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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Dynamical SUSY breaking}
\setcounter{equation}{0}
\setcounter{footnote}{0}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{A rule of thumb for SUSY breaking}

theory with no flat directions
that spontaneously breaks a continuous
global symmetry generally breaks SUSY
\newline$\Rightarrow$ Goldstone boson with a scalar partner
(a modulus),
but if there are no flat directions this is impossible

\vspace{24pt}
rule gives a handful of dynamical SUSY breaking theories 

\vspace{24pt}
With duality we can find
many examples of dynamical SUSY breaking
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The 3-2 model}
\label{Chap12:3-2Model}
Affleck, Dine, and Seiberg 
found the simplest known model
of dynamical SUSY breaking:
\beq
\begin{array}{c|cc|cc}
 & SU(3) & SU(2) & U(1) & U(1)_R\\ \hline
Q & \fund & \fund & 1/3 & 1\\
L &  {\bf 1} &  \fund & -1 & -3 \\
\overline{U}  &  \overline{\fund} &  {\bf 1} & -4/3 & -8 \ \\
\overline{D} & \overline{\fund}  &  {\bf 1} &2/3 & 4 
\end{array}~
\eeq


For $\Lambda_3 \gg \Lambda_2$  instantons  give
the standard ADS superpotential:
\beq
W_{\rm dyn} = {{\Lambda_3^{7}}\over{{\rm det} (\overline{Q} Q) }} ~
\label{Chap12:DSB:dynADS32}
\eeq
which has a runaway vacuum.
Adding a tree-level trilinear term 
\beq
 W =  {{\Lambda_3^{7}}\over{{\rm det} (\overline{Q} Q) }} +
\lambda \, Q \bar{D} L~,
\eeq
removes the classical flat directions and produces a stable
minimum

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The 3-2 model}

$U(1)$ is broken
and we expect (rule of thumb) that SUSY is
broken


\beq
\frac{\partial W}{\partial L_\alpha} = \lambda  \epsilon^{\alpha \beta} 
Q_{m \alpha} \overline{D}^m=0~
\eeq
tries to set 
${\rm det} \overline{Q} Q$ to zero
since
\beq
{\rm det} \overline{Q} Q &=& {\rm det}\left(\begin{array}{cc}
\overline{U} Q_1 & \overline{U} Q_2 \\
\overline{D} Q_1 & \overline{D} Q_2 
\end{array}\right)\nonumber \\
&=& \overline{U}^m Q_{m \alpha}\overline{D}^n
Q_{n \beta}\epsilon^{\alpha \beta}~. 
\eeq

potential cannot have a zero-energy  minimum since the
dynamical term blows up at  ${\rm det} \overline{Q} Q$=0

\vspace{24pt}
\Red{SUSY is indeed broken}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The 3-2 model}
estimate the vacuum energy by taking all the VEVs to
$\sim \phi$
\newline For $\phi \gg \Lambda_3$ and $\lambda \ll 1$ 
 in a perturbative regime\beq
V&=& |\frac{\partial W}{\partial Q}|^2
+|\frac{\partial W}{\partial \overline{U}}|^2
+|\frac{\partial W}{\partial \overline{D}}|^2
+|\frac{\partial W}{\partial L}|^2 \\
&\approx& \frac{\Lambda_3^{14}}{\phi^{10}} 
+ \lambda\frac{\Lambda_3^{7}}{\phi^{3}} + \lambda^2 \phi^4~
\label{Chap12:DSB:32potential}
\eeq
 minimum near
\beq
\langle \phi \rangle \approx \frac{\Lambda_3} {\lambda^{{1}/{7}}}~
\eeq
solution is self-consistent
\beq
V \approx \lambda^{{10}/{7}} \Lambda_3^4~
\eeq
goes to $0$ as $\lambda\rightarrow 0$, $\Lambda_3\rightarrow 0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Duality and the 3-2 model}
Using duality  can also understand
the case where $\Lambda_2 \gg \Lambda_3$ \newline SUSY  broken
nonperturbatively
\newline $SU(3)$ gauge group has two flavors, completely broken
for generic VEVs\newline $SU(2)$ gauge group has
four $\fund$'s $\equiv$
two flavors \newline $\Rightarrow$ confinement with chiral symmetry breaking 



mesons and baryons:
\beq
\begin{array}{ccl}
M & \sim & \left( \begin{array}{cc}
L Q_1 & L Q_2 \\
Q_3 Q_1 & Q_3 Q_2
\end{array} \right)~\\
 B & \sim & Q_1 Q_2 ~\\
 \bar{B} & \sim & Q_3 L~
\end{array}
\eeq


effective superpotential is
\beq
W = X \, \left( {\rm det} M -  B \overline{B} - \Lambda_2^{4} \right)  +
\lambda\left(\sum_{i=1}^2 M_{1{\rm i}} \overline{D}^{\rm i} +
\overline{B} \,\overline{D}^3 \right) ~
\eeq
where $X$ is a Lagrange multiplier field 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Duality and the 3-2 model}
\beq
W = X \, \left( {\rm det} M -  B \overline{B} - \Lambda_2^{4} \right)  +
\lambda\left(\sum_{i=1}^2 M_{1{\rm i}} \overline{D}^{\rm i} +
\overline{B} \,\overline{D}^3 \right) ~
\eeq
$\overline{D}$ eqm tries to force $M_{1i}$ 
and $\overline{B}$ to zero 
\newline constraint means that at least one of $M_{11}$, $M_{12}$, or $\overline{B}$
is nonzero\newline $\Rightarrow$ SUSY is broken at tree-level in the dual 
description
\beq
V\approx \lambda^2 \Lambda_2^4~
\eeq
Comparing the vacuum energies we see that the $SU(3)$
interactions dominate when $\Lambda_3 \gg \lambda^{{1}/{7}} \Lambda_2$


for $\Lambda_2\sim\Lambda_3$ consider the full superpotential
\beq
W = X \, \left( {\rm det} M -  B \bar{B} - \Lambda_2^{4} \right)  +
{{\Lambda_3^{7}}\over{{\rm det} (\overline{Q} Q) }} 
+ \lambda \, Q \bar{D} L~
\eeq
which still breaks SUSY, analysis  more complicated


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$\mbox{SU(5)}$  with $\overline{\fund}  +\raisebox{-3.5pt}{ \Yasymm}$}
\label{Chap12:SU5Model}
chiral gauge theory has no classical flat directions\newline 
ADS tried to match anomalies in  a confined
description \newline only ``bizarre,"  ``implausible''
solutions\newline assume broken $U(1)$ 
$\Rightarrow$ broken SUSY   (using the rule of thumb)

\vspace{24pt}
\noindent Adding flavors
$(\fund +\overline{\fund})$
with masses Murayama showed
 that SUSY is broken, but masses  $\to \infty$
strong coupling

\vspace{24pt}\noindent
With duality   Pouliot
showed that SUSY is broken
at strong coupling

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$\mbox{SU(5)}$  with $\overline{\fund}  +\raisebox{-3.5pt}{ \Yasymm}$}
with 4 flavors  theory s-confines
 \beq
\begin{array}{c|c|ccccc}
& SU(5)  & SU(4) & SU(5) & U(1)_1 & U(1)_2 & U(1)_R \\ \hline
A & \Yasymm & 1 & 1 & 0 & 9 & 0 \\
\overline{Q} & \overline{\Yfund} & 1 & \Yfund & 4 & -3 & 0 \\
Q & \Yfund & \Yfund & 1 & -5 & -3 & \frac{1}{2} 
\end{array}~
\eeq
denote composite meson  by
$(Q\overline{Q})$, spectrum of massless composites is:
\beq
\begin{array}{c|ccccc}
 & SU(4) & SU(5) & U(1)_1 & U(1)_2 & U(1)_R \\ \hline
(Q\overline{Q}) \vphantom{  \Yasymm } &  \Yfund & \Yfund & -1 & -6 & \frac{1}{2} \\
(A\overline{Q}^2) & 1 &   \Yasymm & 8 & 3 & 0 \\
(A^{2}Q)  & \Yfund &  1 & -5 & 15 & \frac{1}{2} \\
(AQ^3) & \overline{\Yfund} &  1 & -15 & 0 & 
\frac{3}{2} \\
(\overline{Q}^{5}) &  1 & 1 & 20 & -15 & 0 \end{array}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$\mbox{SU(5)}$  with $\overline{\fund}  +\raisebox{-3.5pt}{ \Yasymm}$}
with a superpotential
\beq
 W_{\rm dyn} &=& \frac{1}{\Lambda^{9}} \Big[ (A^2Q)(Q\overline{Q})^3 
(A\overline{Q}^2)+(AQ^3)(Q\overline{Q})(A\overline{Q}^2)^{2}\label{Chap12:SU5dualsuperpotential}\\ 
&& \,\,\,\,\,\,\,\,\,\,\,\,+ (\overline{Q}^{5})(A^2 Q)(A Q^3)
\Big]~ \nonumber
\eeq
 first term 
 antisymmetrized in  $SU(5)$ and $SU(4)$
indices\newline  second term  antisymmetrized in just $SU(5)$
indices

add mass terms and Yukawa couplings for the extra flavors:
 \beq
 \Delta W = \sum_{i=1}^4 m Q_i \overline{Q}_i + \sum_{i,j\le 4} 
 \lambda_{ij} A \overline{Q}_i\overline{Q}_j~,
 \eeq
 which lift all the flat directions
 
 eqm  give
 \beq
 {{\partial W}\over{\partial(\overline{Q}^{5})}} &=& (A^2 Q)(A Q^3) =0 ~
\label{Chap12:DSB:SU5eq1} \\
  {{\partial W}\over{\partial(Q\overline{Q})}}&=&3(A^2Q)(Q\overline{Q})^2 
(A\overline{Q}^2)+(AQ^3)(A\overline{Q}^2)^{2}+ m=0~
\label{Chap12:DSB:SU5eq2}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$\mbox{SU(5)}$  with $\overline{\fund}  +\raisebox{-3.5pt}{ \Yasymm}$}
 \beq
 {{\partial W}\over{\partial(\overline{Q}^{5})}} &=&\Red{ (A^2 Q)}\Blue{(A Q^3)} =0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(*)\\
  {{\partial W}\over{\partial(Q\overline{Q})}}&=&3\Red{(A^2Q)}(Q\overline{Q})^2 
(A\overline{Q}^2)+\Blue{(AQ^3)}(A\overline{Q}^2)^{2}+ m=0~~~~~(**)
\eeq
Assuming $\Red{(A^2Q)} \ne 0$ then the first equation of motion 
(*)
requires $\Blue{(A Q^3)}=0$ and multiplying (**) 
by $\Red{(A^2Q)}$ we see that because of the
antisymmetrizations the first term vanishes $\Rightarrow$
\beq
 \Blue{(AQ^3)}(A\overline{Q}^2)^{2}=-m~~~~~~(***)
\label{Chap12:DSB:m5}
\eeq
\Blue{contradiction! }

\noindent Assuming that $\Blue{(A Q^3)} \ne 0$ then (*)
requires $\Red{(A^2Q)}=0$, and plugging into 
 (**) we find eqn (***) directly
Multiplying eqn (***) by $\Blue{(AQ^3)}$ we find that the 
left-hand side vanishes again due to 
antisymmetrizations, so $\Blue{(AQ^3)}=0$, \Blue{contradiction! }

\vspace{12pt}
\Red{ SUSY is broken at 
tree-level in  dual description}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Intriligator--Thomas--Izawa--Yanagida}
\label{Chap12:sec:deformed}


\beq 
\begin{array}{cc|c}
& SU(2)  & SU(4) \\
 \hline
Q  &\Yfund & \Yfund  \\
S  & {\bf 1} &\Yasymm  \\
\end{array}
\eeq
\beq
W=\lambda S^{ij} Q_i Q_j~
\eeq
strong $SU(2)$ enforces a constraint.
\beq
{\rm Pf}(QQ) = \Lambda^4~
\label{Chap12:DSB:PFconstraint}
\eeq
eqm for $S$:
\beq
{{\partial W}\over{\partial S^{ij}}}  = \lambda Q_i Q_j =0~
\eeq
equations incompatible 
\newline \Red{SUSY is broken}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Intriligator--Thomas--Izawa--Yanagida}
 for large $\lambda S$, we
can integrate out the quarks, no flavors $\Rightarrow$ gaugino condensation:
\beq
\Lambda_{\rm eff}^{3N}=\Lambda^{3N-2}\left(\lambda  S\right)^{2} ~
\\
W_{\rm eff}=2 \Lambda_{\rm eff}^{3}= 2 \Lambda^{2}  \lambda S~ \\
\frac{\partial W_{\rm eff}}{\partial S^{ij}}  = 2 \lambda \Lambda^{2}~
\eeq
again vacuum energy is nonzero

%For general values of $\lambda S$ we can write:
%\beq
%W_{\rm eff}=\lambda S^{ij} Q_i Q_j+X({\rm Pf}M-\Lambda^4)~,
%\eeq
%where $X$ is a Lagrange multiplier field.
%For $\lambda \ll 1$ the vacuum is close to the SUSY QCD vacuum given
%by the $X$ equation of motion, and we can treat the first term
%in the superpotential as a small mass perturbation. 
%
%The potential energy is given by:
%\beq
%V=\sum_i|\frac{\partial W_{\rm eff}}{\partial Q_{i}}|^2
% +\sum_{ij}|\frac{\partial W_{\rm eff}}{\partial S^{ij}}|^2 ~.
%\eeq
%A supersymmetric vacuum exists if all the terms vanish.
%Treating $\lambda S$ as a mass perturbation
%we can set the derivatives with respect to $Q$ to zero simply
%by
%solving for the squark VEVs in the standard way.  This gives
%\beq
%Q_{i}Q_{j} = \left( {\rm Pf}(\lambda S) \Lambda^{3N-F}\right)^{{1}\over{N}} 
%\left(
%{{1}\over{\lambda S}}\right)_{ij}~.
%\eeq
%Plugging this back in to the potential gives
%\beq
%V&=&\sum_{ij}|{{\partial W_{\rm eff}}\over{\partial S^{ij}}}|^2 =
%|\lambda|^2\sum_{ij}|M_{ij}|^2 \nonumber ~,\\
%&=&|\lambda|^2 |{\rm Pf}S \Lambda^4| \sum_{ij}|\left({{1}\over{S}}\right)_{ij}%|^2
%~,
%\eeq
%which is minimized at 
%\beq
%S^{ij}=({\rm Pf}S)^{{1}\over{2}} \epsilon^{ij}
%\eeq
%so
%\beq
%V = 4 |\lambda|^2 \Lambda^4
%\eeq
%which agrees with our gaugino condensation calculation.

theory is vector-like, Witten index
 Tr$(-1)^{\bf F}$ is nonzero with mass terms turned on so
there is at least one supersymmetric vacuum\newline index is topological,
does not change under variations of the mass

\noindent
\newline\Red{loop-hole}
potential for large field
values are very different with $\Delta W=m_s S^2$ 
from the theory with $m_s\rightarrow 0$,
 in this limit  vacua can come in from or go out to $\infty$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Pseudo-Flat Direction}

$S$ appears to be a flat direction
but  
%As we saw in the O'Raifeartaigh model 
with SUSY breaking theories becomes pseudo-flat
due to corrections from the \Kahler  function

For large values of 
$\lambda S$   wavefunction
renormalization:
\beq
Z_S= 1+c \lambda \lambda^\dagger \ln\left( {{\mu_0^2}\over{\lambda^2 S^2}} \right)~
\eeq
vacuum energy:
\beq
V = \frac{4 |\lambda|^2}{|Z_S|} \Lambda^4 
\approx |\lambda|^2 \Lambda^4\left[1+c \lambda \lambda^\dagger 
\ln\left( {{\lambda^2 S^2}\over{\mu_0^2}} \right) \right]~
\eeq
potential slopes towards the origin\newline can be stabilized
by gauging a subgroup of $SU(4)$.  Otherwise
low-energy effective theory with local minimum at $S=0$
\newline effective theory non-calculable 
near $\lambda S \approx \Lambda$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Baryon Runaways}
\label{Chap12:DSB:baryonrunaway}

Consider a generalization of the 3-2 model:
\beq \begin{array}{c|cc|ccc}
& SU(2N - 1) & Sp(2N)  & SU(2N - 1) & U(1) & U(1)_R \\
 \hline
Q & \Yfund &\Yfund  & {\bf 1}& 1 & 1 \\
L & {\bf 1} &\Yfund & \Yfund & -1 &-{{3}\over{2N - 1}} \\
\overline{U} & \overline{\Yfund} & {\bf 1}& \overline{\Yfund}& 0&
 {{2N + 2}\over{2N - 1}}\\
\overline{D} & \overline{\Yfund} & {\bf 1}& {\bf 1}& -6& -4N \\
\end{array}~
\eeq
with a tree-level superpotential
\beq
W = \lambda Q L \overline{U}
\eeq
\begin{tabular}{ll}
turn off  $SU(2N - 1)$  and  $\lambda$, 
$Sp(2N)$& \begin{tabular}{l}non-Abelian Coulomb phase for $N \ge 6$ \\
weakly coupled dual description for $N = 4,5$\\ 
s-confines for $N = 3$\\ confines with $\chi$SB for $N = 2$\end{tabular}\\& \\
turn off the $Sp(2N)$  and  $\lambda$, $SU(2N - 1)$&
 \begin{tabular}{l}s-confines for  $N \ge 2$\end{tabular}
\end{tabular}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Baryon Runaways}
consider the case that
$\Lambda_{SU} \gg\Lambda_{Sp}$


 classical moduli space that can be parameterized by:
\beq
 \begin{array}{c|ccc}
 & SU(2N - 1) & U(1) & U(1)_R \\
 \hline
M = (L L)
& \Yasymm & -2& -{{6}\over{2N - 1}}
\\
B = (\overline{U}^{2N - 2} \overline{D})
& \Yfund & -6 & -{{4(N^2 - N + 1)}\over{2N - 1}}
\\
b = (\overline{U}^{2N - 1})
& {\bf 1}& 0& 2N + 2 \\
\end{array}~
\eeq
subject to the constraints
\beq
M_{jk} B_l \epsilon^{k l m_1 \cdots m_{2N - 3}} = 0
\qquad
M_{jk} b = 0
\eeq
two branches: \begin{tabular}{l}
$M = 0$ and $B, b \ne 0$\\
$M \ne 0$ and $B, b = 0$\end{tabular}

% -----------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Baryon Runaways}
branch where $M=0$  (true vacuum ends up here)
\beq
\langle \overline{U} \rangle = \pmatrix{ v \cos\theta & \cr
& v {\bf 1}_{2N - 2} \cr},
\qquad
\langle \overline{D}\rangle
 = \pmatrix{ v \sin\theta \cr 0 \cr \vdots \cr 0 \cr},
\eeq

For $v> \Lambda_{SU}$, $SU(2N-1)$ is generically broken and the superpotential gives masses to $Q$ and $L$
or order $\lambda v$.  The low-energy effective theory is pure $Sp(2N)$ $\Rightarrow$ gaugino condensation
\beq
\Lambda_{\rm eff}^{3(2N+2)}=\Lambda_{Sp}^{3(2N+2)-2(2N-1)}\left(\lambda 
\overline{U}\right)^{2(2N-1)}~
\eeq
\beq
W_{\rm eff}\propto \Lambda_{\rm eff}^{3}\sim \Lambda_{Sp}^{3}\left({{\lambda 
\overline{U}}\over{\Lambda_{Sp}}}\right)^{(2N-1)/(N+1)}~
\label{Chap12:risingsuppot}
\eeq
For $N>2$ this forces $\langle \overline{U} \rangle$ towards zero

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Baryon Runaways}

For  $v < \Lambda_{SU}$, 
then $SU(2N-1)$ s-confines: effective theory
\beq
\begin{array}{c|c|c}
& Sp(2N)  & SU(2N - 1) \\
 \hline
L &\Yfund & \Yfund  \\
(Q \overline{U}) &\Yfund   & \overline{\Yfund} \\
(Q \overline{D}) &\Yfund   & {\bf 1} \\
(Q^{2N-1}) &\Yfund   & {\bf 1} \\
B & {\bf 1}& \Yfund \\
b & {\bf 1}&{\bf 1}\\
\end{array}~
\eeq
with a superpotential
\beq
W_{\rm sc}&=&{{1}\over{\Lambda_{SU}^{4N-3}}}
\left[(Q^{2N-1})(Q \overline{U})B+(Q^{2N-1})(Q \overline{D})b - {\rm det} 
\overline{Q}Q \right] \nonumber \\
&&+\lambda (Q\overline{U})L~.
\eeq
integrated out
 $(Q\overline{U})$ and $L$with $(Q\overline{U})=0$,
\beq
W_{\rm le}={{1}\over{\Lambda_{SU}^{4N-3}}}
(Q^{2N-1})(Q \overline{D})b ~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Baryon Runaways}
On this branch  $\langle b \rangle = \langle \overline{U}^{2N-1}
\rangle \ne 0$, gives a mass to $(Q^{2N-1})$ and $(Q \overline{D})$
\newline leaves  pure $Sp(2N)$ as the low-energy effective theory.
So we again find gaugino condensation 
\beq
\Lambda_{\rm eff}^{3(2N+2)}=\Lambda_{Sp}^{3(2N+2)-2(2N-1)}\left(\lambda 
\Lambda_{SU}\right)^{2(2N-1)} \left({{b}\over{\Lambda_{SU}}}\right)^2~
\eeq
\beq
W_{\rm eff}\propto \Lambda_{\rm eff}^{3}\sim b^{{1}/(N+1)}\left(
\Lambda_{Sp}^{N+4} \lambda^{2N-1}
\Lambda_{SU}^{(2N-2)}\right)^{{1}/(N+1)}~
\eeq
which forces $b \rightarrow \infty$ (this is a baryon runaway vacuum)
\newline effective theory only valid
for scales below $\Lambda_{SU}$
\newline already seen that beyond this point the potential
starts to rise again
\newline vacuum is around 
\beq
\langle b \rangle =\langle \overline{U}^{2N-1}\rangle  \sim \Lambda_{SU}^{2N-1}~
\eeq
With more work  one can also see that SUSY is 
broken when $\Lambda_{Sp}
\gg \Lambda_{SU}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Baryon Runaways: $N=3$ }

$Sp(2N)$ s-confines 
\beq 
\begin{array}{cc|c}
& SU(5)  & SU(5) \\
 \hline
(Q Q) &\Yasymm  & {\bf 1}  \\
(L L)  & {\bf 1} &\Yasymm  \\
(Q L) &\Yfund   &\Yfund \\
\overline{U} & \overline{\Yfund}&  \overline{\Yfund} \\
\overline{D} &  \overline{\Yfund} &{\bf 1}\\
\end{array}
\eeq
with
\beq
W=\lambda (Q L)\overline{U} +Q^{2N-1}L^{2N-1}~
\eeq
global $SU(5) \supset$ SM gauge groups,  candidate
 for gauge mediation 
\newline  integrate  $(Q L)$ and $\overline{U}$ to find $SU(5)$ with
an antisymmetric tensor, an antifundamental, and some gauge singlets,
which we have already seen breaks SUSY


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Baryon Runaways}

other branch  $M= (LL) \ne 0$\newline 
D-flat directions for $L$ break $Sp(2N)$  to $SU(2)$, effective theory is:
\beq
\begin{array}{c|cc}
 & SU(2N - 1) & SU(2) \\
 \hline
Q^\prime     & \Yfund&\Yfund \\
L^\prime   & {\bf 1} &\Yfund \\
\overline{U}^\prime  &\overline{\Yfund} & {\bf 1}\\
\overline{D} &\overline{\Yfund}& {\bf 1}
\end{array}
\eeq
and some gauge singlets with a superpotential
\beq
W=\lambda Q^\prime\overline{U}^\prime L^\prime~
\eeq
This is a generalized 3-2 model


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Baryon Runaways}
For
$\langle L \rangle \gg \Lambda_{SU}$  the vacuum energy is independent
of the $SU(2)$ scale and proportional
to $\Lambda_{SU(2N-1)}^4$ which itself is proportional to
a positive power of 
$\langle L \rangle$, thus the effective potential
in this region drives $\langle L \rangle$ smaller.

\vspace{24pt}
For $\langle L \rangle \ll \Lambda_{SU}$  use the s-confined
description, and find the baryon $b$ runs away.  For
 $\langle L \rangle \approx \Lambda_{SU}$, the vacuum energy is
 \beq
 V \sim \Lambda_{SU}^4~,
 \eeq
 which is larger than the vacuum energy on the other branch
 
 \vspace{24pt}global minimum is on the baryon branch with 
$b= (\overline{U}^{2N - 1})\ne 0$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Direct gauge mediation}

suppose fields
that break SUSY have SM gauge couplings
\newline only need
two sectors  rather than three
\[ \begin{array}{c|cc|c}
& SU(5)_1 & SU(5)_2 & SU(5) \\
 \hline
Y & {\bf 1}& \Yfund & \overline{\Yfund}\\
\phi & \overline{\Yfund} & {\bf 1} & \Yfund   \\
\overline{\phi} &\Yfund &\overline{\Yfund}& {\bf 1} \\
\end{array}
\]
with a superpotential
\beq
W= \lambda Y^i_j \overline{\phi}^j \phi_i~
\eeq
weakly gauge global $SU(5)$ with the SM gauge groups\newline
 $Y \gg \Lambda_1, \Lambda_2$, $\phi$ and $\overline{\phi}$ get a mass, 
matching gives
\beq
\Lambda_{\rm eff}^{3 \cdot 5}=\Lambda_{1}^{3\cdot 5-5}\left(\lambda 
X \right)^{5}~
\eeq
where $X=({\rm  det} Y)^{1/5}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Direct gauge mediation}
effective gauge theory has gaugino condensation
\beq
W_{\rm eff}=\Lambda_{\rm eff}^{3}\sim \lambda X \Lambda_{1}^{2}~
\eeq
SUSY broken a la the Intriligator--Thomas--Izawa--Yanagida\newline
vacuum energy given by
\beq
V \approx\frac{ | \lambda  \Lambda_{1}^{2}|^2}{Z_X}~
\eeq
where $Z_X$ is the wavefunction renormalization for $X$\newline
for large $X$ the vacuum energy grows monotonically\newline
 local minimum occurs where  anomalous dimension
$\gamma=0$\newline for $\langle X \rangle > 10^{14}$~GeV, the Landau pole for
$\lambda$ is above the Planck scale

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Direct gauge mediation}
problem: for small values of $X$, SUSY minimum along a baryonic direction\newline  look at the constrained mesons and baryons of $SU(5)_1$
\beq
W = A( {\rm det} M - B \overline{B} - \Lambda_1^{10})  + \lambda Y M~.
\eeq
SUSY minimum at $B \overline{B}= - \Lambda_1^{10}$, $Y=0$, $M=0$\newline 
SUSY minimum would have to be removed, or the non-supersymmetric minimum
made sufficiently metastable by adding
appropriate terms to the superpotential that force  $B \overline{B}=0$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Direct gauge mediation}
phenomenological problem: 
heavy gauge boson messengers can give negative contributions to 
squark and slepton squared masses.  Consider the general case where
a VEV
\beq
\langle X \rangle &=& M +\theta^2 {\mathcal F}~
\eeq
breaks SUSY and 
\beq
G \times H \rightarrow SU(3)_c \times SU(2)_L \times U(1)_Y~
\eeq
with
\beq
{{1}\over{\alpha(M)}} &=& {{1}\over{\alpha_G(M)}}+{{1}\over{\alpha_H(M)}}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Direct gauge mediation}
Analytic continuation in superspace  gives
\beq
M_\lambda = {{\alpha(\mu)}\over{4 \pi}} (b-b_H -b_G) {{{\mathcal F}}\over{M}}~
\eeq
and
\beq
m_Q^2 &=& 2 C_2(r) {{\alpha(\mu)^2}\over{(16 \pi^2)^2}} \left({{F}\over{M}}\right)^2 
\nonumber \\&& \left[ (b+(R^2-2)b_H -2 b_G) \xi^2
+{{b-b_H-b_G}\over{b}}(1-\xi^2) \right]~
\eeq
where
\beq
\xi = {{\alpha(M) }\over{\alpha(\mu)} }~ \,\,\,
R = {{\alpha_H(M) }\over{\alpha(M)} }~
\eeq
 typically gives a negative mass squared for right-handed sleptons

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Direct gauge mediation}
 if not all the messengers are
heavy, then two-loop RG gives:
\beq
\mu {{d}\over{d \mu}} m_Q^2 \propto -g^2 M_\lambda^2 +c g^4 
{\rm Tr}\left((-1)^{2F} m_{i}^2 \right)~
\label{Chap12:runningsquarkmass}
\eeq
the one-loop term proportional to the gaugino mass squared drives the
scalar mass positive as the renormalization scale is run down\newline 
\Blue{two-loop term
can drive the mass squared negative}\newline effect is maximized when the
gaugino is light\newline when  gluino is
the heaviest gaugino,  \Red{sleptons get dangerous negative contributions}

\vspace{24pt}
also dangerous  in models where the squarks and sleptons of the first two
generations are much heavier that $1$~TeV



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Single sector models}
suppose the strong
dynamics that breaks SUSY also produce composite MSSM particles
\newline rather than having three sectors, there is really just one sector.


\beq
\begin{array}{c|cc|cc}
& SU(k) & SO(10) & SU(10) & SU(2) \\
 \hline
Q &\Yfund & \Yfund  & {\bf 1}& {\bf 1} \\
L &\overline{\Yfund} & {\bf 1}& \Yfund & {\bf 1} \\
\overline{U} & {\bf 1}& \Yfund & \overline{\Yfund}&{\bf 1}\\
S & {\bf 1}& {\bf 16}& {\bf 1}&\Yfund  \\
\end{array}~
\eeq

\beq
W= \lambda Q L \overline{U}~
\eeq
global $SU(10) \supset$  SM or  GUT


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Single sector models}
This is a baryon runaway model \newline
for large ${\rm det}\overline{U}\gg \Lambda_{10}$
\beq
W_{\rm eff} \sim \overline{U}^{10/k}~
\eeq
 for small ${\rm det}\overline{U}\ll \Lambda_{10}$:
\beq
W_{\rm eff} \sim \overline{U}^{10(1-\gamma)/k}~,
\eeq
$\gamma$ is the anomalous dimension of $\overline{U}$

\Red{for $10 \ge k > 10 (1-\gamma)$ SUSY is broken}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Single sector models}
two composite 
generations corresponding to spinor $S$\newline composite
squarks and sleptons have masses of order
\beq
m_{\rm comp}\approx {{{\mathcal F}}\over{\overline{U}}}~
\eeq

 gauge mediation via the strong $SO(10)$ interactions

global $SU(2)$  enforces a degeneracy that suppresses FCNCs

\vspace{24pt}
composite fermions only get couplings to Higgs 
 from higher dim. ops


gaugino and third-gen. scalars masses from
gauge mediation

\vspace{24pt}
superpartners of the first two (composite) generations are 
much heavier than the superpartners of the third generation
\newline \Red{ similar
to ``more minimal" SUSY SM spectrum}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Intriligator---Seiberg---Shih}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.35\hsize]{slideimages/Intriligator} \includegraphics[width=0.3275\hsize]{slideimages/SeibergFace}  \includegraphics[width=0.407\hsize]{slideimages/Shih}   }
\end{figure}
\vspace{12pt}
\Blue{  hep-th/0602239}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Intriligator---Seiberg---Shih}

\beq
\begin{tabular}{c|c|ccccc}
& $SU(N)$ & $SU(F)$  & $SU(F)$  & $U(1)$& $U(1)'$ & $U(1)_R$ \\
\hline
$\phi$ & \fund & $\overline{\fund}$ & {\bf 1} &1 & 1
& $0$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$\overline{\phi}$ & $\overline{\fund}$   & {\bf 1} & \fund
& $-1$ & 1 & $0$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$M$ &{\bf 1}& \fund & $\overline{\fund}$   & 0 & -2 &
$2$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
\end{tabular}~
\eeq

with the superpotential
\beq
W ={\bar  \phi } M   \phi - f^2 {\rm Tr}   M
\eeq

unbroken $SU(N)\times SU(F) \times  U(1)\times U(1)'\times U(1)_R$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{SUSY Breaking}

\beq
\frac{\partial W}{\partial M^j_i}= {\bar  \phi }_j   \phi^i - f^2 \delta_j^i \ne 0
\eeq

${\bar  \phi }_j   \phi^i$ gets VEV $\Rightarrow  SU(N)$ completely broken

\vspace{12pt}
but ${\bar  \phi }_j  \phi^i$ has rank $N< F$



$\Rightarrow  M$ has non-zero ${\mathcal F}$ components

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Wait a Minute}

This is just a dual of SUSY $SU(F-N)$ QCD, quark masses $\propto f^2/\mu$

\vspace{12pt} SUSY vacuum at
\beq
\langle M \rangle
 \propto f^{-2} \left(f^{2F} \Lambda^{3(F-N)-F}\right)^{{1}/{(F-N)}}~
\eeq

\beq
\langle M\rangle
\gg f \,\,\,{\rm if}\,\,\, F>3N
\eeq

\vspace{12pt}
dual is IR free

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Intriligator---Seiberg---Shih}



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

tunnelling $\propto e^{-S}$

\vspace{24pt}
$S \gg 1$ if $F>3N$


\end{document}

