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\draft
\title{ON THE ENHANCEMENT OF ${\Lambda_Q}$ DECAY RATE}
\author{S. Arunagiri}
\address{Department of Nuclear Physics, University of Madras,\\
Guindy Campus, Chennai 600 025, Tamil Nadu, INDIA}
\maketitle
\begin{abstract}
The enhancement of the ${\Lambda_b}$ and ${\Lambda_c}$ decay rates
due to four-quark operators is calculated. Hard gluon exchange
modifies the wave function
for the b${\bar u}$ pair in ${\Lambda_b}$, ${|\Psi(0)|_{bu}^2}$, and
the wave function for the c${\bar d}$ pair in ${\Lambda_c}$,
${|\Psi(0)|_{cd}^2}$. The modified wave function is found to account
for the discrepancy of 0.20 ps${^{-1}}$ among
the lifetimes of ${\Lambda_b}$ and B$^0$ and for the difference of
2.6 ps${^{-1}}$ in the case of
${\Lambda_c}$ and D$^0$.
\end{abstract}

\vskip 0.3cm
%\section{INTRODUCTION}

Heavy flavour hadrons, H, contain a heavy quark, Q (b and c), and a 
light cloud comprising of light quarks (anti-quarks) and gluons. When 
the mass of the heavy quark goes to infinity, the picture of the heavy 
hadron decays are so simplified that the light cloud has no role to
play. Theoretical study of the heavy hadrons is facilitated by the
expansion of the hadronic matrix element, based on the operator
product expansion of QCD, in inverse powers of the heavy quark mass,
m \cite{bigi95}. The leading order of the expansion, corresponding
to  the asymptotic limit of the heavy quark mass, describes the
decay rate of a heavy flavour hadron as if it is of a free heavy
quark decay. This implies the same lifetime for all heavy hadrons
of a given heavy flavour quantum number. The next-to-leading
order terms, appearing at ${1/m^2}$, describe the motion of the
heavy quark inside the hadron and the chromomagnetic interaction,
distinguish the lifetimes of the mesons on one hand and the
baryons on the other (with exception of ${\Omega_Q}$). The term
corresponding to the third order in 1/m of the expansion is the
matrix element of four-quark operators which contains effects
such as W-exchange, weak annihilation and Pauli interference,
coming from the spectator (light) quarks. At this order, the
lifetimes of the various mesons differ among themselves and so
is for baryons. The spectator effects which appear through
four-quark operators, are supposed, yet poorly-understood,
to explain the intricacies of the lifetime differences
and to fix the lifetime hierarchy of the hadrons. 

According to the idea of heavy quark expansion, all hadrons
of a given heavy flavour are expected to have nearly the same
lifetime. The theoretical prediction of the ratio of lifetimes
of ${\Lambda_b}$ and ${B^0}$, obtained to two orders in 1/m,
 is 0.9. But this is found much higher than the observed value
 of the ratio: 0.78 for ${\tau(\Lambda_b)}$ = 1.20 ps and for
${\tau(B^0)}$ = 1.58 ps. On the other hand, the corresponding
charm sector shows an intrinsically different picture due to
the mass of the charm quark, which is not so asymptotically
large as the b quark mass. In the charm case, the dominant
effects come from the four-quark operators rather than the
kinetic and chromomagnetic operators. Experimentally, the
ratio of the lifetimes of ${\tau(\Lambda_c)}$ and ${\tau(D^0)}$
is  0.496 for  ${\tau(\Lambda_c)}$ = 0.206 ps and ${\tau(D^0)}$
= 0.415 ps. Therefore,
for both the cases, the explanation for the suspected small
lifetime, and an
equivalently enhanced decay rate, of ${\Lambda_Q}$ should
come from the
third order term of the heavy quark expansion in 1/m where
the matrix element
involves four-quark operators. Given the present experimental
decay rates
for beauty hadrons: ${\Gamma(\Lambda_b)}$ = 0.83 ${\pm}$
0.02 ps${^{-1}}$                                      
and ${\Gamma(B^0)}$ = 0.63 ${\pm}$ 0.05 ps${^{-1}}$, the
needed enhancement
is 0.2 ps${^{-1}}$, whereas 2.6 ps${^{-1}}$ is needed for
 charmed hadrons of
decay rates: ${\Gamma(\Lambda_c)}$ = 5 ps${^{-1}}$ and
${\Gamma(D^0)}$ = 2.4
ps${^{-1}}$. 

The four-quark operators are estimated using phenomenological
models. Hence
their size is questionable. Nevertheless, it is the only way
now to do. The
four-quark operators are related to the probability of finding
the Q${\bar q}$
pair at the origin simultaneously using quark models, denoted by 
${|\Psi(0)|_{Qq}^2}$. The wave function is evaluated relating
it to the
mass splitting of hadrons arising out of the heavy-light quarks
 interaction.
In Ref.\ \onlinecite{rosner96}, Rosner evaluated the wave function,
 using
the hyperfine splitting in a similarly heavy-flavoured baryon,
utilising the
DELPHI value on ${\Sigma_b^*}$ - ${\Sigma_b}$ splitting
\cite{delphi95}
\footnote{Though the DELPHI value has not hitherto been
confirmed, its
use does not alter the goal anyway since the central value
of the mass
splitting due to hyperfine interaction is expected to be
around 50 MeV.}
Neubert and Sachrajda\cite{neubert97} introduced hadronic
parameters
accounting
for hybrid renormalisation while parameterizing the four-quark
matrix for
b-flavoured hadrons. The hadronic parameters are yet unknown.
Their values
are obtained from QCD sum rules.  As an extension to c-flavoured
hadrons,
Voloshin \cite{vol96} studied the the charmed baryons. In this
approach,
the authors of Ref.\ \onlinecite{guber98} analysed the inclusive
charmed-baryon
decays to fix the hierarchy of charmed lifetimes.  

In the present study,  we take into account the contribution coming
from
the exchange of hard gluons when two different scales, the heavy
quark mass,
m,  and the QCD scale, ${\mu}$, are involved, while estimating
the wave
function in quark model. We treat the coupling constant corresponding to
a meson
and a baryon
differently.
It is found that the wave function,
and hence
the enhanced decay rate, is large in both the cases of b and c and
explains part of the lifetime differences between ${\Lambda_b}$ and
B$^0$
as well as between ${\Lambda_c}$ and D$^0$. 

%\section{FOUR-QUARK MATRIX ELEMENT}
The enhancement in the decay rate of ${\Lambda_b}$ is arising out of the
processes involving four-quarks
\cite{neubert97,volshif85,volur87,bilic84,guber86}: (a) the weak
scattering process bu$\--->$cd in
the ${\Lambda_b}$ involving matrix elements
between hadronic states of  $(\bar bb)(\bar uu)$ operators and
(b) the process contributing to the Pauli
interference involving matrix elements of operators $(\bar bb)(\bar dd)$
operators. Hence, the enhancement
of the ${\Lambda_b}$ decay rate is given by: 
\begin{equation}
\Delta\Gamma(\Lambda_b) = {G_f^2 \over {(2\pi)}} |\Psi(0)|_{bu}^2
|V_{ud}|^2|V_{cb}|^2 m_b^2(1-x)^2[C_-^2-(1+x)C_+(C_--C_+/2)]
\end{equation}
where $x = m_c^2/m_b^2; C_-$ and $C_+ = C_-^{1/2}$ are the short distance
QCD
enhancement and suppression
factors for quarks in a colour antitriplet and sextet respectively:
\begin{equation}
C_- = [\alpha_s(m_b^2)/\alpha_s(m_W^2)]^{4/\beta},~~~\beta = 11-2n_f/3
\end{equation}
where $n_f$ is the active quark flavours between $m_b$ and $m_W$.
The $C_-$ term corresponds to the weak
sattering process $bu \rightarrow cd \rightarrow bu$ and the other
term represents
the destructive interference
between the two intermediate d quarks in the process $bd\rightarrow cudd
\rightarrow cd$.

The wave function for the bu pair (in the initial baryon),
$|\Psi(0)|_{bu}^2$,
in Eq. (1) is of the
form: 
\begin{equation}
|\Psi(0)|_{bu}^2 = {4 \over 3} {\Delta M(B_{ijk}) \over {\Delta M(M_{i\bar
j})}}
\xi|\Psi(0)|_{b\bar u}^2
\end{equation}
where
\begin{equation}
\Delta M(B_{ijk}) = {16\pi \over 9}\alpha_s {\sum_{i>j}} {<S_i.S_j>
\over {m_i.m_j}}|\Psi(0)|_{ij}^2
\end{equation}
and
\begin{equation}
\Delta M(M_{i\bar j}) = {32\pi \over 9}\alpha_s  {<S_i.S_j>
\over {m_i.m_j}}|\Psi(0)|_{i\bar j}^2
\end{equation}
are the hyperfine mass splittings in a baryon and in a meson respectively.
There is the colour factor 1/2
in the baryonic case due to the quark composition: a heavy quark and two
light quarks. Under
isospin symmetry, the effective masses of light quarks are equal. In the
same token, wave
functions for bu and bd pairs are equal. Equation (3) is obtained for the
values of $<S_i.S_j>$ = (1/4, -3/4) with spin (0, 1) for the meson and
$<S_i.S_j>$ = (1/4, -1/2) with
spin (1/2, 3/2) for the baryon
with $S_{qq}$ = 1. In Eq. (3), $\xi$ is the ratio of the coupling
constants governing a baryon and a meson. The
coupling governing a baryon is stronger than that of a meson. Though $\xi$
has been chosen as a free
parameter varying from 0.25 to 1.5, it can be exactly calculated
\cite{deruj75}.
In rigorous sense, it should be more than unity. 
     The wave function on the right hand side of Eq. (3) corresponds to
the matrix element
for B-meson decay into vacuum, parameterised as,
\begin{equation}
|<0|\bar q \gamma_{\mu} \gamma_5 Q|B>|^2 = f_B^2M_B^2
\end{equation}
and to the relation obtained in the non-relativistic limit \cite{shur82}
\begin{equation}
|<0|\bar q \gamma_{\mu} \gamma_5 Q|B>|^2 = 12M_B|\Psi(0)|_{b\bar u}^2
\end{equation}
Both Eqs. (6) and (7) characterise the same process involving a heavy
quark and a light quark
(and gluons) but are normalised at two different scales: the normalisation
point of Eq. (6) is $\mu$($\approx$R$^{-1}$) which is the order of the
virtuality of the quarks inside the meson, and the scale of the Eq.
(7) is the mass of the heavy quark. Therefore, in order to account for the
hard gluon exchange
between the light and heavy quarks, Eqs. (6) and (7) have to be related.
The relation turns out
to be an evolution equation of current operators \cite{vol87,pol88}:
\begin{equation}
<0|\bar q \gamma_{\mu} \gamma_5 Q|B>(m_Q) = 
<0|\bar q \gamma_{\mu} \gamma_5 Q|B>(\mu)[\alpha_s(\mu)/\alpha_s(m_Q)]
^{\gamma/\beta}, ~~~\mu \ll m_Q
\end{equation}
where $\gamma$ (= 2) is the hybrid anomalous dimension. The coupling
constant defined through leading
order renormalisation group equation can be expressed as
\begin{equation}
\alpha_s(m) = {\alpha_s(\mu) \over
{[1-{b\over{2\pi}}\alpha_s(\mu)ln(\mu/m)]}}
\end{equation}
Using eq. (8), one can get the eq. (3) of the form
\begin{equation}
|\Psi(0)|_{bu}^2 = {4 \over 3} {\Delta M(B_{ijk}) \over
{\Delta M(M_{i\bar j})}} {f_B^2M_B \over {12}} \xi \left(1-{{b \over {12}}
\alpha_s(\mu)ln{\mu\over m}}\right)^{2\gamma \over \beta}
\end{equation}
The parameter $\xi$ describes the ratio of the coupling constants of
baryon
and meson. The other term in the bracket represents the logrithmic effects
due to the hard gluon exchange that would be expected when one goes down
to
a scale as small as the hadronic scale from the heavy quark mass. This
obviously
modifies the expectation values of the wave function density at the
origin.

%\section{ENHANCEMENT IN DECAY RATE}

The choice of values of the parameters are given in Table
I. The $\Delta M(B_{ijk})$ for ${\Lambda_b}$ is given by M($\Sigma_b^*$)-
M($\Sigma_b$)
of DELPHI \cite{delphi95}. For c, the $\Delta M(B_{ijk})$ is given by
M($\Sigma_c^*$) = 2517.5  1.4 MeV \cite{caso98,cleo97} and
M($\Sigma_c$) = 2452.2  0.6 MeV \cite{caso98}.

The enhanced decay rate is now given by Eq. (3) alongwith eq. (10). The
results
for the enhanced decay rate of $\Lambda_b$ and $\Lambda_c$ are given in
Tables
II and III respectively. The wave function density is, roughly speaking,
independent of the hadronic scale value. However, its dependence upon
the parameter $\xi$ is important. For $\xi$ is graeter than one, the
enhanced decay rate becomes closer to the enhancement required to explain
the
smaller lifetimes of $\Lambda_b$ and $\Lambda_c$ baryons. 

%\section{CONCLUSION}
In conclusion, it is demonstrated that the four-quark operators appearing
at the third order in 1/m expansion
explains the needed enhancement in the decay rates of $\Lambda_Q$.
Though subtle, the couplings of meson and ${\Lambda_b}$ baryon
make much difference.
On the other hand, if the couplings are considered
equal, the four-quark operators still account for the difference in
lifetimes
the $\Lambda_b$ baryon and B meson.








\acknowledgements
The author wishes to thank Prof. P. R. Subramanian and
for fruitful discussions. He thanks UGC for its support through the
Special
Assistance Programme. 

\references
\bibitem{bigi95}I. I. Bigi, preprint UND-HEP-95-BIG02 .
\bibitem{rosner96}J. L. Rosner, Phys. Lett. {\bf B379}, 267 (1996).
\bibitem{delphi95}DELPHI Collaboration, P. Abreu etal., DELPHI report,
DELPHI 95-107 PHYS 542.
\bibitem{neubert97}M. Neubert and C. T. Sachrajda, Nucl. Phys. {\bf B483},
339 (1997).
\bibitem{vol96} M. B. Voloshin, Phys. Lett. {\bf B385}, 369 (1996).
\bibitem{guber98}B. Guberina and B. Melic, Eur. Phys. J. {\bf C2}, 697
(1998).
\bibitem{volshif85}M. B.Voloshin and M. A. Shifman, Sov. J. Nucl. Phys.
{\bf 41}, 120 (1985); Sov. Phys. JETP {\bf 64}, 698 (1986). 
\bibitem{volur87}M. B.Voloshin, N. G. Uraltsev, V. A. Khoze and M. A.
Shifman,
Sov. J. Nucl. Phys. {\bf 46}, 112 (1987).
\bibitem{bilic84}N. Bilic, B. Guberina and J. Trampetic, Nucl. Phys.
{\bf B248}, 261 (1984).
\bibitem{guber86}B. Guberina, R. Ruckl and J. Trampetic, Z. Phyz. {\bf
C33},
297 (1986).
\bibitem{deruj75}A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev.
{\bf D12}, 147 (1975).
\bibitem{shur82}E. V. Shuryak, Nucl. Phys. {\bf B198}, 83 (1982).
\bibitem{vol87}M. B.Voloshin and M. A. Shifman, Sov. J. Nucl. Phys.
{\bf 45}, 292 (1987).
\bibitem{pol88}H. D. Politzer and M. B. Wise, Phys. Lett. {\bf B206},
681 (1988); {\it ibid} {\bf B208}, 504 (1988).
\bibitem{caso98}Particle Data Group, C. Caso et al.,
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Phys. Rev. Lett. {\bf 78}, 2304 (1997).

\mediumtext
\begin{table}
\caption{Choice of values of parameters \protect\cite{caso98}
($n_f$ = 4; $\Lambda_{QCD}$ = 0.2 GeV).}  
\begin{tabular}{ccccccccc}
Q & $\Delta$ M($B_{ijk}$) &  $\Delta$ M($M_{i \bar j}$)  &
M$_H$ & f$_H$ & m$_Q$ & $\alpha_s$(m$_Q$) & $C_+$ & $C_-$\\
\tableline
b   &   0.056   &   0.046   &   5.2789   &   0.190   &   5.1   &   0.20
&   0.87   &   1.31\\
c   &   0.0653   &   0.142   &   1.8693   &   0.240   &   4.7   & 0.29
&   0.79   &   1.57\\
\end{tabular}
\label{table1}
\end{table}
 

\begin{table}
\caption{Enhanced Decay Rate for $\Lambda_b$. Needed Enhancement is 0.2
ps$^{-1}$.}
\begin{tabular}{c|cc|cc}
 & \multicolumn{2}{c}{$\mu$ = 1 GeV} & \multicolumn{2}{c}{$\mu$ = 0.5 GeV}
\\
$\xi$ & $|\Psi(0)|_{b \bar d}$ (10$^{-2}$GeV$^{3}$) &
$\Delta\Gamma(\Lambda_b)$(ps$^{-1}$) &
$|\Psi(0)|_{b \bar d}$ (10$^{-2}$GeV$^{3}$) &
$\Delta\Gamma(\Lambda_b)$(ps$^{-1}$)\\
\tableline
0.50 & 1.3154 & 0.0763 & 1.3422 & 0.0779 \\
0.75 & 1.9732 & 0.1145 & 2.0133 & 0.1168 \\
1.00 & 2.6309 & 0.1527 & 2.6844 & 0.1558 \\
1.25 & 3.2886 & 0.1909 & 3.3555 & 0.1948 \\
1.50 & 3.9463 & 0.2291 & 4.0266 & 0.2337 \\
\end{tabular}
\label{table2}
\end{table}

\begin{table}
\caption{Enhanced Decay Rate for $\Lambda_c$. Needed Enhancement is 2.6
ps$^{-1}$.}
\begin{tabular}{c|cc|cc}
 & \multicolumn{2}{c}{$\mu$ = 1 GeV} & \multicolumn{2}{c}{$\mu$ = 0.5 GeV}
\\
$\xi$ & $|\Psi(0)|_{c \bar d}$ (10$^{-3}$GeV$^{3}$) &$
\Delta\Gamma(\Lambda_c)$(ps$^{-1}$) &
$|\Psi(0)|_{c \bar d}$ (10$^{-3}$GeV$^{3}$)  &
$\Delta\Gamma(\Lambda_c)$(ps$^{-1}$)\\
\tableline
0.50 & 2.7718 & 1.2005 & 2.8216 & 1.2221 \\
0.75 & 4.1578 & 1.8008 & 4.2325 & 1.8332 \\
1.00 & 5.5437 & 2.4011 & 5.6433 & 2.4443 \\
1.25 & 6.9297 & 3.0015 & 7.0542 & 3.0553 \\
1.50 & 8.3156 & 3.6017 & 8.4650 & 3.6665 \\
\end{tabular}
\label{table3}
\end{table}



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