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\slideheight=7.8 in
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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{The MSSM}
\setcounter{equation}{0}
\setcounter{footnote}{0}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{interactions of particles and  sparticles}
\label{Chap4:mssm:sec:particles}
The field content of the MSSM 
\beq
\begin{tabular}{c|cc|ccr}
 & bosons& fermions & $SU(3)_C$ & $SU(2)_L$  & $U(1)_Y$ \\
\hline
$Q_i$ & $(\wtilde{u}_L$, $\wtilde{d}_L)_i$ & $(u_L, d_L)_i$ &\fund & \fund & 
${{1}\over{6}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$
\\
$\overline{u}_i$ & $\wtilde{u}^*_{Ri}$ & $\overline{u}_i=u^\dagger_{Ri}$
& $\overline{\fund}$     & ${\bf 1}$
 & $-{{2}\over{3}}$ $\vphantom{\raisebox{3pt}{\asymm}}$
\\
$\overline{d}_i$ & $\wtilde{d}^*_{Ri}$ & $\overline{d}_i=d^\dagger_{Ri}$
& $\overline{\fund}$     & ${\bf 1}$
 & ${{1}\over{3}}$ $\vphantom{\raisebox{3pt}{\asymm}}$
\\
$L_i$ & $(\wtilde{\nu}, \wtilde{e}_L)_i$ & $(\nu, e_L)_i$ & ${\bf 1}$ & \fund & $-{{1}\over{2}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$
\\
$\overline{e}_i$ & $\wtilde{e}^*_{Ri}$ & $\overline{e}_i=e^\dagger_{Ri}$
&${\bf 1}$     & ${\bf 1}$
 & $1$ $\vphantom{\raisebox{3pt}{\asymm}}$
\\
$H_u$ & $(H^+_u, H^0_u)$ & $(\wtilde{H}^+_u, \wtilde{H}^0_u)$ & ${\bf 1}$ & \fund & ${{1}\over{2}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$H_d$ & $(H^0_d, H^-_d)$ & $(\wtilde{H}^0_d, \wtilde{H}^-_d)$ &${\bf 1}$ & \fund & $-{{1}\over{2}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\ \hline
$G$ & $G^a_\mu$ & $\wtilde{G}^a$ & {\rm \bf Ad} & ${\bf 1}$ & $0$  $\vphantom{\raisebox{3pt}{\asymm}}$ \\
$W$ & $W^3_\mu$, $W^\pm_\mu$ & $\wtilde{W}^3$, $\wtilde{W}^\pm$  & ${\bf 1}$ & 
{\rm \bf Ad} & $0$ $\vphantom{\raisebox{3pt}{\asymm}}$ \\
$B$ & $B_\mu$ & $\wtilde{B}$ & ${\bf 1}$  & ${\bf 1}$ & $0$ $\vphantom{\raisebox{3pt}{\asymm}}$ 
\end{tabular}
\label{Chap4:MSSMtable}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{interactions of particles and  sparticles}

SM has three generations,  $i$ is a generation label
\beq
u_i &=& (u,c,t),\,\,\,\,\,\,\,\,\,\,\,\,
d_i = (d,s,b),\\
\nu_i &=&(\nu_e, \nu_\mu, \nu_\tau),\,\,\,
e_i = (e,\mu,\tau).
\eeq
Higgs
 VEV breaks $SU(2)_L \times U(1)_Y \rightarrow U(1)$ 
 \beq
Q = T^3_L+Y
\eeq
\beq
\frac{1}{e^2}=\frac{1}{g^2}+\frac{1}{g^{\prime 2}}~.
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Two Higgs Doublets}

Two Higgs doublets with opposite hypercharges
are needed to cancel the $U(1)_Y^3$ and $U(1)_Y SU(2)_L^2$  anomalies from higgsinos
\newline
even number of fermion 
doublets to avoid the Witten anomaly for $SU(2)_L$. 

The superpotential for the  Higgs :
\beq
W_{\rm Higgs}= \overline{u} {\bf Y_u} Q H_u - \overline{d} {\bf Y_d} Q H_d - 
\overline{e}\, {\bf Y_e} L H_d + \mu H_u H_d~.
\label{Chap4:MSSMSuperpot}
\eeq
 In the SM we can
have Yukawa couplings with $H$ or $H^*$ but 
holomorphy requires both $H_u$ and $H_d$ in order to write Yukawa
couplings for both $u$ and $d$ 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Yukawa Couplings}

$m_t \gg m_c, m_u$; $m_b \gg m_s, m_d$; $m_\tau \gg m_\mu, m_e$,  
\beq
{\bf Y_u} \approx \left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & y_t
\end{array}
\right), \, 
{\bf Y_d} \approx \left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & y_b
\end{array}
\right), \,
{\bf Y_e} \approx \left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & y_\tau
\end{array}
\right)
\eeq

\beq
W_{\rm Higgs} &=& y_t(\overline{t}t H^0_u -\overline{t}b H^+_u)
 - y_b(\overline{b}t H^-_d - \overline{b}b H^0_d) \nonumber \\
&& - y_\tau(\overline{\tau}\nu_\tau  H^-_d -\overline{\tau}\tau  H^0_d)
+ \mu (H^+_u H^-_d - H^0_u H^0_d)~.
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.51\hsize]{slideimages/topYukawa}}
\label{Chap4:fig:topYukawa}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$\mu$-term}

gives a mass to the higgsinos and a mixing term between a Higgs and 
the auxiliary ${\mathcal F}$ field of the other Higgs. Integrating out auxiliary fields yields the Higgs mass terms
and the cubic scalar interactions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\hsize]{slideimages/muTrilinear}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Higgs mass terms}

 
\beq
{\mathcal L}_{\mu, {\rm quadratic}} &=& 
-\mu (\wtilde{H}^+_u \wtilde{H}^-_d - \wtilde{H}^0_u \wtilde{H}^0_d) +h.c. 
\nonumber\\
&&- |\mu|^2(|H^0_u|^2 +|H^+_u|^2 + |H^0_d|^2 +|H^-_d|^2 ).
\eeq
The $D$-term potential adds quartic terms with positive curvature, so there is a stable minimum at the origin with $\langle H_u\rangle=\langle H_d\rangle=0$.

EWSB requires  soft SUSY breaking terms. 

without unnatural cancellations we will
need
$\mu \sim {\mathcal O}(m_{\rm soft})\sim {\mathcal O}(M_W)$ rather than $ {\mathcal O}(M_{\rm Pl})$. This is known as the 
$\mu$-problem.  perhaps  $\mu$ is forbidden at
tree-level so $\mu$ is then determined by the SUSY breaking mechanism which also determines
$m_{\rm soft}$.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{cubic scalar}


After integrating out auxiliary fields,
\beq
{\mathcal L}_{\mu, {\rm cubic}} &=& \mu^*\left(
\wtilde{u}_R^* {\bf Y_u} \wtilde{u}_L H_d^{0*} + 
\wtilde{d}_R^* {\bf Y_d} \wtilde{d}_L H_u^{0*} + 
\wtilde{e}_R^* {\bf Y_e} \wtilde{e}_L H_u^{0*} \right.  \\ 
&& \,\,\,\,\,\,\,\,\,\,
 \left.+ \wtilde{u}_R^* {\bf Y_u} \wtilde{d}_L H_d^{-*} + 
\wtilde{d}_R^* {\bf Y_d} \wtilde{u}_L H_u^{+*} + 
\wtilde{e}_R^* {\bf Y_e} \wtilde{\nu}_L H_u^{+*}\right) 
 + h.c.\nonumber
\eeq
The quartic scalar interactions are obtained in a similar fashion.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\myhead{R-Parity}
%\label{Chap4:mssm:sec:rparity}

other holomorphic renormalizable terms :
\beq
W_{\rm disaster} = \alpha^{ijk} Q_i L_j \overline{d}_k 
+\beta^{ijk}L_i L_j \overline{e}_k + 
\gamma^i L^i H_u 
+ \delta^{ijk} \overline{d}_i\overline{d}_j\overline{u}_k~,
\label{Chap4:Wdisaster}
\eeq
$W_{\rm disaster}$ violates lepton and baryon number!


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Rapid Proton Decay}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\hsize]{slideimages/protondecay}}
\label{Chap4:fig:protondecay}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\beq
\Gamma_p &\approx& {{|\alpha \delta|^2}\over{m^4_{\tilde{q}}}} {{m_p^5}\over{8
\pi}}~,
\\
\tau_p &=& {{1}\over{\Gamma}} \approx  
{{1}\over{|\alpha \delta|^2}} 
\left({{m_{\tilde{q}}}\over{1 \, {\rm TeV}}}\right)^4 \, 2 \times 10^{-11}\,{\rm s}.
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Super Kamiokande}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\hsize]{slideimages/superk}
\includegraphics[width=0.35\hsize]{slideimages/kamiokande}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Super Kamiokande}

\begin{figure}[htb]
\centerline{
\includegraphics[width=0.35\hsize]{slideimages/kamiokandedetail2}
\includegraphics[width=0.35\hsize]{slideimages/ideal-peppo}
\includegraphics[width=0.4\hsize]{slideimages/peppomc}
}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Rapid Proton Decay}

\beq\tau_p &=& {{1}\over{\Gamma}} \approx  
{{1}\over{|\alpha \delta|^2}} 
\left({{m_{\tilde{q}}}\over{1 \, {\rm TeV}}}\right)^4 \, 2 \times 10^{-11}\,{\rm s}
\eeq


Experimentally,  $\tau_p > 10^{32}$ years 
$\approx 3 \times 10^{39}$~s

\vspace{24pt}
need $|\alpha \delta| <  10^{-25}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{R-Parity}


invent a new discrete  symmetry called $R$-parity:
\beq
\begin{array}{ccc}
({\rm observed}\,{\rm particle}) & \rightarrow & 
({\rm observed}\,{\rm particle}) ~,\\
({\rm superpartner}) & \rightarrow & - ({\rm superpartner})~.
\end{array}
\eeq
Imposing this discrete $R$-parity forbids $W_{\rm disaster}$ \newline
  $R$-parity $\equiv$ to imposing a discrete subgroup of
$B-L$  \newline (``matter parity")
$P_M= (-1)^{3(B-L)}$ since
\beq
R =  (-1)^{3(B-L) +F}~
\eeq

$R$-parity is part of the definition of the MSSM


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{R-Parity}



$R$-parity has important consequences:
\begin{itemize}
\item{} at colliders superpartners are produced in pairs;
\item{} the lightest superpartner (LSP) is stable, and   thus
(if it is neutral) can be a dark matter candidate;
\item{} each sparticle (besides the LSP) eventually
decays into an odd number of LSPs.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{R-Parity}

\begin{figure}[htb]
\centerline{
\includegraphics[width=0.6\hsize]{slideimages/gluinogluino1}
\includegraphics[width=0.4\hsize]{slideimages/darkmatter}
}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Soft SUSY Breaking}
%\label{Chap4:mssm:sec:softbreaking}
\beq
{\mathcal L}_{\rm soft}^{\rm MSSM} &=& -{{1}\over{2}}
\left( M_3 \wtilde{G} \wtilde{G}
+ M_2 \wtilde{W} \wtilde{W} + M_1 \wtilde{B}\wtilde{B} \right)+ h.c.
\nonumber
\\
&&
-\left ( \wtilde {\overline u} \,{\bf A_u}\, \wtilde Q H_u
- \wtilde {\overline d} \,{\bf A_d}\, \wtilde Q H_d
- \wtilde {\overline e} \,{\bf A_e}\, \wtilde L H_d
\right ) + h.c.
\nonumber
\\
&&
-\wtilde{Q}^*  {\bf m^2_{Q}}\, \wtilde Q
-\wtilde L^* {\bf m^2_{L}}\,\wtilde L
-\wtilde{\overline u}^* {\bf m^2_{{\overline u}}}\, 
{\wtilde{\overline u}}
-\wtilde{\overline d}^* {\bf m^2_{{\overline d}}} \, 
{\wtilde{\overline d}}
-\wtilde{\overline e}^* {\bf m^2_{{\overline e}}}\, 
{\wtilde{\overline e}}
\label{Chap4:softBreaking}\\
&&
- \, m_{H_u}^2 H_u^* H_u - m_{H_d}^2 H_d^* H_d
- \left( b H_u H_d + h.c. \right).\nonumber 
\eeq
to
$m_{\rm soft} \approx 1$ TeV in order to solve the
hierarchy problem by canceling quadratic divergences:
\beq
M_i, {\bf A_f} \sim m_{\rm soft}~, \,\,\,
{\bf m_f^2}, b \sim m_{\rm soft}^2~.
\eeq

105 more parameters than the SM! 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Electroweak symmetry breaking}
\label{Chap4:mssm:sec:ESB}

$D$-term potentials for the Higgs fields.
The $SU(2)_L$  and $U(1)_Y$ $D$-terms are (with other scalars set to zero)
\beq
D^a|_{\rm Higgs}&=& - g\left( H_u^* \tau^a H_u + H_d^* \tau^a H_d\right),\\
D^\prime |_{\rm Higgs}&=& -\frac{g^\prime}{2}
 \left( |H_u^+|^2 + | H_u^0|^2 - | H_d^0|^2 - | H_d^-|^2 \right)
\label{chap4:Dterms}
\eeq
\beq
g= \frac{e}{\sin \theta_W} =\frac{e}{s_W}, \,\, \,\, g^\prime= \frac{e}{\cos \theta_W}= \frac{e}{c_W}~
\eeq

\beq
V(H_u,H_d)&=&
(|\mu|^2 + m^2_{H_u}) (|H_u^0|^2 + |H_u^+|^2)
\nonumber \\ &&
+ (|\mu|^2 + m^2_{H_d}) (|H_d^0|^2 + |H_d^-|^2)
\nonumber \\ &&+\, b\, (H_u^+ H_d^- - H_u^0 H_d^0) + h.c.
+ {{1}\over{2}} g^2 |H_u^+ H_d^{0*} + H_u^0 H_d^{-*}|^2 
\\ &&+ {1\over 8} (g^2 + g^{\prime 2})
( |H_u^0|^2 + |H_u^+|^2 - |H_d^0|^2 - |H_d^-|^2 )^2~\nonumber 
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Electroweak symmetry breaking}

$SU(2)_L$ gauge transformation can  set $\langle H_u^+\rangle =0$.
If we look for a stable minimum along the charged directions we find
\beq
\frac{\partial V}{\partial H_u^+}|_{\langle H_u^+\rangle =0} = b H_d^- +\frac{g^2}{2} H_d^{0*} H_d^- H_u^{0*}
\eeq
will not vanish for nonzero $H_d^-$ for generic values of the parameters.

\beq
V(H_u^0,H_d^0) &=&
(|\mu|^2 + m^2_{H_u}) |H_u^0|^2
+ (|\mu|^2 + m^2_{H_d}) |H_d^0|^2
- (b\, H_u^0 H_d^0 + h.c.)
\nonumber \\ && + {1\over 8} (g^2 + g^{\prime 2})
( |H_u^0|^2 - |H_d^0|^2 )^2 .
\label{chap4:Higgspot}
\eeq

origin is not a stable minimum requires:
\beq
b^2 > (|\mu|^2 + m^2_{H_u} )(|\mu|^2 + m^2_{H_d}).
\eeq
stabilizing $D$-flat direction  $H_u^0=H_d^0$ where the $b$ term is arbitrarily negative requires
\beq
2 b< 2 |\mu |^2 + m^2_{H_u} + m^2_{H_d}.
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Electroweak symmetry breaking}

tight relation between $b$ and $\mu$\newline there is no solution if $m^2_{H_u} = m^2_{H_d}$.  Typically,
choose $m^2_{H_u}$ and  $m^2_{H_d}$ to have opposite signs and different magnitudes

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.51\hsize]{slideimages/muproblem}}
\caption[]{Above the top line
the Higgs VEVs go to $\infty$, while below the bottom line the Higgs VEVs go to zero.}
\label{Chap4:fig:muproblem}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Electroweak symmetry breaking}


\beq
\langle H_u^0\rangle = {{v_u}\over{\sqrt{2}}}~,
\\
\langle H_d^0\rangle = {{v_d}\over{\sqrt{2}}}~.
\eeq
VEVs  produce 
masses for the $W$ and $Z$
\beq
M_W^2={{1}\over{4}} g^2 v^2~,  \\
M_Z^2={{1}\over{4}}(g^2 + g^{\prime 2})v^2~,
\eeq
where we need to have
\beq
v^2 = v_u^2 + v_d^2   \approx (246\>{\rm
GeV})^2~,
\eeq

define an angle $\beta$:
\beq
s_\beta\equiv \sin\beta \equiv {{v_u}\over{v}}~, \,\,\,
c_\beta\equiv  \cos\beta\equiv {{v_d}\over{v}}~,
\eeq
with $0 < \beta < \pi/2$.  From this definition it follows that
\beq
\tan\beta &=& v_u/v_d ~,\\
\cos 2\beta&=&{{v_d^2-v_u^2}\over{v^2}}~.
\eeq



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Electroweak symmetry breaking}

imposing $\partial V/\partial H_u^0=
\partial V/\partial H_d^0 = 0$ gives
\beq
|\mu |^2 + m_{H_u}^2 &=&  b \cot\beta + (M_Z^2/2) \cos 2\beta~.
\\
|\mu |^2 + m_{H_d}^2 &=&  b \tan\beta - (M_Z^2/2) \cos 2\beta~,
\eeq
this is another way of seeing the $\mu$-problem.

Higgs scalar fields consist of 
 eight real scalar degrees of freedom.
three  are eaten by the $Z^0$ and $W^\pm$. This leaves five 
degrees of freedom:
$H^\pm$, the $h_0$ and $H^0$ which are CP even and the $A^0$ is CP odd.


shift the fields by their VEVs:
\beq
H_u^0 \rightarrow {{v_u}\over{\sqrt{2}}}+ H_u^0~,
\\
H_d^0 \rightarrow {{v_d}\over{\sqrt{2}}}+ H_d^0~,
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Higgs spectrum}

\beq
V \supset ({\rm Im} H_u^0,{\rm Im}H_d^0)
\left( \begin{array}{cc}
b \cot\beta & b \\
b & b \tan \beta
\end{array} \right)
\left( \begin{array}{c}
{\rm Im} H_u^0 \\
{\rm Im}H_d^0
\end{array} \right).
\eeq
Diagonalizing, we find the two mass eigenstates:
\beq
\pmatrix{\pi^0 \cr A^0 } = \sqrt{2}
\pmatrix{s_\beta & -c_\beta\cr
          c_\beta &  s_\beta } \pmatrix{{\rm Im}H_u^0 \cr
                                              {\rm Im}H_d^0 }.
\eeq
would-be Nambu--Goldstone boson $\pi^0$ is massless 
\beq
m_{A}^2 = {{b}\over{s_\beta c_\beta}}~.
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Higgs spectrum}


\beq
V \supset (H_u^{+*},H_d^-)
\left( \begin{array}{cc}
b \cot\beta +M_W^2 c^2_\beta& b +M_W^2 c_\beta s_\beta \\
b+M_W^2 c_\beta s_\beta & b \tan \beta+M_W^2 s^2_\beta
\end{array} \right)
\left( \begin{array}{c}
H_u^+ \\
H_d^{-*}
\end{array} \right),
\eeq
mass eigenstates
\beq
\pmatrix{\pi^+ \cr H^+ } =
\pmatrix{s_\beta & -c_\beta\cr
          c_\beta &  s_\beta } \pmatrix{H_u^+ \cr
                                              H_d^{-*} },
\eeq
where 
$\pi^- = \pi^{+*}$ and $H^- = H^{+*}$. 
\beq
m^2_{H^\pm} =& m^2_{A} + M_W^2~.
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Higgs spectrum}



\beq
V \supset ({\rm Re} H_u^0,{\rm Re}H_d^0)
\left( \begin{array}{cc}
b \cot\beta +M_Z^2 s^2_\beta& -b -M_Z^2 c_\beta s_\beta) \\
-b -M_Z^2 c_\beta s_\beta) & b \tan \beta+M_Z^2 c^2_\beta
\end{array} \right)
\left( \begin{array}{c}
{\rm Re} H_u^0 \\
{\rm Re}H_d^0
\end{array} \right),
\eeq
mass eigenstates
\beq
\pmatrix{h^0 \cr H^0 } = \sqrt{2}
\pmatrix{\cos\alpha & -\sin\alpha\cr
          \sin\alpha &  \cos\alpha } \pmatrix{{\rm Re}H_u^0 \cr
                                              {\rm Re}H_d^0},
\eeq
with masses
\beq
m^2_{h, H} &=& {{1}\over{2}}
\left(
m^2_{A} + M_Z^2 \mp \sqrt{(m_{A}^2 + M_Z^2)^2 - 4 M_Z^2 m_{A}^2
\cos^2 2\beta} \right),
\label{Chap4:m2hH}
\eeq
and the mixing angle $\alpha$  is determined given  by
\beq
{\sin 2\alpha\over \sin 2\beta} =
-{m_{A}^2 + m_{Z}^2 \over m_{H}^2 - m^2_{h}},\,\,\,
{\cos 2\alpha\over \cos 2\beta} =
-{m_{A}^2 - m_{Z}^2 \over m_{H}^2 - m^2_{h}}~.
\eeq
By convention,  $h^0$ corresponds to the lighter mass eigenstate


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Higgs spectrum}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\hsize]{slideimages/higgssector}}
\end{figure}

\begin{center}\Blue{Carena, Haber,  hep-ph/0208209}\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Higgs spectrum}



Note that $m_A$, $m_H^\pm$, and $m_H \rightarrow \infty$ as $b \rightarrow \infty$
but $m_h$ is maximized at $m_A = \infty$ so at tree-level there is an upper bound on the Higgs mass
\beq
m_{h} <  |\cos 2\beta | M_Z~,
\eeq
which is  ruled out by experiment
\newline 
There can be large  one-loop  corrections to the Higgs mass



%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The sparticle spectrum}
\label{Chap4:mssm:sec:spectrum}

 gluino, $\wtilde G$, which is a color octet fermion with  mass $|M_3|$
 
for squarks and sleptons masses  have to diagonalize 6$\times$6 matrices\newline neglecting the intergenerational mixing stop mass terms are given by
\beq
\lagr_{\rm stop} = -\pmatrix{\wtilde t_L^* & \wtilde t_R^*}
{\bf m_{\wtilde t}^2} \pmatrix{\wtilde t_L \cr \wtilde t_R}~ 
\eeq

\beq
{\bf m_{\wtilde t}^2} =
\pmatrix{
m^2_{Q33} + m_t^2 +\delta_{u} & v(A_{u33} \, s_\beta - \mu
y_tc_\beta )\cr
v (A_{u33} \, s_\beta - \mu y_tc_\beta ) & m^2_{\overline u 33} + m_t^2 +
\delta_{\overline u}
} ,
\eeq
where
\beq
\delta_f= - g T^3_f \langle D^3\rangle - g^{\prime}Y_f \langle D^\prime\rangle =(T^3_f - Q_f s^2_W)
\cos 2\beta\, M_Z^2~,
\eeq


$m^2_{Q33}$ and $m^2_{\overline u 33}$ and $A_{u33}$ are soft SUSY breaking  terms  \newline
$m_t^2$ terms come from quartic  with two Higgses\newline
 $\delta_f$ terms represent the contributions from quartic $D$-terms  \newline
 terms $\propto$ $\mu$ arise from integrating out the 
Higgs auxiliary fields
 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{stop mixing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\hsize]{slideimages/stopDterm}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.65\hsize]{slideimages/muTrilinear}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The sparticle spectrum}



 for bottom squarks and
tau
sleptons\beq
{\bf m_{\wtilde b}^2} =
\pmatrix{
m^2_{Q33} +m_b^2 + \delta_{d} & v (A_{d33}\, c_\beta - \mu
y_bs_\beta )\cr
v (A_{d33}\, c_\beta - \mu y_bs_\beta ) & m^2_{\overline d33} +m_b^2 +
\delta_{\overline d}
},
\label{Chap4:msbottommatrix}
\eeq
\beq
{\bf m_{\wtilde \tau}^2} =
\pmatrix{
m^2_{L33} +m_\tau^2 +\delta_{e} & v (A_{e33}\, c_\beta -\mu
y_\tau s_\beta )\cr
v (A_{e33}  \, c_\beta - \mu y_\tau s_\beta ) & m^2_{\overline e33}
+m_\tau^2+ \delta_{\overline e}
} 
\eeq
 large Yukawa couplings or  $A$-terms allow for
large mixing and the possibility that the lower mass squared
eigenvalue is driven negative   This would  break $U(1)_{\rm em}$ and/or $SU(3)_c$, and must   be avoided 

 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The sparticle spectrum}


without
soft SUSY breaking mass terms,  6$\times$6 mixing matrices
\beq
{\bf m_{\wtilde u}^2} &=&
\pmatrix{
{\bf m_u}^\dagger {\bf m_u} + \delta_{u}\,{\bf I} & {\bf \Delta}_u\cr
{\bf \Delta_u}^\dagger & {\bf m_u} {\bf m_u}^\dagger +
\delta_{\overline u}\,{\bf I}
},\\
{\bf m_{\wtilde d}^2} &=&
\pmatrix{
{\bf m_d}^\dagger {\bf m_d} +\delta_{d}\,{\bf I} & {\bf \Delta_d} \cr
{\bf \Delta_d}^\dagger & {\bf m_d} {\bf m_d}^\dagger  +
\delta_{\overline d}\,{\bf I}
},
\eeq
where ${\bf m_u}$ and ${\bf m_d}$ are the $3\times 3$ quark mass
matrices, ${\bf I}$ is the  identity matrix Note that $\delta_u +\delta_{\overline u}+\delta_d +\delta_{\overline d}=0$, so at least
one $\delta_f \le 0$  Suppose $\delta_u \le 0$, let $\vec \gamma$ be an eigenvector  with the smallest eigenvalue,
\beq
{\bf m_u} \vec \gamma = m_u \vec \gamma~,
\eeq
squark mass$^2 >0$,
upper bound on the smallest squark eigenvalue, $m_{min}^2$ 
\beq
m_{min}^2 \le ({\vec \gamma}^T,0) {\bf m_{\wtilde u}^2} \pmatrix{ {\vec \gamma} \cr 0} \le m_u^2 
\eeq
So there would be a squark lighter than the $u$ quark 


 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Chargino spectrum}





In the basis $\psi =
(\wtilde W^+,\, \wtilde
H_u^+,\, \wtilde W^- ,\, \wtilde H_d^- )$, the chargino mass terms are
\beq
\lagr_{\rm chargino}= -{{1}\over{2}} \psi^T {\bf M}_{\wtilde C} \psi
+h c 
\eeq
where
\beq
{\bf M}_{\wtilde C}
= \pmatrix{{\bf 0}&{\bf M}^T\cr
         {\bf M} &{\bf 0}},\,\,\,
{\bf M} = \pmatrix{M_2 & \sqrt{2} s_\beta\, M_W\cr
                              \sqrt{2} c_\beta\, M_W & \mu \cr } 
\eeq
mixing comes  from the wino--higgsino--Higgs coupling\newline can be diagonalized by a singular value decomposition:
\beq
{\bf L}^* {\bf M} {\bf R}^{-1} =
\pmatrix{m_{\wtilde C_1} & 0\cr
              0   & m_{\wtilde C_2}},
\label{chap4:SVD}
\eeq
with mass eigenstates given by
\beq
\pmatrix{\wtilde C^+_1\cr
         \wtilde C^+_2} = {\bf R}
\pmatrix{\wtilde W^+\cr
         \wtilde H_u^+},
\pmatrix{\wtilde C^-_1\cr
         \wtilde C^-_2} = {\bf L}
\pmatrix{\wtilde W^-\cr
         \wtilde H_d^-},
\eeq

 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Chargino spectrum}

After diagonalization the elements of ${\bf L}$ and ${\bf R}$ appear in the interaction vertices for chargino mass eigenstates 
\beq
m^2_{{\wtilde C}_{1},{\wtilde C}_{2}}
& =& {1\over 2} 
\left[ (|M_2|^2 + |\mu|^2 + 2 M_W^2)\right.
\nonumber
\\
&&\left.  \mp
\sqrt{(|M_2|^2 + |\mu |^2 + 2 M_W^2 )^2 - 4 | \mu M_2 - M_W^2 \sin 2
\beta |^2 }
\right] 
\eeq
In the limit  that $||\mu| \pm M_2| \gg M_W$
the charginos are approximately a wino and
a higgsino with masses $|M_2|$ and $|\mu|$ 

 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Neutralino spectrum}


$\psi^0 =$ ($\wtilde B\vphantom{^0}$, $\wtilde W^3$, $\wtilde H_d^0$, $\wtilde H_u^0$),
mass terms in the Lagrangian are
\beq
\lagr_{\rm neutralino} -{{1}\over{2}} (\psi^{0})^T {\bf M}_{\wtilde N} \psi^0 + h c 
\eeq
where
\beq
{\bf M}_{\wtilde N} = \pmatrix{M_1 & 0 & - c_\beta\, s_W\, M_Z &
s_\beta\, s_W \, M_Z\cr
0 & M_2 & c_\beta\, c_W\, M_Z & - s_\beta\, c_W\, M_Z \cr
-c_\beta \,s_W\, M_Z & c_\beta\, c_W\, M_Z & 0 & -\mu \cr
s_\beta\, s_W\, M_Z & - s_\beta\, c_W \, M_Z& -\mu & 0 \cr } 
\eeq
mixing terms  come from the wino--higgsino--Higgs and  bino--higgsino--Higgs couplings

Since  ${\bf M}_{\wtilde N} $ is a symmetric complex matrix it can be diagonalized
by a Takagi factorization using a unitary matrix ${\bf U}$
\beq
{\bf M}^{\rm diag}_{\wtilde N} =
{\bf U}^* {\bf M}_{\wtilde N} {\bf U}^{-1}~.
\label{Chap4:diagmN}
\eeq

 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Neutralino spectrum}


In the region of parameter space where
\beq
M_Z \ll |\mu \pm M_{1}|, |\mu \pm M_{2}|
\eeq
then the
neutralino mass eigenstates are very nearly
$ \wtilde B$, $ \wtilde W^0$,
$(\wtilde H_u^0 \pm
\wtilde H_d^0)/\sqrt{2}$, with masses:
$(|M_1|,|M_2|,|\mu|,|\mu|)$.

A``bino-like" LSP
can make a good
dark matter candidate,  $N_1$ is often  arranged to
be the LSP

%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Spectrum}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\hsize]{slideimages/spectrum}}
\end{figure}

\begin{center}\Blue{Martin,  hep-ph/9709356}\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dark Matter}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\hsize]{slideimages/bullet}}
\end{figure}

\begin{center}\Blue{  {\tt astro-ph/0608407}}\end{center}

 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dark Matter Relic Abundance}
Robertson-Walker metric and scale factor $R$
\beq
ds^2=-dt^2+R(t)^2\left(\frac{dr^2}{1-k r^2} + r^2 d\theta^2+r^2 \sin^2 \theta d \phi^2\right)
\eeq
 Friedman equation
\beq
H^2 \equiv \left( \frac{\dot R}{R}\right)^2 = \frac{8}{3} \pi G \rho  -\frac{k}{R^2}+ \ldots~,
\eeq
relates the Hubble parameter $H$  to Newton's constant, $G$, times the energy density, $\rho$, the critical density is for $k=0$ is
\beq
\rho_c = \frac{3 H^2}{8 \pi G} \approx 10^{-29} \,{\rm g}/{\rm cm}^3 \approx 3 \times 10^{-47} \,{\rm GeV}^4~.
\eeq

 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dark Matter Relic Abundance}
Energy conservation
\beq
R^3 \left( \frac{d p}{dt} \right)&=& \frac{d}{dt}\left[R^3\left(\rho+p\right)\right]\\
 \frac{d p}{dt} &=& -3 \frac{\dot R}{R} \left(\rho+p\right)
\eeq

for $p =a \rho$
\beq
\rho \propto R^{-3(1+a)}
\eeq 
\beq
\begin{array}{ccc}
{\rm radiation} & a=1/3&\rho \propto R^{-4} \\
{\rm matter} & a=0&\rho \propto R^{-3} \\
{\rm curvature} & a=0&\rho \propto R^{-2} \\
{\rm vacuum}\,\,{\rm  energy} & a=-1&\rho \propto R^{0} 
\end{array}
\eeq


 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dark Matter Relic Abundance}
a stable weakly interacting dark matter particle $X$ is held in equilibrium by annihilations 
\beq
XX\leftrightarrow p_i {\overline p_i}
\eeq eventually the expansion of the Universe dilutes the particles so  they are too  sparse to maintain equilibrium


equilibrium number density, $n_{eq}$,  thermal average of the annihilation cross section times the relative velocity $\langle \sigma v \rangle$
\newline ${\dot n}_{\rm annihilations}\sim \langle \sigma v \rangle n_{eq}^2$
 \newline ${\dot n}_{\rm 
 expansion}\sim 3 H n_{eq}$
 \newline when  ${\dot n}_{\rm annihilations} \approx {\dot n}_{\rm 
 expansion}$  dark matter `freezes out"
 \newline  after freeze out, number of dark matter particles per comoving volume $N\equiv n/T^3$ remains constant 
 %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Freeze Out}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\hsize]{slideimages/freezeout2}}
\end{figure}


%%%%%%
\myhead{Quantum Stat. Mech.}
Bose-Einstein and Fermi-Dirac
\beq
b(E)&=& \frac{1}{e^{(E-\mu)/T}-1}\\
f(E)&=& \frac{1}{e^{(E-\mu)/T}+1}
\eeq
assume chemical potential $\mu=0$ and relativistic
\beq
N_b&=&\frac{g_s}{2 \pi^2} \int_0^\infty dp \frac{p^2}{e^{p/T}-1}\\
N_f&=&\frac{g_s}{2 \pi^2} \int_0^\infty dp \frac{p^2}{e^{p/T}+1}
\eeq

\beq
\begin{array}{cc}
{\rm scalar} & g_s=1\\
{\rm Dirac} & g_s=2\times 2=4\\
{\rm Majorana} & g_s=2\\
{\rm photon} & g_s=2\\
Z & g_s=3\\
W & g_s=2\times 3=6
\end{array}
\eeq

%%%%%%
\myhead{Quantum Stat. Mech.}

\beq
\int_0^\infty dx \frac{x^{\nu-1}}{e^{ax}-1}&=& a^{-\nu}\,\Gamma(\nu)\zeta(\nu)\\
\int_0^\infty dx \frac{x^{\nu-1}}{e^{ax}+1}&=& (1-2^{1-\nu})\,a^{-\nu}\,\Gamma(\nu)\zeta(\nu)
\eeq

\beq
N_b&=&\frac{g_s}{ \pi^2} \zeta(3) T^3\vphantom{\sqrt{\frac{1}{2}}}\\
N_f&=&\frac{3}{4} \frac{g_s}{ \pi^2} \zeta(3) T^3
\eeq

\beq
\rho_b&=&\frac{g_s}{2 \pi^2} \int_0^\infty dp \frac{p^3}{e^{p/T}-1}=\frac{g_s \pi^2}{ 30}  T^4\\
\rho_f&=&\frac{g_s}{2 \pi^2} \int_0^\infty dp \frac{p^3}{e^{p/T}+1}=\frac{7}{8}\,\frac{g_s \pi^2}{ 30}  T^4
\eeq
where we used $\zeta(4)=\pi^4/90$


%%%%%%
\myhead{Quantum Stat. Mech.}

assume chemical potential $\mu=0$ and non-relativistic $m\gg T$
\beq
N_{f,b}&\approx&\frac{g_s}{2 \pi^2} \int_0^\infty dp \frac{p^2}{e^{m/T+p^2/(2 mT)}\pm1}\\
&\approx&\frac{g_s T^3}{2 \pi^2} \int_0^\infty du \frac{u^2}{e^{m/T+u^2 T/m}\pm1}\\
&\approx&\frac{g_s T^3 e^{-m/T} }{2 \pi^2} \int_0^\infty du \,u^2 \,e^{-u^2 T/m}\vphantom{\sqrt{\frac{1}{2}}}\\
&\approx&\frac{ g_s T^3 e^{-m/T} }{(2 \pi T/m)^{3/2}}
\eeq

  %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Equilibrium}
equilibrium number of nonrelativistic particles
 per comoving volume:
 \beq
N_{eq} = \frac{e^{-m_X/T}}{\left(2 \pi\right)^{3/2}} \left( \frac{m_X}{T}\right)^{3/2}~
 \eeq
above $T\approx 1$~eV the universe is radiation-dominated
\beq 
\rho=  \frac{\pi^2}{15}\,N_*\, T^4
\eeq
\beq
N_*=\frac{1}{2}\left( n_b+\frac{7}{8} n_f\right)
\eeq
so 
\beq
H = \sqrt{  \frac{8}{3} \pi G \rho } =\sqrt{ \frac{8 \pi^3 N_* G}{15} }\, T^2~
\eeq
\beq
\langle \sigma v\rangle = \sigma_0 \left( \frac{T}{m}\right)^\alpha~,
\eeq
 $\alpha=0$ for  Dirac fermion, $\alpha=1$ for a Majorana fermion %%%%%%%%%%%%%%%%%%
   %%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Cross Sections}

Dirac fermion:
 \beq
\langle \sigma v\rangle = \frac{G_F^2}{2 \pi} m_X^2
\eeq

Majorana fermions have no vector current couplings\newline only axial current:
 \beq
\langle \sigma v\rangle \propto \frac{G_F^2}{2 \pi} p^2
\eeq

referred to as p-wave suppression

 \beq
\langle p^2 \rangle = \frac{3}{2} m_X T
\eeq




%%%%%%
\myhead{Freeze Out}

Equating the annihilation rate with the expansion rate  at $T=T_f$ 
\beq
 \langle \sigma v \rangle n_{eq}^2
&= & 3 H n_{eq}\\

 \sigma_0 \left( \frac{T_f}{m_X}\right)^\alpha  \frac{e^{-m_X/T_f}}{\left(2 \pi\right)^{3/2}} \left( \frac{m_X}{T_f}\right)^{3/2} T_f^3 &= & 3 \sqrt{ \frac{8 \pi^3 N_* G}{15} }\, T_f^2
 \eeq
\beq
e^{-m_X/T_f} = 3\sqrt{ \frac{8 \pi^3 N_* G}{15} }\,\frac{(2 \pi)^{3/2} }{\sigma_0 \,m_X} 
\left( \frac{m_X}{T_f}\right)^{\alpha-1/2}~
\eeq
Numerically $m_X/T_f \approx 30$.
So the number  per comoving volume at $T_f$ is
\beq
N_f=  \sqrt{ \frac{8 \pi^3 N_* G}{15} }\,\frac{3}{\sigma_0 \,m_X} 
\left( \frac{m_X}{T_f}\right)^{1+\alpha}~
\eeq
 $\times T^3$ gives the number density,  $\times m_X$ gives the energy density.   weak  annihilation cross section $\sigma_0 =N_A G_F^2 m_X^2/2 \pi$ (where $N_A$  counts final states) with a current temperature of $T=2.7$~K $=  2\times  10^{-13}$~GeV, $\alpha =1$, $N_*=100$, $N_A=20$, that
\beq
\frac{\rho_X}{\rho_c} = 0.6 \left(\frac{100 {\rm GeV}}{m_X}\right)^2
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Stable WIMPS}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\hsize]{slideimages/stablewimp2}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{LSP Dark Matter}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\hsize]{slideimages/LSPDM}}
\end{figure}

\begin{center}\Red{Bino}, \Blue{Higgsino}, \Green{Wino}\end{center}
\begin{center}\Blue{Arkani-Hamed, Delgado, Giudice,  hep-ph/0601041}\end{center}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Xenon Detector}
\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\hsize]{slideimages/Xenon}}
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Xenon 100 and LUX}
\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\hsize]{slideimages/wimplimit_new}}
\end{figure}

	\begin{center}\Blue{Schumann 1405.7600}\end{center}
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dark Matter Searches}
\begin{figure}[htb]
\centerline{\includegraphics[width=0.45\hsize]{slideimages/DMUpdate1}}
\end{figure}


\myhead{Dark Matter Searches}
\begin{figure}[htb]
\centerline{\includegraphics[width=0.66\hsize]{slideimages/DMUpdate2}}
\end{figure}


		

	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$SU(5)$ GUT}

\bc
$SU(5)\rightarrow SU(3) \times SU(2) \times U(1)$
\ec
\beq
{\bf 5} \rightarrow ({\bf 3},{\bf 1})_{-1/3}+({\bf 1},{\bf 2})_{1/2} \sim d_R+(e^c,-\nu^c)_L
\eeq
\beq
{\bf {\bar  5}} \rightarrow ({\bf {\bar 3}},{\bf 1})_{+1/3}+({\bf 1},{\bf 2})_{-1/2} \sim d_R^c+(\nu, e)_L
\eeq

\beq
{\bf 5} \times {\bf 5}= {\bf 10}_A+{\bf 15}_S
\eeq
\beq
{\bf 10} &\rightarrow& ({\bf 3},{\bf 2})_{1/6}+({\bf {\bar 3}},{\bf 1})_{-2/3}+ ({\bf 1},{\bf 1})_{1}\\
&& \sim (u,d)_L + u_R^c+e^c
\eeq

\bc
${\bf {\bar  5}}+{\bf 10} $ is anomaly free
\ec

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$SU(5)\rightarrow SU(3) \times SU(2) \times U(1)$}
\beq
{\bf 5} \rightarrow ({\bf 3},{\bf 1})_{-1/3}+({\bf 1},{\bf 2})_{1/2}
\eeq
\beq
{\bf 5} \times {\bf {\bar  5}}&=& {\bf 1}+{\bf 24}\\
&=& ({\bf 1},{\bf 1})_0+({\bf 8},{\bf 1})_0 +({\bf 1},{\bf 1})_0+({\bf 1},{\bf 3})_0\\
&&+({\bf 3},{\bf 2})_{-5/6}+({\bf {\bar 3}},{\bf 2})_{5/6}
\eeq
\beq
{\bf 24} &\rightarrow& ({\bf 8},{\bf 1})_0 +({\bf 1},{\bf 3})_0+({\bf 1},{\bf 1})_0\\
&&+({\bf 3},{\bf 2})_{-5/6}+({\bf {\bar 3}},{\bf 2})_{5/6}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$SU(5)\rightarrow SU(3) \times SU(2) \times U(1)$}
\beq
G^a_\mu\leftrightarrow T^{1,...,8} =\frac{1}{2}\left( \begin{array}{cc}\lambda^{1,...,8} & 0\\0&0\end{array}\right)
\eeq
\beq
W^a_\mu\leftrightarrow T^{9,10,11} =\frac{1}{2}\left( \begin{array}{cc} 0& 0\\0&\sigma^{1,2,3} \end{array}\right)
\eeq
\beq
X_\mu,Y_\mu \leftrightarrow T^{12,...,23} =\frac{1}{2}\left( \begin{array}{cc}0 & x\\x^\dagger&0 \end{array}\right)
\eeq
\beq
B_\mu \leftrightarrow T^{24} =\frac{1}{2\sqrt{15}}\left( \begin{array}{ccccc}
-2&0&0&0 & 0
\\0&-2&0&0&0\\
0&0&-2&0&0\\
0&0&0&3&0\\
0&0&0&0&3
\end{array}\right)
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$SU(5)\rightarrow SU(3) \times SU(2) \times U(1)$}

\beq
{\rm Tr}\, T^{24} T^{24} = \frac{1}{4 \cdot 15} {\rm Tr} \left( \begin{array}{ccccc}
4&0&0&0 & 0
\\0&4&0&0&0\\
0&0&4&0&0\\
0&0&0&9&0\\
0&0&0&0&9
\end{array}\right)=\frac{1}{2}
\eeq
\beq
Y =\frac{\sqrt{15}}{3}\, T^{24}= \sqrt{\frac{5}{3}} \, T^{24}
\eeq

\beq
g^\prime Y = \left(\sqrt{\frac{5}{3}} g^\prime\right)  \left(\sqrt{\frac{3}{5}} Y\right) =g_1 T^{24}
\eeq
\beq
g_1 = \sqrt{\frac{5}{3}} g^\prime
\eeq
\beq
Q_1= T^{24}=\sqrt{\frac{3}{5}} Y
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Gauge coupling unification}
\label{Chap4:mssm:sec:couplingunification}

for $SU(5)_{GUT}$ \beq
g_1 &\equiv&  \sqrt{{5}\over{3}} g^\prime~,\,\,\,g_2 \equiv g, \,\,\, g_3 \equiv g_C,\,\,\,\
\alpha_i \equiv \frac{g_i^2}{4\pi}~
\eeq
The measured values of gauge couplings renormalized at $M_Z$ are
\beq
\alpha_1(M_Z) &=&0.016830\pm 0.000007~\\
\alpha_2(M_Z) &=&0.03347\pm 0.00003~\\
\alpha_3(M_Z) &=&0.1187\pm 0.002~
\eeq
These couplings run at one-loop according the RG equation:
\beq
\mu\, \frac{d g_a}{d\mu}  = - {1\over 16\pi^2} b_a g_a^3 \,\,\,
\,\, \Rightarrow \,\,\,\,\, \mu \, \frac{d \alpha_a^{-1}}{ d\mu} =
{b_a\over 2\pi}~
\eeq
In the SM and MSSM the $\beta$ function coefficients are
\beq
b^{\rm SM}_a &=&(-41/10, 19/6,7)~\\
b^{\rm MSSM}_a &=&(-33/5,-1,3)
\label{Chap4:bMSSM}
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{SM $\beta$-functions}
\beq
b_1&=&-\frac{2}{3} Q_F^2-\frac{1}{3} Q_S^2= -\frac{3}{5}\left(\frac{2}{3}Y_F^2+\frac{1}{3}Y_S^2\right)\vphantom{\sqrt{\frac{1}{2}}}\\
&=&-\frac{3}{5}\left(\frac{2}{3}N_{\rm gen}\left[ 3\cdot 2 \cdot Y_Q^2+3 Y_u^2+3 Y_d^2+2 Y_L^2+Y_e^2\right]+\frac{1}{3}2 Y_H^2\right)\\
&=&-\frac{1}{5}\left(2 N_{\rm gen}\left[ 3\cdot 2 \cdot \left(\frac{1}{6}\right)^2+3 \left(\frac{2}{3}\right)^2+3 \left(\frac{1}{3}\right)^2+2 \left(\frac{1}{2}\right)^2+1^2\right]+2 \left(\frac{1}{2}\right)^2\right)\vphantom{\sqrt{\frac{1}{2}}}\\
&=&-\frac{1}{5}\left(2N_{\rm gen}\left[ \frac{1}{6}+\frac{4}{3}+\frac{1}{3}+\frac{1}{2}+1\right]+\frac{1}{2} \right)\vphantom{\sqrt{\frac{1}{2}}}\\
&=&-\frac{1}{5}\left(N_{\rm gen}\frac{1+8+2+9}{3}+\frac{1}{2} \right)= -\frac{1}{5}\left(N_{\rm gen}\frac{20}{3}+\frac{1}{2} \right)\vphantom{\sqrt{\frac{1}{2}}}\\
&=& -\frac{41}{10}\\
b_2&=& \frac{11}{3} \cdot N - \frac{2}{3} T(F) - \frac{1}{3} T(S)\vphantom{\sqrt{\frac{1}{2}}}
= \frac{22}{3}- \frac{2}{3} N_{\rm gen}\left( 3 \cdot \frac{1}{2}+\frac{1}{2}\right)- \frac{1}{3} \cdot \frac{1}{2}\\
&= &\frac{22}{3}- \frac{4}{3} N_{\rm gen}- \frac{1}{6} 
= \frac{22}{3}- \frac{4}{3} N_{\rm gen}- \frac{1}{6} = \frac{20-1}{6}\vphantom{\sqrt{\frac{1}{2}}}\\
&=& \frac{19}{6}\\
b_3&=& \frac{11}{3} \cdot 3 - \frac{2}{3} T(F) 
= \frac{33}{3}- \frac{2}{3} N_{\rm gen}\left( 2 \cdot 2 \cdot \frac{1}{2} \right)
= \frac{33}{3}- \frac{4}{3} N_{\rm gen} \vphantom{\sqrt{\frac{1}{2}}}\\
&=&\frac{33-12}{3}=7
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{MSSM $\beta$-functions}
\beq
b_1&=&-\frac{2}{3} Q_F^2-\frac{1}{3} Q_S^2= -Q^2=-\frac{3}{5}Y^2 \vphantom{\sqrt{\frac{1}{2}}}\\
&=&-\frac{3}{5}\left(N_{\rm gen}\left[ 3\cdot 2 \cdot Y_Q^2+3 Y_u^2+3 Y_d^2+2 Y_L^2+Y_e^2\right]+
2\cdot 2 Y_H^2\right)\\
&=&-\frac{3}{5}\left( N_{\rm gen}\left[ 3\cdot 2 \cdot \left(\frac{1}{6}\right)^2+3 \left(\frac{2}{3}\right)^2+3 \left(\frac{1}{3}\right)^2+2 \left(\frac{1}{2}\right)^2+1^2\right]+4 \left(\frac{1}{2}\right)^2\right)\vphantom{\sqrt{\frac{1}{2}}}\\
&=&-\frac{3}{5}\left(N_{\rm gen}\left[ \frac{1}{6}+\frac{4}{3}+\frac{1}{3}+\frac{1}{2}+1\right]+1 \right)\vphantom{\sqrt{\frac{1}{2}}}\\
&=&-\frac{3}{5}\left(N_{\rm gen}\frac{1+8+2+9}{6}+1 \right)= -\frac{3}{5}\left(N_{\rm gen}\frac{20}{6}+1 \right)\vphantom{\sqrt{\frac{1}{2}}}\\
&=& -\frac{33}{5}\\
b_2&=& 3 N - F \vphantom{\sqrt{\frac{1}{2}}}
= 3\cdot 2-N_{\rm gen}\left( 3 \cdot \frac{1}{2}+\frac{1}{2}\right)- 1 \\
&= &6-2N_{\rm gen}-1\\
&=&-1\\
b_3&=& 3 \cdot 3 - 2 N_{\rm gen}  
=9-6\\
&=&3
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Gauge coupling unification}
\label{Chap4:mssm:sec:couplingunification}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.55\hsize]{slideimages/susygut}\includegraphics[width=0.55\hsize]{slideimages/susygutblow}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

common threshold $M_{\rm SUSY}$
\beq
3~{\rm GeV} < M_{\rm SUSY}< 100~{\rm TeV}.
\eeq

\beq
M_U \approx
2 \times 10^{16}~{\rm GeV}.
\label{chap4:unificationscale}
\eeq



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Radiative electroweak symmetry breaking}
\label{Chap4:mssm:sec:radESB}

RG equations for the soft SUSY breaking masses of the Higgs and third-generation scalars 
\newline gaugino terms additive consider separately 
\newline consider  only the running induced by $y_t$ 
\beq
16\pi^2{d\over dt} 
m_{H_d}^2 &=&0~
\\ && \\
16 \pi^2 {d\over dt} 
\left( \begin{array}{c}
m_{H_u}^2 \\
m_{\overline u33}^2 \\
m_{Q33}^2 \end{array}\right)
&=&
2 |y_t|^2 
\left( \begin{array}{ccc}
3 & 3 & 3 \\
2 & 2 & 2 \\
1 & 1 & 1
\end{array}\right)
\left( \begin{array}{c}
m_{H_u}^2 \\
m_{\overline u33}^2 \\
m_{Q33}^2 \end{array}\right)
\label{chap4:RadEWSB}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Radiative electroweak symmetry breaking}
 transform to an eigenbasis: $(1,-1,0)$, $(0,1,-1)$, and $(3,2,1)$ \newline eigenvalues 0, 0 , and 6.
\newline  eigenvector $(3,2,1)$
scaled to zero 

$m_{H_u}^2=m_{\overline u_3}^2=m_{Q_3}^2=m_0^2$ at  high scale\newline decompose initial conditions:
\beq
\left( \begin{array}{c}
1 \\
1 \\
1 \end{array}\right)
= -\frac{1}{2}  \left( \begin{array}{r}
1 \\
-1 \\
0 \end{array}\right)
-\frac{1}{2}  \left( \begin{array}{r}
0 \\
1 \\
-1 \end{array}\right)
+\frac{1}{2}  \left( \begin{array}{c}
3 \\
2 \\
1 \end{array}\right)
\eeq
in IR  masses run to 
\beq
{{m_0^2}\over{2}}\left( \begin{array}{r}
-1 \\
0 \\
1 \end{array}\right)
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Radiative electroweak symmetry breaking}


 $m_{H_u}^2$ runs negative. EWSB may or may not follow depending on the
values of $\mu$ and $b$.  claimed that this
 ``predicted'' a large top mass, but it really only required
a large :
\beq
y_t = \frac{\sqrt{2}\, \, m_t}{v \sin\beta}
\label{chap4:topYuk}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Radiative EWSB}

\vspace{-4pt}
\begin{figure}[htb]
\centerline{\includegraphics[width=0.68\hsize]{slideimages/higgsrunning}}
\end{figure}
\begin{center}\Blue{Giudice, Rattazzi, hep-ph/0606105}\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{One-loop  Higgs mass}
\label{Chap4:mssm:sec:higgsmass}


 tree-level :
\beq
m_{h} <  |\cos 2\beta | M_Z = {{g^2+g^{\prime 2}}\over{4}}|v_d^2-v_u^2|~
\eeq
 Higgs mass is controlled by the quartic Higgs coupling \newline failure of the top-stop cancellation should give the leading correction 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\hsize]{slideimages/higgsSelfCoupling}}
\label{Chap4:fig:higgsSelfCoupling}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beq
\lambda(m_t) &=& \lambda(m_{\wtilde t})+ \int_{m_{\tilde t}}^{m_t} \beta_\lambda  \,d \ln \mu \\
 &=& \lambda_{\rm SUSY} + {{2 N_c |y_t|^4}\over{16 \pi^2}}
\ln\left( \frac{m_{{\tilde t}1}m_{{\tilde t}2}}{m_t^2} \right)
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{One-loop  Higgs mass}

shift in the physical Higgs mass squared 
\beq
\Delta (m^2_{h^0}) &=& 2\, \delta \lambda \, v_u^2 %\nonumber \\
 %&=&
= \frac{3}{4 \pi^2} v^2 y_t^4 \sin^2\beta
\>\> \ln \left( \frac{m_{{\tilde t}1}m_{{\tilde t}2}}{m_t^2} \right) 
\nonumber \\
& \approx& {{(90~{\rm GeV})^2}\over{\sin^2\beta}}~
\eeq
 assuming  $y_t$ does not blowup below
the unification scale:
\beq
m_{h^0} < 130~{\rm GeV}~
\label{Chap4:mssmhiggsbound}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{NMSSM Higgs mass}

add a new singlet field $N$ with coupling 
\beq
W_{\rm NMSSM}= y_N N H_u H_d~
\label{Chap4:NMSSM}
\eeq
so  the VEV of $N$ can generate the $\mu$-term
gives a new contribution,  ${\mathcal O}(y_N^2)$, to the Higgs quartic coupling \newline assuming that $y_N$ remains perturbative up to the unification scale
:
\beq
m_{h^0} < 150~{\rm GeV}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Precision electroweak measurements}
\label{Chap4:mssm:sec:pew}


Below the EWSB scale terms in the effective Lagrangian
like 
\beq
{\mathcal L}_{\rm eff} \subset -{{ \,g g^\prime S}\over{16 \pi}}W^3_{\mu\nu}
B^{\mu\nu}~
\eeq

Experimentally $S$ must be ${\mathcal O}(1/10)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Precision electroweak measurements}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\hsize]{slideimages/SandT}}
\end{figure}

\begin{center}\Blue{Particle Data Group,  {\tt http://pdg.lbl.gov/}}\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Precision electroweak measurements}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\hsize]{slideimages/vacpol}}
\end{figure}

$SU(2)_L$  doublet fermion with $N_c$ colors  that gets a mass from EWSB  contributes
to vacuum polarization $\Pi^{3B}_{\mu\nu}(p^2)$
for $LL$  gauge vertices
\beq
{\rm Tr} \,T^3_L Y =0~
\eeq
for $LR$  gauge vertices
\beq 
{\rm Tr} \, T^3_L Y = {\rm Tr}\,  T^3_L Q ={{1}\over{2}}~
\eeq
by gauge invariance
\beq
\Pi^{3B}_{\mu\nu}(p^2)=\left(g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}\right)\Pi^{3B}(p^2)
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Precision electroweak measurements}

For $m \gg M_Z$,  Taylor series around $p^2=0$:
\beq
\Pi^{3B}(p^2)= m^2 \sum_{n=0}^\infty a_n\,\left(\frac{p^2}{m^2}\right)^n ,
\eeq
 contribution to $S$  proportional to
\beq
{{d}\over{d p^2}}\Pi^{3B}(p^2)|_{p^2=0} \propto N_c\,{{m^2 }\over{m^2}}~.
\eeq
$S$ parameter 
counts the number of fields in the EWSB sector

For a superpartner in the MSSM the masses are of the form $m_{\rm sp}(m_{\rm soft},\mu,v)$.  In
the limit $\mu$, $m_{\rm soft} \rightarrow \infty$ with $v$ fixed
we have $m_{\rm sp} \rightarrow \infty$,  $S \propto$ 
$(v/m_{\rm sp})^n$ \newline superpartners decouple from
EWSB if they are sufficiently heavy 

\vspace{24pt}
\noindent $R$-parity: at low-energy  superpartners  only contribute at loop-level



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Problems with flavor and CP}
\label{Chap4:mssm:sec:flavorCP}

generically the mass matrices
$m^2_{\overline e}$ and $m^2_L$ are not diagonal in the same basis as the 
lepton mass matrix.  This leads to the nonobserved decay $\mu\rightarrow e \gamma$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\hsize]{slideimages/muegamma}}
\label{Chap4:fig:muegamma}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\beq
\Gamma_{\mu\rightarrow e \gamma} &\approx& 8 \sin^2 \theta_W \left( {{\alpha_2}
\over{4 \pi}}\right)^3 {{\pi m_\mu^5}\over{M_{\rm SUSY}^4}} \left( {{\Delta
m_L^2}
\over{M_{\rm SUSY}^2}}\right)^2
\eeq

\beq
\Gamma_{\mu\rightarrow e \nu {\bar\nu}} &=& \left( {{\alpha_2} \over{4 \pi}}\right)^2 {{\pi
m_\mu^5}\over {64 M_W^4}}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Problems with flavor and CP}

\beq
{{\Gamma_{\mu\rightarrow e \gamma}}\over{\Gamma_{\mu\rightarrow e \nu {\bar\nu}}}} &\approx&
3 \times 10^{-4} \, \left( {{500~{\rm GeV}}
\over{ M_{\rm SUSY} }}\right)^4 \left( {{\Delta
m_L^2}
\over{M_{\rm SUSY}^2}}\right)^2~,
\eeq

experimentally  less than 5 $\times 10^{-11}$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{FCNC's}


 $K\overline{K}$ mixing:
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\hsize]{slideimages/KKmixing}}
\label{chap4:fig:KKmix}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

for SM in the limit $m_q\rightarrow0$,  diagram
is proportional to CKM  elements

after diagonalizing the up-type
and down-type quark mass matrices by unitary matrices ${\bf U_u}$ and ${\bf U_d}$ the product 
${\bf V}={\bf U_d}^\dagger{\bf U_u}$ appear in the $W$ couplings 


 $V V^\dagger =I$, so loop is proportional to 
\beq
\left(V_{di}\, V_{is}^*\right)\left( V_{sj}^* V_{jd}\right)= \delta_{ds}\delta_{sd}=0~,
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Glashow, Iliopoulos,  Maiani}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.3\hsize]{slideimages/glashow}\includegraphics[width=0.35\hsize]{slideimages/iliopoulos}\includegraphics[width=0.29\hsize]{slideimages/maiani}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 leading contribution  comes only at ${\mathcal O}(m_{\rm quark}^2)$
 known as the GIM
suppression mechanism


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{FCNC's}


\beq
{\mathcal M}_{K\overline{K}}^{\rm SM} \approx \alpha_2^2 {{m_c^2}\over{M_W^4}}\sin^2\theta_c
\cos^2\theta_c~,
\label{Chap4:KKmixSM}
\eeq
where 
$V_{ud} = \cos \theta_c$.


\beq
{\mathcal M}_{K\overline{K}}^{\rm MSSM} \approx 4 \alpha_3^2 \left(
{{\Delta m_Q^2}\over{M_{\rm SUSY}^2}}\right)^2{{1}\over{ M_{\rm SUSY}^2 }}~.
\eeq
Since the SM amplitude roughly accounts for the observed $K_L$-$K_S$ mass splitting, we require ${\mathcal M}_{K\overline{K}}^{\rm SM}>{\mathcal M}_{K\overline{K}}^{\rm MSSM}$,
so
\beq
 \left( {{\Delta
m_Q^2}
\over{M_{\rm SUSY}^2}}\right) < 4 \times 10^{-3}
{ {M_{\rm SUSY} }\over{500~{\rm GeV}}}~.
\eeq
observed size of CP violation in the $K\overline{K}$ leads to
stringent bounds on the phases of the squark mixing matrix 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{EDM's}

with Higgs  VEV, $A$-terms  introduce off-diagonal squark and
slepton mass mixing\newline
gives rise to an electric dipole moment (EDM) the $d$ quark, and neutron. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\hsize]{slideimages/EDM}}
\label{Chap4:fig:EDM}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 dimension 5 operator in the low-energy effective theory, $d^\dagger_R \sigma^{\mu \nu}d_L F_{\mu\nu}$, the amplitude must have an inverse mass dimension, and it must be proportional to the VEV of $H_d$. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{EDM's}

call the overall phase $\delta$ 
\beq
{\mathcal M}_{\rm EDM}\approx\frac{\alpha_3}{4 \pi} {{e \, v c_\beta \, A_{d11}\, \delta}\over{M_{\rm SUSY}^2}}~.
\eeq
The  experimentally 
EDM of the neutron is $<0.97 \times 10^{-25}$
$e$~cm, which translates into the bound:
\beq
c_\beta A_{d11} \delta \left(\frac{500 {\rm GeV} }{M_{\rm SUSY}^2}\right)^2 <
5 \times 10^{-7}~.
\eeq
for ${\bf A_d} = {\bf Y_d}$ 
\beq
\delta < \left(\frac{M_{\rm SUSY}^2}{500~{\rm GeV} }\right)^2 
10^{-2}~.
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Safe Neighborhoods}

\begin{itemize}
\item{} ``Soft Breaking Universality'' requires 
the soft SUSY breaking squark and slepton masses are proportional to the identity in the same basis where quark and lepton mass matrices are diagonal, the $A$-term $\propto$ Yukawa ,
and  no new nontrivial phases 

\item{} The ``More Minimal Supersymmetric Model''  
only require the leading quadratic divergences in the Higgs mass to cancel.
${\wtilde t}_L$, ${\wtilde t}_R$, ${\wtilde b}_L$, ${\wtilde H}_u$,
${\wtilde H}_d$, ${\wtilde B}$,${\wtilde W}$ must have masses below $1$~TeV,
while first- and second-generation sparticles can be as heavy as $20$~TeV.  
possible danger: two-loop running below the heavy squark threshold
\beq
\frac{d m_{\wtilde t}^2}{d t} = \frac{8 g_3^2}{16\pi^2} C_2 \left[
\frac{3 g_3^2}{16\pi^2} m_{{\wtilde u},{\wtilde d}}^2 - M_3^2 \right],
\eeq
may drive the top squark mass squared negative, depending on  gluino mass
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Safe Neighborhoods}

\item{} The ``Alignment''  scenario requires a particular relation
between squark mass matrices and Yukawa matrices
\beq
{\bf m_Q^2} = {\bf Y_u^*}{\bf Y_u^T}+{\bf Y_d^*}{\bf Y_d^T}~,\\
{\bf m_u^2} = {\bf Y_u^\dagger}{\bf Y_u}~,\\
{\bf m_d^2} = {\bf Y_d^\dagger}{\bf Y_d}~,
\eeq
such that FCNC processes are suppressed.
\end{itemize}


\end{document}
 
 
