%Title: Spin effects in exclusive semileptonic tau decays 
%Author: J.H.Kuehn
%Published: <EM> Act. Phys. Pol. </EM> <B> B 29 </B> (1998) 1371-1383. 
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\begin{document}
\preprint{TTP98-12}

% \eqsec  % uncomment this line to get equations numbered by (sec.num)
\title{Spin effects in exclusive semileptonic $\tau$ decays
\thanks{Presented at the 1998 Cracow Epiphany Conference on Spin
Effects in Particle Physics, January $9-11$, Cracow, Poland.}
% you can use \\ to break lines
}
\author{J.H.K\"uhn}
\address{Institut f\"ur Theoretische Teilchenphysik\\
University of Karlsruhe\\
D--76128, Karlsruhe, Germany}
\maketitle
\begin{abstract}
Tau lepton production and decay to hadronic final states
offer the unique possibility for studying low energy hadron dynamics 
and the weak couplings of the $\tau$ at the same time.
Eventually one may even be sensitive to signals of physics 
beyond the Standard Model, like deviations from the
$V-A$ structure of the charged currents, or even discover CP violation.
Tau polarization and the polarization of the decay products may be
crucial for these investigations.\\  
{\it PACS: 11.30Er, 11.15.-y, 13.35Dx}
\end{abstract}

  
\section{Introduction}
Three topics will be covered in detail during this talk:

i) It will be demonstrated that the $\tau$ polarization can be
recovered on an event by event basis, if the momenta of all
decay products are measured and the relevant squared matrix element is
known from theory \cite{one}.

ii) The formalism of structure functions \cite{two} will be reviewed.
It allows to measure the $\tau$ polarization, at least its average
value, even if the $\tau$ neutrino momentum is unknown.
Structure functions help to separate the contributions 
of $0^+,0^-,1^+$ and $1^-$ states. They have been used to measure 
the sign of $g_V^{\tau}/g_A^{\tau}$ and may also allow for a
determiantion of $g_V^{q}/g_A^{q}$ in $\tau$ decays.

iii) The comparison of structure functions from the decays of 
$\tau ^+$ and $\tau ^-$  allows for nontrivial tests of CP symmetry 
\cite{three}, even with unpolarized beams. Final states with $K\pi$
and $K\pi\pi$ from Cabbibo suppressed decays are particulary 
promising candidates for such an analysis.


\newpage
\section{Tau polarimetry with multi--meson states}

The determination of the polarization of
$\tau$ leptons produced in $Z$
decays has lead to an important determination
of the $\tau$ coupling to
the $Z$ boson, rivaling those from the
forward-backward asymmetry
of leptons or from the left-right asymmetry
measured with longitudinally
polarized beams. It is well known that the
decay mode into a single
pion leads to optimal analyzing power which
expresses itself in an
angular distribution of pions from decays
of polarized $\tau$ 's of the form
\begin{equation}
dN\propto (1+ f \cos \theta)
\end{equation}
where $f=1$. The angle
between $\tau$ spin and pion momentum
is denoted by $\theta$.
Decays into the $\rho$ or $a_1$ meson or higher
excitations exhibit reduced analyzing power, e.g.
$f=(M^2  - 2 Q^2) / (M^2 + 2 Q^2)$
for a spin 1 final state of mass $Q$.
In \cite{KW,three}
it has been argued that significant analyzing
power can be recovered by
exploiting information encoded in the momenta
of the (pseudoscalar) mesons which are the actual decay
products and are observed in the experiment.
Detailed models have been used for the two-
and three-meson channels to
identify various angular distributions which
enhance the sensitivity.

In \cite {one} it was
demonstrated explicitly that maximal
sensitivity, corresponding to $f=1$,
can be recovered for {\it any}
multi-meson final state, once
the dynamics of the decay matrix element is known.
Ingredients are the knowledge of all meson momenta  and
information about the $\tau$ rest frame. The
latter is equivalent to reconstruction of the
actual direction of flight of the $\tau$ and can be
achieved in $e^+e^-$ experiments with the help of
vertex detectors \cite{impact}.

The argument is based on the observation
that the squared matrix element for semileptonic
$\tau$ can always be written (in the $\tau$
restframe) in the form
\begin{equation}
|{\cal M}|^2 \propto 1 - \vec h \vec s .
\end{equation}
The $\tau$-spin direction is denoted by $\vec s$
and the polarimeter vector $\vec h$
is a function of all meson momenta. It has
length $|\vec h|=1$ and its direction therefore
gives the (negative) spin direction of the original $\tau$
with unit probability. This holds true for meson
final states only --- the spin information is
strongly diluted for leptonic $\tau$ decays as a
consequence of averaging over the electron spin
and neutrino momenta corresponding to a reduction
of the length of $\vec h$.
It is,  however, retained in the direction of the $\bar\nu_e$.


In \cite{one} it has been  demonstrated
that the norm of $\vec h$ is in fact equal to
one for all semileptonic decays, which corresponds
to maximal sensitivity.

The matrix element for the semileptonic decay
$\tau\to\nu_\tau + X$ can be written in the form
\begin{equation}
{\cal M} =\frac{G}{\sqrt2}\bar u(N) \gamma^\mu (1-\gamma_5)u(P)J_\mu,
\end{equation}
where $J_\mu\equiv \langle X| V_\mu - A_\mu| 0\rangle$ denotes the
matrix element of the $V-A$ current relevant for the specific final
state $X$. The vector $J_\mu$ depends in general on the momenta of all
hadrons. The squared matrix element for the decay of a $\tau$ with
spin $s$ and mass $M$ then reads
\begin{equation}
|{\cal M}|^2 = G^2 (\omega + H_\mu s^\mu),
\end{equation}
with
\begin{equation}
\omega = P^\mu (\Pi_\mu + \Pi^5_\mu),~~~~~~~~~~~~~~
H_\mu = \frac{1}{M}
(M^2 g_\mu^{\phantom{\mu}\nu} - P_\mu P^\nu) (\Pi_\nu + \Pi^5_\nu)
\end{equation}
and
\begin{equation}
\Pi_\mu = 2\left [ (J^*\cdot N) J_\mu + (J\cdot N) J^*_\mu -
(J^*\cdot J) N_\mu\right], ~~~~
\Pi^5_\mu =2\, {\rm Im}\,
\epsilon_\mu^{\phantom{\mu}\nu\rho\sigma} J_\nu^* J_\rho N_\sigma .
\end{equation}

This formula was derived in \cite {KW} and
constitutes the basis for the simulation of spin
effects in TAUOLA \cite{tauola}.

In the
$\tau$-rest frame, the function $\omega$ coincides
with the time component of the four vector
$\Pi_\mu + \Pi^5_\mu$ (multiplied with $M$) and the vector $\vec
H$ with its space component (multiplied with $M$). The
assertion that $|\vec h| = |\vec H/\omega| =1$ is
therefore equivalent to the statement that $\Pi_\mu +\Pi^5_\mu$
is null-vector. A simple calculation demonstrates that
\begin{equation}
\Pi^5_\mu\Pi^\mu =0,~~~~~~~~~ \Pi^5_\mu \Pi^{5\mu}
= -\Pi_\mu \Pi^\mu
\end{equation}
and hence
\begin{equation}
(\Pi_\mu + \Pi_\mu^5) (\Pi^\mu + \Pi^{5\mu}) = 0
\end{equation}
which proves our assertion and gives, at the same time,
a prescription of how to construct the direction of the
spin on an event-by-event basis.
Obviously, in order to perform this analysis, all
hadron momenta must be measured and the dependence of the
current $J$ on these momenta known --- either from
theoretical considerations or from fits to
experimentally measured distributions.

\section{Structure functions}

The determination of form factors in exclusive decays is not only required
for a precise test of theoretical predictions. The
separation of vector and axial vector amplitudes and their
respective spin zero and one contributions is mandatory for a number 
of improved phenomenological studies like the determination of $\alpha_s$ 
and nonperturbative vacuum condensates based on moments of
vector and axial spectral functions separately, or the
unambigous separation of vector contributions 
(in particular in the $KK\pi$ channels) 
which allows in combination with CVC for an improved determination 
of $\alpha_{QED}(M_Z)$.
Just like the search for CP violation (discussed below) or the measurement
of $g_V^\tau/g_A^\tau$ through parity violation in hadronic decays
this can be performed with the help of a combined analysis of 
angular
and energy distributions of the hadrons even without reconstruction
of the  $\tau$-restframe.
A particulary useful tool to disentangle the various contributions,
to analyse the $\tau$ polarization and to test for new physics
is the technique of structure functions to be discussed in this
section \cite{two}.


The most general ansatz for the matrix element of the
quark current in the three (two) meson case
$
J^{\mu}(q_{1},q_{2}(,q_{3}))= \langle h_{1}(q_{1}) h_2(q_2) (h_{3}(q_3))
|V^{\mu}(0)-A^{\mu}(0)|0\rangle \> 
$
is characterized by four (two) complex form factors $F_i$, which 
are in general functions of
$s_1=(q_2+q_3)^2, s_2=(q_1+q_3)^2$, $s_3=(q_1+q_2)^2$
and $Q^2$ (which is conveniently chosen as an additional variable)
\begin{eqnarray}
J^{\mu}(q_1,q_2)&=& T^{\mu\nu}
\,(q_1-q_2)_{\nu}\,F     + Q^\mu\,F_S \label{f2m}\\
J^{\mu}(q_1,q_2,q_3)&=& T^{\mu\nu}
\left[ \,(q_1-q_3)_{\nu}\,F_1\,+\,(q_2-q_3)_{\nu}\,F_2\,\right]
+
\nonumber \\
&&\,\,i\,
\epsilon^{\mu\alpha\beta\gamma}q_{1\,\alpha}q_{2\,\beta}q_{3\,\gamma}\,F_3
+ Q^\mu\,F_4
\label{f3m}
\end{eqnarray}
$T^{\mu\nu}=  g^{\mu \nu} - (Q^\mu Q^\nu)/Q^2$ denotes a transverse
projector.
The form factors $F_{1}$ and $F_{2}$ in Eq.~(\ref{f3m}) originate from the 
$J^P=1^+$ axial vector hadronic current,
the form factor $F_3$ ($F$ in Eq.~(\ref{f2m})) 
from the $J^P=1^-$ vector current, 
and correspond to a hadronic system in a spin one state,
whereas $F_{4}$ ($F_S$)  is due to the spin zero ($J=0$) part  of the 
axial-vector (vector) current matrix element. 
In specific cases, there are various simplifications. If the two mesons 
in Eq.~(\ref{f2m}) are 
two pions, $h_1 h_2 = \pi^- \pi^0$, then the vector current is conserved
and the scalar form factor vanishes,
$F_S \equiv 0$ for $m_u = m_d$.
In the three pion case, $h_1 h_2 h_3 = \pi^- \pi^- \pi^+$ or
$\pi^0 \pi^0 \pi^-$, Bose symmetry 
relates $F_2$ to $F_1$, via $F_2(Q^2,s_1,s_2) = F_1(Q^2,s_2,s_1)$.
$G$ parity conservation requires $F_3 \equiv 0$ for $m_u = m_d$,
and PCAC requires $F_4 \equiv 0$ for $m_u= m_d = 0$.

The hadronic decay into three (two) mesons is  most easily 
analyzed in the hadronic rest frame
$\vec{q}_{1}+\vec{q}_{2}(+\vec{q}_{3})=~0$ \cite{two}.
In experimental analyses, 
the four (two) complex form factors in Eq.~(\ref{f3m}) (Eq.(\ref{f2m})) 
appear as sixteen (four) real
``structure functions'' $W_X$, which are defined from the hadronic tensor
$H^{\mu\nu}= J^\mu J^{\nu\star}$ in the hadronic rest frame.
For the precise definitions of  $W_X$, we refer the reader 
to Ref.~\cite{two}. The contributions of the different $F_i$ to $W_X$
are summarized in Tab.~\ref{tabsf}.
%-------------------------------------------------------------------------
\begin{table}
\caption{The structure functions} \label{tabsf}
$$
\begin{array}{|c|c|c|c|}
\hline %\hline
%
& \multicolumn{3}{|c|}{J^\mu \longrightarrow}\\
J^{\nu\star} \downarrow   & J^P = 1^+ & J^P = 1^- & J=0 \\
\hline \\[-3mm]
J^P = 1^+ & 
{\bf W_A}
& & \\
 &  W_C W_D W_E & & \\
\hline\\[-3mm]
J^P = 1^- & W_F W_G & {\bf W_B} &  \\
& W_H W_I & & \\
\hline\\[-3mm]
J = 0     & W_{SB}  W_{SC} 
& W_{SF} W_{SG} &
     {\bf W_{SA}}\\
 & W_{SD} W_{SG} & & \\
\hline \hline\\[-6mm]
& & \multicolumn{2}{|c|}{\underbrace{\hspace*{15ex}}_%
{\displaystyle h_1 h_2}}\\
\hline \\[-6mm]
& \multicolumn{3}{|c|}{\underbrace{\hspace*{30ex}}_%
{\displaystyle h_1 h_2 h_3}}\\
\hline
\end{array}
$$
\end{table}
%-------------------------------------------------------------------------
After integration over the unobserved neutrino direction,
the differential decay rate in the hadronic
rest frame  is given in the two-meson case  by \cite{two,kpi}
%
%
\begin{eqnarray}
d\Gamma(\tau^-\rightarrow 2h\nu_\tau)&=&
 \left\{
    \bar{L}_{B} {W}_{B} 
  + \bar{L}_{SA}{W}_{SA}
  + \bar{L}_{SF}{W}_{SF} 
  + \bar{L}_{SG}{W}_{SG}  \right\}\nonumber\\[3mm]
&&
\hspace{-2cm}
\frac{G_F^{2}}{2\mt} 
\sin^2\theta_{c}
\frac{1}{(4\pi)^{3}}
\frac{(\mt^{2}-Q^{2})^{2}}{\mt^{2}}\,
\,|\vec{q}_{1}|\,\,
     \frac{dQ^{2}}{\sqrt{Q^{2}}}  \, \frac{d\cos\theta}{2}\,
     \frac{d\alpha}{2\pi}\,
     \frac{d\cos\beta}{2}\label{gamma}\>.
%gamma1
\end{eqnarray}
%
%
The functions $\bar{L}_X$ depend on the angles 
$\alpha$ and $\beta$
and energy of the
hadronic system only.
For the definition and discussion of the angles and leptonic
coefficients $\bar{L}_X$ 
we refer the reader to Ref.~\cite{two,kpi}.
The hadronic structure functions $W_X$ in the two meson case
depend only on
$Q^2$ and the form factors $F$ and $\tilde{F}_S$ of the hadronic current
and are given in Eq.~(\ref{wsg}) with $\tilde{F}_S\rightarrow F_S$.  
Similarly, the differential decay rate in the
three meson case is given by \cite{two}:
\begin{equation}
   d \Gamma =
   \frac{G_F^2}{4 m_\tau} 
\bigl(^{\cos\theta_{c}^2}_{\sin\theta_{c}^2}\bigr)
   \sum_{X} \bar{L}_X W_X \; 
   \,dPS^{(4)}
 \label{g3m}
\end{equation}
The sum in Eq.~(\ref{g3m}) runs (in
general) over 16 hadronic structure functions $W_{X}$, which
correspond to 16 density 
matrix elements for a hadronic system in a spin one and spin zero state
(nine of them originate from a pure spin one state and the remaining
originate from a pure spin zero state or from interference terms between
spin zero and spin one, see table~\ref{tabsf}).  
These structure functions contain the dynamics of
the hadronic decay and depend only on the form factors $F_i$.
Note that $W_A$, $W_B$ and $W_{SA}$ alone determine $d\Gamma / dQ^2$ through
%
\begin{equation}
\frac{d\Gamma(\tau\to h_1 h_2 h_3 \nu_\tau)}{dQ^2} \propto
   \frac{(m_\tau^2 - Q^2)^2}{Q^4} 
 \times   \int ds_1\, ds_2\, 
  \left\{ (1 + \frac{Q^2}{m_\tau^2})  \frac{W_A + W_B}{6} +
  \frac{W_{SA}}{2} \right\}.
\end{equation}
%

%%%%%%%%%%%

(Almost) all structure functions can be determined 
by studying angular distributions of the hadronic system,
for details see \cite{two}.
This method allows to analyze separately
the contributions from $J^P=0^+,0^-,1^+$ and $1^-$ in a model independent
way (see table~\ref{tabsf}).

First steps towards such analysis have been undertaken by various LEP
experiments for the three pion final state. The measurements
\cite{OPAL}
are in fair agreement with the theoretical predictions
\cite{four} based on chiral Lagrangian providing the 
normalization at vanishing momentum transfer, and vector 
dominance to incorporate the proper resonance 
structure (Fig.\ref{OPAL}). 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\label{OPAL}
\leavevmode               
\epsfxsize=12.0cm
\epsffile[25 0 504 365]{strf.eps}
\caption[]{Measured hadronic structure functions $W_A$, $W_C$, $W_D$ and
$W_E$ (points) in comparison with the model predictions of the KS
model (dashed line) and IMR model (dotted line) 
(from Ref. \cite{OPAL}).}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The terms proportional to the structure function $W_E$ are induced by
parity violating couplings at the $\tau \nu W$ vertex. The analysis
leads, therefore, to a determination of
$\gamma _{VA} = 2g_\tau^V g_\tau^A/({g_\tau^V}^2 +{g_\tau^A}^2)$.

 
The result \cite{OPAL} $\gamma _{VA} = 1.29 \pm 0.26 \pm 0.11$
is consistent with the SM prediction $\gamma _{VA} = 1$
for lefthanded neutrinos. The model dependence of this
analysis can, in principle, be removed through the experimental
determination of the remaining functions $W_A$, $W_C$ and $W_D$.
Also, a potential scalar contribution to three pion channel
has been searched for and an upper limit of $0.84\;\%$
has been obtained by this method.

\section{CP violating in semileptonic $\tau$ decays with unpolarized beams}

\subsection{General considerations}
CP violation has been experimentally observed only in the
$K$ meson system. The effect can be explained by a nontrivial complex
phase in the CKM flavour mixing matrix. 
 However, the fundamental origin of this CP violation
is still unknown.
In particular the CP properties of the third fermion
family are largely unexplored.
Production and decay of $\tau$ leptons might offer a particularly clean
laboratory to study these effects.
In this Section, we investigate the effects of possible
non-Kobayashi-Maskawa-type of CP violation, {\it i.e.} CP violation
effects beyond the Standard Model (SM) on semileptonic $\tau$ decays. 
Such effects could originate for
example from multi Higgs boson models \cite{mhiggs}.

In \cite{three} it has been shown that the structure function
formalism  \cite{two} allows for a systematic analysis
of possible CP violation effects in the two and three
meson cases. Special emphasis is put on the $\Delta S=1$ transition
$\tau\rightarrow K\pi\nu_\tau$ where possible CP violating signals from
multi Higgs boson models would be signalled by a nonvanishing
difference between the structure functions
$W_{SF}[\tau^-\rightarrow (K\pi)^-\nu_\tau]$ and
$W_{SF}[\tau^+\rightarrow (K\pi)^+\nu_\tau]$.
Such a measurement is possible for unpolarized single $\tau$'s without
reconstruction of the $\tau$ rest frame and without
polarized incident $e^+e^-$ beams.
This difference is proportional to
Im(hadronic phases) $\times$ Im(CP-violating phases),
where the hadronic phases arise from the interference of complex 
Breit-Wigner propagators, whereas the CP violating phases could 
arise from an exotic charged Higgs boson.
An additional independent test of CP violation
in the two meson case
would require the knowledge
of the full kinematics and  $\tau$
polarization. 

The subsequent discussion  is organized as follows: 
CP violating terms in the Hamiltonian for 
$\tau$ decays are discussed in a first step. It is shown that
CP violating signals induced through the exchange of an exotic intermediate
vector boson can only arise, if both vector and axial vector
hadronic currents contribute to the same final state,
{\it i.e.} for final state which are not eigenstates of $G$ parity and 
involve three or more mesons.
The kinematics and the relevant form factors and structure functions 
for tests of CP violation in the two meson case are presented,
followed by a brief comment on
CP violation effects in three meson final states.

The Hamiltonian responsible for $\tau$ decays is decomposed into the
conventional term of the SM, 
denoted by $H_{SM}$, a CP violating term 
of similar structure, induced {\it e.g.} by the exchange of a vector boson, 
$H^{(1)}_{{CP}}$ and a CP violating term induced by scalar or pseudo
scalar exchange $H^{(0)}_{{CP}}$:
%
%
\begin{eqnarray}
H_{SM}&=&\cos\theta_c \frac{G}{\sqrt{2}}\,\,
[\bar{\nu}\,\,\gamma_\alpha(g_V-g_A\gamma_5)\,\,\tau]\,
[\bar{d}\,\,\gamma^\alpha(1-\gamma_5)\,\,u]
+h.c.\nonumber\\
H_{{CP}}^{(1)}&=&\cos\theta_c \frac{G}{\sqrt{2}}\,\,
[\bar{\nu}\,\,\gamma_\alpha(g_V^\prime-g_A^\prime\gamma_5)\,\,\tau]\,
[\bar{d}\,\,\gamma^\alpha(\chi_V^d-\chi_A^d\gamma_5)\,\,u]
+ h.c.\label{hamilton}\\
H_{{CP}}^{(0)}&=&\cos\theta_c \frac{G}{\sqrt{2}}\,\,
[\bar{\nu}(g_S+g_P\gamma_5)\,\,\tau] \,
[\bar{d}\,\,(\eta_S^d+\eta_P^d\gamma_5\,)\,u]\nonumber
+ h.c.
\end{eqnarray}
%
%
plus a similar term with the complex parameters
$\eta^d,\chi^d$ replaced by $\eta^s,\chi^s$ for the
$\Delta S=1$ contribution.

The dominance of $(V-A)$ contributions to the leptonic current
in leptonic and semileptonic $\tau$-decays has been
demonstrated experimentally under fairly mild theoretical assumptions.
A pure $(V-A)$ structure of the leptonic current in
$H_{SM}$ will therefore be 
adopted for simplicity. A slight deviation from $(V-A)$ for the
hadronic current can in principle be masked by the form factors.
However, tight restrictions can be derived from the ratio
$\Gamma(\tau\rightarrow\nu\pi)/\Gamma(\tau\rightarrow\nu\pi\pi)$
using as input $f_\pi$ and the pion form factor from $e^+e^-$ 
annihilation\footnote{The relative sign
between hadronic vector and axial vector current is not fixed through this
consideration. It can be determined by interference measurements in the 
$K\pi\pi$ and $KK\pi$ channels, {\it e.g.} of the structure functions
$W_{F,G,H,I}$ \cite{two}.}.
The study of CP violation will entirely
rely on interference terms between $H_{SM}$ and
$H_{{CP}}^{(1)},H_{{CP}}^{(0)}$. Interference
terms between the dominant $(V-A)$ leptonic current in $H_{SM}$
and a possible $V+A$ term in the leptonic current of
$H_{{CP}}^{(0,1)}$ are suppressed with the ratio between the 
mass of the $\tau$ neutrino and $m_\tau$. Therefore
only contributions from left--handed neutrinos will be included,
{\it i.e.} $g_V=g_A=g^\prime_V=g^\prime_A=g_S=-g_P=1$.

The hadronic matrix elements of the currents 
$\bar{d}\gamma^\alpha u$ (and similarly $\bar{d}\gamma^\alpha \gamma_5 u$)  
in $H_{SM}$ and 
$H_{{CP}}^{(1)}$ are of course identical. 
The spin zero part is closely related to the corresponding matrix element
of the scalar current in $H_{{CP}}^{(0)}$
through  the equation of motion:
%
%
\begin{equation}
Q^\alpha\,\,\bar{d}\gamma_\alpha u \,=\,(m_u-m_d)\,\bar{d}u;
\hspace{1cm}
-Q^\alpha\,\,\bar{d}\gamma_\alpha \gamma_5 u \,=\,
(m_u+m_d)\,\bar{d}\gamma_5u
\label{motion}
\end{equation}
%
%
with $Q^\alpha=i\partial^\alpha$.
The Hamiltonian in Eq.~(\ref{hamilton})
can thus be written in the form
%
%
\begin{eqnarray}
H &=&\cos\theta_c\frac{G}{\sqrt{2}}\,\,
\left[\bar{\nu}\gamma_\alpha(1-\gamma_5)\tau\right]\,
\left\{\left(
       (1+\chi_V^d)g^{\alpha\beta}+\frac{Q^\alpha Q^\beta}{m_\tau(m_u-m_d)}\,
       \eta_S^d\right)\,\,\,
       \bar{d}\gamma_\beta u\right. 
      \nonumber \\
  &&- \,\left.\left(
       (1+\chi_A^d)g^{\alpha\beta}+\frac{Q^\alpha Q^\beta}{m_\tau(m_u+m_d)}
       \eta_P^d\right)\,\,\,
       \bar{d}\,\gamma_\beta \gamma_5 \,u \right\}
       + h.c.
        \label{hamilt}
\end{eqnarray}
%
%
and similarly for the Cabibbo suppressed mode.
From this form it is evident that $\chi$
and $\eta$ play a fairly different role.
Effects from $\chi_V$ and/or $\chi_A$ can only arise if both vector
and axial  hadronic currents contribute to the same final state
and hence only for final states which are not eigenstates of $G$
parity and involve three or more mesons.
Conversely,
for all two meson decays and, adopting isospin symmetry, even all multipion
states, CP violation cannot\footnote{In this aspect we disagree
with Ref.~\cite{tsaicp} where it has been claimed that CP 
violation in the $2\pi$  channel can be induced through 
the exchange of an exotic intermediate vector boson.}
arise from a complex $\chi$.
In contrast,  CP violation can arise from a complex
$\eta$, since $J=0$ and $J=1$ partial
waves are affected differently in this case.
For this reason contributions from nonvanishing $\chi$ will be ignored
in the following.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Two Meson Decays: 
 Kinematics,  Form Factors and Structure Functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
Transitions from the vacuum to two pseudoscalar mesons $h_1$ and
$h_2$ 
are induced through vector and scalar currents only, where the latter can 
be related to the former with the help of Eq.~(\ref{motion}).
Expanding this hadronic matrix element along the set of independent momenta
$(q_1-q_2)_\beta$ and $Q_\beta=(q_1+q_2)_\beta$
(see Eq.(\ref {f2m}))
%
%
%\begin{equation}
%J_\beta=\langle h_1(q_1)h_2(q_2)|\bar{u}\gamma_\beta d|0 \rangle=
%(q_1-q_2)^\delta\,T_{\delta\beta}\,\,F(Q^2) \,+\,
%Q_\beta\, F_S(Q^2)
%\label{jdef}
%\end{equation}
%
%
the general amplitude for the strangeness conserving decay 
%
%
\begin{equation}
\tau^-(l,s) \rightarrow \nu(l^\prime,s^\prime) + h_1(q_1,m_1)+h_2(q_2,m_2)\>,
\label{twoh}
\end{equation}
%
%
can be written as\footnote{We suppress the superscript $d$ (or $s$)
in $\eta_S$ in the following.}
%
%
\begin{eqnarray}
{\cal{M}} &=&
  \cos\theta_{c}\frac{\,G}{\sqrt{2}}\,
  \bar{u}(l^\prime,s^\prime)\gamma_{\alpha}
           (1-\gamma_5)u(l,s)
  \left(g^{\alpha\beta}
  +\frac{Q^\alpha Q^\beta}{m_\tau(m_u-m_d)}\,\eta_S\right)
\nonumber \\
 && \times \left[(q_1-q_2)^\delta\,T_{\delta\beta}\, F 
   + 
  Q_\beta\,F_S\right]
\nonumber \\
&=&
\cos\theta_{c}\frac{\,G}{\sqrt{2}}\,
  \bar{u}(l^\prime,s^\prime)\,\,\,\,\gamma_{\alpha}
           (1-\gamma_5)\,u(l,s) \,\,
  \left[\,\,(q_1-q_2)_\beta\,T^{\alpha\beta}\,\,\,  F   \, + \,
  Q^\alpha\,\,\,\tilde{F}_S\right]\hspace{2cm}
\label{me2h}
\end{eqnarray}
with
\begin{equation}
\tilde{F}_S = \left(1+\frac{Q^2}{m_\tau(m_u-m_d)}\,\eta_S\right)\,\,F_S
\label{fs}
\end{equation}
and similarly for the $\Delta{S}=1$ part.
In Eq.~(\ref{me2h}) $s$ 
denotes the polarization 4-vector of the $\tau$ lepton
satisfying 
$
l_{\mu}s^{\mu}=0
$ and 
$
s_{\mu}s^{\mu}= -P^{2}.
$
$P$ denotes the polarization  of the $\tau$
in the $\tau$ rest frame with respect to its direction of flight 
in the laboratory frame. 
%and
%$T^{\alpha\beta}$ is the  projector onto the spin one part
%
%
%\begin{equation}
%T^{\alpha\beta}=  g^{\alpha \beta} - \frac{Q^\alpha Q^\beta}{Q^2}  \>.
%\end{equation}
%
%
As stated before, terms proportioal to $\chi$ do not contribute to CP
violation in the two meson case and have therefore been neglegted.

%{\it CP in $\tau\rightarrow K\nu$ Zerfall?\\
%     Impact on $K\pi$?\\
%     if different phases for $S$ and $P$ waves!
%     watch out for trivial effects?!}

In the subsequent discussion we will not need to rely on the
relation (\ref {fs})
between vector and scalar induced formfactors
and thus define
\begin{equation}
\tilde F_s = F_s + \frac {\eta _s}{m_\tau} F_H.
\end{equation}

The representation of the hadronic amplitude
$\langle h_1h_2|\bar{u}\gamma_\beta d|0\rangle = (q_1-q_2)^\delta
\,T_{\delta\beta}\,\,  F   \, + \, Q_\beta\, \tilde{F}_S$
corresponds to a decomposition into spin one and spin
zero contributions,
{\it e.g.} the vector form factor $F(Q^2)$
corresponds to the $J^P = 1^-$ component
of the  weak charged current, and the scalar form factor 
$F_S(Q^2)$ to the $J^P = 0^+$ component. 
Up to the small isospin breaking terms, induced for example
by the small quark mass difference, CVC implies the
vanishing of $F_S$ for the two pion case. The small
$u$ and $d$ quark masses enter presumably the couplings from 
charged Higgs exchange and thus cancel the apparent enhancement
by the inverse power of $(m_u-m_d)$ in 
Eqs.~(\ref{hamilt},\ref{me2h},\ref{fs}).
The perspectives are more promising for the $\Delta{S}=1$ 
transition $\tau\rightarrow K\pi\nu$. The $J=1$ form factor $F$
is dominated by the $K^*(892)$ vector resonance contribution. 
However, in this case the scalar form factor $F_S$ is expected to
receive a sizable resonance contribution 
($\sim 5\%$ to the decay rate)
from the $K_0^*(1430)$ with $J^P=0^+$
\cite{kpi}.
In the subsequent discussion we will include both $\pi\pi$ and $K\pi$ 
final states. The corresponding $\tau^+$ decay is obtained from 
Eq.~(\ref{me2h})
through the substitutions 
%
%
\begin{equation}
(1-\gamma_5)\, \rightarrow \, (1+\gamma_5), \hspace{1cm}
%\chi      \, \rightarrow \, \chi^*,     \hspace{1cm}
  \eta_S      \, \rightarrow \, \eta_S^*.     \hspace{1cm}
%\eta_S^d      \, \rightarrow \, \eta_S^{d *}.     \hspace{1cm}
\end{equation}
%
%

Reaction (\ref{twoh}) is most easily analyzed in the hadronic rest frame
$\vec{q}_{1}+\vec{q}_{2}=~0$.
After integration over the unobserved
neutrino direction,
the differential decay rate in the
rest frame of $h_{1}+h_{2}$ is given by Eq.(\ref {gamma})
\cite{two,kpi}
%
%
%\begin{eqnarray}
%&&d\Gamma(\tau^-\rightarrow 2h\nu_\tau)=
% \left\{
%    \bar{L}_{B} {W}_{B} 
%%  + \bar{L}_{SA}{W}_{SA}(1+\tilde\eta_S^d)^2
%  + \bar{L}_{SA}{W}_{SA}
%  + \bar{L}_{SF}{W}_{SF} 
%%  + \bar{L}_{SG}{W}_{SG})(1+\tilde\eta_S^d)  \right\}\nonumber\\[2mm]
%  + \bar{L}_{SG}{W}_{SG}  \right\}\nonumber\\[3mm]
%&&\times 
%\frac{G^{2}}{2\mt} 
%\left (^{\cos^2\theta_{c}}_{\sin^2\theta_{c}}\right ) % \cos^{2}\theta_{c}\, 
%\frac{1}{(4\pi)^{3}}
%\frac{(\mt^{2}-Q^{2})^{2}}{\mt^{2}}\,
%\,|\vec{q}_{1}|\,\,
%     \frac{dQ^{2}}{\sqrt{Q^{2}}}  \, \frac{d\cos\theta}{2}\,
%     \frac{d\alpha}{2\pi}\,
%     \frac{d\cos\beta}{2}\label{gamma}\>.
%\end{eqnarray}
%
%
%The coefficients $\bar{L}_X$ contain all  $\alpha, \beta$ and $\theta$
%angular  and $\tau$-polarization dependence and 
%can be found in \cite{two,kpi}. The angles $\alpha$ and $\beta$
%are specified in Fig.\ref{fig1}. 
%
%
The hadronic structure functions $W_X$,
$X\in\{B,SA,SF,SG\}$, 
which appear in Eq.(\ref{gamma})
depend only on
$Q^2$ and the form factors $F$ and $\tilde{F}_S$ of the hadronic current.
The dependence   can be obtained from
Eq.~(34) in \cite{two} with the replacements
$
x_4  \rightarrow 2 \,\vec{q}_1 \>, 
%F_3 \rightarrow -iF\,(1+\chi) \>,
F_3  \rightarrow -iF\, \>,
F_4  \rightarrow   \tilde{F}_S
$.
One has:
%
%
\begin{eqnarray}
W_B[\tau^-]    &=& 4 (\vec{q}_1)^2\,|F|^2 \\
W_{SA}[\tau^-] &=& Q^2\,  |\tilde{F}_S|^2  \label{wsa}         \\
W_{SF}[\tau^-]
 &=& 4\sqrt{Q^2}|\vec{q}_1|\, \mbox{Re}\left[F\tilde{F}_S^*\right]\\
W_{SG}[\tau^-]
 &=&-4\sqrt{Q^2}|\vec{q}_1|\, \mbox{Im}\left[F\tilde{F}_S^*\right]
\label{wsg}
\end{eqnarray}
%
%
where
$|\vec{q}_1|=q_1^z$ is the momentum of $h_1$ in
the rest frame of the hadronic system:
%
%
\begin{equation}
\vec{q}_{1}^{\,\,z}=\frac{1}{2\sqrt{Q^{2}}}\left(
[Q^{2}-m_1^{2}-m_2^{2}]^2-4m_{1}^{2}m_{2}^{2}\right)^{1/2}.
%\quad E^2_1=(\vec{q}_1)^2+m^2_1  \>.
\end{equation}
%
%
The hadronic structure functions $W_X[\tau^+]$  are
obtained by the replacement $\eta_S\rightarrow\eta_S^*$
in $\tilde{F}_S$ in Eqs.~(\ref{wsa}-\ref{wsg},\ref{fs}).
CP conservation implies that all four structure functions are identical
for $\tau^+$ and $\tau^-$. With the ansatz 
for the form factors formulated in Eq.~(\ref{me2h})
CP violation can be present in $W_{SF}$ and $W_{SG}$
only and requires complex $\eta_S$.
As will be shown in the subsequent discussion
CP violation in $W_{SG}$ is maximal for fixed $\eta_S$ in the
absence of hadronic phases whereas $W_{SF}$ in contrast requires
complex $\eta_S$ and hadronic phases simultaneously.

In \cite{three}  it has been demonstrated that $W_{SF}$ can be measured in 
$e^+e^-$ annihilation experiments in the study of single unpolarized
$\tau$ decays even if the $\tau$ rest frame cannot be reconstructed.
In this respect the result differ from earlier studies of the two 
meson modes where either polarized beams and reconstruction
of the full kinematics \cite{tsaicp} or correlated fully reconstructed
$\tau^-$ and $\tau^+$ decays were required \cite{nelson1}.
The determination of $W_{SG}$, however, requires 
$\tau$  polarization and
the knowledge of the full
$\tau$ kinematics.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\setlength{\unitlength}{0.7mm}
%\begin{figure}[t]              
%\leavevmode 
%\epsfxsize =8.0cm
%\epsffile[0 0 447 461]{albet.eps}
%\caption{Definition of the angles $\alpha$ and $\beta$ in two meson
%decays.
%}
%\label{fig1}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The differential rate (Eq.~(\ref{gamma})) for the
CP conjugated process
can be obtained from the previous results by
reversing all momenta $\vec{p}\rightarrow -\vec{p}$, 
the $\tau$ spin vector $\vec{s}\rightarrow -\vec{s}$,
the polarization  $P\rightarrow -P$, and the transition
$\gamma_{VA}\rightarrow-\gamma_{VA}$.
CP therefore  relates the differential decay rates
for $\tau^+$ and $\tau^-$ as follows:
%
%
\begin{equation}
d\Gamma[\tau^-](\gamma_{VA},P,W_X[\tau^-])
\rightarrow
d\Gamma[\tau^+](-\gamma_{VA},-P,W_X[\tau^+])
\end{equation}
%
%
Note that the coefficients $\bar{L}_X$ contain the full
$\gamma_{VA}$ and $P$ dependence.
From the  interference between the spin-zero spin-one terms
(denoted by the subscript $X=SF,SG$)
one can construct 
the following CP-violating quantities:
%
%
\begin{eqnarray}
\Delta_{X} &=& \frac{1}{2}
\left(
\bar{L}_{X}(\gamma_{VA},P)\, W_{X}[\tau^-]
-
\bar{L}_{X}(-\gamma_{VA},-P)\, W_{X}[\tau^+]
 \right)\\
&=&\bar{L}_{X}(\gamma_{VA},P)\,\frac{1}{2}
\left( W_{X}[\tau^-] -  W_{X}[\tau^+]\right)\\
&\equiv& \bar{L}_X\Delta W_X
\end{eqnarray}
%
% 
As mentioned before the hadronic structure functions
$W_X[\tau^-]$ and $W_X[\tau^+]$ differ only in the complex parameter
$\eta_S$ in 
%
%
\begin{equation}
\tilde{F}_S[\tau^-]
 = F_s+\frac {\eta_s}{m_\tau} F_H
%\left(1+\frac{Q^2}{m_\tau(m_u-m_d)}\,\eta_S \right)\,\,F_S
\end{equation}
and
\begin{equation}
\tilde{F}_S[\tau^+]
 =  F_s+\frac {\eta_s^*}{m_\tau} F_H
%\left(1+\frac{Q^2}{m_\tau(m_u-m_d)}\,\eta_S^\ast \right)\,\,F_S
\end{equation}
and one obtains for the
only nonvanishing spin-zero spin-one interference term 
$L_{SF}W_{SF}$  
%
%
\begin{equation}
\Delta W_{SF}
\,\,= \,\,\, 4\sqrt{Q^2}|\vec{q}_1|\, \,
\frac{1}{m_{\tau}}\,\,\,\mbox{Im}\left(FF_H^*\right)\,\,
              \,     \mbox{Im}\left(\,\eta_S \right) \>.
\end{equation}
%
% 
In essence this measurement analyses the difference in the
correlated energy distribution
of the mesons $h_1$ and $h_2$ from $\tau^+$ and $\tau^-$
decay in the laboratory. 
As already mentioned, $\Delta W_{SF}$ is observable for single
$\tau^+$ and $\tau^-$ decays without knowledge of the
$\tau$ rest frame. Any nonvanishing experimental result for 
$\Delta W_{SF}$ would be a clear signal of CP violation.
Note that a nonvanishing $\Delta W_{SF}$ requires nontrivial hadronic phases
(in addition to the CP violating phases $\eta_S$) in the form factors
$F$ and $F_S$. Such hadronic phases in $F$ ($F_S$) originate
in the $K\pi\nu_\tau$ decay mode from complex Breit Wigner
propagators for the $K^*$ ($K_0^*$) resonance. Sizable effects
of these hadronic phases are expected in this decay mode \cite{kpi}.

Once the $\tau$ rest frame is known and a preferred direction
of polarization exists
both angles $\alpha$ and $\beta$ are well defined and
one may proceed further, determine
with the help of $\sin \alpha$
also $W_{SG}$
and thus perform a second independent test for CP violation.


The differential rate  for the
CP conjugated process
can be obtained 
as before, with the additional substitution $\sin \alpha \to -\sin \alpha$.
%by
%reversing all momenta $\vec{p}\rightarrow -\vec{p}$, 
%the $\tau$ spin vector $ \vec{s}\rightarrow -\vec{s}$,
%the polarization $P\rightarrow -P$, $\sin\alpha\rightarrow - \sin\alpha$
%and the transition
%$\gamma_{VA}\rightarrow-\gamma_{VA}$.
%CP therefore  relates the differential decay rates
%for $\tau^+$ and $\tau^-$ as:
%
%
\begin{equation}
d\Gamma[\tau^-](\sin\alpha,\gamma_{VA},P,W_X[\tau^-])
\rightarrow
d\Gamma[\tau^+](-\sin\alpha,-\gamma_{VA},-P,W_X[\tau^+])
\end{equation}
%
%
From the  interference between the spin-zero spin-one terms
one can construct 
the following CP-violating quantities:
%
%
\begin{eqnarray}
\Delta_{X} &=& \frac{1}{2}
\left(
\bar{L}_{X}(\sin\alpha,\gamma_{VA},P)\, W_{X}[\tau^-]
-
\bar{L}_{X}(-\sin\alpha,-\gamma_{VA},-P)\, W_{X}[\tau^+]
 \right) \nonumber \\
&=& \bar{L}_{X}(\sin\alpha,\gamma_{VA},P)\,\frac{1}{2}
\left( W_{X}[\tau^-] -  W_{X}[\tau^+]\right)\\
&\equiv& \bar{L}_X\Delta W_X
\end{eqnarray}
%
% 
\begin{equation}
\Delta W_{SF}
\,\,= \,\,\, 4\sqrt{Q^2}|\vec{q}_1|\,\,\,
   \frac{1}{m_{\tau}}\,\,\, \mbox{Im}\left(FF_H^*\right)
  \,\,    \mbox{Im}\left( \eta_S \right) \>.
\label{dwsf}
\end{equation}
%
% 
\begin{equation}
\Delta W_{SG}
\,\,= \,\,\, 4\sqrt{Q^2}|\vec{q}_1|\,\,\,
\frac{1}{m_{\tau}}
\,\,\, \mbox{Re}\left(FF_H^*\right)
              \,\,     \mbox{Im}\left(\eta_S \right) \>.
\label{dwsg}
\end{equation}
%
% 
Any observed nonzero value of these quantities would again signal a 
true CP violation. Eqs.(\ref{dwsf}) and (\ref{dwsg}) show that the sensitivity
to CP violating effects in $\Delta W_{SF}$ and $\Delta W_{SG}$
can be fairly different depending on the hadronic phases.
Whereas $\Delta W_{SF}$ requires nontrivial hadronic phases
$\Delta W_{SG}$ is maximal for fixed $\eta_S$ in the absence of hadronic
phases.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Three  Meson Decays}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The structure function formalism \cite{two}
allow also for a systematic analysis of possible CP
violation effects in the three meson case.
Some of these effects have already been briefly discussed in
\cite{argonne}. The $K\pi\pi$ and $KK\pi$ decay modes
with nonvanishing vector \underline{and} axial vector
current are of particular importance for the detection of possible
CP violation originating from exotic intermediate vector bosons.
This would be signalled by a nonvanishing difference between
the structure functions $W_X(\tau^-)$ and $W_X(\tau^+)$
with $X\in\{F,G,H,I\}$. A difference in the structure functions
with $X\in\{SB,SC,SD,SE,SF,SG\}$ can again be induced through
a CP violating scalar exchange.


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\end{thebibliography}

\end{document}





