\documentclass[landscape]{seminar} \pagestyle{empty} \usepackage{amsmath,amssymb,graphicx} %use drftcite in draftmode otherwise \usepackage{color} \usepackage{amsbsy} \def\lagr{{\mathcal L}} \def\sigmabar{\overline\sigma} \def\eps{\epsilon} \def\bea{\begin{eqnarray*}} \def\eea{\end{eqnarray*}} \def\wtilde{\widetilde} \def\half{\frac{1}{2}} \newcommand{\sfrac}[2]{{\textstyle\frac{#1}{#2}}} \definecolor{purple}{cmyk}{0.0,1.0,0,0} \definecolor{dpurple}{rgb}{0.5,0,1} \definecolor{turq}{rgb}{0,1,1} \definecolor{orange}{cmyk}{0,0.31,0.87,0} \definecolor{grn}{rgb}{0.18,0.66,0.24} \newcommand\Red[1]{{\color{red}#1}} \newcommand\Orange[1]{{\color{orange}#1}} \newcommand\Green[1]{{\color{green}#1}} \newcommand\Grn[1]{{\color{grn}#1}} \newcommand\Blue[1]{{\color{blue}#1}} \newcommand\Purple[1]{{\color{purple}#1}} \newcommand\Cyan[1]{{\color{cyan}#1}} \newcommand\Magenta[1]{{\color{magenta}#1}} \newcommand\DPurple[1]{{\color{dpurple}#1}} \newcommand\Turq[1]{{\color{turq}#1}} \newcommand{\BL}{} \newcommand{\bc}{\begin{center}} \newcommand{\ec}{\end{center}} \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} \setcounter{equation}{0} \setcounter{footnote}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter{Introduction to Supersymmetry} \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \myhead{Unreasonable effectiveness of the SM} \label{Chap1:intro:sec:sm} \beq \lagr_{\rm Yukawa} = - \frac{y_t}{\sqrt{2}} H^0 \overline{t_L} t_R + h.c. \label{Chap1:topYukawa} \eeq \beq H^0=\langle H^0 \rangle+ h^0=v+h^0 \eeq \beq m_t = \frac{y_t\, v}{\sqrt{2}} \eeq %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centerline{\includegraphics[width=0.4\hsize]{slideimages/yukawadiv}} \caption[]{The top loop contribution to the Higgs mass term.} \label{Chap1:fig:yukawadiv} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \beq - i \delta m_h^2|_{\rm top} &=& (-1) N_c \int \frac{d^4 k}{(2 \pi)^4} {\rm Tr}\left[ \frac{-i y_t}{\sqrt{2}} \frac{i}{\not\!k - m_t}\left(\frac{-i y_t^*}{\sqrt{2}}\right)\frac{i}{\not\!k - m_t}\right] \nonumber\\ &=& - 2 N_c |y_t|^2 \int \frac{d^4 k}{(2 \pi)^4} \frac{k^2+m_t^2}{(k^2-m_t^2)^2} \eeq \bc $k_0\rightarrow i k_4$, $k^2 \rightarrow -k_E^2$ \ec \beq - i \delta m_h^2|_{\rm top} &=& \frac{i N_c |y_t|^2}{8 \pi^2} \int_0^{\Lambda^2} d k_E^2 \, \frac{k_E^2(k_E^2-m_t^2)}{(k_E^2+m_t^2)^2} \eeq \bc $x= k_E^2+m_t^2$ \ec \beq \delta m_h^2|_{\rm top} &=& - \frac{ N_c |y_t|^2}{8 \pi^2} \int_{m_t^2}^{\Lambda^2} d x \left(1- \frac{3 m_t^2}{x}+ \frac{2 m_t^4}{x^2}\right)\nonumber\\ &=&- {{ N_c |y_t|^2}\over {8 \pi^2}} \left[\Lambda^2 -3\, m_t^2\, \ln\left({{\Lambda^2+ m_t^2}\over{m_t^2}} \right) + \ldots \right] \label{Chap1:yukawadiv} \eeq \newpage \beq \lagr_{\rm scalar} &=& - {{\lambda}\over{2}} (h^0)^2(|\phi_L|^2 + |\phi_R|^2)- h^0(\mu_L |\phi_L|^2 +\mu_R |\phi_R|^2)\nonumber\\ && - m_L^2 |\phi_L|^2-m_R^2 |\phi_R|^2 \label{Chap1:scalarYukawa} \eeq %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centerline{\includegraphics[width=0.18\hsize]{slideimages/quarticdiv}} \caption[]{Scalar boson contribution to the Higgs mass term via the quartic coupling.} \label{Chap1:fig:quarticdiv} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centerline{\includegraphics[width=0.4\hsize]{slideimages/trilineardiv}} \caption[]{Scalar boson contribution to the Higgs mass term via the trilinear coupling.} \label{Chap1:fig:trilineardiv} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \beq - i \delta m_h^2|_{2} = -i \lambda N \int \frac{d^4 k}{(2 \pi)^4} \left[ \frac{i}{ k^2 - m_L^2}+\frac{i}{k^2 - m_R^2}\right] \eeq \beq \delta m_h^2|_{2}= {{\lambda N}\over {16 \pi^2}} \left [2 \Lambda^2 - m_L^2 \ln \left({{\Lambda^2+ m_L^2}\over{m_L^2}} \right) - m_R^2 \ln \left({{\Lambda^2+ m_R^2}\over{m_R^2}} \right)+ \ldots \right ]. \label{Chap1:quarticdiv} \eeq \beq - i \delta m_h^2|_{3} = N \int \frac{d^4 k}{(2 \pi)^4} \left[ \left( -i \mu_L\, \frac{i}{ k^2 - m_L^2}\right)^2+\left(-i \mu_R\,\frac{i}{k^2 - m_R^2}\right)^2\right] \eeq \beq \delta m_h^2|_{3} =- {{ N}\over {16 \pi^2}} \left [\mu_L^2\,\ln \left({{ \Lambda^2+ m_L^2}\over{m_L^2}} \right) + \mu_R^2 \,\ln \left({{\Lambda^2+ m_R^2}\over{m_R^2}} \right)+ \ldots \right ]. \label{Chap1:trilineardiv} \eeq If $N=N_c$ and $\lambda = |y_t|^2$ then $\Lambda^2$ cancels \vspace{24pt} If $m_t = m_L = m_R$ and $\mu_L^2=\mu_R^2= 2 \lambda m_t^2$ $\log \Lambda$ are canceled as well \vspace{24pt} SUSY will guarantee these relations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \myhead{Coleman-Mandula} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centerline{\includegraphics[width=0.4\hsize]{slideimages/Coleman} \includegraphics[width=0.25\hsize]{slideimages/Mandula} } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \myhead{Golfand-Lichtman} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centerline{\includegraphics[width=0.35\hsize]{slideimages/Golfand} \includegraphics[width=0.247\hsize]{slideimages/Lichtmann} } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \myhead{Haag-Lopuszanski-Sohnius} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centerline{\includegraphics[width=0.3\hsize]{slideimages/Haag} \includegraphics[width=0.4\hsize]{slideimages/Lopuszanski} } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \myhead{SUSY algebra} \label{Chap1:intro:sec:susyalgebra} \beq \{ Q_\alpha , Q^\dagger_{\dot{\alpha}} \} = 2\sigma^\mu_{\alpha\dot{\alpha}} P_\mu, \label{Chap1:susyalg2} \eeq \beq \sigma^\mu_{\alpha\dot{\alpha}}&=&(1,\sigma^i)\,\,\, \sigmabar^{\mu \dot{\alpha}\alpha}=(1,-\sigma^i) \label{Chap1:sigma} \eeq \beq \begin{array}{ccc} \sigma^1=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) & \sigma^2=\left(\begin{array}{cc} 0 & -i \\ i & 0\end{array} \right) & \sigma^3=\left(\begin{array}{cc} 1 & 0\\ 0 & -1\end{array} \right) \end{array} \label{Chap1:pauli} \eeq \beq [P_\mu,Q_\alpha] = [P_\mu, Q^\dagger_{\dot{\alpha}} ] =0 \label{Chap1:susyalg3} \eeq \beq \begin{array}{cc} [Q_\alpha, R] = Q_\alpha & [Q^\dagger_{\dot{\alpha}}, R ] = - Q^\dagger_{\dot{\alpha}} \end{array} \label{Chap1:susyalgR} \eeq \beq H = P^0 = {{1 }\over{4}} (Q_1 Q^\dagger_1 +Q^\dagger_1 Q_1+ Q_2 Q^\dagger_2 +Q^\dagger_2 Q_2 ) \label{Chap1:energy} \eeq \newpage \beq (-1)^{\bf F}\, |{\rm boson}\rangle &= &+1\, |{\rm boson}\rangle\\ (-1)^{\bf F}\, |{\rm fermion}\rangle &=& -1\, |{\rm fermion}\rangle \label{Chap1:fermionOp} \eeq \beq \{(-1)^{\bf F}\, , Q_\alpha \}=0 \eeq \beq \sum_i | i \rangle \langle i | =1 \eeq so \beq \sum_i \langle i | (-1)^{\bf F} P^0 | i \rangle &=& \frac{1}{4}\left( \sum_i \langle i | (-1)^{\bf F} Q Q^\dagger |i\rangle +\sum_i\langle i | (-1)^{\bf F} Q^\dagger Q | i \rangle \right) \nonumber\\ &=&\frac{1}{4}\left( \sum_i \langle i | (-1)^{\bf F} Q Q^\dagger | i \rangle + \sum_{ij} \langle i | (-1)^{\bf F} Q^\dagger |j \rangle \langle j | Q | i \rangle \right) \nonumber\\ &=&\frac{1}{4}\left( \sum_i \langle i | (-1)^{\bf F} Q Q^\dagger | i \rangle + \sum_{ij} \langle j | Q | i \rangle\langle i | (-1)^{\bf F} Q^\dagger |j \rangle \right) \nonumber\\ &=&\frac{1}{4}\left( \sum_i \langle i | (-1)^{\bf F} Q Q^\dagger | i \rangle + \sum_j \langle j | Q (-1)^{\bf F} Q^\dagger | j \rangle \right) \nonumber\\ &=&\frac{1}{4}\left( \sum_i \langle i | (-1)^{\bf F} Q Q^\dagger | i \rangle - \sum_j \langle j | (-1)^{\bf F} Q Q^\dagger | j \rangle \right) \nonumber \\ &=& 0. \label{Chap1:bosonfermiondeg} \eeq \newpage \mysubhead{SUSY:} \beq Q_\alpha |0\rangle = 0 \eeq implies that the vacuum energy vanishes \beq \langle 0|H|0\rangle = 0 \eeq \vspace{24pt} \mysubhead{SUSY breaking:} \beq Q_\alpha |0\rangle \ne 0 \eeq and the vacuum energy is positive \beq \langle 0|H|0\rangle \ne 0 \eeq \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centerline{\includegraphics[width=0.78\hsize]{slideimages/potentials}} \label{Chap1:fig:potentials} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \myhead{SUSY representations} \label{Chap1:intro:sec:susyreps} massive particle rest frame: $p_\mu = (m,\vec{0})$. \beq \{ Q_\alpha , Q^\dagger_{\dot{\alpha}} \} &=& 2 \, m \,\delta_{\alpha{\dot{\alpha}}}\\ \{ Q_\alpha , Q_\beta \} &=&0\\ \{ Q^\dagger_{\dot{\alpha}} , Q^\dagger_{\dot{\beta}} \} &=&0 \eeq Clifford vacuum: \beq |\Omega_s\rangle = Q_1 Q_2 |m,s^\prime,s_3^\prime \rangle, \\ Q_1 |\Omega_s\rangle = Q_2 |\Omega_s\rangle =0 \eeq massive multiplet: \beq \begin{array}{c} |\Omega_s\rangle \\ Q^\dagger_1 |\Omega_s\rangle , Q^\dagger_2 |\Omega_s\rangle \\ Q^\dagger_1 Q^\dagger_2 |\Omega_s\rangle \end{array} \eeq \newpage massive ``chiral" multiplet: \beq \begin{array}{rc} {\rm state} & s_3 \\ |\Omega_0\rangle & 0 \\ Q^\dagger_1 |\Omega_0\rangle , Q^\dagger_2 |\Omega_0\rangle & \pm {{1}\over{2}} \\ Q^\dagger_1 Q^\dagger_2 |\Omega_0\rangle & 0 \end{array} \eeq massive vector multiplet: \beq \begin{array}{rl} {\rm state} & s_3 \\ |\Omega_{{1}\over{2}}\rangle & \pm {{1}\over{2}} \\ Q^\dagger_1 |\Omega_{{1}\over{2}}\rangle , Q^\dagger_2 |\Omega_{{1}\over{2}}\rangle & 0,1,0,-1 \\ Q^\dagger_1 Q^\dagger_2 |\Omega_{{1}\over{2}}\rangle & \pm {{1}\over{2}} \end{array} \eeq \myhead{Massless particles} frame: $p_\mu = (E,0,0,-E)$ \beq \{ Q_1 , Q^\dagger_1 \} &=& 4 E\\ \{ Q_2 , Q^\dagger_2 \} &=& 0\label{Chap1:anti2} \\ \{ Q_\alpha , Q_\beta \} &=&0\\ \{ Q^\dagger_{\dot{\alpha}} , Q^\dagger_{\dot{\beta}} \} &=&0 \eeq Clifford vacuum: \beq |\Omega_\lambda\rangle = Q_1 |E,\lambda^\prime \rangle , \\ Q_1 |\Omega_\lambda\rangle =0 \eeq \beq \langle\Omega_\lambda|Q_2 Q^\dagger_2|\Omega_\lambda\rangle +\langle\Omega_\lambda|Q^\dagger_2 Q_2|\Omega_\lambda\rangle = 0 \eeq \beq \langle\Omega_\lambda|Q_2 Q^\dagger_2|\Omega_\lambda\rangle = 0 \eeq \myhead{massless supermultiplet} \beq \begin{array}{rc} {\rm state} & {\rm helicity} \\ |\Omega_\lambda\rangle & \lambda \\ Q^\dagger_1 |\Omega_\lambda\rangle & \lambda + {{1}\over{2}} \end{array} \eeq CPT invariance requires: \beq \begin{array}{rc} {\rm state} & {\rm helicity} \\ |\Omega_{-\lambda-{{1}\over{2}}}\rangle & -\lambda-{{1}\over{2}} \\ Q^\dagger_1 |\Omega_{-\lambda-{{1}\over{2}}}\rangle & -\lambda \end{array} \eeq \myhead{massless chiral multiplet} \beq \begin{array}{rc} {\rm state} & {\rm helicity} \\ |\Omega_0\rangle & 0 \\ Q^\dagger_1 |\Omega_0\rangle & \frac{1}{2} \end{array} \eeq include CPT conjugate states: \beq \begin{array}{rc} {\rm state} & {\rm helicity} \\ |\Omega_{-\frac{1}{2}}\rangle &-\frac{1}{2} \\ Q^\dagger_1 |\Omega_{-\frac{1}{2}}\rangle & 0 \end{array} \eeq \myhead{massless vector multiplet} \beq \begin{array}{rc} {\rm state} & {\rm helicity} \\ |\Omega_{{1}\over{2}}\rangle & {{1}\over{2}} \\ Q^\dagger_1 |\Omega_{{1}\over{2}}\rangle & 1 \end{array} \eeq and its CPT conjugate: \beq \begin{array}{rc} {\rm state} & {\rm helicity} \\ |\Omega_{-1}\rangle & -1 \\ Q^\dagger_1 |\Omega_{-1}\rangle & -{{1}\over{2}} \end{array} \eeq \myhead{Superpartners} \beq\begin{array}{lcl} {\rm fermion} & \leftrightarrow & {\rm sfermion} \\ {\rm quark} & \leftrightarrow & {\rm squark} \\ {\rm gauge}\,{\rm boson} & \leftrightarrow & {\rm gaugino} \\ {\rm gluon} & \leftrightarrow & {\rm gluino} \end{array} \eeq %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \myhead{Extended SUSY} \label{Chap1:intro:sec:extendedsusy} \beq \{ Q_\alpha^a , Q^\dagger_{\dot{\alpha}b} \} &=& 2\sigma^\mu_{\alpha\dot{\alpha}} P_\mu \delta^a_b\\ \{ Q_\alpha^a , Q_\beta^b \} &=&0\\ \{ Q^\dagger_{\dot{\alpha}a} , Q^\dagger_{\dot{\beta}b} \} &=&0 \eeq where \beq a,b= 1, \ldots, {\mathcal N} \eeq \bc $U({\mathcal N})_R$ R-symmetry\ec massless multiplets: $p_\mu = (E,0,0,-E)$ \beq \{ Q^a_1 , Q^\dagger_{1b} \} &=& 4 E \delta^a_b,\\ \{ Q^a_2 , Q^\dagger_{2b} \} &=& 0. \eeq \myhead{general massless multiplet} \beq \begin{array}{rcc} {\rm state} & {\rm helicity} & {\rm degeneracy}\\ |\Omega_\lambda\rangle & \lambda & 1\\ Q^\dagger_{1a} |\Omega_\lambda\rangle & \lambda + {{1}\over{2}}& {\mathcal N} \\ Q^\dagger_{1a}Q^\dagger_{1b} |\Omega_\lambda\rangle & \lambda + 1 & {\mathcal N}({\mathcal N}-1)/2\\ \vdots & \vdots & \vdots\\ Q^\dagger_{11}Q^\dagger_{12}\ldots Q^\dagger_{1{\mathcal N}} |\Omega_\lambda\rangle & \lambda + {\mathcal N}/2 & 1 \end{array} \eeq \myhead{ ${\mathcal N}=2$ massless vector multiplet} \beq \begin{array}{rcc} {\rm state} & {\rm helicity} & {\rm degeneracy}\\ |\Omega_{-1}\rangle & -1 & 1 \\ Q^\dagger |\Omega_{-1}\rangle & -{{1}\over{2}} & 2 \\ Q^\dagger Q^\dagger |\Omega_{-1}\rangle & 0 & 1 \end{array} \label{Chap1:Neq2Vector} \eeq with the addition of the CPT conjugate: \beq \begin{array}{rcc} {\rm state} & {\rm helicity} & {\rm degeneracy}\\ |\Omega_0\rangle & 0 & 1\\ Q^\dagger |\Omega_0\rangle & {{1}\over{2}} & 2 \\ Q^\dagger Q^\dagger |\Omega_0\rangle & 1 & 1 \end{array} \eeq built from one ${\mathcal N}=1$ vector multiplet and one ${\mathcal N}=1$ chiral multiplet. \myhead{${\mathcal N}=2$ Hypermultiplet} \beq \begin{array}{rccl} {\rm state} & {\rm helicity} & {\rm degeneracy}\\ |\Omega_{-{{1}\over{2}}}\rangle & -{{1}\over{2}}& 1& \chi_\alpha \\ Q^\dagger |\Omega_{-{{1}\over{2}}}\rangle & 0 & 2 & \phi \\ Q^\dagger Q^\dagger |\Omega_{-{{1}\over{2}}}\rangle & {{1}\over{2}} & 1 & \psi^{\dagger {\dot\alpha}} \end{array} \eeq \vspace{24pt} \bc gauge-invariant mass term: $\psi^\alpha \chi_\alpha $\ec \vspace{24pt} \bc ${\mathcal N}=2$ is vector-like \ec \myhead{${\mathcal N}=3$ massless supermultiplet} \beq \begin{array}{rcl} {\rm state} & {\rm helicity} & {\rm degeneracy}\\ |\Omega_{-1}\rangle & -1 & 1\\ Q^\dagger |\Omega_{-1}\rangle & -{{1}\over{2}} &3 \\ Q^\dagger Q^\dagger |\Omega_{-1}\rangle & 0 & 3 \\ Q^\dagger Q^\dagger Q^\dagger |\Omega_{-1}\rangle & {{1}\over{2}} & 1 \end{array} \eeq plus CPT conjugate \beq \begin{array}{rcl} {\rm state} & {\rm helicity} & {\rm degeneracy}\\ |\Omega_{-{{1}\over{2}}}\rangle & -{{1}\over{2}}& 1 \\ Q^\dagger |\Omega_{-{{1}\over{2}}}\rangle & 0 & 3 \\ Q^\dagger Q^\dagger |\Omega_{-{{1}\over{2}}}\rangle & {{1}\over{2}} & 3 \\ Q^\dagger Q^\dagger Q^\dagger |\Omega_{-{{1}\over{2}}}\rangle & 1 & 1 \end{array} \eeq \bc ${\mathcal N}=3$ is vector-like \ec \myhead{${\mathcal N}=4$ massless vector supermultiplet} \beq \begin{array}{rcl} {\rm state} & {\rm helicity} & ${\bf R}$\\ |\Omega_{-1}\rangle & -1 & {\bf 1}\\ Q^\dagger |\Omega_{-1}\rangle & -{{1}\over{2}} & {\bf 4} \\ Q^\dagger Q^\dagger |\Omega_{-1}\rangle & 0 & {\bf 6} \\ Q^\dagger Q^\dagger Q^\dagger |\Omega_{-1}\rangle & {{1}\over{2}} & {\overline{\bf 4}} \\ Q^\dagger Q^\dagger Q^\dagger Q^\dagger |\Omega_{-1}\rangle & 1 & {\bf 1} \end{array} \label{Chap1:N4mult} \eeq \vspace{24pt} \bc vector-like theory \ec \myhead{Massive Supermultiplets} \beq \{ Q_\alpha^a , Q^\dagger_{\dot{\alpha}b} \} &=& 2 \, m \,\delta_{\alpha{\dot{\alpha}}}\delta^a_b \eeq \beq \begin{array}{cc} {\rm state} & {\rm spin}\\ |\Omega_s\rangle & s \\ Q^\dagger_{\dot{\alpha}a} |\Omega_s\rangle & s+\frac{1}{2} \\ Q^\dagger_{\dot{\alpha}a} Q^\dagger_{\dot{\beta}b} |\Omega_s\rangle& s+1\\ \vdots \\ Q^\dagger_{11}Q^\dagger_{21}Q^\dagger_{12}Q^\dagger_{22}\ldots Q^\dagger_{1{\mathcal N}} Q^\dagger_{2{\mathcal N}} |\Omega_\lambda\rangle & s \end{array} \eeq \myhead{${\mathcal N}=2$ massive supermultiplet} \beq \begin{array}{rc} {\rm state} & (d_R,2j+1) \\ |\Omega_0\rangle & (1,1) \\ Q^\dagger |\Omega_0\rangle& (2,2)\\ Q^\dagger Q^\dagger |\Omega_0\rangle & (3,1)+(1,3)\\ Q^\dagger Q^\dagger Q^\dagger |\Omega_0\rangle & (2,2)\\ Q^\dagger Q^\dagger Q^\dagger Q^\dagger |\Omega_0\rangle & (1,1) \end{array} \label{Chap1:massiveNeq2} \eeq 16 states: five of spin 0, four of spin $\frac{1}{2}$, and one of spin 1. \myhead{${\mathcal N}=4$ massive supermultiplet} \beq \begin{array}{rc} {\rm state} & ({\bf R},2j+1) \\ |\Omega_0\rangle & ({\bf 1},1) \\ Q^\dagger |\Omega_0\rangle& ({\bf 4},2)\\ Q^\dagger Q^\dagger |\Omega_0\rangle & ({\bf 10},1)+({\bf 6},3)\\ Q^\dagger Q^\dagger Q^\dagger |\Omega_0\rangle & ({\overline {\bf 20}},2)+({\overline {\bf 4}},4)\\ Q^\dagger Q^\dagger Q^\dagger Q^\dagger |\Omega_0\rangle &({\bf 20'},1)+ ({\bf 15},3)+({\bf 1},5)\\ Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger |\Omega_0\rangle & ({\bf 20},2)+({\bf 4},4)\\ Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger |\Omega_0\rangle & ({\overline {\bf 10}},1)+({\bf 6},3)\\ Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger |\Omega_0\rangle & ({\overline {\bf 4}},2)\\ Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger Q^\dagger |\Omega_0\rangle & ({\bf 1},1) \end{array} \eeq which contains 256 states, including eight spin $\frac{3}{2}$ states and one spin 2 state %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \myhead{Central Charges} \label{Chap1:intro:sec:centralcharges} \beq \{ Q_\alpha^a , Q^\dagger_{\dot{\alpha}b} \} &=& 2\sigma^\mu_{\alpha\dot{\alpha}} P_\mu \delta^a_b\\ \{ Q_\alpha^a , Q_\beta^b \} &=& 2 \sqrt{2} \epsilon_{\alpha \beta} Z^{a b}\\ \{ Q^\dagger_{\dot{\alpha}a} , Q^\dagger_{\dot{\beta}b} \} &=&2 \sqrt{2} \epsilon_{\dot{\alpha} \dot{\beta}} Z^*_{a b} \eeq where \beq \epsilon= i \sigma^2 \eeq for ${\mathcal N}=2$ \beq \{ Q_\alpha^a , Q^\dagger_{\dot{\alpha}b} \} &=& 2\sigma^\mu_{\alpha\dot{\alpha}} P_\mu \delta^a_b\\ \{ Q_\alpha^a , Q_\beta^b \} &=& 2 \sqrt{2} \epsilon_{\alpha \beta} \epsilon^{a b} Z\\ \{ Q^\dagger_{\dot{\alpha}a} , Q^\dagger_{\dot{\beta}b} \} &=&2 \sqrt{2} \epsilon_{\dot{\alpha} \dot{\beta}} \epsilon_{a b} Z \eeq \newpage Defining \beq A_\alpha &=& \frac{1}{2}\left[ Q_\alpha^1 +\epsilon_{\alpha \beta}\left(Q^2_\beta\right)^\dagger \right]\\ B_\alpha &=& \frac{1}{2}\left[ Q_\alpha^1 -\epsilon_{\alpha \beta}\left(Q^2_\beta\right)^\dagger \right] \eeq reduces the algebra to \beq \{ A_\alpha, A_\beta^\dagger \} &=& \delta_{\alpha \beta}(M+ \sqrt{2} Z)\\ \{ B_\alpha, B_\beta^\dagger \} &=& \delta_{\alpha \beta}(M- \sqrt{2} Z) \label{Chap1:antiB} \eeq \beq \langle M,Z | B_\alpha B_\alpha^\dagger | M,Z\rangle +\langle M,Z | B_\alpha^\dagger B_\alpha | M,Z\rangle = (M- \sqrt{2} Z)~ , \label{Chap1:BPSstate} \eeq \beq M\ge \sqrt{2} Z \eeq for $M =\sqrt{2} Z$ (short multiplets): $B_\alpha$ produces states of zero norm \vspace{24pt} $M > \sqrt{2} Z$ (long multiplets) \newpage short (BPS) multiplet: \beq \begin{array}{rc} {\rm state} & 2j+1\\ |\Omega_0\rangle & 1\\ A^\dagger |\Omega_0\rangle &2\\ (A^\dagger )^2 |\Omega_0\rangle &1 \end{array} \eeq \beq \begin{array}{rc} {\rm state} & 2j+1\\ |\Omega_\frac{1}{2}\rangle & 2\\ A^\dagger |\Omega_\frac{1}{2}\rangle &1+3\\ (A^\dagger )^2 |\Omega_\frac{1}{2}\rangle &2 \\ \end{array} \eeq short multiplet has 8 states as opposed to 32 states for the corresponding long multiplet BPS state: \beq M =\sqrt{2} Z \eeq \bc is exact \ec %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}