%Title: Parton Hadron Duality in nonleptonic B hadron decays
%Author: Boris Blok and Thomas Mannel
%Published: <EM>Mod. Phys. Lett.</EM> <B>A11</B> (1996) 1263-1272. 
%<EM>Mod. Phys. Lett. </EM> <B>A 11 </B> (1996) 1263-1272
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\begin{document}
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%---------------- TTP Titlepage <---------------------------
\thispagestyle{empty}
\begin{titlepage}
\begin{flushright}
{\bf TTP 95--22} \\
     Technion-PH-95-3 \\
     Mai 1995  \\
     hep-ph/yymmnnn
\end{flushright}
\vspace{1cm}

%\boldmath
\begin{center} 
{\Large\bf PARTON HADRON DUALITY IN\vspace*{4mm} \\ 
           NONLEPTONIC $B$ HADRON DECAYS }
\end{center}
%\unboldmath
\vspace{0.8cm}

\begin{center}
{\sc Boris Blok} \vspace*{2mm} \\
{\sl Department of Physics, \\ 
     Technion - Israel Institute of Technology, Haifa 32000, Israel} \vspace*{5mm} \\ 
{\sc Thomas Mannel}  \vspace*{2mm} \\
{\sl Institut f\"{u}r Theoretische Teilchenphysik, \\
     Universit\"at Karlsruhe \\
     D -- 76128 Karlsruhe, Germany.} 
\end{center}
\vfill
\begin{abstract}
\noindent
Assuming so called global duality we argue that it is very likely 
that local duality needed to obtain results for the hadronic width 
of heavy meson decays within the $1/m_Q$ expansion holds. Hence, if 
the discrepancy between experiment and the theory concerning 
charm counting, the semileptonic branching fraction and the lifetimes
of $b$ hadrons persist, it may be taken as a hint at some 
qualitatively new effect 
in (nonperturbative) QCD or even as a new physics.  
\end{abstract}
\end{titlepage}
%---------------- END TTP Titlepage <---------------------------
\newpage
\section{Introduction}
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The $1/m_Q$ expansion allows us to perform calculations of inclusive 
heavy hadron decay rates in a model independent and QCD based framework
\cite{Bigi}.
Unlike for exclusive decays we are even able to deal with purely hadronic
processes using this method based on operator product expansion and 
Heavy Quark Effective Theory. This in turn allows us to calculate 
lifetimes and branching fraction for $b$ hardrons within this QCD 
based approach. 

The leading order of the $1/m_Q$ expansion is generically simply the 
parton model expression. The corresponding decay rates in the parton 
model are by now known for more than twenty years, and two problems 
with the rates calculated in this way have been noticed since then. 
Firstly, in the pure parton model, the semileptonic branching fraction 
comes out to be too large as compared to the measurements and secondly
in the parton model all $b$ hadron lifetimes are the same and equal the 
$b$ quark lifetime.

The hope was to fix these problems by including perturbative 
and nonperturbative corrections originating from terms of order 
$\alpha_s (m_Q)$ and $1/m_Q^2, 1/m^3_Q$.  It turned out 
soon that the nonperturbative contributions are way too small to explain 
the semileptonic branching fraction \cite{Baffling}
 as well as the lifetime difference 
between the $B$ mesons and the $\Lambda_b$ baryon
\cite{BStalk}. In fact the 
nonperturbative contributions are so small that they can safely be 
ignored given the present experimental situation. 

The purely perturbative corrections to the semileptonic decays have been
calculated some time ago \cite{Cabibbo}
 and were also found to be too small to explain
the data. The perturbative corrections to the hadronic decays 
have also been calculated in the late seventies for the case of 
vanishing masses of the final state quarks; again they are not sufficient
to explain the semileptonic braching fraction (see e.g. refs.
\cite{Petrarca,Baffling} for the recent reviews). 

This was the motivation for Bagan and coworkers \cite{Bagan}
 to perform the full 
${\cal O} (\alpha_s)$ calculation including the mass effects of the 
final state $c$ quarks in the channel $b \to c \bar{c} s$. In fact, 
Bagan et al.\ find a substantial enhancement for this channel, which,
together with the more modest enhancement in $b\rightarrow \bar cud$ channel
due to the same mechanism 
almost explains the semileptonic branching fraction. This channel 
has also been discussed previously in this context in \cite{Falketal}, 
where it was argued that the operator product expansion could fail due 
to the small energy release in this channel, since the $c$ quarks are 
heavy, or that parton hadron duality could fail due to resonances that 
are close to the point where the hadronic spectral function is replaced 
by the partonic one. 

However, one cannot blame the problem with the semileptonic branching 
fraction completely to the $b \to c \bar{c} s$ channel, since this would
increase this channel by something like (30 - 40)\%. This on the other 
hand contradicts charm counting in $B$ decays, since this would give 
for the number $n_c$ of $c$ and $\bar{c}$ quarks created per $B$ decay 
$n_c \sim 1.3$ which is about two standard deviations away from the 
experimentally observed value of $n_c = 1.07 \pm 0.07$ \cite{Glasrap}.
The QCD radiative corrections calculated in \cite{Bagan} also enhance
this channel; the full ${\cal O} (\alpha_s)$ calculation also gives 
$n_c \sim 1.3$, again in  disagreement to charm counting.  

Another problem for the $1/m_Q$ expansion is the lifetime ratio of the 
$\Lambda_b$ baryon and the $B$ meson. While the $1/m_Q$ expansion predicts 
a value in excess of 0.9 for the ratio $\tau(\Lambda_b) / \tau(B^+)$, 
recent measurements yield
$\tau(\Lambda_b) / \tau(B^+) = 0.7 \pm 0.1$ \cite{Lambdab} which is a 
deviation at the level of two standard deviations.

These two problems, namely the semileptonic branching fraction in 
combination with charm counting and the lifetime of the $\Lambda_b$ 
baryon, have attracted considerable attention over the last time, although they 
are not yet of convincing significance. Neither the nonperturbative nor
the perturbative corrections seem to be big enough to explain the problems
\cite{Baffling}. In particular, as it was mentioned above, although
the recently completed calculation of the complete ${\cal O} (\alpha_s)$ 
correction \cite{Bagan} is close to explain the discrepancy in the 
semileptonic branching fraction, it does it by the price of generating  
disagreement with charm counting. 
 
All the arguments used in the $1/m_Q$ expansion are based on parton-hadron 
duality and it has been suggested that a breakdown of duality is the 
explanation of these discrepancies. It has been pointed out that unlike in the 
calculation of the semileptonic branching fraction, where {\it global}
duality is needed, in the calculation of the hadronic width of the $B$ 
meson {\it local} duality needs to be invoked: In the calculation 
of the semileptonic width only a weighted average of the sepectral function 
of hadronic operators is replaced by the corresponding average of the 
parton model expression, while for the hadronic width one replaces the  
hadronic spectral function at a fixed kinematic point (i.e.\ locally) 
by the partonic expression. 

In this short note we shall assume that global duality holds and investigate
to what extend the hadronic spectral function may deviate from the paron model
expressions. Our arguments are necessarily qualitative, since it involves 
the nonperturbative sector of QCD.  Intuitively one expects that the 
final state quarks my form resonance states which may lead to a hadronic spectral 
function different from the partonic one. We shall use simple resonance 
models to study the possible effects. 


\section{Parton Hadron Duality}
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%
We shall start from the effective Hamiltonian 
describing nonleptonic $B$ decays, neglecting $b \to u$ 
transitions and penguins
\begin{eqnarray} \label{heff}
H_{eff} = \frac{G_F}{\sqrt{2}} V_{cb} \sum_{U = u,c} && \left[ C_2
  (\bar{b} \gamma_\mu (1-\gamma_5) c) 
  (\bar{U} \gamma_\mu (1-\gamma_5) D_U) \right. \\ 
&& \left. + C_1
  (\bar{b} \gamma_\mu (1-\gamma_5) D_U) 
  (\bar{U} \gamma_\mu (1-\gamma_5) c)\right] \nonumber 
\end{eqnarray}
where $D$ is the Cabbibo-rotated down type quark field
\begin{eqnarray}
D_u &=& V_{ud}^* d + V_{us}^* s \\
D_c &=& V_{cd}^* d + V_{cs}^* s \nonumber 
\end{eqnarray}
and $C_1$ and $C_2$ are the Wilson coefficients in the leading log 
approximation
\begin{eqnarray}
C_1 &=& \frac{1}{2} \left[ 
        \left(\frac{\alpha_s(M_W)}{\alpha_s(m_b)}\right)^{\gamma_+} -
        \left(\frac{\alpha_s(M_W)}{\alpha_s(m_b)}\right)^{\gamma_-} \right] \\
C_2 &=& \frac{1}{2} \left[ 
        \left(\frac{\alpha_s(M_W)}{\alpha_s(m_b)}\right)^{\gamma_+} +
        \left(\frac{\alpha_s(M_W)}{\alpha_s(m_b)}\right)^{\gamma_-} \right] 
\nonumber
\end{eqnarray}
Here 
$$\gamma_+=\frac{6}{33-2N_f}=-\frac{1}{2}\gamma_-.$$
Using this Hamiltonian we may define the forward matrix element
\begin{equation}
T(q^2,vq) = \int d^4x \, e^{-iqx} \langle B(v) | 
            H_{eff} (x) H_{eff} (0) | B(v) \rangle 
\end{equation}
where $v = P_B/m_B$ is the velocity of the $B$ meson. 

We shall use the correlator $T$ to study parton hadron duality. The 
analytical structure of $T$ for complex $vq$ and fixed $q^2$ 
is given by the intermediate states 
$| X \rangle$ that may be excited by $H_{eff}$
\begin{equation}
T(q^2,vq) = \int d^4 x \, (2\pi)^4 \delta^4 (m_B v - q - P_X)  
| \langle B(v) | H_{eff} | X  \rangle | ^2  . 
\end{equation}  

The support of the function $T$ may easily be calculated. For a fixed 
$q^2$, $T$ is nonvanishing in the interval 
\begin{equation}
vq < \frac{1}{2m_B} (q^2 + m_B^2 - \mu^2) \mbox{ and }
vq > \frac{1}{2m_B} (q^2 + 2 m_B^2 + \mu^2)
\end{equation}
where $\mu$ is the mass of the lightest hadronic state which may appear in the 
sum. In our case this state is a $D\pi$ state where both the $D$ and the 
$\pi$ are at rest, so $\mu = m_D + m_\pi$. 

In the parton model these intermediate states are three quark states 
at the tree level, and the function $T$ may be calculated to be 
\begin{eqnarray}
T(q^2,vq) &=& \left(\frac{G_F^2 m_b^4}{96 \pi^3} \right)
\langle B(v) | \bar{b} b | B(v)  \rangle N_c 
\left( C_1^2 + C_2^2 + \frac{2C_1 C_2}{N_c} \right) V_{cb} \\ 
\nonumber 
&& \left[ V_{ud}^* F_2 (m_c^2/Q^2) + V_{cs}^* F_3 (m_c^2/Q^2) \right]
\end{eqnarray}
where $Q = m_B v - q$. Furthermore, 
we have neglected CKM suppressed terms proportional to $V_{us}$ and
$V_{cd}$ and 
\begin{eqnarray}
F_2 (x) &=&(1-x^2)(1-8x+x^2)-12x^2 \ln (x)\Theta(1-x) \\
F_3 (x) &=& v(1-14x-2x^2-12x^3)
+24x^2(1-x^2)\ln\frac{1+v}{1-v}\Theta (1-4x),\\
&&  v=\sqrt{1-4x}
\end{eqnarray}
Note that the functions $F_2$ and $F_3$ have support in the intervals 
\begin{equation}
vq < \frac{1}{2m_B} (q^2 + m_B^2 - m_c^2) .
\end{equation}
and
\begin{equation}
vq<\frac{1}{2m_B}(q^2+m_B^2-4m_c^2)
\end{equation}
respectively.  

Parton hadron duality means that the correlator $T_{parton}$ calculated 
in the parton model is ``on the average'' equal to the true hadronic 
correlator $T_{hadron}$
\begin{equation} \label{ave}
\int d(vq) \, T_{parton} (q^2, vq) w(vq) =      
\int d(vq) \, T_{hadron} (q^2, vq) w(vq), \quad q^2 \mbox{ fixed}
\end{equation}
where $w$ is some smooth weighting function having support over 
some averaging interval. Close to the regions where 
only the $D\pi$ or $J/\Psi \, K$ states
contribute the partonic and the hadronic results 
have in general a quite different shape, and the interval over 
which one needs to average in (\ref{ave}) is larger than if we consider 
the two functions at higher invariant masses. THis situation has been 
analyzed for the case of the total cross section $e^+ e^- \to $ hadrons 
in \cite{PQW}. 

For the calculation of 
the total hadronic width we have to consider the correlator at the point where
the invariant mass of the final state hadrons is equal to the $B$ meson mass 
\begin{equation}
\Gamma (B \to \mbox{hadrons}) \propto  T(0,0) ,
\end{equation} 
and the common folklore  is that at this high invariant
mass one may savely shrink the interval of averaging in (\ref{ave}) to a point
and simply equate the partonic with the hadronic correlator. In other words, 
we shall assume that ``global'' duality in the sense of a large 
averaging interval holds and discuss under which circumstances we may shrink
the averaging interval to a single point, thereby going from ``global'' 
to ``local'' duality.
 
The situation in the nonleptonic decays has to be compared with the one
in semileptonic decays \cite{CDG,Bigi2}, where 
the hadronic contribution is the product of the two hadronic currents. 
In this case $q$ may be identified with the momentum transferred to the 
leptons. While the analytical structure is quite similar (the lowest state
being a $D$ meson at rest) the integration over the neutrino momentum 
introduces a smooth averaging such that parton hadron duality is guaranteed
in almost all phase space for the charged lepton, except for the endpoint, 
where the energy of the charged lepton becomes maximal. Only if we would 
consider
doubly differential distributions such that the invariant mass of the 
final hadronic state would be fixed, a similar situation would occur as 
in the nonleptonic decays, namely that the interval over which we average 
shrinks to a point. In this sense one expects that the calculation of 
the lepton spectrum as well as of the total semileptonic rate is more 
reliable than the calculation of the total nonleptonic rate, since in the 
semileptonic case only global duality is needed, while the nonleptonic 
case rests on the assumption of global duality.


\section{Resonance Model}
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Motivated by this reasoning we have considered the correlator for the 
nonleptonic decays in the context of very simple model. The idea is to 
identify possible contributions which could generate some structure of 
the hadronic result which could result in a difference between 
$T_{parton} (0,0)$ and $T_{hadron} (0,0)$ hence a violation of local
duality, although global duality holds.
As pointed out above, the 
partonic tree level result is a smooth function of 
$Q^2 = m_b^2 + q^2 - 2m_b(vq)$, 
since it is obtained from a three particle phase space for the three 
final state quarks. One possibility one may consider is that two of 
these final state quarks form a resonance, corresponding e.g. to the 
semiinclusive final processes $B \to D_{(s)}$+ anything or 
$B \to J/\Psi$+ anything. In order to study the qualitative features of 
this idea, we consider as a simple model for such a situation the 
effective Hamiltonian 
\begin{equation} \label{resmod}
H_{eff}^\prime = \frac{G_F}{\sqrt{2}} f_\phi 
                 (\bar{b} \gamma_\mu (1-\gamma_5) q) \phi^\mu 
\end{equation}
where $\phi_\mu$ is the field of a vector like resonance  
of mass $M$ made of a quark-antiquark
pair, and $f_\phi$ is its coupling strength to the left handed 
color singlet $b \to q$ current, including all the CKM matrix elements.
A similar model has been introduced by Stech and Palmer \cite{Stech}
to analyze semiinclusive nonleptonic $B$ decays.    

It is a simple exercise to calculate the corresponding correlator. 
The resonance contribution to $T$ near threshold  turns out to be 
\begin{eqnarray} \label{rescon}
T(q^2, vq ) &=& \frac{G_F^2}{16 \pi} f_\phi^2   
\langle B(v) | \bar{b} b | B(v) \rangle  \left(\frac{1}{Q^2}\right)^2 \\
\nonumber 
&& \quad \sqrt{(Q^2 - (M - m_q)^2)(Q^2 - (M + m_q)^2)} 
\Theta(Q^2 - (M - m_q)^2) \\ \nonumber 
&& \quad \left[Q^2 - M^2 + m_q^2 + \frac{1}{M^2}
      ((Q^2 - m_q^2)^2 - M^4) \right] 
\end{eqnarray}
Contributions of other resonances to the spectral function are smooth
since these resonances lie away from threshold, and we do not write 
it explicitly.
Generically the phase space contributes the factor involving the square 
root, and in the limit in which $M, Q^2 
\gg m_q$ the r.h.s. of (\ref{rescon}) is 
a smooth function even at threshold
\begin{eqnarray}
T(q^2, vq ) &=& \frac{G_F^2}{16 \pi} f_\phi^2   
\langle B(v) | \bar{b} b | B(v) \rangle  \left(\frac{1}{Q^2}\right)^2 \\
\nonumber 
&& \quad (Q^2 - M^2)^2 (Q^2 + M^2)  
\Theta(Q^2 - M^2) 
\end{eqnarray}

In our case $m_q$ is just the mass of the light strange quark.
This means that a scenario of a resonance as modelled in (\ref{resmod}) 
such that the thereshold is 
close to the mass of the $B$ meson does not introduce any rapid variations
in the hadronic spectral function. Furthermore, since we assume global 
duality in the sense of (\ref{ave}), one finds that the hadronic and 
partonic spectral functions cannot be drastically different, at least 
not in a scenario as moddelsed by the Hamiltonian (\ref{resmod}).
Hence we conclude that it is quite likely that the partonic 
correlator $T$ is a good approximation to the hadronic,  
even for the case of a resonance as we introduced above fairly 
close to the mass of the $B$ meson. 

Typically the correlators obtained from intermediate states with 
two or more partons or hadrons are smooth functions of the variable
$Q^2 = m_B^2 + q^2 -2m_b (vq)$ and any resonance 
formation involving at least a two particle intermediate state does not 
introduce rapid variations of the correlator. For all these states we 
may apply the same argument as above, leading to the statement that 
it is quite likely that local duality holds. In this framwork, the only way  
to introduce strong variations is if there is a relatively narrow
resonance $R_c$ leading to 
a single particle intermediate state in the correlator $T$. 

In order  not to get into conflict with charm counting, this resonance 
state has to appear in the channel $b \to c \bar{u} d$, thereby enhancing 
this channel by some amount, which then would explain the semileptonic
branching fraction without enhancing the charm created in $B$ decays. 
In this case, the correlator would take the form 
\begin{equation}
T(q^2,vq) = 2M\Gamma \frac{\langle B(v) | H_{eff} |R_c \rangle 
\langle R_c | H_{eff} | B(v) \rangle}{(Q^2 - M^2)^2 + M^2\Gamma^2}
\end{equation}
where $M$ and $\Gamma$ are the mass and the width of $R_c$ respectively. 
If this resonance would be sufficiently narrow and its mass sufficiently 
close to the mass of the $B$ meson,  parton hadron duality 
could still hold on the average, but the interval of averaging would simply 
be too large to justify the calculation of the total hadronic width using 
local duality. 

However, this possibility is quite unlikely. First of all, it would not help 
to explain the lifetime problem with the $\Lambda_b$ baryon, since such a 
resonance state would increase the width of the $B$ mesons due to an increase 
in the channel $b \to c \bar{u} d$,  and hence would make 
their lifetime shorter compared to the parton model value. Assuming that there
is not such a resonance in the channel involving the baryons, this would increase
the ratio $\tau(\Lambda_b) / \tau(B)$, which will make the discrepancy even 
stronger. 

Furthermore, the effect of such a resonance would be visible as well in the 
semileptonic channel, if the resonance state would have isospin $I=1/2$. 
However, one may avoid any influence on the semileptonic decays, if the 
isospin of $R_c$ is $I = 3/2$.  This would
mean that in a quark model picture this state is a four quark state 
consisting of a two quarks and two antiquarks. 

However, the $b \to c\bar{u}d$ piece of the effective Hamiltonian (\ref{heff}) 
is an isospin triplet with $I_3 = -1$, and hence the hadronic decays of the 
charged $B$ mesons will have only final states with isospin $I = 3/2$, while the 
final states of the nonleptonic decays of the neutral $B$ meson will be a 
mixture of $I = 3/2$ and $I = 1/2$. Consequently an enhancement of the 
$I = 3/2$ contribution through a resonance would make the lifetime of 
the charged $B$ meson shorter than the one of the neutral states, which 
is in conflict with the data on the lifetime ratio between charged and neutral 
$B$ mesons, which indicate that the lifetimes are equal 
$\tau (B^+) / \tau (B^0) = 1.009 \pm 0.069$ \cite{Glasrap}. Hence there is 
not much room for such a resonance state and even if it would exist, 
its contribution would be not sufficient to explain the semileptonic 
branching fraction problem.  


\section{Conclusions} 
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Even after more detailed calculations of the QCD radiative corrections 
the situation concerning charm counting and the semileptonic branching 
fraction remains inconclusive. A failure of local parton hadron duality 
as it is needed to obtain results within the $1/m_Q$ expansion has been 
considered as a possible solution of the problem, but we have argued 
based on models that local parton hadron duality likely to hold, if 
global duality is assumed. 

Global duality, on the other hand, is a concept without which we could 
not relate many theoretical results to data, which in turn are compatible 
with the assumption of global duality. Hence the semileptonic branching 
fraction problem and the charm counting remain open questions in view of 
the present data. Either the data on charm counting 
change, such that $n_c \sim 1.3$. In this case the problem is solved, 
and such a result would also be in agreement with the recent calculation 
of the full ${\cal O} (\alpha_s)$ corrections \cite{Bagan}. However, 
if data remain as they are and the errors become smaller, than there have 
to be either yet unknown effects in nonperturbative QCD, or the 
discrepancies find an explanation in terms of physics beyond the 
standard model \cite{kagan,shifman}. However, we find an explanation 
in terms of QCD effects more likely, since physics beyond the standard 
model involves scales much larger than the $B$ meson mass and hence 
will affect the decays of the $b$ {\it quark} rather than the ones of the 
hadrons, in other words, such an explanation cannot be the source of the 
$\Lambda_b$ lifetime problem.



\section*{Acknowledgements}
The authors thank M. Beneke, M. Shifman and  A. Vainshtein for very 
usefull discussions. T.M. acknowledges the hospitality of the particle 
theory group of the Techion, where this work has been initiated. 

\begin{thebibliography}{99}

\bibitem{Bigi}
I. Bigi, N. Uraltsev and  A. Vainshtein, {\it Phys. Lett.} {\bf B293}
(1992)
430; (E) B297 (1993) 477.\\
B. Blok and M. Shifman, {\it Nucl. Phys.} {\bf B399} (1993) 441; 459\\
I. Bigi, B. Blok, M. Shifman, N. Uraltsev and  A. Vainshtein, {\it
The Fermilab Meeting} Proc. of
the 1992 DPF meeting of APS, C. H. Albright {\it et al.}, Eds. World
Scientific, Singapore 1993, vol. 1, page 610.

\bibitem{Baffling} I. Bigi et al, Phys. Lett., B323 (1994) 408.

\bibitem{BStalk}
B. Blok and M. Shifman, in  Proc. of the Third Workshop on the  {\em 
Tau/Charm 
Factory}, Eds. J. Kirkby and R. Kirkby, 
Marbella, May 1993 (Editions Frontieres, Gif-Sur-Yvette 1994),
page 247.

\bibitem{Petrarca} A. Altarelli and  S. Petrarca, Phys. Lett., B263 (1991) 303.

\bibitem{Cabibbo}
N. Cabbibo and  L. Maiani, {\it Phys. Lett.} 
{\bf 79B} (1978) 109.

\bibitem{Bagan} E. Bagan et al, Nucl. Phys. B432 (1995) 3; preprint
CERN-TH/95-25, hep-ph/9502338.

\bibitem{Falketal} A. Falk et al, Phys. Rev., D51 (1995) 1183; I . Dunietz 
et al, Phys. Rev. Lett., 73 (1994) 1075.

\bibitem{five} I. Bigi et al, in B-decays, p. 132, World Scientific,
 Second edition, 1995. Ed. S. Stone. 

\bibitem{Glasrap} P. Roudeau, Rapporteur 
                  talk at the ICHEP 94 Conference, Glasgow, Scottland. 

\bibitem{Lambdab}see e.g. P.S. Wells, preprint CERN-PPE/94-203 (December 
1994).

\bibitem{PQW}{E. Poggio, H. Quinn and S. Weinberg, 
         Phys. Rev. D13 (1976) 1958.} 

\bibitem{CDG} J. Chay, H. Georgi, B. Grinstein, Phys. Lett., B247 (1990) 
399.

\bibitem{Bigi2} I. Bigi et al, Phys. Rev. Lett., 71 (1993) 496. 

\bibitem{Stech} B. Stech and W. Palmer, Phys Rev D48 (1993) 4174.

\bibitem{kagan} A. Kagan, preprint SLAC-PUB-6626, September 1994/hep-ph 
9409215. 

\bibitem{shifman} L. Roszkowski and M. Shifman, TPI-MINN-95-04/T/hep-ph 
9503358, March 1995. 

\end{thebibliography}

\end{document}

exclusive resonanes (old)

Such a $I = 3/2$ resonance state $R_c$ could decay like $R_c \to D \pi$, 
and hence it would effect the nonleptonic decays $B \to D \pi$. The 
$b \to c \bar{u} d$ part of the effective Hamiltonian (\ref{heff}) allows
for $I = 3/2$ and $I = 1/2$ final states in the decays $B \to D \pi$. 
Performing a simple isospin analyis we obtain 
\begin{eqnarray}
\Gamma (B^+ \to \bar{D}^0 \pi^+) &=& |{\cal A}_{3/2}|^2 \\
\Gamma (B^0 \to      D ^- \pi^+) &=& \frac{1}{3} |{\cal A}_{3/2}|^2 
                                  +\frac{2}{3} |{\cal A}_{1/2}|^2
\end{eqnarray}
where ${\cal A}_I$ is the amplitude with isospin $I$ of the final state.
If we were to calculate these decays in pure factorization, we would obtain 
\begin{equation}
\Gamma (B^+ \to \bar{D}^0 \pi^+) = 
\Gamma (B^0 \to      D ^- \pi^+) 
\end{equation}
and hence both isospin amplitudes should be the same. This is, however, 
not what is observed. Using the data of \cite{PDG94}
\begin{eqnarray}
\Gamma (B^+ \to \bar{D}^0 \pi^+) &=& (5.3 \pm 0.5) \times 10^{-3} \\
\Gamma (B^0 \to      D ^- \pi^+) &=& (3.0 \pm 0.4) \times 10^{-3}
\end{eqnarray}
which shows indeed a significant enhancement of the charged $B$ decay. This 
translates into an enhancement of the amplitude ${\cal A}_{3/2}$ by 
a factor 1.7  relative to the $I = 1/2$ amplitude
\begin{equation}
\frac{|{\cal A}_{1/2}|^2}{|{\cal A}_{3/2}|^2} = \frac{3}{2} 
\frac{\Gamma (B^0 \to      D ^- \pi^+)}{\Gamma (B^+ \to \bar{D}^0 \pi^+)}
-\frac{1}{2} \sim 0.35  .
\end{equation}
This may be interpreted in our context as an enhancement of the 
$I = 3/2$ amplitude due to a nonfactorizable 
contribution to the two-body exclusive decay $B \to D \pi$ due to 
a nearby resonance $R_c$. 
Similarly one may analyse the decays $B \to D \rho$ which also show an 
enhancement of the $I = 3/2$ amplitude over the $I = 1/2$ one of the 
same order. 

However, the existence of the exotic state in the I=3/2 channel will 
imply more. All two particle final states in $B^+$ decays have isospin 3/2.
The degenerate resonance will lead to the uniform enhancement of the 
3/2 over 1/2 amplitude in all the channels. In the simplest model of 
inclusive decays of B-s we may assume that 
they go via two particle channels. The resonances that are created 
in two particle decays then decay further strongly into multiparticle 
states. Such model was discussed in ref. \cite{Isgur}. 
Then the lifetimes and the widths of $B^+$ and $B^0$ mesons will be 
different. Since the current ratio of the widths of charged and neutral
B-mesons is currently restricted by $0.98\pm 0.09$ \cite{PDG94}, it is 
clear that the existence of the exotic state described above is highly 
unlikely. 




conclusions (old)

We conclude that we can safely use quark-hadron duality in the 
calculation of nonleptonic widths of heavy mesons, and whatever is the 
origin of the problem of the semileptonic branching ratio, it is not in   
the violation of the local quark-hadron duality.

This means that if we believe in the present day experimental data,
it is highly unlikely that there is any conventional explanation of the 
problem of semileptonic branching ratio
(e.g. breaking of quark parton duality in nonleptonic B-decays,
large $\alpha_s$ corrections in $b\rightarrow c\bar c s$ channel, 
large $\alpha_s^2$ corrections ) 
 that does not enter into the 
conflict with the charm counting problem. In particular we have seen that 
at least if there is no significant difference of lifetimes of
neutral and charged  B-mesons the quark-hadron duality works in nonleptonic
decays as good as in the semileptonic case, meaning that the duality 
based calculations are quite reliable for B-mesons. The only way 
to break duality is to assume that there is an exotic state in the isospin 
3/2 channel and mass close to the masses of B-mesons. However such state 
will also mean unusually big difference of lifetimes of charged and neutral
B-mesons, that is highly  unlikely in view of the
of the current experimental data.

This leaves us with two possibilities: either there is 
new unknown physics in QCD or there is new physics beyond standard model.
The latter explanation was pursued in several recent papers
\cite{kagan,shifman}.
The former explanation  is yet to be found.

