%Title: Large Tau Decays into Four Pions
%Authors: R. Decker, M. Finkemeier, P. Heiliger and H. H. Jonsson
%Published: <EM> Z. Phys. </EM> <B> C70 </B> (1996) 247-254. 
%hep-ph/9410260
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\begin{document}
%
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\makebox[14cm][r]{TTP 94--13} \par
\makebox[14cm][r]{September 1994}\par
\vspace{.7cm}
\centerline{\Large Tau Decays into Four Pions}
\par
\vspace{1.cm}
\centerline{R. Decker, M. Finkemeier\footnote{%
Now at: INFN - Laboratori Nazionali di Frascati, Gruppo Teorico,
P.O. Box 13, I-00044
Frascati, Italy%
}, P. Heiliger and H. H. Jonsson}
\par 
\centerline{Institut f\"ur Theoretische Teilchenphysik} \par
\centerline{Universit\"at Karlsruhe}\par
\centerline{D--76128 Karlsruhe, Germany}\par
\normalsize
\vspace{2.cm}

\begin{abstract}
We compute branching ratios and invariant mass distributions of the
tau decays into four pions. The hadronic matrix elements are 
obtained by starting from the structure of the hadronic current in 
chiral limit and then implementing low-lying resonances in the 
different channels with slowly varying coupling constants. Reasonable
agreement with experiment is obtained both for the $\tau \to 
\nu_{\tau} + (4 \pi \!) $ decay rates and the $e^+e^- \to (4 \pi \!)
$  cross sections. We derive a new prediction for the decay rate 
of $\tau^-\to\nu_\tau 2\pi^- \pi^+ \pi^0 $ which is significantly 
lower than a previous one. Furthermore we supply an interface to use
our matrix elements within the Tauola Monte-Carlo program.
\end{abstract}

\newpage
\noindent
\section{Introduction}
The last two years have provided us with a vast of new data on the 
physics of the $\tau $ lepton. (See \cite{schw} for a review of the
most recent experimental results.) The overall branching ratio 
deficit that has been a problem for many years is getting smaller
\cite{schw} and considerable improvement in the detection of small
exclusive branching ratios involving $\pi^0 $'s and kaons has been 
achieved. In general agreement between theory and experiment is
satisfactory. \par
\bigskip
In this paper we concentrate on a specific hadronic final state, 
namely the $\tau $ decays into four pions, $\tau^- \to \nu_{\tau}
\pi^- \pi^- \pi^+ \pi^0 $ and $\tau^- \to \nu_{\tau} \pi^0 \pi^0
\pi^0 \pi^- $. Of course these decays are related through CVC to 
corresponding $e^+ e^- $ data. The predictions \cite{tsai,gil} from
$e^+ e^- $ data for the total branching ratio are in good agreement
with the $\tau $--data \cite{arg,cleo}. 

However, the prediction for the integrated rate for $ \tau^- \to 
\nu_{\tau} \pi^- \pi^- \pi^+ \pi^0 $ from the $e^+e^-$ cross sections
carries a rather large uncertainty due to the rather poorly known 
$e^+e^- \to \pi^+\pi^- 2\pi^0$ cross section. What is more important,
this ansatz allows only to calculate the integrated rates.  
In this paper we will obtain a parametrization of the hadronic matrix
element which allows us to describe differential decay rates.
Furthermore, we are able to fix the parameters of our model without
having to use the $e^+e^- \to \pi^+\pi^- 2\pi^0$ cross sections.
There have been several similar attempts in the past, let us quote 
a typical example \cite{pham}. \par
In our discussion we follow along the lines set in the 
phenomenological description for $\tau $ decays into three
pseudoscalars \cite{rog}. Lorentz--invariance dictates the number of
relevant formfactors. Then the resonances dominating these form 
factors are identified (in our case namely the $\rho$, $\rho'$, 
$\rho''$, $a_1$ and $\omega $ mesons) and an expression for the 
form factors is constructed. Furthermore the expressions are 
restricted by assuming that only on mass-shell vertices 
contribute and that different vertices are related by Breit--Wigner 
propagators. In the case of the $\tau$ decays into three 
pseudoscalars, the coupling constants at the vertices were assumed 
to be constant, and the products of the coupling constants were 
fixed by the requirement that the formfactors have the correct 
chiral limit \cite{wess}. \par
In the case of the four pion final states, we are
forced to assume that these products are slowly varying functions of
the momentum, because otherwise we cannot reproduce the experimental
data.

An expression constructed from the chiral limit and including $\rho 
$ resonances in all possible ways had been constructed and included 
in TAUOLA \cite{tauo}. Unfortunately it yields a branching ratio off
by about a factor of 2 to 4. In this paper we improve this result by
including the $3 \pi $ resonance, namely the $a_1 $ meson, and 
obtain reasonable agreement with experiment. \par
\bigskip
In section 2 we discuss the general structure of the $\tau $ decay
into four pions and we implement the low--lying resonances according
to the different $4 \pi $ final states into the hadronic matrix 
element. We compare the model predictions with a model--independent 
determination of the relative probabilities of the channels 
corresponding to the alternatives of charged and neutral $\pi $'s 
\cite{pais}. In section 3 we discuss the two possible $4 \pi $ final
states in $e^+ e^- $ annihilation and the relationship of these
processes to the $\tau $ decays into four pions. We discuss the 
numerical results as well as the invariant mass--distributions in 
the various $2 \pi $--subsystems in comparison to experiment in 
section 4. 



\section{The Four Pion Decay Mode of the Tau}
There exist two $4 \pi $ final states in $\tau^- $ decay:
\begin{itemize}
\item[i)] $\tau^- (P) \to \nu (l) \pi^0 (p_1) \pi^0 (p_2) \pi^0 (p_3)
           \pi^- (p_4) $
\item[ii)] $\tau^- (P) \to \nu (l) \pi^- (p_1) \pi^- (p_2) \pi^0 
           (p_3) \pi^+ (p_4) $
\end{itemize}  
In the standard model \cite{sm} the following effective lagrangian is
responsible for the semileptonic decays of the $\tau $ lepton 
\cite{rog}
\begin{equation}
L_{eff} = {G_F U^{ij} \over \sqrt{2}} {\overline u}(l') \gamma_{\mu }
(1 - \gamma_5) u(l) H_{ij}^{\mu }
\end{equation}
with $G_F $ the Fermi constant, $H_{ij}^{\mu } $ and $U^{ij} $ are 
the charged quark current and the quark mixing matrix respectively.
In the following we present a model for the matrix element of the
quark current, which reveals itself as being in agreement with 
general properties of a system of multi--pions. These considerations
are consequences of a classification of various pion--subsystems
according to isospin \cite{pais}. In our model the charged current
consists of three parts which we explain in detail below.\par
\smallskip
Our model is based on an amplitude derived \cite{wess} in the chiral 
limit, supplemented by vector dominance to correctly describe the
resonances in the $4 \pi $ mass distributions and a slow $Q^2$
dependence of the form factors. In this frame it is quite natural to
calculate $\tau \to 4 \pi \nu_{\tau} $ via sequential processes like
$\tau \to \nu_{\tau} \rho, \; \rho \to a_1 \pi, \; a_1 \to \rho \pi,
\; \rho \to \pi \pi $ and $\tau \to \nu_{\tau} \rho, \; \rho \to 
(\pi \pi)_{s-wave} \rho, \; \rho \to \pi \pi $. To construct the 
covariant amplitudes corresponding to Fig. 1 we need the appropriate
vertices and propagators. Following the successful reasoning in a 
similar analyses concerning tau decay into three pseudoscalar mesons
\cite{rog} we assume the vertex functions to be given by their on 
shell structure and to be transverse, e.g.
\begin{equation}
p_{a_1}^{\nu} \Gamma_{\nu \sigma} (a_1 \to \rho \pi ) = 
p_{\rho}^{\sigma} \Gamma_{\nu \sigma} (a_1 \to \rho \pi) =
p_{\rho}^{\tau} \Gamma_{\tau} (\rho \to \pi \pi) = 0
\end{equation}
With the pions all on mass shell, the vertex factors may thus be
written \cite{isg}
\begin{eqnarray}
\Gamma^{\mu \nu} \left (a_1(q) \to \rho(k) \pi(p) \right ) & = &
f_{a_1 \rho \pi} (q^2,k^2) \left ( - g^{\mu \nu} + {k^{\mu} k^{\nu}
\over k^2} + {q^{\mu} q^{\nu} \over q^2} - {k \cdot q \over k^2 q^2}
q^{\mu} k^{\nu} \right ) \nonumber \\
\Gamma^{\nu} \left ( \rho (k) \to \pi_1 (p_1) \pi_2 (p_2) \right )
& = & -i f_{\rho \pi \pi} (k^2) (p_1^{\nu} - p_2^{\nu })
\end{eqnarray}
where the formfactors are real and assumed to be only slowly
varying with the momentum.  
In the first case, we have neglected a further possible 
formfactor which is of higher order in momenta. \par
\noindent
The resonances are in general described by normalized Breit--Wigner 
resonance propagators $BW_R(Q^2=s) $ with $BW(0) = 1 $ and 
\begin{equation}
BW_R(s) = {-M_R^2 \over s - M_R^2 + i \sqrt{s} \Gamma_R(s) }
\end{equation}
where the energy dependent width $\Gamma_R(s) $ is computed from
its usual definition
\begin{equation}
\Gamma_R(s) = {1 \over 2 \sqrt{s}} |M_{R \to f_i}|^2 d\Phi 
\delta(Q - \sum p_i) \;\;\;\; \Gamma_R(s)_{s = M_R^2} = \Gamma_R 
\end{equation}
with $\Gamma_R(s) = 0 $ for $s < $ threshold. For $R $ decaying into
two particles with masses $M_1, M_2 $, this formula is simplified to
\begin{eqnarray}
\Gamma_R(s) & = & \Gamma_R {M_R^2 \over s} \left ( {p \over p_R}
\right )^{2n+1} \nonumber \\
p & = & {1\over 2 \sqrt{s}} \sqrt{(s - (M_1 + M_2)^2)(s - (M_1 -
M_2)^2)} \nonumber \\
p_R & = & {1\over 2 M_R} \sqrt{(M_R^2 - (M_1 + M_2)^2)(M_R^2 -
(M_1 - M_2)^2)}
\end{eqnarray}
where $n $ is the power of $|p| $ in the matrix element. In the 
Breit--Wigner of the $a_1 $ we use
\begin{eqnarray}
m_{a_1} = 1.251\; \mbox{GeV}, \;& &  \Gamma_{a_1} = 0.599 \; 
\mbox{GeV} \nonumber
\\
\sqrt{s}\, \Gamma_{a_1} (s) & = & m_{a_1} \Gamma_{a_1} {g(s) \over
g(m_{a_1}^2)}
\end{eqnarray}
where the function $g(s) $ is given by \cite{kuhn}
\begin{equation}
g(s) = \left ( \begin{array}{cc} 4.1(s - 9 m_{\pi}^2)^3(1 - 3.3(s -
9 m_{\pi}^2) + 5.8 (s - 9 m_{\pi^2}^2)^2) & \mbox{if} \; s < 
(m_{\rho} + m_{\pi})^2 \\
s(1.623 + {10.38 \over s} - {9.32 \over s^2} + {0.65 \over s^3})
& \mbox{else} \end{array} \right )
\end{equation}
The vector resonances for more meson channels allow a richer 
structure, containing the $\rho $--meson and some exitations.
With energy dependent widths we take 
\begin{equation}
T^{(i)}_{\rho} (s) = \left \{ {\sum_R \beta_R^{(i)} BW_R^{(i)}(s) 
\over \sum_R \beta_R^{(i)} 
} \right \}
\end{equation}
The index $i$ indicates the fact that for rho like resonances 
which couple to different particles, the relative contributions
$\beta_R^{(i)}$ of the
different radial exitations need not be the same. 
For $a_1 \to \rho \pi$ we use the parametrization which was derived 
in
\cite{kuhn} and also used in \cite{tauo,rog}:
\begin{eqnarray}
m_\rho = 0.773\; \mbox{GeV}, \; & & \Gamma_\rho = 0.145\;
\mbox{GeV}\nonumber 
\\
m_{\rho'} = 1.370\; \mbox{GeV}, \; & & \Gamma_{\rho'} = 0.510 \;
\mbox{GeV}\nonumber
\\
\beta_{\rho'}^{a_1\to\rho\pi} = -0.145
\end{eqnarray}
For all other rho-like resonances we take for the moment a common 
$T_\rho$. We fix its parameters (masses, widths and $\beta_R$'s) by
fitting the cross section for the process $e^+ e^- \to 2\pi^+ 2\pi^-
$ to experimental data (see Sec.~4). 


\bigskip
These effective vertices and propagators allow to construct the full
amplitude for the $\tau $ decays into $4 \pi $ final states using the
structure of the chiral currents \cite{wess}, and assuming a 
dominant role of the $a_1 $ meson in these decays, as indicated by
the data. Explicitly we have for this part of the hadronic current
in eq.(1) the following matrix element:
\begin{eqnarray}
<4 \pi|H^{\mu (1)}|0> & \sim & T_{\rho }(Q^2)\, \Gamma^{\mu \nu} 
\left (\rho (Q) \to a_1(k) \pi \right ) \, BW_{a_1}(k^2) \nonumber \\
& & \cdot \Gamma^{\nu \lambda} \, \left (a_1(k) \to \rho(l) \pi 
\right ) \, BW_{\rho}(l^2)\, \Gamma^{\lambda} (\rho(l) \to \pi \pi)
\end{eqnarray}
In the chiral limit, where the meson masses tend to zero
and then the limit $s \to 0 $ is performed, the hadronic current 
should have the form \cite{wess}
\begin{eqnarray}
J_{\mu}(\pi^- \pi_1^0 \pi_2^0 \pi_3^0) & = & {2\sqrt{3} \over 
f_{\pi}^2} \sum_{i=1}^3 (q_{0i} - q_- )^{\nu} A_{\mu \nu}^{i-}
\nonumber \\
J_{\mu}(\pi_1^- \pi_2^- \pi^+ \pi^0) & = & {2\sqrt{3} \over f_{\pi}^2
} [ 2(q_+ - q_0)^{\nu} A_{\mu \nu}^{+0} + (q_{-1} - q_+)^{\nu}
A_{\mu \nu}^{1+} + (q_{-2} - q_+)^{\nu} A_{\mu \nu}^{2 +}] \nonumber 
\\
\mbox{with} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; & & \\
A_{\mu \nu}^{ik} & = & g_{\mu \nu} - \sum_{l \neq i,k} 
{(Q - 2 q_l)_{\mu} (Q - q_l)_{\nu} \over (Q - q_l)^2 } \nonumber
\end{eqnarray}
Explicit calculation shows that the decay chain $\tau \to \nu_{\tau}
\rho, \rho \to a_1 \pi, a_1 \to \rho \pi, \rho \to \pi \pi $ alone
does not have the correct chiral limit. This can be corrected by
adding additional contributions, see Fig. 1 (b), which can be 
interpreted as being induced by two pions in the $s $--wave which
possibly induce $H^{\mu (2)} $ given by
\begin{equation}
<4 \pi | H^{\mu (2)} |0> \; \sim \; T_{\rho}(Q^2) \left ( g^{\mu 
\nu} - {Q^{\mu} Q^{\nu} \over Q^2 } \right ) T_{\rho} (k^2) 
\Gamma^{\nu} (\rho (k) \to \pi \pi ) (BW_{f_0}((Q-k)^2))
\end{equation}
Thus a second decay chain of the type $\tau \to \nu_{\tau} \rho,
\rho \to (\pi \pi)_s \rho, \rho \to \pi \pi $ is built
up; where the total charge of the two pions is zero,
we introduce a Breit--Wigner for $(\pi \pi)_s $
resonance: $f_0 $.
In its Breit-Wigner we use
\begin{equation}
   m_{f_0} = 1300 \, \mbox{MeV}  
   \qquad  \Gamma_{f_0} = 600 \, \mbox{MeV}
\end{equation}
In the chiral limit, the product of the coupling constants and 
the relative contributions of the two decay chains are fixed. 
However, in order to reproduce the experimental data, we are forced 
to multiply the hadronic currents with an additional function 
$F(Q^2)$, which describes the energy dependence of the form factors. 
From the low energy theorem, we know that it is unity at zero 
momentum squared, $F(0) =1$. However, we have found that $F(Q^2)$ 
can no longer be unity at $Q^2 \sim (1.5 \, \mbox{GeV})^2$. In order
to be able to reproduce the experimental $e^+ e^- \to 4 \pi$ cross 
sections, we have to assume that $F(Q^2)$ is considerably lower than
unity at high $Q^2$. Now the question arises which functional 
dependence of $F(Q^2)$ should be assumed. From experimental data 
\cite{arg} we know that in about $75 \%$ of the decays $\tau^-\to 
\pi^-\pi^-\pi^+ \pi^0\nu_\tau$, the invariant mass of the four pions
falls into the relatively small range of $ Q^2 = (1.15 \cdots
1.55\, \mbox{GeV})^2 $. Therefore we make the approximation that 
$F(Q^2)$ is constant in the relevant energy intervall:
\begin{equation}
    F(Q^2) = \left\{ \begin{array}{cl} 
        1 & \mbox{for\ \ }Q^2 = 0 \\
        \gamma & \mbox{for\ \ } Q^2 \approx (1.15 \cdots 1.55\, 
        \mbox{GeV})^2
    \end{array} \right.
\end{equation}
So in fact we multiply the hadronic current by a constant $\gamma$, 
but we have to keep in mind that this might be a bad approximation 
outside the range indicated above. Furthermore the two decay chains 
need not be multiplied by the same function $F(Q^2)$. So we could 
multiply the first decay chain with the $a_1 \pi$ intermediate state
by $\gamma_1$ and the second decay chain with the $(\pi \pi)_s \rho$
intermediate state by a different $\gamma_2$. For simplicity, 
however, we use
\begin{equation}
   \gamma = \gamma_1 = \gamma_2
\end{equation}
In addition to these two decay chains, the decay $\tau^- \to 2 
\pi^- \pi^+ \pi^0 $ has contributions from an anomalous part ($\rho
\omega \pi $--coupling) \cite{rog2} in Fig. 2; the matrix element of
the corresponding hadronic current reads \cite{tauo}
\begin{eqnarray}
<4 \pi |H^{\mu\; anom} |0> & = & \left ( {1\over m_{\rho}^2 - Q^2 - 
i m_{\rho}\Gamma_{\rho} } + \sigma  {1\over m_{\rho'}^2 - Q^2 - i 
m_{\rho'} \Gamma_{\rho'} } \right ) \nonumber \\
& & \cos {\Theta_c} G_{\omega 3 \pi} g_{\rho \omega \pi} F_{\rho}
\left \{ {1\over m_{\omega}^2 - (Q-p_2)^2 - i m_{\omega} 
\Gamma_{\omega} } \right. \nonumber \\
& & \left [ p_{1 \mu} \left ( (Q - p_2) \cdot p_3 p_4 \cdot p_2
- (Q - p_2) \cdot p_4 p_3 \cdot p_2 \right ) \right. \nonumber \\
& & + p_{3 \mu} \left ( (Q - p_2) \cdot p_4 p_1 \cdot p_2
- (Q - p_2) \cdot p_1 p_4 \cdot p_2 \right ) \nonumber \\
& & \left. + p_{4 \mu} \left ( (Q - p_2) \cdot p_1 p_3 \cdot p_2 
- (Q - p_2) \cdot p_3 p_1 \cdot p_2 \right ) \right ] \nonumber \\
& & \left. + (1 \leftrightarrow 2) \right \} 
\end{eqnarray}
As for the parameters $\sigma$, $m_{\rho'}$ and $\Gamma_{\rho'}$, we
stick to the parametrization used in \cite{tauo}. 
The product $G_{\omega 3 \pi} g_{\rho \omega \pi}$ was taken to be
\begin{equation}
   G_{\omega 3 \pi} g_{\rho \omega \pi}  
   = 1476\, \mbox{GeV}^{-3} 12.924\, \mbox{GeV}^{-1}
\end{equation}
in \cite{tauo},
where the coupling constants have been extracted from experimental 
data. Note, however, that both numbers have some uncertainty. 
Especially the extraction of $g_{\rho\omega\pi}$ from the
$\omega\to\gamma\pi$ decay rate is not free from theoretical
uncertainties. The rate for the decay $\tau\to\pi\omega\nu_\tau$ 
obtained with the above product disagrees somewhat with the 
experimental number \cite{arg}.
Therefore we write
\begin{equation}
   G_{\omega 3 \pi}\, g_{\rho \omega \pi}  
   =   g_\omega \, \cdot ( 1476 \cdot  12.924)\, \mbox{GeV}^{-4}
\end{equation}
where $g_\omega$ is a number of order one to be determined below.

\section{Relation between the Tau Decays and the $e^+e^-$ 
Annihilation Cross Sections}
%
There are two possible $4 \pi $ final states in $e^+ e^- $ 
annihilation, $2\pi^- 2 \pi^+ $ and $\pi^+ \pi^- 2 \pi^0 $ (the 
$4 \pi^0 $ channel is forbidden by charge--conjugation invariance).
They are accessible by a different $I_3 $ component of the same 
$I = 1 $ weak current describing $\tau $ decay. This allows to relate
these  processes to the $\tau $ decays \cite{gil}
\begin{eqnarray} \label{eqnee}
{\Gamma (\tau^- \to \nu_{\tau} \pi^- 3\pi^0) \over \Gamma (\tau^-
\to \nu_{\tau} e^- \overline{\nu}_e )} & = &  {3 \cos^2 \theta_c
\over 2 \pi \alpha^2
m_{\tau}^8} \int_0^{m_{\tau}^2} dQ^2 Q^2 (m_{\tau}^2 - Q^2)^2 
(m_{\tau}^2 + 2 Q^2) \nonumber \\
& & \cdot \left [{1\over 2} \sigma_{e^+e^- \to 2\pi^- 2\pi^+} (Q^2) 
\right ] \nonumber \\
{\Gamma (\tau^- \to \nu_{\tau} 2\pi^- \pi^+ \pi^0) \over \Gamma
(\tau^- \to \nu_{\tau} e^- \overline{\nu}_e) } & = & {3 \cos^2 
\theta_c \over 2 \pi
\alpha^2 m_{\tau}^8} \int_0^{m_{\tau}^2} dQ^2 Q^2 
(m_{\tau}^2 - Q^2)^2 (m_{\tau}^2 + 2Q^2) \nonumber \\
& & \cdot \left [{1\over 2} \sigma_{e^+e^- \to 2\pi^- 2\pi^+} (Q^2)
+ \sigma_{e^+ e^- \to \pi^+ \pi^- 2\pi^0}(Q^2) \right ]
\end{eqnarray}

These relations can be inverted  (in the range of $Q^2 $ covered
by the tau ) to
$$
%\begin{eqnarray}
  \sigma_{e^+e^- \to 2\pi^+2\pi^-}(Q^2)  = 
  {2 \pi \alpha^2\over 3 \cos^2 \theta_c}
  {m_\tau^2 \over Q^3 (1 - Q^2/m_\tau^2)^2
  (1 + 2 Q^2/m_\tau^2)}  \mbox{\ \hspace*{3cm}\ } 
$$
%\nonumber \\  
$$ 
  \cdot
  \left[ {\Gamma_{\pi^- 3\pi^0} \over 
   \Gamma_e}
  {1\over N_{\pi^- 3\pi^0} }
  {dN_{\pi^- 3\pi^0}(Q)\over dQ} \right]
$$
$$
  \sigma_{e^+ e^- \to \pi^+ \pi^- 2\pi^0}(Q^2)  = 
  {\pi \alpha^2\over 3 \cos^2 \theta_c}
  {m_\tau^2 \over Q^3 (1 - Q^2/m_\tau^2)^2
  (1 + 2 Q^2/m_\tau^2)} \mbox{\ \hspace*{3cm}\ }
%\nonumber \\ 
$$
\begin{equation}
\cdot
  \left[ 
   {\Gamma_{\pi^+2\pi^-\pi^0} \over 
   \Gamma_e}
  {1\over N_{\pi^+2\pi^-\pi^0} }
  {dN_{\pi^+2\pi^-\pi^0}(Q)\over dQ}
   - {\Gamma_{\pi^+2\pi^-\pi^0} \over 
   \Gamma_e}
  {1\over N_{\pi^+2\pi^-\pi^0} }
  {dN_{\pi^+2\pi^-\pi^0}(Q)\over dQ} \right]
%
%
%
\end{equation}
(Here $Q$ denotes the square root of the four 
momentum squared, $Q := \sqrt{Q_\mu Q^\mu}$.)

Thus any measurement of, and any model for differential 
distributions of the tau decays into four pions has implications for
the electron positron annihilation cross sections. Whereas from a 
measurement of the tau decays there are of course only implications 
for cross sections up to $Q^2 < m_\tau^2$, within a model for tau 
decays one can formally assume a larger mass of the tau and deduce 
predictions for the cross sections even at higher $Q^2$.

%%%%%%%%%%%%%%%%%%%%%%%%
\par
\bigskip
Before confronting the model with experiment and making more
detailed predictions, it is worth noting that any model for the 
description of a multipion final state has to satisfy quite general
conditions of charge correlations following from isospin 
considerations \cite{pais}. They fix the relative probabilities of
the channels corresponding to the various alternatives of charged and
neutral $\pi $'s. In the case of the tau decays into four pions 
these considerations amount to the statement that
\begin{equation}
{\Gamma (\tau^- \to 2 \pi^- \pi^+ \pi^0) \over \Gamma (\tau^- \to
3 \pi^0 \pi^- )} \ge {3\over 2}
\end{equation}
which is well satisfied in our model, as will be seen in the next 
section.

\section{Numerical Results in Comparison to Experiment}

%\subsection{Determination of the Parameters of the Model}

We still have to determine the following parameters of our model,
$g_\omega$, $\beta_{\rho'}$, $\beta_{\rho''}$, 
$m_{\rho'}$, $\Gamma_{\rho'}$ and $\gamma \equiv F(Q^2 \approx 
m_{\rho'}^2)$. (The parameters of the $\rho'' \equiv \rho(1700)$, 
which turns out to contribute only very little, are kept at
their particle data book values \cite{pdg}.) The $\omega$ does not 
contribute to $e^+e^- \to  2\pi^+ 2\pi^-$, so the cross section for 
this process is independent of $g_\omega$. We determine the other 
parameters by fitting the cross-section obtained with our model to 
experimental data, see Fig.~3. Our best fit is
\begin{eqnarray}
   \beta_{\rho'} = 0.08 & \qquad \qquad & \beta_{\rho''} = -0.0075
\nonumber \\
   m_{\rho'} = 1.35\, \mbox{GeV} & & \Gamma_{\rho'} = 0.3\, 
\mbox{GeV} \nonumber \\
   m_{\rho''} \equiv 1.70\, \mbox{GeV} & & \Gamma_{\rho''} 
\equiv 0.235\, \mbox{GeV}
\nonumber \\
   \gamma = 0.38
\end{eqnarray}
where the $\rho''$ parameters have been stated for completeness.
If we fix $\gamma=1$ in order to retain the assumption of constant 
coupling constants, it is not possible to obtain a reasonable fit 
within our model. We have tried various modifications of our model 
in order to be able to obtain a good fit with $\gamma=1$. For 
example, we allowed for different resonance parameters for the 
different $T_\rho^{(i)}$ involved, modified the $T_\rho^{a_1 \to
\rho\pi}$, or assumed that the dominance by the $a_1$ occurs only 
for the decay of the $\rho(770)$, whereas for the $\rho'$ we fixed 
the $a_1$ resonance factor equal to unity. However, in all these
cases, the best fit for the cross section is by about a factor of
two or three too large. Furthermore, when keeping $\gamma=1$, the 
fits tend to prefer unphysical values for the resonance parameters. 
Therefore we are convinced that when going from $Q^2 = 0$ to $Q^2 = 
(1.5\, \mbox{GeV})^2$, the couplings at the vertices can not be 
assumed to be constant.
 
The result of our fit in Fig.~3 lies somewhat below the data in the 
range $\sqrt{Q^2} = 1.9\mbox{--}2.2\, \mbox{GeV}$. This could 
possibly be cured by including the $\rho(2150)$ in the fit. As our 
main goal is to describe the tau decays, where $Q^2$ can of course 
not be larger than $m_\tau^2$, we do not do this. Furthermore one has
to remember that at such high $Q^2$, $F(Q^2)$ is presumably no 
longer constant. 
There is also some disagreement between our fit and the Nowosibirsk 
data \cite{nowo} in the region $\sqrt{Q^2} \approx 1.3\, \mbox{GeV}
$, but as the Nowosibirsk and the new Orsay data \cite{orsay} do not
match well, this might be due to errors in the experimental data. 
\par
\noindent
Next we have fixed $g_\omega$ by requiring that the experimental
results for $\tau \to \nu_{\tau} \omega \pi $ \cite{arg} are 
reproduced reasonably (see Tab.~\ref{tabbrs} below). This results in 
\begin{equation}
   g_\omega = 1.4
\end{equation}
Having fixed the parameters from $e^+e^- \to 2\pi^+ 2\pi^-$ and from
$\tau\to \nu_\tau \omega \pi $, we obtain predictions for the cross 
section for $e^+e^-\to \pi^+\pi^-2\pi^0$ and for the other tau 
decays into four pions.

Our prediction for the $e^+e^-\to \pi^+\pi^-2\pi^0$ cross section is
shown in Fig.~4. The experimental data for this state are much poorer
than those for the other mode, and the data from different 
experiments do not agree very well. Note that our prediction is 
rather low, but still compatible with the data. It is interesting 
that our prediction favours the new Orsay data \cite{orsay}, which 
is in fact also true for the process $e^+e^- \to 2\pi^+ 2\pi^-$, 
see Fig.~3.

Now we will discuss the $\tau$ decays. Our results for the integrated
decay rates are given in Tab.~\ref{tabbrs}, where we compare with the
version 2.4 of Tauola and with experimental data. 
The branching ratios have been calculated assuming a tau lifetime of
\begin{equation}
   \tau_\tau = 295.7\, \mbox{fs}
\end{equation}
Note that we use physical masses for the pions, $m_{\pi^+} 
\neq m_{\pi^0}$, in the phase space, and  a tau mass of
\begin{equation}
   m_\tau = 1.7771\, \mbox{GeV}
\end{equation}
This explains why the numbers in Tab.~\ref{tabbrs} for Tauola 2.4 do
not agree exactly with those in Tabs.~4 and 5 of \cite{tauo}. 
Whereas the numbers for the $\tau\to\nu_\tau\omega\pi$ mode have 
been fitted by adjusting $g_\omega$, the other values are 
predictions, and they agree quite well with experiment.
 
Our predictions are also to be compared to the earlier
predictions in \cite{gil},
\begin{eqnarray}
  \frac{\Gamma(\tau^-\to\nu_\tau 2\pi^-\pi^+\pi^0)}{\Gamma_e} 
   & = & 0.275 \; (0.185 \mbox{--} 0.305)
\nonumber \\
  \frac{\Gamma(\tau^-\to\nu_\tau \pi^-3\pi^0)}{\Gamma_e}
   & = &  0.055  
\end{eqnarray}
Whereas our prediction for the channel $\pi^-3\pi^0$ is practically 
identical to
that of \cite{gil}, our prediction for the $2\pi^-\pi^+\pi^0$ mode is
significantly lower than theirs. This is due to our rather small
prediction for the $e^+e^-\to \pi^+\pi^- 2 \pi^0$ cross section, 
which is supported, as we have stated before, by the new Orsay data 
which where not available for the analysis in \cite{gil}.

Note that actually this lower prediction does not depend on details 
of our model. 
According to Eqn.~\ref{eqnee}, this decay rate depends strongly on 
the cross section for $e^+e^- \to \pi^+ \pi^- 2 \pi^0$. The 
calculation by Gilman and Rhie used a fit to this cross section 
which reproduced the Nowosibirsk data at low $Q$, and estimated the 
uncertainty from variations of the fit in the high $Q$ region. 
However, as can be seen from Fig.~3, the Nowosibirsk data do not fit
well to the Frascati data. The new Orsay data now strongly support 
the impression that the Nowosibirsk data are probably about a factor
of two too large at $Q = 1.4\, \mbox{GeV}$. Taking this into account
brings the prediction for the tau decay rate down to a value much 
below the central value given by Gilman and Rhie. 

A detailed recent analysis of $e^+e^- $ cross sections and $\tau $
decay rates has been performed in \cite{clegg} where data have been 
analyzed in terms of partitions as defined by Pais \cite{pais} 
rather than by a dynamical model. From these results, the authors 
deduce values of the rates into definite charge states in $\tau $ 
decay: For $\Gamma (\pi^{\pm}3 \pi^0)/\Gamma_e $ they obtain a value
of $0.050 \pm 0.002 $, for $\Gamma (\pi^{\pm}\pi^+ \pi^- \pi^0)/
\Gamma_e $ their result is $ 0.190 \pm 0.009 $ or $0.223 \pm 0.011 
$, according to different data sets.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{table}
\caption{Numerical results for the integrated 
decay rates of our model, compared to Tauola 2.4 and to experimental
data}
\label{tabbrs}
$$
\begin{array}{|c|c|c|c|}
\hline\hline
  \mbox{observable} & \mbox{present model} & \mbox{Tauola 2.4} 
& \mbox{experiment}
 \ \\
\hline
  \Gamma(\tau^-\to\nu_\tau 2\pi^-\pi^+\pi^0) / \Gamma_e
   & 0.172 & 0.0832 & 
\\ \hline
 \mbox{BR}(\tau^-\to\nu_\tau 2\pi^-\pi^+\pi^0) 
   & 3.11\% & 1.51 \% & (4.4 \pm 1.6) \% \cite{tau8}
\\ \hline
  \Gamma(\tau^-\to\nu_\tau \pi^- 3\pi^0) / \Gamma_e  
  & 0.0540 & 0.0172 & 
\\ \hline
  \mbox{BR}(\tau^-\to\nu_\tau \pi^- 3\pi^0)   & 
  0.98 \% & 0.31\% & 
  \begin{array}{c}
      \tau\to h^- 3 \pi^0 \cite{schw}: 
      \\ 1.00 \pm 0.15 \% 
  \end{array}
\\ \hline
  \Gamma(\tau^-\to\nu_\tau \pi^- \omega(\pi^-\pi^+\pi^0)) / \Gamma_e
 & 
  0.0664 & 0.0339 & 
\\ \hline
  \mbox{BR}(\tau^-\to\nu_\tau \pi^- \omega(\pi^-\pi^+\pi^0))   & 
   1.20 \% & 0.61 \% & 1.65 \pm 0.36 \% \cite{arg}
\\ \hline
\frac{\displaystyle  \Gamma(\tau^-\to\nu_\tau \pi^- \omega(\pi^-
\pi^+\pi^0))}   
{\displaystyle \Gamma( \tau^-\to\nu_\tau 2\pi^-\pi^+\pi^0) }
 & 0.39 & 0.41 & 0.33 \pm 0.05 \cite{arg}
\\ \hline
\end{array}
$$
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Our results for the differential distributions in the decay $\tau\to
\nu_\tau 2\pi^-\pi^+\pi^0$ are given in Figs.~5--7. We compare with 
the Argus data \cite{arg} and with the 2.4 version of Tauola.
For the Argus data we have performed a rebinning into larger bins.  

Note that we display distributions
\begin{equation}
   \frac{1}{N} \frac{dN}{dm}
\end{equation}
which are  normalized to the total number of events $N$. We run the 
Monte-Carlos with very high statitics. Therefore the statistical 
errors of the Monte-Carlo results are very small and not shown in 
the figures. At this point it is important to stress that the
predicted total rate by TAUOLA 2.4 is much too low, which is
not visible in these normalized shapes.

Overall we find a reasonable agreement between our model and the
experimental data. 
In the invariant mass of the four pion system (Fig.~5), our 
prediction for the peak of the distribution is a bit on the high 
side, but we do not consider this as very pronounced. We have tried 
to improve the fit by lowering the mass of the $\rho'$. This, 
however, inevitably leads to predictions for the $e^+e^-$ cross 
sections which badly disagree with the experimental data.

In comparing the experimental three pion invariant mass distribution
in Fig.~6 with our model one should remember the limited 
experimental mass resolution. Furthermore, we have chosen 
$g_\omega = 1.4$ such that both the ratio $\frac{\displaystyle  
\Gamma( \tau^-\to\nu_\tau \pi^- \omega(\pi^-\pi^+\pi^0))}   
{\displaystyle \Gamma( \tau^-\to\nu_\tau 2\pi^-\pi^+\pi^0) }$ and the
absolute number for the branching ratio agree reasonably with the
experimental numbers. However, in the line shapes obviously only the
relative contribution of the $\omega$ channel is important, which 
comes out a bit on the high side with $g_\omega = 1.4$. A somewhat 
lower value of about 1.2 or 1.3 would result in a better fit here.

In the case of the two pion invariant mass distributions (Fig.~7
(a)--(e)), we get good agreement for the charged opposite sign
combination (containg the $\rho^0$)(Fig.~7(a)) and the charged like 
sign one (Fig.~7(e)).  In the case of the charged-neutral 
combinations, however, we find that we predict too many $\rho^+$ 
(Fig.~7(d)). 


\section{Conclusions}
In this paper we have analyzed the four pion decay modes of the
$\tau $--lepton and the corresponding $e^+e^- $ data related through
$CVC $. We improved the hadronic matrix element in comparison to
previous work by implementing low--lying resonances like $\rho, 
\rho', \rho'', a_1 $ and $\omega $ mesons in the different
channels into the chiral structure. In particular we respected the
dominant role of the $a_1 $ meson as indicated by the data. We fixed
our parameters from $e^+e^- \to 2 \pi^+ 2 \pi^- $ and $\tau \to
\nu_{\tau} \omega \pi $ and obtained predictions for the other 
four--pion decay modes of the $\tau $--lepton (Table 1). We have 
given a new prediction for the normalized decay rate
\begin{equation}
  \frac{\Gamma(\tau^-\to\nu_\tau 2\pi^-\pi^+\pi^0)}{\Gamma_e} 
   = 0.172
\end{equation}
which is significantly lower than the prediction by Gilman and Rhie 
of $ 0.275 \; (0.185 $ $\,\mbox{--} \, 0.305) $ \cite{gil}. This 
prediction is supported both by the prediction of our model and 
independently by the new data \cite{orsay} on the $e^+e^- \to \pi^+
\pi^- 2\pi^0$ cross section. \par
We have studied in detail two-- three-- and four--pion mass 
distributions based on a detailed dynamical model with slowly varying
form factors and compared our predictions with available data. In 
general we find reasonable agreement between the TAUOLA Monte
Carlo with our matrix elements and the experimental data.Fine tuning
of the parameters is left to the experimentalists and requires more 
accurate data. \par

\vspace{3.cm}
\noindent
{\large \bf Acknowledgement} \par
\bigskip
\noindent
We thank J. K\"uhn for his interest and informative discussions. \par
This work has been supported by the HCM program under EEC contract
number CHRX-CT920026. \par

\newpage
\section{Appendix}
Our model is available in the form of a FORTRAN code suitable to be 
used with the Monte-Carlo Tauola. The file HCURR4PI.F contains three
COMPLEX FUNCTIONs, viz.\ BWIGA1, BWIGEPS, FRHO4 and the SUBROUTINEs
CURINF, CURINI and CURR. The latter is the main subroutine, which 
calls the other functions and routines and which replaces the 
original subroutine CURR of Tauola 2.4. Therefore the user has to 
delete the subroutine CURR in Tauola 2.4 and link our HCURR4PI 
instead. The subroutine CURINF prints some general information about
the new routine CURR. The subroutine CURINI initializes the COMMON
block /TAU4PI/ with the values we obtained in our fit. The meaning 
of the individual parameters is explained in Tab.~\ref{tabcommon}. 
Alternatively, the user may choose to perform a new fit of these 
parameters to experimental data. Note that in the routine $\gamma_1 
$ and $\gamma_2 $ may be chosen differently.

\begin{table}[h] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{Input parameters in the common block TAU4PI}
\label{tabcommon}
$$
\begin{array}{ccc}
\hline
\qquad \mbox{Parameter} \qquad & \mbox{Meaning} & \mbox{Our fit}
\\
\hline
\mbox{GOMEGA} & g_\omega                      & 1.4 \\
\mbox{GAMMA1} & \gamma_1                      & 0.38 \\
\mbox{GAMMA2} & \gamma_2                      & 0.38 \\
\mbox{ROM1}   & m_{\rho'}\,[\mbox{GeV}]       & 1.35 \\
\mbox{ROG1}   & \Gamma_{\rho'}\,[\mbox{GeV}]  & 0.3\\
\mbox{BETA1}  & \beta_1                       & 0.08\\
\mbox{ROM2}   & m_{\rho''}\,[\mbox{GeV}]      & 1.70\\
\mbox{ROG2}   & \Gamma_{\rho''}\,[\mbox{GeV}] & 0.235\\
\mbox{BETA2}  & \beta_2                       & -0.0075\\
\hline
\end{array}
$$
\end{table}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To obtain a copy of the file HCURR4PI.F, send an e-mail to M.F.: \par
finkemeier@vaxlnf.lnf.infn.it \par



\newpage
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\end{document}
