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\def\sigmabar{\overline\sigma}
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\begin{document}

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\chapter{Introduction to Supersymmetry}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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


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\end{document}
