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\newcommand{\bc}{\begin{center}} \newcommand{\ec}{\end{center}} \newcommand{\Kahler}{K\"ah\-ler~} \newcommand{\Poincare}{Poincar\'e~} \newcommand{\beq}{\bc $\begin{array}{rcl}}% can be used as {equation} or {eqnarray} \newcommand{\eeq}{\end{array}$ \ec} \newcommand{\bh}{\begin{huge}} \newcommand{\eh}{\begin{huge}} \newcommand\myhead[1]{\newpage\begin{huge}\begin{center}\Purple{#1}\end{center}\end{huge}} \newcommand\mysubhead[1]{\begin{huge}\begin{center}\Purple{#1}\end{center}\end{huge}} \newcommand\chapter[1]{\newpage\begin{center}\begin{huge}\begin{center}\Purple{\bf #1}\end{center}\end{huge}\end{center}} \slidewidth=11.9 in \slideheight=7.8 in \pdfpagewidth=11.9 truein \pdfpageheight=9 truein \def\slidetopmargin{-2pt} \def\slideleftmargin{200pt} \def\slidebottommargin{0pt} \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}