%Title: Polarized Beams and the Top Threshold
%Author: J.H. K"uhn
%Published:     TESLA Technical Design Report, LC-TH-2001-004
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\begin{document}
\noindent
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\begin{flushright}
{\bf TTP01-02}
%{\bf hep-ph/xxxxxxx}\\
{\bf LC-TH-2001-04}\\ 
{\bf January 2001}\\
\end{flushright}

\begin{center}
\begin{Large} 
Polarized Beams and the Top Threshold\\
\end{Large}

\vspace{0.5cm}

\begin{large}
J.H.~K\"uhn\\[2mm]
\end{large}
Institut f\"ur Theoretische Teilchenphysik, Universit\"at Karlsruhe, 
D-76128 Karlsruhe, Germany\\  
\vspace{4mm}
{\bf Abstract}\\
\vspace{0.5cm}

\noindent
\begin{minipage}{13.0cm}
\begin{small}
The strong impact of
longitudinally polarized electron and positron beams on precision
studies of top quark production and decay close to the production
threshold is demonstrated. 
The right-left asymmetry of the production cross section
can be measured with a precision of one to two percent. 
The top quarks will be highly polarized which allows to measure the
couplings  responsible for the top quark decay. 
As characteristic examples the decay chains $t\to bW(\to e^+\nu)$ and
$t\to H^+b$ will be considered.

\end{small}
\end{minipage}

\end{center}

\setcounter{footnote}{0}
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\vspace{1.cm}
\noindent {\it The Right Left Asymmetry}

\vspace{0.5cm}
\noindent
The right-left asymmetry
\begin{equation}
A_{RL}=  {\sigma_R - \sigma_L \over \sigma_R + \sigma_L}
\label{defasy1}
\end{equation}
can be measured with polarized beams. 
(Here R and L correspond to fully polarized right- and lefthanded
beams, respectively.)
Close to threshold the total cross section for $t\bar t$ production is
dominated by the $S$-wave contribution. It is therefore given by the 
ratio of $S$-wave amplitudes
induced by electromagnetic and neutral current, respectively, and thus
independent of uncertainties associated with
the hadronic interaction of the $t\bar t$ system.
In fact, the result for $A_{RL}$ close to threshold \cite{GK,HJKT,HJKP} 
is identical to the
one on a ``would be'' toponium resonance \cite{KZ,GKKS}. 
The small $P$ wave admixture to the rate
which amounts to about 1\% is numerically well under control \cite{KT}.
A precise measurement of $A_{LR}$ will thus lead to a correspondingly
precise determination of the vector coupling of the top quark, assuming,
of course that the weak and electromagnetic couplings of the electron
are known.

In Born approximation the asymmetry $A_{RL}$ at threshold is given by
\begin{equation}
A_{RL}= - {a_2 \over a_1}
\label{defasy2}
\end{equation}
with the coefficients
\begin{equation}
a_1 = \left( q_e q_t + v_e v_t d\right) ^2 + \left( a_e v_t d\right) ^2 \,, 
\qquad 
a_2 = 2 a_e v_t d \left( q_e q_t  + v_e v_t d\right) \,,
\label{eqcouplingsa}
\end{equation} 
\begin{equation}
d = {1\over 16 \sin^2\theta_W\cos^2\theta_W}\,{s\over s - M_Z^2}
\end{equation}
and the electromagnetic and weak charges
\begin{eqnarray}
q_e = -1 \,,\quad & v_e = -1 + 4 \sin^2\theta_W\,,\quad 
& a_e = -1\,,\nonumber\\[1mm]
q_t = 2/3 \,,\quad & v_t = 1 - 8/3\, \sin^2\theta_W\,,\quad
& a_t = 1\,.
\label{eqcharges}
\end{eqnarray}
At threshold $s=4m_t^2$.

Numerically one finds (for $\sin^2\theta = 0.23$ and $m_t=175$~GeV)
$A_{RL}= -0.426$. 
Electroweak corrections have been evaluated in
\cite{GKKS,GK} and modify this result to
$A_{LR}= -0.405$. The corresponding results for the Two-Higgs-Doublett
Model can be found in \cite{DGK}.

For experiments with polarized electron and positron beams the effective
polarization is given by
\begin{equation}
P_{eff} = {P_+ - P_- \over 1 - P_+ P_-}\,.
\label{Peff}
\end{equation}
The values $|P_-|=0.8$ and $|P_+|=0.6$ have been
quoted for TESLA,  corresponding to $P_{eff} = 0.946$.

The relative contamination of the top signal by the background is denoted by
$\eta = \sigma_{bg} / \sigma_{t\bar t}$ and the effective asymmetry 
thus given by
\begin{equation}
\langle A_{RL}\rangle = { A_{RL} + A_{RL}^{bg} \eta \over 1 + \eta }\,.
\label{Abgdef}
\end{equation}

For the analysis below we shall assume that the background is dominated
by $W$-pairs and for simplicity take $A_{RL}^{bg}=-1$, the worst case.
(The results are hardly affected by this assumption, as long as
$\eta$ remains small.)
The statistical error expected for a measurement of $A_{RL}$ is given by
\cite{LEP}

\begin{equation}
\Delta A_{RL} = {(1 + \eta) \over \sqrt {N} } 
\sqrt{ \left ( {1 \over P_{eff}^2} - \langle A_{RL}^2 \rangle\right) + 
        \eta^2 (\langle A_{RL}\rangle - A_{RL}^{bg} )^2 } \,.
\label{staterr}
\end{equation}
with
\begin{equation}
N = N_{RL}+N_{LR} = (\sigma_{bg} + \sigma_{t\bar t}) 
            {1 + |P_+ P_-| \over 2} \int {\cal L} 
\label{Nevts}
\end{equation}

We assume that positive and negative polarizations of electrons are
chosen with equal probability, similarly those of positrons.
Electron and positron polarisations are chosen uncorrelated. 
The asymmetry measurement, however, is only based on the two samples
with dominantly righthanded electrons and lefthanded positrons ($RL$)
and vice versa ($LR$). Operating only with these properly correlated
combinations the rate would be enhanced by a factor two. 

In addition one has to take into account the systematic error from the
uncertainty in the beam polarization
\begin{equation}
\Delta \langle A_{RL}\rangle = \langle A_{RL}\rangle 
                               {\Delta  P_{eff} \over  P_{eff} }
\label{polerr}
\end{equation}
and from the uncertainty in the signal to background ratio


\begin{equation}
\Delta  A_{RL} = |\langle A_{RL}\rangle - A_{LR}^{bg}| 
                                \Delta \eta 
\label{etaerr}
\end{equation}
 
The subsequent analysis will be based on the extraction of the top
signal  described in \cite{MM}.
Using appropriate cuts the efficiency for extracting the top signal 
is quoted to be 41.4 \%. The remaining background cross section of 8.4~fb is
constant throughout the threshold region, the signal to background ratio
varies between  6 at the lowest scan point and  60 at the highest scan point
At $E_{cm} = 2 m_t$ it amounts to 21. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%[Comment: 
%
%The value 21 is consistent with the cross section of 0.42 pb
%($=(8.4\times  21 / 0.414)~fb$) on threshold, as shown in 
%Fig 1 of A. Juste et
%al.. I don't understand the value 6 and in particular not the value 60
%quoted below and ``far'' above threshold, which would
%correspond to a top cross section of about 1.2 pb. However, I will not
%use the values 6 and 60.
%
%End of Comment]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For simplicity I will use the value 20 for the signal to background
ratio throughout (whence $\langle A_{RL}\rangle = 0.433$,)
and assume, furthermore, an integrated luminosity of
$100~fb^{-1}$. With these assumptions, and using eq.\ref{Nevts}
\begin{equation}
N_{t\bar t} = 8.4 fb \times 20 \times (1 + 0.8 * 0.6) /2
\times 100~fb^{-1} = 12400 \,.
\end{equation}
from the $RL$ plus $LR$ samples.
We do not employ the potential improvement in background
rejection from the difference in the polarization asymmetries of signal
and background (dominantly $W$-pairs with $A_{RL}$ close to -1).

For the evaluation of the statistical error (eq. \ref{staterr}) the second
term under the squareroot can safely be neglected. In total one finds
\begin{equation}
\Delta  A_{RL}\approx {1\over \sqrt{N}} \approx 0.0093 \,. 
\end{equation}

With a difference of 0.02 between the prediction of $A_{RL}$ at tree
level and in one loop approximation the measurement is evidently
sensitive to the quantum corrections.

In order to match this statistical error, a relative precision in
$P_{eff}$ better than 2\% is required (cf.\ eq.\ \ref{polerr}) and 
an accuracy in $\eta$ of
$\Delta \eta \le  \Delta A_{RL}/ (\langle A_{RL}\rangle -
                       A_{LR}^{bg})\approx 1.64\% $ (cf.\ eq.\ \ref{etaerr}),
corresponding to a relative error of $\Delta \eta /\eta =30\%$ 
in the signal to background ratio.

The relative errors of $A_{RL}$ and of the top vector coupling $v_t$ are
related through

\begin{equation}
{\Delta  A_{RL} \over A_{RL}} = \left[ 1 + 
                    v_t d {(v_e + a_e A_{RL}) \over (q_e q_t + v_e v_t d)}
                   +{v_e\over a_e}  A_{RL}\right] {\Delta v_t \over v_t}
                            \approx 0.87 {\Delta v_t \over v_t}\,.
\end{equation}
  
In total one thus expects a measurement  of $v_t$ with a relative
precision of about 2\%. Adopting more optimistic assumptions,
e.g. an integrated luminosity of $300~fb^{-1}$ invested for properly
correlated  electron and positron polarisations only, 
precisions in $A_{RL}$ and $v_t$ of about
0.4\% and 1\% respectively can be achieved without  
drastic requirements on the systematic error of the polarization.
%==========================================================================

\vspace{1cm}
\noindent
{\it Decays of Polarized Top Quarks at Rest: 
Measuring the Lorentz Structure of Charged Currents and the Handedness
of a Charged Higgs Boson}

Close to threshold the orbital angular momentum of top quarks is
dominantly $J=0$ and the helicity of electrons and positrons is
transferred to the spins of top quark and antiquark \cite{K}. 
[The angular dependent transverse and normal components of the top quark,
resulting from S-P-wave interference, have been studied in
\cite{HJKT,HJKP}. They are sensitive to the axial coupling of the top
quark and to $t$-$\bar t$ rescattering.
Beam polarization  plays an important role also in
this case. A detailed analysis of the statistical and systematical error
in such an analysis
is beyond the scope of this note.]
Fully polarized beams would thus allow to study the decay of fully polarized
top quarks at rest. The formation time of top mesons is far longer than
the life time of the quark. Angular distributions of its decay
products thus reflect those predicted for free quarks \cite{K}.

In practice one will deal with only partially polarized beams. For
dominantly left handed electrons ($P_-=-0.8$) and right handed positrons
($P_+=+0.6$) the effective beam polarization $P_{eff}= 0.946$ is close to
one. The quark polarizations are thus given by 
\begin{equation}
P_{t}=P_{\bar t} = {P_{eff} + A_{RL} \over 1 + P_{eff} A_{RL} } =- 0.978\,,
\end{equation}
and are thus very close to the maximal value.
For both beam polarizations reversed, $P_{eff}$ changes its sign and with
\begin{equation}
P_{t}=P_{\bar t} = 0.877\,,
\end{equation}
the quark polarizations are only slightly degraded.

For an integrated luminosity $\int {\cal L} = 100 fb^{-1}$ distributed
evenly among the four helicity combinations one obtains the event rate
\begin{eqnarray}
N_{RL} =  {1 + |P_{eff}| A_{RL} \over 2}
\sigma_{t\bar t} {1 + |P_+ P_-| \over 2} \int {\cal L}\nonumber\\
N_{LR} =  {1 - |P_{eff}| A_{RL} \over 2}
\sigma_{t\bar t} {1 + |P_+ P_-| \over 2} \int {\cal L}\nonumber\\
\end{eqnarray}
with a relative rate $N_{RL}/N_{LR}=0.446$.

Two decay channels will be discussed in the following, which illustrate
the power of measurements with polarized beams: the angular distribution
of leptons from the decay chain 
$t \to b W^+ (\to e^+\nu)$ and the angular distribution of a charged
Higgs boson from the decay $t\to H^+b$.
The former will allow to determine the Lorentz structure of
the top quark coupling to the charged current, the latter gives the
relative strength of the charged Yukawa coupling to right handed versus
left handed $b$ quarks, which is a direct measure of $\tan\beta$.

In the rest frame of the top quark
the distribution of leptons from the decay $t \to b W^+(\to e^+\nu)$
factorizes into an angular and an energy dependent part
\cite{angular}, with
\begin{equation}
{{\rm d} N \over {\rm d}\cos\theta} = {1+P_t\alpha\cos\theta \over 2}\,.
\end{equation}
and $\alpha=1$ in the Standard Model. This angular dependence is hardly
affected by QCD corrections\cite{QCDdis}. 
The smearing of this angular distribution from the momentum
distribution \cite{momdis}
of the top quark due to the interaction between $t$ and
$\bar t$ is relatively unimportant ($\beta_t\approx 0.1$) and will be
ignored. It can be either unfolded by measuring the top momentum or
included in a more detailed Monte Carlo simulation.



In the case where the background contribution can be neglected,
the statistical error for a measurement of the coefficient $\alpha$ is
given by
\begin{equation}
\Delta \alpha = {1 \over \sqrt{N}}\sqrt{C/P_t^2-\alpha^2}
\label{Deltaalpha}
\end{equation}
where $C=3$ if the measurement is based on an evaluation of the moment
\begin{equation}
\alpha = 3 \langle\cos\theta\rangle / P_t
\end{equation}
and $C=4$ if it is based on the lepton forward backward asymmetry
\begin{equation}
\alpha = {2\over P_t}{N_F - N_B \over N_F + N_B}\,.
\end{equation}

For the channel where one quark decays semileptonically and the other
one into four jets, a signal to background ratio of $S/B = 55$ has been
quoted \cite{MM}, with a signal cross section of 77~fb.
For an integrated luminosity of 100 $fb^{-1}$, distributed evenly among
the four combinations of beam polarizations, one finds $N_{RL} = 1760$
and $N_{LR}=3940 fb^{-1}$. The statistical precision
\begin{equation}
\Delta\alpha = 0.02
\end{equation}
is largely dominated by the $LR$-sample. A precision of 
$\delta\,P_{eff}/P_{eff} = 2\%$ 
is required to match this accuracy. Investing even
an integrated luminosity of 300 $fb^{-1}$ specifically into the $LR$ and
$RL$ combinations, $\Delta\alpha =0.008$ can be achieved, again without
drastic requirements on the control of $P_{eff}$. 
In addition one may also study the (energy dependent) angular
distribution of the neutrino, whose missing momentum can well be
reconstructed in these measurements. 
A precision of one to two percent in the coefficient $\alpha$ 
in combination with the neutrino analysis will indeed severly restrict
anomalous $tbW$ couplings \cite{JezK}.

The analysis of ``Non-Standard-Decays'' of polarized top quarks allows
to explore in a fairly direct way the Lorentz structure of the couplings
responsible for these unconventional modes. Let us discuss, as a
characteristic example, the transition $t\to H^+ b$ in the framework of
the two-Higgs doublet model. From the amplitude

\begin{equation}
{\cal A} = \sqrt{G_f \over \sqrt{2}}\, \bar b [m_t\cot\beta (1-\gamma_5)
           + m_b \tan\beta (1 + \gamma_5) ] t
\end{equation}
one obtains \cite{KZ}
\begin{equation}
{{\rm d} N(H^+) \over {\rm d}\cos\theta} = {1\over2} (1 +
P_t\,\alpha_H\cos\theta) 
\end{equation}
with
\begin{equation}
\alpha_H =
{(m_t^2\cot^2\beta - m_b^2 \tan^2\beta)\over
 (m_t^2\cot^2\beta + m_b^2 \tan^2\beta)}
\end{equation}

The determination of $\alpha_H$ is thus complementary to the measurement
of the branching ratio proportional to $(m_t^2\cot^2\beta +
m_b^2 \tan^2\beta)$ . In principle the anlysis proceeds just as in the
case of semileptonic decay. Let us concentrate on the case of large
$\tan\beta$, which allows for a sizeable branching ratio ${\rm Br} (t\to
H^+b) = {\cal O} (10\%)$, if the charged Higgs boson is sufficiently
light \cite{Guasch}. For $\tan\beta\ge3$ the mode $H^+\to \tau^+\nu$
dominates the charged Higgs decays. With good vertex detectors it should
be possible to identify the $\tau$ lepton and thus a large fraction of
these final states. The signal characteristics are similar to those of
the semileptonic decay $t\to b W(\to\tau\nu)$, albeit with a different
energy of the $b$ jet and a different missing momentum. Hence we shall
assume that the background can similarly be rejected to a large extent,
and the signal retained with an efficiency of 30\%. From
eq.\ref{Deltaalpha} and with $N_{RL}+N_{LR}=2000$ (corresponding again
to ${\cal L}=100 fb^{-1}$), using furthermore $C=3$ and $P_{t}^2\approx1$ 
one finds
\begin{equation}
\Delta\alpha\approx 0.04\,.
\end{equation}
For the optimistic asumptions on luminosity and correlated beam
polarizations even $\Delta\alpha =0.016$ seems within reach.
From the relation between the errors in $\alpha_H$ and $\tan\beta$
\begin{equation}
\Delta \alpha_H = { 8 {m_b^2\over m_t^2} \tan^4\beta \over
                     (1 + {m_b^2\over m_t^2} \tan^4\beta)^2}
                     {\delta\tan\beta \over \tan \beta}
\end{equation}
one may expect a fairly precise determination of $\tan\beta$ if
$m_b^2/m_t^2\tan\beta$ is of order one. Otherwise one would 
confirm the dominance of right- or of lefthanded couplings and thus provide
an important test of the Lorentz structure of the Yukawa couplings.

\vspace{1cm}
\noindent
{\bf Acknowledgments}\\[1mm]
Work partly supported by BMBF Contract 05HT9VKA8 and the European Union
under Contract  HPRN-CT-2000-00149.

\def\prep#1#2#3{{\it Phys.~Rep.~}{\bf #1} (#2) #3}
\def\app#1#2#3{{\it Acta~Phys.~Polon.~}{\bf B #1} (#2) #3}
\def\apa#1#2#3{{\it Acta Physica Austriaca~}{\bf#1} (#2) #3}
\def\npb#1#2#3{{\it Nucl.~Phys.~}{\bf B #1} (#2) #3}
\def\plb#1#2#3{{\it Phys.~Lett.~}{\bf B #1} (#2) #3}
\def\prd#1#2#3{{\it Phys.~Rev.~}{\bf D #1} (#2) #3}
\def\pR#1#2#3{{\it Phys.~Rev.~}{\bf #1} (#2) #3}
\def\prl#1#2#3{{\it Phys.~Rev.~Lett.~}{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{\it Sov.~J.~Nucl.~Phys.~}{\bf #1} (#2) #3}
\def\yadfiz#1#2#3{{\it Yad.~Fiz.~}{\bf #1} (#2) #3}
\def\jetp#1#2#3{{\it JETP~Lett.~}{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z.~Phys.~}{\bf C #1} (#2) #3}
\def\epjc#1#2#3{{\it Eur. Phys. J.~}{\bf C #1} (#2) #3} 
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\end{thebibliography}

\end{document}




