%Paper: 
%From: Roland Hoepker <hoepker@HP-Cluster.desy.de>
%Date: Fri, 13 Jan 95 10:07:14 MEZ


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\begin{document}
\thispagestyle{empty}

\hfill\vbox{\hbox{\bf DESY 94-235}
            \hbox{December 1994}
                                }
\vspace{1.in}
\begin{center}
{\large\bf Gluon Radiation Off Scalar Stop Particles} \\
\vspace{0.5in}
W.~Beenakker, R.~H\"opker and P.~M.~Zerwas \\
\vspace{0.5in}
Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, FRG \\
\end{center}
\vspace{2in}


\begin{center}
ABSTRACT \\
\end{center}
We present the distributions for gluon radiation off stop-antistop
particles produced in $e^+e^-$ annihilation: $e^+e^- \to
\tilde t\, \bar{\tilde t} \,g$.  For high energies the splitting
functions of the
fragmentation processes $\tilde t \to \tilde t \,g$ and $g \to \tilde t
\,\bar{\tilde t}$ are derived; they
are universal and apply also to high-energy stop particles produced at
hadron colliders.

\pagebreak

\paragraph{Introduction.}
Stop particles are exceptional among the supersymmetric partners of
the standard-model fermions. Since the top quarks are heavy, the
masses of the two stop particles $\tilde t_1$ and $\tilde t_2$,
mixtures of the left (L) and right (R)
squarks, may split into two levels separated by a large gap
\cite{ellis}-\cite{olive}. The mass of the lightest
eigenstate $\tilde t_1$
could be so low that the particle may eventually be accessible at
the existing $p\bar p$ and even $e^+e^-$ storage rings. So far the result
of search experiments at $e^+e^-$ colliders \cite{tristan,opal} has
been negative and a lower limit of $45.1$ GeV has been set at
LEP \cite{opal} for the L/R mixing angle outside the band of
$\cos^2\theta_t$ between 0.17 and 0.44
and for a mass difference between the
$\tilde t_1$ and the lightest neutralino $\tilde\chi_1^0$ of more
than $5$ GeV. The higher energy at LEP2 and
dedicated efforts at the Tevatron will open the mass range beyond the
current limits soon.

To begin, we briefly summarize the well-known theoretical predictions for
the cross section  of the production process [Fig.\ref{feyn}(a)]
\begin{displaymath}
  e^+\,e^- \to \tilde t_1\, \bar{\tilde t}_1
\end{displaymath}
For a given value $\theta_t$ of
the L/R mixing angle, the vertices of the
$\tilde t_1$ pair with the photon and the $Z$ boson may be written as
$i e_0 \tilde Q [p_{\tilde t_1} - p_{\bar{\tilde t}_1}]_\mu$,
where $p_{\tilde t_1}$ and $p_{\bar{\tilde t}_1}$ are the 4-momenta of
the stop and antistop squarks, and the charges read
\begin{eqnarray*}
  \tilde Q_\gamma & = & - e_t \\
  \tilde Q_Z & = & (\cos^2\theta_t -2 e_t \sin^2\theta_W)/\sin 2\theta_W
\end{eqnarray*}
respectively. $\theta_W$ is the standard electroweak mixing
angle and $e_0=\sqrt{4\pi\alpha}$ is the electromagnetic coupling to be
evaluated with $\alpha^{-1}(M_Z) = 129.1$ in the
improved Born approximation \cite{swartz}. The
$Z$ boson coupling vanishes for the L/R mixing angle
$\cos^2 \theta_t \to 2 e_t \sin^2 \theta_W \approx 0.30$. Defining the
$\gamma$ and $Z$ vector/axial-vector charges of the electron, as
usual, by $e_e = -1$, $v_e = -1 +4 \sin^2 \theta_W$ and $a_e =
-1$, the cross section can be expressed in the compact form \cite{hikasa}
\begin{eqnarray}
  \sigma_B[e^+e^- \to \tilde t_1 \bar{\tilde t}_1] &=&
  \frac{\pi\alpha^2}{s} \left[\tilde Q_\gamma^2 + \frac{(v_e^2
    +a_e^2)\tilde Q_Z^2}{4\sin^2 2\theta_W}
  \frac{s^2}{(s -M_Z^2)^2 + M_Z^2 \Gamma_Z^2}\right. \nonumber\\
  & &\hphantom{\frac{\pi\alpha^2\beta^3}{s}a}\left.+\,\frac{v_e\tilde
    Q_\gamma \tilde Q_Z}{\sin2\theta_W} \frac{s (s-M_Z^2)}{(s
    -M_Z^2)^2 + M_Z^2 \Gamma_Z^2}\right] \beta^3
\label{born}
\end{eqnarray}
where $\sqrt{s}$ is the center of mass
energy and $M_Z,\,\Gamma_Z$ are the mass and the total
width of the $Z$ boson, respectively.
The $P$-wave excitation near the threshold gives rise to the familiar
$\beta^3$ suppression, where $\beta = (1 -4 m^2_{\tilde t_1}/s)^{1/2}$ is
the velocity of the stop particles. Angular momentum conservation
enforces the $\sin^2\theta$ law, $\sigma_B^{-1} d\sigma_B/d\cos\theta
= \frac{3}{4}\sin^2\theta$, for the angular distribution of the stop
particles with respect to the beam axis.

\paragraph{QCD corrections.}
Gluonic corrections modify the cross section \cite{schwinger,
drees}\footnote{Since we focus on QCD gluon effects for light stop
          particles
          in the LEP range, we do not take into account quark-gluino
          loop effects, assuming the gluino to be heavy; these
          loop effects have been discussed for squark production at the
          Tevatron in Ref.\cite{been} and at $e^+e^-$ colliders in
          Ref.\cite{djou}.}.
The virtual
corrections, Fig.\ref{feyn}(b), can be expressed by the form factor
\newpage
\begin{eqnarray}
  F(s) &=& \frac{4}{3}\frac{\alpha_s}{\pi}\left\{\frac{s -2 m_{\tilde
      t_1}^2}{s\beta}\left[2\, \mbox{Li}_2(w)
  + 2\log(w)\log(1-w) - \frac{1}{2}\log^2(w)
  +\frac{2}{3}\pi^2 - 2\log(w) \right.\right.\nonumber \\
  & & \hphantom{\frac{4}{3}\frac{\alpha_s}{\pi}a}\left.\left.
  -\,\log(w)\log\left(\frac{\lambda^2}{m_{\tilde t_1}^2}\right)\right]
   -2 - \log\left(\frac{\lambda^2}{m_{\tilde t_1}^2}\right)\right\}
\end{eqnarray}
where $\alpha_s$ is the strong coupling constant and  the
kinematical variable $w$ is defined as $w =(1-\beta)/(1+\beta)$.
The form factor is infrared (IR) divergent. We have
regularized this divergence by introducing a small parameter
$\lambda$ for the gluon mass. The IR
singularity is eliminated by adding the contribution of the soft gluon
radiation [Fig.\ref{feyn}(c)], with the scaled gluon energy
integrated up to a cut-off value $\epsilon_g = 2
E_g^{cut}/\sqrt{s} \ll 1$. The sum of the virtual correction
($V$) and the soft-gluon radiation ($S$) depends
only on the physical energy cut-off $\epsilon_g$,
\begin{eqnarray*}
  \sigma_{V+S} & = & \sigma_B\frac{4}{3}\frac{\alpha_s}{\pi}\left\{\frac{s -2
      m_{\tilde t_1}^2}{s\beta}\left[\vphantom{\frac{1}{3}}
    4\,\mbox{Li}_2(w)  -2\log(w)\log(1+w)
     + 4\log(w)\log(1-w) \right. \right. \\
    & &\left. \left.  + \,\frac{1}{3}\pi^2
    -2\log(w)\log(\epsilon_g) \right]  +
    \frac{4m_{\tilde t_1}^2 - 3 s}{s \beta}\log(w) +
 \log\left(\frac{m_{\tilde t_1}^2}{s}\right) -2\log(\epsilon_g) -2 \right\}
\end{eqnarray*}

After including the hard gluon radiation, the
dependence on the cut-off $\epsilon_g$ disappears from the total cross
section. The total QCD corrections
can finally be summarized in a universal
factor \cite{drees}
\begin{equation}
  \sigma[  e^+e^- \to \tilde t_1 \,\bar{\tilde t}_1\, (g)]
 = \sigma_B \left[1 +\frac{4}{3}\frac{\alpha_s}{\pi} f(\beta)\right]
\label{sigma}
\end{equation}
with (Fig.\ref{fbeta})
\begin{eqnarray*}
  f(\beta) &=& \frac{1+\beta^2}{\beta}\left[ \vphantom{\frac{1}{3}}
  4\,\mbox{Li}_2(w) +
  2\,\mbox{Li}_2(-w) +2\log(w)\log(1-w) +\log(w)\log(1+w)\right] \nonumber\\
 & & -\,4\log(1-w) -2\log(1+w) +\left[3 + \frac{1}{\beta^3}\left(2
 -\frac{5}{4}(1 + \beta^2)^2\right)\right]\log(w)
 +\frac{3}{2}\frac{1+\beta^2}{\beta^2}
\end{eqnarray*}
Very close to the threshold the Coulombic gluon exchange between the
slowly moving stop particles generates the universal Sommerfeld
rescattering singularity \cite{sommer} $f \to \pi^2/2\beta$, which damps the
threshold suppression, yet does not neutralize it entirely.
Employing methods based on non-relativistic Green's functions,
an adequate
description of stop pair production near threshold has been given in
Ref.\cite{bigi1}, which also
takes into account screening effects due to the finite decay width of
the stop particles. In the high-energy limit \cite{drees} the
correction factor in eq.(\ref{sigma}) approaches the value $( 1 +
4\alpha_s/\pi)$.

In this note we present a general analysis of hard gluon radiation. We
also include stop fragmentation due to collinear gluon emission in the
perturbative regime at high energies and we give
an account of non-perturbative fragmentation effects.

\newpage
For unpolarized lepton beams the cross section for gluon radiation
off $\tilde t_1$ squarks
\begin{displaymath}
  e^+\,e^- \to \tilde t_1\, \bar{\tilde t}_1 \,g
\end{displaymath}
depends on four variables: the polar angle $\theta$ between the
momentum of the $\tilde t_1$ squark and the $e^-$ momentum, the
azimuthal angle $\chi$ between the $\tilde t_1 \bar{\tilde t}_1 g $ plane
and the plane spanned by the $e^\pm$ beam axis with the $\tilde t_1$
momentum [see Ref.\cite{laer1}], and two of the scaled
energies $x(\tilde t_1)$, $\bar{x}(\bar{\tilde t_1})$, $z(g)$ in units of
the beam energy. The energies are related through $x +\bar x +z
=2$ and vary
over the intervals $\mu \leq x,\bar x \leq 1$ and $0\leq z \leq 1 -\mu^2$,
where $\mu = 2m_{\tilde t_1}/\sqrt{s}$ denotes the squark
mass in units of the beam energy. For the angles between the squark
and gluon momenta we have
\begin{eqnarray*}
  \cos\theta_{\tilde t_1 \bar{\tilde t}_1} & = & \frac{2 -2(x +\bar x)
    + x \bar
  x +\mu^2}{\sqrt{(x^2-\mu^2)(\bar x^2 -\mu^2)}}\\
  \cos\theta_{\tilde t_1 g} & = &\frac{2 -2 (x +z) +x
    z}{z\sqrt{x^2-\mu^2}}
\end{eqnarray*}
The spin-1 helicity analysis of the cross
section results in the following well-known angular decomposition \cite{laer2}
\begin{eqnarray}
  \frac{d\sigma}{dx\,d\bar x\,d\cos\theta\,d\chi/2\pi} &=&
  \frac{3}{8}( 1+ \cos^2\theta) \frac{d\sigma^U}{dx\,d\bar x}
  +\frac{3}{4}\sin^2\theta \frac{d\sigma^L}{dx\,d\bar x} \nonumber\\
  & & -\frac{3}{2\sqrt{2}}\sin2\theta\cos\chi
  \frac{d\sigma^I}{dx\,d\bar x}
  +\frac{3}{4}\sin^2\theta \cos 2\chi \frac{d\sigma^T}{dx\,d\bar x}
\end{eqnarray}
[$U$ = transverse (no flip), $L$ = longitudinal, $I$ = trv$\ast$long,
$T$ = trv$\ast$trv (flip)].
If the polar and azimuthal angles are integrated out, the cross
section is given by $\sigma = \sigma^U +\sigma^L$.

It is convenient to write the helicity cross sections as
\begin{equation}
  \frac{\beta^3}
       {\sigma_B}\frac{d\sigma^j}{dx\,d\bar x} = \frac{\alpha_s}{4\pi}
       \frac{S^j + \mu^2 N^j}{(1-x)(1-\bar x)}
\label{d2sigma}
\end{equation}
The densities $S^j$ and $N^j$ are summarized in Table \ref{sandn};
$p$ is the momentum of the $\tilde t_1$ squark, $\bar p$
and $k$ are the longitudinal momenta of
$\bar{\tilde t}_1$ and $g$ in the $\tilde t_1$ direction, and $p_T$
is the modulus of the transverse $\bar{\tilde t}_1$, $g$ momentum with
respect to this axis [all momenta in units of the beam energy].
Since $I,T$ correspond to $\gamma,Z$ helicity flips by 1
and 2 units, they are of order $p_T$ and $p_T^2$, respectively.
Note that the threshold suppression is absent in the $U$, $I$, $T$
components and attenuated in the leading longitudinal $L$ term as
expected from eq.(\ref{sigma}).

\begin{table}[t]
  \begin{center}
    \leavevmode
    \begin{tabular}{|c|c|c|}
      \hline
      & & \\[-0.2cm]
      & $S^j$ & $N^j$ \\
      & & \\[-0.2cm]
      \hline
      & & \\[-0.2cm]
      $\D  U $ & $\D \frac{\T32}{\T3}\,(1-x)\,(1-\bar x)$  &
      $\D -\frac{\T4}{\T3}\,p_T^2\,\frac{\T1-x}{\T1-\bar x}$ \\ [0.4cm]
      $\D  L $ & $\D \frac{\T16 \beta^2}{\T3}\,(1-z) $ &
      $\D \frac{\T4}{\T3}\,\left[p_T^2\,
      \frac{\T1-x}{\T1-\bar x} - \beta^2 \left(\frac{\T1-x}{\T1-\bar x}
      +\frac{\T1-\bar x}{\T1-x} +2\right)\right] $ \\ [0.4cm]
      $\D  I $ & $\D  -\frac{\T4\sqrt{\T2}}{\T3}\,p_T \,p$ &
      $\D \frac{\T2\sqrt{\T2}}{\T3}\,p_T \left(p - \bar p\,
      \frac{\T1-x}{\T1-\bar x}\right) $\\ [0.4cm]
      $\D  T $ & $\D 0$& $\D \frac{\T2}{\T3}\,p_T^2\,
      \frac{\T1-x}{\T1-\bar x} $ \\[-0.2cm]
      & & \\
      \hline
    \end{tabular}
  \end{center}
  \caption{Coefficients of the helicity cross sections in
    eq.(\protect\ref{d2sigma}). The energy and momentum variables are
    defined in the text.}
  \label{sandn}
\end{table}

\paragraph{Fragmentation.}
In the limit where the gluons are emitted from fast moving squarks
with small angles, the gluon radiation
\begin{displaymath}
  \label{brems}
  \tilde t_1 \to \tilde t_1\, g
\end{displaymath}
can be interpreted as a perturbative fragmentation process. From
the form of the differential cross section $d\sigma/dz\,dp_T^2 $ we
find in this limit for the splitting functions, in
analogy to the Weizs\"acker-Williams \cite{weiz} and Altarelli-Parisi
splitting functions \cite{altarelli},
\begin{eqnarray}
  P[\tilde t_1 \to \tilde t_1; x] & = & \frac{\alpha_s}{2\pi}\,
  \frac{8}{3}\, \frac{x}{1-x}\, \log\frac{Q^2}{m_{\tilde t_1}^2}  \\
  P[\tilde t_1 \to g; z] & = & \frac{\alpha_s}{2\pi}\, \frac{8}{3}\, \frac{
    1-z}{z}\, \log\frac{Q^2}{m_{\tilde t_1}^2}
                                                                 \nonumber
\end{eqnarray}
As usual, $x$ and $z$ are the fractions of energy
transferred from the $\tilde t_1$ beam to the squark $\tilde t_1$ and
the gluon $g$ after
fragmentation, respectively; $Q$ is the
evolution scale of the elementary process, normalized by the squark
mass rather than the QCD $\Lambda$ parameter [in contrast to the light
quark/gluon sector]. As a consequence of angular-momentum
conservation, the gluon cannot pick up the total momentum of
the squark beam. [Similar zeros have been found for helicity-flip
fragmentation functions in QED/ QCD \cite{altarelli,falk}.]

By using the crossing
rules \{$z \to 1, 1 \to x \}\mbox{ and }\{ 1-x \leftrightarrow 1-x$\},
familiar from the analogous splitting functions in QED \cite{chen}, we
derive for the elementary gluon splitting process into a
squark-antisquark pair
\begin{displaymath}
g \to \tilde t_1 \bar{\tilde t_1}
\end{displaymath}
the distribution
\begin{equation}
  P[g \to \tilde t_1; x] =  \frac{\alpha_s}{2\pi}\, \frac{1}{2} \,x \,(1-x)
  \,\log\frac{Q^2}{m_{\tilde t_1}^2}
\end{equation}
after adjusting color and spin coefficients properly. This splitting
function is symmetric under the $\tilde t_1 \leftrightarrow
\bar{\tilde t_1}$ exchange, i.e. $\{x \leftrightarrow
1-x\}$. The probability is maximal for the splitting into equal
fractions $x = 1/2$ of the momenta, in contrast to spinor QED/QCD
where the splitting into a quark-antiquark pair is proportional
to $x^2 + (1-x)^2$ and hence asymmetric configurations are preferred.

The above splitting functions provide the kernels for the
shower expansions in
perturbative QCD Monte Carlos for $e^+e^-$ annihilation such as Pythia
\cite{sjos} and Herwig \cite{webb}. They serve the same purpose in the
hadron-hadron versions of these generators as well as Isajet
\cite{baer}. Of course, the interpretation of the
radiation processes as universal fragmentation processes becomes
increasingly
adequate with rising energy of the fragmenting squarks/gluons.

If the $\tilde t_1$ squark is lighter than the top quark, the
lifetime will be long, $\tau \ge 10^{-20}$sec, since the dominant decay
channel $\tilde t_1  \to t +\tilde\chi_1^0$ is shut off
$[\tilde\chi_1^0 = LSP]$. The decay widths corresponding to
the 2-body decay $\tilde t_1 \to c +\tilde\chi_1^0$ and 3-body
slepton decays involve the
electroweak coupling twice and hence will be very small
\cite{hikasa}. As a result, the lifetime is much
longer than the typical non-perturbative fragmentation time of order
$1$ fm [i.e. ${\cal O}\, (10^{-23}$sec)] so that the squark has got
enough time to form $(\tilde t_1\bar q)$
and $(\tilde t_1 q q)$ fermionic and bosonic hadrons. However, the
energy transfer due to the non-perturbative
fragmentation, evolving after the early perturbative
fragmentation, is very small as a result of Galilei's law of
inertia. Describing this last step in the hadronization process of a
$\tilde t_1$ jet by the non-perturbative fragmentation function {\it
  \`a la} Peterson et al. \cite{pete}
(which accounts very well for the heavy-quark analogue), we find
\begin{equation}
  D (x)^{NP} \approx \frac{4 \sqrt{\epsilon}}{\pi}\,\frac{1}{x\,[1 - 1/x
    -\epsilon/(1-x)]^2}
\end{equation}
with the parameter $\epsilon \sim 0.5\,\mbox{GeV}^2/m_{\tilde
  t_1}^2$. Here, $x = E[(\tilde t_1\,\bar q)]/E[\tilde t_1]$ is the
energy fraction transferred from the $\tilde t_1$ parton to the $(\tilde
t_1\,\bar q)$ hadron etc. The resulting average non-perturbative energy loss
\begin{displaymath}
<1-x>^{NP}\sim\frac{2\sqrt{\epsilon}}{\pi}\left[
\log\left(\frac{1}{\epsilon}\right) -3\right]
\end{displaymath}
is numerically at the level of a few percent.

Monte Carlo programs for the hadronization of $\tilde t_1$ squarks
link the early perturbative fragmentation with the subsequent
non-perturbative hadronization. The relative weight of perturbative
and non-perturbative fragmentation can be characterized by the
average energy loss in the two consecutive steps. The overall retained
average energy
of the $\tilde t_1$ squarks factorizes into the two components,
\begin{equation}
<x> = <x>^{NP} <x>^{PT}
\end{equation}
Summing up the energy loss due to multiple gluon radiation
at high energies, we
find in analogy to heavy-quark fragmentation \cite{bigi2}
\begin{displaymath}
  <x>^{PT} = \left[\frac{\alpha_s(m_{\tilde
      t_1}^2)}{\alpha_s(E^2)}\right]^{-8/3b}
\end{displaymath}
with $ b = (11 -2 n_f/3) +(-2 - n_f/3)$ being the LO QCD $\beta$
function including the colored supersymmetric particle spectrum.
At high energies, the perturbative multi-gluon radiation has a
bigger impact than the final non-perturbative
hadronization mechanism, e.g. $<x>^{PT} \approx 0.93$ for a $\tilde
t_1$ beam energy $E=1\,\mbox{TeV}$ and $m_{\tilde t_1}=200\,\mbox{GeV}$
as compared to $<x>^{NP}\approx 0.98$. At low energies the two
fragmentation effects are of comparable size.

After finalizing the manuscript, we received a copy of Ref.\cite{djou}
in which the total cross sections for squark pair production in $e^+e^-$
annihilation have been discussed including squark-gluon and quark-gluino
loops, yet not the gluon-jet distributions analysed in the present note.

We thank our colleagues at the LEP2 Workshop who demanded the analysis
presented here to refine the experimental stop search techniques.

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\pagebreak
\begin{figure}[b]
  \begin{center}
    \leavevmode
    \epsfig{file=desy94-235-1.ps,height=12cm,bbllx=3.1cm,bblly=6cm,%
bburx=18cm,bbury=18cm}
  \end{center}
  \caption{Generic diagrams for $\tilde t_1 \bar{\tilde t}_1$
    production in $e^+e^-$ collisions. (a) Born level; (b) virtual
    QCD corrections; (c) gluon radiation.}
  \label{feyn}
\end{figure}

\pagebreak
\begin{figure}[b]
  \begin{center}
    \leavevmode
    \epsfig{file=desy94-235-2.ps,height=12cm,bbllx=3.1cm,bblly=5cm,%
bburx=18cm,bbury=18cm}
   \end{center}
  \caption{Coefficient of the QCD correction to the total cross
    section; shown is $\beta f(\beta)$, cf. eq.(\protect\ref{sigma}),
    with $\beta = (1 -4m_{\tilde t_1}^2/s)^{1/2}$.}
  \label{fbeta}
\end{figure}


\end{document}





