%Title: Light-Cone Sum Rule Results for Semileptonic  Heavy-to-Light Decays 
%Author: Alexander Khodjamirian 
%Published:     <EM> Proceedings of the First Workshop on the CKM Unitarity Triangle, CERN 13-16 February 2002, CERN Yellow Report; Editors - M.Battaglia, A.J. Buras, P. Gambino, A. Stocchi 
%hep-ph/0304132
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\begin{document}
\begin{flushright}
TTP02-14\\
\end{flushright}
\begin{center}
{\Large\bf Light-Cone Sum Rule Results for 
Semileptonic Heavy-to-Light Decays}\\
\vspace{0.5cm} 

{\large\bf
Alexander~Khodjamirian \footnote{\it on leave from Yerevan Physics 
Institute, 375036 Yerevan, Armenia}} 
\vskip 0.7true cm
{\it Institut f\"ur Theoretische Teilchenphysik, Universit\"at
Karlsruhe, \\ D-76128 Karlsruhe, Germany } \\
\begin{abstract}

\noindent
I summarize the current status of the QCD light-cone sum rule predictions 
for the $B\to \pi$ form factor $f^+_{B\pi}$ that is needed to extract 
$|V_{ub}|$  from experimental data on $B\to \pi l\nu$ decays.


\end{abstract}




\bigskip
{presented at the Workshop on the CKM Unitarity Triangle, CERN, February
  2002,\\ to be published in the Workshop report}

\end{center}



\newpage

QCD light-cone sum rules (LCSR) \cite{lcsr,cz}
provide estimates  of various 
heavy-to-light transition form factors. In particular, 
$B\to P,V$ form factors ($P=\pi,K$ and
$V=\rho,K^*,\phi$) have been calculated at small and 
intermediate momentum transfers, typically 
at $0<q^2 \leq m_b^2-2m_b\Lambda_{QCD}$. 
The upper part of this interval overlaps with the 
region accessible to the lattice calculations
of the same form factors, allowing one to compare the results 
of two methods. 
In what follows we will concentrate on the LCSR prediction 
for the $B\to \pi$  form factor $f^+_{B\pi}$ \cite{Bpi,Bpialphas,Ball1}.
Its accuracy has been recently improved in Ref.~\cite{BallZ}.
For the LCSR $B\to V$ form factors we refer to the NLO 
calculation in Ref.~\cite{BallBraun} and to the resulting parametrization
in Ref.~\cite{AliB}. 

The form factor $f^+_{B\pi}$ entering  the 
$B\to \pi$ hadronic matrix element,  
$
\langle\pi(p)|\bar{u} \gamma_\mu b |B(p+q)\rangle=
2f^+_{B\pi}(q^2)p_\mu +O(q_\mu)\,, 
$
is at  $0<q^2<(m_B-m_\pi)^2$ the {\em only} 
hadronic parameter one needs to extract $|V_{ub}|$ 
from the measured $B\to \pi l \nu$ semileptonic width: 
\be
\frac{d\Gamma(B\to \pi l \bar{\nu})}{dq^2}=
\frac{G^2|V_{ub}|^2}{24\pi^3}(E_\pi^2-m_\pi^2)^{3/2}[f^+_{B\pi}(q^2)]^2\,.
\label{eq:width}
\ee
The LCSR approach to calculate $f^+_{B\pi}$ 
employs a specially designed theoretical object, 
the  vacuum-to-pion correlation function 
\bea
F_\mu (p,q)=
i\!\! \int \!d^4xe^{iqx}\langle \pi^+(p)\!\mid T\{\bar{u}\gamma_\mu b(x),
m_b\bar{b}i\gamma_5 d(0)\}\mid\! 0\rangle= F((p+q)^2,q^2)p_\mu+
O(q_\mu)\,,
\label{eq:lcsrcorr}
\eea
where the $b \to u$ weak current is correlated with  
the quark current which has the $B$-meson 
quantum numbers, and $p^2=m_\pi^2$.
Writing down  the dispersion relation 
for the invariant amplitude $F$:
\be
F((p+q)^2,q^2)=
\frac{2f_B f^+_{B\pi}(q^2)m_B^2}{m_B^2-(p+q)^2}
+\sum\limits_{B_h}\frac{2f_{B_h} f^+_{B_h\pi}(q^2)m_{B_h}^2}{m_{B_h}^2-(p+q)^2}~,
\label{eq:displcsr}
\ee
one represents the correlation function (\ref{eq:lcsrcorr})
in terms of  hadronic degrees of freedom in  the $B$ channel.
The ground-state contribution 
in Eq.~(\ref{eq:displcsr}) contains a product
of the $B$ meson decay constant $f_B$ and the form factor 
$f^+_{B\pi}(q^2)$ we are interested in, whereas the 
sum over $B_h$ accounts for  the contributions
of excited and continuum $B$ states. 

The dispersion relation is then matched 
to the result of QCD calculation of $F((p+q)^2,q^2)$ 
at large virtualities, that is, at 
$\mid (p+q)^2-m_b^2 \mid \gg \Lambda_{QCD}^2$
and $q^2\ll m_b^2$. In this region the 
operator-product expansion (OPE) near the light-cone $x^2=0$
is employed:
\begin{equation}
F((p+q)^2,q^2)= \sum\limits_{t=2,3,4}
\int Du_i \sum\limits_{k=0,1}\left(\frac{\alpha_s}{\pi}\right)^k
~T^{(t)}_k((p+q)^2,q^2,u_i,m_b,\mu)
\varphi^{(t)}_\pi(u_i,\mu)\,. 
\label{eq:sr}
\end{equation}
This generic expression is a convolution 
of calculable short-distance
coefficient functions $T^{(t)}_k$ and universal pion light-cone 
distribution amplitudes (DA) $\varphi^{(t)}_\pi(u_i,\mu)$ of twist $t$. 
Here, $m_b$ is the one-loop $b$-quark pole mass, 
$\mu$  is the factorization scale and the integration 
goes over the pion momentum fractions $u_i={u_1,u_2,...}$
distributed among quarks and gluons, so that 
$Du_i\equiv du_1du_2...\delta(1-\sum_i u_i)$. 
In particular, $\varphi^{(2)}_\pi(u_1,u_2,\mu)=
f_\pi\varphi_\pi(u,\mu)$, ($u_1=u,u_2=1-u$)
where $\varphi_\pi$ is the lowest twist 2, 
quark-antiquark pion DA normalized to unit: 
\be
\varphi_\pi(u,\mu)=6u(1-u)\left(1+\sum\limits_n a_{2n}(\mu)
C_{2n}^{3/2}(2u-1)\right)\,.
\label{eq:phipi}
\ee
In the above, $C_{2n}$ are Gegenbauer polynomials and   
the coefficients $a_2(\mu)>a_4(\mu)>a_6(\mu)....$ 
determine the deviation of $\varphi_\pi(u)$ from its asymptotic
form.
Importantly, the contributions to Eq.~(\ref{eq:sr})
corresponding to higher twist and/or higher multiplicity 
pion DA are suppressed by inverse powers of the $b$-quark 
virtuality $ ((p+q)^2-m_b^2)$, allowing one to retain  
a few low twist contributions in this expansion.
Furthermore, one uses quark-hadron duality to approximate the sum over 
$B_h$ in Eq.~(\ref{eq:displcsr})
by a dispersion integral over 
the quark-gluon spectral density, introducing  
a threshold parameter  $s_0^B$. The final 
step involves  a Borel transformation $(p+q)^2 \to M^2$,
where  the scale of  the Borel parameter $M^2$ reflects  the 
characteristic virtuality at which the correlation function is calculated. 

The resulting sum rule relation obtained by matching 
Eqs.~(\ref{eq:displcsr}) and (\ref{eq:sr})
can be cast in the following form:
\bea
f_Bf_{B\pi}^+(q^2)=\frac{1}{m_B^2}
\exp\left (\frac{m_B^2}{M^2}\right)
\sum\limits_{t=2,3,4}~~\sum\limits_{k=0,1} \left(
\frac{\alpha_s}{\pi}\right)^{k}
{\cal F } ^{(t)}_{k}(q^2,M^2;m_b,s_0^B,\mu;\mbox{\{DA\}}^{(t)}) \,,
\label{eq:lcsrfBpi}
\eea
where the double expansion (in twists and in $\alpha_s$) 
and the dependence on the relevant parameters are made explicit.
In particular, $\{ DA \}^{(t)}$ denotes the nonperturbative 
normalization constant and nonasymptotic coefficients
for each given twist component, e.g., for $\varphi_\pi$:
$\{DA\}^{(2)}=\{f_\pi,a_i\}$.   
The sum rule (\ref{eq:lcsrfBpi}) includes all zeroth order in
$\alpha_s$, twist 2,3,4 contributions containing quark-antiquark and 
quark-antiquark-gluon DA of the pion. The perturbative expansion
has NLO accuracy, including the 
$O(\alpha_s)$ corrections to the twist 2 ~\cite{Bpialphas} 
and twist 3  coefficient functions, the latter recently calculated  
in Ref.~\cite{BallZ}. 
More details on the derivation
of LCSR (\ref{eq:lcsrfBpi}) and the explicit expressions  
can be found in the review papers \cite{KR,Brauntalk,CK}.

For the $B$-meson decay constant entering LCSR (\ref{eq:lcsrfBpi})
one usually employs the conventional
SVZ sum rule \cite{SVZ} for the two-point correlator
of $\bar{b}i\gamma_5 q$ currents with $O(\alpha_s)$ 
accuracy (a recent update of this sum rule 
is discussed in these proceedings \cite{Jamintalk}): 
\be
f_B=
\sum\limits_{d=0,3 \div 6}~\sum\limits_{k=0,1}
\left(\frac{\alpha_s}{\pi}\right)^k
C_k^{(d)}(\overline{M}^2,m_b,s_0^B,\mu)\langle 0| \Omega_d(\mu)|0\rangle\,,
\label{eq:fBsr}
\ee
where the expansion contains the perturbarive term 
with dimension $d=0$ ($\Omega_0=1$), and, at $d\geq 3 $, goes over 
condensates, the vacuum averages of 
operators $\Omega_d=\bar{q}q, 
G^{a}_{\mu\nu}G^{a\mu\nu}, ....$, multiplied by calculable   
short-distance coefficients $C_k^{(d)}$. The Borel parameter 
$\overline{M}$ is correlated with $M$. 
The LCSR prediction for the $B\to \pi$ form factor is finally obtained 
dividing Eq.~(\ref{eq:lcsrfBpi}) by Eq.~(\ref{eq:fBsr}):
\be
f^+_{B\pi}(q^2)=(f_Bf^+_{B\pi}(q^2))_{LCSR}/(f_B)_{2ptSR}\,.
\label{eq:div}
\ee
In order to demonstrate that the expansion
in this relation works quite satisfactory in both twist and $\alpha_s$, 
we present the approximate percentage  of various 
contributions to the  resulting form factor (\ref{eq:div}) in the
following table:\\
\begin{center}
\begin{tabular}{ccccc}
%\begin{table}
\hline
&twist& DA & $LO$& $O(\alpha_s)$ NLO\\
\hline
& 2   & $\bar{q}q$ & $\sim$  50 \% &  $\sim$ 5\%\\
\hline
& 3   & $\bar{q}q$ & $\sim $ 40 \% &   $\sim$  1\%\\
\hline
& 4   & $\bar{q}q$ &            &          \\
&     &            & $\sim$ 5 \%              & -\\
& 3+4 & $\bar{q}qG$&           &     \\
\hline
%\hline 
%\end{table}
\end{tabular}\\
\bigskip
\end{center}

The input parameters used in the numerical analysis of  
the sum rules (\ref{eq:lcsrfBpi}) and (\ref{eq:fBsr}) have a limited accuracy.
The theoretical uncertainty is estimated  
by varying these inputs within the allowed regions and 
 adding up linearly the separate uncertainties induced by these
variations in the numerical prediction for $f^+_{B\pi}$. The resulting total
uncertainties are given below, together with  the parametrizations
of the form factor. A detailed 
``theoretical error'' analysis can be found 
in Ref.~\cite{KRWWY}. 
To summarize it briefly, one source of uncertainty 
is the value of the $b$-quark one-loop pole mass.
The two most recent LCSR analyses use 
$m_b= 4.7\pm 0.1 $ GeV\cite{KRWWY} and $m_b=4.6\pm 0.1$ GeV
\cite{BallZ}. In both studies, the threshold $s_0^B$ is not 
an independent parameter, being  determined 
by stabilizing $f_B$ calculated from Eq.~(\ref{eq:fBsr})
at a given $b$-quark mass. The uncertainty induced
by varying the factorization scale $\mu$ (adopted 
simultaneously as the normalization scale for $\alpha_s$) is very small,  
firstly, because the NLO approximation is implemented 
for both dominant twist 2 and 3 terms, and, secondly, because the 
relatively large $O(\alpha_s)$ corrections
to the twist 2 contribution and to the $f_B$ sum rule 
cancel in the ratio (\ref{eq:div}).
Another source of uncertainty is our limited knowledge of the 
nonasymptotic part in the pion DA
(determined by the coefficients $a_{2n}$ and the 
analogous coefficients in twist 3,4 DA). In Ref.~\cite{KRWWY} 
these coefficients were varied  
from a certain nonasymptotic  ansatz 
of DA (motivated by QCD sum rules) 
to purely asymptitic DA. Such a substantial variation 
covers the existing constraints 
on nonasymptotic coefficients obtained from LCSR for pion 
form factors. The latter constraints have been used 
in Ref.~\cite{BallZ}. In fact, LCSR involve integration over 
normalized DA, therefore it is natural that 
the results only moderately depend on the  
nonasymptotic coefficents.
Finally, to assess the reliability of the LCSR procedure 
one has to comment on the use of quark-hadron duality, 
which is the most sensitive point in the sum rule approach. 
We expect that the sensitivity to 
the duality approximation is substantially reduced: 1) by 
restricting the Borel parameter at not too large values
and 2) by dividing out the $f_B$ sum rule which 
depends on the same threshold. The fact that the QCD sum rule 
prediction for $f_B$ is in a good agreement 
with the lattice results (see e.g. \cite{Jamintalk})
indicates that quark-hadron duality is indeed valid in the $B$ channel.

For a convenient use in the experimental analysis, 
the LCSR results for $B\to \pi$ form factor 
are usually fitted to simple parametrizations. 
One of them, suggested in Ref.~ \cite{KRWWY} employs
the ansatz \cite{BecirevicKaidalov} based
on the dispersion relation for $f_{B\pi}^+(q^2)$.
The latter is fitted to the LCSR predictions for $f^+_{B\pi}$ 
in its validity region $0< q^2 \leq 14-16$ GeV$^2$. For 
the  $B^*$-pole term the $B^*B\pi$  coupling \cite{coupling} is used 
determined from the same correlation function (\ref{eq:sr}). The result is: 
\be
f_{B\pi}^+(q^2)=\frac{0.23\div 0.33 }{(1-q^2/m_{B^*}^2)
(1-\alpha_{B\pi}q^2/m_{B^*}^2)}\,.
\label{eq:KRWWY}
\ee
where the values of the slope parameter correlated with the 
lower and upper limits of the interval for $f_{B\pi}^+(0)$ are almost equal:
$\alpha_{B\pi}=0.39\div 0.38$.
A different parametrization 
was suggested recently in Ref.~\cite{BallZ}:
\bea
f_{B\pi}^+(q^2)&&=\frac{0.26 \pm 0.06 \pm 0.05}{1-a(q^2/m_B^{2})+
b(q^2/m_B^2)^2},~~~ q^2<q_0^2,
\nonumber
\\
&&= \frac{c}{1- q^2/m_{B^*}^2}, ~~~q^2>q_0^2\,,
\label{eq:Ball}
\eea
where the LCSR result is extrapolated to large $q^2$ 
using the $B^*$-pole form. In Eq.~(\ref{eq:Ball}) the ranges 
of fitted parameters, 
$a=2.34 \div 1.76$, $b=1.77\div 0.87 $, 
$c= 0.384\div 0.523$, and $q_0^2= 14.3\div 18.5$ GeV$^2$,
are correlated with the first error in $f_{B\pi}^+(0)$,
whereas the second error is attributed to the 
uncertainty of the quark-hadron duality  approximation.
Note that all values within the uncertainty intervals 
in Eqs.~(\ref{eq:KRWWY}) and (\ref{eq:Ball})
have to be considered as equally acceptable 
theoretical predictions, without any ``preferred central value''. 
The  numerical differences between the form factors
(\ref{eq:KRWWY}) and (\ref{eq:Ball}) are smaller than the 
estimated  uncertainties and are caused by slightly different inputs and
by the small  $O(\alpha_s)$ correction to the twist 3 term taken 
into account in Eq.~(\ref{eq:Ball}) but not in Eq.~(\ref{eq:KRWWY})
(where an additional uncertainty was attributed to this missing 
correction).


Having at hand the form factor one may predict 
the $B\to \pi l \nu$ decay distribution using Eq.~(\ref{eq:width}).
It is shown in Fig.~\ref{fig:lcsr1} for the form factor 
(\ref{eq:KRWWY}). The corresponding 
integrated semileptonic width 
is 
\be
\Gamma(B^0\to \pi^- l^+ \nu_l)=(7.3 \pm 2.5)|V_{ub}|^2 \mbox{ps}^{-1}\,,
\label{gamma}
\ee
where the indicated error is mainly caused 
by the uncertainty of $f^+_{B\pi}(0)$, whereas 
the uncertainty of the form factor shape is insignificant.
This prediction  was recently used by Belle collaboration 
in their preliminary analysis of $B\to \pi l \nu$ decay
yielding \cite{Belle}: 
$|V_{ub}|= (4.09\pm 0.17(stat) \pm 0.33(syst.) \pm 0.76(th))\cdot 10^{-3}$. 
A close estimate was obtained in Ref.~\cite{KRWWY} using the older
CLEO measurement \cite{CLEO} of the   $B\to \pi l \nu$ width.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure}[htb]
\begin{center}
\begin{picture}(600,60)(0,0)
\Text(150,0)[l]{$\frac{1}{|V_{ub}|^2}d\Gamma/
dq^2[\mbox{ps}^{-1}\mbox{GeV}^{-2}]$} 
\Text(280,-240)[l]{$q^2$ [GeV$^2$]}
\end{picture}
\end{center}
\centerline{
\epsfig{file=figBpilnu.ps,scale=0.5,%
clip=}}
\vspace{0.3cm}
\caption{{\it LCSR prediction for the $B\to \pi l \nu$ 
decay distribution \cite{KRWWY} at the nominal values of inputs (solid),
with the interval of theoretical
uncertainties (dashed), compared with  some of the 
recent lattice calculations taken from Refs.~\cite{UKQCD} (solid points) and 
\cite{Abada} (open points).}}
\label{fig:lcsr1}
\end{figure}

The advantage of LCSR is that one can easily switch from 
the $B\to \pi $ to $D\to \pi $ form factor replacing $b$ quark by 
$c$ quark in the underlying correlation function (\ref{eq:sr}).
The LCSR prediction for $f_{D\pi}^+$ obtained in Ref.~\cite{KRWWY} and 
parametrized in the form analogous to Eq.~(\ref{eq:KRWWY}) 
yields a  $D^*$-pole dominance:
\be
f_{D\pi}^+= \frac{065\pm 0.11}{1- q^2/m_{D^*}^2}\,,
\label{eq:KRWWYDpi}
\ee
at $0<q^2<(m_D-m_\pi)^2$. The corresponding semileptonic width
$\Gamma( D^0\to\pi^- e^+ \nu_e)/|V_{cd}|^2=0.13 \pm 0.05 ~\mbox{ps}^{-1}$
calculated with the known value of $|V_{cd}|$ 
is, within errors,  in agreement with the experimental number 
$0.177 \pm 0.038~\mbox{ps}^{-1}$ \cite{PDG}. To make this comparison
more decisive it would be very important to have new, more accurate 
measurements of  the decay distribution and integrated  width of
$D^0\to\pi^- e^+ \nu_e$. 

Are further improvements of the LCSR result for $f^+_{B\pi}$ and 
$f^+_{D\pi}$ possible? As we have seen, the accuracy of  
OPE for the correlation function is quite sufficient.  
The  $O(\alpha_s^2)$ level recently
achieved in the sum rule for $f_B$ \cite{Jamintalk} is certainly 
not an immediate task for LCSR, being also technically very difficult.
More important is to improve the accuracy of the input parameters by
1) narrowing the interval of the $b$ quark mass and 
2) gaining a better control over the parameters of pion DA.
For the latter, in particular, one needs more precise data on 
pion form factors,
especially on $\gamma^*\gamma \to \pi^0$ (the latter form factor 
can in principle be measured at the same $e^+e^-$ $B$-factories)  
and, eventually, lattice QCD simulations of $\varphi_\pi(u)$ and other DA.     
A better control over duality approximation in the $B$ and $D$ channels
can be achieved  if radially excited $B$ and $D$ states  
are accurately identified with their masses and widths. 
Optimistically, one may hope to reduce the overall 
accuracy of the LCSR prediction for $f^+_{B\pi}$ and other  
heavy-to-light form factors 
to the level of $\pm 10 \%$ which is a natural limit for any 
QCD sum rule prediction.

Concluding let us emphasize that, in addition to 
approximate estimates of the form factors themselves, LCSR 
help in understanding important physical aspects of heavy-to-light 
transitions. First of all, LCSR allow  to {\em quantitatively} assess 
the role of the soft (end-point) vs hard (perturbative gluon exchange) 
contributions to the form factors, because both contributions 
are taken into account in this approach. 
Secondly, with LCSR one is able to predict \cite{cz} the 
$m_b\to \infty $ limit, $f_{B\pi}^+(0)\sim 1/m_b^{3/2}$, 
which is used in some lattice extrapolations.
Last but not least, LCSR can be expanded in powers of $1/m_b$ 
and $1/E_\pi$  assessing the size of $1/m_b$ and $1/E$ corrections 
to various relations predicted in effective theories for 
heavy-to-light decays.

This work is supported by BMBF (Bundesministerium
f\"ur Bildung und Forschung). 


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%%CITATION = HEP-PH 0001297;%%


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%%CITATION = HEP-PH 9904490;%%



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%%CITATION = PRLTA,77,5000;%%




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%%CITATION = EPHJA,C15,1;%%




\end{thebibliography}

\end{document}

