%Title: Higgs Effects in Top Production
%Authors: M. Je\.zabek and J. H. Kuehn
%Published:  <EM>Phys. Lett.</EM><B> B316 </B><EM>(1993) 360--364.</EM>
%TTP93--22\\
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\begin{document}
\begin{titlepage}
\pagestyle{empty}
\noindent
\hfill TTP93--22\\
\mbox{}
\hfill  June  1993   \\   %                 \hfill \today \\
\vspace{0.5cm}
\begin{center}
  \begin{Large}
  \begin{bf}
  \vspace{3cm}
Higgs Effects in Top Production
 \footnote{\normalsize Supported by BMFT Contract 056KA93P and by EEC
Contract ERB-CIPA-CT-92-2077}
\\
  \end{bf}
  \end{Large}

  \vspace{0.8cm}
  \begin{large}
   M. Je\.zabek\\
  \end{large}
{\it Institute of Nuclear Physics, Kawiory 26a,
PL-30055 Cracow, Poland}\\
and\\
{\it Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe\\
D-76128 Karlsruhe, Germany}\\[5mm]
  \begin{large}
and
  \end{large}
\\[5mm]
  \begin{large}
   J.H. K\"uhn \\
  \end{large}
{\it    Institut f\"ur Theoretische Teilchenphysik,
 Universit\"at Karlsruhe\\
 D-76128 Karlsruhe, Germany}\\
  \vspace{2.5cm}
  {\bf Abstract}
\end{center}
%
\noindent
Toponium and open top production are strongly enhanced by the
exchange of a light Higgs boson.  This enhancement exhibits a
rapid energy variation.  It is shown that the leading
energy dependent term can be
absorbed in an instantaneous Yukawa potential.  The remainder
can again be evaluated in the static limit.
\end{titlepage}

\pagestyle{normal}
%\section{Introduction}
Electroweak
corrections to toponium production have been calculated a long
time ago \cite{GKKS}.  Since $W$ and $Z$ boson masses are far above the
characteristic momentum scale of the bound state wave function,
the amplitudes depicted in Fig.\ref{Feyn} give rise to effective
pointlike vertex corrections.  In this approximation they can be
considered as momentum independent and hence absorbed in the weak
coupling constants.  Most of the corrections differ for the
$\gamma t \bar t$ and the $Z t\bar t$  vertex such that both the
cross section and the right-left asymmetry $A_{RL}$ are affected.  The
latter can be measured with polarized beams, or --- for unpolarized
beams --- with the help of lepton angular distributions in single
quark decay \cite{KS}.  Diagrams which influence only the
normalization are vertex corrections from photon and from Higgs
exchange.  Photon exchange is evidently absorbed in the
quarkonium potential, apart from the contribution of transverse
photons which provides an additional factor $(1-4\alpha/\pi)$.
Both the QED
potential and the transverse photon piece are completely overwhelmed
by the corresponding contributions from gluon exchange and are therefore
neglected.  Also the Higgs exchange affects the total cross
section only and has no influence on the left-right asymmetry.
However, as a consequence of the large Yukawa coupling it might
be possible to disentangle this effect from the QCD potential, in
particular if the Higgs mass is relatively low and $m_t$ close to
200 GeV.
\pictup{12.cm}{17cm}{Feyn}%
{Feynman diagrams relevant for the evaluation of one-loop corrections.
     Permutations have been omitted.}

Two strategies have been proposed to implement Higgs
exchange.  The dominant effect from a light Higgs boson can be
described by an instantaneous Yukawa potential
\cite{Ina,Hagi,Feig,Strass}
\begin{equation}
V_{\it Yuk}(r)=-\kappa\frac{e^{-m_H r}}{r}
\end{equation}
with
\begin{equation}
\kappa=\sqrt 2 G_F m_t^2/4\pi
\end{equation}
Retardation effects are absent in this case, as are contributions
from highly virtual top quarks.  Alternatively one may invoke a
purely perturbative treatment which leads to an overall
correction factor
\begin{equation}
1+\frac{\kappa}{\pi} f_{thr}(m_H^2/m_t^2)
\end{equation}
to the amplitude \cite{GKKS,guth,Fadin} with
\begin{equation}\label{fVthr}
f_{thr}(r)=-\frac{1}{12}\left[-12+4r+(-12+9r-2r^2)\ln r +
                  \frac{2}{r}(-6+5r-2r^2)l_4(r) \right]
\end{equation}
where
\begin{equation}
l_4(r)=\cases{\sqrt{r(4-r)}\,\arccos(\sqrt r /2) & if $r\le4$\cr
   -\sqrt{r(r-4)}\,\frac{1}{2}\ln\frac{1+\sqrt{1-4/r}}{1-\sqrt{1-4/r}}
        & if $r>4$\cr} \label{defl4}
\end{equation}
In the limit of small Higgs mass one obtains
\begin{equation}
f_{thr}\to \pi\frac{m_t}{m_H} + 2 \ln \frac{m_H}{m_t}+ \dots
\label{fthrlim}
\end{equation}
As stated before,
the $m_t/m_H$ term reflects the Coulomb singularity and leads to a
large enhancement of the production cross section close to threshold.
The correction (\ref{fVthr}) has been evaluated exactly at threshold,
that is for quarks with velocity $\beta=0$
and with the $t\bar t$ system in a ${}^3S_1$ configuration.

The perturbative calculation for nonvanishing velocity,
that is for open $t\bar t$ production, is conveniently
expressed in terms of two and three point functions B and C \cite{GDK}
\newcommand{\mts}{m_t^2}
\newcommand{\mhs}{m_H^2}
\begin{eqnarray}\label{fVpert}
f_{pert}(s,\mts,\mhs) = { \frac{\scriptsize 1}{\scriptsize 4}}
\left\{
\left[ - \mts\,(4C_0 - C_{21}) +
   s\,(C_{22} - C_{23}) + 2 C_{24} \right]
   (\mts,s,\mts,\mhs,\mts,\mts)\right. \nonumber\\
\left.  + B_1(\mts,\mts,\mhs)+2\mts\,\left[B_1'(\mts,\mts,\mhs)-
B_0'(\mts,\mts,\mhs)\right] -1/2 \right\}
\end{eqnarray}
with the property
\begin{equation}
f_{pert}(4\mts,\mts,\mhs)= f_{thr}(\mhs/\mts)
\end{equation}
The enhancement, which is particularly pronounced at threshold,
drops rapidly \cite{ben,GDK}
with increasing $\beta$ resembling Sommerfeld's
rescattering correction familiar from one photon or one gluon
exchange.  This
connection can be made more explicit by considering the integral
equation for the vertex correction in the nonrelativistic limit.

Let us relate the vertex function to the Green's function through
\begin{equation}
\tilde\Gamma(\vec k,E) = (E- k^2/m_t) \tilde G(\vec k,E)
\end{equation}
where
\begin{equation}
E= \sqrt{s} -2 m_t
\end{equation}
is the kinetic energy.
$\tilde\Gamma$ fulfills the following integral equation:
\begin{equation}
\tilde\Gamma(\vec p,E)= 1+ \int\frac{d^3k}{(2\pi)^3}\tilde V(\vec p-\vec
k)\frac{1}{E - k^2/m_t + i\epsilon}\tilde\Gamma(\vec k,E)
\end{equation}
where $\tilde V(\vec p)$ denotes
the Fourier transform of the Yukawa potential
\begin{equation}
\tilde V(\vec p)=-\frac{4\pi\kappa}{p^2 + m_H^2}
\end{equation}
For open top production $p$ is fixed through the kinetic energy
$p^2\ =\ E\,m_t$  or the velocity $\beta\ =\ p/m_t$~.
In perturbation theory
\begin{equation}
\tilde\Gamma=1+\tilde\Gamma^{(1)} + \dots
\end{equation}
$\tilde\Gamma^{(1)}$ can be obtained from a simple integration
\begin{equation}
\tilde\Gamma^{(1)}=-4\pi\kappa m_t\,\int\frac{d^3k}{(2\pi)^3}\,
\frac{1}{(\vec p - \vec k)^2 +\mhs}\,\,
                             \frac{1}{( p^2 -  k^2)+i\epsilon}
\end{equation}
and the corresponding correction function
$f_{\it Yuk}(\beta,m_t^2,m_H^2)= \frac{\pi}{\kappa} \tilde\Gamma^{(1)}$
reads
\newcommand{\rat}{\left({2m_t\beta/m_H}\right)}
%\newcommand{\rat}{\left(\frac{2m_t\beta}{m_H}\right)}
\begin{equation}
f_{\it Yuk}(\beta,m_t^2,m_H^2)
= \frac{\pi}{2\beta}\left[ \arctan\rat +
             \frac{i}{2}\log\left(1+\rat^2\right)
             \right] \label{fVnr}
\end{equation}
In the nonrelativistic regime $f_{\it Yuk}$ and $f_{pert}$
are closely related. Their absorptive parts are approximately equal.
The limiting behavior of Re~$f_{pert}$ for $\beta=0$
at fixed $m_H/m_t<<1$ , c.f. Eq.(\ref{fthrlim})~,
as well as the behavior
for $m_H\to 0$  at fixed $\beta<<1$
both coincide with the corresponding nonrelativistic results
\begin{equation}
{\rm Re}\,f_{\it Yuk} =
\cases{\pi\,m_t/m_H & for $\beta=0$\cr
       \pi^2/4\beta & for $m_H=0$\cr}
\end{equation}
We define the correction function
\begin{equation}
f_{nr}(s,\mts,\mhs)= f_{\it Yuk}(\beta,m_t^2,m_H^2)
- f_{\it Yuk}(\beta=0,m_t^2,m_H^2) + f_{thr}(m_H^2/m_t^2)
\label{fV}
\end{equation}
which by construction is equal to $f_{pert}$ at the threshold
$(\beta=0)$
for arbitrary $m_H$~. Thus, $f_{nr}$ and $f_{pert}$
are equal
up to ${\cal O}(\beta)$
subleading terms and $f_{nr}$
should approximate well
the full one loop result (\ref{fVpert}) in the region where
the nonrelativistic approximation is valid.

\pictup{15.1cm}{7.5cm}{curves}%
{a) Comparison of Re$f_{pert}(s,\mts,\mhs)$
-- solid lines -- and Re$f_{nr}(s,\mts,\mhs)$
-- dashed lines --  for $m_t$ = 150 GeV and $m_H$ = 50, 100
and 150 GeV (lower values correspond to higher $m_H$~)
 b) Energy dependence of Re$f_{V}(s,\mts,\mhs)$ for $m_H$ = 50 GeV
and $m_t$ = 120, 150 and 180 GeV -- solid, dashed and dotted lines.}

This is demonstrated in Fig.\ref{curves}a where the
nonrelativistic approximation $f_{nr}(s,\mts,\mhs)$ is
compared\footnote{In our numerical calculations we use the
relativistic
$\beta = \sqrt{1 - {4 m_t^2/s}}$~.}
to the full one loop result $f_{pert}(s,\mts,\mhs)$.

After subtracting of the  Yukawa potential contribution
the remainder
\begin{equation}
f_{V}(s,m_t^2,m_H^2)
= f_{pert}(s,m_t^2,m_H^2) - f_{\it Yuk}(\beta,m_t^2,m_H^2)
\end{equation}
is fairly energy independent, c.f. Fig.\ref{curves}b,
but clearly different from zero.
Although the energy dependence of $f_{pert}$ is dictated
by the instantaneous Yukawa potential it would be
a rather poor approximation to replace $f_{pert}$ simply by
$f_{\it Yuk}$~. The absolute value of $f_V$ can be greater than
$f_{pert}$ for typical $m_t$ and $m_H$~. However, since $f_{V}$
is nearly energy independent it
can be well approximated by its value at the threshold $s= 4m_t^2$~:
\begin{equation}\label{fVdef}
\hat f_{V}(m_t^2,m_H^2) = f_{thr}(m_H^2/m_t^2) - \pi \frac{m_t}{m_H}
\end{equation}

These  results suggest subtracting from the perturbative result
the piece which is identified with the instantaneous
potential contribution and including this
term by adding $V_{\it Yuk}$ to the QCD potential:
\begin{equation}\label{subtr}
          V_{eff} = V_{QCD} + V_{\it Yuk}
\end{equation}
Then, $V_{eff}$ can be used instead of $V_{QCD}$ in numerical
calculations of the total \cite{Strass} and differential
\cite{Sumino} cross sections for $e^+e^-$ annihilation
near $t\bar t$ threshold.
The dominant contribution to the remainder
can be included as an overall correction factor
\begin{equation} \label{fVhat}
1+\frac{\kappa}{\pi} \hat f_{V}(m_H^2/m_t^2)
\end{equation}
to the amplitude.  This factor
is energy independent and can be used below as well as
above the threshold.\\
This procedure is quite similar to the derivation of the
relativistic corrections from one gluon exchange \cite{buch}.
The calculation for the exchange of a massive gluon at threshold
leads to a correction of the form
\begin{equation}
\psi(0)\to \psi(0)\left(1+\frac{4}{3}
\frac{\alpha_s}{\pi}\left[-1 - \frac{2}{3}r - \frac{r^2}{3}\ln r +
\left(\frac{4}{3} +\frac{2}{3}r +\frac{1}{r} +\frac{3}{r-4}\right) l_4
\right]
\right)
\end{equation}
where $r=m_{\rm gluon}^2/\mts$. Employing the approximation
$m_{\rm gluon} \ll m_t$
and subtracting the $1/m_{\rm gluon}$ term one arrives in this case at
the familiar correction factor $(1-\frac{8}{3}\frac{\alpha_s}{\pi})$.

Our treatment of electroweak effects to the cross section can be
summarized as follows:  \\
i)  Corrections from the exchange of $W$ and $Z$ plus the
corresponding Higgs ghosts and from photon and $Z$ self energy
diagrams plus corrections to the electron and $Z$ vertex exhibit a
fairly moderate energy dependence for a given top mass.  Their
contribution is conveniently parameterized by \cite{guth}
\newcommand{\mrat}{\frac{m_t}{100\mbox{GeV}}}
\begin{equation}
\hat\rho=0.5838 - 0.0824 \mrat + 0.0403 \left(\mrat\right)^2 -
         0.0056\left(\mrat\right)^3
\end{equation}
which replaces the charge factor $e_t^2$ in the pure photon case.
It includes the lowest order $\gamma$ and $Z$ contribution as well as
the radiative corrections discussed above.  These corrections are
independent of the Higgs mass apart from diagram d) which is tiny and
can be neglected in this context.  \\
ii) The vertex correction from Higgs exchange
can be calculated by including $V_{\it Yuk}$ in the potential
calculation according to (\ref{subtr}) and
multyplying the amplitude by the correction factor
(\ref{fVhat}) which is evaluated purely
perturbatively and in the static limit, c.f. (\ref{fVdef})
and (\ref{fVthr})~.

The left right-asymmetry $A_{LR}$ for toponium and
top production close to threshold can be cast into a similar form
\begin{eqnarray}
A_{RL}&=&-2.336 - 15.044 \mrat + 6.008 \left(\mrat\right)^2 -
       \nonumber\\
   & &\left[3.403\mrat + 0.761\left(\mrat\right)^2 \right] \log\frac
          {m_t}{1\mbox{GeV}}
\end{eqnarray}
For fixed $m_W$ and $m_Z$ it
is fairly insensitive towards the Higgs mass.
\begin{thebibliography}{99}
\sloppy
\raggedright
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\sloppy
\raggedright
\bibitem{GKKS} B. Grzadkowski, P. Krawczyk, J.H. K\"uhn and
    R.G. Stuart, \npb{281}{1987}{18}.
\bibitem{KS} J.H. K\"uhn and K.H. Streng, \npb{198}{1982}{71};\\
    A. Czarnecki, M. Je\.zabek and J.H. K\"uhn, \npb{351}{1991}{70}.
\bibitem{Ina}H. Inazawa, T. Morii and J. Morishita,
                  \plb{203}{1988}{279}, \zpc{42}{1989}{569}.
\bibitem{Hagi} K. Hagiwara et al., \npb{344}{1990}{1}.
\bibitem{Feig} J. Feigenbaum, \prd{43}{1991}{264}.
\bibitem{Strass} M.J. Strassler, M. Peskin, \prd{43}{1991}{1500}.
\bibitem{guth} R.H. Guth and J.H. K\"uhn, \npb{368}{1992}{38}.
\bibitem{Fadin} V.S. Fadin and O.I. Yakovlev, \sovnp{53}{1991}{1053}.
\bibitem{GDK} R.H. Guth, A. Denner and J.H. K\"uhn, \npb{377}{1992}{3}.
\bibitem{ben} W. Beenakker and W. Hollik, \plb{269}{1991}{425}.
\bibitem{Sumino} Y. Sumino, K. Fujii, K. Hagiwara, H. Murayama
   and C.-K. Ng, \prd{47}{1992}{56};\quad M. Je\.zabek, J.H. K\"uhn
   and T. Teubner, \zpc{56}{1992}{653}.
\bibitem{buch} see the detailled discussion in the appendix of
    W. Buchm\"uller and S.-H. H. Tye, \prd{24}{1981}{132}.
\end{thebibliography}
%==============================================================
\end{document}



