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\documentstyle[aps,preprint]{revtex}
\begin{document}

\title{Squark-Chargino Production in Polarized Gamma-Proton
Collisions at TeV Energy Scale}
\author{Z. Z. Aydin and O. Yilmaz}
\address{ Ankara University,Faculty of Sciences,Department of
Engineering Physics\\
06100 Tando\u gan,Ankara, Turkey}
\maketitle

\begin{abstract}
The associated production of squarks and charginos in high energy collisions
of the polarized real photons and protons is discussed. We give the cross
sections for different initial beam polarizations and the polarization
asymmetries which can be used to anticipate the masses of squarks and charginos.
\end{abstract}

\section{introduction}

Although the Standard Model (SM) of elementary particles has been successful
with high precision up to the scale of 100 GeV, there are many theoretical
reasons that new physics beyond the SM should exist at TeV scale.
Among the models of new physics, the supersymmetry (SUSY) seems to be one of
the most promising candidates for TeV scale. A number of TeV energy machines
have been proposed or constructing, such as the Large Hadron Collider (LHC),
Next Linear $e^{+}e^{-}$ Collider (NLC) and Linac-Ring type ep and $\gamma$p
machines. The latter one can be realized by using the beam of high energy
photons produced through the Compton backscattering of laser photons off a TeV
energy linear $e^{-}$ (or $e^{+}$) beam. We have already proposed some
Linac-Ring type ep and $\gamma$p machines \cite{Aydin}. Here we concentrate
ourselves only on three of them, i.e., the HERA+ LC, LHC+Linac 1 and LHC+TESLA.
Their calculated center of mass energies and luminosities are given in Table 1.

A supersymmetric SM has a new spectrum of particles called SUSY particles
which are the partners of all the known particles with the spins
differing by $\frac{1}{2}$. Some of the SUSY partners are scalar
leptons (sleptons, $\tilde{l}$), scalar quarks (squarks, $\tilde{q}$),
wino ($\tilde{w}^\pm$), Higgsino ($\tilde{H}^\pm_{1,2}$)
(or mixing of the latter ones, charginos, $\chi^\pm_{1,2}$),
photino ($\tilde{\gamma}$), zino ($\tilde{z}^0$), Higgsino ($\tilde{H}
^0_{1,2}$) (or their mixed states, neutralinos, $\chi^{0}_{1,2,3,4}$ ),
gluino ($\tilde{g}$) etc. It is commonly believed that these SUSY particles
should have masses below 1 TeV. Experiments at the existing colliders have
already put the lower mass limits as $m_{\tilde{q}} > 176$ GeV and
$m_{\chi^\pm}> 99$ GeV.

In this paper we study the associated production of the squarks and
charginos at TeV energy $\gamma$p colliders with polarized beams. We have
already discussed this process with unpolarized beams \cite{Alan}. Many
other processes such as ${\gamma}p\to\tilde{q}\tilde{q}^\ast$X, ${\gamma}p\to
\tilde{q}\tilde{g}$X, ${\gamma}p\to\tilde{q}\tilde{\gamma}( or \tilde{q}
\tilde{z})$X, have already been discussed \cite{buch}.

\section{polarized high energy gamma beam}

A beam of laser photons ($\omega_{0}\approx 1.26$ eV, for example) with high
intensity, about $10^{20}$ photons per pulse, is Compton-backscattered off
high energy electrons ($E_e$=250 GeV, for example) from a linear accelerator
and turns into hard photons with a conversion coefficient close to unity.
The energy of the backscattered photons, $E_{\gamma}$, is restricted by the
kinematic condition $y_{max}= 0.83$ (where $y=E_{\gamma}/E_{e}$) in order
to get rid of background effects, in particular $e^{+}e^{-}$ pair production
in the collision of a laser photon with a backscattered photon in the
conversion region.

The details of the Compton kinematics and calculations of the cross section can
be found in ref \cite{ginz}. The energy spectrum of the high energy
real (backscattered) photons, $f_{{\gamma}/e}(y)$, is given by

\begin{eqnarray}
f_{{\gamma}/e}(y)=\frac{1}{D(\kappa)}\biggr[1-y+\frac{1}{1-y}
-4r(1-r)-\lambda_e\lambda_0 r\kappa(2r-1)(2-y)\biggr]
\end{eqnarray}
where $\kappa=4E_e\omega_0/m_e^2$ and $r=y/\kappa(1-y)$.
Here $\lambda_0$ and $\lambda_e$ are the laser photon and
the electron helicities respectively, and $D(\kappa)$ is

\newpage
\begin{eqnarray}
D(\kappa)&=&\biggr(1-\frac{4}{\kappa}-\frac{8}{\kappa^2}\biggr)
ln(1+\kappa)+\frac{1}{2}+\frac{8}{\kappa}-\frac{1}{2(1+\kappa)^2}\nonumber\\
&+&\lambda_e\lambda_0\biggr[(1+\frac{2}{\kappa})ln(1+\kappa)-
\frac{5}{2}+\frac{1}{1+\kappa}-\frac{1}{2(1+\kappa)^2}\biggr]
\end{eqnarray}
In our numerical calculations, we assume $E_e\omega_0=0.3$ $MeV^2$ or
equivalently $\kappa=4.8$ which corresponds to the optimum value of $
y_{max}=0.83$, as mentioned above.

The energy spectrum, $f_{\gamma /e}(y)$, does essentially depend on the value
$\lambda _e\lambda _0$. In the case of opposite helicities ($\lambda
_e\lambda _0=-1$) the spectrum has a very sharp peak at the high
energy part of the photons.
This allows us to get a highly monochromatic high energy gamma
beam by eliminating low energy part of the spectrum \cite{ginz,borden}.
On the contrary, for the same helicities ($\lambda _e\lambda
_0=+1 $) the spectrum is flat.

The average degree of linear polarization of the photon is proportional to
the degree of linear polarization of the laser. In our calculations, we
assume that the degree of linear polarization of the laser is zero so that
the final photons have only the degree of circular polarization $(\lambda
(y)=<\xi _2>\neq 0$ and $<\xi _1>$$=<\xi _3>=0$ $)$. The circular
polarization of the backscattered photon is given as follows

\begin{eqnarray}
<\xi _2>=\lambda(y)=\frac {(1-2r)(\frac{1}{1-y}+1-y)\lambda_0
+\lambda_e r\kappa \biggr(1+(1-2r)^2(1-y)\biggr)}{\frac{1}{1-y}+1-y-4r(1-r)
-\lambda_0\lambda_e r\kappa(2r-1)(2-y)}
\end{eqnarray}

For the same initial polarizations ($\lambda_0\lambda_e=+1$) , it is seen
that $\lambda(y)\approx +1$, as nearly independent of $y$;
while for the case of the opposite
polarizations ($\lambda_0\lambda_e=-1$),
the curve $\lambda(y)$
smoothly changes from $-1$ to $+1$ as $y$ increases from zero to $0.83$
 \cite{ginz,borden}.

\section{polarized cross-sections for the reaction}


The subprocess contributing to our physical process ${\gamma}p\to
\tilde{w}\tilde{q}$X  is  ${\gamma}q\to\tilde{w}\tilde{q}$.
The invariant amplitude for the specific subprocess ${\gamma}u\to
\tilde{w}^{+}\tilde{d}$ is the sum of the three terms corresponding
to the s-channel $u$ quark exchange, the t-channel $\tilde{w}$ wino
exchange and the u-channel $\tilde{d}$ squark exchange interactions:
\begin{eqnarray}
{\cal M}_a &=&\frac{-iee_q g}{2\hat{s}}\bar{u}(p^{\prime})(1-\gamma_5)
(\not\! p+\not\! k)\not\! \epsilon u(p)\nonumber\\
{\cal M}_b &=&\frac{-iege_{\tilde{w}}}{2(\hat{t}-m^2_{\tilde{w}})}
\bar{u}(p^{\prime})\not\! \epsilon (\not\! p-\not\! k+m_{\tilde{w}})
(1-\gamma_5)u(p)\\
{\cal M}_c &=&\frac{-iee_{\tilde{q}}g}{2(\hat{u}-m^2_{\tilde{q}})}
\bar{u}(p^{\prime})(1-\gamma_5)u(p)(p-p^\prime +k^\prime ).\epsilon\nonumber
\end{eqnarray}
where $e_q$, $e_{\tilde{q}}$ and $e_{\tilde{w}}$ are the quark,
squark and wino charges, and $g=e/sin{\theta_{w}}$ is the weak coupling
constant. Note that we ignore the quark masses.

Since we are interested in the polarized cross-section, we use the following
density matrices for the initial photon and quark:

\begin{eqnarray}
\rho^{(\gamma)}&=&\frac{1}{2}(1+\vec{\xi}.\vec{\sigma})\nonumber\\
\rho^{(q)}&= &u\bar{u} =\not\! p[1+\gamma_{5}(\lambda_{q}+\vec{\xi}_{\perp}.
\vec{\gamma}_{\perp})]
\end{eqnarray}
where ${\xi}_1$, ${\xi}_2$ and ${\xi}_3$ are Stokes parameters. We take into
account only circular polarization for the photon which is defined
by ${\xi}_2$, as has been already mentioned in the previous section.
$\lambda_q$ stands for the helicity of the parton-quark that is
$+1 (-1)$ for the spin directions parallel (anti-parallel) to its
momentum. The last term in the quark density matrix does not contribute,
because after the integration over the azimuthal angle it vanishes.

One can easily obtain the differential cross section for the subprocess
${\gamma}u\to\tilde{w}^+\tilde{d}$  as follows
\begin{eqnarray}
\frac{d\hat{\sigma}}{d\hat{t}}=\frac{\pi \alpha^2}{\hat{s}^2 sin^2
\theta_w}(1+\lambda_{q})\biggr[\frac{d\hat{\sigma_0}}{d\hat{t}}
+\lambda(y)\frac{d\hat{\sigma}_1}{d\hat{t}}\biggr].
\end{eqnarray}
Performing the $d\hat{t}$ integration from $t_{min}$ to $t^{max}$ which
are given by
\begin{eqnarray}
t^{max}_{min}=\frac{1}{2}(m^2_{\tilde{w}}+m^2_{\tilde{q}}-\hat{s})
\biggr[1\mp\sqrt{1-4m^2_{\tilde{w}}m^2_{\tilde{q}}/(m^2_{\tilde{w}}
+m^2_{\tilde{q}}-\hat{s})^2}\biggr]
\end{eqnarray}
we immediately get the total cross section as
\begin{eqnarray}
\hat{\sigma} (m_{\tilde{w}},m_{\tilde{q}},\hat{s},\lambda{(y)})
=\frac{\pi \alpha^2}{\hat{s}^2 sin^2 \theta_w}(1+\lambda_q)
\biggr[\hat{\sigma_{0}}(m_{\tilde{w}},m_{\tilde{q}},\hat{s})
+{\lambda(y)} \hat{\sigma_{1}}(m_{\tilde{w}},m_{\tilde{q}},\hat{s}) \biggr].
\end{eqnarray}
Note that the cross sections ( Eqs(6) and (8)) are zero for $\lambda_{q}
=-1$ because of the fact that we ignore the quark mass. Integrating the
subprocess cross-section $\hat{\sigma}$ over the quark and photon distributions
we obtain the total cross-section for the physical process
$\gamma p\to\tilde{w} \tilde{q}X$ (the new variables are defined by
$\hat{s}\equiv s_{\gamma q}=xys$, $xy=\tau$ and $s\equiv s_{e p}$):


\begin{eqnarray}
\sigma=\int^{0.83}_{(m_{\tilde{w}}+m_{\tilde{q}})^{2}/s} d\tau
\int^{1}_{\tau/0.83} \frac{dx}{x} f_{\gamma /e}(\tau/x) f_{q}(x)
\hat{\sigma}(m_{\tilde{w}},m_{\tilde{q}},\hat{s},\lambda(\tau/x))
\end{eqnarray}
where the photon distribution function, $f_{{\gamma}/e}(y)$, is actually the
normalized differential cross-section of the Compton backscattering, Eq.(1)
; $f_q (x)$ is the distribution of quarks inside the proton. We set
$\lambda_q=+1$ and $f_q (x)\to u^{+} (x)=\frac{1}{2}(u_{unp}+{\Delta}u_{pol})$
for the $u$-type valence quark distribution. In our numerical calculations,
we use the distribution functions given in Ref. \cite{alta}
and Ref. \cite{kell} for the unpolarized and polarized up-quarks, respectively:
\begin{eqnarray}
u_{unp}(x)= 2.751 x^{-0.412}(1-x)^{2.69}
\end{eqnarray}
\begin{eqnarray}
\Delta u_{pol}(x)= 2.132 x^{-0.2}(1-x)^{2.40}
\end{eqnarray}
Performing the integrations in Eq.(9) numerically we obtain
the total cross-section for the associated wino-squark production.
We plot the dependence of the total cross-sections on the masses of
the SUSY particles for various proposed $\gamma p$ colliders in
Figs. 1(a-c) for $\lambda_0 \lambda_e =+1$ and $-1$.
By taking 100 events per running year
as observation limit for a SUSY particle, one can easily find the upper
discovery mass limits from these figures using the luminosities of
the proposed $\gamma p$ colliders given in Table 1. These discovery
limits are tabulated in the same table.

\section{asymmetry}
It may be more interesting to use a polarization asymmetry in determining
the masses of SUSY particles. Such an asymmetry can be defined with
respect to the product of the polarizations of the laser photon and
the electron as follows
\begin{eqnarray}
A=\frac{\sigma_{-}-\sigma_{+}}{\sigma_{-}+\sigma_{+}}
\end{eqnarray}
where $\sigma_{+}$ and $\sigma_{-}$ are the polarized total cross-sections
given in Figs. 1(a-c).
The results of the polarization asymmetry are shown in Figs. 2(a-c) for
three colliders.

\section{conclusion}
If one compares the curves $\sigma_{+}$ and $\sigma_{-}$ in Figs. 1(a-c)
for each collider one sees that the
polarized cross-sections for different polarization do not differ much
from each other and also from the unpolarized ones. Therefore, the discovery
mass limits for SUSY partners obtained with polarized beams are nearly
equal to those obtained with unpolarized beams. But the polarization
asymmetry is highly sensitive to the wino
and the squark masses and as high as 0.4 for all cases.
Especially in the case of $m_{\tilde{q}}=250$ $GeV$,
the asymmetry parameter A is around 0.6 for the higher wino masses.

The signature of the associated $\tilde{w}^{+} \tilde{d}$ production
will depend on the mass spectrum of SUSY particles. It is generally assumed
that the photino and sneutrino are the lightest SUSY particles and
that the hierarchy of the squark masses is similar to that of quarks.
With these assumption we have the following decays for the case
$m_{\tilde{q}}=m_{\tilde{w}}$:

$\tilde{d}\to d\tilde{\gamma}$, $d\tilde{g}$   and
  $\tilde{w}\to l^{+}\tilde{\nu}$,  $\nu\tilde{l}^{+}$,  $W^{+}\tilde{\gamma}$

By taking into account the further decays $\tilde{l}^{+}\to l\tilde{\gamma}$
and $W^{+}\to l^{+}\nu, q \bar{q}$ we arrive at the ultimate final states
as

$l^{+}+n$ $jets(n=1,3,5)+$ large missing energy and missing $P_{T}$

The main background for the final state $l^{+}+jet+P_{T}^{miss}$ will come
from the process $\gamma q\to W q \to l^{+}\nu q$; but this background
may be reduced, in principle,by the cut $P_{T}^{miss}>45$ $GeV$ if
$m_{\tilde{w}} \gg m_{W}$.



\begin{references}
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Z.Z. Aydin, V.M. Maniev and S.F. Sultansoy, Particle World 4(1994) 22

Z.Z. Aydin, A.K. \c{C}ift\c{c}i and S. Sultansoy, Nucl. Instr.
Meth. A351(1994)261

A.K. \c{C}iftc\c{c}i, S. Sultansoy, S. T\"{u}rk\"{o}z and
\"{O}. Yava\c{s}, Nucl.Instr. Meth. A365(1995)317

Z.Z. Aydin $et$ $al$., Int.J. Mod. Phys. A11 (1996) 2012.

\bibitem{Alan} A.T. Alan, Z.Z. Aydin and S. Sultansoy, Phys. Lett.
B327 (1994)70.

\bibitem{buch}W. Buchm\"{u}ller and Z. Fodor, Phys. Lett. B316(1993)510

A.T. Alan, S. Ata\u{g} and Z.Z. Aydin, J. Phys. G20 (1994)1399

A.Kandemir and A.U. Y{\i}lmazer, Phys. Lett. B385(1996)143.

\bibitem{ginz} I.F. Ginzburg $et$ $al$., Sov.J. Nucl. Phys. 38(1983)614

I.F. Ginzburg $et$ $al$., Nucl. Instr. Meth.219(1984)5.

\bibitem{borden} D.L. Borden, D.A. Bauer and D.O. Caldwell, SLAC preprint
SLAC-PUB-5715, Stanford(1992).

\bibitem{alta} G. Altarelli and W.J. Stirling, Particle world 1 (1989)40.

\bibitem{kell} S. Keller and J.F. Owens, Phys. Rev. D49(1994)1199

A. Sch$\ddot{a}$fer, Phys.Lett. B208 (1988)175.
\end{references}
                               
\figure{FIG.1.(a-c) Squark-wino production cross-sections as
a function of the mass. Two thin curves on the left
stand for the collider HERA+LC, middle curves for
LHC+Linac 1 and two curves on the right for LHC+TESLA.
A little bit higher (lower) curve of each twin is for
$\lambda_{0} \lambda_{e}=-1$
($\lambda_{0} \lambda_{e}=+1$), i.e., $\sigma_{-}$ ($\sigma_{+}$)}.

\figure{FIG.2.(a-c) Polarization asymmetries as a function of the mass.
Thin curve stands for the collider HERA+LC, middle curve for LHC+Linac 1
and thick curve for LHC+TESLA}.


\begin{table}
\caption{Energy and luminosity values of different $\gamma$p
colliders. The discovery mass limits
for squarks and winos are given in last three columns for
$\lambda_0 \lambda_e=+1$ ($\lambda_0 \lambda_e=-1$).}

\begin{tabular}{|l|l|l|l|l|l|}
\hline
{\bf Machines} &
\begin{tabular}{l}
$\sqrt{s_{ep}}$ \\ 
$(TeV)$%
\end{tabular}
&
\begin{tabular}{l}
${\cal L}_{\gamma p}$ \\
$(10^{30} cm^{-2}s^{-1})$%
\end{tabular}
&
\begin{tabular}{l}
$m_{\tilde{q}}=m_{\tilde{w}}$ \\
$(TeV)$%
\end{tabular}
& 
\begin{tabular}{l}
$m_{\tilde{q}}=0.25$ \\
$m_{\tilde{w}}\quad (TeV)$%
\end{tabular}
& 
\begin{tabular}{l}
$m_{\tilde{w}}=0.10$ \\
$m_{\tilde{q}}\quad (TeV)$%
\end{tabular}
\\ \hline
{\bf HERA+LC} & $1.28$ & $25$ &$0.23(0.25)$ & $0.18(0.23)$ & $0.35(0.40)$ \\ \hline
{\bf LHC+Linac 1} & $3.04$  & $500$ & $0.65(0.70)$ & $1.00(1.15)$ & $1.33(1.48)$ \\ \hline
{\bf LHC+TESLA} & $5.55$ & $500$ & $0.95(1.05)$ & $1.55(1.70)$ & $2.15(2.50)$ \\ \hline
\end{tabular}
\end{table}

\begin{center}
\input{fig1a}
\smallskip

Fig.1.a

\input{fig1b}
\smallskip

Fig.1.b

\input{fig1c}
\smallskip

Fig.1.c

\input{fig3a}
\smallskip

Fig.2.a

\input{fig3b}
\smallskip

Fig.2.b

\input{fig3c}
\smallskip

Fig.2.c

\end{center}
\end{document}

