

\ifnum\wg=0
\subsection{Angular correlations and CP violation}
\subsubsection{Angular correlations of $t\bar t$ decay products}
As stated in the previous chapter, top quarks are produced through the
virtual photon and $Z$. In the threshold region the are polarized to a
degree
\begin{equation}
P_t= A_{RL}\approx - 0.4
\end{equation}
Assuming for the distribution of leptons from the decay of polarized
top quarks
\begin{equation}
\frac{dN}{dx\,d\cos\theta}=f(x)+ g(x) P_t\cos\theta
\end{equation}
(with $g(x)=f(x)$ in the \sm, see eq.\ref{eq:15})
the angular distribution allows to test for the chirality of the $tb$
current. Implicitly it was assumed that hadronization does not affect
the top spin degrees of freedom \cite{acta,jk41}. This assumption can
be tested independently through the study of correlations between t and
$bar t$ decay products. In the threshold region the spins are correlated
$\propto (1+\frac{1}{3} \vec{s_+}\cdot\vec{s_-})$. This leads to the
following correlated $\ell_+\ell_-$ distribution:
\begin{equation}
\frac{dN}{dx_+dx_-d\cos_{=-}}= f_+(x_+)f_-(x_-) +
                      \frac{1}{3} g_+(x_+)g_-(x_-)\cos\theta_{+-}
\end{equation}
where $f_+=f_-$ and $g_+= -g_-$. $\theta_{+-}$ denotes the angle between
$\ell_+$ and $\ell_-$. After averaging the lepton energies,
\begin{equation}
\frac{dN}{d\cos\theta_{+-}}=1+\frac{1}{3} h_+ h_- \cos\theta_{+-}
\end{equation}
Note that the coefficient of the correlation term is $-h^2_+/3$ and
hence always negative (assuming CP conservation). Since
$|h_+|\leq1$ it ranges between 0 and $-1/3$. This limiting value is
assumed in the \sm.
For illustration we give in Fig.\ref{Fwas} the Monte Carlo prediction
obtained with the Monte Carlo program TIPTOP \cite{TIPTOP} for the \sm
and for a pure V+A current,  corresponding to correlation coefficients
of -1/3 and $\approx.09$.
\begin{figure}\label{Fwas}
\vspace{12cm}
\caption[]{Monte Carlo prediction for the distribution of the angle
between $\ell_+$ and $\ell_-$ in the \sm (solid) and for V+A (dotted).
Parameters: $m_t=150GeV$, $\sqrt s = 302 GeV$.\cite{Was}}
\end{figure}
%
\subsubsection{CP violating dipole form factors}
\paragraph{Search for CP violation}
One of the remarkable properties of a heavy top is that
to a good approximation it
decays before it can form hadronic bound states.
This implies in particular that the information about
the spin polarisation of the $t$ is not
diluted, but reveals itself in the angular distributions of its
weak decay products. This was already
discussed and applied in the previous sections. Moreover
this phenomenon also allows for searches of CP-violating
effects in $t\bar t$ production and decay by means of
CP-odd spin--momentum correlations.
Although effects due to the Kobayashi--Maskawa phase
are very small, observable CP violation could arise
from new interactions. Performing
tests of the CP symmetry at high energies would be of
interest, and data
from an $e^+e^-$ collider (with unpolarized or
transversely polarized beams) are suitable for that purpose
\cite{Donog,Bern1,Bern2}.
Specific investigations of CP symmetry tests for the $t\bar t$
system were recently made also in \cite{Nelson,Kane}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{CP-odd correlations}
For definiteness we consider unpolarized $e^+e^-$ collisions
above the $t\bar t$ threshold with $t$ decay as in the
Standard Model (SM):
\begin{eqnarray}
e^+ e^- \to t\bar t\to W^+b+W^-\bar b
\label{eetoWbWb}
\end{eqnarray}
Most useful for the following are
the cannels where both or at least one
of the $W$'s decay leptonically:
$W \to \ell \nu_\ell$,
where $\ell=e,\mu,\tau$.
Various CP-odd correlations --- which cannot be faked by
CP-even unitarity corrections --- involving the
momenta of $\ell^+,\ell^-$ and/or $b,\bar b$
can be studied. Using, for instance, the (unit) three momenta
${\bf q_\pm}({\bf\hat q_\pm})$ of $\ell^\pm$ in the laboratory
frame one
may form the CP--odd Cartesian tensors
$(i,j = x,y,z)$:
\begin{eqnarray}
T_{ij}=({\bf q_-}-{\bf q_+})_i({\bf q_-}\times{\bf q_+})_j+
(i\leftrightarrow j) ,\nonumber\\
\hat T_{ij}=({\bf \hat q_-}-{\bf \hat q_+})_i{({\bf \hat q_-}
\times{\bf \hat q_+})_j
 \over |{\bf \hat q_+}\times{\bf \hat q_-}|}+
(i\leftrightarrow j).
\label{Tij}
\end{eqnarray}
Analogously one can form these tensors by replacing one or both
lepton momenta by the momenta of the $b$ and/or $\bar b$ jets.
The two "non--diagonal" correlations between
$\ell^+({\bf q_+})\bar b({\bf q_-})$ and
$\ell^-({\bf q_-})b({\bf q_+})$ must be added in order to get
an expectation value which has definite properties with respect to CP.
If the phase space cuts are C-- and P--blind then it is
straightforward to show \cite{Bern3} that
\begin{eqnarray}
<T_{ij}>_{a \bar b} + <T_{ij}>_{b \bar a} & \neq & 0
\label{3}
\end{eqnarray}
(and likewise for $<\hat T>$)
would be a signal of CP violation. Two comments are in order:\\
a) In the approximation where the amplitude for (\ref{eetoWbWb})
can be factorized into $t\bar t$ production and $t,\bar t$ decay
amplitudes it can be shown that the tensors (\ref{Tij}) project
onto CP-violating terms in the $t\bar t$ production density matrix
only. Whether there is CP violation in $t$ decay can be
investigated with other correlations (see below).\\
b) The expectation values of
(\ref{Tij}) can be traced back to several
correlations involving the spins and momenta of $t\bar t$.
In particular the CP--odd correlation
$({\bf p_+}\times{\bf k_+})\cdot(\vec{\sigma_+}-\vec{\sigma_-})$
appears (${\bf p_+},{\bf k_+}$ are the momenta of $e^+$ and $t$,
respectively) which measures a possible asymmetry in the transverse
polarisations of $t$ and $\bar t$.
(The analogous case of $\tau^+\tau^-$ production was investigated
in detail in \cite{Bern3}.)\\
In various models, in particular Higgs models
of CP violation the most
important CP-violating contributions to the $e^+e^-\to t\bar t$
scattering amplitude arise from terms in the
$\gamma t\bar t$ and $Zt\bar t$ vertex functions. Therefore
we shall take into account only such contributions in the following.
>From the usual form factor decomposition of these vertex functions
(see, e.g. \cite{Grimus}) one sees that only
the chirality flipping electric and
weak dipole form factors $d_\gamma(s)$ and $d_Z(s)$, respectively,
are relevant. They may be represented by the effective interaction
\begin{eqnarray}
{\cal L}=-i d_\gamma \bar t\sigma_{\mu\nu}\gamma_5tF^{\mu\nu}/2
         -i d_Z \bar t\sigma_{\mu\nu}\gamma_5tZ^{\mu\nu}/2.
\label{EffLag}
\end{eqnarray}
 
The expectation values of (\ref{Tij}) can be calculated in terms
of the SM couplings and those of (\ref{EffLag}). For rotationally
invariant cuts one obtains to Born approximation
\begin{eqnarray}
<\hat T_{ij}>_{a \bar b} &=& <\hat T_{ij}>_{b \bar a}\\
<\hat T_{ij}>_{a \bar b} &=& (c_{\gamma,ab} \hat d_\gamma
                           +c_{Z,ab} \hat d_Z) s_{ij}
\label{c}
\end{eqnarray}
where
\begin{eqnarray}
\hat d_{\gamma,Z}(s)&=&{\sqrt s\over e} \mbox{Re}[d_{\gamma,Z}(s)]\\
s_{ij}&=&\Big({\hat p_{+i}}{\hat p_{+j}}-{1\over3}\delta_{ij}\Big)/2.
\end{eqnarray}
Analogous formulae hold for $<T_{ij}>$.
 The form factors $d_\gamma,d_Z$
could have absorptive parts, too. However, because the expectation
values of (\ref{3}) are CPT--even, Im$[d_{\gamma,Z}]$ can contribute to
(\ref{c}) only in the next order;
through interference with absorptive
parts of 1--loop SM amplitudes. The calculation of the correlations
were made using standard $V-A$ charged weak currents and putting
$m_t=150$ GeV.
\begin{figure}\label{FCP1}
\vspace{12cm}
\caption[]{Coefficients $c_\gamma,c_Z$ defined in eq.(\ref{c}) for
  correlations among (a) lepton--lepton--momenta,
  (b) lepton--$b$--momenta, (c) $b$--$b$--momenta.}
\end{figure}
We discuss here only the results for the
dimensionless tensor $\hat T$. In Figs.\ref{FCP1}a,b,c
the coefficients
$c_{\gamma,Z}$ are shown as functions of $\sqrt s$.
These results imply that the tensors (\ref{Tij}) are primarily
sensitive to $d_\gamma$. Moreover the lepton--lepton momentum
correlations are largest which is due to the fact that in the SM
the angular distribution of the charged lepton has a higher analyzing
power of the spin polarisation
of the $t$ quark than the
angular distribution of the $b$ quark. Close to the $t\bar t$
threshold $<\hat T_{ij}>\sim\beta^2$ ($\beta=\sqrt{1-4m_t^2/s}$)
because this expectation value is dominated in this region by the
interference of the SM amplitudes containing the $\gamma t\bar t$
and the vector part of the $Zt\bar t$ couplings with the $p$--wave
amplitudes due to the couplings (\ref{EffLag}). QCD corrections,
which render the production rate nonzero at threshold
(see, e.g.\cite{KZ}) because of the Coulomb singularity, may
change this behaviour. This requires further study.\\
Another useful correlation is the CP--odd asymmetry
\begin{eqnarray}
A={\bf\hat p_+}\cdot{({\bf\hat q_+}\times{\bf\hat q_-})
                \over|{\bf\hat q_+}\times{\bf\hat q_-}|}
\end{eqnarray}
where $\bf p_+$ is the $e^+$ momentum.
The expectation value is
\begin{eqnarray}
<A>_{a\bar b}+<A>_{b\bar a}=r_{\gamma,a\bar b}\hat d_\gamma
                           +r_{Z,a\bar b}\hat d_Z
                           +(a\leftrightarrow\bar b)
\label{r}
\end{eqnarray}
In Born approximation $r_{\gamma,Z;a\bar b}=r_{\gamma,Z;b\bar a}$
holds. The values of these coefficients are given in Fig.\ref{FCP2}
for correlations among the lepton--lepton momenta. Values,
respectively upper bounds on $d_\gamma$ and $d_Z$ can be obtained
by measuring $<\hat T>$ and $<A>$ simultaneously at a given c.m.
 energy.\\
\begin{figure}\label{FCP2}
\caption[]{Coefficients $r_\gamma,r_Z$ defined in eq.(\ref{r}) for
  correlations among lepton--lepton--momenta.}
\end{figure}
Assuming the SM decay mode $t\to W^+b$ to be the dominant one
a Monte--Carlo estimate was made \cite{PIK} of the efficiency
to tag the two charged leptons from $t\bar t\to \ell^+{\ell'}^-
b\bar b\nu_\ell\bar\nu_{\ell'}$ and the $b$ and $\bar b$ jets from
$t\bar t\to b\bar b X$, respectively; both at $\sqrt s=310$ GeV
and $\sqrt s=500$ GeV (using $m_t=150$ GeV).
Various selection criteria and cuts were applied
to suppress the the background from $q\bar q
 (q\neq t$), $W^+W^-$, and $ZZ$ production. It was found that with
an integrated luminosity of 10 (fb)$^{-1}$ $N(\ell^+{\ell'}^-)
\approx180 (90)$ and $N(b\bar b)\approx 240(110)$
events could be isolated at $\sqrt s=310(500)$ GeV. Assuming
the tagging efficiency for "off--diagonal" events, i.e. lepton +
$b$ jet, to be about 1/10 we obtain $N(\ell^+\bar b)=N(\ell^-b)
\approx 300(200)$ events at $\sqrt s=310(500)$ GeV. With the
calculated variances of $\hat T_{ij}$ and $A$ we estimate that
a 1 s.d. statistical sensitivity of $\delta \hat d_\gamma,
\delta \hat d_Z\approx0.2$ may be obtained by measuring these
correlations. In dimensionful units this corresponds to $\delta
d_\gamma,\delta d_Z\approx9\cdot10^{-18}$ ecm ($5\cdot10^{-18}$ ecm)
at $\sqrt s=310(500)$ GeV.\\
\indent
Various other CP--odd asymmetries are also of interest. For instance
one may consider the CP--odd and "CPT--odd" energy asymmetry
\begin{eqnarray}
{N(E_+>E_-)-N(E_->E_+)\over N(E_+>E_-)+N(E_->E_+)}
\label{NE+E-}
\end{eqnarray}
where $E_\pm=E_{\ell^\pm}$, which requires besides CP violation
also absorptive parts to be nonzero. If one wants
to check (by means of momentum correlations) whether there is CP
violation in $t$ and $\bar t$ decay one must correlate at least two
of the particle momenta from one decay vertex. For a CP-odd but CPT-even
correlation the obvious choice is
(see also \cite{Kane})
\begin{eqnarray}
w\
{\bf\hat p_+}\cdot({\bf\hat q_{\ell^+}}\times{\bf\hat q}_b
                  -{\bf\hat q_{\ell^-}}\times{\bf\hat q}_{\bar b})
\end{eqnarray}
where $w$ is an arbitrary CP-even scalar weight function depending only
on the energies and momenta of the particles involved.
Finally one should keep in mind that appropriately chosen
momentum correlations can be used to investigate $t\bar t$ final state
interactions (more generally: unitarity corrections)
arising from CP--conserving (non) SM interactions --- for instance
the transverse polarisation of $t$ and $\bar t$ quarks due to
one--loop gluon contributions \cite{Kane}.
(In the work of ref. \cite{Nelson} the effect of final state interactions
on energy--energy correlations was investigated.)
A suitable observable for this purpose is the CP--even but T--odd
tensor \cite{Bern2}
\begin{eqnarray}
U_{ij}=({\bf q_+}+{\bf q_-})_i({\bf q_+}\times{\bf q_-})_j
       +(i\leftrightarrow j)
\label{Uij}
\end{eqnarray}
and the corresponding dimensionless one (analogous to $\hat T$).
By measuring the correlations (\ref{Tij}) and (\ref{Uij})
simultaneously one can disentangle possible CP--violating effects from
CP--even unitarity corrections. A more detailed discussion of the
observables (\ref{NE+E-}) --(\ref{Uij}) will be given elsewhere
\cite{Bern4}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Form factors $d_\gamma,d_Z$ in Higgs models
            of CP violation}
CP--violating effects in $t\bar t$ production and decay will
only be observable if there are additional sources of CP violation
besides the Kobayashi--Maskawa (KM) phase.
In extensions of the SM various extra CP--violating interactions
appear in a natural way. (In passing we recall that attempts
to explain the baryon asymmetry of the universe along the lines of
the Sakharov scenario seem to indicate the necessity for additional
CP--violating forces. For a recent investigation see \cite{Turoc}.)
Of particular interest are Higgs models of CP violation where effects
can grow drastically with the mass of the fermion. We consider here
only the simplest models: the two-Higgs doublet extensions of the SM.
The doublets $\Phi_1,\Phi_2$ are coupled to fermions
in such a way that no flavour--changing neutral Higgs couplings
appear at tree level. (We follow the convention $t_R\leftrightarrow
\Phi_2$. There are various possibilities to couple $\Phi_{1,2}$ to
$b_R$ and $\ell_R$.)
In these models there can be additional CP--violating interactions
--- apart from those due to the KM mechanism --- resulting from
CP--violating terms in the Higgs potential (see, e.g. \cite{Weinberg}).
If such terms are absent --- which is often assumed or imposed
by a discrete symmetry --- there are two CP=+1 and one CP=--1
physical neutral Higgs particles. If $V(\Phi_1,\Phi_2)$ contains
CP--violating terms then the CP=+1 and CP=--1 states mix
--- which is described by an orthogonal mixing matrix $d_{ij}$ ---
and the resulting mass eigenstates $\varphi_j$ no longer have a definite
CP parity. That is, they couple both to scalar and pseudoscalar
fermion currents.
In particular, the couplings of the $\varphi_j$ to the top quark
are
\begin{eqnarray}
{\cal L}_Y=-(\sqrt2 G_F)^{1/2}\sum_{j=1}^3
   [a_j m_t\bar tt+\tilde a_j m_t\bar ti\gamma_5t]\varphi_j
\label{Yukawa}
\end{eqnarray}
where
\begin{eqnarray}
a_j=d_{2j}/\sin\beta,\ \ \ \ \tilde a_j=-d_{3j}\cot\beta ,
\label{aj}
\end{eqnarray}
$\tan\beta=v_2/v_1$ is the ratio of the vacuum expectation values,
and $d_{2j},d_{3j}$ are the matrix elements of the mixing matrix
$d$. (The CP--violating parameters are $d_{31},d_{32}\neq0$.)
These Higgs couplings induce CP violation in flavour diagonal
amplitudes; in particular
they generate CP--violating respectively T--violating
dipole form factors for quarks and leptons. For the electron
and for light quarks the dominant effects
actually occur at two--loop order \cite{Barr}.
The size of these contributions to the electric dipole moments of the electron
and of the neutron is --- for a large range of parameters --- smaller than
the present experimental upper bounds. For heavy fermions the dominant
CP--violating effects appear at one--loop order. For $e^+e^-\to t\bar t$
with $m_e=0$ CP violation arises
from the amplitudes Fig.\ref{FCP3} which contain the form factors
$d_\gamma,d_Z$. One gets \cite{Bern5}
\begin{figure}\label{FCP3}
\vspace{10cm}
\caption{CP--violating contributions to $e^+e^-\to t\bar t$.}
\end{figure}
\begin{eqnarray}
d_\gamma(s)&=&\sum_{j=1}^3 d_\gamma^j(s) g_j ,\\
d_Z(s)&=&{3v_t\over4\sin\Theta_W\cos\Theta_W} d_\gamma(s)
       +\sum_{j=1}^3{d'}_Z^j(s) {g'}_j
\end{eqnarray}
where
\begin{eqnarray}
g_j&=&d_{2j}d_{3j}\cot\beta/\sin\beta,\\
{g'}_j&=&(\cos\beta d_{1j}+\sin\beta d_{2j})d_{3j}\cot\beta
\label{gj}
\end{eqnarray}
%
\begin{figure}\label{FCP4}
\vspace{10cm}
\caption{Rescaled real parts of the form factors $d_\gamma^j$
  and ${d'}_Z^j$ for various Higgs masses and $m_t=150$ GeV.}
\end{figure}
%
and $v_t=1/2-4\sin^2\Theta_W/3$. The rescaled real parts
of these form factors are plotted in Figs.\ref{FCP4}a,b for various
Higgs masses and for $m_t=150$ GeV. Fig.\ref{FCP4}a shows that in the
threshold region the contribution from the amplitude Fig.\ref{FCP3}a
can become quite sizeable: this is due to the Coulomb--like
singularity at threshold if $m_\varphi\to 0$. This behaviour
also shows up in the CP--conserving (SM) Higgs contribution to the $t\bar t$
production amplitude \cite{KG,BeHo}.
If the mass of the lightest Higgs particle, say $\varphi_1$,
is less than about 100 GeV and the other two neutrals are
considerably heavier then one gets for $\sqrt s\le 310$ GeV:
\begin{eqnarray}
|\hat d_\gamma|\ge 0.013 |g_1|, \ \ \ \ \hat d_Z\approx0.34 \hat d_\gamma.
\end{eqnarray}
This is about one order of magnitude below the sensitivity limits discussed
above. However, if $m_t=200$ GeV $\hat d_\gamma$ and $\hat d_Z$
would be enhanced by a factor of about
2.4. Moreover, if $\tan\beta<1$ then
$|g_1|>1$ is possible.
Finally we only mention that more exotic models are conceivable
which would yield larger effects. For instance in models with
an extra, heavier fermion generation and a richer Higgs sector
 (in particular with CP-violating mixing among charged Higgs
particles) the form factors $d_\gamma ,d_Z$ can be larger
and can have an energy dependence being
different from that of Fig.\ref{FCP4}a,b. \\
 
In conclusion, there are various observables ( momentum correlations,
energy asymmetries, etc.) with which experimentalists
will be able to check whether there is CP violation in
$t \bar t$ production and/or decay. Effects are observble only if
new interactions exist. In view of the sizeable Yukawa couplings
the $t \bar t$ system is
very well suited for detecting CP-violating interactions
of the Higgs type. The production of $t \bar t$
in unpolarized or transversely polarized $e^+ e^-$ collisions has
the advantage that there is a theoretically clean way to
discriminate between CP-violating effects and CP-even unitarity
corrections faking T violation. Our study shows that events
where both $t$ and $\bar t$ decay semileptonically are most
useful for measuring momentum correlations. The tensors
$T, \hat T$ and the asymmetry $A$ allow the determination
of CP-violating form factors of the top quark. The sensitivity
to these form factors is highest far above the $t \bar t$ threshold.
Nevertheless, investigation of $<T>$ and $<A>$ in the
threshold region is also of interest in view of the results from
2-Higgs models of CP violation.
\newpage
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