Sensitivity to the supersymmetric scalar states $\phi$ at the future linear $\ee$ and photon colliders is discussed. In particular it is illustrated a search strategy for massive sgoldstinos, the supersymmetric partners of the goldstino. %Comparisons with other machine sensitivity is also given.
sgoldstino_lc.tex
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\newcommand {\eefg}      {e^+e^-\rightarrow \phi \gamma}
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\begin{document}
%\rightline{\bf DELPHI }
\rightline{ LC-TH-2001-015.}
\vspace*{0.2cm}
\rightline{\today}
\vspace*{0.3cm}
\begin{center}
\Huge {\bf Sensitivity to sgoldstino states 
at the future 
%$\sqrt{s}$=500 GeV 
linear $\ee$ and photon colliders  
\\}
%at $\sqrt{s}$=500 GeV\\}
%\vspace*{0.6cm}
%\Large {Preliminary}\\
%\vspace*{0.5cm}
%\Large {DELPHI Collaboration}\\
\vspace{.5 cm}
\large{\bf   Paolo Checchia \\ 
\it I.N.F.N sezione di Padova and Dipartimento di Fisica G.Galilei,\\ Padova, Italy\\ 
\vspace*{0.3cm}

  and\\
  \vspace*{0.3cm}
  
\bf   Enrico Piotto \\
\it CERN, European Organization for Nuclear Research, Geneva, Switzerland.\\

\rm}
\vspace{.5 cm}
\end{center}
%--------------------------------------------------------------------------------
%--------------------------------------------------------------------------------
%--------------------------------------------------------------------------------
\vspace{1. cm}
\begin{abstract}
Sensitivity to the supersymmetric scalar states $\phi$
at the future linear $\ee$ and photon colliders is discussed.
In particular it is illustrated a search strategy  for massive sgoldstinos, the supersymmetric
 partners of the goldstino.
%Comparisons with other machine sensitivity is also given.


\end{abstract}
%--------------------------------------------------------------------------------
%--------------------------------------------------------------------------------
%--------------------------------------------------------------------------------
\newpage


\section{ Introduction}


In the Supersymmetric extension of the Standard Model, once Supersymmetry is
spontaneously broken the gravitino $\grav$ can acquire a mass absorbing the
degrees of freedom of the goldstino.
% with a 
The mechanism is analogous to
the spontaneous breaking of the electro-weak 
symmetry in the Standard Model, when Z and W bosons 
acquire mass absorbing the goldstone bosons.

A very light gravitino $\grav$ as predicted by supersymmetric 
models ~\cite{ref:GMSB} has been searched for at LEP and Tevatron 
%experiments~\cite{ref:l3bound,ref:alephbound}.
experiments~\cite{ref:lepbound,ref:cmsbound} and the sensitivity
to its signatures of an experiment at a future linear collider 
has been studied ~\cite{ref:gravmio}.
% and from cosmological 
%and astrophysical arguments~\cite{cosmo}. 
Limits on the $\grav$ mass are related to 
the supersymmetry-breaking scale $\sqrt{F}$.

%Recently
It has been pointed out \cite{ref:prz} that in such
supersymmetric extensions  
of the  Standard Model 
with a  light gravitino,
the effective theory at the weak scale
must contain also the supersymmetric partner of the goldstino,
called sgoldstino. The production of this particle, 
which could be massive,
may be relevant at the LEP and Tevatron energies \cite{ref:przhad}
if the supersymmetry-breaking scale and the sgoldstino mass are not too large.
Two states are considered in \cite{ref:prz,ref:przhad}, S CP-even and P
CP-odd. Assuming R-parity conservation, it has to be noticed that, while
the goldstino is R-odd,  the sgoldstino is R-even and therefore 
it can be produced  together with Standard Model particles. 

At  LEP 2 
sgoldstino signatures have been searched for
by  the DELPHI experiment 
\cite{ref:delphisgold}
and preliminary results from CDF
\cite{ref:cdfsgold} show the
higher sensitivity of hadron colliders.
None of the two searches found an evidence for such states.

At an $\ee$ collider
one of the most interesting channels for the production of such  scalars (from now on the symbol
$\phi$ will be used to indicate a generic state) is
the process  $\eefg$ which depends on the $\phi$ mass $m_{\phi}$ and on
$\sqrt{F}$:

\begin{equation}
\frac{d \sigma} {dcos\theta} (e^+e^-\rightarrow \phi \gamma  )
=\frac{\left|\Sigma\right|^2 s}{64 \pi F^2} 
\left( 1- \frac{m_{\phi}^2}{s} \right)^3 (1+cos^2\theta)
\label{dsigma}
\end{equation}
where $\theta$ is the scattering angle in the centre-of-mass and

\begin{equation}
\left|\Sigma\right|^2=\frac{e^2 M_{\gamma\gamma}^2}{2s}+
                      \frac{g_Z^2(v_e^2+a_e^2) M_{\gamma Z}^2 s}{2(s-m_Z^2)^2}+
                      \frac{e g_Z v_e M_{\gamma\gamma}M_{\gamma Z}}{s-m_Z^2}
\end{equation}
with $v_e=sin^2 \theta_W -1/4$, $a_e=1/4$ and 
$g_Z=e/(sin \theta_W cos \theta_W)$.
The parameters $M_{\gamma\gamma}$ and  $M_{\gamma Z}$
are related to the diagonal mass term for the $U(1)_Y$ and $SU(2)_L$ gauginos
$M_1$ and $M_2$:

\begin{equation}
M_{\gamma\gamma}= M_1 cos^2 \theta_W+ M_2 sin^2 \theta_W,~
M_{\gamma Z}= (M_2-M_1) sin \theta_W cos \theta_W.
\end{equation}

Other interesting processes are due to $\gamma \gamma$- or gg-fusion
occurring, respectively, at $\ee$ and hadron colliders.
In both cases the production cross sections are proportional to the corresponding widths:
\begin{equation}
\sigma(\ee \rightarrow \ee  \phi)\propto \sigma_0^{\gamma \gamma}=\frac{4 \pi^2}{ m^3_{\phi}}\Gamma(\phi \rightarrow \gamma \gamma),
\sigma(p\bar{p} \rightarrow \phi) \propto \sigma_0^{g g}=\frac{\pi^2}{8 m^3_{\phi}}\Gamma(\phi \rightarrow g g) 
\end{equation}  
and they can be obtained, respectively, from the photon and gluon distribution functions.

The  decay modes  $\phi \rightarrow \gamma \gamma$ and 
$\phi \rightarrow gg$ widths 
are 
\begin{equation}
\Gamma(\phi\rightarrow \gamma \gamma)=\frac{m_{\phi}^3 M_{\gamma\gamma}^2}{32 \pi F^2}
\label{phitogam}
\end{equation}
and 
\begin{equation}
\Gamma(\phi \rightarrow g g)= \frac{m_{\phi}^3 M_3^2}{4 \pi F^2}
\end{equation}
where $M_3$ is the gluino mass. 
%The corresponding branching ratios range

As noticed in \cite{ref:przhad}
the production formulae are  
similar in form to those for a light SM Higgs production in Born approximation where 
$\Gamma(H\rightarrow\gamma \gamma)$ and  $\Gamma(H\rightarrow g g)$
substitute the $\phi$ widths.
%\begin{equation}
%\sigma_0=
%\end{equation} 
It is straightforward to apply the same correspondence between these two different physical 
cases to the $\phi$ production on photon colliders. With a reverse substitution, an
effective  production cross section in the narrow-width approximation can be deduced
from the studies of Higgs Physics at a $\gamma \gamma$ collider~\cite{ref:Telnov}:
\begin{equation}
\sigma^{eff}=\frac{dL_{\gamma \gamma}}{dW_{\gamma \gamma}}\frac{m_{\phi}}{L_{\gamma\gamma}}  \times
             \frac{4\pi^2 \Gamma(\phi\rightarrow \gamma \gamma)}{m_{\phi}^3}
\label{sggphi}
\end{equation}
where $dL_{\gamma \gamma}/dW_{\gamma \gamma}$ is the luminosity spectrum in the 
two photon center-of-mass $W_{\gamma \gamma}$  and $L_{\gamma\gamma}$ 
is defined as the luminosity   
at the high  $\gamma \gamma$ energy peak.
%The formula \{sggphi} clearly applies to the case of a narrow state   

All the above formulae depend on model dependent mass parameters.
%In this note two sets for these parameters as suggested 
In \cite{ref:prz} two sets for these parameters  
are considered to give numerical examples.
They are reported in Table. \,\ref{tab:param}. 

\begin{table}[bth]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
        & $M_1$ & $M_2$        & $M_3$ \\
\hline
1)      & 200   & 300         & 400  \\
\hline
2)      & 350   & 350         & 350  \\ 
\hline
\end{tabular}
\caption[]{Two choices for the gaugino 
mass parameters (in GeV) relevant for the sgoldstino production and decay. }
\label{tab:param}
\end{center}
\end{table}


The total width
%depending on the considered S mass and $\sqrt{F}$,
for  a large interval of the parameter space is dominated by
$\Gamma(\phi \rightarrow g g)$ and  it is  narrow 
(below the few GeV order) except  for the region with small $\sqrt{F}$
where the production cross section is expected to be very large.

%$F$, defining the supersymmetry-breaking scale $\Lambda_S~=~ |F|^{1/2}$,
%is related to the gravitino mass by
%\begin{equation}
% |F|=\sqrt{3} m_{3/2}M_P,
% ~M_P\equiv (8 \pi G_N)^{-1/2} \simeq 2.4\times 10^{18} GeV.
%\end{equation}


In this note the sensitivity to these states  of an experiment 
at a $\ee$ linear collider with a center-of mass energy of 500 GeV
and the sensitivity of an experiment at a photon collider obtained 
from the same energy primary $\ee$ 
beams are evaluated. 
An integrated  luminosity of 500 fb$^{-1}$ for the $\ee$ collisions
is considered with a reduction factor for the $\gamma \gamma$ interactions.

\section{$\ee$ collider}

The search for these scalars at a future linear collider can be
an upgrade of the analysis done at LEP
%In particular 
where the two decay channels $\phi \rightarrow \gamma \gamma$ 
and $\phi \rightarrow gg$
were considered \cite{ref:delphisgold}.  For the present sensitivity evaluation
only the dominant channel is considered. 
%produce events 
%with very different 
%topologies in the final state.
%\begin{enumerate}
% \item{ $S\rightarrow \gamma \gamma$ give rise to
%events with three high energy photons one of which is expected to be 
%monochromatic with energy $E_{\gamma}=\frac{s-ms^2}{2 \sqrt{s} }$ for 
%the large fraction of the parameter space producing S with a negligible
%width. Despite of the lower S-decay branching ratio
%(4 and 11$\%$ for the two sets of Table  \,\ref{tab:param}, respectively),
%this final state has an unique background source from the $\eeggg$ QED 
%events which are expected not to be very frequent especially if cuts  on the 
%polar angular of the production (monochromatic) photon are used.
%}
%\item{
%In particular the 
The
 $\phi \rightarrow  g g$ decay gives rise to events with one
% monochromatic 
photon 
%(with the same
%exceptions as in the previous case)
and two jets.
An irreducible background from 
$e^+ e^- \rightarrow q \bar{q} \gamma$ events is associated to this topology
and therefore the signal must be searched for as an excess of events 
over the  background expectations for every mass hypothesis.
%}
%\end{enumerate}

%This note describes the results obtained with the DELPHI detector
%at LEP  centre-of-mass energies of 189, 192, 196, 200 and 202 GeV, 
%corresponding to a total integrated luminosity of  380 pb$^{-1}$.
%In the present analysis the 192 GeV data were not used for
%the  $S\rightarrow  g g$ topology.

%\subsection{ $S\rightarrow \gamma \gamma$ channel}
%
%%The analysis was similar to the one reported in \cite{ref:gampap}.
%Events with energetic clusters in the electromagnetic calorimeters
%were selected.
%Charged particle final states
%(mainly $\ee$  final state events)
%were rejected using
%a Vertex Detector track search which required that there were hits
%in at least two of the three VD layers, which were aligned with the
%mean beam crossing point.
%Although suppressing background, vetoing tracks using a VD track search
%also removed a small number of signal events where a
%photon converted into $\ee$ before or in the region of sensitivity of
%the Vertex Detector.
%
%Events were selected as $\gamma \gamma \gamma$ candidates if they 
%satisfied the following criteria :  
%\begin{itemize}
%\item{ at least two electromagnetic energy clusters  
%with $0.219<E / \sqrt{s}<0.713$ GeV; 
%}
%\item{ at least another one with $E>5 $ GeV }
%%both in the HPC or in the FEMC; 
%\item{ the two most energetic electromagnetic energy clusters in the HPC region 
%  $42^{\circ}<\theta<89^{\circ}$ 
%     or in the FEMC region
%% if  both energy clusters  in the region $42^{\circ}<\theta<89^{\circ}$ or
%   $25^{\circ}<\theta<32.4^{\circ}$; }
%\item{ the third most energetic cluster in the region $42^{\circ}<\theta<88^{\circ}$ 
%or $25^{\circ}<\theta<32.4^{\circ}$;}
%\item{ no tracks reconstructed in the Vertex Detector
%associated to the HPC
%%corresponding to the HPC
%or FEMC clusters
%($\pm 2^{\circ}$ in the azimuthal angle $\phi$);}
%\item{ two hemispheres were defined by the direction of the most energetic
%cluster.  In the barrel region, one hemisphere was
%required to have no TK with a
%momentum greater than 1 GeV$/c$ which
%extrapolated to within 5 cm of the mean beam crossing point.
%In the forward region, the requirement was strengthened to suppress
%the larger $e^+e^-$ background further, by demanding that
%both hemispheres have no such TK.}
%\end{itemize}
%
%The events obtained after this selection are three-body final state events 
%if there is no additional radiation lost in the  detector (mainly
%initial state radiation lost along the beam pipe). A simple way to check
%if an event is, within a reasonable approximation, a three-body final state,
%is to look at the distribution of the quantity 
%$\Delta= \left| \delta_{12}\right| +\left| \delta_{13}\right| +\left| \delta_{23}\right| $ where $\delta{ij}$
%is the angle between the particle $i$ and $j$. In a three-body final state 
%event the particles lay in a plane and therefore $\Delta$ should be $360^{\circ}$. 
%If only the events with $\Delta>358^{\circ}$  are accepted, the energies of the 
%particles  can be determined with very good precision on the basis of the
%photon measured directions:
%
%
%
%\begin{equation}
% E_1= \frac{sin \delta_{23}}{\delta};~E_2= \frac{sin \delta_{13}}{\delta};~
% E_3= \frac{sin \delta_{12}}{\delta}
%\label{ecorr}
%\end{equation}
%with $\delta = sin \delta_{12}+ sin \delta_{13}+  sin \delta_{23}$. The error on the energy 
%evaluation was further minimized by requiring 
%$min(\delta_{12},\delta_{13},\delta_{23})>2^{\circ}$.
%
%
%In $S\gamma$ events the $S$ decay products are expected to be isotropically
%distributed in the $S$ centre-of mass system. This fact implies that 
%the distribution of $cos \alpha$, where $\alpha$ is the angle between the $S$ 
%direction (opposite to the production photon) and the direction of one of the
%two  $S$ decay products, in the $S$ centre-of mass system, should be flat. 
%On the other hand, in the QED 
%background, $\left| cos \alpha \right|$  peaks at 1
%and therefore only the combinations giving $\left | cos \alpha<0.9 \right |$ were accepted.
%
%The number of selected event giving up to three combinations and the expected
%background are listed in  
%Table \,\ref{tab:evlist}.
%
%The  acceptance for a $S\gamma$ signal produced as in (\ref{dsigma}) after the described 
%geometrical
%cuts was $50 \%$ independently from $m_s$. The selection efficiency inside
%the acceptance region was evaluated by means of the QED background events
%generated according to $\cite{ref:kleiss}$ and processed through the full 
%DELPHI simulation $\cite{ref:delsim}$ and reconstruction chain. 
%The efficiency was reweighted according to the background and signal 
%polar angle distributions. It was $76.6 \%$.
%
%The energy resolution for the values obtained with (\ref{ecorr}), was also evaluated
%by means of QED events as shown in 
%Fig.\ref{resggg}. It was smaller than 0.5 $\%$ in the whole photon energy
%range.
%
%\begin{figure}[th]
%\begin{center}\mbox{\epsfxsize 9cm\epsfbox{resggg.eps}} \end{center}
%\caption{
% The  energy resolution for the photons of the
%$\gamma \gamma \gamma$ candidates in QED $\eeggg$ simulated sample.
%The photon energy was obtained according to (\ref{ecorr}).
%A fit to two gaussians  with about the same area give two resolution components 
% $\sigma_1= 0.12$ GeV and
% $\sigma_2= 0.35$ GeV }
%% of the parameter $\alpha_L$ (here $\alpha_R=\alpha_L$) of the equation (\ref{gen}).}
%\label{resggg}
%\end{figure}

%\subsection{ $S\rightarrow  g g$ channel.}
%This channel is expected to give a final state with one photon 
%and two jets. 
To  select    $g g \gamma$ candidate events 
the following selection criteria can be defined:
\begin{itemize}
\item
% there must be 
an electromagnetic energy  cluster identified as photon with 
%$E> E_{cut}$ and 
a polar angle 
% than 20 degrees
 $\theta>20^{\circ}$;
%The events were rejected also if 
the angle between 
the photon
and the nearest jet must be greater than $10^{\circ}$;
%to remove off momentum electrons
%\item  events with the most forward em cluster below 5 degrees are removed
\item  no electromagnetic cluster with $\theta< 5^{\circ}$;
\item to remove $\gamma \gamma$ fusion  events:
%\item 
the total multiplicity  $>10$;
%\item 
the charged multiplicity  $> 5$;
%\item 
the energy in transverse plane $> 0.12\cdot \sqrt{s}$;
%\item 
the sum of absolute values of track momentum 
along thrust axis $>0.20 \cdot\sqrt{s}$;
%\item remove  Bhabha background by rejecting
\item to  remove  Bhabha background:
% by rejecting
%\item
reject the events with electromagnetic cluster with $E> 0.45 \cdot \sqrt{s}$ and
low track multiplicity;
% degrees
% are removed;
%to remove qqg 
\item to reduce $\qqb \gamma$ events:  $ |cos(\theta_p)|<.995 $ where $\theta_p$ is the polar angle of
missing momentum;
%\item 
the visible energy  greater than $0.60 \cdot \sqrt{s}$;
%\item 
%total btag of the events must be less than 0;
reject events with c or b tag;
\item to remove WW background
%\item 
the events are reconstructed forcing into 2 jets topology 
      but removing from jetization the tracks associated to the photon
      cluster. Events are removed if $ y_{cut}>0.02$.
%       of jetization is greater
%      than 0.02.
%      The events are rejected also if the angle between photon
%      and nearest jet is less than $10^{\circ}$;

%\item $|\cos{\alpha}|<0.9$;
%\item        $\Delta >350 ^{\circ}$.     


\end{itemize}


The  polar angle acceptance for a $\phi \gamma$ signal produced as in (\ref{dsigma})
is about $80 \%$. It has been evaluated by generating 4-vectors corresponding 
to the prompt photon and to the $\phi$ decay products.  
%almost independently from $m_{\phi}$
Considering the DELPHI results \cite{ref:delphisgold}, the selection efficiency inside
the acceptance region 
is assumed to be
% about 
%was evaluated by means of the $ q \bar{q} \gamma$ background events
%generated according to \cite{ref:pythia} and processed through the full 
%DELPHI chain. The efficiency was reweighted according to the background and signal 
%polar angle distributions. It was 
of the order of 50 $\%$.

The associated photon is monochromatic (except for the region  with small $\sqrt{F}$
where the production cross-section is expected to be very large) 
for a given center-of-mass energy. 
Therefore the signal  can be detected as a peak in the photon energy distribution 
of the selected events. In addition, the photon energy could be determined very precisely
by means of kinematic constraints if a final state three body topology is assumed.
However, the presence of the beamstrahlung ($2.8 \%$ of mean beam-energy 
loss~\cite{ref:brink}) induces a smearing on the photon energy 
which is comparable with or larger than
the experimental resolution.
On the other hand, the signal can be searched for directly in the jet-jet  
invariant mass distribution. Clearly the detector performance plays a crucial role
in the optimal search strategy. Here a jet energy resolution following the
$\sigma_E^{jet}/E=40\%/\sqrt{E} \oplus 2\%$ dependence and an error of about one degree
in the jet angle reconstruction is assumed. With these assumptions
% it results that 
the direct mass search is convenient or comparable w.r.t. the
recoil photon search. The mass resolution is given in Fig. \ref{m_res}.
% in the mass region below 300 GeV/c$^2$.
%
\begin{figure}[th]
\begin{center}\mbox{\epsfxsize 9cm\epsfbox{m_res.eps}} \end{center}
\caption{
Mass resolution as function of the considered $\phi$ mass hypothesis. 
The full line corresponds to a linear fit.}
\label{m_res}
\end{figure}

%The value of $E_{cut}$ may depend on the calorimeter.....

%The events obtained after this selection have a three-body final state kinematics
%if there is no significant additional radiation lost in the  detector (mainly
%initial state radiation lost along the beam pipe). A simple way to check
%if an event is, within a reasonable approximation, a three-body final state,
%is to look at the distribution of the quantity
%$\Delta= \left| \delta_{12}\right| +\left| \delta_{13}\right| +\left| \delta_{23}\right| $
% where $\delta_{ij}$
% is the angle between the particle $i$ and $j$. 
%%  (Fig. \ref{delta}). 
%In a three-body final state
%event the particles lie in a plane and therefore $\Delta$ should be $360^{\circ}$.
% If only the events with $\Delta>350^{\circ}$  are accepted, the energies of the
% particles  can be determined with very good precision on the basis of the
% photon and jet measured directions:
%\begin{equation}
%E_1= \sqrt{s} \frac{sin \delta_{23}}{\delta};~E_2= \sqrt{s} \frac{sin \delta_{13}}{\delta};~
%E_3= \sqrt{s} \frac{sin \delta_{12}}{\delta}
%\label{ecorr}
%\end{equation}
%with $\delta = sin \delta_{12}+ sin \delta_{13}+  sin \delta_{23}$.
%% The error on the energy
%%evaluation can be further minimised by requiring
%%$min(\delta_{12},\delta_{13},\delta_{23})>2^{\circ}$.
  
%In $S\gamma$ events the $S$ decay products are expected to be isotropically
%distributed in the $S$ centre-of-mass system. This fact implies that
%the distribution of $cos \alpha$, where $\alpha$ is the angle between the $S$
%direction (opposite to the prompt photon) and the direction of one of the
%two  $S$ decay products, in the $S$ centre-of-mass system, should be flat.
%On the other hand, in the QED
%background, $\left| cos \alpha \right|$  peaks at 1
%and therefore only the combinations giving $\left | cos \alpha  \right |<0.9$ were accepted.



%The energy resolution  was  $\sigma_E = 2.x$ GeV. also evaluated
%%Fig.\ref{resggg}. It was smaller than 0.5 $\%$ in the whole photon energy
%range.

%\section{Results} 
%The obtained photon energy spectra for the two decay channels 
%are shown in Fig. \ref{eggg} and Fig. \ref{egglgl} for the
%different energy points. In Fig. \ref{msggg} and Fig. \ref{msgglgl} the photon 
%recoil mass spectra are shown. 
%The data are superimposed to the expected background distributions. 
%In the case of the $S\rightarrow \gamma \gamma$ channel, the QED  background generator
%included corrections  only to the order $\alpha^3$ and 
%therefore no additional radiation was simulated.
%The effect of the missing higher order corrections within the described selection
%was evaluated from the $\Delta$ distribution and a normalization correction of $(-10 \pm10 \%)$ 
%was applied to the simulated sample.

%\begin{figure}[th]
%\begin{center}\mbox{\epsfxsize 10cm\epsfbox{eggg.eps}} \end{center}
%\caption{
% Photon energy spectrum for the
%%  $S\rightarrow \gamma \gamma $ decay channel.
%$\gamma \gamma \gamma $ candidates.
% The photon energy was obtained according to (\ref{ecorr}). }
%\label{eggg}
%\end{figure}




\begin{figure}[th]
\begin{center}\mbox{\epsfxsize 9cm\epsfbox{ms_lc.eps}} \end{center}
\caption{
 Jet-jet invariant mass spectrum for the 
%%$S\rightarrow \gamma \gamma $ decay channel.
$\qqb \gamma $ events.   }
\label{ms_lc}
\end{figure}

%The number of detected events, 
The  background rate depends 
%and the detection efficiency depend
on the considered $\phi$ mass hypothesis as it  can be seen in Fig. \ref{ms_lc} where
the reconstructed jet-jet invariant mass of 
 $\qqb \gamma$ events generated with PYTHIA \cite{ref:pythia} in the acceptance region is shown.
%In order to obtain  the photon and jet momenta from eq. \ref{ecorr}, an angular smearing of
%0.6 to 1. degree was applied to the photon an quark directions.    
%A typical resolution of a few GeV is obtained for the photon energy.  
The  events are scaled in order to reproduce the number of expected events with an integrated
$L_{\ee}$ luminosity of 500 fb$^{-1}$. However the statistical fluctuations are not reproduced.   

%%, the data are compared with the background events
%in a region corresponding to
%% about
%the  $80 \%$ of the signal area. 
%As a consequence, the limit on the signal cross section depends 
%on $m_{\phi}$ and $\sqrt{F}$. However since .....
%Furthermore, to take into account  the different sensitivity of the two analyzed
%channels, the likelihood ratio technique \cite{ref:lrat} was used. 
%Since the expected S  branching ratio and total width depends on the mass 
%parameters as explained above, the 
%$95\%$ Confidence Level  cross section limit shown in Fig. \ref{cslim} was computed as
%function of $m_S$ and $\sqrt{F}$ for the two set of parameters listed 
%in Tab. \,\ref{tab:param}. 
Given the background event distribution as function of $m_{\phi}$ and  
the detection efficiency for any  $\phi$ mass hypothesis it is possible to estimate 
a $95\%$ Confidence Level  cross section limit for the  $\phi$ production cross section.
Only statistical fluctuations are considered here. The bin to bin 
fluctuations on the number of background events due to the reduced Monte Carlo statistics
are removed by a spline function. 

By comparing the experimental limits with the production cross section 
computed from (\ref{dsigma}) 
%taking into account the beamstrahlung effects 
%which are more relevant than the,
it is possible to determine a  $95 \%$ Confidence Level
excluded region on the parameter space and a $5~\sigma$ discovery region. 
The beamstrahlung effects which are more relevant than the 
Initial State Radiation one's
are taken into account.
The limit  and the $5~\sigma$ regions are shown in Fig. \ref{excl}. 
% In addition, when the expected total
The $\phi$ width for all the considered  $m_{\phi}$ values is 
smaller than the 
experimental resolution in all the points corresponding to the   
limit curves. Therefore 
the limit has been computed integrating the signal only over the
experimental resolution.
The region where the expected width is larger than the experimental resolution
is indicated in  Fig. \ref{excl}.
For $m_{\phi}<420$ it is
 possible to cover this region of parameter space given the high cross section.
%  also where width is
%  larger than the experimental resolution.
This is no more true for $m_{\phi}> 420$ GeV where the decreasing cross section   and
the increasing  width   result in a drop of experimental sensitivity. 
% In this region
%the signal is distributed on several
%mass bins but it can be detected as well
%given the large production cross-section.
%On the other hand, for $m_{\phi}> 420$ GeV the production cross section is sufficiently
%large only for low $\sqrt{F}$ values where the width dominates 
%over the experimental resolution and consequently the experimental sensitivity 
%drops quickly. 

%In the high mass region for low $\sqrt{F}$ values the width 
%can be larger  

%As explained in
%\cite{ref:prz},
%in order not to  conflict with unitarity bounds,
%the region with $m_S>2 \sqrt{F}$ was not considered.

\begin{figure}[th]
\begin{center}\mbox{\epsfxsize 8cm\epsfbox{limits_ee_lc.eps}\epsfxsize 8cm\epsfbox{disc_ee_lc.eps}} \end{center}
%\begin{center}\mbox{\epsfxsize 9cm\epsfbox{limits_ee_lc.eps}} \end{center}
\caption{
 Exclusion region at the 95$\%$ Confidence Level
and $>5~\sigma$ signal discovery region 
 in  $m_{\phi}$  $\sqrt{F}$ space 
 for the two
 sets of parameters of Tab. \ref{tab:param}. The thick lines indicate the region
 where the decay width $\Gamma$ is larger than the experimental resolution.  
}
\label{excl}
\end{figure}
    

%
In the near future the Fermilab Tevatron Collider is expected to increase the luminosity 
by a factor $\sim 20$ \cite{ref:tdrcdf} and consequently an increase of  about 1.5 in their $\sqrt{F}$ 
limits can be envisaged.
The limits shown in Fig. \ref{excl} are then competitive with the future 
improved Tevatron results.

At $\ee$ colliders,
additional information can be obtained by  
searching for  the associated $\phi \Zzero$ production as described in \cite{ref:prz}. 
%$\phi \rightarrow \gamma \gamma$ 
%Since a larger cross section is expected, 
As far as the  production cross section is considered, competitive results are expected
in the $m_{\phi}<\sqrt{s}-m_Z$ region. However, since this channel has  a
%However giving the
%a 4-body
different final state topology requiring a more sophisticated analysis, it is  
not considered here.   

%

\section{$\gamma \gamma$ collider}
The effective  cross section given in eq. (\ref{sggphi}) depends on the luminosity 
factor  
$f_L=\frac{dL_{\gamma \gamma}}{dW_{\gamma \gamma}}\frac{m_{\phi}}{L_{\gamma \gamma }} $.
In the photon collider projects \cite{ref:gamgampro} there are several possible scenarios 
concerning the photon energy spectra. It may be desirable a photon energy 
distribution peaked as much as possible toward the primary electron/positron  energy.
In \cite{ref:Telnov}  $f_L=7$ is assumed and $L_{\gamma \gamma}$ is taken as the integral luminosity
for $z>z_{min}=0.65$ where $z=W_{\gamma \gamma}/2E_e$ and $E_e$ is the primary electron beam energy. 
The luminosity high energy peak 
is expected to have a FWHM of 
$\sim 10-15 \%$ with a sharp edge at $z\sim0.8$.
Therefore the unexcluded $m_{\phi}-\sqrt{F}$  parameter space achievable
at these machines with $2E_e=500$ GeV ensures that the $\phi$ width is negligible.

The effective cross section obtained with $f_L=7$ is much higher 
(several orders of magnitude, depending on $m_{\phi}$) than the  
$\ee \rightarrow \phi \gamma$ cross section with the same parameters. 
Considering the photon and gluon decay channels, 
the signal would appear as a peak of two high energy photons 
or  jets with no transversal missing energy. The two jets 
final state has to compete 
with large Standard Model background which can be suppressed using polarized
photon beams  with polarizations $\lambda_1, \lambda_2$:
$\sigma({\gamma\gamma \rightarrow \qqb}) \propto 1-\lambda_1 \lambda_2$
while $\sigma({\gamma\gamma \rightarrow \phi}) \propto 1+\lambda_1 \lambda_2$.  
However, taking into account QCD corrections \cite{ref:bord,ref:jikia2}, the $\qqb g$
final state with unresolved gluon jet gives rise to 
a   sizeable background which may be   hard to reject. 
Therefore, despite of the smaller decay branching ratio,   
only the two photons final state which has a very little Standard Model 
background is considered here. 




The selection of events with two collinear high energetic photons
is rather simple and the LEP experience can be used \cite{ref:lepgg}.
An efficient way to select photons and to reject electrons is to require two 
energy clusters in the electromagnetic calorimeter not associated to hits in the 
vertex detector. Events with tracks detected in the other tracking devices only in one 
hemisphere  can be accepted to recover photon conversions.
Other requirements are:
\begin{itemize}
\item acollinearity  between the e.m. clusters smaller than $30 ^{\circ}$;  
\item acoplanarity
% between the e.m. clusters 
smaller than $5^{\circ}$; 
\item polar angle $\theta>30^{\circ}$; 
\item $E_{\gamma}>0.9 \cdot z_{min} \cdot E_e$.
\end{itemize}

The detection efficiency is very high ($> 90\%$) in the region $W_{\gamma \gamma}/2E_e> z_{min}$
and the acceptance for the decay of a scalar particle is $86\%$.
 
The irreducible Standard Model background of $\gamma \gamma \rightarrow \gamma \gamma$
events has been discussed in \cite{ref:Jikia,ref:Gounaris}.    
In the $W_{\gamma \gamma}$ region above   200-250 GeV 
the cross section is in the range  8-14 fb for $\theta > 30 \mydeg$
%and it is  dominated by the $W$ loop contribution.
and then,  
assuming $L_{\gamma \gamma}\sim0.15 \cdot L_{\ee}$, the number of expected events 
is of the order of 600-1000. As a consequence any New Physics signal has
to exceed the corresponding statistical error ( which is of the order of 3 to 4 $\%$) and
the systematic uncertitude including the precision on the background calculation.
For the present sensitivity study an overall background uncertitude of 5 $\%$ is assumed, 
leaving more detailed analysis of the   
signal and background including the comparison of 
their angular distributions to a later stage. 
With these assumptions, the sensitivity to a scalar state decaying in two photons is given
by the expected 95 $\%$  Confidence Level limit on the cross section times 
branching ratio and it is 
\[
\sigma ({\gamma \gamma \rightarrow \phi}) 
\times B.R.(\phi \rightarrow \gamma \gamma) < 1 ~\rm{fb}
\] 
at the 95 $\%$ Confidence Level 
for $m_{\phi} \sim 400$ GeV.
% scalar state.

This value is obtained following the hypothesis that
the whole luminosity is collected at the maximal energy spectrum
available with
%  $2E_e=500$ 
250 GeV electron beams. The actual sensitivity for 
several $\phi$ mass hypothesis depends on the machine run strategy, on the available
energy spectrum and on the photon beam polarization. 
Nevertheless, taking the given limit just
as an evaluation of the order of magnitude for the sensitivity, it is 
worth   investigating the 
effect on the supersymmetry breaking scale from 
 (\ref{phitogam}) and (\ref{sggphi}). In particular, defining as  a reference 
cross-section-branching-ratio-product the value $\sigma B$ obtained
with $M_{\gamma \gamma} =350$ GeV
and a 10$\%$ branching ratio to two photons, the limit on
$\sqrt{F}$ and the $5~\sigma$ signal can be expressed in terms of the ratio 
$R=\sigma^{eff} \times B.R.(\phi \rightarrow \gamma \gamma)/\sigma B$.
They are
% on $\sqrt{F}$
 then proportional to $R^{\frac{1}{4}}$  
as shown in Fig. \ref{lim_gamgam}.
The sensitivity is clearly much larger than the one expected at the $\ee$ machines.
\begin{figure}[th]
%\begin{center}\mbox{\epsfxsize 9cm\epsfbox{lim_gamgam.eps}} \end{center}
\begin{center}\mbox{\epsfxsize 9cm\epsfbox{lim5sig_gamgam.eps}} \end{center}
\caption{
 Limit at 95 $\%$ Confidence Level 
 on the supersymmetry breaking scale and $5~\sigma$ signal   (thick line)
 for the production of a $\sim 400$ GeV
$\phi$ scalar state as function of the ratio $R$.   }
\label{lim_gamgam}
\end{figure}

% as function of 
%exceeds by a factor xx
%the corresponding number at $\ee$ colliders. 

%which is much wider than the $\phi$ width and therefore the
%Defining $z=W_{\gamma \gamma}/2E_e$, the $\gamma \gamma$ 
%luminosity can have a  peak at $z>0.65$
%with a FWHM of  sharp 




\section{Conclusions}
The sensitivity to the supersymmetric scalar $\phi$ at the future 
linear $\ee$ and photon colliders is such that unexplored parameter space regions
can be investigated. The $\ee$ machines with center of mass energy of  500 GeV
can set limits for the production of sgoldstino scalars up to about 420 GeV. 
These limits are competitive w.r.t. the expected future results from the Tevatron RUN II.
The sensitivity at the photon colliders obtained from the same electron-positron beam energy
is expected to be much higher for 
$m_{\phi}\sim 400$ GeV. 
 
%For the first time, a search for the production of $S \gamma$
%where S is a CP-even state of the sgoldstino, the goldstino supersymmetric partner,
%was done using the LEP 1998 and 1999 data collected by the DELPHI detector. 
%Centre-of-mass energies from 189 to 202 GeV were considered for a total integrated 
%luminosity of 380 pb$^{-1}$. Final state $\gamma \gamma \gamma $ and $ \gamma g g $ events 
%were studied corresponding to  $S\rightarrow \gamma \gamma$ and  $S\rightarrow g g$
%decay channels, respectively. No evidence of such a signal was found 
%in neither of the two channels and therefore limits on the $m_S \div \sqrt{F} $ space 
%are given.

\subsection*{Acknowledgements}
\vskip 3 mm
We want to thank  F. Zwirner for useful explanations on the
theoretical framework and 
for suggestions on the experimental possibilities, A. Castro for discussions 
concerning the future CDF results and M. Mazzucato for comments
and for reading the manuscript.


\newpage
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hep-ph0102208.extracted_bib
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\bibitem{ref:GMSB}
%                   P.Fayet, Phys. Lett. B69(1977)489;\\
%                   P.Fayet, Phys. Lett. B70(1977)461;\\
%                   P.Fayet, Phys. Lett. B84(1979)421;\\
                   P.Fayet, Phys. Lett. {\bf B86} (1979) 272;\\
%                   J.Ellis, K.Enqvist and D.Nanopoulos, Phys. Lett. B147(1984) 99;\\
                   J.Ellis, K.Enqvist and D.Nanopoulos, Phys. Lett. {\bf B151} (1985) 357;\\
%                   D.Dicus, S.Nandi, and J.Woodside, Phys. Rev. D41(1990) 2347;\\
                   D.Dicus, S.Nandi, and J.Woodside, Phys. Rev. {\bf D} 43(1991) 2951\\
                    A. Brignole, F. Feruglio and F. Zwirner  Nucl. Phys. B  {\bf 516} (1998) 13 \\
                   and references therein.
\bibitem{ref:lepbound}
                     L3 collab., O. Adriani et al., Phys. Lett. {\bf B297} (1992) 469; \\
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                     DELPHI  collab., P. Abreu et al., Eur. Phys. J. {\bf C17} (2000)53.
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{\it Sensitivity to the Gravitino mass from single-photon spectrum at TESLA Linear Collider
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%\bibitem{ref:theo} 
%                   P.Fayet, Phys. Lett. {\bf B70} (1977)461  and {\bf B86} (1979)272;\\
%                   R. Casalbuoni,et al., Phys. Lett. {\bf B215} (1988)313 and  
%                   Phys. Rev. {\bf D} 39(1989) 2281.

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