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\def\lagr{{\mathcal L}}
\def\sigmabar{\overline\sigma}
\def\eps{\epsilon}
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\begin{document}

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\chapter{SUSY Lagrangians}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Wess-Zumino}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.35\hsize]{slideimages/Wess} \includegraphics[width=0.41\hsize]{slideimages/Zumino}   }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The free Wess--Zumino model}
\label{Chap2:lagrang:sec:freeWessZumino}

\beq
&&S = \int d^4x\>
\left (\lagr_{\rm s} + \lagr_{\rm f}\right ) 
 \\
&&\lagr_{\rm s} = \partial^\mu \phi^* \partial_\mu \phi, \,\,\,
\lagr_{\rm f} = i {\psi^\dagger} \sigmabar^\mu \partial_\mu \psi .
\eeq
\beq
g^{\mu\nu}=\eta^{\mu\nu}= {\rm diag}(1,-1,-1,-1) 
\eeq

\beq
\phi \rightarrow \phi +\delta \phi  \\
\psi \rightarrow \psi+ \delta \psi 
\eeq
\beq
\delta \phi &=&  \eps^\alpha \psi_\alpha \\
& = & \eps^\alpha {\boldsymbol  \eps}_{\alpha \beta} \psi^\beta \equiv \eps \psi 
\label{Chap2:phitrans}
\eeq
\beq
{\boldsymbol \eps}_{\alpha \beta} = \left(
\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right) , \,
{\boldsymbol\eps}^{\alpha \beta} = \left(
\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) 
\eeq
\beq
\eps \psi = - \psi^\beta {\boldsymbol \eps}_{\alpha \beta} \eps^\alpha =
\psi^\beta{\boldsymbol  \eps}_{ \beta \alpha} \eps^\alpha = \psi \eps 
\eeq
\newpage

\beq
\delta \phi^* =\eps^\dagger _{\dot{\alpha}} {\psi^\dagger}^{\dot{\alpha}}
\equiv {\eps^\dagger} {\psi^\dagger} 
\eeq

\beq
\delta
\lagr_{\rm s} = \eps \partial^\mu \psi \> \partial_\mu \phi^*
+{\eps^\dagger} \partial^\mu {\psi^\dagger} \> \partial_\mu \phi 
\label{Chap2:dLscalar}
\eeq

\beq
\delta\psi_\alpha
=
-i (\sigma^\nu {\eps^\dagger})_\alpha\> \partial_\nu \phi \,\,\,
\delta{\psi^\dagger}_{\dot{\alpha}}
=
i (\eps\sigma^\nu)_{\dot{\alpha}}\>   \partial_\nu \phi^* 
\eeq

\beq
\delta \lagr_{\rm f} =
-\eps \sigma^\nu \partial_\nu \phi^* \sigmabar^\mu \partial_\mu \psi\> 
+{\psi^\dagger} \sigmabar^\mu \sigma^\nu {\eps^\dagger} \>
\partial_\mu \partial_\nu \phi 
\eeq

Pauli identities:
\beq
&&
\bigl[ \sigma^\mu \sigmabar^\nu + \sigma^\nu \sigmabar^\mu
\bigr ]_\alpha^\beta  =
2 \eta^{\mu\nu} \delta_\alpha^\beta \,\,\,
\bigl[ \sigmabar^\mu \sigma^\nu + \sigmabar^\nu \sigma^\mu \bigr
]_{\dot{\alpha}}^{\dot{\beta}}=
2 \eta^{\mu\nu} \delta_{\dot{\alpha}}^{\dot{\beta}} 
\label{Chap2:Pauliident}
\eeq

\beq
\delta \lagr_{\rm f} & =
&-\eps\partial^\mu\psi\> \partial_\mu\phi^*
-{\eps^\dagger} \partial^\mu{\psi^\dagger}\> \partial_\mu \phi
\nonumber\\
&& +\partial_\mu \left (
\epsilon \sigma^\mu \sigmabar^\nu \psi \> \partial_\nu \phi^* -
\epsilon \psi\> \partial^\mu \phi^*
+{\eps^\dagger} {\psi^\dagger} \> \partial^\mu \phi \right ).
\label{Chap2:dLfold}
\eeq
total derivative so:
\beq
\delta S = 0 
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Commutators of SUSY transformations}
\label{Chap2:lagrang:sec:commutators}


\beq
(\delta_{\eps_2} \delta_{\eps_1} -
\delta_{\eps_1} \delta_{\eps_2}) \phi =
-i (\eps_1 \sigma^\mu {\eps_2^\dagger} -
   \eps_2 \sigma^\mu {\eps_1^\dagger})\> \partial_\mu \phi  
\eeq

\beq
(\delta_{\eps_2} \delta_{\eps_1} -
\delta_{\eps_1} \delta_{\eps_2}) \psi_\alpha =
-i(\sigma^\nu{\eps_1^\dagger})_\alpha\>  \eps_2 \partial_\nu\psi +
i(\sigma^\nu{\eps_2^\dagger})_\alpha \> \eps_1 \partial_\nu\psi 
\eeq
Fierz identity:
\beq
\chi_{\alpha}\> (\xi\eta) =
- \xi_{\alpha}\> (\chi\eta) -  (\xi\chi)\eta_\alpha\>
\label{Chap2:fierz}
\eeq

\beq
(\delta_{\eps_2} \delta_{\eps_1} -
\delta_{\eps_1} \delta_{\eps_2}) \psi_\alpha & = &
-i (\eps_1 \sigma^\mu {\eps_2^\dagger} -
   \eps_2 \sigma^\mu {\eps_1^\dagger}) \> \partial_\mu \psi_\alpha
\nonumber \\
& &
+i (\eps_{1\alpha} \> {\eps_2^\dagger} \sigmabar^\mu \partial_\mu\psi
- \eps_{2\alpha} \> {\eps_1^\dagger} \sigmabar^\mu \partial_\mu\psi)
{}.
\label{Chap2:naivefermioncomm}
\eeq
SUSY algebra closes on-shell.

\newpage
on-shell the fermion EOM reduces DOF by  two

 $p_\mu=(p,0,0,p)$ 
\beq
\overline{\sigma}^\mu p_\mu \psi = \left(\begin{array}{cc} 0 & 0 \\ 0 & 2 p \end{array}\right)
\left( \begin{array}{c}
\psi_1 \\\psi_2 \end{array}\right) 
\label{Chap2:ondhellprojector}
\eeq
projects out half of DOF  
\beq
\begin{array}{rcc}
& \textrm{ off-shell}\,\,\, & \textrm{ on-shell}  \\
\phi, \phi^* & 2 \textrm{ d.o.f.}& 2 \textrm{ d.o.f.}\\
\psi_\alpha, {\psi^\dagger}_{\dot{\alpha}} & 4 \textrm{ d.o.f.}& 2 \textrm{ d.o.f.}
\end{array}
\eeq
SUSY is not manifest off-shell

 trick: add an auxiliary boson field ${\mathcal F}$
\beq
\begin{array}{rcc}
& \textrm{ off-shell}\,\,\,  & \textrm{ on-shell}  \\
{\mathcal F}, {\mathcal F}^* & 2 \textrm{ d.o.f.}& 0 \textrm{ d.o.f.}\\
\end{array}
\eeq

\beq
\lagr_{\rm aux} = {\mathcal F}^* {\mathcal F}  
\label{Chap2:Laux}
\eeq
\newpage


\beq
\delta {\mathcal F} = -i {\eps^\dagger} \sigmabar^\mu \partial_\mu \psi,
\,\,\,
\delta {\mathcal F}^* = i\partial_\mu {\psi^\dagger} \sigmabar^\mu \eps 
\label{Chap2:dF}
\eeq

\beq
\delta \lagr_{\rm aux} = 
i \partial_\mu {\psi^\dagger} \sigmabar^\mu \eps \> {\mathcal F}
-i {\eps^\dagger} \sigmabar^\mu \partial_\mu
\psi \> {\mathcal F}^* 
\label{Chap2:dLaux}
\eeq
modify the transformation of the fermion:
\beq
\delta \psi_\alpha =
-i (\sigma^\nu {\eps^\dagger})_{\alpha}\> \partial_\nu
\phi + \eps_\alpha {\mathcal F},\,\,\,
\delta {\psi^\dagger}_{\dot{\alpha}}=
+i (\eps\sigma^\nu)_{\dot{\alpha}}\> \partial_\nu
\phi^* + \eps^\dagger_{\dot{\alpha}} {\mathcal F}^* 
\label{Chap2:dpsi}
\eeq

\beq
\delta^{\rm new}\lagr_f&=&\delta^{\rm old}\lagr_f+i {\eps^\dagger} \sigmabar^\mu \partial_\mu \psi {\mathcal F}^*
+i {\psi^\dagger} \sigmabar^\mu \partial_\mu \eps {\mathcal F}\\
&=&\delta^{\rm old}\lagr_f+i {\eps^\dagger} \sigmabar^\mu \partial_\mu \psi {\mathcal F}^*
-i \partial_\mu {\psi^\dagger} \sigmabar^\mu  \eps {\mathcal F}
+\partial_\mu (i {\psi^\dagger} \sigmabar^\mu \eps {\mathcal F}) 
\label{Chap2:dLferm}
\eeq
 last term is a total derivative
\beq
S^{\rm new} = \int d^4x \> \lagr_{\rm free}=\int d^4x \> (\lagr_{\rm s} +\lagr_{\rm f} + \lagr_{\rm aux}) 
\label{Chap2:Lfree}
\eeq
is invariant under SUSY transformations:
\beq
\delta S^{\rm new} = 0 
\eeq
\newpage


\myhead{Commutator of two SUSY transformations acting on the fermion}
\beq
(\delta_{\eps_2} \delta_{\eps_1} -
\delta_{\eps_1} \delta_{\eps_2}) \psi_\alpha & = &
-i (\eps_1 \sigma^\mu {\eps_2^\dagger} -
   \eps_2 \sigma^\mu {\eps_1^\dagger}) \> \partial_\mu \psi_\alpha
\nonumber \\
& &
+i (\eps_{1\alpha} \> {\eps_2^\dagger} \sigmabar^\mu \partial_\mu\psi
- \eps_{2\alpha} \> {\eps_1^\dagger} \sigmabar^\mu \partial_\mu\psi)\nonumber\\
&&+\delta_{\eps_2}  \eps_{1\alpha} {\mathcal F} - \delta_{\eps_1} \eps_{2\alpha} {\mathcal F}
 
\eeq

\beq
\delta_{\eps_2}  \eps_{1\alpha} {\mathcal F} - \delta_{\eps_1} \eps_{2\alpha} {\mathcal F}
= \eps_{1\alpha}(-i {\eps_2^\dagger}  \sigmabar^\mu \partial_\mu\psi)-
\eps_{2\alpha} ( -i {\eps_1^\dagger} \sigmabar^\mu \partial_\mu\psi) 
\eeq

\beq
(\delta_{\eps_2} \delta_{\eps_1} -
\delta_{\eps_1} \delta_{\eps_2}) \psi_\alpha & = &
-i (\eps_1 \sigma^\mu {\eps_2^\dagger} -
   \eps_2 \sigma^\mu {\eps_1^\dagger}) \> \partial_\mu \psi_\alpha 
   \label{Chap2:fermioncomm}
\eeq
SUSY algebra closes for off-shell fermions
\newpage

\myhead{Commutator acting on the auxiliary field}
\beq
(\delta_{\eps_2} \delta_{\eps_1} -
\delta_{\eps_1} \delta_{\eps_2}) {\mathcal F} & = &
\delta_{\eps_2} (-i {\eps_1^\dagger} \sigmabar^\mu \partial_\mu \psi)
-\delta_{\eps_1}( -i {\eps_2^\dagger} \sigmabar^\mu \partial_\mu \psi)\nonumber\\
& = &
-i {\eps_1^\dagger} \sigmabar^\mu \partial_\mu(-i \sigma^\nu {\eps_2^\dagger} \> \partial_\nu
\phi + \eps_2  {\mathcal F})\\ &&
+ i {\eps_2^\dagger} \sigmabar^\mu \partial_\mu (-i \sigma^\nu {\eps_1^\dagger} \> \partial_\nu
\phi + \eps_1  {\mathcal F})\nonumber\\
&=&-i (\eps_1 \sigma^\mu {\eps_2^\dagger} -
   \eps_2 \sigma^\mu {\eps_1^\dagger}) \> \partial_\mu {\mathcal F}\nonumber\\
&&
- {\eps_1^\dagger} \sigmabar^\mu  \sigma^\nu {\eps_2^\dagger} \>\partial_\mu  \partial_\nu \phi
+ {\eps_2^\dagger} \sigmabar^\mu  \sigma^\nu {\eps_1^\dagger} \>\partial_\mu  \partial_\nu \phi 
\eeq

Thus for 
 \beq
X=\phi, \phi^*, \psi, \psi^\dagger, {\mathcal F}, {\mathcal F}^*
\label{Chap2:listoffields}
\eeq

\beq
(\delta_{\eps_2} \delta_{\eps_1} -
\delta_{\eps_1} \delta_{\eps_2}) X =
-i (\eps_1 \sigma^\mu {\eps_2^\dagger} -
   \eps_2 \sigma^\mu {\eps_1^\dagger}) \> \partial_\mu X 
   \label{Chap2:generalcommt}
\eeq



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Noether}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.43\hsize]{slideimages/Noether}   }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Noether's Theorem}
\label{Chap2:lagrang:sec:supercurrent}
 Noether theorem: 
 \newline corresponding to every continuous symmetry is a conserved
current. 

infinitesimal symmetry $(1+\eps\, T)X= X+\delta X$ 
\beq
\delta {\mathcal L} = {\mathcal L}(X+\delta X) -{\mathcal L}(X)=\partial_\mu V^\mu 
\eeq
EOM:
\vspace{-14pt}
\beq
\partial_\mu \left( \frac{\partial  {\mathcal L}} {\partial(\partial_\mu X)} \right) =\frac{\partial {\mathcal L}} {\partial X},
\eeq

\beq
\partial_\mu V^\mu=\delta {\mathcal L} &=& \frac{\partial {\mathcal L}} {\partial X} \delta X+ \left( \frac{\partial  {\mathcal L}} {\partial(\partial_\mu X)} \right) \delta (\partial_\mu X) \\
&=& \partial_\mu \left( \frac{\partial  {\mathcal L}} {\partial(\partial_\mu X)} \right) \delta X+ \left( \frac{\partial  {\mathcal L}} {\partial(\partial_\mu X)} \right)  \partial_\mu \,\delta X\\
&=& \partial_\mu \left( \frac{\partial  {\mathcal L}} {\partial(\partial_\mu X)} \delta X \right) 
\eeq

\beq
\eps \partial_\mu J^\mu= \partial_\mu  \left( \frac{\partial  {\mathcal L}} {\partial(\partial_\mu X)} \delta X -V^\mu\right)  
\eeq

\myhead{Conserved SuperCurrent}

conserved
supercurrent, $J_\alpha^\mu$:

\beq
\eps J^\mu + {\eps^\dagger} {J^{\dagger \mu}}
\equiv  \,\frac {\partial \lagr}{\partial (\partial_\mu X)} \,  \delta X-
V^\mu 
\label{Chap2:Noether}
\eeq

\beq
\eps J^\mu + {\eps^\dagger} {J^{\dagger \mu}}
= \delta \phi \partial^\mu \phi^* + \delta \phi^* \partial^\mu \phi
+i {\psi^\dagger} \sigmabar^\mu \delta \psi - V^\mu 
\eeq

\beq
\eps J^\mu + {\eps^\dagger} {J^{\dagger \mu}}
&=& \eps \psi \partial^\mu \phi^* + {\eps^\dagger} {\psi^\dagger} \partial^\mu \phi
+i {\psi^\dagger} \sigmabar^\mu (-i \sigma^\nu {\eps^\dagger}\> \partial_\nu
\phi + \eps {\mathcal F}) \nonumber \\
&&- \epsilon \sigma^\mu \sigmabar^\nu \psi \> \partial_\nu \phi^* +
\epsilon \psi\> \partial^\mu \phi^*
-{\eps^\dagger}{\psi^\dagger} \> \partial^\mu \phi -i {\psi^\dagger} \sigmabar^\mu  \eps {\mathcal F} \nonumber\\
&=& 2\eps \psi \partial^\mu \phi^* 
+{\psi^\dagger} \sigmabar^\mu \sigma^\nu {\eps^\dagger}\> \partial_\nu
\phi \nonumber - \epsilon \sigma^\mu \sigmabar^\nu \psi \> \partial_\nu \phi^* 
 
\eeq
\newpage
Using the Pauli identity:
\beq
J^\mu_\alpha = (\sigma^\nu\sigmabar^\mu\psi)_\alpha\> \partial_\nu \phi^*
,\,\,\,\,
{J^{\dagger \mu}}_{\dot{\alpha}}
=  ({\psi^\dagger} \sigmabar^\mu \sigma^\nu)_{\dot{\alpha}}
\> \partial_\nu \phi .
\label{Chap2:WZsupercurrent}
\eeq
 
conserved supercharges:
\beq
Q_\alpha = {\sqrt{2}}\int d^3x\, J^0_\alpha,\,\,\,\,
{Q^\dagger}_{\dot{\alpha}} = {\sqrt{2}} \int d^3x \, {J^{\dagger 0}}_{\dot{\alpha}} 
\label{Chap2:WZsupercurrent-charge}
\eeq
generate SUSY transformations 
\beq
\left [ \eps Q + {\eps^\dagger} {Q^\dagger}, X \right ]
= -i{\sqrt{2}} \> \delta X 
\eeq


Commutators of the supercharges acting on fields give:
\beq
\Bigl [
\eps_2 Q + {\eps_2^\dagger} {Q^\dagger},\,
 \bigl [\eps_1 Q + {\eps_1^\dagger} {Q^\dagger}
,\, X
\bigr ] \Bigr ]
-
\Bigl [
\eps_1 Q + {\eps_1^\dagger} {Q^\dagger},\,
\bigl [
\eps_2 Q + {\eps_2^\dagger} {Q^\dagger}
,\, X
\bigr ] \Bigr ]
\qquad{}
\nonumber
\\
 = 2(\eps_2 \sigma^\mu {\eps_1^\dagger} -
         \eps_1 \sigma^\mu {\eps_2^\dagger})\, i\partial_\mu X \,\,\,\,\,\,\,\,
\eeq

\beq
\Bigl [ \bigl [
\eps_2 Q + {\eps_2^\dagger} {Q^\dagger},\,
\eps_1 Q + {\eps_1^\dagger} {Q^\dagger}\bigr ]
,\, X
\Bigr ]
=  2(\eps_2 \sigma^\mu {\eps_1^\dagger} -
         \eps_1 \sigma^\mu {\eps_2^\dagger})\,  [ P_\mu , X
 ] 
\eeq

Since this is true for any $X$, we have
\newpage
\beq
 \bigl [
\eps_2 Q + {\eps_2^\dagger} {Q^\dagger},\,
\eps_1 Q + {\eps_1^\dagger} {Q^\dagger}\bigr ]
=  2(\eps_2 \sigma^\mu {\eps_1^\dagger} -
         \eps_1 \sigma^\mu {\eps_2^\dagger})\,  P_\mu  
\eeq
Since $\eps_1$ and $\eps_2$ are arbitrary, we have
\beq
\bigl [\eps_2 Q, {\eps_1^\dagger} {Q^\dagger}\bigr ] &=& 2 \eps_2 \sigma^\mu {\eps_1^\dagger} 
P_\mu  \\
\bigl [ {\eps_2^\dagger} Q, \eps_1 {Q^\dagger}\bigr ] &=& -2 \eps_2 \sigma^\mu {\eps_1^\dagger} 
P_\mu  \\
\bigl [ \eps_2  Q, \eps_1 Q \bigr ] &=&\bigl [ {\eps_2^\dagger}  {Q^\dagger}, {\eps_1^\dagger} {Q^\dagger}\bigr ] =0 
\eeq
Extracting the arbitrary $\eps_1$ and $\eps_2$:
\beq
&&\{ Q_\alpha , {Q^\dagger}_{\dot{\alpha}} \} =
2\sigma^\mu_{\alpha\dot{\alpha}} P_\mu,
\label{Chap2:nonschsusyalg1}\\
&&\{ Q_\alpha, Q_\beta\} =
\{ {Q^\dagger}_{\dot{\alpha}}, {Q^\dagger}_{\dot{\beta}} \} = 0 
\eeq
which is just  the SUSY algebra

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The interacting Wess--Zumino model}
\label{Chap2:lagrang:sec:WessZumino}


\beq
\lagr_{\rm free} = \partial^\mu \phi^{*j} \partial_\mu \phi_j
+i \psi^{\dagger j} \sigmabar^\mu \partial_\mu \psi_j
+ {\mathcal F}^{*j} {\mathcal F}_j
\label{Chap2:Lfree2}
\eeq

\beq
\delta \phi_j = \eps\psi_j &&
\delta \phi^{*j} = {\eps^\dagger} \psi^{\dagger j}  \nonumber\\
\delta \psi_{j \alpha} =
-i (\sigma^\mu {\eps^\dagger})_{\alpha}\, \partial_\mu
\phi_j + \eps_\alpha {\mathcal F}_j  &&
\delta \psi^{\dagger j}_{\dot{\alpha}}=
i (\eps\sigma^\mu)_{\dot{\alpha}}\, \partial_\mu
\phi^{*j} + \eps^\dagger_{\dot{\alpha}} {\mathcal F}^{*j} 
\\
\delta {\mathcal F}_j = -i {\eps^\dagger} \sigmabar^\mu\partial_\mu
\psi_j  &&
\delta {\mathcal F}^{* j} = i\partial_\mu \psi^{\dagger j} \sigmabar^\mu \eps   \nonumber 
\label{Chap2:WZSUSYtrans}
\eeq


 most general set of renormalizable
interactions:
\beq
\lagr_{\rm int} = -\half W^{jk} \psi_j \psi_k
+ W^j {\mathcal F}_j
+ h.c. ,
\eeq

$\psi_j \psi_k = \psi_j^\alpha \eps_{\alpha \beta} \psi^\beta_k$ is  symmetric under $j\leftrightarrow k$, $\Rightarrow$ $W^{jk}$

potential  $U(\phi_j,\phi^{*j})$  breaks SUSY, since a 
SUSY transformation gives
\beq
\delta U = \frac{\partial U}{\partial \phi_j} \eps\psi_j +  \frac{\partial U}{\partial \phi^{*j}} {\eps^\dagger}\psi^{\dagger j} 
\eeq
which is linear  in $\psi_j$ and $\psi^{\dagger j}$ with no derivatives or ${\mathcal F}$ dependence and cannot be canceled by any other term  in $\delta \lagr_{\rm int}$

\myhead{require SUSY}

\beq
\delta \lagr_{\rm int} |_{\rm 4-spinor} \! &=&\!
-\half \frac{\partial W^{jk}}{\partial \phi_n} (\eps \psi_n)(\psi_j \psi_k)
-\half \frac{\partial W^{jk}}{ \partial \phi^{*n}} ({\eps^\dagger}
\psi^{\dagger n})
(\psi_j \psi_k)
+ h.c.\,\,\,\,\,\,
\eeq
Fierz identity $\Rightarrow$
\beq
(\eps \psi_j) (\psi_k \psi_n) + (\eps \psi_k) (\psi_n \psi_j)
+ (\eps \psi_n) (\psi_j\psi_k) = 0 ,
\eeq
 $\delta\lagr_{\rm int} |_{\rm 4-spinor} $
vanishes  iff $\partial W^{jk}/\partial \phi_n$
is totally symmetric under the interchange of $j,k,n$.
We also need 
\beq
\frac{\partial W^{jk}}{\partial \phi^{*n}}=0 
\eeq
so $W^{jk}$ is {\it analytic} ( {\it
holomorphic})


define superpotential $W$:
\beq
W^{jk} = {\partial^2 \over \partial\phi_j\partial\phi_k} W 
\eeq
\newpage

for renormalizable interactions
\beq
W = E^j \phi_j + {1\over 2} M^{jk} \phi_j \phi_k 
+ {1\over 6} y^{jkn} \phi_j \phi_k
\phi_n 
\label{Chap2:superpotential}
\eeq




and $M^{jk}$, $y^{jkn}$ are 
are symmetric under interchange of indices.

take $E^j=0$ so SUSY is unbroken

\beq
\delta \lagr_{\rm int} |_\partial &=&
- i W^{jk}\partial_\mu \phi_k \, \psi_j \sigma^\mu {\eps^\dagger}
- i W^j\, \partial_\mu \psi_j\sigma^\mu {\eps^\dagger}
+ h.c.
\label{Chap2:partials}
\eeq

\beq
W^{jk} \partial_\mu \phi_k =
\partial_\mu \left ( {\partial W\over \partial{\phi_j}}\right ) 
\eeq
so $\delta \lagr_{\rm int} |_\partial $ will be a total derivative
iff
\beq
W^j = \frac{\partial W}{ \partial \phi_j} 
\eeq
 remaining terms:
\beq
\delta \lagr_{\rm int}|_{{\mathcal F},{\mathcal F}^*} = -W^{jk}{\mathcal F}_j \eps \psi_k +\frac{\partial W^j}{\partial \phi_k} \eps \psi_k {\mathcal F}_j 
\eeq
identically cancel if previous conditions are satisfied


proof did not rely on the functional form
of $W$, only that it was holomorphic

\myhead{integrate out auxillary fields}
action is quadratic in ${\mathcal F}$
\beq
\lagr_{\mathcal F}=
{\mathcal F}_j {\mathcal F}^{*j} + W^j {\mathcal F}_j + W^{*}_j {\mathcal F}^{*j} 
\eeq
perform the corresponding Gaussian path integral exactly by solving its algebraic equation of motion:
\beq
{\mathcal F}_j = -W_j^*, \,\,\, {\mathcal F}^{*j} = -W^j
\label{Chap2:Feqnmotion}
\eeq
without auxiliary fields  SUSY transformation $\psi$  would be different for each choice of $W$

plugging in to $\lagr$:
\beq
\lagr &=& \partial^\mu \phi^{*j} \partial_\mu \phi_j
+i \psi^{\dagger j} \sigmabar^\mu \partial_\mu \psi_j \nonumber \\
&&-\half \left (W^{jk} \psi_j \psi_k  + W^{*jk} \psi^{\dagger j}
\psi^{\dagger k} \right )
- W^j W^{*}_j
\eeq


\newpage
\myhead{WZ Lagrangian}
\beq
&&V(\phi,\phi^*) = W^j W_j^* = {\mathcal F}_j {\mathcal F}^{*j} =
M^*_{jn} M^{nk} \phi^{*j} \phi_k
\\  && +
{1\over 2} M^{jm} y_{knm}^* \phi_j \phi^{*k} \phi^{*n}
+{1\over 2} M_{jm}^{*} y^{knm} \phi^{*j} \phi_k \phi_n
+{1\over 4} y^{jkm} y_{npm}^{*} \phi_j \phi_k \phi^{*n} \phi^{*p} 
\nonumber\eeq
as required by SUSY:
\beq
V(\phi,\phi^*)  \ge 0 
\eeq

 interacting Wess--Zumino model:
\beq
\lagr_{\rm WZ} &=&
\partial^\mu \phi^{*j} \partial_\mu \phi_j
+i \psi^{\dagger j} \sigmabar^\mu \partial_\mu \psi_j
\nonumber \\ &&
- \half M^{jk} \psi_j\psi_k - \half M_{jk}^{*} \psi^{\dagger j}
\psi^{\dagger k}
- V(\phi,\phi^*)
\nonumber\\
&& - \half y^{jkn} \phi_j \psi_k \psi_n - \half y_{jkn}^{*} \phi^{*j}
\psi^{\dagger k}{\psi^{\dagger n}}.
\label{Chap2:wesszuminolagr}
\eeq
quartic coupling is $|y|^2$  as
 required to cancel the $\Lambda^2$ divergence  in $\phi$ mass
 
 \vspace{18pt}
 \noindent
$| $cubic coupling$|^2 \propto$ quartic coupling $\times |M|^2$
as required to cancel the $\log \Lambda$ divergence

\newpage
\myhead{ linearized equations of motion}
\beq
\partial^\mu\partial_\mu \phi_j &=& -M_{jn}^{*} M^{nk} \phi_k
+ \ldots; \\
i\sigmabar^\mu\partial_\mu\psi_j &=& M_{jk}^{*}
\psi^{\dagger k}+\ldots; 
\label{Chap2psiEQM} \\
i\sigma^\mu\partial_\mu\psi^{\dagger j} &=& M^{jk} \psi_k
+\ldots 
\label{Chap2psidagEQM}
\eeq

Multiplying $\psi $ eqns  by $i \sigma^\nu \partial_\nu$, and $i \sigmabar^\nu \partial_\nu$, and
using the Pauli identity we  obtain
\beq
\partial^\mu\partial_\mu \psi_j &=&  -M_{jn}^{*} M^{nk} \psi_k
+ \ldots ; \\
\partial^\mu\partial_\mu \psi^{\dagger k} &=&
- \psi^{\dagger j}
M_{jn}^{*} M^{nk}+\ldots  
\eeq
 scalars and fermions have the same mass eigenvalues,
as required by SUSY

diagonalizing gives a collection of massive
chiral supermultiplets.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Yang-Mills}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.335\hsize]{slideimages/Yang} \includegraphics[width=0.41\hsize]{slideimages/Mills}   }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{SUSY Yang--Mills}
\label{Chap2:lagrang:sec:yangmills}

under a gauge transformation 
 gauge field, $A^a_\mu$, and gaugino field, $\lambda^a$, transform as:
\beq
&&\delta_{\rm gauge} A^a_\mu = -\partial_\mu \Lambda^a + g f^{abc}
A^b_\mu \Lambda^c 
\label{Chap2:Agaugetr}
\\
&&\delta_{\rm gauge} \lambda^a = g f^{abc} \lambda^b \Lambda^c 
\label{Chap2:lamgaugetr}
\eeq
where $\Lambda^a$ is an infinitesimal gauge transformation parameter,
$g$ is the gauge coupling, and $f^{abc}$ are the
antisymmetric structure constants of the gauge group which satisfy 
\beq
[T_{\bf r}^a,T_{\bf r}^b]=i f^{abc} T_{\bf r}^c 
\eeq
for the generators $T^a$  for any representation ${\bf r}$. For the adjoint representation :
\beq
(T_{\bf Ad}^b)_{ac} = i f^{abc} 
\eeq

\myhead{degrees of freedom}
Gauge invariance removes one degree of freedom from the 
gauge field, while the eqm projects out another,  fermion eqm  project out half the degrees of freedom, so
\beq
\begin{array}{rcc}
& \textrm{ off-shell}\,\,\,  & \textrm{ on-shell}  \\
A^a_\mu & 3 \textrm{ d.o.f.}& 2 \textrm{ d.o.f.}\\
\lambda^a_\alpha, {\lambda^{\dagger a}}_{\dot{\alpha}} & 4 \textrm{ d.o.f.}& 2 \textrm{ d.o.f.}
\end{array}
\eeq
 for SUSY to be manifest off-shell 
add a real auxiliary boson field $D^a$.
\beq
\begin{array}{rcc}
& \textrm{ off-shell}\,\,\,  & \textrm{ on-shell}  \\
D^a & 1 \textrm{ d.o.f.}& 0 \textrm{ d.o.f.}\\
\end{array}
\eeq


\myhead{SUSY Yang--Mills Lagrangian}
\beq
\lagr_{\rm SYM} = -{1\over 4} F_{\mu\nu}^a F^{\mu\nu a}
+ i {\lambda^{\dagger a}} \sigmabar^\mu D_\mu \lambda^a
+ {1\over 2} D^a D^a 
\label{Chap2:LSYM}
\eeq
where the gauge field strength is given by
\beq
F^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu -
g f^{abc} A^b_\mu A^c_\nu 
\label{Chap2:FieldStrength}
\eeq
 and the gauge covariant derivative of the gaugino is 
\beq
D_\mu \lambda^a = \partial_\mu \lambda^a - g f^{abc} A^b_\mu \lambda^c 
\eeq
auxiliary field has dimension $[D^a]= 2$

\myhead{infinitesimal SUSY transformations}
\begin{itemize}
\item
 should be linear in $\eps$ and ${\eps^\dagger}$
 \item transform
$A^a_\mu$ and $\lambda^a$ into each other 
\item keep $A^a_\mu$ real 
\item maintain the correct dimensions of fields with $[\eps]=-\sfrac{1}{2}$
\item 
infinitesimal change in
$D^a$ should vanish when the equations of motion are satisfied
\item infinitesimal change in the  $\lambda^a$  should involve the derivative of the $A^a_\mu$ so that the infinitesimal changes in the two kinetic terms cancel
\end{itemize}

but gauge transformation of $\partial_\mu A^a_\nu$ different from $\lambda^a$ and $F^a_{\mu\nu}$  

%\myhead{correct SUSY transformations} 
\beq
&& \delta A_\mu^a = -{1\over \sqrt{2}} \left [{\eps^\dagger}
\sigmabar_\mu
\lambda^a + {\lambda^{\dagger a}} \sigmabar_\mu \eps \right ]
\label{Chap2:YMSUSYtrans1}\\
&& \delta \lambda^a_\alpha =
-{i\over 2\sqrt{2}} (\sigma^\mu \sigmabar^\nu \eps)_\alpha
\> F^a_{\mu\nu} + {1\over \sqrt{2}} \eps_\alpha\> D^a \label{Chap2:YMSUSYtrans2}\\
&& \delta {\lambda^{\dagger a}}_{\dot{\alpha}} =
{i\over 2\sqrt{2}} ({\eps^\dagger} \sigmabar^\nu \sigma^\mu)_{\dot{\alpha}} 
\> F^a_{\mu\nu} + {1\over \sqrt{2}} \eps^\dagger_{\dot{\alpha}} \> D^a \label{Chap2:YMSUSYtrans3}\\
&& \delta D^a = {-i\over \sqrt{2}} \left [
{\eps^\dagger} \sigmabar^\mu D_\mu \lambda^a -
D_\mu {\lambda^{\dagger a}} \sigmabar^\mu \eps \right ] 
\label{Chap2:YMSUSYtrans4}
\eeq





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{SUSY gauge theories}
\label{Chap2:lagrang:sec:gaugetheories}

add a set of  chiral  supermultiplets 
\beq
\delta_{\rm gauge}X_j = ig \Lambda^a T^a X_j 
\eeq
for $X_j = \phi_j,\psi_j,{\mathcal F}_j$.
gauge covariant derivatives are:
\beq
&&
D_\mu \phi_j =
\partial_\mu \phi_j + i g A^a_\mu \, T^a\phi_j 
\\
&&
 D_\mu \phi^{*j} =
\partial_\mu \phi^{*j} - i g A^a_\mu \, \phi^{*j} T^a 
\\
&&
 D_\mu \psi_j =
\partial_\mu \psi_j + i g A^a_\mu\, T^a\psi_j 
\eeq

new allowed renormalizable
interactions:
\beq
(\phi^* T^a \psi)\lambda^a,\qquad
{\lambda^{\dagger a}} ({\psi^\dagger} T^a
\phi) 
 \qquad
(\phi^* T^a \phi) D^a 
\label{Chap2:extrater}
\eeq
all are required by SUSY with particular  couplings. The first two are required
to cancel pieces of the SUSY transformations of the gauge interactions of $\phi$ and $\psi$. The third  is needed to cancel pieces of the SUSY transformations  of the first two terms.
 
\myhead{ Lagrangian for a SUSY gauge theory}
\beq
\lagr & = & \lagr_{\rm SYM} + \lagr_{\rm WZ} 
         - \sqrt{2} g \left [
(\phi^* T^a \psi)\lambda^a + {\lambda^{\dagger a}} ({\psi^\dagger} T^a \phi)
\right ]
 + g  (\phi^* T^a \phi) D^a.\,\,\,\,\,\,\,\,\,\,
\label{Chap2:gensusylagr}
\eeq
 $\lagr_{\rm WZ}$ is general Wess-Zumino with gauge-covariant
derivatives.  superpotential must be gauge invariant:
\beq
\delta_{\rm gauge}W =ig \Lambda^a  \frac{\partial W}{\partial \phi_i} T^a \phi_i = 0.
\label{Chap2:Wgaugeinv}
\eeq 


infinitesimal SUSY transformations of $\phi$ and $\psi$ are have  derivatives promoted
 to  gauge covariant derivatives, ${\mathcal F}$ has an additional term required bythe gaugino interactions:
\beq&&
\delta \phi_j = \eps\psi_j 
\label{Chap2:gphitran}\\
&&\delta \psi_{j\alpha} =
-i (\sigma^\mu {\eps^\dagger})_{\alpha}\, D_\mu
\phi_j + \eps_\alpha {\mathcal F}_j \\
&&\delta {\mathcal F}_j = -i {\eps^\dagger} \sigmabar^\mu D_\mu \psi_j
\, + \, \sqrt{2} g (T^a \phi)_j\, {\eps^\dagger} {\lambda^{\dagger a}}  
\eeq

\myhead{Integrating out auxiliary fields}
eqm  for the auxiliary field $D^a$:
\beq
D^a = -g \phi^* T^a \phi  
\eeq
scalar potential is given by
``${\mathcal F}$-terms" and ``$D$-terms":
\beq
V(\phi,\phi^*) = {\mathcal F}^{*i} {\mathcal F}_i + {{1}\over{2}}  D^a D^a = W_i^* W^i +
 {{1}\over{2}}  g^2 (\phi^* T^a \phi)^2 
\label{Chap2:fdpot}
\eeq
 as required by SUSY,  is positive definite:
 \beq
V(\phi,\phi^*) \ge 0 
\eeq 
for the vacuum to preserve SUSY  $V=0$ $\Rightarrow$  ${\mathcal F}_i=0$ and $D^a=0$.


\myhead{Feynman vertices}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\hsize]{slideimages/yangmills}}
\label{Chap2:fig:yangmills}
\caption[]{}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Cubic and quartic Yang--Mills interactions; wavy lines denote gauge fields.
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\hsize]{slideimages/gaugeint}}
\label{Chap2:fig:gaugeint}
\caption[]{}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Interactions required by gauge invariance. Solid lines denote fermions,
dashed lines denote scalars, wavy lines denote gauge bosons, wavy/solid lines denote
gauginos.


\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\hsize]{slideimages/susygaugeint}}
\label{Chap2:fig:SUSYrelgaugeint}
\caption[]{}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Additional interactions required by gauge invariance and 
SUSY: (a) $\phi^* \psi \lambda$, (b)  $\phi^* \phi D$
 coupling, 
Note that
these three vertices all have the same gauge index structure, being proportional to the gauge
generator $T^a$. Integrating
out (c) the auxiliary field  gives (d) the quartic scalar coupling proportional to $T^a T^a$.

\newpage


\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\hsize]{slideimages/yukint}}
\caption[]{}
\label{Chap2:fig:yukint}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dimensionless non-gauge interaction vertices
in a renormalizable supersymmetric theory: (a) $\phi_i \psi_j \psi_k$ Yukawa
interaction vertex  $-i y^{ijk}$, (b) $\phi_i \phi_j {\mathcal F}_k$ interaction vertex $i y^{ijk}$, (c) integrating out the auxiliary field
yields, (d) the quartic scalar interaction $-i y^{ijn}y^*_{kln}$ required for cancelling  the $\Lambda^2$
divergence in the Higgs mass


\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.44\hsize]{slideimages/massint}}
\label{Chap2:fig:massint}
\caption[]{}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dimensionful couplings: (a) $\psi\psi$ mass
 insertion $-i M^{ij}$, (b) $\phi {\mathcal F}$ mixing term insertion $+i M^{ij}$, (c) integrating out ${\mathcal F}$ in cubic term, (d) integrating out ${\mathcal F}$ in mass term,
(e) $\phi^2\phi^*$interaction vertex $-i M^*_{in} y^{jkn}$, (f) $\phi^*\phi$ mass insertion $-i M^*_{ik}M^{kj}$
ensuring cancellation of  the $\log \Lambda$ in the Higgs mass



\myhead{Supercurrent}
 using the Noether theorem one finds that the conserved supercurrent is:
\beq
J_\alpha^\mu &=& {i\over {\sqrt{2}}} D^a 
\, (\sigma^\mu {\lambda^{\dagger a}})_\alpha
+ {\mathcal F}_i  \, i (\sigma^\mu \psi^{\dagger i})_\alpha\, 
\nonumber
\\ &&+(\sigma^\nu\sigmabar^\mu \psi_i)_\alpha\, D_\nu \phi^{*i}
- {1\over 2 \sqrt{2}}
(\sigma^\nu \sigmabar^\rho \sigma^\mu
{\lambda^{\dagger a}})_\alpha\, F^a_{\nu\rho} 
\label{Chap2:SUSYgaugesupercurrent}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Salam}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.335\hsize]{slideimages/Salam}  }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Superspace Notation}
\label{Chap2:lagrang:sec:superspace}


anticommuting (Grassmann) spinors:
$\theta_\alpha$, ${\bar \theta}_{\dot  \alpha}=\theta^\dagger_{\alpha}$.
for a single component Grassmann variable $\eta$ we have:
\beq
\int d\eta = 0, \int \eta \,  d\eta =1
\label{Chap2:Grassman}
\eeq
two-component Grassmann spinor:
\beq
\{\theta_\alpha,{\bar \theta}_{\dot  \alpha}\}=0 
\eeq
define:
\beq
d^2 \theta &\equiv& -{{1}\over{4}} d\theta^\alpha d\theta^\beta {\boldsymbol  \eps}_{\alpha
\beta} \\
d^2 {\bar \theta} &\equiv& -{{1}\over{4}} d{\bar \theta}_{\dot \alpha}
 d{\bar \theta}_{\dot \beta} {\boldsymbol  \eps}^{{\dot \alpha}
{\dot \beta}}  \\
d^4 \theta &\equiv& d^2 \theta d^2 {\bar \theta} 
\eeq
Then 
\beq
\int d^2\theta \, \theta^2 &=& \int d^2\theta \,
\theta^\sigma \theta_\sigma 
= -{{1}\over{4}}\int d\theta^\alpha d\theta^\beta {\boldsymbol  \eps}_{\alpha
\beta}\theta^\sigma {\boldsymbol  \eps}_{\sigma \tau} \theta^\tau \nonumber\\
&=& -{{1}\over{4}}({\boldsymbol  \eps}_{\alpha \beta}\delta^{\beta \sigma}{\boldsymbol  \eps}_{\sigma \tau}
\delta^{\tau \alpha} -{\boldsymbol  \eps}_{\alpha \beta}\delta^{\alpha \sigma}{\boldsymbol  \eps}_{\sigma
\tau} \delta^{\tau \beta}) \\
&=& -{{1}\over{4}}({\boldsymbol  \eps}_{\alpha \beta}{\boldsymbol  \eps}_{\beta \alpha} + {\boldsymbol  \eps}_{\beta \alpha}
{\boldsymbol  \eps}_{\alpha \beta}) 
= -{{1}\over{2}}{\boldsymbol  \eps}_{\alpha \beta}{\boldsymbol  \eps}_{\beta \alpha}\nonumber \\
&=&1 \nonumber
\eeq

\newpage
and for arbitrary spinors $\chi$ and $\psi$ we have
\beq
\int d^2\theta \, (\chi\theta) (\psi \theta) =-{{1}\over{2}}(\chi \psi) 
\eeq
define a ``superspace coordinate"
\beq
y^\mu = x^\mu - i \theta \sigma^\mu {\bar \theta} 
\eeq
Then we can assemble the fields of a chiral supermultiplet into a
{\em chiral superfield}:
\beq
\Phi(y) &\equiv& \phi(y) +\sqrt{2} \theta \psi(y)+ \theta^2 {\mathcal F}(y) \nonumber \\
&=&\phi(x) - i\theta \sigma^\mu {\bar \theta} \partial_\mu \phi(x)
-{{1}\over{4}}\theta^2 {\bar \theta}^{2} \partial^2 \phi(x) \nonumber \\
&&+\sqrt{2} \theta \psi(x) +{{i}\over{\sqrt{2}}} \theta^2 \partial_\mu \psi(x) 
\sigma^\mu {\bar \theta} + \theta^2 {\mathcal F}(x) 
\eeq
second line follows by  Taylor expanding in Grassmann variables $\theta$ and ${\bar \theta}$

\myhead{Superspace SUSY Lagrangians }

\vspace{24pt}
\beq
\int d^4\theta \, \Phi^\dagger \Phi &=&
\int d^4\theta \, \left(
\begin{array}{l}
\phi^* +i\theta\sigma^\mu
{\bar \theta} \partial_\mu \phi^* -{{1}\over{4}}{\bar \theta}^{2} 
\theta^2 \partial^2 \phi^*\\
+\sqrt{2} {\bar \theta} {\psi^\dagger}
-{{i}\over{\sqrt{2}}} \theta \sigma^\mu 
\partial_\mu {\psi^\dagger}{\bar \theta}^{2}  +{\bar  \theta}^{2} {\mathcal F}^*
\end{array}
\right)\nonumber \\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\left(
\begin{array}{l}
\phi -i\theta\sigma^\mu
{\bar \theta} \partial_\mu \phi -{{1}\over{4}} 
\theta^2 {\bar \theta}^{2}\partial^2 \phi\\
+\sqrt{2} \theta \psi
+{{i}\over{\sqrt{2}}} \theta^{2}  
\partial_\mu \psi  \sigma^\mu {\bar \theta} + \theta^2 {\mathcal F}
\end{array}
\right) \nonumber \\
&=& {\mathcal F}^*{\mathcal F} + \partial^\mu \phi^* \partial_\mu \phi
+ i {\psi^\dagger} \sigmabar^\mu \partial_\mu \psi  \\
&&-{{1}\over{4}}\partial^\mu
(\phi^*\partial_\mu \phi+ \partial_\mu \phi^* \phi)+{{i}\over{2}}
\partial_\mu
({\psi^\dagger} \sigmabar^\mu \psi)  \nonumber
\eeq
so
\beq
\int d^4 x \, d^4 \theta \, \Phi^\dagger \Phi = \int d^4 x\, {\mathcal L}_{\rm free} 
\eeq

\myhead{Superpotential}
\vspace{24pt}
\beq
\int d^2\theta \, W(\Phi) &=& \int d^2\theta \,(W(\Phi)|_{\theta=0} +
\theta W_1 + \theta^2 W_2)
= \int d^2\theta \,\theta^2 W_2  \\
&=&W_a {\mathcal F}^a -{{1}\over{2}} W^{ab} \psi_a \psi_b
-\partial_\mu({{1}\over{4}}
W^a {\bar \theta}^{2} \partial^\mu \phi_a -{{i}\over{\sqrt{2}}} W^a \psi_a
\sigma^\mu {\bar \theta})  \nonumber
\eeq

\beq
\int d^4 x \,  d^2 \theta \,W(\Phi) +h.c. = \int d^4 x \,{\mathcal L}_{\rm int} 
\eeq
product of chiral superfields is also a chiral superfield

\vspace{12pt}
\noindent SUSY variation of  the $\theta^2$ component of a chiral superfield
is a total derivative

\vspace{12pt}
$\int d^4x \,d^2\theta W$ is always  SUSY invariant 

\myhead{\Kahler function}
\beq
\int  d^4 x \,  d^4 \theta K(\Phi^\dagger, \Phi) 
\eeq
where $K$  is real give kinetic and other (in general non-renormalizable) interactions

\vspace{12pt}
\noindent
 SUSY variation of the $\theta^2 {\bar \theta}^{2}$ component of any
superfield is
a total derivative

\myhead{Real Superfield}

\vspace{24pt}
real superfield contains,
in addition to the vector supermultiplet, an auxiliary  field $D^a$, plus three real scalar fields and
an additional Weyl spinor. In  Wess--Zumino gauge  the extra scalars and spinor vanish, simplifies  to:
\beq
V^a = \theta \sigma^\mu {\bar \theta} A_\mu^a+ \theta^2 {\bar \theta} 
{\lambda^{\dagger a}}+  {\bar \theta}^{2} 
\theta \lambda^{a} +{{1}\over{2}} \theta^2 {\bar \theta}^{2} D^a~,
\label{Chap2:vectorsuperfield}
\eeq
In the Wess--Zumino gauge we have:
\beq
V^a V^b &=& {{1}\over{2}} \theta^2 {\bar \theta}^{2} A^{a\mu} A^b_\mu \\
V^a V^b V^c&=&0 
\eeq
\myhead{Extended gauge invariance}
gauge parameter becomes a chiral superfield, $\Lambda^a$

\vspace{12pt}
\noindent
extended gauge transformation
can reintroduce (or remove) the extra
 scalar and spinor fields:
\beq
\exp (T^a V^a)\rightarrow\exp (T^a \Lambda^{a\dagger} ) \exp (T^a V^a)  \exp (T^a \Lambda^a) 
\eeq\
so
\beq
V^a\rightarrow  V^a +  \Lambda^a +\Lambda^{a\dagger} + {\mathcal O}(V^a \Lambda^a)
\eeq

chiral superfield $\Phi$ transforms under the extended gauge invariance as
\beq
\Phi \rightarrow  e^{- g T^a  \Lambda^a } \Phi 
\eeq

\myhead{``field strength" chiral superfield}
superspace 
derivatives  defined by
\beq
D_\alpha &=& \frac{\partial}{\partial \theta^\alpha} - 2 i (\sigma^\mu {\bar \theta})_\alpha
\frac{\partial}{\partial y^\mu}  \\
{\bar D}_{\dot \alpha} &=& -\frac{\partial}{\partial {\bar \theta}^{\dot \alpha}}  
\label{Chap2:SUSYDeriv}
\eeq
Then the ``field strength" chiral superfield is given by
\beq
T^a W^a_\alpha &=&  -\frac{1}{4} {\bar D}_{\dot \alpha} {\bar D}^{\dot \alpha} e^{-T^aV^a}D_\alpha e^{T^aV^a} \\
W^a_\alpha&=&-i\lambda^a_\alpha(y) +\theta_\alpha D^a(y) 
-(\sigma^{\mu \nu} \theta)_\alpha F^a_{\mu\nu}(y)
-(\theta \theta)\sigma^\mu D_\mu {\lambda^{\dagger a}}(y),\,\,\,\,
\label{Chap2:FieldStrengthchiralsuper}
\eeq
where 
\beq
\sigma^{\mu\nu} &=& {{i}\over{4}}(\sigma^\mu {\overline \sigma}^\nu - 
\sigma^\nu {\overline \sigma}^\mu) 
\label{Chap2:sigmamunu}
\eeq

\myhead{SUSY Yang--Mills action} 
\vspace{24pt}
\beq
\int d^4 x \, \lagr_{SYM}&=&\frac{1}{4} \int d^4 x  \,d^2 \theta \, \, W^{a \alpha}
W^a_\alpha + h.c.\\
&=&\frac{1}{4} \int d^4 x \,d^4 \theta \, \, {\rm Tr} T^a W^{a \alpha} e^{-T^aV^a}D_\alpha e^{T^aV^a} \!\!+ h.c.
\label{Chap2:SuperspaceSYM}
\eeq

standard gauge invariant kinetic terms:
\beq
\int d^4 \theta \> \Phi^\dagger  e^{ g T^a V^a } \Phi 
\eeq
or more generally (to include non-renormalizable interactions):
\beq
\int d^4 \theta \> K(\Phi^\dagger , e^{ g T^a V^a } \Phi) 
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{ ${\mathcal N}=0$ SUSY}
\label{Chap2:break:sec:soft}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.25\hsize]{slideimages/Weisskopf}  Weisskopf }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


chiral symmetry $\Rightarrow$ multiplicative mass renormalization

\beq
m_f = m_0 
+c_f\, {{\alpha}\over{16 \pi^2}} m_0 \ln\left({{\Lambda}\over{m_0}} \right) 
\eeq
where $\Lambda$ is the cutoff

\vspace{12pt}
\noindent
SUSY ensures that the scalar mass is given by the same
formula

\newpage
\myhead{ SUSY dim.less couplings
$\Rightarrow$ no  $\Lambda^2$ divergences}


\vspace{12pt}
\noindent
SUSY must be broken in the real world, eg.
\beq
W= E^a \phi_a 
\eeq
 gives  a scalar potential 
 \beq
 V=W_a^* W^a = E^a E_a^* \ne 0 
 \eeq
 which
breaks SUSY.  


\vspace{12pt}
\noindent
We want to break SUSY such that Higgs -- top squark
quartic coupling  $\lambda=|y_t|^2$.  If not we reintroduce a $\Lambda^2$
divergence
in the Higgs mass:
\beq
\delta m^2_h  \propto (\lambda -|y_t|^2) \Lambda^2 
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Effective Theory}
\label{Chap2:break:sec:soft}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.25\hsize]{slideimages/Grisaru} \includegraphics[width=0.25\hsize]{slideimages/girardello}   Grisaru, Girardello}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent
We want an {\em effective theory} of
broken SUSY with only soft breaking  terms (operators with dimension $<4$).
Girardello and Grisaru  found:
\beq
{\mathcal L}_{\rm soft} &=&-{{1}\over{2}}(M_\lambda \lambda^a \lambda^a + h.c.)
-(m^2)^i_j \phi^{*j} \phi_i \nonumber \\
&&-({{1}\over{2}}b^{ij} \phi_i \phi_j +{{1}\over{6}} a^{ijk} \phi_i \phi_j
\phi_k + h.c.) \label{Chap2:SoftSUSY} \\
&&-{{1}\over{2}} c_i^{jk} \phi^{i*} \phi_j \phi_k + e^i \phi_i +h.c. \nonumber
\eeq
$e^i \phi_i$ is only allowed if $\phi_i$ is a gauge singlet
The  $c_i^{jk}$ term may introduce quadratic divergences
if there is a gauge singlet multiplet in the model.


\newpage
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\begin{figure}[htb]
\centerline{\includegraphics[width=0.99\hsize]{slideimages/softsusy}}
\label{Chap2:fig:softsusy}
\caption[]{}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Additional soft SUSY breaking interactions: (a) 
gaugino mass $M_\lambda$, (b) non-holomorphic mass $m^2$, (c) holomorphic mass
$b^{ij}$, (d) holomorphic trilinear coupling $a^{ijk}$, 
(e) non-holomorphic trilinear coupling $c_i^{jk}$, and (f) tadpole $e^i$.


\myhead{Spurions}

fictitious background fields that transform under a symmetry group

\vspace{12pt}
\noindent
``spontaneously" breaks symmetry

\vspace{12pt}
\noindent
chiral superfield $\Phi$ with a wavefunction renormalization factor $Z$

\vspace{12pt}
\noindent
Take $Z$  to be a real SUSY breaking spurion field
\beq
Z=1+b\, \theta^2
+b^* {\bar \theta}^2
+c \,\theta^2 {\bar \theta}^2 
\label{Chap2:spurionZ}
\eeq
\beq 
\int  d^4 \theta \, Z \Phi^\dagger \Phi =  \lagr_{\rm free} +
b {\mathcal F}^*  \phi+
b^* \phi^* {\mathcal F}+c \phi^* \phi 
 \eeq
 integrate out the auxiliary field ${\mathcal  F}$:
\beq 
\int  d^4 \theta \, Z \Phi^\dagger \Phi =
\partial^\mu \phi^* \partial_\mu \phi+ i {\psi^\dagger} \sigmabar^\mu \partial_\mu \psi 
+(c-|b|^2) \phi^* \phi 
\eeq
\vspace{12pt}
\noindent
soft SUSY breaking mass: $m^2=|b|^2-c$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Superpotential spurions}
 $\theta^2$ component spurions in  Yukawa couplings, masses,
and the coefficient
  of $W_\alpha W^\alpha$ generate  $a$, $b$, and $M_\lambda$
soft SUSY breaking terms   \beq
{\mathcal L}_{\rm soft} &=&-{{1}\over{2}}(M_\lambda \lambda^a \lambda^a + h.c.)
-(m^2)^i_j \phi^{*j} \phi_i \nonumber \\
&&-({{1}\over{2}}b^{ij} \phi_i \phi_j +{{1}\over{6}} a^{ijk} \phi_i \phi_j
\phi_k + h.c.) \label{Chap2:SoftSUSY} \\
&&-{{1}\over{2}} c_i^{jk} \phi^{i*} \phi_j \phi_k + e^i \phi_i +h.c. \nonumber
\eeq

The $c$ term requires a term like
\beq 
\int  d^4 \theta \, C^{jk}_i \Phi^{i \dagger} \Phi_j \Phi_k +h.c. 
\eeq
where $C^{jk}_i$ has a nonzero $\theta^2 {\bar \theta}^2$ component


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\end{document}
 
