to note that
the smallest values of $\delew$ in the figure are no smaller
than for the NUHM2 model. 
The gap at small values of $m_0(3)$ is an artifact of the upper limit on
$m_{1/2}$ in the scan: for small values of $m_0(3)$ the lighter
$t$-squark is often driven to tachyonic masses by two-loop contributions
of heavy first/second generation squarks. With
larger values of $m_{1/2}$ in the scan, solutions fill in the entire gap.
Even though the GUT
scale value of $m_0(3)$ is in the multi-TeV regime, the $\tst_2$ and
especially $\tst_1$ physical masses are considerably lower -- in the few
TeV regime -- due to radiative effects from RGE running and also large
mixing.

The key advantage of the NUHM3 model is seen in Fig.~\ref{fig:nuhm3}{\it
b}), where $\delew$ versus $m_0(1,2)$ is plotted. In this case, the GUT scale first/second generation scalar masses can easily
range between $1-30$~TeV while still maintaining low $\delew$.
The solutions with $m_0(1,2)$ in the multi-TeV region will also produce
first/second generation squark and slepton masses which are comparable
to $m_0(1,2)$.  This allows for a much more robust solution to the SUSY
flavor/$CP$ problems. 
%

\begin{figure}[htbp!] 
\begin{center}
\includegraphics[width=.7\textwidth,clip]{delew_m03_nuhm2split.eps}
\includegraphics[width=.7\textwidth,clip]{delew_m012_nuhm2split.eps}
\end{center}
\caption[The value of $\delew$ versus $m_0(3)$ and $m_0(1,2)$ from
a scan over NUHM3 model with split first/second and third generations.]{The value of $\delew$ versus $m_0(3)$ and $m_0(1,2)$ from
a scan over NUHM3 model with split first/second and third generations. As in
  Fig.~\ref{fig:nuhm2}, the red pluses denote the distributions from
  the complete scan, whereas the blue crosses depict the results for
  the dedicated low $\mu$ scan. The line at $\delew=30$ is to
  guide the eye.}
\label{fig:nuhm3}
\end{figure}

Plots of $\delew$ versus sparticle masses are not shown since
these are very similar to results shown in Fig.~\ref{fig:mass1} and
Fig.~\ref{fig:mass2} except for the fact that NUHM3 scans allow for much
heavier first/second generation squark and slepton masses in the 10-30
TeV range, whereas in the NUHM2 model the squarks and sleptons are
typically constrained to be below 8~TeV due to the imposed relation
$m_0(3)=m_0(1,2)$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Rare $B$ decay constraints on RNS}
\label{ssec:b}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{BF($b\to s\gamma$)}
\label{sssec:bsg}

The combination of several measurements of the $b\to s\gamma $ decay rate finds that
$BF(b\to s\gamma )=(3.55\pm 0.26)\times 10^{-4}$~\cite{Asner:2010qj}.
This is slightly higher than the SM prediction\cite{Misiak:2006zs} of
$BF^{SM}(b\to s\gamma )=(3.15\pm 0.23)\times 10^{-4}$. SUSY contributions to the
$b\to s\gamma$ decay rate come mainly from chargino-stop loops and
the W-charged Higgs loops, and so are large when these particles are light
and when $\tan\beta$ is large\cite{Baer:1996kv}. Thus, in generic natural SUSY where
$m_{\tst_{1,2},\tb_1}\alt 600$~GeV, one finds generally large deviations from the SM value for
$BF(b\to s\gamma )$, as shown in Ref.~\cite{bbht}. In contrast, in radiative natural SUSY
where third generation squarks are in the~TeV range, SUSY contributions to
$BF(b\to s\gamma )$ are more suppressed. The situation is shown in Fig.~\ref{fig:BF}{\it a})
along with the measured central value (green solid line) and errors.
The red points all have $\delew<30$ and qualify as RNS points. The bulk of RNS points
are consistent with the measured $BF(b\to s\gamma )$, although there are outliers.

%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=.7\textwidth,clip]{DeltaEWBsgma.eps}
\includegraphics[width=.7\textwidth,clip]{DeltaEWBmumu.eps}
\end{center}
\caption[The values of $\delew$ versus {\it a}) $BF(b\to s\gamma )$
and {\it b}) $BF(B_s\to\mu^+\mu^- )$.]{The values of $\delew$ versus {\it a}) $BF(b\to s\gamma )$
and {\it b}) $BF(B_s\to\mu^+\mu^- )$. 
The vertical lines represent the experimental measurements with uncertainties.
}
\label{fig:BF}
\end{figure}


\subsection{$B_s\to \mu^+\mu^-$}

Recently, the LHCb collaboration has discovered an excess over the
background for the decay $B_s\to\mu^+\mu^-$\cite{lhcb}.  They find a
branching fraction of $BF(B_s\to\mu^+\mu^- )=3.2^{+1.5}_{-1.2}\times
10^{-9}$ in accord with the SM prediction of $(3.2\pm 0.2)\times
10^{-9}$\cite{bmm_sm}.  In supersymmetric models, this flavor-changing decay occurs
through pseudoscalar Higgs $A$ exchange\cite{Babu:1999hn}, and the
contribution to the branching fraction from SUSY is proportional to
$\frac{(\tan\beta)^6}{m_A^4}$. The value of
$BF(B_s\to\mu^+\mu^- )$ from RNS is shown in Fig.~\ref{fig:BF}{\it b}).  The
decay is most constraining at large $\tan\beta\sim 50$ as occurs in
Yukawa-unified models\cite{bkk} and low $m_A$. In the case of RNS with
lower $\tan\beta$ and heavier $m_A$, the constraint is less important.
The bulk of the RNS points in Fig.~\ref{fig:BF}{\it b}) fall well within
the newly measured error bands although there are some outlier red
points, mainly at larger values of the branching fraction.

\subsection{$(g-2)_\mu$}

In addition, the well-known $(g-2)_\mu$ anomaly has been reported as a
roughly $3\sigma$ deviation from the SM value: $\Delta a_\mu=(28.7\pm
8.0)\times 10^{-10}$\cite{gm2_ex}.  In RNS, since the
$\tmu_{1,2}$ and $\tnu_\mu$ masses are expected to be in the multi-TeV range,
only a tiny non-standard contribution to the $(g-2)_\mu$ anomaly is expected,
and alternative explanations for this anomaly would have to be sought.
%%
%--------------------------------------------------------------------
\newpage
\section{Radiative Natural Supersymmetry at the LHC \cite{lhc,lhcltr}}
\label{sec:lhc}
%--------------------------------------------------------------------
In this chapter, the prospects for CERN LHC operating
at $\sqrt{s}=14$~TeV and $30-300$~fb$^{-1}$ (or even higher) to discover
supersymmetry within the RNS context are assessed, along with a detailed treatment
of a variety of different signatures expected at LHC14 for the RNS
model. In Sec.~\ref{ssec:mline}, a RNS model line
which contains all the generic features of RNS models is constructed, but with a
variable gluino mass.  In Sec.~\ref{ssec:prod1}, one may find sparticle
production cross sections and branching fractions along the RNS model
line.  Section~\ref{ssec:gg} examines prospects for discovering gluino pair
production via signals from their cascade decays.  If a signal is found,
then the shape of the mass distribution of opposite sign, same flavor
dileptons from $\tz_2\to\tz_1\ell\bar{\ell}$ decays of neutralinos
produced via cascade decays (or directly, see Sec.~\ref{ssec:3l}),
characterizes models with light higgsinos, as emphasized in
Ref.~\cite{kn,heaya,kadala}.  In Sec.~\ref{ssec:SSdB}, aspects
of the characteristic same-sign diboson signature from SUSY models with
light higgsinos are examined. Sec.~\ref{ssec:WZ}, presents the
prospects for LHC to detect the clean trilepton signal arising from
wino pair production followed by decay to $WZ+\eslt$.
Section~\ref{ssec:4l} examines a novel $4\ell+\eslt$ signal from wino
pair production.  Section~\ref{ssec:3l} examines the possibility of
detecting directly produced higgsinos -- whose decays have a very low
energy release in the RNS framework -- in the soft trilepton channel
with low jet activity.

\subsection{Simulation of Collider Events}
\label{ssec:simulation}

The simulation of collider scattering events through computer packages
is generally broken into several steps. The first step is to calculate the hard scattering cross sections
within the framework of the parton model. The partons
are the pointlike constituents of the particles that are being
collided, such as the quarks and gluons in protons. For example, if $a$ is a
parton in hadron $A$, and the fraction of longitudinal momentum it
carries is $x_a$, then the parton distribution function (PDF) may be
denoted by $f_{a/A}(x_a)$. Then, for a hadronic collision between two
particles $A$ and $B$
\be
A\hspace{.25cm}+\hspace{.25cm}B\hspace{.25cm}\to\hspace{.25cm}c\hspace{.25cm}+\hspace{.25cm}X
\label{eq:process}
\ee
where $c$ is the produced quark or lepton and $X$ is anything, the
hard scattering ``subprocess'' is actually
\be
a\hspace{.25cm}+\hspace{.25cm}b\hspace{.25cm}\to\hspace{.25cm}c.
\ee
The cross section for this subprocess reaction may be calculated from the
Lagrangian. However, it is then necessary to convolute the subprocess
cross section $d\hat{\sigma}$ with the PDFs, and sum over all partons to
obtain the final cross section for the process.\footnote{In Chapter~\ref{sec:ilc} where production at an $e^+e^-$ collider is
discussed, electron PDFs are used to
incorporate beamstrahlung and bremstrahlung effects. However, these
have sharp peaks at the electron (or photon) pole, but may result in
energy loss of the beam. It is also necessary to take into account
polarized beams, which is one of the advantages of $e^+e^-$ over hadron colliders.} For
Eq.~\ref{eq:process} this becomes:
\be
d\sigma(AB\to cX)=\displaystyle\sum\limits_{a,b}
\int^1_0dx_a\int^1_0dx_b f_{a/A}(x_a) f_{b/B}(x_b)d\hat{\sigma}(ab\to c).
\ee

Next, parton showers must be included for both initial and final
state colored particles. These can be the result of QCD radiation
effects and are difficult to calculate exactly. Instead, they are
usually incorporated approximately via clever algorithms. Multiple
quark and gluon bremstrahlung can lead to effects such as jet broadening. In
addition, the products of the hard scattering subprocess may
also decay. These {\it cascade decays} can be very complex, and it is
possible to produce a variety of final states which depend on the
various branching fractions of the decay chain. 

Once the cascade decays have been performed and the parton showers
have commenced, the quarks and gluons will begin to hadronize due to
the strong force. Hadronization models descride how mesons and baryons
may be formed from these particles. The hadronized products can decay
into quasi-stable particles such as $\pi$ and $K$ mesons, which are
often detected. Finally, all other remnants of the beams must be taken
into account to achieve a complete picture of the physics. These can
be described as non-perturbative processes, and lead to hadronic
showers in the forward beam direction.

Simulations here are performed
mainly with Isajet~7.83 \cite{isajet}. However, some processes are
calculated using other software packages as noted in the text.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A radiative natural SUSY model line}
\label{ssec:mline}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

NUHM2 model parameter values leading to low $\delew\sim 10$ (RNS solutions) 
were found in the previous chapter. Here, those results are used to
construct a RNS model line which features a variable gluino mass, via
%
\bea
m_0 &=&5\ {\rm TeV}, \nonumber \\
m_{1/2}&:&\ {\rm variable\ between\ 0.3-2\ TeV}, \nonumber \\
A_0&=&-1.6 m_0, \nonumber \\
\tan\beta &=& 15 ,  \\
\mu &=& 150\ {\rm GeV}, \nonumber \\
m_A &=& 1\ {\rm TeV} \nonumber .
%\label{eq:param2}
\eea 
%
The variation in $m_{1/2}$ corresponds to variation in $m_{\tg}$
from about $0.9$~TeV to $\sim 5$~TeV.

Shown in Fig.~\ref{fig:delew} is the value of $\delew$ along the
RNS model line.  One finds that $\delew$ begins around 11 at
$m_{1/2}\sim 300$~GeV and increases only mildly with $m_{1/2}$, reaching
$\delew\sim 20$ for $m_{1/2}$ as high as 1000~GeV.  This
corresponds to EWFT of $\sim 9\%$ at the low end of $m_{1/2}$ and $\sim
5\%$ at around $m_{1/2}\sim 1$~TeV where $m_{\tg}\sim 2.5$~TeV. 

In Fig.~\ref{fig:mass}, the various sparticle masses from the RNS
model line versus $m_{1/2}$ are plotted. Along the model line, the value of $m_h$
varies from $124.4-125.2$~GeV, quite compatible with the recent
ATLAS/CMS Higgs resonance discovery\cite{atlas_h,cms_h}.  Also, since
$\mu$ is fixed at 150~GeV, one obtains a spectrum of higgsino-like
$\tw_1^\pm,\ \tz_1$ and $\tz_2$ states with mass $\sim
150$~GeV. However, along the model line, the mass gap
$m_{\tz_2}-m_{\tz_1}$ varies from 55.6~GeV for very low $m_{1/2}$ to just
under $\sim 10$~GeV if $m_{1/2}$ nears the values allowed
by $\delew\lesssim 30$, as shown in Fig.~\ref{fig:gap1}.  
The behaviour of light chargino/neutralino masses
is understood since for low $m_{1/2}$ the weak scale gaugino mass
$M_1\simeq 136$~GeV and so the $\tz_1$ state is really a bino-higgsino
admixture, while at $m_{1/2}\sim 1$ TeV then $M_1\simeq 444$~GeV so that
$\tz_1$ is more nearly a pure higgsino state.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{model_line_smooth.eps}
\end{center}
\caption{Plot of $\delew$ versus $m_{1/2}$ along the RNS model line.}
\label{fig:delew}
\end{figure}
%
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{spectrum2.eps}
\end{center}
\caption{Various sparticle masses versus $m_{1/2}$ for the RNS
model line.
}
\label{fig:mass}
\end{figure}
%

After the higgsinos, the next lightest sparticles are the bino-like
$\tz_3$ -- whose mass varies between $160-900$~GeV -- and the wino-like $\tw_2^\pm$
and $\tz_4$ states -- whose masses vary between $300-1700$~GeV -- for the range of $m_{1/2}$ shown in the figure.
The solid red curve denoting $m_{\tg}$ varies between $900-4500$~GeV.
% though of
%course the gluinos can be significantly heavier for yet larger values
%of $m_{1/2}$. 
The red-dashed $m_{\tst_1}$ contour varies between
$1360-2500$~GeV over the $m_{1/2}$ range shown in the figure; the line crosses the
$m_{\tg}$ curve at $m_{1/2}\sim 520$~GeV.  The first/second generation
squarks and sleptons inhabit the multi-TeV range, and are far beyond the
reach of LHC14.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{z12diff_model.eps}
\end{center}
\caption{The $m_{\tz_2}-m_{\tz_1}$ mass gap versus $m_{1/2}$ along the RNS
model line.
}
\label{fig:gap1}
\end{figure}
%

%%
\subsection{Sparticle production at LHC}
\label{ssec:prod1}

In Fig.~\ref{fig:xsec}, various sparticle pair production cross sections
at LHC for {\it a}) $\sqrt{s}=8$~TeV and {\it b}) $\sqrt{s}=14$~TeV versus $m_{1/2}$
along the RNS model line are shown. 
Prospino\cite{prospino} is used to generate the cross sections at NLO in QCD.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{xsec8}
\includegraphics[width=\textwidth,clip]{xsec_14_2TeV}
\end{center}
\caption{Plot of various NLO sparticle pair production cross sections 
versus $m_{1/2}$ along the RNS model line for $pp$ collisions at
{\it a}) $\sqrt{s}=8$~TeV and {\it b}) $\sqrt{s}=14$~TeV.
}
\label{fig:xsec}
\end{figure}
%

Figure~\ref{fig:xsec} shows that the four higgsino pair production
reactions -- $pp\to \tw_1^\pm\tz_1$, $\tw_1^\pm\tz_2$, $\tw_1^+\tw_1^-$
and $\tz_1\tz_2$ -- all occur at comparable rates of $\sim 10^3$~fb at
LHC8 and of $\sim (2-4)\times 10^3$~fb at LHC14. These cross sections
are nearly flat with increasing $m_{1/2}$ since they mainly depend on
$\mu$ which is fixed at 150~GeV along the model line.

The gluino pair production cross section -- $\sigma (pp\to\tg\tg X)$ -- is
denoted by the red curve with pluses (note that
$m_{\tg}\simeq 2.5 m_{1/2}$).  While the $\tg\tg$ production cross
section is large at $m_{1/2}\sim 300$~GeV (corresponding to $m_{\tg}\sim 900$~GeV), it
drops off rapidly with increasing values of $m_{1/2}$: it is likely to be inconsequential 
for even LHC14 searches for the upper range of $m_{1/2}\gtrsim 1$~TeV unless extremely high integrated 
luminosities are attained. 

Also of importance are the gaugino pair production reactions: wino pair
production $pp\to\tw_2^\pm\tz_4$ and $\tw_2^+\tw_2^-$, and also
$\tw_1^\pm\tz_3$ which proceeds via the higgsino component of
the bino-like $\tz_3$.  Wino pair production can be large due to the
large $SU(2)$ triplet gauge coupling. The cross section for this drops
off much less sharply than that for $\tg\tg$ production since the
wino masses are much smaller than the gluino mass. The cross section for
$\tw_1\tz_3$ production falls off faster than the wino production 
cross section because the higgsino content of $\tz_3$ drops off with
increasing $m_{1/2}$. These reactions constitute the largest observable SUSY cross
sections over most of the range of $m_{1/2}$.

For comparison, also shown are cross sections for the pair production of
top squarks, the lightest sfermions in RNS.  The tiny $\tst_1\bar{\tst}_1$ 
production cross section at $\sqrt{s}=8$ TeV precludes any 
possibility of stop detection at LHC8. Detection of top-squark pairs 
at LHC14-- which occurs at a lower rate than gluino production unless
$m_{1/2} > 1$~TeV-- may be possible if the stop signal can be sorted
from gluino pair production; detection in this case will likely 
require several hundred fb$^{-1}$ of integrated luminosity.
%
%We see that stop pair production occurs at 
%lower rates than $\tg\tg$ production: it is impossible to observe at LHC8 
%and will likely require $\gtrsim 300\ {\rm fb}^{-1}$ 
%of integrated luminosity at LHC14.

In Fig.~\ref{fig:xsecmu}, selected electroweak-ino cross sections
versus $\mu$ for $m_{1/2}=750$~GeV along the RNS model line are shown. Here, $\tw_1\tz_2$, $\tz_1\tz_2$ and $\tw_1^+\tw_1^-$ production are all
comparable and as high as $\sim 10^4$~fb at $\mu\sim 100$~GeV. They drop
to the vicinity of $\sim 10^2$~fb at $\mu\sim 300$~GeV. 
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{xsec_14.eps}
\end{center}
\caption{Plot of various NLO electroweak-ino pair production cross sections 
versus $\mu$ for the RNS model line with $m_{1/2}=750$~GeV 
for $pp$ collisions at 14~TeV.
}
\label{fig:xsecmu}
\end{figure}
%


\subsection{Sparticle branching fractions}

Shown in Fig.~\ref{fig:bf} are various sparticle branching fractions
for the
sparticles most accessible at the LHC, {\it i.e.}
{\it a}) $\tg$, {\it b}) $\tst_1$, {\it c}) $\tz_2$, {\it d}) $\tz_3$, {\it e}) $\tz_4$,
and {\it f}) $\tw_2$.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=7cm,clip]{br_glu.eps}
\includegraphics[width=7cm,clip]{br_t1.eps}
\includegraphics[width=7cm,clip]{br_z2.eps}
\includegraphics[width=7cm,clip]{br_z3.eps}
\includegraphics[width=7cm,clip]{br_z4.eps}
\includegraphics[width=7cm,clip]{br_w2.eps}
\end{center}
\caption{Plot of various sparticle branching fractions versus $m_{1/2}$ along the RNS model line.
}
\label{fig:bf}
\end{figure}
%
From frame {\it a}), it is apparent that for the lower portion of $m_{1/2}$
corresponding to $m_{\tg}\lesssim 1.8$~TeV, the gluino decays via 3-body
modes into $tb\tw_1$ and $t\bar{t}\tz_{1,2,3}$ states. For heavier
$m_{\tg}\gtrsim 1.8$~TeV, the 2-body modes $\tg\to t\tst_1$ open up
and dominate the decays. Thus, one expects the gluino pair production
events to be rich in $b$-jet activity\cite{btw1,btw2}. In the case where
$\tg\to t\tst_1$, it is important to know how $\tst_1$ decays. This is
shown in frame {\it b}). For the very lowest $m_{1/2}$ values, 
the $\tst_1\to t\tg$ decay mode is open and is dominant. However, as $m_{1/2}$
increases, this mode quickly closes and instead $\tst_1$ decays into
$b\tw_1$ or $t\tz_{1,2,3}$.

In frame {\it c}), the $\tz_2$ decay modes are shown. Since the
$m_{\tz_2}-m_{\tz_1}$ mass gap ranges from $\sim 55$~GeV 
(already excluded for this model line by LHC8 gluino searches) 
to $\sim 10$~GeV along the model line, then $\tz_2$ always decays dominantly to
3-body modes $\to\tw_1 f\bar{f}'$ or $\to\tz_1 f\bar{f}$, where $f$
stands for kinematically accessible SM fermions. As mentioned earlier,
since the $\tz_2-\tz_1$ mass gap is small and the released decay energy
is shared between three particles, then the decay products from $\tz_2$ decay
are usually very soft -- in the few~GeV range. 
The light chargino (branching fractions not shown) decays into $\tz_1
f\bar{f}'$ mainly via $W^*$, where the $f$ and $\bar{f}'$ are again
typically rather soft.

In frame {\it d}) the bino-like $\tz_3$ decays are shown. Here $\tz_3\to
\tw_1 f\bar{f}'$ or $\tz_{1,2}f\bar{f}$ for $m_{1/2}\lesssim
500$~GeV. For heavier $m_{\tz_3}\gtrsim 220$~GeV (this value
depends on the choice of $\mu$), the 2-body decays
$\tz_3\to \tw_1^\pm W^\mp$ and $\tz_{1,2}Z$ and $\tz_1 h$ turn on,
leading to production of vector bosons and Higgs bosons in the SUSY
events. 

Frames {\it e}) and {\it f}) show the neutral $\tz_4$ and charged
$\tw_2^\pm$ wino branching fractions. One sees that
$\tz_4\to \tw_1^\pm W^\mp$ mode dominates over the entire range of
$m_{1/2}$. The subdominant decay modes $\tz_4\to \tz_{1,2}Z$ and
$\tz_{1,2} h$ can also be important and occur at significant rates.  
The sizeable branching ratio for the decay $\tz_4 \to Z \tz_1$
may be surprising at first glance since $\tz_4$ is dominantly a wino
while $\tz_1$ is mostly a higgsino, so
that the $Z\tz_4\tz_1$ coupling should be suppressed by the small
higgsino content $\sim M_Z/M_2$ (assuming $M_2\gg |\mu|$) of $\tz_4$. For
heavy $\tz_4$, this suppression is compensated for by the fact that the
amplitude for decay to the longitudinally polarized $Z$ boson is
enhanced by $\sim |\mu|/M_Z$. As a result, for
$M_2 \gg M_Z, |\mu|$, the branching fractions for decays to $Z$ and to $h$
become comparable. This is discussed in
detail in Ref.~\cite{han}. In the
case of $\tw_2$ decay shown in frame {\it f}), one sees that $\tw_2\to
\tw_1 Z$ or $\tz_{1,2} W$ or $\tw_1 h$ over the entire range of
$m_{1/2}$, leading again to production of gauge and Higgs bosons in
wino pair production events.  The dominant sparticle branching
fractions for $m_{1/2}=1$~TeV along the RNS model line are shown in
Table~\ref{tab:BF}.

%
\begin{table}
\begin{center}
\begin{tabular}{|l|r|r|}
\hline
Particle & dom. mode  &  BF \\
\hline
\hline
$\tg$    & $\tst_1 t$      &  $\sim 100\%$ \\
$\tst_1$ & $b\tw_1$        &  $\sim 50\%$ \\
$\tz_2$  & $\tz_1 f\bar{f}$ &  $\sim 100\%$ \\
$\tz_3$  & $\tw_1^\pm W^\mp$ &  $\sim 50\%$ \\
$\tz_4$  & $\tw_1^\pm W^\mp$ &  $\sim 50\%$ \\
$\tw_1$  & $\tz_1 f\bar{f}'$ &  $\sim 100\%$ \\
$\tw_2$  & $\tz_i W$       &  $\sim 50\%$ \\
\hline
\end{tabular}
\caption{Dominant branching fractions of various sparticles 
along the RNS model line for $m_{1/2}=1$~TeV.}
\label{tab:BF}
\end{center}
\end{table}
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gluino cascade decay signatures}
\label{ssec:gg}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

First, the $pp\to\tg\tg X$ reaction followed by gluino cascade
decays\cite{cascade} are examined.
This can be searched for in multi-lepton plus multi-jet $+ \eslt$ events. Squark pair production and gluino-squark associated
production which occur at very low rates are neglected because squarks are heavy.

Isajet~7.83~\cite{isajet} is used for the generation of signal events at LHC14.
For event generation, a toy detector simulation is used with calorimeter cell size
$\Delta\eta\times\Delta\phi=0.05\times 0.05$ and $-5<\eta<5$. The HCAL
(hadronic calorimetry) energy resolution is taken to be
$80\%/\sqrt{E}+3\%$ for $|\eta|<2.6$ and FCAL (forward calorimetry) is
$100\%/\sqrt{E}+5\%$ for $|\eta|>2.6$, where the two terms are combined
in quadrature. The ECAL (electromagnetic calorimetry) energy resolution
is assumed to be $3\%/\sqrt{E}+0.5\%$. The cone-type Isajet
 jet-finding algorithm~\cite{isajet} is used to group the hadronic
final states into jets. Jets and isolated leptons are defined as follows:
%
\bi
\item Jets are hadronic clusters with $|\eta| < 3.0$,
$R\equiv\sqrt{\Delta\eta^2+\Delta\phi^2}\leq0.4$ and $E_T(jet)>50$~GeV.
\item Electrons and muons are considered isolated if they have $|\eta| <                          
2.5$, $p_T(l)>10 $~GeV with visible activity within a cone of $\Delta                             
R<0.2$ about the lepton direction, $\Sigma E_T^{cells}<5$~GeV.
\item  Hadronic clusters are identified as
$b$-jets if they contain a B hadron with $E_T(B)>$ 15~GeV, $\eta(B)<$ 3 and
$\Delta R(B,jet)<$ 0.5. A tagging efficiency of 60$\%$ is assumed, and
light quark and gluon jets can be mis-tagged
as a $b$-jet with a probability 1/150 for $E_{T} \leq$ 100~GeV,
1/50 for $E_{T} \geq$ 250~GeV, with a linear interpolation
for intermediate $E_{T}$ values.
\ei

Gluino pair production cascade decay signatures have been previously
calculated and compared against backgrounds in Ref.~\cite{heaya}.  In that
paper, it was advocated that in models where gluino pair production
signatures are dominant above background (such as the focus point region
of mSUGRA), if one can suppress the background entirely, then the
remaining total cross section may be used to extract the gluino mass to
10-15\% precision.  The cuts from that paper are adopted and compared RNS signal rates
along the model line against previously calculated backgrounds using the
exact same set of cuts.

In Ref.~\cite{heaya}, the following pre-cuts set {\it C1} are first invoked:
%
\\
\textbf{C1 Cuts:}
\bea
\eslt & >& \max (100\ {\rm GeV},0.2 M_{eff}),\nonumber \\
n(jets) &\ge & 4, \nonumber\\
E_T(j_1,j_2,j_3,j_4)& > & 100,\ 50,50,50\ {\rm GeV}, \label{c1cutsend}\\
S_T &>&0.2,\nonumber \\
p_T(\ell)& >& 20 \ {\rm GeV}.\nonumber
\eea
%
Here, $M_{eff}$ is defined as in Hinchliffe {\it et al.}\cite{frank} as
$M_{eff}=\eslt +E_T(j_1)+E_T(j_2)+E_T(j_3)+E_T(j_4)$, where $j_1-j_4$
refer to the four highest $E_T$ jets ordered from highest to lowest
$E_T$, $\eslt$ is missing transverse energy and $S_T$ is transverse
sphericity.  The SM cross sections in fb after C1 cuts are listed in
Table~III of Ref.~\cite{heaya}.  It is found that the signal with these
cuts is swamped by various SM backgrounds (BG), especially those from
QCD multi-jet production and $t\bar{t}$ production.  After inspection of
a variety of distributions including jet multiplicity $n(jets)$, $b$-jet
multiplicity $n(b-jets)$ and augmented effective mass $A_T$ (here,
$A_T=\eslt +\sum_{leptons}E_T +\sum_{jets}E_T$), for $0\ell$ and $1\ell$
events, the {\it C1} cuts are amended to \\ \textbf{C2 Cuts:} \beas & apply\
cuts\ set\ C1 & \\ & n(jets)\ge 7 & \\ & n(b-jets)\ge 2 & \\ & A_T\ge
1400\ {\rm GeV}.  \eeas For multi-lepton events (opposite sign dileptons
$OS$, same sign dileptons $SS$ and trileptons $3\ell$), somewhat
softer cuts are used: \\ \textbf{C3 Cuts:} \beas & apply\ cuts\ set\ C1 & \\ &
n(isol.\ leptons)\ge 2 & \\ & n(jets)\ge 4 & \\ & n(b-jets)\ge 2 & \\ &
A_T\ge 1200\ {\rm GeV} .  \eeas

After {\it C2} cuts, it is found that 1~fb of BG remains in the $1\ell +jets$ channel and 
0.5~fb of BG remains in the $0\ell +jets$ channel. No BG was found in the 
$OS+jets$, $SS+jets$ or $3\ell +jets$ channels after cuts {\it C3}.
The signal rates along the RNS model line are shown in
Fig.~\ref{fig:multi}.
From the plot, one can read off the $5\sigma$ discovery level for
various integrated luminosity choices for different signal channels.
For the $0\ell +jets$ channel with $300$~fb$^{-1}$ a reach to
$m_{1/2}\sim 650$~GeV is expected, corresponding to $m_{\tg}\sim 1.7$~TeV.The reach in the lower background multilepton channels is not projected
as these would depend on the residual background that remains. 

%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{reach_mu150_gg.eps}
\end{center}
\caption[Plot of gluino cross section in $fb$ after cuts {\it C2} for
$1\ell+jets$ and $0\ell+jets$ channel and cuts {\it C3} for $OS$, $SS$
and $3\ell+jets$ channels from gluino cascade decays along the RNS model
line at LHC14.]{Plot of gluino cross section in $fb$ after cuts {\it C2} for
$1\ell+jets$ and $0\ell+jets$ channel and cuts {\it C3} for $OS$, $SS$
and $3\ell+jets$ channels from gluino cascade decays along the RNS model
line at LHC14. The horizontal lines denote the corresponding backgrounds
estimated in Ref.~\cite{heaya}. }
\label{fig:multi}
\end{figure}
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{OS/SF dilepton mass distribution from cascade decays}
\label{sssec:mll}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Within the OS dileptons plus jets channel, one expects a large fraction of signal
events to contain an OS dilepton pair arising from
$\tz_2\to\ell^+\ell^-\tz_1$ decay. For these events, the $m(\ell^+\ell^-
)$ distributions will be bounded by the kinematic mass difference
$m_{\tz_2}-m_{\tz_1} < M_Z$.  In Fig.~\ref{fig:mll}, the invariant
mass of opposite-sign/same-flavor dilepton pairs from the $OS+jets$
events which survive cuts {\it C3} is shown. In the figure,
$m_{1/2}=450$~GeV for which $m_{\tg}=1250$~GeV and
$m_{\tz_2}-m_{\tz_1}=32$~GeV.  A mass edge at 32~GeV is clearly visible
from the plot, as is the $Z$ peak. A detection of an excess of events
with a cut-off on the dilepton mass could readily be attributed to
neutralinos of SUSY.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{mll_C3_lhc14.eps}
\end{center}
\caption{Distribution of the invariant mass of opposite-sign/same-flavor
dileptons after cuts {\it C3} at LHC14 from the RNS benchmark model line
with $m_{1/2}=450$~GeV.}
\label{fig:mll}
\end{figure}
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Same-sign diboson signature}
\label{ssec:SSdB}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Presented in this section are details on the same-sign
diboson (SSdB) signature. In models where $|\mu|$ is
smaller than the magnitude of gaugino mass parameters -- as exemplified
by the RNS model line -- wino pair production provides a
novel signature with a final state characterized by two same sign $W$
bosons and $\eslt$ but accompanied by just modest jet activity. 
The most promising reaction appears to be $pp\to \tw_2^\pm\tz_4$, where $\tw_2^\pm\to
W^\pm\tz_{1,2}$ and $\tz_4\to W^\pm\tw_1^\mp$ although $\tw_2^+\tw_2^-$ pair
production also provides a non-negligible signal contribution. 
It is seen in Fig.~\ref{fig:bf} that the winos have substantial branching 
fractions for decays to $W$ bosons.
For these decays, half the time the final states consist of $W^\pm W^\pm +\eslt$. 
The analysis here is focused on the SS dilepton signal from the leptonic decays of both $W$s.
The jet activity in these events is relatively limited  since 
the daughter higgsinos $\tw_1$ and $\tz_{1,2}$ usually yield only soft decay products. 
This serves to distinguish the wino-pair induced SSdB signature from the
SS dilepton signal from gluino pair production -- the latter is expected to be accompanied by
several hard jets. 

The SM physics backgrounds to the SSdB signal come from $uu\to            
W^+W^+ dd$ or $dd\to W^-W^- uu$ production with a cross section $\sim 350$~fb. 
These events will be characterized by high rapidity
(forward) jets and rather low $\eslt$. $W^\pm W^\pm$ pairs may also occur
via two overlapping events: such events will mainly have low $p_T$ $W$s
and possibly distinct production
vertices. Double parton scattering will also lead to SSdB events at a
rate somewhat lower than the $qq \to W^\pm W^\pm q'q'$ process\cite{stirling}. 
Additional physics backgrounds come from
$t\bar{t}$ production where a lepton from a daughter $b$ is
accidentally not isolated, from $t\bar{t}W$ production and from $4t$ production. 
SM processes such as $WZ\to 3\ell$ and $t\bar{t}Z\to 3\ell$ production,
where one lepton is missed, constitute reducible backgrounds to the
signal.

Here, it is assumed that the $2\to 4$ processes as well as the double parton
scattering processes, which have different characteristics from the
signal, can be readily eliminated by suitable cuts and do not simulate
these.   AlpGen~\cite{alpgen} is used for the simulation of the remaining background events
and MadGraph~5~\cite{madgraph} is used to generate the hard
scattering events. Those events are then passed to Pythia~6.4~\cite{pythia}
via the LHE interface~\cite{lhe} for showering and hadronization.  For 
the $2\to 4$ ``$WZ$''
process, the full matrix element for $pp\rightarrow l^+ l^-
l' \nu'$ that includes contributions from on- and off-shell $Z$
and $\gamma$ as well as from interference diagrams is computed.  Signal and background are normalized to NLO cross sections obtained with
Prospino~\cite{prospino} and MCFM~\cite{mcfm}, respectively. The
procedure to reconstruct jets and isolated leptons that was followed is described in
Sec.~\ref{ssec:gg}.

The following cuts are imposed:
%                                                                              
\bi
\item {\it exactly} 2 isolated same-sign leptons with 
$p_T(\ell_1 )>20$~GeV and $p_T(\ell_2 )>10$~GeV,
\item $n(b-jets)=0$ (to aid in vetoing $t\bar{t}$ background).
\ei
After these cuts, the event rate is dominated by $WZ$ and $t\bar{t}$
backgrounds.  

To distinguish signal from background, the transverse mass of
each lepton is constructed with $\eslt$: 
%
\be 
\mtmin\equiv \min\left[
m_T(\ell_1,\eslt),m_T(\ell_2 ,\eslt )\right] \nonumber \, .  
\ee 
The signal gives rise to a continuum distribution, while the dominant
backgrounds have a kinematic cut-off around $\mtmin \simeq M_W$ (as long
as the $\eslt$ dominantly arises from the leptonic decay of a single
$W$).  The situation is seen in Fig.~\ref{fig:ssdist}, where in
{\it a}) the $\mtmin$ distribution is shown, while in {\it b}) the
$\eslt$ distribution is shown.  The bulk of $t\bar{t}$ and $WZ$ backgrounds can
be eliminated by requiring
%{\it e.g.} 
\bi
\item $\mtmin >125$~GeV and
\item $\eslt >200$~GeV.  
\ei 
%
After these cuts, no background events were generated from
$t\bar{t}$ and $WZ$ production, where the 1-event level in the
simulation was 0.05~fb and 0.023~fb, respectively.  The dominant SM
background for large $\mtmin$ then comes from $Wt\bar{t}$ production for
which (including a QCD $k$-factor $k=1.18$ obtained from
Ref.~\cite{garzelli}) a cross section of $0.019$ ($0.006$)~fb is found after the
cuts $\mtmin > 125$~(175)~GeV and $\eslt>200$~GeV; the harder cuts serve
to optimize the signal reach for high $m_{1/2}$ values.
%\footnote{We have
%ignored detector-dependent backgrounds from jet-lepton misidentification
%in our analysis, but are optimistic that these can be controlled by the
%$\mtmin$ and $\eslt$ cuts. Estimates of the background from charge
%mis-identification, which could be important, especially for electrons, are
%also beyond the scope of this analysis.}
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=12.5cm,clip]{mT_2l0b_lhc14.eps}
\includegraphics[width=12.5cm,clip]{MET_2l0b_lhc14.eps}
\end{center}
\caption[Transverse mass and missing energy distributions for SSdB 
events after cuts at LHC14.]{Transverse mass and missing energy distributions for SSdB 
events after cuts at LHC14.  The open black and red histograms
represent the signal from winos -- via $\tw_2\tz_4$ and $\tw_2^+\tw_2^-$
pair production -- for the RNS model-line points with $m_{1/2}=400$~GeV and 700~GeV, respectively.}
\label{fig:ssdist}
\end{figure}
%

The calculated signal rates after cuts along the RNS model line from
just $\tw_2^\pm\tz_4$ and $\tw_2^\pm\tw_2^\mp$ production are shown
vs. $m_{1/2}$ in Fig.~\ref{fig:ss} where the upper (blue) curves require
$\mtmin>125$~GeV and the lower (orange) curve requires $\mtmin>175$~GeV.
The $\tw_2\tz_4$ and $\tw_2\tw_2$ cross sections are normalized to those
from Prospino\cite{prospino}.  For observability with an assumed value
of integrated luminosity, it is required: 1)~significance $> 5\sigma$,
2)~Signal/BG$>0.2$ and 3)~at least 5 signal events.  The LHC signal
(blue dashed curve) and reach lines for integrated luminosity values 25
and 100~fb$^{-1}$ with a soft $\eslt >75$~GeV cut are shown first. The
25~fb$^{-1}$ reach is to $m_{1/2}\simeq 450$~GeV corresponding to
gluinos of $\sim 1300$~GeV. As greater integrated luminosity is
accumulated, harder cuts can be applied. The solid blue line shows signal
for $\eslt >200$~GeV and reach for 100, 300 and 1000~fb$^{-1}$.  With
harder cuts, the 100~fb$^{-1}$ reach extends to $m_{1/2}\simeq 680$~GeV
corresponding to $m_{\tg}\sim 1.75$~TeV in a model with gaugino mass unification. 
The direct search for $\tg\tg$ gives a projected reach of
$m_{\tg}\sim 1.6$~TeV as seen in Sec.~\ref{ssec:gg}; see also
Ref.~\cite{bblt2012}. Thus, with ${\cal O}(100)$~fb$^{-1}$ of integrated
luminosity, the SS diboson
signal offers a comparable reach to that for gluino cascade decays.  For
300~(1000)~fb$^{-1}$ of integrated luminosity, the reach is improved
with a harder $\mtmin >175$~GeV cut. In this case, the LHC14
reach for SS dibosons extends to $m_{1/2}\sim 840$~(1000)~GeV,
corresponding to $m_{\tg}$ of 2.1 and 2.4~TeV.  For the RNS model-line
where gaugino mass unification is assumed, these reach numbers extend
well beyond the LHC14 reach for direct gluino pair
production\cite{bblt1014}. Regardless of this, the
SSdB signal is a new independent signal, and detection of
signals in multiple channels will be essential to unravel the underlying
origin of any new physics that is found. 
% 
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{reach_mu150.eps}
\end{center}
\caption[Same-sign dilepton cross sections (in {\it fb}) at LHC14 after cuts vs. $m_{1/2}$
along the RNS model line from $\tw_2^\pm\tz_4$ and
$\tw_2^\pm\tw_2^\mp$ production and
calculated reach for 100, 300 and 1000~fb$^{-1}$.]{Same-sign dilepton cross sections (in {\it fb}) at LHC14 after cuts vs. $m_{1/2}$
along the RNS model line from $\tw_2^\pm\tz_4$ and
$\tw_2^\pm\tw_2^\mp$ production and
calculated reach for 100, 300 and 1000~fb$^{-1}$.  The upper solid and
dashed (blue) curves requires $\mtmin >125$~GeV while the lower solid
(orange) curve requires $\mtmin >175$~GeV. The signal is observable
above the horizontal lines.
}
\label{fig:ss}
\end{figure}
%

The low jet activity associated with the SSdB
signal from SUSY models with light higgsinos makes it quite distinct
from the usual SS dilepton signal arising from gluino pair production,
which is usually accompanied by numerous hard jets and high $\eslt$.
Recent CMS searches for SS dileptons from SUSY\cite{cms_ss} required the
presence of multiple jets (some $b$-tagged jets) or large $H_T$ in the
events; these cuts greatly reduce or even eliminate the SSdB
signal. Likewise, the cuts $n_j\ge 3$ high $p_T$ jets (possibly
$b$-tagged) along with $\eslt>150$~GeV and large $m_{\rm eff}$ required
by a recent ATLAS search for SS dileptons from gluinos\cite{atlas_ss}
would have eliminated much of the SSdB signal from SUSY with light
higgsinos.

Hard trilepton production from winos (discussed in the next section) can
lead to clean, same-sign dilepton events if a lepton is not isolated or
fails to be identified. The CMS collaboration used this channel to
extend the search for electroweak-inos to portions of parameter space
not accessible via the trilepton search, requiring 120~GeV $< \eslt <
200$~GeV \cite{cmsdilep}.  They do not, however, impose the $m_T^{\rm
min}$ cut that was crucial for the SSdB analysis. The CMS search is
thus not optimized for the clean SS dilepton signal in the RNS
scenario. In any case, with just $\sim 20$~fb$^{-1}$ at LHC8, this
channel should have a lower reach than that via multi-jet plus
multi-lepton events from gluino pair production.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hard trileptons from wino pair production}
\label{ssec:WZ}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, prospects for detection of reactions such as
\be 
pp\to\tw_2\tz_4\to (\tw_1 Z)+(\tw_1 W)\to WZ+\eslt\to
\ell^+\ell^-\ell^\prime+\eslt. \nonumber\ 
\ee 
are examined. The trilepton channel
where the neutralino decays via the three-body decay $\tz_2\to
\ell^+\ell^-\tz_1$ because the two-body decay $\tz_2\to Z\tz_1$ is
kinematically forbidden (so that SM trileptons from $WZ$ production can
be eliminated via a mass cut on the opposite-sign, same flavor dilepton
pair) has long been regarded as a golden channel in the search for
gauginos from supersymmetry~\cite{trilep}.
More recently, it has been pointed out~\cite{wz} that at least within
mSUGRA the trilepton search for gauginos is viable even when the
neutralinos decay to on-shell $Z$ bosons. Indeed, the CMS and ATLAS
experiments have searched in this channel and found that there is no
excess above SM expectations~\cite{lhctrilepZ}.  For a recent assessment
of multilepton signals, see Ref.~\cite{han}. 
Here, prospects for this signal for the RNS model line are analysed,
for the most part 
following the cuts of Ref.~\cite{wz} which required:
%
\bi
\item[] \underline{Pre-Selection Cuts:}
\item $n(b-jets)=0$ (to aid in vetoing $t\bar{t}$ background),
\item 3 isolated leptons with $p_T(\ell )>20$~GeV and
\item $|m(\ell^{+}\ell^{-}) - M_Z| < 10$~GeV (leptonic $Z$), 
\ei 
%
where two of the leptons in the event must form an OS/SF pair. If more
than one OS/SF pairing is possible, the pair which minimizes
$|m(\ell^{+}\ell^{-}) - M_Z|$ is chosen.  The remaining lepton is
labeled $\ell^{'}$.  In the case of the RNS model line, the $WZ+\eslt$ signal
also receives a smaller, though non-negligible contribution, from
$\tw_2^+\tw_2^-$ where one of the winos decays via $\tw_2\to W\tz_{1,2}$ and
the other via $\tw_2\to Z\tw_1$ mode. 
At this point, a large background from the $2\to 4$
process $pp\to (\ell^+\ell^-)+(\ell^{\pm\prime}\nu_{\ell^\prime})$ which
occurs via various on- and off-shell processes -- including $W^*Z^*$ and
$W^*\gamma^*$ production -- tends to dominate the signal. Here, the $2\to 4$ process is evaluated using MadGraph with no restriction on
the invariant mass around the $Z$ and $W$ resonances.
%\footnote{The $WZ$
%background after all cuts in Table~1 of Ref.~\cite{wz} has been
%underestimated by a factor of about 2.5 because the virtual $W$-mass
%did not extend to large enough values, leading to an under-estimate of
%the tail of the $m_T$ distribution.} 
For $t\bar{t}$, $Z(ll)+jets$,
$W(l\nu)+jets$, $Z(ll)+t\bar{t}$ and $W(l\nu)+t\bar{t}$ (all summed over 3
lepton flavors) at least two additional partons in the
final state are allowed, and the MLM matching scheme~\cite{mlm} is
used to avoid double
counting.  Also included are $ZZ$, $W(l\nu)+tb$ and $Z(ll)+b\bar{b}$
backgrounds.  The signal and background distributions in
$m_{T}(\ell^{'},\eslt )$ and $\eslt$ are shown in Fig.~\ref{fig:mT}.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=12.5cm,clip]{mT_0b3l_lhc14.eps}
\includegraphics[width=12.5cm,clip]{MET_0b3l_lhc14.eps}
\end{center}
\caption[Transverse mass and missing energy distributions for hard trilepton
  events after the preliminary cuts at LHC14.]{Transverse mass and missing energy distributions for hard trilepton
  events after the preliminary cuts at LHC14.  The open red histograms
  represent the signal from winos, $\tw_2$ and $\tz_4$, for the RNS
  point with $m_{1/2}=350$~GeV.  }
\label{fig:mT}
\end{figure}
%

To enhance the signal relative to background, it is required that,
%
\bi
\item $m_{T}(\ell^{'},\eslt ) > 125$~GeV,
\item $\eslt >150$~GeV.
\ei
%
The $m_T$ cut is as in Ref.~\cite{wz}, but for the larger integrated
luminosity and concomitantly higher wino masses that are considered
here, one finds that stiffening the $\eslt$ cut yields a better
signal-to-background ratio.  The background from various SM sources
along with RNS signal for $m_{1/2}=350$~GeV is shown after cuts in
Table~\ref{tab:WZ} for LHC14.
%

\begin{table}
\small
\begin{center}
\begin{tabular}{|l|r|r|r|r|r|r|r|}
\hline
& $t\bar{t}$ & $WZ$ & $ZZ$ & $Z+t\bar{t}$ &  $W+t\bar{t}$ & Total BG & Signal\\
\hline
Events Generated & 12M & 1.5M & 1M & 1.2M & 10M & & 200K \\
\hline
$n(b) = 0, n(l) = 3$  & 6.96 & 211.94 & 26.07 & 4.26 & 1.84 & 247.29 & 2.88 \\
OS/SF pair            & 5.25 & 211.51 & 26.02 & 4.21 & 1.37 & 251.97 & 2.57 \\
$m(\ell^{+}\ell^{-})$ cut & 0.95 & 186.90 & 25.55 & 3.99 & 0.24 & 221.20 & 1.52 \\
$m_{T} > 125$~GeV   & 0.03 & 1.64 & 0.05  & 0.20 & 0.07 & 1.99 & 0.43 \\
$\eslt > 150$~GeV    & 0.006 & 0.24 & $< 0.00085$ & 0.0058 & 0.016 & 0.32 & 0.22 \\
\hline
\end{tabular}
\normalsize
\caption[Number of events generated and
cross section after cuts for the dominant backgrounds in the hard trilepton
channel and for the RNS signal with $m_{1/2}=350$~GeV.]{Number of events generated and
cross section after cuts for the dominant backgrounds in the hard trilepton
channel and for the RNS signal with $m_{1/2}=350$~GeV. 
All cross sections are in {\it fb}. 
The total BG values include all processes listed in the text, including the
subdominant ones not shown in the Table.}
\label{tab:WZ}
\end{center}
\end{table}
%
\normalsize
Shown in Fig.~\ref{fig:WZ} is the $3\ell +\eslt$ signal cross section after
all cuts versus $m_{1/2}$ along the RNS model line. The turn-over at
the left end of the curve is because of the efficiency loss resulting 
from the stiff $\eslt$ cut which is optimized to yield the best
reach for high wino masses. For 100~fb$^{-1}$ of
integrated luminosity, there is no reach, while the reach in $m_{1/2}$
is shown for 300 and 1000~fb$^{-1}$. The 300~fb$^{-1}$ reach extends to
$m_{1/2}=500$~GeV while the 1000~fb$^{-1}$ reach extends to
$m_{1/2}=630$~GeV. 
These values correspond to gluino masses of
$m_{\tg}=1.3$~TeV and 1.65~TeV, respectively. These reaches are smaller
than those obtained from the $\tg\tg$ and SSdB signals. They
would, nevertheless, offer corroborative evidence for any SUSY discovery at the
lower range of allowed $m_{1/2}$ values.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{reach_mu150_3l}
\end{center}
\caption{Tri-lepton cross sections (in {\it fb}) at LHC14 after cuts
vs. $m_{1/2}$ along the RNS model line from wino pair production processes
$pp\to\tw_2\tz_4 / \tw_2\tw_2 \to WZ+\eslt\to 3\ell+\eslt$ events.}
\label{fig:WZ}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Four leptons from heavy gaugino production} 
\label{ssec:4l}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In Fig.~\ref{fig:bf} it was seen that the wino-like $\tw_2$ and $\tz_4$ have
significant branching fractions to $W$ and $Z$ bosons resulting in the
dilepton and trilepton signals already discussed. A small fraction of
the time, there may be two $Z$ bosons in these events, leading to the
possibility of a four-lepton signal. Additional leptons can arise from
the leptonic decays of daughter $\tw_1$ and $\tz_2$. Although the decay
products are generally soft, the ubiquity of these light higgsino-like
states within the RNS framework often results in additional detectable
leptons ($e$ and $\mu$) in would-be trilepton events. This
characteristic feature of low $|\mu|$ models such as RNS is absent in
models such as mSUGRA, and leads to the possibility of four-lepton plus $\eslt$
signal, even in $R$-parity conserving SUSY.
%\footnote{It is well-known
%that high lepton multiplicities are obtained if the LSP decays via
%lepton-number-violating interactions that do not conserve $R$-parity.}
A study of this new signal for which it is required that
%
\bi
\item 4 isolated leptons with $p_T(\ell)> 10$~GeV within $|\eta(\ell)|<
  2.5$,
\item $n_b=0$, to veto backgrounds from top decays,
\item $\eslt > \eslt({\rm cut})$, where $\eslt({\rm cut})$ is chosen
  to select signal events above SM backgrounds,
\ei
%
is the subject of this section.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{MET_0b4l_lhc14.eps}
\end{center}
\caption{The $\eslt$ distributions for $4\ell$ events with $n_b=0$ from 
various SM sources and for two signal points on the RNS model-line.  }
\label{fig:4l}
\end{figure}
%

Within the SM, the main sources of $4\ell + \eslt$ events are
$ZZ$, $Zt\bar{t}$, $ZWW$, $ZZW$, $ZZZ$ and $Zh(\to WW^*)$,
followed by leptonic decays of tops, and of the electroweak vector bosons. 
The bulk of
the background from $ZZ$ production is eliminated by requiring a large
$\eslt$. Nevertheless, this background remains significant since $\eslt$
can arise via $Z\to \tau^+\tau^- \to \ell^+\ell'^- +\eslt$. 


The RNS signal, along with backgrounds from $ZZ$,
$t\bar{t}Z$ and $VVV$ ($V=W,Z$), is simulated using AlpGen and Pythia. 
The cross sections for the most important of these backgrounds
are listed in the second column of Table~\ref{tab:4l}, together with that for the
signal for three model-line points. The last two columns list these
signal and background calculations for $\eslt({\rm cut}) = 100$ and
200~GeV, the choice being motivated by the $\eslt$ distributions shown
in Fig.~\ref{fig:4l}. The numbers in bold-face show the statistical
significance for an integrated luminosity of 300~fb$^{-1}$.
%
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
cuts & $n(b) = 0, n(l) = 4$ & $\eslt > 100$~GeV & $\eslt > 200$~GeV \\
\hline
\hline
$ZZ$        & 18.02 & 0.0611 & 0.0094 \\
$Zt\bar{t}$ & 0.450 & 0.158  & 0.0232 \\
$ZWW$       & 0.155 & 0.0516 & 0.0134 \\
\hline
Total BG    & 18.66 & 0.280  & 0.0483 \\
\hline
$m_{1/2}=400$~GeV & 0.527 & 0.343 \ ({\bf 11.2}) & 0.122 \ ({\bf 3.8})\\
$m_{1/2}=500$~GeV & 0.195 & 0.157 \ ({\bf 5.1}) & 0.0769 \ ({\bf 6.1}) \\
$m_{1/2}=600$~GeV & 0.084 & 0.0728 \ ({\bf 2.3}) & 0.0467 \ ({\bf 3.7}) \\
\hline
\end{tabular}
\caption[Background and signal rates in $fb$ for 4-lepton events at
LHC14 after cuts.]{Background and signal rates in $fb$ for 4-lepton events at
LHC14 after cuts. The bold-faced numbers in parenthesis in the last two
columns show the statistical significance of the signal with
300~fb$^{-1}$ of integrated luminosity at LHC14. The signal comes from
wino pair production for points on the RNS model line introduced in the text.}
\label{tab:4l}
\end{center}
\end{table}
%

It is estimated that $Zh(\to W^\pm \ell\nu)$ yields a $4\ell$ cross section
 $\sim 1300$~fb $\times 0.06 \times 0.22 \times 0.03$ (where the last
 factor is the branching fraction for $h\to W^\pm \ell^\mp\nu$ decay)
 $\simeq 0.5$~fb, before any lepton acceptance cuts which further reduce
 the cross section by factor about 5-10. After the hard $\eslt$
 requirement, one may expect
 expect this to make a relatively unimportant contribution to the
 background. Backgrounds from $ttWW$ and $4V$ processes should also be
 small. 
 

Several comments are worth noting:
%
\bi
\item there is no benefit, and in fact a loss of significance, by requiring pairs of
  leptons to reconstruct to $M_Z$. This is largely because the largest
  $4\ell$ backgrounds also all have a $Z$ in them, and both signal and
  backgrounds drop roughly equally due to this requirement. 

\item the softer $\eslt > 100$~GeV cut works better for
  $m_{1/2}=400$~GeV for which a 6$\sigma$ signal is obtained
  even with just  100~fb$^{-1}$ of integrated luminosity.

\item the $5\sigma$ reach for 300~fb$^{-1}$ (1000~fb$^{-1}$) extends
 to  $m_{1/2} = 500$~GeV (beyond $m_{1/2}=600$~GeV) with the harder
  $\eslt({\rm cut})=200$~GeV.
\ei 
%
In conclusion, the $4\ell$ channel
  would serve to confirm a the SSdB signal pointing to light higgsinos
  out to $m_{1/2}$ values $\lesssim500-650$~GeV, depending on the
  integrated luminosity that is ultimately available.

Current ATLAS~\cite{atlas_4l} and CMS~\cite{cms_4l} 4-lepton searches 
are optimized for the signal from the cascade decays of gluinos 
(and so do not veto hadronic activity)  
with the high lepton multiplicity originating in R-parity violating leptonic decays of $\tz_1$. 
In contrast, the signal described here is hadronically quiet and would
stand out over SM backgrounds with veto on b-jets as described in the text.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Soft trileptons from direct higgsino pair production}
\label{ssec:3l}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this Section, an attempt to exploit the large cross sections for higgsino
pair production from the RNS model at the LHC is presented: 
$pp\to\tw_1\tz_1,\ \tw_1^+\tw_1^-,\ \tz_1\tz_2$ and $\tw_1^\pm\tz_2$.
The purely hadronic$+\eslt$ final states from higgsino pair production
are expected to be buried beneath prodigious QCD backgrounds since the
signal yields only soft, low $p_T$ jets and soft $\eslt$
spectra. Likewise, most single and dilepton signals are expected to be
buried under $W\to \ell\nu_\ell$ and $WW$, $t\bar{t}$ backgrounds
respectively. Examined here is the clean trilepton signal from
higgsino pair production by searching for the $pp\to\tw_1\tz_2\to
(e\nu_e\tz_1)+(\mu^+\mu^-\tz_1)$ topology where a
dilepton trigger with $p_T(e)>10$~GeV and $p_T(\mu )>5$~GeV is
assumed. It is required that:
\bi
\item $10\ {\rm GeV}<p_T(e)<50$~GeV, 
\item $5\ {\rm GeV}<p_T(\mu_{1})<50$~GeV,
\item $5\ {\rm GeV}<p_T(\mu_{2})<25$~GeV.
\ei
Scrutiny of a variety of distributions suggests the following cuts:
\begin{enumerate}
\item $10\ {\rm GeV}<m(\mu^+\mu^- )<75$~GeV,
\item $n(jets) =0$ (jet-veto),
\item electron transverse mass $m_T(e ,\eslt )<60$~GeV,
\item $25\ {\rm GeV}<\eslt <100$~GeV.
\end{enumerate}

The signal and four background processes are shown in Table~\ref{tab:3l}. 
%
\begin{table}
\begin{center}
\begin{tabular}{|l|r|r|r|r|r|}
\hline
cuts & $t\bar{t}$ & $W^*Z^*$ &  $ZZ$ & $Wt\bar{t}$ & signal \\
\hline
\hline
cut 1  & 12.4 & 7.6  & 0.15 & 0.1 & 0.42 \\
cut 2  & 2.4  & 7.1  & 0.09 & 0.006 & 0.42 \\
cut 3  & 1.3  & 4.4  & 0.08 & 0.003 & 0.42 \\
cut 4  & 0.9  & 2.0  & 0.03 & 0.002 & 0.28 \\
\hline
\end{tabular}
\caption[Background and signal rates in $fb$ for soft $3\ell +\eslt$ 
events at LHC14 after cuts.]{Background and signal rates in $fb$ for soft $3\ell +\eslt$ 
events at LHC14 after cuts. The signal comes from higgsino pair production at $m_{1/2}=400$~GeV point 
on the RNS model line. The $2\to 4$ process labelled $W^*Z^*$ includes $\gamma^*,\ Z^*\to\tau\bar{\tau}$.}
\label{tab:3l}
\end{center}
\end{table}
%
Shown in Fig.~\ref{fig:softe} are distributions of electron $p_T$ before cuts for two sample 
points on the RNS model line. The energy release is very small, less than $\sim
25$~GeV, and quickly decreases with $m_{1/2}$. For $m_{1/2}=1$~TeV most of electrons have
$p_T$ less than 10~GeV, the trigger threshold.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{pTe_sl_lhc14_1GeV.eps}
\end{center}
\caption{$p_T(e)$ distribution for soft tri-leptons from higgsino pairs 
before cuts for two RNS points with 
$m_{1/2}=400$~GeV (red) and 1000~GeV (blue) at LHC14.
}
\label{fig:softe}
\end{figure}
%
\begin{table}
\begin{center}
\begin{tabular}{|l|r|r|r|r|}
\hline
 Int. lum. (fb$^{-1}$) & $\tg\tg$ &  SSdB & $WZ\to 3\ell$ &$4\ell$ \\
\hline
\hline
10   & 1.4 &  --  & -- & --\\
100  & 1.6 &  1.6 & -- & $\sim 1.2$\\
300  & 1.7 &  2.1 & 1.4& $\gtrsim 1.4$ \\
1000 & 1.9 &  2.4 & 1.6& $\gtrsim 1.6$ \\
\hline
\end{tabular}
\caption[Reach of LHC14 for SUSY in terms of gluino mass, $m_{\tg}$ (TeV),  
assuming various integrated luminosity values along the RNS model line.]{Reach of LHC14 for SUSY in terms of gluino mass, $m_{\tg}$ (TeV),  
assuming various integrated luminosity values along the RNS model line.
Presented is each search channel considered except soft
$3\ell$.}
\label{tab:reach}
\end{center}
\end{table}
%
After cut 4, the background exceeds signal by a factor of 10.  The dimuon
invariant mass distribution after cuts is shown in
Fig.~\ref{fig:m(mm)}{\it a}) for $m_{1/2}=400$~GeV ($\tz_2-\tz_1$ mass
gap at 38~GeV), {\it b}) $m_{1/2}=550$~GeV (mass gap at 25~GeV) and {\it
c}) $m_{1/2}=700$~GeV (mass gap at 18~GeV). The shapes
of the dilepton mass distribution for the signal+background in frame
{\it a})  differs from that of the background alone. A shape analysis
using the data at large $m_{\ell\ell}$ to normalize the background may
allow one to claim a signal, given sufficient integrated luminosity, since 
an excess of events should be found in
bins with $m(\mu^+\mu^- )<38$~GeV as compared to higher mass bins where
a theory-experiment match is expected.  For a counting analysis alone, 
invoking a cut
$m(\mu^+\mu^-)<38$~GeV, a $5\sigma$ signal over background (without any
requirement on the $S/B$ ratio) would require
about 700~fb$^{-1}$ of integrated luminosity.  In the other frames with
the smaller mass gap, an excess only appears in the lowest mass bin(s) and
the possibility of extracting a signal appears even more difficult.
%

%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=9cm,clip]{mll_s3l_lhc14_mhf400.eps}
\includegraphics[width=9cm,clip]{mll_s3l_lhc14_mhf550.eps}
\includegraphics[width=9cm,clip]{mll_s3l_lhc14_mhf700.eps}
\end{center}
\caption[The dimuon invariant mass distributions after cuts for $e\mu\mu$ events.]{The dimuon invariant mass distributions after cuts for $e\mu\mu$ events. 
Open red histograms represent signals from higgsino pair productions for three points 
with $m_{1/2}=400,\ 550$ and 700~GeV along the RNS model line.
}
\label{fig:m(mm)}
\end{figure}
%


The final reach situation for CERN LHC is summarized in Table~\ref{tab:reach}, where
the $5\sigma$ reach in terms of $m_{\tg}$ is given for various discovery
channels and integrated luminosity values.  While LHC14 can explore RNS
up to $m_{\tg}\sim 2$~TeV for 300~fb$^{-1}$, a large swath of parameter
space with $m_{\tg}\sim 2-5$~TeV seemingly lies beyond LHC14 reach.

%%
%--------------------------------------------------------------------
\newpage
\section{Physics at a higgsino factory \cite{rnsilc}}
\label{sec:ilc}
%--------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As seen in the last chapter, light higgsinos can be produced at large rates at LHC8 and LHC14.  However, the compressed higgsino spectrum leads to only small
visible energy release from $\tw_1\to\tz_1 f\bar{f}'$ and $\tz_2\to
\tz_1 f\bar{f}$ decays (where $f$ denotes SM fermions).
As demonstrated in the previous section, LHC14 should probe gluinos with mass up to $m_{\tg}\sim 2$~TeV, assuming an
integrated luminosity of $\sim 1000$~fb$^{-1}$. This means that LHC14 probes
about half of the gluino mass range allowed by RNS. 

This chapter addresses the detectability of the light higgsinos of RNS
at the International Linear Collider (ILC), a proposed $e^+e^-$
collider\cite{tdr,snowmass} designed to operate at $\sqrt{s}\sim 0.25-1$
TeV, with an added capability of electron beam polarization.  While such
a machine is often touted as a Higgs boson factory due to the capacity
to study the reaction $e^+e^-\to Zh$, for the case of models with light
higgsinos that are required for naturalness,
the ILC would also become a higgsino factory
and a SUSY discovery machine\cite{bbh}.

Early pioneering studies of sparticle production at linear colliders were performed by the Japan Linear Collider
group on mixed higgsino-wino type of chargino pair production where mass
gaps were around 50~GeV\cite{jlc}.  Additional studies incorporating
cascade decays were performed in Ref.~\cite{bmt}, and in
Ref. \cite{tadas1,tadas2} chargino pair production in the hyperbolic
branch/focus point region \cite{CCN,fp} was examined also with $\sim
40$~GeV mass gaps.  Very recent studies include those in
Ref.~\cite{Han:2013kza}.

Recently, studies of higgsino pair production with mass gaps of order 1
GeV have been performed\cite{Berggren:2013vfa}.  In these studies, use
is made of initial state photon radiation and exclusive one-or-two
particle hadronic decays of the charginos which have large branching
fractions because the $Q$-value is limited at the GeV-level.  These
studies were relevant for string-motivated high-scale gauge-mediation
models where the very large gaugino masses lead to large
values of $\Delta_{EW}\sim 275$\cite{ilcbm}. The current discussion
examines the case of models with $\Delta_{EW}\sim 10-30$ where mass gaps
of 10-20~GeV are typical, and for which the techniques of
Ref.~\cite{Berggren:2013vfa} are not needed.

Section~\ref{ssec:bm} presents two RNS benchmark models labeled as
ILC1 and ILC2.  In Sec.~\ref{ssec:prod}, sparticle production
and decay for the two RNS benchmark models at a higgsino factory is discussed.  In
Sec.~\ref{ssec:evgen}, some details of the signal and
background event generation calculations are presented. Finally,
Sec.~\ref{ssec:det} and Sec.~\ref{ssec:det2}
discuss how ILC acts as a natural SUSY discovery machine for light
higgsino pair production and show how it can make precision measurements
of the associated sparticle masses and mass gaps.

\subsection{Simulation of Events at an $e^+e^-$ collider}
\label{ssec:epem}

A linear collider offers several advantages over hadron colliders,
such as:
\bi
\item the energy is no longer limited to the colliding partons,
  essentially all of the electron and positron energy goes into the
  hard scattering subprocess.
\item such a collider allows for a polarized electron beam. These
  beams are valuable in separating signal from background and
  distinguishing similar signals. 
\item the energy is tunable, allowing experimentalists to tune into a
  specific process.
\item the events are far cleaner than hadron collider events.
\ei
As mentioned in the footnote of Sec.~\ref{ssec:simulation}, simulation at an $e^+e^-$
collider follows a similar process as a hadron collider. For these
types of collisions there are electron PDFs which describe
bremstrahlung \cite{brem} and beamstrahlung \cite{beam} effects. These PDFs
are described in \cite{tadas2}. Bremstrahlung allows for initial state
photons which have the capacity to affect the background and
signals, while beamstrahlung describes the beam-beam interactions which can
result in energy loss and similar effects on the background and
signal. 

The polarization dependence
can be incorporated directly into the cross section
calculations. Figure~\ref{fig:pol} illustrates these effects for
several processes. The degree of polarization can be quantified as
\bea
P_{\rm L}(e^-)&=&f_{\rm L}-f_{\rm R},\hspace{1cm} {\rm with}\\
f_{\rm L}&=&\frac{n_{\rm L}}{n_{\rm L} + n_{\rm R}}=\frac{1+P_{\rm L}}{2}\\
f_{\rm R}&=&\frac{n_{\rm R}}{n_{\rm L} + n_{\rm R}}=\frac{1-P_{\rm L}}{2}
\eea
where $n_{\rm L,R}$ is the number of left (right) polarized electrons
in the beam, and $f_{\rm L,R}$ is the corresponding fraction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Two RNS benchmark points}
\label{ssec:bm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Two benchmark points were selected for study of light higgsinos in RNS
models.  To generate spectra for models with low $\Delta_{EW}$,
the Isasugra spectrum generator\cite{isasugra} from
Isajet~7.84\cite{isajet} was used. Point ILC1 is similar to benchmark RNS2 of
Ref.~\cite{ltr}, except a lower $\mu =115$~GeV value has been selected
to yield a spectrum of light higgsinos which would already be accessible
at ILC250 (ILC with $\sqrt{s}=250$~GeV). The mass gaps for ILC1 are
$m_{\tw_1}-m_{\tz_1}=14.6$~GeV and $m_{\tz_2}-m_{\tz_1}=21.3$~GeV. While
safe from LHC8 bounds, gluino and also wino production will lead to
observable signals at LHC14\cite{lhc}. 

Also examined is the much more challenging case of benchmark ILC2 which
will likely be beyond the reach of LHC14. This point is chosen from the
RNS model-line with $\mu=150$~GeV introduced in the previous chapter, with
$m_{1/2}$ adjusted to obtain as small a mass gap as possible, consistent
with $\delew^{-1}$ of no more than 3\%. For this challenging case, the
mass gaps are rather small, with $m_{\tw_1}-m_{\tz_1}=10.2$~GeV and
$m_{\tz_2}-m_{\tz_1}=9.7$~GeV.  This point is not accessible to ILC250,
and so examines the feasibility of detection at a centre-of-mass
energy just below the top pair threshold. 


Within the RNS framework, $m_{\tz_2}-m_{\tz_1} =10$~GeV
is close to the minimum of the mass gap if one requires that
$\delew^{-1}>3$\%. This can be seen from Fig.~\ref{fig:gap2} where the
neutralino mass gap is shown in the $m_{1/2}-\mu$ plane, with the other
NUHM2 parameters fixed at the same values as for the ILC2 case.  From
the figure it is seen that the mass gap ranges from about 10~GeV (for large
$m_{1/2}$, where the $\Delta_{EW}$ contours become vertical because the
top squarks become too heavy) to over 100~GeV in the region where the
gaugino and higgsino states are strongly mixed. 
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{muvmhf_tb15_n16_m05000_DmZ_fi.eps}
\caption[Contours of the mass gap (green curves) $m_{\tz_2}-m_{\tz_1}$
  in the $m_{1/2}-\mu$ mass plane of the NUHM2 model for $m_0=5$~TeV,
  $A_0=-1.6m_0$, $\tan\beta=15$ and $m_A=1$~TeV. The red curves show
  contours of $\Delta_{EW}$.]{Contours of the mass gap (green curves) $m_{\tz_2}-m_{\tz_1}$
  in the $m_{1/2}-\mu$ mass plane of the NUHM2 model for $m_0=5$~TeV,
  $A_0=-1.6m_0$, $\tan\beta=15$ and $m_A=1$~TeV. The red curves show
  contours of $\Delta_{EW}$. The blue (gray) shaded regions are 
  excluded by the absence of a chargino signal at LEP2 (LEP1). The
  region to the left of the dot-dashed line is excluded by the LHC8 limit
  $m_{\tg} > 1.2$~TeV, obtained assuming squarks are very heavy. The
  dashed line is where $m_{\tw_1}=300$~GeV. 
  The crosses denote $\mu$ and $m_{1/2}$ values for ILC1 and ILC2 benchmark points. 
  Note that the other parameters for ILC1 differ from those in the figure, 
  but the mass difference is insensitive to these.}
\label{fig:gap2}
\end{center}
\end{figure}

%
For these large mass
gaps, LHC experiments should be awash in clean multilepton signals from
electroweak-ino production, including signals from $\tw_2$ and $\tz_4$
events.
The concern here is the difficult region
with a small mass gaps $m_{\tz_2}-m_{\tz_1}$ and $m_{\tw_1}-m_{\tz_1}$
where there may well be no detectable signals even at LHC14.  To the
extent that the difficulty of extracting ILC SUSY signals (without using
kinematic properties particular to exclusive chargino decays
\cite{Berggren:2013vfa}) increases with decreasing mass gap, the ILC2
point represents nearly the most challenging case that may be encountered
in the examination of linear colliders as a definitive probe of
naturalness.

Listed at the bottom of Table~\ref{tab:bm} are the neutralino relic density,
some $B$-decay branching fractions
and WIMP detection rates along with the value of $\Delta_{EW}$.
WIMP detection sensitivities should be multiplied by a factor $\xi\equiv
\Omega_{\tz_1}h^2/0.12$ since the higgsino-like WIMPs could make up only
a fraction of the local DM density, while {\it e.g.} axions
might make up the remainder\cite{dfsz}. This will be investigated in the
next chapter.
%
\begin{table}\centering
\begin{tabular}{lcc}
\hline
parameter & ILC1 & ILC2 \\
\hline
$m_0$      & 7025.0 & 5000  \\
$m_{1/2}$  &  568.3 & 1200  \\
$A_0$      & -11426.6 & -8000  \\
$\tan\beta$&  10 & 15  \\
$\mu$      & 115 & 150  \\
$m_A$      &  1000 & 1000  \\
\hline
$m_{\tg}$   & 1563.5 & 2832.6   \\
$m_{\tu_L}$ &  7021.3 & 5440.4  \\
$m_{\tu_R}$ &  7254.2 & 5565.6  \\
$m_{\te_R}$ &  6758.6 & 4817.1  \\
$m_{\tst_1}$&  1893.3 & 1774.3  \\
$m_{\tst_2}$&  4919.4 & 3877.9  \\
$m_{\tb_1}$ &  4959.2 & 3902.8  \\
$m_{\tb_2}$ &  6893.3 & 5204.5  \\
$m_{\ttau_1}$ & 6656.6 & 4652.3  \\
$m_{\ttau_2}$ & 7103.1 & 5072.5  \\
$m_{\tnu_{\tau}}$ & 7114.0 & 5078.7 \\
$m_{\tw_2}$ & 513.0 & 1017.5  \\
$m_{\tw_1}$ & 117.3 & 158.3  \\
$m_{\tz_4}$ & 524.2 & 1031.1  \\ 
$m_{\tz_3}$ & 267.0 & 538.7   \\ 
$m_{\tz_2}$ & 124.0 & 157.8  \\ 
$m_{\tz_1}$ & 102.7 & 148.1  \\ 
$m_h$      & 125.3 & 125.4  \\ 
\hline
$\Omega_{\tz_1}^{std}h^2$ & 0.009 & 0.007  \\
$BF(b\to s\gamma)\times 10^4$ & 3.3 & $3.3$  \\
$BF(B_s\to \mu^+\mu^-)\times 10^9$ & 3.8 & $3.9$  \\
$\sigma^{SI}(\tz_1 p)$ (pb) & $1.3\times 10^{-8}$ & $2.9\times 10^{-9}$ \\
$\Delta_{EW}$ & 13.9 & 28.5 \\
\hline
\end{tabular}
\caption{NUHM2 input parameters and masses in GeV units for the two RNS benchmark points introduced in the text.
Here $m_t=173.2$~GeV.}
\label{tab:bm}
\end{table}
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sparticle production and decay}
\label{ssec:prod}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Sparticle production}

Shown in Fig.~\ref{fig:eexsec}, are sparticle and Higgs boson production rates for
unpolarized beams at the ILC versus $\sqrt{s}$ for the ILC1 benchmark
point.  Rates were computed at leading order, with leading order
spectra, using formulae from Ref.~\cite{bbkmt}. Also shown for
comparison is the rate for muon pair production. Around
$\sqrt{s}\sim 220-230$~GeV, the threshold for production of $Zh$,
$\tw_1^\pm\tw_1^\mp$ and $\tz_1\tz_2$ is crossed so that production
rates rise rapidly.  Whereas one might expect ILC at these energies to
be a Higgs boson factory, ILC would also be a higgsino factory,
where the higgsino pair production rates exceed $Zh$ production by
factors of 5-10.  While the higgsino decay debris may be too soft to be
picked out above SM backgrounds at LHC, the clean environment of a linear
$e^+e^-$ collider allows for straightforward discovery, as
discussed in Sec.~\ref{ssec:det} and~\ref{ssec:det2}. Thus, although RNS might well elude LHC searches, it cannot elude searches at
ILC provided that $\sqrt{s}> 2m(higgsino)$.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{xsec_light_higgsinos_ilc_with_mumu.eps}
\end{center}
\caption{Sparticle production cross sections vs. $\sqrt{s}$ for
  unpolarized beams at an $e^+e^-$ collider for the ILC1 benchmark point
  listed in Table \ref{tab:bm}.}
\label{fig:eexsec}
\end{figure}
%

While the reactions $e^+e^- \to \tw_1^+\tw_1^-$ and $e^+e^- \to \tz_1\tz_2$ will be the
first sparticle production processes accessed at ILC250, 
the discovery prospects do not end there.
As $\sqrt{s}$ is increased beyond $2m(higgsino)$, further thresholds will be
passed, including those for $\tz_2\tz_3$, $\tw_1\tw_2$ and $\tz_2\tz_4$
production. 
These occur at somewhat lower but still measureable rates. 
Even reactions with much lower production rates -- such as
$\tz_2\tz_2$, $\tz_1\tz_3$, $\tz_3\tz_3$ and $\tz_3\tz_4$ -- might
ultimately be detectable, depending on the machine energy and integrated
luminosity that is ultimately attained.


Shown in Fig.~\ref{fig:pol}, are the $\tw_1^+\tw_1^-$ and  $\tz_1\tz_2$
%$Zh$ 
production rates for the ILC1 benchmark case at $\sqrt{s}=250$~GeV,
but as a function of the electron beam polarization $P_L(e^-)$.  Whereas $\tw_1^+\tw_1^-$
production has the largest rate for unpolarized beams ($P_L(e^-)=0$),
for the case of right polarized electron beam, $\sigma (\tw_1^+\tw_1^-)$
diminishes by a factor of about 4 and instead $\sigma (\tz_1\tz_2)$,
which is much less sensitive to beam polarization, is dominant.  
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{sigvpol_ilc1_250.eps}
\end{center}
\caption{Sparticle production cross sections vs. $P_L(e^-)$ at an
$e^+e^-$ collider for the ILC1 benchmark point with
$\sqrt{s}=250$~GeV. The positrons are taken to be unpolarized. For
comparison, shown is a point with a wino-like chargino of similar
mass. For the wino-like case with $m_{1/2}=120$~GeV, then the $\sigma
(e^+e^-\to \tz_1\tz_2)\sim 0.1$~fb, while $\sigma(\tz_2\tz_2)$ is even
smaller, and so is far below the cross section values shown.  }
\label{fig:pol}
\end{figure}
%
The
comparable rates (within an order of magnitude) for both both chargino
and neutralino pair production (solid curves), together with the
relatively mild polarization is characteristic of the production of
higgsino-like charginos and neutralinos. For wino-like gauginos in the
kinematically accessible range, chargino production would occur at a
high rate, but neutralino pair production would be strongly suppressed
because $SU(2)_L\times U(1)_Y$ gauge symmetry forbids couplings of the
$Z$ and $\gamma$ to both binos and (neutral) winos.
This can be seen in the dashed curve in Fig.~\ref{fig:pol} which shows
the cross section for $\tw_1\tw_1$ production for the ILC1 model
point except that $m_{1/2}$ and $\mu$ are now chosen so that the weak
scale values of $M_2$ and $\mu$ are essentially exchanged. In this case,
the masses of the wino-like $\tw_1$ and $\tz_2$ is about the same as for
the higgsinos of the ILC1 point.  The neutralino-pair production cross
sections for this wino-like case are below 0.1~fb and do not show up in
this frame.  This observation will be important in Sec.~\ref{ssec:det}
where
the analysis is described.  The polarization dependence of the chargino pair
production cross section provides an independent handle that may enable
us to argue the higgsino-like nature of the charginos of the ILC1
point. For a right-handed electron beam the amplitude for charged wino
pair production is suppressed by a factor of $M_W^2/s$ relative to that
for charged higgsino pair production, accounting for the strong drop of
the dashed curve at $P_L(e^-) = -1$.

%production rate varies much more steeply with $P_L(e^-)$ and $\sigma
%(\tz_1\tz_2)$ (bino-wino) production occurs at only tiny rates $\sim
%0.1$~fb.


\subsubsection{Higgsino decays}

Since the inter-higgsino mass gaps are so small, for the case of RNS one
expects the following three-body decays to be dominant: 
%
\bea
\tw_1^-&\to & \tz_1 f\bar{f}'\;,\\ \tz_2& \to & \tz_1 f\bar{f}\;, \eea 
%
where the $f$ stand for SM fermions.  Despite the larger phase space
suppression for the three body decays,
the branching fraction for the loop decay $\tz_2\to \tz_1\gamma$ is
still small because of the large $Z\tz_1\tz_2$ coupling \cite{loop}.
Moreover, squark and slepton masses are expected very large within the
RNS framework, and  the $\tw_1$ and $\tz_2$ three-body
decay amplitudes are dominated by $W^*$ and $Z^*$ exchange,
respectively. The branching fractions into specific modes will thus
closely follow the corresponding $W$ and $Z$ decay branching fractions,
{\it i.e.}  one obtains $B(\tw_1^-\to \tz_1 e^-\bar{\nu}_e) \simeq 11\%$,
$B(\tz_2 \to e^+e^-\tz_1) \simeq 3\%$, {\it etc.}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{SUSY event generation} %at a Higgsino Factory}
\label{ssec:evgen}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

Within the RNS framework, higgsino pair production at the ILC will be
signalled by events with low visible energy from the relatively
soft daughter leptons and jets from $\tw_1$ and $\tz_2$ decays, and
modest $\eslt$. One needs, therefore, to pay particular attention to
SM sources of low visible energy events. Since the bulk of $2\to 2$
events lead to large visible energy, the most important backgrounds
come from two photon processes, $e^+e^- \to e^+e^- f\bar{f}$, where the
energetic final state electron and positron are lost down the
beam-pipe, and the visible energy in the detector arises only from 
$f$ and $\bar{f}$. 

ISAJET v7.84 was used for the SUSY event simulation as well as simulation
of $2 \to 2$ and $\gamma\gamma$-induced SM backgrounds.
The  $2\to 2$ SM background sources include
\be
e^+e^-\to f\bar{f},\ W^+W^-\ \ {\rm and}\ \ Z^0Z^0\;,
\label{eq:2to2}
\ee
while $\gamma\gamma$ processes include, 
\be
\gamma\gamma\to \tau^+\tau^-,\ \ c\bar{c}\ \ {\rm and}\ \ b\bar{b}.
\label{eq:2to4}
\ee
%
The reaction $e^+e^-\to Zh$ is included in the signal sample, but plays
no role here.
 

The Isajet toy calorimeter was used, covering
$-4<\eta <4$ with cell size $\Delta\eta\times\Delta\phi =0.05\times 0.05$.
Energy resolution for electromagnetic and hadronic depositions is taken to be
$\Delta E_{em}/E_{em} =0.15/\sqrt{E_{em}}\oplus 1\%$ and 
$\Delta E_h/E_h=0.5/\sqrt{E_h}/E_h\oplus 2\%$, respectively (where
$\oplus$ denotes addition in quadrature).
Jets are found using fixed cones of size
$R=\sqrt{\Delta\eta^2+\Delta\phi^2} =0.6$ using the ISAJET routine
GETJET (modified for clustering on energy rather than transverse energy).
Clusters with $E>5$~GeV and $|\eta ({\rm jet})|<2.5$ are labeled as jets.
Muons and electrons are classified as isolated if they have $E>5$~GeV,
$|\eta (\ell )|<2.5$, and the visible activity within a cone of $R=0.5$
about the lepton direction is less than 
$min({E_{\ell}\over 10},\ 1\ {\rm GeV})$. 

The production reactions are run using electron PDFs which include a
convolution of bremsstrahlung~\cite{brem} and beamstrahlung~\cite{beam}
contributions. For $\sqrt{s}=250$~GeV, 
a beamstrahlung parameter $\Upsilon =0.02$ was used and for
$\sqrt{s}=340$~GeV, $\Upsilon =0.03$ was used.  For both cases,
the beam bunch length $\sigma_z =0.3$~mm~\cite{tdr3} was used.

For processes with low visible energy, the two-photon processes 
$\gamma\gamma\to f\bar{f}$ can be the most relevant.
These processes also give rise to  $\eslt$ when the decay products of $f$
include neutrinos. This analysis therefore includes only 
\bi
\item $\gamma\gamma\to\tau^+\tau^-,\ c\bar{c}$ and $b\bar{b}$
\ei
contributions from Isajet using a photon PDF which again
includes a beam/bremsstrahlung convolution\cite{drees,tadas2}.


\subsection{Benchmark ILC1 at $\sqrt{s}=250$~GeV}
\label{ssec:det}

This section begins by discussing higgsino pair production for the ILC1
benchmark point with $\sqrt{s}=250$~GeV, the nominal turn-on energy of
the ILC.  Once
threshold for pair production is passed, then the two higgsino pair
production reactions occurring at the highest rates are 
\bi
\item $e^+e^-\to \tw_1^+\tw_1^-\to (f\bar{f}'\tz_1)+(F\bar{F}'\tz_1$)\ \  and 
\item $e^+e^-\to \tz_1\tz_2\to \tz_1+(f\bar{f}\tz_1)$ 
\ei 
where $f$ and $F$ are SM fermions. For models where $|\mu|\ll M_{1,2}$, the two
lightest neutralinos are well approximated by $(\th_u\pm \th_d)\over
\sqrt{2}$, and the coupling of $Z$ to $\tz_1\tz_1$ and $\tz_2\tz_2$
pairs is dynamically suppressed~\cite{wss}. Thus, though the phase space
for $\tz_2\tz_2$ production is qualitatively similar to that for
$\tz_1\tz_2$ production, $\sigma(\tz_2\tz_2)$ is much smaller in the RNS
framework: see Fig.~\ref{fig:eexsec}.

Since $m_{\tz_1}$ is only slightly smaller than $m_{\tw_1,\tz_2}$, most
of the collision energy ends up in the rest mass $2m_{\tz_1}$ of the
LSPs, and the visible final state fermions are relatively soft.  To
illustrate this, shown in Fig.~\ref{fig:Evis} is the visible
energy distribution expected at ILC250 for benchmark ILC1.  From the
figure, one sees the bulk of SM background from $e^+e^-$ annihilation
 processes (green curve) in (\ref{eq:2to2}) peaks around $E_{vis}\sim
250$~GeV, with some spillover to higher values due to detector energy
mis-measurement. A continuous $E_{vis}$ tail occurs at lower values due
to production of $WW$, $ZZ$, $b\bar{b}$ {\it etc.} where substantial
energy is lost due to decays to neutrinos. There are also two small
bumps at $E_{vis}\sim 140$~GeV and $250$~GeV arising from $Zh$ production
(blue curves). The 140~GeV bump occurs due to $e^+e^-\to Zh\to
(\nu\bar{\nu})+(b\bar{b})$.  The SUSY signal distribution is depicted by
the bounded (red) histogram with $E_{vis}\sim 0-40$~GeV is already
well-separated from the $2\to 2$ SM backgrounds.  However, as
anticipated, backgrounds from the $2\to 4$ processes, $\gamma\gamma\to
c\bar{c},\ b\bar{b}$ and $\tau\bar{\tau}$, shown as the black histogram
overwhelm the signal by a factor of $\sim 100-1000$.  
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{etot_250.eps}
\end{center}
\caption[Distribution in visible energy measured in $e^+e^-$ events at
$\sqrt{s}=250$~GeV for ILC1 signal and SM backgrounds from
$e^+e^-$ and $\gamma\gamma$ collisions.]{Distribution in visible energy measured in $e^+e^-$ events at
$\sqrt{s}=250$~GeV for ILC1 signal and SM backgrounds from
$e^+e^-$ and $\gamma\gamma$ collisions.
Beamstrahlung parameters are $\Upsilon =0.02$ and $\sigma_z=0.3$~mm.
}
\label{fig:Evis}
\end{figure}
%

To select signal events, the first requirement is:
\bi
\item $20$~GeV $<E_{vis}<50$~GeV.  
\ei 
%
The $\gamma\gamma$ background yields mainly soft visible energy events
with a tail extending to higher values. To differentiate signal from
this background, plotted in Fig.~\ref{fig:ETM} is the missing transverse
energy distribution $d\sigma /d\eslt$ after the visible energy cut. The $\gamma\gamma$ background falls very rapidly since $\eslt$
occurs mainly due to neutrinos from the decays of the relatively light
$c,b$ and $\tau$, and the signal emerges from $\gamma\gamma$ background
if one requires $\eslt > 10$~GeV.  The bulge of events with low $E_{vis}$
but modest $\eslt$ would herald the discovery of new physics.  
This also explains why $\gamma\gamma \to f\bar{f}$ processes with
$f=e,\mu,u,d,s$ were not included in the analysis. These
yield  back-to-back events in the transverse plane, with
essentially no $\eslt$, and are efficiently eliminated by a $\eslt$ cut.
Thus, for
the new physics event sample, it is also required that
%
\bi
\item $\eslt >10$~GeV.
\ei
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{ptm_250.eps}
\end{center}
\caption[Distribution of missing transverse energy from $e^+e^-$ collisions at
$\sqrt{s}=250$~GeV for ILC1 signal along with SM background
from $e^+e^-$ and $\gamma\gamma$ collisions.]{Distribution of missing transverse energy from $e^+e^-$ collisions at
$\sqrt{s}=250$~GeV for ILC1 signal along with SM background
from $e^+e^-$ and $\gamma\gamma$ collisions.
It is required that $20$~GeV$<E_{vis}<50$~GeV.
Beamstrahlung parameters are $\Upsilon =0.02$ and $\sigma_z=0.3$~mm.
}
\label{fig:ETM}
\end{figure}
%

To understand the expected event topologies, the multiplicity
of isolated leptons and identified jets is examined. These distributions are shown in
Fig.~\ref{fig:nlepjet}. One sees that the most lucrative signal channels
from the perspective of the signal to background ratio appears to
be the $n(\ell)=0$ and $n(jet)=1-3$ bins to which neutralino and chargino 
production can contribute. To cleanly separate chargino and neutralino
contributions so that each particle can be studied in detail, it is
also useful to examine other channels. 

Before turning to this, note that the observation of an excess above
SM in the multi-jet plus multi-lepton channels would suggest the production of charginos and
neutralinos. The small energy release in these events would point to a
small mass gap between the parent particles and the undetected LSP. In
the simplest models with gaugino mass unification, this would indicate
the production of higgsino-like states, with $|\mu| \ll m_{1/2}$ where
$\tw_1$, $\tz_2$ and $\tz_1$ are roughly degenerate, and the bino and
winos are substantially heavier. However, it is also possible that such
events may arise from wino pair production in models with heavy
higgsinos, and a bino only slightly lighter than the wino-states. It
should, however, be possible to distinguish between these possibilities
since, as mentioned in the discussion of Fig.~\ref{fig:pol}, with
unpolarized beams neutralino production is smaller than 0.1~fb ({\it
i.e.} three orders of magnitide below expectations for higgsino-like
neutralinos) in the latter case. Moreover, the chargino signal from the
production of wino-like charginos will reduce much more sharply for
right-handed electron beams than for higgsino-like charginos. It should
thus be possible to unambiguously conclude that the signal is from
higgsino-like, and not gaugino-like super-partners. 

With this in mind, the goal is to find strategies that will help obtain essentially pure samples
of chargino and of neutralino events for the ILC1 point under
examination.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=12.5cm,clip]{Nlep_250.eps}
\includegraphics[width=12.5cm,clip]{Njet_250.eps}
\end{center}
\caption[Distribution of {\it a}) isolated lepton multiplicity and {\it
b}) jet multiplicity from $e^+e^-$ collisions at $\sqrt{s}=250$~GeV for
higgsino signals from the ILC1 case study
along with corresponding SM backgrounds from $e^+e^-$ and
$\gamma\gamma$ collisions.]{Distribution of {\it a}) isolated lepton multiplicity and {\it
b}) jet multiplicity from $e^+e^-$ collisions at $\sqrt{s}=250$~GeV for
higgsino signals from the ILC1 case study
along with corresponding SM backgrounds from $e^+e^-$ and
$\gamma\gamma$ collisions.  It is required that $20$~GeV$<E_{vis}<50$~GeV and
$\eslt >10$~GeV.  Beamstrahlung parameters are $\Upsilon =0.02$ and
$\sigma_z=0.3$~mm.  }
\label{fig:nlepjet}
\end{figure}
%


\subsubsection{Chargino pair production for ILC1}
\label{sssec:w1w1}

To select out a nearly pure sample of chargino pair events where the jets all
arise from the same chargino, first selected are events with the
$E_{vis}$ and $\eslt$ cuts introduced above, but also require 
\bi
\item $n(\ell)=1$
\ei
 and
\bi 
\item $n(jet)=2$.
\ei
After these requirements, the signal cross section is 6.43~fb. 
Just one background event passes cuts, leading to $\sigma_{BG}\sim 0.018$~fb, 
{\it i.e.} there is a nearly pure sample of chargino pair events, 
where one chargino decays leptonically and the other decays hadronically.

A scatter plot of these selected events in the $E(jj)$
vs. $m(jj)$ plane is shown in Fig.~\ref{fig:ejjmjj}.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{ejj_mjj_250.eps}
\end{center}
\caption[Scatter plot in the $E(jj)$ vs. $m(jj)$ plane for $1\ell
+2-jets$ events from the ILC1 point in $e^+e^-$ collisions at
$\sqrt{s}=250$~GeV.]{Scatter plot in the $E_{jj}$ vs. $m(jj)$ plane for $1\ell
+2-jets$ events from the ILC1 point in $e^+e^-$ collisions at
$\sqrt{s}=250$~GeV. It is required that $20$~GeV$<E_{vis}<50$~GeV and $MET>10$
GeV.  Beamstrahlung parameters $\Upsilon =0.02$ and
$\sigma_z=0.3$~mm.  }
\label{fig:ejjmjj}
\end{figure}
%
The $m(jj)$ distribution is expected to be bounded from above by
$m_{\tw_1}-m_{\tz_1}=14.6$~GeV up to energy mis-measurement corrections;
this cut-off is seen in Fig.~\ref{fig:ejjmjj}, from which it is apparent the
$m_{\tw_1}-m_{\tz_1}$ mass gap is $\sim 15$~GeV.

The sparticle masses $m_{\tw_1}$ and $m_{\tz_1}$  can be obtained 
from fits of the $E(jj)$ data distribution\cite{jlc,bmt,tadas1} 
%(as shown in Fig.~\ref{fig:ejj}) 
to various expected theory distributions which vary depending on
$m_{\tw_1}$ and $m_{\tz_1}$. The lower endpoint of $E(jj)$ is determined
largely by the $E(j)>5$~GeV jet requirement but the upper endpoint is
quite sensitive to $m_{\tw_1}$ and $m_{\tz_1}$ values.

To assess the precision which can be attained, a synthetic
``data'' set is generated, assuming 100~fb$^{-1}$ of integrated luminosity, along with
expected statistical error bars. Also generated is a theory sample of
distributions run over a large grid of $\mu$ and $m_{1/2}$ points (which
yields a corresponding grid of $m_{\tw_1}$ and $m_{\tz_1}$ points) where
each theory sample is run with 10 times the statistics of data. This
analysis ignores any sensitivity to other parameters, and implicitly
assumes that one can distinguish between higgsino- and wino-like chargino
events (which should be possible as noted just before the start of
Sec.~\ref{sssec:w1w1}).  The $E(jj)$ ``data'' distribution
 (with 1~GeV bins) is then compared to these theory templates and one
 may obtain the values of
$\chi^2$ between the ``data'' and the theory. The
normalization of theory curves is fixed to match ``data'' so that
only the shape of the distribution is fitted. To obtain the $\chi^2$, the 
appropriately weighted statistical errors for the theory and data sets
are added in quadrature.

This procedure enables one to obtain a grid of $\Delta\chi^2
=\chi^2-\chi^2_{min}$ values in the $m_{\tw_1}-m_{\tz_1}$ plane.  The
reader should keep in mind that the theory calculation  is also subject
to statistical fluctuations that will be reflected in the distribution 
of $\Delta\chi^2$ values. To enable the reader to personally assess the
reliability of the computation, shown in Fig.~\ref{fig:chi2}{\it a}
are these
$\Delta\chi^2$ values binned by $\Delta\chi^2 < 2.3$ ($1\sigma$ CL),
$\Delta\chi^2 < 4.6$ (90\% CL) and $\Delta\chi^2 > 4.6$. Also shown
are the corresponding $1\sigma$ and 90\% CL error ellipses that were obtained as 
conservative fits to the $\Delta\chi^2$ data.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=13.5cm,clip]{chisq_cont_ejj_ilc1_pts.eps}
\includegraphics[width=11.5cm,clip]{ejj_250_bestfit.eps}
\end{center}
\caption[Shown in {\it a}) are values of $\Delta\chi^2$ found from
matching 100~fb$^{-1}$ of ILC1 ``data'' to various ``theory''
distributions generated from a scan over $\mu\ vs.\ m_{1/2}$ space. In
{\it b}), is the distribution in $E(jj)$ from 100~fb$^{-1}$ of
``data'' along with best fit distribution.]{Shown {\it a}), are values of $\Delta\chi^2$ found from
matching 100~fb$^{-1}$ of ILC1 ``data'' to various ``theory''
distributions generated from a scan over $\mu\ vs.\ m_{1/2}$ space. Each
``theory'' point is run with ten times the events contained in the
``data'' distribution.  Also shown are fitted error ellipses corresponding
to $1\sigma$ and 90\% CL measurements.  In
{\it b}), is the distribution in $E(jj)$ from 100~fb$^{-1}$ of
``data'' along with best fit distribution.  }
\label{fig:chi2}
\end{figure}
%
From these error ellipses, the 2-3\% mass measurements 
\bi
\item $m_{\tw_1}=117.8\pm 2.8$~GeV ($1\sigma$),
\ei
 and also,
\bi
\item $m_{\tz_1}=103.1\pm 2.2$~GeV ($1\sigma$), \ei 
%
should be possible
for the ILC1 point.  The  synthetic ``data'',
together with statistical error bars corresponding to an integrated
luminosity of 100~fb$^{-1}$, are
shown in Fig.~\ref{fig:chi2}{\it b}) along with the best fit
distribution shown as the solid curve.

Note that if instead the time is taken to perform various total cross
section measurements around the higgsino pair threshold -- which will
require a much higher integrated luminosity investment at several
$\sqrt{s}$ values \cite{threshold} -- even better precision on the
masses may be expected.

\subsubsection{Neutralino pair production for ILC1}
\label{sssec:z1z2}

For the case of $\tz_1\tz_2$ pair production, events where
$\tz_2\to q\bar{q}\tz_1$ that yield an  $n(\ell)=0,\ n(j)=2$ sample as well as
events where $\tz_2 \to \ell^+\ell^-\tz_1$, for which $n(\ell)=2$ and
$n(j)=0$ were examined. 
While one might expect the dijet sample to yield more events due to
the large $\tz_2\to\tz_1 q\bar{q}$ branching fraction, in fact one finds
after cuts that the $\ell^+\ell^-$ sample is larger.  This is because
frequently the two possible quark jets merge to yield only a single
resolvable jet given the jet finding algorithm, or else one of the
possible jets becomes too soft or too forward to be identified.

For the opposite-sign/same flavor (OS/SF) dilepton signal that is focused
on, a polarized electron beam with $P_L(e^-)=-0.9$ is used since this 
helps to reduce potential backgrounds from $WW$ production, and also limits 
contamination from chargino production to around 10\%. 
The signal is required to have: 
%
\bi
\item exactly 2 OS/SF isolated leptons with no jets, 
\item $E_{vis}<35$~GeV,
%\item $\eslt > 15$~GeV
\item transverse plane angle between the two leptons
$\Delta\phi(\ell^+\ell^-)< \pi /2$.  
\ei 
%
After these cuts, the
$\gamma\gamma$ background is eliminated but some SM background -- mainly
$WW$ production -- remains.  
The $E(\ell^+\ell^- )$ distribution after these cuts is shown in
Fig.~\ref{fig:Ell}.  This leaves a OS/SF dilepton signal of 19.55~fb
while SM background is $0.44$~fb. 
The signal has a negligible contribution from
chargino production.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{Ell_250.eps}
\end{center}
\caption{Distribution in $E(\ell^+\ell^- )$ from 100~fb$^{-1}$ of
``data'' of OS/SF dilepton events with $P_L(e^-)=-0.9$ from the signal
for the ILC1 case, and from SM background (which only comes from $2\to 2$
processes), after the $E_{vis}$ and $\Delta\phi(\ell\ell)$ cuts
discussed in the text.}
\label{fig:Ell}
\end{figure}
%

With the clean sample of OS/SF dilepton signal events from
essentially $\tz_1\tz_2$ production, next examined is the $m(\ell^+\ell^-
)$ distribution for the ILC1 case. One expects that this distribution is
kinematically bounded from above by the $m_{\tz_2}-m_{\tz_1}$ mass
difference and relatively insensitive to the absolute masses of the
particles.  The theory templates generated with 10 times the
statistics, as described in Sec.~\ref{sssec:w1w1} were used to obtain a map of
$\chi^2$ vs. $m_{\tz_2}-m_{\tz_1}$, shown by the jagged (black) line in
Fig.~\ref{fig:chi2_gap}{\it a}. As before, the fit is to the shape, allowing
the normalization to float. While statistical fluctuations do
contribute to the jaggedness, it was checked that the points with the
largest $\chi^2$ values come from theory templates where the mass scale
of the neutralinos is very different. Also shown in the figure
is a parabolic fit to the values of $\chi^2$. From this, one
sees that the mass gap should be
measureable at the percent level: 
%
\bi
\item $m_{\tz_2}-m_{\tz_1} =21.0\pm 0.2$~GeV  ($1\sigma$). 
\ei
%This illustrates that the $m_{\tz_2}-m_{\tz_1}$ mass gap can be much more 
%precisely measured than either of the individual masses.
The best fit line and the dilepton mass ``data'' used to obtain the fit
 are shown in the lower frame in the figure. The shape of
 this mass distribution is indicative of an opposite sign of the $\tz_1$
 and $\tz_2$ mass eigenvalues~\cite{kadala}, completely compatible with
 expectation~\cite{kn} from the decay of a higgsino-like $\tz_2$ to a
 higgsino-like $\tz_1$.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=13cm,clip]{mass_diff_chisq_ilc1_ptsandfit.eps}
\includegraphics[width=13cm,clip]{mll_250_bestfit.eps}
\end{center}
\caption{Shown in {\it a}) are values of $\chi^2$ vs. $m(\ell^+\ell^- )$
from 100~fb$^{-1}$ of OS/SF dilepton ILC1 ``data'' from $\tz_1\tz_2$
production fit to the shapes from various ``theory'' templates, as
described in the text.  In {\it b}), are the ILC1 ``data'' for the
$m(\ell^+\ell^-)$ distribution from $\tz_1\tz_2$ events along with
statistical error for 100~fb$^{-1}$. The solid curve shows the best fit
to these ``data''.  }
\label{fig:chi2_gap}
\end{figure}
%
%The $m(\ell^+\ell^- )$ data distribution is shown in
%Fig.~\ref{fig:chi2_gap}{\it b}) along with the best fit.


Once the mass gap is known, it is possible to extract the neutralino
mass value via a fit to the $E(\ell^+\ell^-)$ distribution because the
energy of the daughter leptons (but not their invariant mass) depends 
on the boost of the parent $\tz_2$. 
The same procedure described above was used to fit the 100
fb$^{-1}$ OS/SF dilepton $E(\ell^+\ell^- )$ using theory templates
with different values of $m_{\tz_2}$, but with
$m_{\tz_2}-m_{\tz_1}$ fixed at 21~GeV.  The corresponding values of 
$\chi^2$ along with the parabolic fit
is shown in Fig.~\ref{fig:chi2ll}{\it a}. It is found that
$m_{\tz_2}$ is measured as 
%
\bi
\item $m_{\tz_2}=123.7\pm 0.2$~GeV ($1\sigma$),
\ei
a 0.2\% measurement.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=12cm,clip]{ell_chisq_ilc1_ptsandfit.eps}
\includegraphics[width=12cm,clip]{ell_250_bestfit.eps}
\end{center}
\caption{Shown in {\it a}), are fitted values of $\chi^2$ found from
matching 100~fb$^{-1}$ of OS/SF dilepton ``data'' from $\tz_1\tz_2$
production to various ``theory'' distributions generated from varying
$m_{\tz_2}$ while keeping $m_{\tz_2}-m_{\tz_1}$ fixed at 21~GeV.  In
{\it b}) is the distribution in $E(\ell^+\ell^- )$ from a 100~fb$^{-1}$ 
of OS/SF dilepton ILC1 ``data'' from $\tz_1\tz_2$ production
along with best fit.  }
\label{fig:chi2ll}
\end{figure}
%
Combining the $m_{\tz_2}-m_{\tz_1}$ and $m_{\tz_2}$ measurements also
gives $m_{\tz_1}$: 
\bi
\item $m_{\tz_1}=102.7\pm 0.3$~GeV (ILC1-dileptons).  
\ei 
%
This value
serves as a consistency check against the measurement of $m_{\tz_1}$
from chargino pair production, and most importantly, lends support to the
SUSY interpretation of these events. 
The distribution of $E(\ell^+\ell^- )$ data are shown in
Fig.~\ref{fig:chi2ll}{\it b}) along with the corresponding best fit 
depicted by the solid curve.



\subsection{Benchmark ILC2 at $\sqrt{s}=340$~GeV}
\label{ssec:det2}

Benchmark point ILC2 is more challenging for ILC studies because the
mass gap between $\tw_1/\tz_2$ and the $\tz_1$ is just about 10~GeV,
resulting typically in softer energy release from three-body $\tw_1$ and
$\tz_2$ decays. This mass gap is close to the minimum for RNS models
where $\delew^{-1}> 3$\%. Since charginos and neutralinos
are heavier, its exploration requires a higher $\sqrt{s}$ to reach
higgsino pair production threshold.  In this case, studies are performed at
$\sqrt{s}=340$~GeV, enough for $\tw_1^+\tw_1^-$ and $\tz_1\tz_2$
production, but just below $t\bar{t}$ threshold. For these higher
$\sqrt{s}$ values, the expected beamstrahlung parameter $\Upsilon$ is
expected to increase to $0.03$, while $\sigma_z$ remains at $0.3$~mm~\cite{tdr3}.

The $E_{vis}$ distribution from signal and background is shown in 
Fig.~\ref{fig:EvisILC2}. Here, the ILC2 signal is
restricted to the region with $E_{vis}\alt 30$~GeV, while the
background from  $\gamma\gamma$ collisions is more severe than for
the $\sqrt{s}=250$~GeV case. Therefore, a cut of
\bi
\item $E_{vis}<30$~GeV.
\ei
%
is imposed.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{etot_340ilc2.eps}
\end{center}
\caption{Distribution in $E_{vis}$ from benchmark ILC2 signal and SM
backgrounds at ILC with $\sqrt{s}=340$~GeV and $\Upsilon =0.03$.  }
\label{fig:EvisILC2}
\end{figure}
%


Following the earlier analysis, the $\eslt$ is examined
in Fig.~\ref{fig:ptm340}.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{ptm_340ilc2.eps}
\end{center}
\caption[Distribution in missing transverse energy from $e^+e^-$
collisions at $\sqrt{s}=340$~GeV for signal from the ILC2 benchmark
case, along with SM backgrounds from $e^+e^-$ and $\gamma\gamma$
collisions.]{Distribution in missing transverse energy from $e^+e^-$
collisions at $\sqrt{s}=340$~GeV for signal from the ILC2 benchmark
case, along with SM backgrounds from $e^+e^-$ and $\gamma\gamma$
collisions. $E_{vis}<30$~GeV is required.  Beamstrahlung
parameters are $\Upsilon =0.03$ and $\sigma_z=0.3$~mm.  }
\label{fig:ptm340}
\end{figure}
%
Unlike the case of ILC1, the signal never emerges from the
$\gamma\gamma$ background. Clearly additional cuts are necessary for
observability of the signal.



\subsubsection{Chargino pair production for ILC2}
\label{sssec:w1w1ilc2}

To extract a chargino pair production signal from the SM background
for benchmark ILC2, it is required that:
%
\bi
\item $E_{vis}<30$~GeV,
\item $\eslt>10$~GeV,
\item exactly one isolated lepton with $E>5$~GeV and one jet with $E(j)>5$~GeV.  
\ei 
%
For the case of ILC2, the hadronic energy release from
$\tw_1\to q\bar{q}'\tz_1$ is so small that almost never are two
resolvable jets produced, making chargino mass extraction difficult via continuum
production (although perhaps still possible via threshold scans with sufficient
integrated luminosity). Hence, instead
the $n(\ell)=1,\ n(jet)=1$ signal is focused on. The transverse plane lepton-jet opening angle which is shown in
Fig.~\ref{fig:thlj}.  Most of the SM background comes from
$\gamma\gamma\to\tau^+\tau^-$ followed by one leptonic and one hadronic
tau decay.  This may be mostly eliminated by requiring 
%
\bi
\item $\Delta\phi(\ell ,jet)<120^\circ$.
\ei
%
After this cut, a signal of 7.1~fb is left while SM background
is at the 2.8~fb level, and all of the $\gamma\gamma$ background, which
arises from tau pair production, is eliminated. This should not be
surprising because most taus that decay into 5~GeV jets/leptons will be
significantly boosted, and hence tend to have their decay products nearly
back-to-back in the transverse plane.  The discovery of new
physics might be possible with a data set of just a few fb$^{-1}$ at
ILC340 even in this difficult case.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{philj_340.eps}
\end{center}
\caption[Distribution in transverse plane 
opening angle between the isolated lepton and jet 
for the ILC2 signal point at $\sqrt{s}=340$~GeV, and for SM backgrounds.]{Distribution in transverse plane 
opening angle between the isolated lepton and jet 
for the ILC2 signal point at $\sqrt{s}=340$~GeV, and for SM backgrounds.
Here, $E_{vis}<30$~GeV and $\eslt>10$~GeV.
Beamstrahlung parameters are $\Upsilon =0.03$ and $\sigma_z=0.3$~mm.
}
\label{fig:thlj}
\end{figure}
%


\subsubsection{Neutralino pair production for ILC2}
\label{sssec:z1z2ilc2}

As mentioned earlier, if any signal in the $1\ell 1j$ channel just
discussed is to be attributed to higgsino-like charginos of SUSY, one
may also expect a signal from $\tz_1\tz_2$ production as this
reaction must have a
comparable production cross section. Therefore,
$\tz_1\tz_2$ production is examined for the ILC2 point
with $\sqrt{s}=340$~GeV, where
$\tz_2\to\ell^+\ell^-\tz_1$. This acoplanar dilepton signal may also
allow for neutralino mass reconstruction via continuum production.  
It is required of the signal to have
\bi 
\item $E_{vis}<30$~GeV,
\item a pair of OS/SF leptons, with $n(j)=0$,
\item $\eslt >5$~GeV.
\ei 
%
For this channel, mainly right polarized electron beams
with $P_L(e^-)=-0.9$ are used to reduce backgrounds from $W^+W^-$ and
contamination from $\tw_1^+\tw_1^-$ production.  The
transverse OS/SF dilepton opening angle is plotted in Fig.~\ref{fig:thll}.  To
eliminate the $\gamma\gamma$ background which is once again nearly
back-to-back in the transverse plane, and to improve the
signal-to-background ratio,  it is required that: 
%
\bi
\item $\Delta\phi(\ell^+\ell^-)<90^\circ$.
\ei 
%
At this point, the signal has a cross section of $2.6$~fb while SM background is at the $0.15$~fb level
with no $\gamma\gamma$ background. Once again, discovery of new physics
is possible with just a few~fb$^{-1}$ of integrated luminosity. 
%Such a sample should be amenable
%to neutralino mass extraction via continuum production.
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\textwidth,clip]{phill_340.eps}
\end{center}
\caption[Distribution in transverse opening angle between isolated OS/SF
leptons for ILC2 signal at $\sqrt{s}=340$~GeV and for SM backgrounds.]{Distribution in transverse opening angle between isolated OS/SF
leptons for ILC2 signal at $\sqrt{s}=340$~GeV and for SM backgrounds.
It is required that $E_{vis}<30$~GeV and $\eslt> 5$~GeV.  Beamstrahlung
parameters are $\Upsilon =0.03$ and $\sigma_z=0.3$~mm.  }
\label{fig:thll}
\end{figure}
%
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=13cm,clip]{mass_diff_chisq_ilc2_ptsandfit.eps}
\includegraphics[width=13cm,clip]{mll_bestfit_ilc2.eps}
\end{center}
\caption{Shown in {\it a} are $\chi^2$ values vs. $m(\ell^+\ell^- )$
from 100~fb$^{-1}$ of OS/SF dilepton ILC2 ``data'' from $\tz_1\tz_2$
production fit to theory along with a best-fit parabola.  In {\it b})
is the distribution in $m(\ell^+\ell^- )$ from a 100~fb$^{-1}$ of
OS/SF dilepton ILC2 ``data'' from $\tz_1\tz_2$ production along with
best fit.  }
\label{fig:chi2_gap_ilc2}
\end{figure}
%
For benchmark ILC2, the same procedure is used as for the ILC1 
case study to extract the neutralino masses. First the normalized theory templates (generated with 1000~fb$^{-1}$ each)
with varying $m_{\tz_2}-m_{\tz_1}$ mass gaps are fit to a 100~fb$^{-1}$ ``data''
distribution.  The various $\chi^2$ values along with a parabolic fit
are shown in Fig.~\ref{fig:chi2_gap_ilc2}{\it a}. As before, it was checked that the very large $\chi^2$ values for a mass gap
near the bottom of the parabola arise from extreme values of $m_{\tz_2}$
in the templates. For the ILC2 case, the $m_{\tz_2}-m_{\tz_1}$ mass gap is measured to be 
%
\bi
\item $m_{\tz_2}-m_{\tz_1} = 9.7\pm 0.2$~GeV ($1\sigma$) 
\ei 
a 2\% measurement.  The data along with best theory fit are shown in
Fig.~\ref{fig:chi2_gap_ilc2}{\it b}.


Keeping the mass gap fixed near 9.7~GeV, theory
templates are generated for the $E(\ell^+\ell^- )$ distributions
from $\tz_1\tz_2$ production with 10 times the statistics of
``data'' but with varying $m_{\tz_2}$ values and  fit the shapes of
these to the corresponding ``data'' distribution as before. 
Shown in Fig.~\ref{fig:chi2ll_ilc2}, are the $\chi^2$ values along with the  parabolic fit. 
From this, a measurement of 
%
\bi
\item $m_{\tz_2}=158.5\pm 0.4$~GeV ($1\sigma$).
\ei
%
is found.
The $E(\ell^+\ell^-)$ distribution for ``data'' along with best fit
theory are shown in Fig.~\ref{fig:chi2ll_ilc2}{\it b}.  
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=13cm,clip]{ell_chisq_ilc2_ptsandfit.eps}
\includegraphics[width=13cm,clip]{ell_bestfit_ilc2.eps}
\end{center}
\caption{Shown in {\it a}) are values of $\chi^2$ found from matching 100~fb$^{-1}$ 
of OS/SF dilepton ILC2 ``data'' from $\tz_1\tz_2$ production
to various ``theory'' distributions generated from varying $m_{\tz_2}$
while keeping $m_{\tz_2}-m_{\tz_1}$ fixed at 9.7~GeV.  In {\it b}) is the distribution in $E(\ell^+\ell^- )$ from a 100~fb$^{-1}$ of
OS/SF dilepton ILC2 ``data'' from $\tz_1\tz_2$ production along with
best fit.  }
\label{fig:chi2ll_ilc2}
\end{figure}
%
By combining the $m_{\tz_2}$ and mass gap measurements, one finds
\bi
\item $m_{\tz_1}=148.8\pm 0.5$~GeV (ILC2-dileptons).
\ei

Although this analysis was performed using the RNS model as a guide,
the results should be applicable to all models with light
higgsinos. It is encouraging that even the difficult point with the smallest mass gap
for 150~GeV higgsinos allows not only detection, but also precision mass
measurements, even at a centre-of-mass energy just modestly above the
production threshold. This leads one to infer that an electron-positron
collider will serve as a definitive probe of the idea of naturalness in
all SUSY models where the $\mu$-term is the dominant
contribution to higgsino masses. In particular, from the dashed contour
in Fig.~\ref{fig:gap2}, one may conclude that ILC600 will either discover or
decisively exclude models where fine-tuning is worse than 3\%.
Precision measurements that can be made
at ILC will definitively show that higgsino pair production is indeed
occuring, and will allow measurement with high precision at least the
higgsino mass scale and associated mass gaps.


%--------------------------------------------------------------------
\newpage
\section{Radiative Natural Supersymmetry and Dark Matter \cite{bbm}}
\label{sec:dark}
%--------------------------------------------------------------------
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Dark matter (DM) has been postulated to account for
several unexplained phenomena, such as anisotropies in cosmic
microwave background (CMB) and discrepencies between luminosity and
gravitational effects in mass measurements of stellar
objects. Possibly the strongest evidence for DM came with the study of
rotation curves of spiral galaxies. Assuming Newtonian dynamics and
spherical symmetry, the rotational velocity $v(r)$ at a distance $r$
can be related to the mass of the galaxy $M(r)$ by
\be
v(r)=\sqrt{\frac{GM(r)}{r}}
\ee
where G is Newton's gravitation constant. Therefore, outside of a
large enough radius, $M(r)$ is expected to be constant and $v(r)$
should fall off as $r^{-1/2}$. However, this is not the case in
measurement, where $v(r)$ flattens out and remains constant.
This measurement suggests that a halo of DM surrounds each galaxy. Many different types
of DM have been theorized, such as black holes, hot (relativistic)
dark matter, and various forms of cold
dark matter (CDM). However, of these
candidates CDM is required by experimental evidence. 

One possibility for a CDM candidate is the
weakly-interacting massive particle, or WIMP. In addition to collider
searches, as addressed previously, SUSY might be found much earlier by
discovery of higgsino-like WIMPs. As seen above, the LSP in RNS is the lightest higgsino-like
neutralino $\tz_1$. The conservation of R-parity implies
$\tz_1$ is stable; in addition, it does not interact via the
electromagnetic or strong forces. Therefore $\tz_1$ is an excellent WIMP
candidate. 

Big bang nucleosynthesis (BBN) and data from WMAP
\cite{wmap} and other experiments suggest that
the baryonic matter density of the universe is $\Omega_bh^2\simeq
0.0224$, whereas the total matter density estimate is
$\Omega_mh^2\simeq 0.135$. Here, $\Omega$ is the ratio of the density
to the critical density of the universe $\Omega=\rho/\rho_c$, $h$
is a dimensionless scaling constant that parameterizes the Hubble
constant $H_0\equiv 100h$ km s$^{-1}$ Mpc$^{-1}$, and the critical density
$\rho_c=\frac{3H^2_0}{8\pi}$ is the density for which the the spatial
geometry is flat. These values imply that the cold dark matter density is
$\Omega_{CDM}h^2 \simeq 0.11$. For the lightest neutralino to be a
viable CDM particle, one would expect the {\it relic density}  of
neutralinos leftover from the big bang to make up a portion of
$\Omega_{CDM}h^2$, possibly supplemented by other CDM particles, such
as axions. The predicted relic density of neutralinos $\Omega_{\tz_1}h^2$ can be found by
using the Boltzmann equation in the Freidmann-Robertson-Walker metric,
\be
\frac{dn}{dt}=-3Hn-\left<\sigma_{ann}v\right>(n^2-n^2_{eq}),
\label{eq:boltz}
\ee
Here, $n$ is the number density of neutralinos, $t$ is time, $n_{eq}$
is the thermal equilibrium number density of neutralinos, and
$\left<\sigma_{ann}v\right>$ is the thermally averaged neutralino
annihilation cross section times the relative velocity.

In the early universe, the WIMPs are in thermal equilibrium and follow
a Maxwell-Boltzmann distribution with the number density decreasing
exponentially with the temperature. However,
as the universe cools and expands the number density of WIMPs
\be
n_{eq}=g(\frac{mT}{2\pi}^{3/2})e^{-(m-\mu)/T}
\ee
decreases until the
{\it freeze-out temperature} $T_{fr}$ is reached, at which time the
density of WIMPs only reduces due to the expansion. Here, $g$ is the
number of degrees of freedom and $\mu$ is the chemical potential. Therefore, the
number of neutralinos would be larger than expected from thermal
equlibrium and becomes effectively constant per co-moving volume. 

At the freeze-out temperature, the Hubble term and the scattering term in
Eq.~\ref{eq:boltz} become comparable, {\it i.e.}
\be
\left<\sigma_{ann}v\right>n\simeq H(T_{Fr}),
\ee
which defines the freeze-out temperature.
Using the Friedmann relation for a radiation dominated universe (it is
assumed here that freeze-out occurs in a radiation dominated phase),
\be
H^2=\frac{8\pi G\rho_r}{3}=\frac{\rho_r}{3M_P^2},
\label{eq:fried}
\ee
where 
\be
\rho_r=\frac{\pi^2}{30} g_{*}(T) T^4
\label{eq:radden}
\ee 
is the
radiation density, $H(T)$ is the Hubble parameter, $M_P$ is the
reduced Planck mass, and $g_{*}$ is the number of relativistic degrees of
freedom at a temperature $T$. One may now define the yield variable as 
\be
Y\equiv \frac{n}{s}=\frac{H}{\left<\sigma_{ann}v\right>s},
\label{eq:yie1}
\ee
with the entropy density $s$ given by
\be
s=\frac{2\pi^2}{45}g_{*}(T)T^3.
\label{eq:ent}
\ee
Combining Eq.'s~\ref{eq:radden} and \ref{eq:ent} into
Eq.~\ref{eq:yie1} gives the yield as
\be
Y_{Fr}=\frac{(90/\pi^2 g_{*}(T_{Fr})(T_{Fr}))^{1/2}}{4
  \left<\sigma_{ann}v\right>M_P T_{Fr}}
\ee
It is assumed that the yield is conserved from $T=T_{Fr}$ to
$T=T_0$ where $T_0$ is the present day temperature of radiation. Now
the neutralino relic density (also known as the 'standard thermal
neutralino abundance' can be written in terms of the yield,
accounting for the entropy density for {\it today} as
\be
\Omega^{std}_{\tz_1} h^2=\frac{2\pi^2 g_{*}(T_0)T_0^3}{45\rho_c/h^2}
m_{\tz_1}Y_{Fr}
\label{eq:relic}
\ee
In Eq.~\ref{eq:relic}, $\rho_c$ and $h^2$ appear from the definition
of $\Omega h^2$ and $m_{\tz_1}$ is from writing the energy density of
the neutralinos as $\rho_{\tz_1}=m_{\tz_1}Y_{Fr}s$.

While this calculation made several simplifications, it is a good
estimate and gives the general behavior of the relic density.
The relic density and annihilation cross section calculations are
often calculated in real-time numerically using computer packages,
such as in the IsaReD\cite{red} relic density subroutine.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp!]
  \begin{center}
\includegraphics[width=\textwidth,clip]{omh2.eps}
  \end{center}
  \caption[Plot of standard thermal neutralino abundance
  $\Omega^{std}_{\tz_1}h^2$ versus higgsino mass.]{Plot of standard
    thermal neutralino abundance $\Omega^{std}_{\tz_1}h^2$ versus
    higgsino mass from a scan over NUHM2 parameter space with $\delew<50$ (red crosses) and
  $\delew<100$ (blue dots). Green points are excluded by current
  direct and indirect search experiments.}
\label{fig:omh2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%

Using the same NUHM2 parameter space as Section~\ref{sec:rns}, RNS
models are generated and the IsaReD\cite{red} relic
density subroutine is used to
calculate the `standard thermal neutralino abundance'
$\Omega^{std}_{\tz_1}h^2$ which is the relic density expected from the
model point. Only models with
$\Omega^{std}_{\tz_1}h^2<0.12$ and $m_{\tw_1}>103.5$ GeV were accepted. The relic
abundance is shown in Fig.~\ref{fig:omh2}, where the red crosses have
$\delew<50$ and blue dots have $\delew<100$. Green points are excluded
by current direct and indirect search limits. In the figure, one sees a high
density band of parameter space points from
$\Omega^{std}_{\tz_1}h^2\sim 0.004$ for $\tz_1\sim 100$ GeV to
$\Omega^{std}_{\tz_1}h^2\sim 0.02$ for $\tz_1\sim 300$ GeV. Thus, there is
typically an {\it underabundance} of higgsino dark matter compared to
measurement from WMAP9 by a factor ranging from 3-25. There
is some spread in these values above and below the main band from
cases where $\mu$ is quite large and $m_{1/2}$ is small so that one
has instead a mixed higgsino-bino LSP state. The bulk of points above
the band are already excluded as will be seen. The neutralino
abundance is expected to be low, as the annhililation rate for
higgsino-like WIMPs tends to be large since $\Omega h^2 \propto \left<\sigma
v\right>^{-1}$. This implies that a mainly higgsino-like neutralino by
itself is not an ideal CDM candidate. The main annihilation channels are
$\tz_1\tz_1\rightarrow W^+W^- (\sim 60\%)$, and $ZZ (\sim 26\%)$ with
the remainder coming from annihilation in $Zh$ and $\tz_1\tw_1$
co-annihilation. These annihilation channels are as expected from
mainly higgsino-like WIMP states \cite{jungman}.

To address the underabundance of WIMPs, it is possible to consider the
Peccei-Quinn-Weinberg-Wilczek solution to the strong CP problem
\cite{pq1,pq2,weincp,wilccp} which introduces a PQ symmetry
and the resultant axions from PQ symmetry breaking. The SUSY partners of the axions are the
R-parity-even spin-0 {\it saxions s} and R-parity-odd
spin-$\frac{1}{2}$ {\it axinos $\tilde{a}$}\cite{axions}. In models of gravity
mediation, as is the case in RNS, the saxion and axino are expected to
have masses of $m_s\sim m_{\tilde{a}}\sim m_{3/2}\sim 5-20$
TeV\cite{axmass}. The axion itself will be much lighter ($\sim\mu$eV), but depends
on the scale of PQ symmetry breaking. In this case, the dark matter
would consist of both axions and higgsinos acting as co-dark matter
particles. 


The relic abundance of mixed axion-neutralino CDM has been addressed
in Refs \cite{bbl,bbl2}. In \cite{bbl2}, it was found that SUSY models with a
standard overabundance of dark matter are still excluded in the PQMSSM
by a combination of dark matter overabundance constraints, big bang
nucleosynthesis (BBN) constraints and dark radiation constraints. However, SUSY models with
a standard underabundance of neutralinos are still allowed over large
ranges of PQMSSM parameters. In RNS models, for any particular
parameter set, one expects the relic higgsino abundance to lie
somewhere between the standard value $\Omega^{std}_{\tz_1}h^2$ (which
would correspond to axion domination) up to the measured value 0.11,
in which case CDM would be higgsino-dominated. The question then arises: what are the prospects for direct/indirect detection of relic higgsinos in WIMP detection experiments?

%
%
\begin{figure}[htbp!]
\begin{center}
\includegraphics[trim=0cm 0cm 2cm
      0cm,width=.6\textwidth]{dir_feyn.eps}
\end{center}
\caption{Two of the Feynman diagrams that contribute to the
  nucleus-neutralino interactions. On the left is Higgs exchange, the
  light Higgs exchange will dominate, since the heavy Higgs will not
  contribute in the decoupling limit. On
  the right is squark exchange.}
\label{fig:dirdet}
\end{figure}

%
\begin{figure}[htbp!]
  \begin{center}
\includegraphics[width=\textwidth]{sc_si_xsec.eps}
  \end{center}
  \caption[Plot of rescaled WIMP spin-independent detection rate $\xi\sigma^{SI}(\tz_1p)$
  versus m(higgsino).]{Plot of rescaled WIMP spin-independent
    detection rate $\xi\sigma^{SI}(\tz_1p)$ versus m(higgsino) from a
    scan over NUHM2 parameter space with $\delew<50$ (red crosses) and
  $\delew<100$ (blue dots). Green points are excluded by current
  direct and indirect search experiments. Also shown is the reach of
  Xe-100 experiement, the LUX experiment after 83.5 days of running,
  and the projected reaches of LUX300, SuperCDMS 150 kg and Xe-1 ton.}
\label{fig:scsi}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Direct Detection of higgsino-like WIMPs}
\label{ssec:dd}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There are currently several experiments hoping to detect dark matter directly through
scattering from nuclei. This is attempted far underground to
reduce induced radioactivity background from high energy cosmic
rays. The idea is that as the Earth moves through the galaxy, DM
particles will pass through a media (such as liquid Xenon or Germanium
crystals) and interact with nuclei, releasing energy which can be
detected. A typical neutralino-nucleus elastic scattering event will
involve energies of just 10's of keV. Because of the low energies involved, these are approximated as effective operators of four particle interactions.

There are two types of
interactions, spin-{\it independent} and spin-{\it dependent}. Which one
dominates will depend on the type of nucleus of the interaction media. The
spin-independent interaction will typically be larger, because the
interaction will sum for all available nucleons and the neutralino
will scatter via the entire mass of the particle. The diagrams in
Fig.~\ref{fig:dirdet} contribute to spin-independent scattering. In spin-dependent
scattering, the neutralino will scatter only via the spin of the
nucleus, such as an interaction with a hydrogen atom in the Sun's
core. The spin-dependent scattering can occur through
$Z$-exchange and squark exchange, however, the squark
decouples and $Z$-exchange will dominate.

As an example, the direct detection interaction
diagrams between the neutralinos and quarks via Higgs and squark
exchange are illustrated in
Fig.~\ref{fig:dirdet} (not shown, but allowed, is the neutralino-gluon
interaction). On the left is the Higgs exchange. Both the
light and heavy Higgs scalars are shown, however, due to their large masses,
the heavy Higgs and squarks will decouple and not contribute. 

Shown in Fig.~\ref{fig:scsi}
are the spin-independent higgsino-proton scattering rates
calculated by IsaReS \cite{res}. These
rates are adjusted by a rescaling factor $\xi=$ to account for the
fact that the local WIMP density may be far below the usual assumed
value $\rho_{local}\simeq 0.3$ GeV/cm$^3$, as suggested by
Bottino {\it et al.}\cite{bott} (the remainder would be composed of
axions). As discussed above, the WIMP in the present case scatters
from quarks and gluons mainly via $h$ exchange. The $\tz_1-\tz_1-h$
coupling involves a product of both higgsino and gaugino
components. In the case of RNS models, the $\tz_1$ contains enough
gaugino component that the coupling is never small: in the notation of \cite{wss}
\be
\l -X^h_{11}\bar{\tz_1}\tz_1h
\ee
where
\be
X^h_{11}=-\frac{1}{2}\left(v_2^{(1)}\sin\alpha-v_1^{(1)}\cos\alpha\right)\left(gv^{(1)}_3-g'v^{(1)}_4\right),
\ee
and where $v^{(1)}_1$ and $v^{(1)}_2$ are the higgsino components, $v^{(1)}_3$ and
$v^{(1)}_4$ are the gaugino components of the lightest neutralino, $\alpha$
is the Higgs mixing angle, and $g$ and $g'$ are the $SU(2)_L$ and
$U(1)_Y$ gauge couplings.
As can be seen in the figure, the Xe-100 \cite{xenon100} and LUX \cite{lux} experiments
are already probing the parameter space. The Xe-1-ton \cite{xenon1t} (which is
currently being deployed) or other comparable
noble liquid detectors can make a complete exploration of RNS
parameter space.
%
\begin{figure}[htbp!]
  \begin{center}
\includegraphics[width=\textwidth,clip]{sc_sd_xsec.eps}
  \end{center}
  \caption[Plot of rescaled spin-dependent higgsino-like WIMP
    detection rate $\xi\sigma^{SD}(\tz_1 p)$ versus m(higgsino) from a
    scan over NUHM2 parameter space.]{Plot of rescaled spin-dependent higgsino-like WIMP
    detection rate $\xi\sigma^{SD}(\tz_1 p)$ versus m(higgsino) from a
    scan over NUHM2 parameter space with $\delew<50$ (red crosses) and
    $\delew<100$ (blue dots). Green points are excluded by current
    direct/indirect WIMP search experiments. Also shown is the current
  reach of the COUPP detector.}
\label{fig:scsd}
\end{figure}

Figure~\ref{fig:scsd} shows the rescaled spin-dependent
higgsino-proton scattering cross section $\xi\sigma^{SD}(\tz_1
p)$. Also shown are recent limits from the COUPP \cite{coupp}
detector. COUPP is a `bubble detector' that uses purified water along
with the chemical CF3I, kept at a temperature near boiling point. When a particle passes through
the liquid, bubbles will form along the ionized recoil trail, which can
be observed. Current limits
are still about an order of magnitude away from reaching the predicted rates from RNS models. 

%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Indirect Detection of higgsino-like WIMPs}
\label{ssec:id}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Indirect detection focuses on discovering dark matter through either
decays or annihilation and searching for the decay/annihilation
product. This differs from direct detection, where the detection media interacts
directly with the dark matter as described in the previous
section. One such detector is the IceCube detector at the South Pole
Neutrino Observatory \cite{icecube}. In Fig.~\ref{fig:sd} the value of
the non-rescaled spin-dependent scattering cross section is shown,
along with the reach of the IceCube experiment. 
\begin{figure}[htbp!]
  \begin{center}
\includegraphics[width=\textwidth,clip]{sd_xsec.eps}
  \end{center}
  \caption[Plot of (non-rescaled) spin-dependent higgsino-like WIMP
    detection rate $\sigma^{SD}(\tz_1 p)$ versus m(higgsino) from a scan over NUHM2 parameter
  space.]{Plot of (non-rescaled) spin-dependent higgsino-like WIMP
    detection rate $\sigma^{SD}(\tz_1 p)$ versus m(higgsino) from a scan over NUHM2 parameter
  space with $\delew<50$ (red crosses) and $\delew<100$ (blue dots). Green points are excluded
  by current direct/indirect WIMP search experiments. Also shown is
  the current reach from IceCube.}
\label{fig:sd}
\end{figure}

As the Sun moves through the galaxy and the DM halo that permeates it,
WIMPs can be swept up and scatter within the Sun's core. If enough
energy is lost in the scattering, the WIMP can then become captured
within the Sun and accumulate at
high density in the solar core. Within the core, WIMPs can annihilate
with each other into SM particles. Most are absorbed by the solar
material, but high energy GeV scale neutrinos can escape. IceCube
tries to detect these by their conversion to muons in the ice. The
detection rates therefore do not include the rescaling factor $\xi$
because the IceCube detection rates depend on whether the Sun has
equilibrated its core abundance between capture rate and annihilation
rate \cite{jungman} and not the local WIMP density. Typically for the
Sun, equilibration is reached for almost all of SUSY parameter space \cite{bott3}. The IceCube limits have entered the RNS parameter space and exclude the larges values of $\sigma^{SD}(\tz_1 p)$.

Shown in Fig.~\ref{fig:scsv} is the thermally averaged neutralino annihilation
cross section times relative velocity in the limit as $v\rightarrow 0:
\xi^2\left<\sigma v\right>_{v\rightarrow 0^+}$. Here the rescaling factor $\xi$ is
squared since limits depend on the square of the local WIMP abundance
\cite{bott2}.  Space-based satellite experiments search for
anomalously high rates for antimatter or photon production, e.g. the
Fermi-LAT satellite or AMS-02 aboard the space station. WIMPs in space
may annihilate one against another into matter-antimatter pairs into
SM particles which then decay to photons. 
On the plot, the limit derived from the Fermi LAT gamma ray obervatory \cite{fermilat} for WIMP annihilations into $WW$ is shown. These
limits have not yet reached the RNS parameter space due in part to the
squared rescaling factor. Anomalies in the positron and $\gamma$
spectra have been reported, although the former may be attributed to
pulsars \cite{barggao}, while the latter 130 GeV gamma line may be
instrumental. In Ref.~\cite{belanger}, constraints on the pMSSM model have
been derived from lack of anti-proton signal in PAMELA data
\cite{pamela}. The anti-proton and gamma-ray constraints from \cite{belanger} occur
at $\left<\sigma v\right>$ values $\sim 10^{-25}$cm$^3$/s. In the
current case where the galactic annihilation rates are suppressed by
the $\xi^2$ factor, the rates lie a couple orders of magnitude below
these limits.
%
\begin{figure}[htbp!]
  \begin{center}
\includegraphics[width=\textwidth]{sc_sv.eps}
  \end{center}
  \caption[Plot of $\xi^2\left<\sigma v\right>_{v\rightarrow 0}$
  versus m(higgsino).]{Plot of rescaled $\xi^2\left<\sigma v\right>_{v\rightarrow 0}$ versus m(higgsino) from a scan over NUHM2 parameter space. Green points are excluded by the current direct and indirect WIMP search experiments.}
\label{fig:scsv}
\end{figure}

In addition to the discovery of higgsino-like WIMPs, discovery of
axions is expected in co-dark matter scenarios. The
Axion Dark Matter Experiement (ADMX) \cite{admx} searches for axions by exploiting
their coupling with photons. Axions would necessarily exist in the dark matter halo
that permeates our galaxy (which would also include the WIMP
neutralinos). ADMX uses a microwave cavity with strong magnetic field
which, if the axions exist at the resonant frequency of the cavity,
will decay into microwave photons via the $a-\gamma-\gamma$ coupling. So far, no positive results have
been reported. 
%--------------------------------------------------------------------
\newpage
\section{Summary}
%--------------------------------------------------------------------
%%
%%
Recent results from LHC7 and LHC8, setting the limits  $m_{\tg}\agt
1.8$ TeV (for $m_{\tg}\simeq m_{\tq}$) and $m_{\tg}\agt 1.3$ TeV (for $m_{\tg}\ll
m_{\tq}$), have resulted in heightened concern
for reconciling electroweak naturalness with lack of SUSY signals and
the rather large value of $m_h=125$~GeV. 
It has been argued that this reconciliation can occur within the context of
radiatively-driven natural supersymmetry (or RNS) where $m^2_{H_u}$ is
driven to a small negative value and $\mu \sim m_Z$.

RNS is a SUSY model based on the MSSM,
which may be valid all the way up to the GUT scale. Thus, it maintains
the desirable features of gauge coupling unification and radiative
electroweak symmetry breaking while avoiding the introduction of extra
possibly destabilizing gauge singlets or other forms of exotic matter.
The main features of the RNS model include 1)~a low value of
superpotential higgsino mass $|\mu |\sim 100-300$~GeV, and 2)~a weak
scale value of $-m_{H_u}^2\sim m_Z^2$: both these qualities are required
to fulfill electroweak naturalness at the tree level. The term
$m_{H_u}^2$ is driven to low values radiatively by the same mechanism
leading to REWSB and depends on a large top quark Yukawa coupling.  

The EWFT is evaluated at the 1-loop level. In this case,
top squark masses enter the computation of $\delew$ and are
also driven radiatively to few-TeV values.  By allowing for large
top-squark mixing ($|A_0|\sim (1-2) m_0$), top-squark contributions to
EWFT are suppressed at the same time as the light Higgs boson mass is
uplifted: thus, the model reconciles electroweak fine-tuning with
$m_h\simeq 125$~GeV all in the context of the MSSM.

RNS may be realized in the two-parameter non-universal Higgs models
NUHM2, but not the more common mSUGRA/CMSSM model. In this case, low
EWFT with $\delew\alt 10$, which corresponds to 10 \% EWFT, can be
attained for model parameters which lead to a distinctive mass spectrum: 
\bi
\item light higgsino-like $\tw^{\pm}_1$ and $\tz_{1,2}$ with mass $\sim 100-300$~GeV,
\item gluinos with mass $m_{\tg}\sim 1-4$~TeV,
\item heavier top squarks than generic NS models: $m_{\tst_1}\sim 1-2$~TeV 
      and $m_{\tst_2}\sim 2-5$~TeV,
\item first/second generation squarks and sleptons with mass
$m_{\tq,\tell}\sim 2-10$~TeV. The $m_{\tell}$ range can be pushed up to
20-30~TeV if non-universality of generations with $m_0(1,2)> m_0(3)$ is
allowed.  
\ei
%
The RNS model with the above spectra also fulfills limits from rare
$B$-decay measurements, which can (along with the value of $m_h$) be an Achilles heel for generic NS
models with much lighter third generation squarks.

The RNS model can be tested at LHC for a range of parameter space. An RNS model line
was constructed which allows variable $m_{1/2}$.  It was found that $\tg\tg$
production followed by cascade decays leads to the expected
leptons+jets+$\eslt$ events. These should allow values of $m_{\tg}$ up
to $\sim 1.7$~TeV to be discovered by LHC14 with about
300~fb$^{-1}$ of integrated luminosity. 
%The
%distribution of $m(\ell^+\ell^- )$ in OS/SF dilepton+jets+$\eslt$ events
%may display a characteristic mass edge at $m_{\tz_2}-m_{\tz_1}$.

A qualitatively new signal, endemic to SUSY models with light
higgsinos, arises mainly from wino pair production $pp\to\tw_2\tz_4\to (W^\pm
\tz_{1,2})+(W^\pm\tw_1^\mp)$ which leads to same sign-diboson production
accompanied by minimal jet activity.  After cuts, the largest background
comes from $t\bar{t}W$ production.  This channel seems to offer the best
reach for RNS for higher integrated luminosity values
$>100$~fb$^{-1}$. The SSdB signal from wino pair production may be confirmed
if the decays of $\tw_2$ and $\tz_4$ yield a final state with $WZ\to 3\ell +\eslt$ 
at high integrated luminosity. Wino pair
production also leads to observable $4\ell+\eslt$ signals
%(high lepton multiplicity is usually thought to be the hallmark of $R$-parity violating SUSY with
%lepton number violating interactions) 
for $m_{1/2} \lesssim 500$~GeV (up to $\sim 650$~GeV at the high-luminosity LHC). 
Signals in the soft $3\ell$ channel arising from direct
higgsino pair production $pp\to\tw_1^\pm\tz_2$ were also explored. This channel should be
visible over the lower $m_{1/2}$ range, which provide a large enough
$m_{\tz_2}-m_{\tz_1}$ mass gap such that one may avoid the $2\to 4$ process 
$W^*Z^*/W^*\gamma^*$ which contains an obstructing virtual photon contribution at
the lower portion of the $m(\ell^+\ell^-)$ distribution.  Detection will
likely be possible via the analysis of the shape of the dimuon invariant
mass distribution for $e^\pm\mu^+\mu^-$ events where there should be a
distortion due to an excess for $m(\mu^+ \mu^-)<m_{\tz_2}-m_{\tz_1}$.

To probe the possibilites at a linear collider,  two benchmark
scenarios were investigated:
ILC1 with lighter higgsinos $\sim 120$~GeV and mass gap $\sim 15-22$~GeV
relative to the LSP, and ILC2 with heavier higgsinos $\sim 150$~GeV but
with a mass gap of just 10~GeV, close to the minimum possible in  models
with no worse than 3\% fine-tuning.

For both these cases, the chargino pair and neutralino pair signals
should be seen above usual SM $2\to 2$ background and $\gamma\gamma$
induced events via a combination of specially devised $E_{vis}$,
$\eslt$, angle and topology cuts with an integrated luminosity of just a
few fb$^{-1}$.  The signal will be characterized by low $E_{vis}$
($\lesssim 30-50$~GeV)  plus $\eslt$ events indicating the production
of heavy parents that decay into an invisible partner with a mass just
10-20~GeV lighter.  Observation of a signal in both $jet(s) + \ell$ and OS/SF
dilepton channels at the expected rates will point to the production of
higgsinos that are the hallmark of natural SUSY models.  For ILC1, the
$\ell+jets$ signal allows for a continuum measurement of $m_{\tw_1}$ and
$m_{\tz_1}$ to $\sim 2\%$ accuracy assuming a canonical value of
100~fb$^{-1}$ of integrated luminosity.  
%Dedicated threshold scans should improve the precision. 
The neutralino pair production reaction
can be seen above the background in the OS/SF, same-side dilepton signal
which allows for the mass gap $m_{\tz_2}-m_{\tz_1}$ to be measured via
the $m(\ell^+\ell^-)$ distribution to $\sim 1\%$ accuracy while
$m_{\tz_2}$ can be measured to sub-GeV precision.

The more challenging ILC2 point allows for chargino pairs to be seen
above background, but the smaller $m_{\tw_1}-m_{\tz_1}$ mass gap
makes dijet resolution very difficult so that a continuum mass
measurement via $E(jj)$ is not possible with the simplified methods
used here. The OS/SF same-side dilepton measurement still remains viable in the case of
ILC2 where the mass gap $m_{\tz_2}-m_{\tz_1}$ can be measured to $\sim
2\%$ accuracy and $m_{\tz_2}$ can be again measured to sub-GeV precision.

In RNS, one also expects the presence of higgsino-like WIMPs which have
large rates for direct and indirect WIMP detection.  With the ILC over
a decade away, and signals possibly hidden at the LHC, these higgsino-like WIMPs may provide the first evidence of RNS. Since higgsinos are
thermally underproduced, one expects them to constitute only a portion of
the measured dark matter abundance, with perhaps axions comprising the
remainder. Detectability via WIMP searches will depend on the higgsino
fraction of the dark matter. Noble liquid detectors such as the 1-ton
scale Xenon detector have the opportunity to probe the entirety of RNS
parameter space. Detection of the co-dark matter axions offer
complementary evidence to WIMP searches. 

In conclusion, the possibilities for detection of RNS at
colliders and dark matter detectors, along with its desirable
theoretical features, merits a high level of
scrutiny. The many elegant features presented above impel one to regard RNS as the new paradigm SUSY model.  
%
%
%\vspace{20pt}
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%
%--------------------------------------------------------------------
%	BibTex References
%--------------------------------------------------------------------
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{singlespace}
\begin{thebibliography}{99}
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%
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%
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%
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%
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%
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%
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%
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%
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%
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%
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%
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%
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%
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\end{thebibliography}
\end{singlespace}
%
\end{document}
