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\setlength{\parindent}{0cm}

\begin{document}

\title{\Large\bf\sc Weak Decays Of Heavy Mesons In A Covariant Quark Model}

\author{Dirk Merten, Ralf Ricken, Matthias Koll, Bernard Metsch, Herbert Petry\\\vspace*{0.25cm}}

\address{Institut f\"ur Theoretische Kernphysik,\\ Nu{\ss}allee 14--16,\\D-53115 Bonn,\\Germany\\e-mail: merten@itkp.uni-bonn.de\\}


\date{\today}

\maketitle


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\input{semilep}
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\end{document}


\begin{thebibliography}{99}

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\bibitem{Koll} M. Koll, R. Ricken, D. Merten, B. Metsch and H. Petry, Eur. Phys. J. {\bf A9}, 73 (2000).

\bibitem{Ralf} R. Ricken, M. Koll, D. Merten, B. Metsch and H. Petry, Eur. Phys. J. {\bf A9}, 221 (2000).

\bibitem{Andre} A. Kaufmann, diploma thesis, Universit\"at Bonn TK-99-01 (1999).

\bibitem{ZHMB} G. Z\"oller, S. Hainzl, C.R. M\"unz and  M. Beyer, Z. Phys.  {\bf C68}, 103 (1995).

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\bibitem{Hooft} G. 't Hooft, Phys. Rev. {\bf D14}, 3432 (1976).

\bibitem{Shifman} M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. {\bf B163}, 46 (1980).

\bibitem{Mandelstam} S. Mandelstam, Proc. Roy. Soc. {\bf 233}, 248 (1955).

\bibitem{slrev} J.D. Richman and P.R. Burchat, Rev. Mod. Phys. {\bf 67}, 893 (1995).

\bibitem{Isgur} N. Isgur and M.B. Wise, Phys. Lett. {\bf B232}, 113 (1989).

\bibitem{Isgur2} N. Isgur and M.B. Wise, Phys. Lett. {\bf B237}, 527 (1990).

\bibitem{ISGW2} D. Scora, N. Isgur, Phys. Rev. {\bf D52}, 2783 (1995).

\bibitem{Godfrey} S. Godfrey and N. Isgur, Phys. Rev. {\bf D32}, 189 (1985).

\bibitem{Beneke}  M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, .

\bibitem{WSB} M. Wirbel, B. Stech and M. Bauer, Z. Phys. {\bf C29}, 637 (1985).

\bibitem{MS} D. Melikhov and B. Stech, .

\bibitem{caprini} I. Caprini, L. Lellouch and M. Neubert, Nucl. Phys. {\bf B530}, 153 (1998).

\bibitem{Neubert} M. Neubert and B Stech, in: Heavy Flavours II, edited by A.J. Buras and M. Lindner (World Scientific, Singapore, 1998), p. 294.

\bibitem{Ciulli} V. Ciulli, talk presented at 8th International Symposium on Heavy Flavour Physics,
 University of Southampton, July 1999, JHEP Conf. Proc. PRHEP-hf8/044, .

\bibitem{CLEODstar} {\sc Cleo} Collaboration, B. Barish {\sl et al.}, Phys. Rev. {\bf D51}, 1014  (1995).

\bibitem{CLEOD}  {\sc Cleo} Collaboration, M. Athanas {\sl et al.}, Phys. Rev. Lett. {\bf 79}, 2208 (1997).

\bibitem{CLEODstp} {\sc Cleo} Collaboration, S. Anderson {\sl et al.}, Nucl.Phys. {\bf A663-664}, 647 (2000).

\bibitem{CLEOAvery} {\sc Cleo} Collaboration, P. Avery {\sl et al.}, CLEO CONF 94-7.

\bibitem{CLEO2k} {\sc Cleo} Collaboration, J.P. Alexander {\sl et al.},  .

\bibitem{E791DK} {\sc E791} Collaboration, E.M. Aitala {\sl et al.},  Phys. Lett. {\bf B440}, 435 (1998).

\bibitem{E791Dsphi} {\sc E791} Collaboration, E.M. Aitala {\sl et al.},  Phys. Lett. {\bf B450}, 294 (1999).

\bibitem{L3} {\sc L3} Collaboration, M. Acciarri {\sl et alt.}, Phys. Lett. {\bf B465},  323 (1999).

\bibitem{OPAL} {\sc Opal} Collaboration, G. Abbiendi {\sl et al.},  Phys.Lett. {\bf B482},  15 (2000).


\end{thebibliography}
\section*{FIGURES}
\vspace*{-0.5cm}
\begin{center}
\begin{figure}
\input{Weak_Decays_isovector.pstex_t}\\[1ex] %40%
\caption{The spectrum of the isovector mesons. The centric column shows the experimental data from \protect\cite{PDG2k}. Errors are indicated by  shadowed boxes. The left and right columns show our results with model $\cal A$ and $\cal B$, respectively. }\label{isovec}
\end{figure}
\begin{figure}
\hfill
\input{Weak_Decays_D.pstex_t}%\\[1ex] %40%
\hfill
\input{Weak_Decays_Ds.pstex_t}%\\[1ex] %40%
\hspace*{1cm}\\[1ex]
\caption{The spectra of the charmed $D$ and $D_s$ mesons.  The centric column shows the experimental data from \protect\cite{PDG2k}. Errors are indicated by  shadowed boxes. The levels marked with {\tiny $^\dagger$} are taken from \protect\cite{CLEODstp}. The left and right columns show our results with model $\cal A$ and $\cal B$, respectively. Note that the $1^+$ states are twofold degenerated in our calculation corresponding to the  total spin $S=0$ and $S=1$.}\label{DDs}
\end{figure}

\begin{figure}
\hfill
\input{Weak_Decays_B.pstex_t} %40%
\hfill
\input{Weak_Decays_Bs.pstex_t} %40%
\hspace*{1cm}\\[2ex]
\caption{The spectra of the bottom $B$ and $B_s$ mesons.  The centric column shows the experimental data from \protect\cite{PDG2k}. Errors are indicated by  shadowed boxes. The levels marked with {\tiny $^\dagger$} are taken from \protect\cite{Ciulli}. The left and right columns show our results with model $\cal A$ and $\cal B$, respectively. Note that the $1^+$ states are twofold degenerated in our calculation corresponding to the  total spin $S=0$ and $S=1$.}\label{BBs}
\end{figure}

\begin{figure}
\input{Weak_Decays_Charmonium.pstex_t} %40%
\hfill
\input{Weak_Decays_Bottomonium.pstex_t}\\[2ex] %40%
\caption{The spectra of heavy quarkonia. The centric column shows the experimental data from \protect\cite{PDG2k}. Errors are indicated by  shadowed boxes. The left and right columns show our results with model $\cal A$ and $\cal B$, respectively. }\label{onium}
\end{figure}
\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\input{BD.tex}%\\[1ex]
\vspace*{2ex}
\caption{The differential decay rate for $B\to D\ell\bar\nu$. The data are taken from \protect\cite{CLEOD}. The values of $|V_{cb}|$ corresponding to our calculations are  $|V_{cb}| = 0.034$  and $|V_{cb}| = 0.035$ for model $\cal A$ and $\cal B$, respectively.\label{CLEO_BD} }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\input{BDstar.tex}
\vspace*{2ex}
\caption{The differential decay rate for $B\to D^*\ell\bar\nu$. The data are taken from \protect\cite{CLEODstar}. The values of $|V_{cb}|$ corresponding to our calculations are  $|V_{cb}| = 0.034$  and $|V_{cb}| = 0.035$ for model $\cal A$ and $\cal B$, respectively.\label{CLEO_BDstar} }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section*{TABLES}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\table
\begin{table}
\begin{tabular}{cldd}
\\
 		& \bf Parameter				& \bf Model $\cal A$	&  \bf Model $\cal B$       \\
\\\hline\\
 		&$m_n$ \space [MeV] 			& 306 			& 380                   \\[0,5ex]
\sl Constituent	&$m_s$ \space [MeV] 			& 503 			& 550                   \\[0,5ex]
\sl quark masses	&$m_c$ \space [MeV]		& 1835			& 1780                  \\[0,5ex]
		&$m_b$ \space [MeV]			& 5240			& 5092                  \\
\\
		&$g_{nn}$ \space [GeV$^{-2}$]		& 1.73	 		& 1.62                 \\[0,5ex]
		&$g_{ns}$ \space [GeV$^{-2}$]		& 1.54 			& 1.35                 \\[0,5ex]
\sl Residual	&$g_{nc}$ \space [GeV$^{-2}$]      	& 1.11 			& 1.58                \\[0,5ex]
\sl   interaction	&$g_{nb}$ \space [GeV$^{-2}$]   & 0.53  		& 1.07                 \\[0,5ex]
	 	&$g_{sc}$ \space [GeV$^{-2}$]      	& 0.65  		& 1.27                 \\[0,5ex]
		&$g_{sb}$ \space [GeV$^{-2}$]      	& 0.00			& 0.76                 \\[0,5ex]
		&$\Lambda$ \space [fm] 			& 0.30			& 0.42                  \\ 
\\
\sl Confinement	&$a_c$ \space [GeV]			& $-$1.751 		& $-$1.135              \\[0,5ex]
\sl parameters	&$b_c$ \space [GeV fm$^{-1}$]     	&   2.076		& 1.300                 \\
\\
\sl Spin structure	&$\mit\Gamma\otimes\mit\Gamma$	& $\frac12\left(\id\otimes\id - \gamma^0\otimes\gamma^0\right)$	&  $\frac12\left(\id\otimes\id - \gamma^5\otimes\gamma^5-\gamma_\mu\otimes\gamma^\mu\right)$	\\
\\
\end{tabular}
\vspace*{0.5ex}
\caption{The parameters of the confinement force, the 't Hooft interaction and the constituent quark masses in the models $\cal A$ and $\cal B$.\label{params}}
\end{table}
\endtable
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\table
\renewcommand{\arraystretch}{1}
\begin{tabular}{lcdd@{\hspace{2em}}|lcdd}
&&&&&&&\\
\bf Meson ($J^{\pi}$)	& \bf  n	& \bf Model $\cal A$	& \bf Model $\cal B$	& \bf Meson ($J^{\pi}$)	&  \bf n	& \bf Model $\cal A$	& \bf Model $\cal B$\\
&&&&&&&\\\hline&&&&&&&\\
$D(0^-)$		&  0	& 1869			& 1869			& $D_s(0^-)$		&  0	& 1969			& 1969\\
			&  1	& 2677			& 2578			& 			&  1	& 2794			& 2683\\
			&  2	& 3258			& 3041			&			&  2	& 3388			& 3152\\
&&&&&&&\\												    
$D^*(1^-)$		&  0	& 1993			& 2034			& $D_s^*(1^-)$		&  0	& 2049			& 2116\\
			&  1	& 2769			& 2648			&			&  1	& 2852			& 2746\\
			&  2	& 2822			& 2679			& 			&  2	& 2905			& 2776\\
			&  3    & 3327			& 3079			&			&  3	& 3432			& 3192\\
			&  4    & 3344			& 3101			&			&  4	& 3454			& 3210\\
&&&&&&&\\
$D_0^*(0^+)$		&  0	& 2519			& 2375			& $D_{s0}^*(0^+)$	&  0    & 2563			& 2464\\
			&  1	& 3115			& 2884			&            		&  1    & 3196			& 2986\\
&&&&&&&\\												        
$D_1(1^+)$		&  0	& 2464			& 2420			& $D_{s1}(1^+)$		&  0    & 2532			& 2506\\
			&  1	& 2464	 		& 2420			&   			&  1    & 2532			& 2506\\
			&  2	& 3075			& 2908			&            		&  2    & 3172			& 3007\\
			&  3	& 3075			& 2908 			&            		&  3    & 3172			& 3007\\
&&&&&&&\\												        
$D_2^*(2^+)$		&  0	& 2475			& 2469			& $D_{s2}^*(2^+)$      	&  0    & 2541			& 2552\\
			&  1	& 3086			& 2930			& 			&  1    & 3182			& 3032\\
			&  2	& 3128			& 2957			&			&  2    & 3223			& 3057\\
&&&&&&&\\
%			&	&			&			&			&	&			&\\
\end{tabular}
\vspace*{0.5ex}
\caption{Masses of the charmed $D$ and $D_s$ mesons in MeV, calculated in model $\cal A$ and $\cal B$; n denotes the radial excitation.  Note that the $1^+$ states are twofold degenerated  corresponding to the  total spin $S=0$ and $S=1$.}

\vfill
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\arraystretch}{1}
\begin{tabular}{lcdd@{\hspace{2em}}|lcdd}
&&&&&&&\\
\bf Meson ($J^{\pi}$)	& \bf  n	& \bf Model $\cal A$	& \bf Model $\cal B$	& \bf Meson ($J^{\pi}$)	&  \bf n	& \bf Model $\cal A$	& \bf Model $\cal B$\\
&&&&&&&\\\hline&&&&&&&\\
$B(0^-)$		&  0	& 5279			& 5279			& $B_s(0^-)$		&  0	& 5368			& 5369\\
			&  1	& 6002			& 5869			& 			&  1	& 6101			& 5960\\
			&  2	& 6517			& 6279			&			&  2	& 6628			& 6376\\
&&&&&&&\\			  			      						    	  			  
$B^*(1^-)$		&  0	& 5325			& 5346			& $B_s^*(1^-)$		&  0	& 5369			& 5425\\
			&  1	& 6035			& 5900			&			&  1	& 6102			& 5986\\
			&  2	& 6108			& 5947			& 			&  2	& 6172			& 6031\\
			&  3    & 6543			& 6296			&			&  3	& 6629			& 6392\\
			&  4    & 6575			& 6322			&			&  4	& 6664			& 6417\\
&&&&&&&\\			  			      							  			  
$B_0^*(0^+)$		&  0	& 5796			& 5675			& $B_{s0}^*(0^+)$	&  0    & 5822			& 5750\\
			&  1	& 6341			& 6120			&            		&  1    & 6401			& 6208\\
&&&&&&&\\			  			      						          			  
$B_1(1^+)$		&  0	& 5770			& 5696			& $B_{s1}(1^+)$         &  0    & 5822			& 5774\\
			&  1	& 5770	 		& 5696			&   			&  1    & 5822			& 5774\\
			&  2	& 6322			& 6136			&            		&  2    & 6401			& 6224\\
			&  3	& 6322			& 6136 			&            		&  3    & 6401			& 6224\\
&&&&&&&\\			  			      						          			  
$B_2^*(2^+)$		&  0	& 5771			& 5711			& $B_{s2}^*(2^+)$	&  0    & 5823			& 5788\\
			&  1	& 6324			& 6140			& 			&  1    & 6402			& 6230\\
			&  2	& 6391			& 6186			&			&  2    & 6466			& 6273\\
&&&&&&&\\
\end{tabular}
\vspace*{0.5ex}
\caption{Masses of the bottom $B$ and $B_s$ mesons in MeV, calculated in model $\cal A$ and $\cal B$; n denotes the radial excitation.  Note that the $1^+$ states are twofold degenerated  corresponding to the  total spin $S=0$ and $S=1$.}
\endtable

\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\table
\renewcommand{\arraystretch}{1}
\begin{tabular}{lrdd@{\hspace{2em}}|lcdd}
&&&&&&&\\
\bf Meson ($J^{\pi c}$)	& \bf  n	& \bf Model $\cal A$	& \bf Model $\cal B$	& \bf Meson ($J^{\pi c}$)	&  \bf n	& \bf Model $\cal A$	& \bf Model $\cal B$\\
&&&&&&&\\\hline&&&&&&&\\
$\eta_c(0^{-+})$	&  0	& 3037			& 3114			& $\eta_b(0^{-+})$       &  0	& 9497			& 9562\\
			&  1	& 3789			& 3708			&         		 &  1	& 10055			& 9983\\
			&  2	& 4363			& 4147			&                        &  2	& 10503			& 10318\\
&&&&&&&\\
$J/\Psi(1^{--})$	&  0	& 3039			& 3133			& $\Upsilon(1^{--})$     &  0	& 9497			&  9565\\
			&  1	& 3792			& 3719			&                        &  1	& 10055			&  9985\\
			&  2	& 3846			& 3754			&     			 &  2	& 10103			& 10020\\
			&  3    & 4366			& 4155			&                        &  3	& 10503			& 10319\\
			&  4    & 4398			& 4176			&                        &  4	& 10534			& 10342\\
			&      	& 			& 			&                        &  5	& 10892			& 10609\\
			&      	& 			& 			&                        &  6	& 10914			& 10625\\
			&      	& 			& 			&                        &  7	& 11250			& 10871\\
			&      	& 			& 			&                        &  8	& 11261			& 10882\\
&&&&&&&\\																 
$h_c(1^{+-})$		&  0	& 3482			& 3483			& $h_b(1^{+-})$          &  0   & 9824			& 9811\\
			&  1	& 4110			& 3965			&                        &  1   & 10304			& 10171\\
&&&&&&&\\
$\chi_{c0}(0^{++})$	&  0	& 3482			& 3449			& $\chi_{b0}(0^{++})$    &  0   & 9824			& 9805\\
			&  1	& 4110			& 3943			&                        &  1   & 10304			& 10166\\
&&&&&&&\\
$\chi_{c1}(1^{++})$	&  0	& 3482			& 3483	  		& $\chi_{b1}(1^{++})$    &  0   & 9824			& 9811\\
			&  1	& 4110	 		& 3965			&			 &  1   & 10304			& 10171\\
&&&&&&&\\
$\chi_{c2}(2^{++})$	&  0	& 3485			& 3508			& $\chi_{b2}(2^{++})$    &  0   & 9824			& 9816\\
			&  1	& 4114			& 3983			&		         &  1   & 10304			& 10175\\
			&  2	& 4164			& 4015			&                        &  2   & 10352			& 10210\\
&&&&&&&\\
%			&	&			&			&			&	&			&\\
\end{tabular}
\vspace*{0.5ex}
\caption{Masses of the heavy quarkonia  in MeV, calculated in model $\cal A$ and $\cal B$; n denotes the radial excitation.}
\endtable

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\arraystretch}{1}
\table
\begin{table}
\renewcommand{\arraystretch}{1}
\begin{tabular}{lcddddd}
&&&&&&\\
\bf Parameter			& \bf Exp. \cite{PDG2k}	&\bf Model $\cal A$	& \bf Model $\cal B$	& \bf ISGW2 \cite{ISGW2}& \bf WSB \cite{WSB} &\bf MS\cite{MS}\\
&&&&&&\\
\hline
&&&&&&\\
$\Gamma (B\to D)\; \left[10^{13} |V_{cb}|^2 s^{-1}\right]$	& 	---			& 1.05	& 0.93  & 1.19	& 0.808 & 0.86 \\[0,5 ex]
$\Gamma (B\to D^*)\; \left[10^{13} |V_{cb}|^2 s^{-1}\right]$	& 		---		& 2.78	& 2.64  & 2.48	& 2.19  & 2.28  \\[0,5ex]
$\Gamma_L/\Gamma_T$						& $1.24 \pm 0.16$\cite{CLEOAvery}& 1.14	& 1.20  & 1.04 	& ---   & 1.11  \\[0,5ex]
$\Gamma_+/\Gamma_-$						&	---			& 0.23  & 0.27  & ---	& ---   &  ---      \\
&&&&&&\\
$R_1$						& $1.18 \pm 0.30 \pm 0.12$	& 1.18	& 1.10  & 1.27	& 1.09$^{*}$  &  ---  \\[0,5ex]
$R_2$						& $0.71 \pm 0.22 \pm 0.07$      & 0.94 	& 0.87 & 1.02	& 1.06$^{*}$  &  ---  \\[0,5ex]
$\rho_{A_1}^2$					& $0.91 \pm 0.15 \pm 0.06$      & 0.75	& 1.02 & ---		& ---   &  ---  \\
&&&&&&\\
\end{tabular}
\hfill {\tiny * taken from \cite{slrev}}
\caption{$B\to D^{(*)}\ell\bar\nu$ decay observables and form factor
parameters. We use $|V_{cb}| = 0.034 ({\cal A})$ resp. $|V_{cb}| =
0.035 ({\cal B})$, as described in section \ref{semilep}.}\label{semilepBD2}

\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{tabular}{lcddddd}
&&&&&&\\
\bf Parameter			& \bf Exp. \cite{PDG2k}	&\bf Model $\cal A$	& \bf Model $\cal B$	& \bf ISGW2 \cite{ISGW2}& \bf WSB \cite{WSB} &\bf MS\cite{MS}\\
&&&&&&\\
\hline
&&&&&&\\
$\Gamma (D\to K) \; \left[10^{10}s^{-1}\right]$		& $7.97 \pm 0.36$& 7.51	& 7.26        & 10.0	& 8.26 	& 9.7\\[0,5 ex]
$\Gamma (D\to K^*) \; \left[10^{10}s^{-1}\right]$	& $4.55 \pm 0.34$& 7.64	& 10.08       & 5.4	& 9.53 	& 6.0\\[0,5 ex]
$\Gamma_L/\Gamma_T$					& $1.14 \pm 0.08$& 1.29& 1.48        & 0.94	& 0.91 	& 1.28\\[0,5 ex]
$\Gamma_+/\Gamma_-$					& $0.21 \pm 0.04$& 0.23& 0.34        & ---	& ---	& ---\\
&&&&&&\\
$A_1(0)$				& $0.56 \pm 0.04$ \cite{slrev}   & 0.69 & 0.81		&  ---	& 0.88	& 0.66\\[0,5 ex]
$r_V$							& $1.82 \pm 0.09$& 1.54  & 1.18        	& 2.0$^{*}$& 1.44& 1.56\\[0,5 ex]
$r_2$							& $0.78 \pm 0.07$& 0.81 & 0.62         & 1.3$^{*}$& 1.31& 0.74\\
&&&&&&\\
\end{tabular}
\hfill {\tiny * taken from \cite{E791DK}}
\caption{$D\to K^{(*)}\ell\bar\nu$ decay observables and form factor
parameters. The experimental decay rates are averaged over isospin. We use $|V_{cs}| = 0.975$ \protect\cite{PDG2k}. \label{DK}}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{tabular}{lcddddd}
&&&&&&\\
\bf Parameter			& \bf Exp. \cite{PDG2k}	&\bf Model $\cal A$	& \bf Model $\cal B$	& \bf ISGW2 \cite{ISGW2}& \bf WSB \cite{WSB} &\bf MS\cite{MS}\\
&&&&&&\\
\hline
&&&&&&\\
$\Gamma (D_s\to \eta) \; \left[10^{10}s^{-1}\right]$		& $5.24 \pm 1.41$& 4.05 & 3.11       & 3.5 $^{ a)}$	& --- 	& 5.0\\[0,5 ex]
$\Gamma (D_s\to \eta') \; \left[10^{10}s^{-1}\right]$		& $1.80 \pm 0.69$& 1.27 & 1.75       & 3.0 $^{ a)}$	& --- 	& 1.85\\[0,5 ex]
$\Gamma (D_s\to \phi) \; \left[10^{10}s^{-1}\right]$		& $4.03 \pm 1.01$& 7.89 & 9.67       & 4.6	& --- 	& 5.1\\[0,5 ex]
$\Gamma_L/\Gamma_T$						& $0.72 \pm 0.18$& 1.20 & 1.42    & 0.96	& ---	& --- \\[0,5 ex]
$\Gamma_+/\Gamma_-$						& 	---	 & 0.20 & 0.33       & ---	& ---	& ---\\
&&&&&&\\
%\hline
$A_1(0)$							&	---	 & 0.66  & 0.79		&  ---		& ---	& 0.65\\[0,5 ex]
$r_V$								& $1.92 \pm 0.32$& 1.77  & 1.30       	& 2.1$^{*}$	& --- 	& 1.71\\[0,5 ex]
$r_2$								& $1.60 \pm 0.24$  & 0.85& 0.63     	& 1.3$^{*}$	& --- 	& 0.72\\
&&&&&&\\
\end{tabular}
{\hfill \tiny * taken from \cite{E791Dsphi}}
\caption{$D_s\to \eta/\eta'/\phi\ell\bar\nu$ decay observables and form factor parameters. The experimental decay rates are averaged over isospin. We use $|V_{cs}| = 0.975$ \protect\cite{PDG2k}.\\ $^{a)}$ A $\eta-\eta'$ mixing angle of $-20^\circ$ is
assumed. An angle of $-10^\circ$ would lead to 5.3 and 2.3.\label{Dseta}}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{tabular}{lcddddd}
&&&&&&\\
\bf Parameter			& \bf Exp. \cite{PDG2k}	&\bf Model $\cal A$	& \bf Model $\cal B$	& \bf ISGW2 \cite{ISGW2}& \bf WSB \cite{WSB} &\bf MS\cite{MS}\\
&&&&&&\\
\hline
&&&&&&\\
$\Gamma (D^+\to \pi^0) \; \left[10^{9 }s^{-1}\right]$	& $2.9 \pm 1.4$ & 1.03  & 0.99   & 2.4  & 3.6 & 4.8 \\[0,5 ex]
$\Gamma (D^0\to \pi^-) \; \left[10^{9 }s^{-1}\right]$	& $9.0 \pm 1.5$ & 2.06  & 1.99   & 4.8  & 7.1 & 9.6 \\[0,5 ex]
$\Gamma (D^+\to \rho^0) \; \left[10^{9 }s^{-1}\right]$	& $2.1 \pm 0.8$ & 2.25  & 3.36   & 1.2  & 3.4 & 2.1 \\[0,5 ex]
$\Gamma_L/\Gamma_T$					& --- 		& 1.30	& 1.55   & 0.67 & 0.91& 1.16 \\[0,5 ex]
$\Gamma_+/\Gamma_-$					& --- 		& 0.15	& 0.26   & ---  & --- & ---\\
&&&&&&\\						  
$A_1(0)$						& --- &  0.58	& 0.72	& ---  & 0.78	& 0.6 \\[0,5 ex]
$r_V$							& --- &  1.68 	& 1.26 	& ---  & 1.58	& 1.48 \\[0,5 ex]
$r_2$							& --- &  0.77	& 0.59  & ---  & 1.18	& 0.82\\
%$A_1(0)$						& --- &  0.58(8\%)	& 0.72(7\%)*	& ---  & 0.78	& 0.6 \\[0,5 ex]
%$r_V$							& --- &  1.68(5\%) 	& 1.26(2\%)    	& ---  & 1.58	& 1.48 \\[0,5 ex]
%$r_2$							& --- &  0.77(5\%)	& 0.59(4\%)*      & ---  & 1.18	& 0.82\\
&&&&&&\\
\end{tabular}
\vspace*{0.5ex}
\caption{$D\to \pi/\rho\ell\bar\nu$ decay observables and form factor
parameters. We use $|V_{cd}| = 0.222$ \protect\cite{PDG2k}. \label{Dpi}}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{tabular}{lcddddd}
&&&&&&\\
\bf Parameter			& \bf Exp. \cite{PDG2k}	&\bf Model $\cal A$	& \bf Model $\cal B$	& \bf ISGW2 \cite{ISGW2}& \bf WSB \cite{WSB} &\bf MS\cite{MS}\\
&&&&&&\\
\hline
&&&&&&\\
$\Gamma (D\to \eta) \; \left[10^{9 }s^{-1}\right]$	& --- & 0.79 	& 0.95   & 1.5 $^{ a)}$ & --- & --- \\[0,5 ex]
$\Gamma (D\to \eta') \; \left[10^{9 }s^{-1}\right]$	& --- & 0.14	& 0.16   & 0.3 $^{ a)}$ & --- & --- \\[0,5 ex]
$\Gamma (D\to \omega) \; \left[10^{9 }s^{-1}\right]$	& --- & 2.23	& 3.35   & 1.2 & --- & --- \\[0,5 ex]
$\Gamma_L/\Gamma_T$					& --- & 1.31	& 1.55 	 & 0.68& --- & --- \\[0,5 ex]
$\Gamma_+/\Gamma_-$					& --- & 0.15	& 0.26 	 & --- & --- & --- \\
&&&&&&\\						  
$A_1(0)$						& --- & 0.41 	& 0.51 	& --- &	--- & --- \\[0,5 ex]
$r_V$							& --- & 1.68  	& 1.27    	& --- &	--- & --- \\[0,5 ex]
$r_2$							& --- & 0.77 	& 0.59      & --- &	--- & --- \\
%$A_1(0)$						& --- & 0.41(8\%) 	& 0.51(6\%) 	& --- &	--- & --- \\[0,5 ex]
%$r_V$							& --- & 1.68(5\%)  	& 1.27(2\%)    	& --- &	--- & --- \\[0,5 ex]
%$r_2$							& --- & 0.77(5\%) 	& 0.59(5\%)*      & --- &	--- & --- \\
&&&&&&\\
\end{tabular}
\vspace*{0.5ex}
\caption{$D\to \eta/\eta'/\omega\ell\bar\nu$ decay observables and form factor
parameters. We use $|V_{cd}| =0.222$ \protect\cite{PDG2k}. \\ $^{a)}$ A $\eta-\eta'$ mixing angle of $-20^\circ$ is
assumed. An angle of $-10^\circ$ would lead to 1.1 and 0.4.\label{Deta}}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{tabular}{lcddddd}
&&&&&&\\
\bf Parameter			& \bf Exp. \cite{PDG2k}	&\bf Model $\cal A$	& \bf Model $\cal B$	& \bf ISGW2 \cite{ISGW2}& \bf WSB \cite{WSB} &\bf MS\cite{MS}\\
&&&&&&\\
\hline
&&&&&&\\
$\Gamma (D_s\to K  ) \; \left[10^{9 }s^{-1}\right]$	& --- & 3.42	& 3.15   & 4.4 & --- & 6.4 \\[0,5 ex]
$\Gamma (D_s\to K^* ) \; \left[10^{9 }s^{-1}\right]$	& --- & 2.71	& 4.54   & 2.2 & --- & 3.9 \\[0,5 ex]
$\Gamma_L/\Gamma_T$					& --- & 1.24	& 1.51   & 0.76& --- & 1.21\\[0,5 ex]
$\Gamma_+/\Gamma_-$					& --- & 0.14	& 0.29   & --- & --- & --- \\
&&&&&&\\						      
$A_1(0)$						& --- & 0.43 	& 0.58	& --- &	--- & 0.57 \\[0,5 ex]
$r_V$							& --- & 1.86  	& 1.26  	& --- &	--- & 1.83 \\[0,5 ex]
$r_2$							& --- & 0.76 	& 0.56    & --- &	--- & 0.74\\
%$A_1(0)$						& --- & 0.43(6\%) 	& 0.58(5\%)	& --- &	--- & 0.57 \\[0,5 ex]
%$r_V$							& --- & 1.86(4\%)  	& 1.26(0.5\%)  	& --- &	--- & 1.83 \\[0,5 ex]
%$r_2$							& --- & 0.76(3\%) 	& 0.56(8\%)*    & --- &	--- & 0.74\\
&&&&&&\\
\end{tabular}
\vspace*{0.5ex}
\caption{$D_s\to K/K^*\ell\bar\nu$ decay observables and form factor
parameters. We use $|V_{cd}| = 0.222$ \protect\cite{PDG2k}. \label{DsK}}
\end{table}
\endtable
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\renewcommand{\arraystretch}{1}
\begin{tabular}{lcddd}
\\
\bf Decay ratio      & \bf Exp. \cite{PDG2k}& \bf Model $\cal A$ & \bf Model $\cal B$  &\bf NRSX \cite{Neubert}\\
\\
\hline
\\
$\frac{\Gamma(B^0\to D^-\pi^+)}{\Gamma(B^0\to D^{*-}\pi^+)}$     & $1.09 \pm 0.17$  & 1.00 & 1.05  & 1.04 \\
\\
$\frac{\Gamma(B^0\to D^-\rho^+)}{\Gamma(B^0\to D^{*-}\rho^+)}$    & $1.16 \pm 0.62$  & 0.87 & 0.92 & 0.88\\
\\
\begin{tabular}{l}$\frac{\Gamma(B^0\to D^-D_s^+)}{\Gamma(B^0\to D^{*-}D_s^+)}$\\[2 ex] $\frac{\Gamma(B^+\to \overline{D}^0D_s^+)}{\Gamma(B^+\to \overline{D}^{*0}D_s^+)}$ \end{tabular}   & \begin{tabular}{c}$0.83 \pm 0.43$\\[2 ex]$1.08 \pm 0.56$\end{tabular} & 1.42  & 1.28 & 1.47 \\
\\
\begin{tabular}{l}$\frac{\Gamma(B^0\to D^-D_s^{*+})}{\Gamma(B^0\to D^{*-} D_s^{*+})}$\\[2 ex]$\frac{\Gamma(B^+\to \overline{D}^0D_s^{*+})}{\Gamma(B^+\to \overline{D}^{*0} D_s^{*+})}$ \end{tabular} & \begin{tabular}{c}$0.50 \pm 0.31$\\[2 ex]$0.33 \pm 0.19$\end{tabular} & 0.41  & 0.36 & 0.39 \\
\\
\end{tabular}
\vspace*{0.5ex}
\caption{Ratios of non-leptonic B meson decay rates}\label{nonlept_ratios}
\widetext
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{tabular}{lcddd}
\\
\bf Decay mode      & \bf Exp. \cite{PDG2k} & \bf Model $\cal A$ & \bf Model $\cal B$  & \bf NRSX\cite{Neubert}\\
\\
\hline
\\
$B^0\to D^-\pi^+$     & $1.94 \pm 0.26$  & 2.21 & 1.97 & 1.94 \\[0,5 ex]
$B^0\to D^-\rho^+$    & $5.10 \pm 0.90 $ & 5.68 & 6.13 & 4.84 \\[0,5 ex]
$B^0\to D^{*-}\pi^+$  & $1.78 \pm 0.14 $ & 2.22 & 1.88 & 1.87 \\[0,5 ex]
$B^0\to D^{*-}\rho^+$ & $4.4 \pm 2.2$  	 & 6.57 & 6.67 & 5.48 \\
\\
$B^0\to D^-K^+$	      & --- & 0.17 & 0.15  & 0.15 \\[0,5 ex]
$B^0\to D^-K^{*+}$    & --- & 0.31 & 0.30  & 0.26 \\[0,5 ex]
$B^0\to D^{*-}K^+$    & --- & 0.16 & 0.14  & 0.14 \\[0,5 ex]
$B^0\to D^{*-}K^{*+}$ & --- & 0.36 & 0.34  & 0.32 \\
\\					  
$B^0\to D^-D^+$	      & --- & 0.53 & 0.49  & 0.23  \\[0,5 ex]
$B^0\to D^-D^{*+}$    & --- & 0.37 & 0.34  & 0.23  \\[0,5 ex]
$B^0\to D^{*-}D^+$    & --- & 0.39 & 0.41  & 0.17  \\[0,5 ex]
$B^0\to D^{*-}D^{*+}$ & $0.40 {+0.26 \atop -0.20}$ & 0.86 & 0.89 & 0.54  \\
\\
$B^+\to \overline{D}^0D_s^+$	& $7.9 \pm 2.4$  & 8.73  & 8.27 & 6.61 \\[0,5 ex]
$B^0\to D^-D_s^+$       	& $5.2 \pm 1.9$  & 8.73  & 8.27 & 6.61 \\[0,5 ex]
$B^+\to \overline{D}^0D_s^{*+}$ & $5.4 \pm 2.4 $ & 6.05  & 5.49 & 6.15  \\[0,5 ex]
$B^0\to D^-D_s^{*+}$    	& $6.5 \pm 3.2 $ & 6.05  & 5.49 & 6.15  \\[0,5 ex]
$B^+\to \overline{D}^{*0}D_s^+$ & $7.3 \pm 3.0 $ & 6.14  & 6.47 & 4.49 \\[0,5 ex]
$B^0\to D^{*-}D_s^+$    	& $6.2 \pm 2.2 $ & 6.14  & 6.47 &  4.49 \\	[0,5 ex] 
$B^+\to \overline{D}^{*0} D_s^{*+}$& $16.3 \pm 6.0$  & 14.68 & 15.24 & 15.8 \\[0,5 ex]
$B^0\to D^{*-} D_s^{*+}$ 	& $12.9 \pm 4.5$ & 14.68     & 15.24 & 15.8 \\
\\
\end{tabular}
\vspace*{0.5ex}
\caption{Non-leptonic B decay rates $\Gamma  \;[ns^{-1}]$. We use $|V_{cs}| = 0.975$, $|V_{ud}| = 0.975$, $|V_{us}| = 0.223$ \protect\cite{PDG2k} and  $|V_{cb}| = 0.034 \;({\cal A})$ resp. $|V_{cb}| =
0.035 \;({\cal B})$, as described in section \ref{semilep}. The results of \protect\cite{Neubert} have been adapted to the decay constants used in our calculation.}\label{nonlept}
%\widetext
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{tabular}{lcdd}
\\
\bf Meson      & \bf Exp. \cite{PDG2k}  & \bf Model $\cal A$ & \bf Model $\cal B$ \\
\\\hline\\
$\pi$      & $130.7 \pm  0.1 \pm  0.36$  & 212 & 219\\[0,5 ex]
$K$       &  $159.8 \pm  1.4 \pm  0.44$  & 248 & 238\\[0,5 ex]
$D$        & $300 {+180+80 \atop -150-40}$   & 293 & 263\\[0,5 ex]
$D_s$      & $280 \pm 19 \pm 28 \pm 34$     & 315 & 284\\
\\
$\rho$     & $216 \pm 5$ \cite{Neubert}  & 470 & 717 \\[0,5 ex]
$\omega$   & $195 \pm 3$ \cite{Neubert}  & 472 & 726\\[0,5 ex]
$\phi$     & $237 \pm 4$ \cite{Neubert}  & 475 & 685\\[0,5 ex]
$K^*$      & $214 \pm 7$ \cite{Neubert}  & 467 & 695\\[0,5 ex]
$D^*$      & ---  & 339 & 409\\[0,5 ex]
$D_s^*$    & ---  & 378 & 445\\
\end{tabular}
\vspace*{0.5ex}
\caption{Decay constants of pseudoscalar and pseudovector mesons in MeV}\label{weakdec}
\end{table}

