\documentclass[aps,prl,twocolumn]{revtex4}

\newcommand{\cL}{{\cal L}}
\newcommand{\kapppa}{{\kappa^\prime}}
\newcommand{\wt}{\widetilde}
\begin{document}

%\title{Is Lepton flavour conservation an accidental symmetry in the MSSM
%without R-parity symmetry?}
%\title{A Note on R-parity violation in N=1 Supersymmetry}
\title{The Rotation of the R-parity violating bilinear term in
N=1 Supersymmetry with a minimal content of fields.} 
\date{\today}
\author{A. Dedes, H.~K.~Dreiner and H.~P.~Nilles}
%\email{dedes@th.physik.uni-bonn.de}
\affiliation{Physikalisches Institut der Universitat Bonn, 
 Nussallee 12, 
 D-53115 Bonn, Germany}

\begin{abstract}
%We prove that the lepton number conservation in the R-parity violating
%MSSM is an accidental symmetry.
We prove that once the billinear R-parity violating term is set 
to zero at some scale, it will never reappeared at any other one 
in perturbation theory. Consequently, we show how to rotate away 
the bilinear R-parity violating term in N=1 Supersymmetry 
with Standard Model (super)fields.
 
\end{abstract}

\maketitle

Consider the most general renormalizable, gauge invariant and 
holomorphic superpotential involving the indistinguisable 
lepton $L_i$, and Higgs $H_1$ superfields in an N=1 minimal supersymmetric
model,
%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\cal W} \ &=& \ (Y_E)_{ij} L_i H_1 \bar{E}_j + (Y_D)_{ij}Q_i H_1 \bar{D}_j
\nonumber \\
&+& \frac{1}{2} (\Lambda_{E^k})_{ij} L_i L_j \bar{E}_k 
+(\Lambda_{D^k})_{ij} L_i Q_j \bar{D}_k  \nonumber \\
&+& \mu H_1 H_2 + \kappa_i L_i H_2 \;.
\label{sp}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%
The latin indices $i,j=1,..3$ are the flavour indices while 
the SU(2) and SU(3) indices are supressed. 
It is well known~\cite{add} that even if we set the billinear 
$\kappa$-term equal to zero this will be generated from loop 
corrections due to the appearence of the non-zero anomalous
dimensions $\gamma_{H_1}^{L_i}$. The same also will happens
for vanishing (at tree level) Yukawa couplings-this time due to
the non-vanishing anomalous dimension, $\gamma_{L_i}^{L_j}, i\ne j$
in addition to $\gamma_{H_1}^{L_i}$. Thus it seems an ``apparent'' 
violation of the celebrated nonrenormalization theorem~\cite{nonrenorm}.
Of course this is not happend because in this framework  the kinetic
terms of the superfields are {\it not} canonically normalized.
Thus,
the most general Kahler potential involving the renormalized lepton 
superfileds $L_i$ and the ``Higgs'' doublet $H_1$ can be written in
the form,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
K(L,L^*)\ =\ \sum_{\alpha,\beta=1}^4 g_{\alpha\beta}L^{*\alpha} L^\beta \;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
with $\alpha=\{i,4 \}$ i.e., $L_4\equiv H_1$ . The matrix 
$g_{\alpha\beta}$ must be hermitian (because the action must be real)
 with a positive definite norm\footnote{This is required to make the canonical
 commutation relations consistent and it is not an assumption.}. 
So that, the  general form of $g_{\alpha\beta}$ to all orders in perturbation
theory is
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
g_{\alpha\beta}=\biggl ( 
\begin{array}{cc}  g_{ij} & g_{i4} \\ g_{i4}^* & g_{44} 
\end{array} \biggr ) \;,
\label{metric1}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
with $g_{ij}=g_{ji}^*$ and $g_{44}\in R_+$. 
More explicitly, we can write this matrix in the form,
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
g_{\alpha\beta}&=&\biggl ( 
\begin{array}{cc}  \delta_{ij}+\gamma_{ij} & \gamma_{i4} \\ 
\gamma_{i4}^* &1+ \gamma_{44} 
\end{array} \biggr ) \;,  \\
& & \nonumber
\label{metric2}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
where $\gamma_{ij}=\gamma_{L_i}^{L_j}$,  
$\gamma_{i4}=\gamma_{L_i}^{H_1}=\gamma_{H_1}^{L_i*}$ and 
$\gamma_{44}=\gamma_{H_1}^{H_1}$ are the wave function renormalization
matrices.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since $g_{\alpha\beta}$
is a hermitian matrix it can always be diagonilized by a matrix $N$,
%%%%%%%%%%%%%
\begin{eqnarray}
(N^\dagger g N)_{\alpha\beta} \ =\ \delta_{\alpha\beta} \;.
\label{dia}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This can be done by decomposing $N$ into a unitary matrix $U_{\alpha\beta}$
 which 
diagonilizes the hermitian $g_{\alpha\beta}$ times a diagonal
matrix whose elements are the inverse square roots of the eigenavalues 
$\omega_\alpha$
of $g_{\alpha\beta}$, i.e, 
$N_{\alpha\beta}=\frac{U_{\alpha\beta}}{\sqrt{\omega_\alpha}}$, (\rm no sum
over $\alpha$). The diagonilization of the metric means also a transformation
of the superfields,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
L_i &=& \sum_{\alpha} N_{i\alpha} {\cL_{\alpha}} \;, \\
L_4 &=& \sum_{\alpha} N_{4\alpha} {\cL_{\alpha}} \;.
\label{redefin}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%
under which the Kahler potential is diagonal, i.e. $K(L,L^*)=\sum_\alpha
\cL_{\alpha}^*\cL_{\alpha}$. The renormalized superpotential (\ref{sp})
also changes into 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\cal W}=
\frac{1}{2}
\biggl [ (Y_E)_{ik}N_{i\alpha}N_{4\beta}-
(Y_E)_{ik}N_{i\beta}N_{4\alpha}&+&(\Lambda_{E^k})_{ij}
N_{i\alpha}N_{j\beta} \biggr ] \cL_\alpha \cL_\beta \bar{E}_k
+\biggl [-(Y_D)_{jk}N_{4\alpha}+(\Lambda_{D^k})_{ij}N_{i\alpha} \biggr ]
\cL_\alpha Q_j \bar{D}_k \nonumber \\  
&+&\biggl [\mu N_{4\alpha}+\kappa_i N_{i\alpha} \biggr ] \cL_{\alpha} H_2
 \;,
\label{fterms}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%
Now from (\ref{fterms}) and 
at 1-loop order the anomalous dimension for the superfields
$\cL_\alpha$ and $\cL_\beta$ with $\alpha \ne \beta$ is given by
(we use here the  relation $\sum_\gamma N_{\gamma\alpha}^*N_{\gamma\beta}=
\frac{\delta_{\alpha\beta}}{Tr(\omega)}$ which is followed from the fact that
$U$ is unitary in the decomposition of the matrix $N$)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16\pi^2
\gamma^{\cL_\beta (1-loop)}_{\cL_\alpha} &=& 3 \biggl [ Tr(Y_DY_D^\dagger)
N_{4\alpha}N_{4\beta}^*-(Y_D)_{jk}(\Lambda_{D^k}^*)_{lj}N_{4\alpha}
N_{l\beta}^* -(Y_D^*)_{jk}(\Lambda_{D^k})_{ij}N_{i\alpha}N_{4\beta}^*+
(\Lambda_{D^k}\Lambda_{D^k}^\dagger)_{il}N_{i\alpha}N_{l\beta}^* \biggl ]
\nonumber \\
&+&
\frac{1}{Tr(\omega)}
\biggl [(Y_EY_E^\dagger)_{il}N_{i\alpha} N_{l\beta}^*
+Tr(Y_E Y_E^\dagger)N_{4\alpha}N_{4\beta}^*
+(\Lambda_{E^k}\Lambda_{E^k}^\dagger)_{il}N_{i\alpha}N_{l\beta}^*
\nonumber \\
&-&(\Lambda_{E^k}^*Y_E)_{lk}N_{4\alpha}N_{l\beta}^*
-(\Lambda_{E^k}Y_E^*)_{lk}N_{l\alpha}N_{4\beta}^* \biggr ] \;.
\label{wf}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
But the similarity transformation (\ref{dia}) implies that
for ($\alpha \ne \beta$)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\sum_{ij}N_{i\alpha}^*\gamma_{ij}N_{j\beta}+\sum_i N_{i\alpha}^*\gamma_{i4}
N_{4\beta} + \sum_i N_{4\alpha}^*\gamma_{i4}^*N_{i\beta} +
\gamma_{44}N_{4\alpha}^*N_{4\beta} =0 \;
\label{st}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
Substituting the explicit formulae for the anomalous dimensions of
the superfields appeared in the superpotential (\ref{sp})
at 1-loop from Ref.~\cite{add}  and for ($i\ne j$)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
16\pi^2\gamma_{ij} &=& (Y_EY_E^\dagger)_{ji}+
(\Lambda_{E^q}\Lambda_{E^q}^\dagger)_{ji}+
3(\Lambda_{D^q}\Lambda_{D^q}^\dagger)_{ji} \;, \nonumber \\ & & \\
16\pi^2\gamma_{i4} &=& 16\pi^2\gamma_{4i}^*=
-3(\Lambda_{D^q}^*Y_D)_{iq}-(\Lambda_{E^q}^*Y_E)_{iq} \;, \nonumber \\ & &
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
into (\ref{st}) we get 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\gamma^{\cL_\beta (1-loop)}_{\cL_\alpha} =0 \;\;\; (\alpha \ne \beta) \;.
\label{constraint}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%
Since this is valid at 1-loop it valids also to {\it all} orders
in perturbation theory because of the non-renormalization theorem.
One could also see that neither of $\mu$ or $\kappa_i$ could
be generated in the superpotential. Their RGE's are now decoupled
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{d\mu}{dt} &=& \mu \biggl [ \gamma_{{\cal H}_1}^{{\cal H}_1}
+ \gamma_{H_2}^{H_2} \biggr ] \;, \\ 
\frac{d\kappa_i}{dt} &=& \kappa_i \biggl [ \gamma_{\cL_i}^{\cL_i}+
\gamma_{H_2}^{H_2} \biggr ]  \;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
where ${\cal H}_1\equiv \cL_4$ is the Higgs doublet responsible in
giving masses to down quarks and leptons.
Thus the billinear $\kappa$-term can be set to zero at some scale
and it will never reappeared at any other one.  
Note that, soft supersymmetry breaking {\it does not} affect the proof
since the non-renormalization theorem is still valid and thus the
superfield wave functions do not depend on the soft breaking parameters. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{Rotate away all $\kappa$'s}
One should notice that the superpotential (\ref{fterms}) has the
same form like in (\ref{sp}) subject to the constraint (\ref{constraint})
and $L \rightarrow \cL$ and $\Lambda\rightarrow \Lambda', 
\kappa \rightarrow \kappa', 
\mu \rightarrow  \mu'$. 
We show that it is possible to rotate away the bilinear interactions
$\kappa_i'\cL_iH_2$ by a unitary transformation thus preserving the
canonical form of the K\"ahler potential. In order to do this we
introduce new fields $L_1',L_2',L_3',$ and $H_1'$. The old fields
$L_1,L_2,L_3,H_1$ are given in terms of the new fields by the 
rotation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\cL_\alpha = {\cal O}_{\alpha\beta}\cL_\beta' \;,
\label{rot}
\end{equation}
%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\left(
\begin{array}{c}
\cL_1 \\ \cL_2 \\ \cL_3 \\ \cL_4
\end{array}\right) =
\left(
\begin{array}{cccc}
c_2c_3&-s_2&0&c_2s_3 \\
c_1s_2c_3 & c_1c_2 & -s_1 & c_1s_2s_3 \\
s_1 s_2 c_3 & s_1c_2 & c_1 & s_1s_2s_3 \\
-s_3 & 0 &0 & c_3
\end{array}
\right)
\left(
\begin{array}{c}
\cL_1' \\ \cL_2'\\ \cL_3' \\ \cL_4'
\end{array}
\right)
\label{transf}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $c_i=\cos\theta_i$ and $s_i=\theta_i$ are rotation angles.
They are given explicitly by 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
c_1&=&\frac{\kapppa_2}{\sqrt{\kapppa_2^2+\kapppa_3^2}},\quad
s_1=\frac{\kapppa_3}{\sqrt{\kapppa_2^2+\kapppa_3^2}}\nonumber\\
c_2&=&\frac{\kapppa_1}{\sqrt{\vec{\kapppa}^2}},\quad
\;\;\;\;\;\;\;\;s_2=\frac{\sqrt{\kapppa_2^2+\kapppa_3^2}}
{\sqrt{\vec{\kapppa}^2}}\\
c_3&=&\frac{\mu}{\sqrt{\mu^2+\vec{\kapppa}^2}},\quad
s_3=\frac{{\sqrt{\vec{\kapppa}^2}}}
{\sqrt{\mu^2+\vec{\kapppa}^2}} \nonumber
\label{angles}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where we have introduced the notation $\vec{\kapppa}^2=\kapppa_1^2+
\kapppa_2^2+\kapppa_3^2$. After the above field redefinition the
only remaining superfield bi-linear term is
\begin{equation}
\mu''H_1'H_2
\end{equation}
with $\mu''=\sqrt{\mu'^2+\vec{\kapppa}^2}$ and $H_1'\equiv \cL_4'$.
Notice that the orthogonal transformation (\ref{transf}) is not
the Hall-Suzuki~\cite{hall} transformation. It is easy to show that 
the constraint (\ref{constraint}) is not violated by the transformation
(\ref{transf}). From \cite{add} and from (\ref{constraint}) we obtain
at 1-loop :
%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\gamma_{\cL_a^{'}}^{\cL_\beta^{'}} 
\propto Tr(\Lambda^{''}\Lambda^{''\dagger})
&=&Tr(\Lambda'{\cal O}{\cal O}^\dagger \Lambda^{'\dagger})\nonumber \\ &=&
Tr(\Lambda'\Lambda^{\dagger'})=\gamma_{\cL_\alpha}^{\cL_\beta}=0
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%
where $\Lambda^{''}\equiv (\Lambda^{''}_{\cL^\alpha})_{Q_j\bar{D}_k}$
or $\Lambda^{''}\equiv (\Lambda^{''}_{\cL^\alpha})_{\cL_\beta\bar{E}_k}$
are the new superpotential couplings after the transformation 
(\ref{transf}). Thus one can rotate away the billiear R-parity
violating term to all orders in prturbation theory and that completes
the proof.
%%%%%%%%%%%%%%%%%%%SOFT SUSY BREAKING TERMS%%%%%%%%%%%%%%%%%%
In what follows we prove that it is possible to rotate 
at the same time away the R-parity violating bilinear superpotential term 
toghether with the R-parity violating soft SUSY breaking bilinear term 
in the Lagrangian. Thus, the bilinear terms (we remove the primes now)
are written in the form
%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\cal L}^{bil} \ =\ \int d^2\theta \biggl \{ \mu \cL_4 H_2 +
\kappa_i \cL_i H_2 \biggr \} + {\rm H.c} 
+\int d^2\theta \eta \biggl \{ \wt{B} \cL_4 H_2 + \wt{D} \cL_i H_2
\biggr \} + {\rm H.c} \;,
\label{bil}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%
where $\eta\equiv \theta^2$ is the spurion superfield and 
$\theta_\alpha, a=1,2$ (as well as ${\bar \theta}$) are the spinorial
Gransmann components of the superspace. The transformation which rotates away
the terms proportional to $\kappa_i$ and $\wt{D}$ is 
the orthogonal matrix (\ref{transf}) with the replacement in (\ref{angles}) 
%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\mu &\rightarrow& \mu -\eta \wt{B} \nonumber \\
\kappa_i &\rightarrow& \kappa_i -\eta \wt{D}_i \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
Therefore, we end up with the Lagrangian of the form
%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\cal L}^{'bil} \ =\ \int d^2\theta \biggl \{ \mu' \cL_4' H_2 -
\eta {\wt{B}'}\cL_4' H_2 \biggr \} + {\rm H.c} \;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
where\footnote{We have made use of  the relation : $\sqrt{a+b\eta}=\sqrt{a}+
\eta \frac{b}{2\sqrt{a}}$.} 
%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\mu' &=& \sqrt{\mu^2+{\vec{\kappa}}^2} \;, \nonumber \\[1mm]
{\wt{B}'} &=& \frac{\mu \wt{B}+
\vec{\kappa}\cdot \wt{D}}{\sqrt{\mu^2+{\vec{\kappa}}^2}} \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
 













%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
%%%%%%%%%%%%%%%%%%%%
\bibitem{add}
B.~C.~Allanach, A.~Dedes and H.~K.~Dreiner,
%``2-loop supersymmetric renormalisation group equations including  R-parity violation and aspects of unification,''
Phys.\ Rev.\  {\bf D60}, 056002 (1999)
[hep-ph/9902251].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{nonrenorm}
M.~T.~Grisaru, W.~Siegel and M.~Rocek,
%``Improved Methods For Supergraphs,''
Nucl.\ Phys.\  {\bf B159} (1979) 429;
N.~Seiberg,
%``Naturalness versus supersymmetric nonrenormalization theorems,''
Phys.\ Lett.\  {\bf B318} (1993) 469
[hep-ph/9309335].
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{hall}
L.~J.~Hall and M.~Suzuki,
%``Explicit R Parity Breaking In Supersymmetric Models,''
Nucl.\ Phys.\  {\bf B231}, 419 (1984). The transformation 
presented in this paper is $U=\biggl (\begin{array}{cc}
1& -\delta_i \\
\delta_i & I 
\end{array} \biggr )$ where it is assumed that 
$\delta_i\equiv \frac{\kappa_i}{\mu}$ is less than a unity. The 
transformation is unitary only to the first order in $\delta_i$,
and thus does not preserve the K\"ahler metric in general.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{thebibliography}









\end{document}


















We begin our proof by discussing the tree level forms of the
Kahler and superpotenetial functions.  Since we can always 
follow the same procedure for the term $\sum_{i,j}g_{ij}L^{*i}L^j$ of
the Kahler potential, we obtain that at tree level $g_{ij}$ must be
proportional to the unit matrix, i.e., $g_{ij}=a\delta_{ij}$ with 
$a$ being a positive constant. In addition, and without loss of generality 
but for simplicity reasons we can have $a=g_{44}$. Thus the metric
at tree level becomes,
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
g_{\alpha\beta}=\biggl ( 
\begin{array}{cc}  a\delta_{ij} & g_{i4} \\ g_{i4}^* & a 
\end{array} \biggr ) \;,
\label{metrictree}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
The requirement of a positive definite metric implies 
$\sum_i|g_{i4}|^2< a^2$. The eigenvalues of (\ref{metrictree}) are
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\omega_1 &=& a \;, \\
\omega_2 &=& a \;, \\
\omega_3 &=& a-\sqrt{\sum_i|g_{i4}|^2} \;, \\
\omega_1 &=& a+ \sqrt{\sum_i|g_{i4}|^2}\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%
It is also not difficult to find the eigenvectors of (\ref{metrictree}),
however, we follow rather general arguments which will be useful for
completing the proof to {\it all} orders in perturbation theory.
Thus, from (\ref{dia}) we obtain,
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
a\delta_{\alpha\beta}&+&\sum_i(g_{i4}U_{i\alpha}^*U_{4\beta}+
g_{i4}^*U_{4\alpha}^*U_{i\beta} ) =\omega_\alpha \delta_{\alpha\beta}\;, \\
& &({\rm no~ sum~ over~} \alpha) \;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%
which implies the following relations :
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
U_{4\rho}&=&0 \;\;, \rho=1,2 \\
\sum_i \Re e(g_{i4}U_{i3}^*U_{43})&=&-\frac{1}{2}\sqrt{\sum_i|g_{i4}|^2} \;,\\
\sum_i \Re e(g_{i4}U_{i4}^*U_{44})&=&\frac{1}{2}\sqrt{\sum_i|g_{i4}|^2} \;,\\
\sum_ig_{i4}^*U_{i\rho}&=&0 \;, \rho=1,2 \\
\sum_i(g_{i4}U_{i3}^*U_{44}&+&g_{i4}^*U_{43^*}^*U_{i4})=0 \;.
\label{relations}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In addition, explicitly formulae for the eigenvectors give us also 
%%%%%%%%%%%%%%%%
\begin{eqnarray}
U_{43}=U_{44}=\frac{1}{\sqrt{2}} \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%
Now,  under the transformation (\ref{redefin}) 
 the $F$-terms\footnote{We ignore here the gauge kinetic
 terms since they are irrelevant in our
discusson.} in the Lagrangian can be written as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\cal L}=2\Re e \int d^2\theta \biggl \{
\biggl [ (Y_E)_{ik}N_{i\alpha}N_{4\beta}&+&\frac{1}{2}(\Lambda_{E^k})_{ij}
N_{i\alpha}N_{j\beta} \biggr ] \cL_\alpha \cL_\beta \bar{E}_k
+\biggl [-(Y_D)_{jk}N_{4\alpha}+(\Lambda_{D^k})_{ij}N_{i\alpha} \biggr ]
\cL_\alpha Q_j \bar{D}_k \nonumber \\  
&+&\biggl [\mu N_{4\alpha}+\kappa_i N_{i\alpha} \biggr ] \cL_{\alpha} H_2
\biggr \} \;,
\label{fterms}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%
If one wants to rotate away the vector biliear term proportional to
$\kappa_i$ in (\ref{fterms}) then it is not hard to see that,
%%%%%%%%%%%%%%%%
\begin{equation}
g_{i4}=\frac{kappa_i^*}{\mu^*} \;.
\label{kapparot}
\end{equation}
%%%%%%%%%%%%%%%%%%%
Following the relations (\ref{relations}) we arrive at 
%%%%%%%%%%%%%%
 



\end{document}