%Title: Hadronic Decays of Excited Heavy Quarkonia
%Author: Thomas Mannel and Res Urech
%Published: <EM>Z. Phys.</EM> C73 (1997) 541 - 546 
%
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\begin{document}

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%--------------------------------
%---------------- TTP Titlepage <---------------------------
\thispagestyle{empty}
\begin{titlepage}
\begin{flushright}
{\bf TTP95--36} \\
     hep-ph/9510406
\end{flushright}
\vspace{1cm}

%\boldmath
\begin{center} 
{\Large\bf HADRONIC DECAYS OF } \vspace*{3mm} \\
{\Large\bf EXCITED HEAVY QUARKONIA}
\end{center}
%\unboldmath
\vspace{0.8cm}

\begin{center}
{\sc Thomas Mannel}\footnote{e-mail: Thomas.Mannel@physik.uni-karlsruhe.de}
and {\sc Res Urech}\footnote{e-mail: ru@ttpux2.physik.uni-karlsruhe.de}
\vspace*{2mm} \\ 
{\sl Institut f\"{u}r Theoretische Teilchenphysik \\
     Universit\"at Karlsruhe \\
     D -- 76128 Karlsruhe, Germany} 
\end{center}
\vspace{4cm}
\begin{abstract}
\noindent
We construct an effective Lagrangian for the hadronic decays of a heavy
excited $s$-wave-spin-one quarkonium $\Psi'$ into a lower $s$-wave-spin-one 
state $\Psi$. We show that reasonable fits to the measured invariant mass
spectra in the charmonium and 
bottomonium systems can be obtained within this framework. The mass
dependence of the various terms in the Lagrangian is discussed on the
basis of a quark model.
\end{abstract}
\end{titlepage}
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\section{Introduction}
%
%
%
%
Bound state systems consisting of a heavy quark and a heavy antiquark 
are fairly well described in terms of wave function models, where the 
potential among the two heavy quarks is determined from fitting a 
phenomenological ansatz to the data \cite{Buch}. 
In this way many static properties 
of these systems may be understood even quantitatively. 

As far as decays of heavy quarkonia are concerned there has been also some 
theoretical progress recently \cite{BBL,MS95}. For inclusive decays in 
which the two heavy quarks annihilate an effective theory approach 
has been developed allowing for a systematic treatment of these processes. 
This method puts the  description of this type of decays on a model
independent basis. 

The theoretical framework of exclusive hadronic decays is in general 
still quite model dependent. However, for a large class of decay modes 
of excited quarkonia $\Psi '$ we may use chiral symmetry 
arguments to construct 
an effective Hamiltonian for decays such as $\Psi' \to \Psi \pi$ or 
$\Psi' \to \Psi \pi \pi$ where $\pi$ is a member of the Goldstone boson
octet. This ansatz has been discussed already in the 
literature \cite{Dono}, however, without giving a systematic derivative
expansion 
in the spirit of chiral perturbation theory.    

Such an expansion may be performed starting from the infinite mass limit 
for the heavy quarkonium\footnote{% 
   The idea we are proposing here is along the lines
   as in \cite{Jenkins}, where this 
   method is applied to $\phi$ decays.}.
The momentum of the heavy quarkonium scales with 
the heavy quark masses and hence it may not be used as an expansion 
parameter. In order to define a systematic derivative expansion 
we decompose the momentum $p (p')$ of the heavy 
quarkonium in the initial (final) state as 
$p' (p) = (m_Q + m_{\overline{Q}}) v + k'(k)$, 
where $k'$ and $k$ are small residual momenta of the order of the binding 
energy of the heavy quarkonium,
and $m_Q$ and $m_{\overline{Q}}$ are the masses of the heavy quark and
antiquark respectively. Note that unlike in Heavy Quark Effective Theory
where systems with a single heavy quark are considered 
we cannot use the static limit for quarkonia states. Rather one has 
to employ the non-relativistic limit and the expansion in $k/m_Q$ 
and $k'/m_{\overline{Q}}$ becomes an expansion in $|\vec{v}_{rel}|/c$, 
where $\vec{v}_{rel}$ is the relative (three)velocity of the two heavy quarks.

The velocity $v$ is the velocity of the initial state 
quarkonium and hence the momenta of the final state heavy quarks will  
be of the order of the mass difference of the initial and final state 
quarkonia, which we shall assume to be small compared to the masses 
$m_Q$ and $m_{\overline{Q}}$.

In the spirit of chiral perturbation theory one may now formulate a
derivative expansion, which after Fourier transformation becomes an 
expansion in the pion momenta and the residual momenta $k'$ and $k$. 
In the next section we shall use chiral symmetry to write down the leading 
as well as the chiral symmetry breaking terms in such an expansion. 
In section 3 we show that one may obtain 
a good fit to the data, total rates as well as invariant mass spectra 
of the two pions in the decay $\Psi' \to \Psi \pi \pi$. We shall 
compare the charmonia and the bottomonia and discuss the dependence of the
parameters in the effective Lagrangian on the bottom and charm mass in
section 4. Conclusions are given in section 5.  



\section{Effective Lagrangian for Heavy Quarkonia \newline Decays}
%
%
%
%
%
We construct an effective Lagrangian for the decays of a heavy excited
$s$-wave-spin-one quarkonium $\Psi'$ into a lower $s$-wave-spin-one
state $\Psi$. Let $A_\mu$
be the field of the initial excited $s$-wave $1^-$ state and $B_\mu$ the
one of the final $1^-$ state. We shall consider here only quarkonia where 
$m_Q = m_{\overline{Q}}$, hence we look at charmonium or bottomonium
systems.  

The momenta of the heavy quarkonia are split into a large piece which 
scales with $m_Q$ and a small residual part $k$ which depends only weakly on 
the heavy quark mass 
\begin{eqnarray*}
 p' &=& 2 m_Q v + k' \mbox{ for } A_\mu  \\
 p &=& 2 m_Q v + k \mbox{ for } B_\mu
\end{eqnarray*}
In coordinate space this is achieved by a phase redefinition of the fields
\begin{eqnarray}
A_\mu (x) = \exp (-i2m_Q v \cdot x) A_\mu^{(v)} (x) \nonumber \\
B_\mu (x) = \exp (-i2m_Q v \cdot x) B_\mu^{(v)} (x) 
\end{eqnarray}
such that the derivative acting on the fields with superscript $(v)$
yields now $k$ (or $k'$) which is small compared to the heavy mass. 
The fields $A^{(v)}$ and $B^{(v)}$ obey the equations of motion
corresponding to static fields and are transverse with respect to 
the velocity vector $v$
\begin{eqnarray}
iv \cdot \partial A_\mu^{(v)} (x) = 0,  \quad v^\mu A_\mu^{(v)} (x) = 0 \\
iv \cdot \partial B_\mu^{(v)} (x) = 0, \quad v^\mu B_\mu^{(v)} (x) = 0 
\end{eqnarray}

We are interested in an effective Lagrangian for the hadronic decays 
of the type $A \to B \pi$ or $ A \to B \pi \pi$, and we construct this 
Lagrangian using chiral symmetry. Higher order terms may be included by 
chiral perturbation theory, i.e.\ by a systematic expansion in the derivatives
(these correspond to pion momenta or residual momenta for quarkonia) and 
in the light quark masses. The heavy quarkonia are singlets under the chiral 
symmetry, and hence the decays $A \to B \pi$ are forbidden in the chiral 
limit. 

In the chiral limit only decays to an even number of pions are allowed, 
and the leading term in the derivative expansion obeying chiral symmetry
consists of three terms 
\begin{eqnarray} \label{l0}
{\cal L}_0&=& g A_\mu^{(v)} B^{(v)\mu *} 
             \mbox{ Tr}[(\partial_\nu U) (\partial^\nu U)^\dagger] 
           + g_1 A_\mu^{(v)} B^{(v)\mu *} 
             \mbox{ Tr}[(v\cdot\partial U) (v\cdot\partial U)^\dagger]
             \nonumber\\
         &&+ g_2 A_\mu^{(v)} B_\nu^{(v)*}
             \mbox{ Tr}[(\partial^\mu U) (\partial^\nu U)^\dagger +
                        (\partial^\mu U)^\dagger (\partial^\nu U)]
           + \mbox{ h.c.}
\end{eqnarray}
where $U$ is a unitary $3\times 3$ matrix that contains the Goldstone fields
\begin{equation}
U = \exp\,(i\Phi/F_0)\hspace{1cm}
\Phi= \sqrt{2}\left( 
\begin{array}{ccc}
\frac{1}{\sqrt{2}}\pi^0+\frac{1}{\sqrt{6}}\eta_8  & \pi^+ & K^+\\
\pi^- & -\frac{1}{\sqrt{2}}\pi^0+\frac{1}{\sqrt{6}}\eta_8 & K^0\\
K^- & \overline{K^0} & -\frac{2}{\sqrt{6}}\eta_8
\end{array}
      \right)
\end{equation}
and $F_0$ is the pion decay constant\footnote{%
    We use the convention in which the pion decay constant is 
    $ F_0 \simeq 93$ MeV.} 
in the chiral limit.              
In previous considerations the second term in (\ref{l0}) has been often
omitted, but  
since a consistent definition of the derivative expansion forces us to 
introduce the vector $v$, the second term is a perfectly allowed one. 
We shall show in the next section that the magnitude of the coupling 
$g_1$ is not small compared to $g$ and that a reasonable fit to the
invariant mass spectra of the pions may be obtained already from
(\ref{l0}). 

The quark mass matrix breaks chiral symmetry, and the leading terms 
of this kind are
\begin{eqnarray}\label{lsb}
{\cal L}_{S.B.} &=& g_3 A_\mu^{(v)} B^{(v)\mu *} 
                        \mbox{ Tr}[{\cal M} (U + U^\dagger-2)]\nonumber \\
                && + i g' \epsilon^{\mu \nu \alpha \beta} \left[
                     v_\mu A_\nu^{(v)} \partial_\alpha B_\beta^{(v)*} -  
                     (\partial_\mu A_\nu^{(v)}) v_\alpha B_\beta^{(v)*}
                     \right]
                     \mbox{ Tr}[{\cal M} (U - U^\dagger)]\nonumber \\
                && + \mbox{ h.c.}
\end{eqnarray}
 where 
\begin{equation}
{\cal M} = \left( \begin{array}{ccc}
                    m_u & 0   & 0 \\ 
                      0 & m_d & 0 \\ 
                      0 & 0   & m_s 
                     \end{array} \right)
\end{equation}
The two operators proportional to the $\epsilon$ tensor are relevant for
decays involving an odd number of pions. The relative sign of the these
terms is fixed by reparameterization invariance \cite{luke92}. 

The Lagrangian in (\ref{l0}) has been considered also by Casalbuoni
{\it et al.} \cite{casa93}, as well as the part proportional to $g'$ in
(\ref{lsb}) \cite{casa932}. In the latter case the authors include in
addition the mixing and direct coupling to the singlet $\eta'$, whereas we
concentrate on the contributions from the octet according to 
\cite{ioffe80}. 
  
 
\section{Fit to the Data}
%
%
\subsection{$\Psi' \to \Psi \pi \pi$}
%
%
Starting from the Lagrangians given in (\ref{l0}) and (\ref{lsb}) we fit
the experimental data on the decays $\psi(2S) \rightarrow J/\psi\pi^+\pi^-$
in the charmonium system, and $\Upsilon(aS)\rightarrow
\Upsilon(bS)\pi^+\pi^-$ in the bottomonium system, where 
$(a,b)$ takes the values $(2,1),(3,2)$ and $(3,1)$. The amplitude has the
form 
\begin{eqnarray}\label{ampl}
\lefteqn{{\cal A}(\Psi' \to \Psi \pi^+\pi^-) =}\nonumber \\
&& -\frac{4}{F_0^2}\left\{\;\left[\,
              \frac{g}{2}(m_{\pi\pi}^2 - 2 M_\pi^2) 
            + g_1 (v \cdot p_{\pi^+}) (v \cdot p_{\pi^-}) 
            + g_3 M_\pi^2\right] 
              \epsilon_\Psi^* \cdot \epsilon_{\Psi '} \right. \nonumber \\
&& \hspace{1.8cm} + g_2 \left[ p_{\pi^+\mu} p_{\pi^-\nu} 
                       + p_{\pi^+\nu} p_{\pi^-\mu}\right]
                       {\epsilon_\Psi^*}^\mu {\epsilon_{\Psi '}}^\nu \biggr\}
\nonumber \\
\end{eqnarray} 
where $m^2_{\pi\pi}= (p_{\pi^+} + p_{\pi^-})^2$ and $\epsilon_\Psi, \,
\epsilon_{\Psi '}$ are the polarization vectors of the heavy spin-one
quarkonia . In the rest frame of the decaying particle $\Psi'$ we have 
$v_\mu=(1,\vec{0})$ and the amplitude (\ref{ampl}) reduces to the
non-relativistic expression given by Brown and Cahn \cite{brown75} a long
time ago. In this frame the heavy final particle $\Psi$ 
moves non-relativistically. If we neglect the motion of the final state 
heavy quarkonium, 
the amplitude proportional to $g_2$ describes the $d$-wave part
of the dipion 
system, which is found to be suppressed experimentally to a few percent in
the charmonium decay \cite{abrams75} as well as in the
bottomonium decays \cite{albrecht87,wu93,butler94}. This observation is
supported in addition by some qualitative arguments that we will give in
Sec.\ref{discuss}. Thus we set $g_2=0$.  

The remaining coupling constants $g,g_1$ and $g_3$ can be fitted to the
total decay rate and the invariant mass spectra of the two pions. The data
for the (normalized) differential rates for $\psi(2S) \rightarrow
J/\psi\pi^+\pi^-$ and $\Upsilon(2S)\rightarrow \Upsilon(1S)\pi^+\pi^-$
are taken from Fig.7 in \cite{albrecht87}, for the $\Upsilon(3S)$ decays we
consider the data from \cite{butler94}. Our results are listed in
Table \ref{tab1} and the fits are shown in Fig.1 and Fig.2\,. It is worth
discussing briefly the results decay by decay.\\ 

$\bullet \hspace{3mm} \psi(2S) \to J/\psi\pi^+\pi^-$: We obtain already a
good fit working 
in the chiral limit, i.e. by ignoring $g_3$. The  ratio $g_1/g$ is then
given by  
\begin{equation}
\left(\frac{g_1}{g}\right)_{c\bar{c}}=-0.35 \pm 0.03
\end{equation}
where the error is given by the fitting program, 
including the experimental uncertainties. 

Varying $g_3$ in the range $g_3=0\ldots2.1$ and fitting again $g$ and $g_1$
gives an approximate linear relation among the triple $(g,g_1,g_3)$,
\begin{equation}
(g,g_1,g_3) = \left(0.30 + \frac{1}{20}x,- 0.11 - \frac{3}{10}x,x\right),
\hspace{1cm} x=0\ldots2.1
\end{equation}
where $\chi^2_{min}$ is found at $g_3=1.55$. The fit for $\chi^2_{min}$ 
is shown in the upper plot of Fig.1\,.\\

$\bullet \hspace{3mm} \Upsilon(2S) \to\Upsilon(1S)\pi^+\pi^-$: A good fit
is again  
obtained by putting $g_3=0$. Moreover, allowing for all
three parameters to be free, this fit is included in the set
of solutions for which $\chi^2 \leq  (\chi^2_{min} + 0.01)$ holds. 
However, $\chi^2$ as a function of the couplings $g_1$ and $g_3$
is very flat around $\chi^2_{min}$ and thus the 
errors of $g_1$ and $g_3$ in
the all-parameter-fit are an order of magnitude larger than the values
themselves. Therefore we do not indicate this fit here. 
In the case of $g_3=0$,
$g$ is of the same order as in the corresponding charmonium decay, whereas
the ratio $g_1/g$ differs roughly by a factor of 2,
\begin{equation}\label{ups21}
\left(\frac{g_1}{g}\right)_{b\bar{b}}=-0.19 \pm 0.04
\end{equation}
\vspace*{5mm}
The fit corresponding to $g_3 = 0$ is shown in the lower plot of Fig.1.

$\bullet \hspace{3mm} \Upsilon(3S) \to\Upsilon(2S)\pi^+\pi^-$: In this
decay, the 
experimental data have rather large errors. A reasonable
fit is obtained by using the current algebra amplitude, i.e. by putting
$g_1=g_3=0$. If we allow in addition for a variation of $g_1$ the $\chi^2$
decreases a little bit, however, the error of $g_1$ is of the same order as
the value itself. Altough $g$ is larger by a factor of $\sim (5 - 6)$ in
comparison 
to the $\Upsilon(2S)$ decay, the ratios $g_1/g$ are close to each other
(within a large error)
\begin{equation}\label{ups32}
\left(\frac{g_1}{g}\right)_{b\bar{b}}=-0.23 \pm 0.22
\end{equation} 
In the upper plot of Fig.2 we show the fit corresponding to $g_1 \neq 0$.\\
\begin{table}[t] 
$$
\begin{array}{c|c|c|c|c}
    & g             & g_1            & g_3           & \chi^2/d.f. \\[2mm]
\hline &&&&\\[-3mm]
\psi(2S) \to J/\psi\pi^+\pi^- 
    & 0.30 \pm 0.01 & -0.11 \pm 0.01 & 0^*           & 25.3/24     \\[2mm]
\cline {2-5}
&&&&\\[-3mm]
    & 0.38 \pm 0.03 & -0.59 \pm 0.19 & 1.55 \pm 0.58 & 20.1/23     \\[2mm]   
\hline&&&&\\[-3mm]
\Upsilon(2S) \to\Upsilon(1S)\pi^+\pi^- 
    & 0.25 \pm 0.01 & -0.05 \pm 0.01 & 0^*           & 25.3/19     \\[2mm]
\hline &&&&\\[-3mm]
\Upsilon(3S) \to\Upsilon(2S)\pi^+\pi^- 
    & 1.19 \pm 0.04 & 0^*            & 0^*           & 26.2/21     \\[2mm]
\cline {2-5}
&&&&\\[-3mm]
    & 1.53 \pm 0.33 & -0.36 \pm 0.34 & 0^*           & 25.0/20     \\[2mm]
\hline &&&&\\[-3mm]
\Upsilon(3S) \to\Upsilon(1S)\pi^+\pi^-\hspace{2mm} 
    &\hspace{2mm}0.026 \pm 0.001\hspace{2mm}
    &\hspace{2mm}-0.087 \pm 0.003\hspace{2mm}
    &\hspace{2mm}0.54 \pm 0.02\hspace{2mm}&\hspace{2mm} 87.6/29     \\[2mm]
\end{array}
$$
\caption[]{Values of the coupling constants obtained from fits to the data 
as indicated in the text. The $0^*$ means that this parameter is ignored in
this fit.}
\label{tab1}
\end{table}

$\bullet \hspace{3mm} \Upsilon(3S) \to\Upsilon(1S)\pi^+\pi^-$: Here we are
confronted with 
two aspects which are absent in the other decays. Firstly the phase space
is considerably larger than in all the other decays discussed so far. Since
chiral perturbation theory is valid only for momenta small compared to the
scale $\Lambda_{\chi SB} = 4\pi F_0 \simeq 1.2$ GeV, the fits obtained
with the amplitude (\ref{ampl}) should be considered with some
caution. Secondly the experimental data indicate that the dipion invariant
mass distribution is double peaked \cite{wu93,butler94}, whereas all the
other decays have a single peak only.  

The three parameter fit is indeed rather poor, however, it implies at least
that there is a double 
peak (shown in the lower plot of Fig.2 as the solid line). 
In order to improve the situation, we pick up 
the idea of \cite{belanger89} by taking $\pi^+ \pi^-$ final
state interactions into account. Working in the approach where
both the heavy particles are at rest, the first line of the amplitude in
(\ref{ampl}) represents the dipion system in an $s$-wave. The amplitude
(for $g_2=0$) can therefore be improved by introducing the 
Omn\`{e}s function \cite{barton65}
\begin{eqnarray}\label{omnes}
\lefteqn{{\cal A}(\Psi' \to \Psi \pi^+\pi^-)|_{g_2=0} \longrightarrow 
\Omega(m^2_{\pi\pi}){\cal A}(\Psi' \to \Psi \pi^+\pi^-)|_{g_2=0} }
\nonumber \\[2mm]
&&\Omega(s) = \exp \left( \frac{s}{\pi} \int^{\infty}_{4M^2_{\pi}}\;   
              ds'\frac{\delta_0^0(s')}{s'(s' - s)} \right)
\end{eqnarray}
where $\delta_0^0(s)$ (denoted as $\delta$ from here on) is the
energy dependent phase shift of the isospin $I=0$ $s$-wave $\pi\pi$
scattering amplitude. We decompose $\delta$ in a low  energy phase shift
$\delta_{LE}$ (``low'' with respect to the integration range of
$\Omega$) and in a part  coming from a (sharp) scalar resonance,
$\delta_R$. For the low energy tail we use the parametrization of Schenk
\cite{schenk91}, which reproduces the measured data up to $s^{1/2} = 0.9$
GeV,  
\begin{eqnarray}
\delta &=& \delta_{LE} + \delta_R \quad , \nonumber \\[2mm]
\tan \delta_{LE}(s) &=&  \left( 1 - \frac{4M_\pi^2}{s} \right)^{1/2}
\frac{4M_\pi^2 - s_0}{s - s_0} \nonumber \\
               && \times \left[a_0 - \tilde{b}_0 
                  \left( 1 - \frac{s}{4M_\pi^2} \right) 
                  + c_0 \left( 1 - \frac{s}{4M_\pi^2} \right)^2 \right] 
\quad ,\nonumber \\[2mm] 
\tilde{b}_0 &=& b_0 + a_0 \frac{4M_\pi^2}{4M_\pi^2 - s_0} + a_0^3 
\quad ,\nonumber \\[2mm]
\delta_R(s) &=& \pi \theta (s - M_R^2)
\end{eqnarray}
with the numerical values
\begin{equation}
a_0 = 0.2, \hspace{1cm} b_0 = 0.24, \hspace{1cm} c_0 = 0, 
\hspace{1cm} s_0^{1/2} = 0.865\;{\mbox GeV}
\end{equation}
and the resonance mass $M_R$ is left as a free parameter. We repeat the fit
procedure and get the following result,
\begin{eqnarray}
\begin{array}{c}
\Upsilon(3S) \to\Upsilon(1S)\pi^+\pi^- : \\[3mm]
\begin{array}{c|c|c|c|c}
g               & g_1            & g_3           & M_R\;\left[ {\mbox GeV}
\right]         &\chi^2/d.f. \\[2mm]
\hline &&&&\\[-3mm]
0.014 \pm 0.001\hspace{2mm} &\hspace{2mm}-0.040 \pm 0.002\hspace{2mm}
&\hspace{2mm} 0.21 \pm 0.01\hspace{2mm} &\hspace{2mm} 1.17 \pm 0.07 
\hspace{2mm}  &\hspace{2mm} 26.6/28  \\[2mm]
\end{array}
\end{array}\nonumber \\
\end{eqnarray}
The fit is in good agreement with the data, see lower plot of Fig.2
(dashed line). The parameters $g_1$ and $g_3$ are larger than
$g$, which has decreased (compared to the $\Upsilon(2S)$ decay) by a factor
of $\sim (10 - 15)$. The resonance mass $M_R$ is around 1 GeV, as it is
expected from experimental data on $\delta$ \cite{martin76} and from the
corresponding 
scalar resonances $f_0(980)$ and $f_0(1300)$ (or a mixture of them)
listed by the PDG \cite{pdg94}.
 
In summary we find that the data on the decays $\Psi' \to \Psi \pi
\pi$ can be fitted in most cases with two parameters $g$ and $g_1$
to a good accuracy. In the charmonium system we find the ratio $g_1/g =
0.35$, whereas for bottomonia we have $g_1/g \simeq 0.2$. 
The observation, that the data can be fitted accurately with operators
allowed in the chiral limit, has first been made by Novikov
and Shifman \cite{novikov81}. An exception to this rule is
found in the transition 
$\Upsilon(3S) \to\Upsilon(1S)\pi^+\pi^-$, where the amplitude (\ref{ampl})
does not give a good description. The situation can be improved by taking
$\pi \pi$ rescattering into account using the parametrization of
\cite{schenk91} 
and a  scalar resonance, the mass of which is preferably around 1 GeV. 
 
\subsection{$\Psi' \to \Psi \pi$}
%
%
In the case of the decay amplitudes of $A\rightarrow B \pi$ where $\pi=
\pi^0, \eta$ experimental values exist in the charmonium system, upper
limits are known for the $\Upsilon(2S)$ decays and for the transition
$\Upsilon(3S) \to\Upsilon(1S)\eta$ \cite{pdg94}.  For the $\Psi'(2S)$
decays we consider 
the coupling $g'$ in the Lagrangian (\ref{lsb}) with respect to the same
decay channel in the different systems,
\begin{eqnarray}\label{ratiog}
\left|\frac{g'_{b\bar{b}}}{g'_{c\bar{c}}}\right|_{\pi} &=& 
           \frac{M_{\Upsilon(2S)}}{M_{\psi(2S)}} \left[ \frac{\Gamma 
           \left( \Upsilon(2S) \to \Upsilon(1S)\pi \right)}
           {\Gamma \left( \psi(2S) \to J/\psi \pi \right)} 
           \right]^{1/2} \left( 
           \frac{|\stackrel{\rightarrow}{p_\psi}|}
           {|\stackrel{\rightarrow}{p_\Upsilon}|} \right)^{3/2} 
\nonumber \\[2mm]
|\stackrel{\rightarrow}{p_\Psi}| &=&\frac{1}{2M_{\Psi'}} \left\{ 
           \left[ M^2_{\Psi'} - (M_\Psi + M_\pi)^2 \right] 
           \left[ M^2_{\Psi'} - (M_\Psi - M_\pi)^2 \right] \right\}^{1/2}
\end{eqnarray}
where the stronger limit is found in the $\eta$ decay channel,  
\begin{equation}
\left|\frac{g'_{b\bar{b}}}{g'_{c\bar{c}}}\right|_{\pi^0} < 3.08\;,
\hspace{2cm}
\left|\frac{g'_{b\bar{b}}}{g'_{c\bar{c}}}\right|_\eta < 0.57
\end{equation} 
For $\Upsilon(3S) \to \Upsilon(1S)\eta$ the phase space is much larger
than in the other decays, i.e. the $\eta$ is by no means soft. The present
upper limits for bottomonia in the $\eta$ decay channel are similar to each
other
\begin{equation}
\frac{\Gamma(\Upsilon(3S) \to \Upsilon(1S)\eta)}{\Gamma_{total}}< 2.2
\times 10^{-3}\;,\hspace{1.5cm}
\frac{\Gamma(\Upsilon(2S) \to \Upsilon(1S)\eta)}{\Gamma_{total}}< 2
\times 10^{-3}
\end{equation}
However, equal branching ratios lead to completely different $g'$,
\begin{equation}
\left|g'_{\Upsilon(3S)}\right|\simeq 
\frac{\left|g'_{\Upsilon(2S)}\right|}{14} \hspace{2cm}
(equal\; branching\; ratios)
\end{equation}
mainly due to the difference in phase space.

\begin{figure}[p]
 \begin{center}
   \vspace*{-1.5cm}
   \epsfxsize=13.1cm
   \leavevmode
   \epsffile[0 0 624 482]{jpsi.ps}\\
   \epsfxsize=13.1cm
   \leavevmode
   \epsffile[0 0 624 482]{upsilon.ps}
 \caption[]{The curves show the fit as indicated in the text to the
normalized invariant mass 
spectra of the two pions with $x=(m_{\pi\pi} - 2M_\pi)/(M_{2S} - M_{1S} -
2M_\pi)$ and $m_{\pi\pi}=(p_{\pi^+} + p_{\pi^-})$. The  experimental data
are taken from \cite{albrecht87}.} 
 \end{center}
\end{figure}
\begin{figure}[p]
 \begin{center}
   \vspace*{-2cm}
   \epsfxsize=12.2cm
   \leavevmode
   \epsffile[0 0 510 425]{upsilon32.ps}\\
   \epsfxsize=12.2cm
   \leavevmode
   \epsffile[0 0 510 425]{upsilon31.ps}
 \caption[]{The dots represent the data from the exclusive
deacys, the circles mark the inclusive distributions. The latter are
slightly shifted to the right for clarity. The straight lines show the fit
to the invariant mass spectra of the two pions, the dashed curve in the
case of $\Upsilon(3S)\rightarrow \Upsilon(1S)\pi^+\pi^-$ represents the fit
with  $\pi^+ \pi^-$ final state interactions taken into account (see
text). The experimental data are taken from \cite{butler94}.}
 \end{center}
\end{figure}

\section{Discussion of the Results}\label{discuss}
%
%
%
%
In this section we shall discuss our results, in particular we
shall try to give some arguments concerning 
the mass dependence of the coefficients $g$, $g_1$ and $g'$ (neglecting
contributions driven by the pion mass, i.e. $g_3=0$). We 
achieve an at least qualitative understanding by representing the
coefficients in the Lagrangian by matrix elements of heavy quark
operators. Hence we consider a model in which the heavy quarks and the
light pseudoscalar mesons are the relevant degrees of freedom; in this
respect this resembles the chiral quark model of Georgi and
Manohar \cite{ChiQua}. Note that we omit in the following discussion the
decay $\Upsilon(3S) \to 
\Upsilon(1S) \pi^+ \pi^-$ which is somewhat beyond the scope of our method
due to the large phase space for the pions.

First of all, this model easily explains the fact that the
pion-$d$-wave contribution is suppressed, since this at least requires a
dimension-6-operator for the heavy quarks, which would be contained
in the operator 
\begin{equation}
{\cal L}_{d-wave} = \frac{\gamma_2}{\Lambda^3} (\bar{Q}_v \gamma_\mu Q_v)
                    (\bar{Q}_v \gamma_\nu Q_v)
\mbox{Tr } \left[
           (\partial^\mu U)(\partial^\nu U^\dagger)
          +(\partial^\nu U)(\partial^\mu U^\dagger) \right]
\end{equation}
while the leading contribution to the pion-$s$-wave is proportional to a
dimension-3-operator for the heavy quarks,
\begin{equation}
{\cal L}_{s-wave} \propto (\bar{Q}_v Q_v)
\end{equation}
Here $Q_v$ are the operators of heavy quarks which have been rescaled with
a phase according to 
$$
Q_v = \exp (imvx ) Q
$$
and the scale $\Lambda$ is at least of the order of the chiral symmetry
breaking scale $\Lambda_{\chi SB}$ or maybe even the heavy quark 
mass; in the latter
case the $d$-wave contribution to this decay would be more strongly
suppressed in the $\Upsilon$ decays as compared to the $J/\psi$
transitions. 

We have argued that the $d$-wave contribution is suppressed and hence
we have only two operators left in the chiral limit. Both operators
contribute to the $s$-wave of the decays $\Psi' \to \Psi \pi^+ \pi^-$. 
Employing the model discussed
above, we have the corresponding two operators on the quark level
\begin{eqnarray}
{\cal L}_{quark} &=& \gamma (\bar{Q}_v Q_v)
\mbox{Tr } \left[
           (\partial^\mu U)(\partial_\mu U^\dagger) \right] \\
\nonumber  &+& \gamma_1 (\bar{Q}_v Q_v)
\mbox{Tr } \left[
           (v \cdot \partial U)(v \cdot \partial U^\dagger) \right]
\end{eqnarray}
The coupling constants $g$ and $g_1$ are related to the couplings 
in $\gamma$ and $\gamma_1$ 
the quark Lagrangian ${\cal L}_{quark}$ through the relations
\begin{eqnarray}
&& g \langle \Psi | A^{(v)\mu} B_\mu^{(v)*} | \Psi ' \rangle =
   g \epsilon_\Psi^* \cdot \epsilon_{\Psi '}
=  \gamma \langle \Psi | \bar{Q}_v Q_v | \Psi ' \rangle  \\
&& g_1 \langle \Psi | A^{(v)\mu} B_\mu^{(v)*} | \Psi ' \rangle =
   g_1 \epsilon_\Psi^* \cdot \epsilon_{\Psi '}
=  \gamma_1 \langle \Psi | \bar{Q}_v Q_v | \Psi ' \rangle
\end{eqnarray}
From the combined chiral and
heavy mass expansion one would expect that both coupling constants
$\gamma$ and $\gamma_1$ do not depend on the heavy quark mass. As
far as the mass dependence of $g$ and $g_1$ is concerned, one has to
take into account the mass dependence of the matrix element $\langle \Psi |
\bar{Q}_v Q_v | \Psi ' \rangle$. For the operators itself there exists
a heavy mass limit, while the mass dependence of the states may not be
accessed within the framework of the $1/m_Q$ expansion \cite{MS95}. 
To this end we have to rely on simple dimensionality arguments. 
Assuming that the states are normalized to unity we may write 
the matrix elements as 
\begin{equation} 
\langle \Psi | \bar{Q}_v Q_v | \Psi ' \rangle \propto E^3   
\end{equation}
where $E$ is a scale related to the binding of the heavy 
quarkonia. Here we may consider two extreme cases. The first one 
is the purely coulombic scenario, in which the parameter 
$E$ is the heavy mass times the fine structure 
constant
\begin{equation}
E = \alpha_s (m_Q) m_Q
\end{equation}
and therefore one would expect a strong scaling with the heavy mass. 
This is, however, not supported by the energy differences in the known 
quarkonia, where e.g. the level spacing $M_{2S} - M_{1S}$
is practically the same in the $J/\psi$ system and the $\Upsilon$ system. 
On the other hand, the leptonic decay widths are determined by the 
value of the wave function at the origin and for this case one 
has a significant mass dependence. 

Without considering the dynamics of the binding mechanism in detail, 
one cannot say much about the mass dependence of the matrix elements 
of the quark operators. For definiteness we consider  
the scenario in 
which $E$ is independent of the mass of the heavy quark; 
consequently we would expect that $g$ and $g_1$ are independent of 
the heavy quark mass, since the 
parameters $\gamma$ and $\gamma_1$ in the 
quark level Lagrangian ${\cal L}_{quark}$ are mass independent.
At least for the coupling $g$ this is supported by data, since  
$g_{b\bar{b}} / g_{c\bar{c}}$ is practically unity for 
$\psi(2S) \to J/\psi \pi^+ \pi^-$ and $\Upsilon(2S) \to \Upsilon(1S) 
\pi^+ \pi^-$.

However, this scenario also implies that 
$g_1 / g$ should not scale with the heavy mass. From our fits we find  
that the coupling $g_1$ is larger in the
$J/\psi$ system by a factor between two or three compared to the
$\Upsilon(2S)$ decay, while $g_1 / g$ for the different excited $\Upsilon$
states is (within the errors) constant, see eqs. (\ref{ups21},\ref{ups32}). 

The only consistent interpretation of this factor between 
$g_1 / g$ in the charm and bottom systems (given the scenario we
are considering) is that the difference is induced by the QCD 
running of the corresponding coupling 
\begin{equation}
\frac{g_{1 \, b\bar{b}}}{g_{1 \, c\bar{c}}} =
\left[ \frac{\alpha_s (m_b)}{ \alpha_s (m_c)}\right]^{\kappa}, 
\hspace{2cm} \kappa \simeq 2 - 3
\end{equation}
Consequently the second term of our effective Lagrangian must be governed
at the level of QCD by an operator with a sufficiently large anomalous
dimension. However, if our assumption that the matrix elements of the 
quark operators are mass independent is not valid, the mass dependence
of $g_1 / g$ can be also due to the mass dependence of these matrix 
elements. 

Let us finally also consider the chiral symmetry breaking contribution
relevant for the decays $\psi(2S) \to J/\psi \pi$ and $\Upsilon(2S) \to
\Upsilon (1S)\pi$. Interpreting this transition again in terms of a
dimension-3 operator for the heavy quarks one writes
\begin{equation}
{\cal L}^\prime_{quark} = \gamma^\prime (\bar{Q}_v \gamma_5 Q_v)
\mbox{Tr } \left[
           (U - U^\dagger) \right]
\end{equation}
The leading term in the heavy mass expansion will vanish for this
operator, since
$$
\bar{Q}_v \gamma_5 Q_v = {\cal O} (1/m_Q)
$$
and hence one expects the coefficients $g^\prime$ to scale as
\begin{equation}
\frac{g'_{b \bar{b}}}{g'_{c \bar{c}}} = \frac{m_c}{m_b}\simeq 0.3
\end{equation}
assuming again that the mass dependence of the matrix elements is weak. 


\section{Conclusions}
%
%
%
%
%
We have formulated an effective theory approach for the hadronic decays 
of heavy quarkonia based on the chiral limit for the light degrees of 
freedom and on the $1/m_Q$ expansion for the heavy ones. The leading 
term in both the inverse of the heavy quark mass and the inverse of 
the chiral symmetry breaking scale consists of three terms multiplied
by coupling constants which we have fitted from experiment. The fit 
to the invariant mass spectrum of the two pions in the decays 
$\psi(2S) \rightarrow J/\psi\pi^+\pi^-$ and $\Upsilon(2S)\rightarrow
\Upsilon(1S)\pi^+\pi^-$ shown in Fig.1, as well as $\Upsilon(3S)\rightarrow
\Upsilon(2S)\pi^+\pi^-$ depicted in the upper plot of Fig.2, is quite
satisfactory. In addition we have fitted the decay $\Upsilon(3S)\rightarrow
\Upsilon(1S)\pi^+\pi^-$, where however, unlike in the other cases, the fit
is rather poor (solid line in the lower plot of Fig.2). The situation can
be improved by taking $\pi\pi$ final state interactions into account, see
the dashed line in the lower plot of Fig.2.  

We also have discussed the mass dependence of the various terms in the
Lagrangian which unfortunately may not be considered entirely in the 
effective theory. We model the leading terms of the effective Lagrangian
by switching to a description in which the relevant degrees of freedom
are pions and heavy quarks. This model resembles the chiral quark model
as proposed by Georgi and Manohar \cite{ChiQua}. In this picture the 
coupling constants 
of the effective Lagrangian for the quarkonia and the pions may be
reexpressed in terms  
of the ones of the underlying quark Lagrangian and certain matrix elements 
of heavy quark operators between heavy quarkonia states. The mass dependence
of the matrix elements may not be evaluated from chiral and heavy quark 
symmetry and thus requires additional input. Using a specific assumption 
on the mass dependence of these matrix elements we arrive at definite
predictions for the ratios  $g_{b\bar{b}}/g_{c\bar{c}}$ and 
$g'_{b\bar{b}}/ g'_{c\bar{c}}$. The predictions are consistent with
what is found experimentally, although for $g'_{b\bar{b}}/ g'_{c\bar{c}}$ 
only an experimental bound exists and more data is needed to confirm or
falsify our values.\\[8mm] 
\section*{Acknowledgements}
R.U. thanks H.Leutwyler for discussions and H.Genz for
providing unpublished notes. This work is partially supported by the 
Graduiertenkolleg ``Elementarteilchenphysik an Beschleunigern''.         
 
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\end{document}




