%Title: Subleading Shape Functions and the Determination of $V_{ub}$
%Author: T. Mannel
%Published: <EM> Proceedings of ICHEP 2002, Elsevier Science B.V., Bentvelsen et al.  Eds.  </EM> <B>  </B> (2002) 472-474. 
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\title{Subleading Shape Functions and the Determination of $V_{ub}$}
%
\author{Thomas Mannel \address{Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe, D--76128 Karlsruhe, Germany}}
\begin{document}

\begin{abstract}
The endpoint region of the inclusive lepton energy spectrum
in $B \to X_u \ell \bar\nu$ can be used for a determination of
$V_{ub}$ by comparing it to the endpoint region of the
inclusive photon spectrum of $B \to X_s \gamma$. We consider
the subleading twist contributions to this $V_{ub}$ determination.
%\vspace{1pc}
\vspace*{-5mm}
\end{abstract}

% typeset front matter (including abstract)
\maketitle

% \section{INTRODUCTION}
\noindent
A precise determination of the CKM matrix element $|V_{ub}|$ is one of
the most important measurements in the current quest to overconstrain
the unitarity triangle to test the standard model.  It is currently
believed that the theoretically
cleanest determination of $|V_{ub}|$ comes from the inclusive rate for
$B \to X_u \ell \bar \nu$, which may be calculated model-independently
via an operator product expansion in terms of local operators
\cite{HQETOPE}.   Such a measurement is hampered by the experimental
necessity to suppress the overwhelming charm background, which usually
requires tight kinematic cuts.  With a few exceptions \cite{qsqspec},
such cuts invalidate the local OPE usually used to calculate  the
inclusive branching ratio, and a twist expansion has to be performed
instead \cite{twistexp}. At leading order in this twist expansion,
and ignoring perturbative corrections, the lepton energy
spectrum in inclusive $B \to X_u \ell \bar \nu$ decay is given by
\begin{eqnarray}\label{Ellead}
\frac{d\Gamma}{dE_\ell} &=& \! \frac{2 \Gamma_0}{m_b} \! \! \! 
\int\limits_{-\infty}^{m_b-2E_\ell} \! \! \! d\omega\, f(\omega)\, , \,\,
\Gamma_0 = \frac{G_F^2 |V_{ub}|^2 m_b^5}{192 \pi^3} \, ,
\end{eqnarray}
where $f(\omega)$ is the nonperturbative light cone structure function
of the $B$ meson
\begin{eqnarray}
f(\omega) = \frac{1}{2m_B} \langle B | \bar h \,
\delta(i n \cdot D + \omega) \, h | B \rangle\,.
\end{eqnarray}
Since this function is a property of the $B$ meson, it is process
independent and can thus be measured elsewhere. The best way to
measure this structure function is from the photon energy spectrum
of the inclusive decay $B \to X_s \gamma$ \cite{shapeuniversal}.
Up to perturbative and subleading twist corrections, this spectrum
is directly proportional to the structure function,
\begin{eqnarray}\label{diffEgamma}
\frac{d\Gamma}{dE_\gamma} &=&  \frac{G_F^2 |V_{tb}V_{ts}^*|^2
  \alpha |C_7^{\rm eff}|^2m_b^5}{32 \pi^4} f(E_\gamma)\,.
\end{eqnarray}
Thus, combining data on $B \to X_s \gamma$ with data from
$B \to X_u \ell \bar \nu$, one can eliminate the dependence
on the structure function and therefore determine $|V_{ub}|$
with no model dependence at leading order
\cite{twistexp,shapeuniversal,ira}. The relation is
\begin{eqnarray}\label{treeVub}
\left|\frac{V_{ub}}{V_{tb}V_{ts}^*}\right|^2 =
\frac{3 \alpha}{\pi} |C_7^{\rm eff}|^2 \frac{\Gamma_u(E_c)}{\Gamma_s(E_c)}
+ \cdots 
\end{eqnarray}
where the elipses denote terms of order $1/m_b$ and $\alpha_s$ and 
\begin{eqnarray}\label{GuGsdefs}
\Gamma_u(E_c) &\equiv& \int_{E_c}^{m_B/2} d E_\ell
\frac{d \Gamma_u}{d E_\ell}\nonumber\\
\Gamma_s(E_c) &\equiv& \frac{2}{m_b}
\int_{E_c}^{m_B/2} d E_\gamma (E_\gamma - E_c)\frac{d \Gamma_s}{d E_\gamma} \,.
\end{eqnarray}
In this talk I describe the work done in collaboration with
C. Bauer and M. Luke \cite{subleading,BLMb2u} (see also \cite{LLW}),
where we calculate the subleading twist corrections to Eq.~(\ref{treeVub}).
While perturbative corrections to this relation have been studied in
detail \cite{shapeuniversal,ira}, much less is known about nonperturbative
corrections suppressed by powers of $\lqcd/m_b$. In \cite{subleading} we
introduced a formalism  which allows us to parametrize these subleading
twist corrections in terms of subleading shape
functions. The subleading twist corrections to the
photon energy spectrum in $B \to X_s \gamma$ read 
\begin{eqnarray}\label{subleadingbsg}
&& \frac{m_b}{\Gamma_0^s}\frac{d\Gamma}{dE_\gamma} =
(4 E_\gamma-m_b)F(m_b-2E_\gamma) \nonumber \\
&& \qquad
 + \frac{1}{m_b} \left[h_1(m_b-2E_\gamma) + H_2(m_b-2E_\gamma) \right]
\end{eqnarray}
where 
\begin{eqnarray}\label{Gamma0s}
\Gamma_0^s = \frac{G_F^2 |V_{tb}V_{ts}^*|^2
             \alpha |C_7^{\rm eff}|^2 m_b^5}{32 \pi^4}
\end{eqnarray}
and the functions $F$, $h_1$ and $H_2$ are defined in (\ref{ME}) below.

The subleading twist corrections to the lepton energy spectrum in
$B \to X_u \ell \bar \nu$ may be calculated in much the same way as
the photon spectrum for $B\to X_s\gamma$.
The matching calculation is performed by expanding full QCD matrix
elements to subleading twist and identifying the result with non-local
operators introduced in \cite{subleading}. The calculation is standard
and uses the optical theorem to
obtain  the lepton energy spectrum from the imaginary
part of a forward scattering matrix element. This matrix element is
rewritten using $p_b^\mu = m_b v^\mu + k^\mu $ and
$Q^\mu = m_b v^\mu - p_\ell^\mu$ and the light cone vectors 
$n^\mu$,  $\bn^\mu$
($n^2=\bn^2 = 0$, $n\cdot \bn = 2$, $2v = n + \bn$), defined  by
$p_\ell^\mu = E_\ell \bn^\mu$. The endpoint region is defined by 
\begin{eqnarray}
\bn \cdot Q = m_b \, \quad  
n \cdot Q = m_b - 2 E_\ell \sim O(\lqcd) \nonumber 
\end{eqnarray}
and the matrix element is expanded in powers of $1/m_b$, assuming
$k^\mu \sim O(\lqcd) $.

The leading twist contribution is the first term of this expansion. 
It can be written as a convolution of a Wilson coefficient function
$C_0 (\omega)$ with an operator $O_0(\omega)$
\begin{eqnarray}
\int \! d \omega \, C_0(\omega) O_0(\omega) \!\!\! &=& \!\!\!
\int \! d \omega \, C_0(\omega) \bar h \delta(i n \cdot D + \omega) h
\end{eqnarray}
and one finds at tree level
\begin{eqnarray}
C_0(\omega) = 2 \, \Gamma_0 \, \theta(m_b - 2 E_\ell - \omega)\,,
\end{eqnarray}
leading to Eq.~(\ref{Ellead}). 

At subleading twist there are five nonlocal operators contributing to
$B$ meson decays \cite{subleading}
\begin{eqnarray}
&& O_1^\mu(\omega) = 
\bar h_v \left\{i D^\mu,\delta(in \mcdot D + \omega)\right\} h_v
\nn\\
&& P_2^{\mu\alpha}(\omega) =
\bar h_v s^\alpha \left[i D^\mu,\delta(in \mcdot D + \omega)\right] h_v \nn\\
&& O_T (\omega) = i \int\! d^4y \, \frac{dt}{2 \pi} \,
  e^{-i\omega t} T \left( \bar{h}_v (0) h_v (t) O_{1/m} (y)
  \right)\nn \\
&& O_3(\omega) =
\!\!\int \!d \omega_1 d \omega_2 \,
\frac{\delta(\omega-\omega_1)- \delta(\omega-\omega_2)}{\omega_1 - \omega_2}
\nn \\ &&  
\Big[ h_v \delta(in \mcdot D \!+\! \omega_2) (i D_\perp)^2
     \delta(in \mcdot D \!+\! \omega_1) h_v\Big]\nn\\
&& P_4^\alpha (\omega) = g
\!\!\int \!d \omega_1d \omega_2\,\frac{\delta(\omega-\omega_1)- \delta(\omega-\omega_2)}{\omega_1 - \omega_2} \nn \\ \nn && 
\Big[ h_v s^\alpha \delta(in \mcdot D \!+\! \omega_2) \epsilon_\perp^{\mu\nu} G^\perp_{\mu\nu} \delta(in \mcdot D \!+\! \omega_1) h_v\Big]\,,\nonumber
\end{eqnarray}
where $s^\alpha = P_+ \gamma^\alpha \gamma^5 P_+$ and $O_{1/m}$ are the
subleading contributions to the Lagrangian. 

To match onto these operators we expand the full theory calculations to one
order higher in $k^\mu/m_b$ and $n\cdot Q/m_b$. The
result for the coefficients for the subleading twist operators
$O_1^\mu(\omega)$ and $O_3(\omega)$ is
\begin{eqnarray}
C_1^\mu(\omega)  &=& \frac{\Gamma_0}{m_b}n^\mu \theta(m_b-2 E_\ell-\omega) \\
C_3(\omega) &=& -\frac{2 \Gamma_0}{m_b} \theta(m_b-2 E_\ell-\omega)\,.
\end{eqnarray}
The result for the the coefficient $C_3(\omega)$ agrees with the prediction
from reparameterization invariance, which relates this coefficient to
the leading order Wilson coefficient to all orders
in perturbation theory
\cite{mannel_RPI}. For the operators $P_2^{\mu\alpha}(\omega)$ and
$P_4^\alpha(\omega)$
we obtain the matching coefficients 
\begin{eqnarray}
D_2^{\mu,\alpha}(\omega)  &=& -\frac{\Gamma_0}{m_b} i
\epsilon_\perp^{\mu\alpha}  \theta(m_b-2E_\ell-\omega)\\
D_4^\alpha(\omega) &=& \frac{3\Gamma_0}{m_b} n^\alpha
\theta(m_b-2E_\ell-\omega)\,.
\end{eqnarray}

Finally we have to parametrize the matrix elements of these operators
between $B$ meson states. We define
\begin{eqnarray} \label{ME}
\langle H | O_0( \omega) | H \rangle &=& 2m_B \, f(\omega) \nn\\
\langle H | O_1^\mu( \omega) | H \rangle &=& -4m_B\, \omega
f(\omega)\left( v^\mu - n^\mu \right) \nn\\
\langle H | O_3( \omega) | H \rangle &=& 2m_B \, G_2(\omega)\\
\langle H | O_T( \omega) | H \rangle &=& 2m_B \, t(\omega) \nn\\
\langle H | P_2^{\alpha,\mu}( \omega) | H \rangle &=& 
- 2m_B \, i \epsilon_\perp^{\mu\alpha}h_1(\omega)  \nn\\
\langle H | P_4^\alpha( \omega) | H \rangle &=& 4m_B \, (v^\alpha - n^\alpha)H_2(\omega)\nn
\end{eqnarray}
Due to reparametrization invariance and the general structure of
the $1/m_b$ corrections we find  that we only can have
the combination 
\begin{eqnarray}
F(\omega) = f(\omega) + \frac{1}{2m_b}
\left[ t(\omega) - 2  \, G_2(\omega) \right]\,.
\end{eqnarray}
which is a universal function for any decay rate.  
Putting all this information together, we therefore find for the lepton
energy spectrum in $B \to X_u \ell \bar\nu$ 
\begin{eqnarray}\label{finalbtou}
&& \frac{d \Gamma}{d E_\ell} =
\frac{4\,\Gamma_0}{m_b} \int \!d \omega \, \theta(m_b-2E_\ell-\omega) \nn \\ 
&& \left[ F(\omega) \left(1-\frac{\omega}{m_b} \right)
- \frac{1}{m_b} h_1(\omega) 
+ \frac{3}{m_b} H_2(\omega) \right] 
\end{eqnarray}
up to terms of order $\lqcd^2 / m_b^2 $.
We can now use these results to get the ${O}(\lqcd/m_B)$ corrections to the
determination of $|V_{ub}|$ given in Eq.~(\ref{treeVub}). We find
for the observables defined in Eq.~(\ref{GuGsdefs})
\begin{eqnarray} \label{rel1}
&& \Gamma_u(E_c) = \frac{4\, \Gamma_0}{m_b}\int_{E_c}^{m_B/2}\!\!dE
   \int_{-\infty}^{m_b-2E} \!\!d \omega
\nn \\ 
&& \left[ F(\omega) 
\left(1 - \frac{\omega}{m_b}\right) 
- \frac{1}{m_b} h_1(\omega) 
+ \frac{3}{m_b} H_2(\omega) \right] \\
&& \Gamma_s(E_c) = \frac{2\Gamma_0^s}{m_b}\int_{E_c}^{m_B/2}\!\!dE
   \int_{-\infty}^{m_b-2E} \!\!d \omega \nn \\
&& \left[ F(\omega) 
   \left(1 - \frac{2\omega}{m_b}\right) 
+ \frac{1}{m_b} h_1(\omega) 
+ \frac{1}{m_b} H_2(\omega) \right] \label{rel2}
\end{eqnarray}
where $\Gamma_0^s$ and $\Gamma_0$ are 
defined in Eqs.~(\ref{Gamma0s}) and (\ref{Ellead}).
This finally allows us to obtain 
the correction to the determination of $V_{ub}$  
\begin{eqnarray}\label{Vubrelation}
\left|\frac{V_{ub}}{V_{tb}V_{ts}^*}\right|
&=&\left( \frac{3 \alpha}{\pi} |C_7^{\rm eff}|^2
\frac{\Gamma_u(E_c)}{\Gamma_s(E_c)}\right)^\frac12 \! \! \! 
\left(1 + \delta(E_c)\right),
\end{eqnarray}
where $\delta(E_c)$ can be calculated form (\ref{rel1}) and (\ref{rel2}). 


To get a quantitative estimate for these corrections, we have to model the
subleading structure functions. Using a simple model for the leading
order structure function given in \cite{fmodel}, we take the subleading 
functions as the derivative of the leading-order shape function, normalized
such that they reproduce the known first moments.
%
\begin{eqnarray}\label{definefmodel}
f_\model(\omega) &=& \frac{32}{\pi^2\Lambda}
\left(1+\frac{\omega}{\Lambda}\right)^2
\exp \left[ -\frac{4}{\pi} \left(1+\frac{\omega}{\Lambda}\right)^2 \right]
\nn \\ 
&& \theta\left(1+\frac{\omega}{\Lambda}\right)
\end{eqnarray}
where $\Lambda=m_B-m_b$.  This gives
\begin{eqnarray}\label{Ffunct}
&& F_\model(\omega) = f_\model(\omega) + \frac{\lambda_1 -9 \lambda_2}{6\,m_b}
            f_\model'(\omega)  \\ \nonumber 
&& h_{1\,\model} =  \lambda_2\, f_\model'(\omega) \quad  
H_{2\,\model} = - \lambda_2\, f_\model'(\omega)  
\end{eqnarray}
%
and allows us to compute the correction $\delta (E_c)$,
which is plotted in Fig.~\ref{fig1}.
%\begin{figure}[htb]
%\vspace*{-9mm}
%\includegraphics[scale=0.8]{deltaplot.eps}
%\vspace*{-11mm}
%\caption{$\delta (E_c)$ as a function of $E_c$.}
%\label{fig1}
%\end{figure}
%
%\vspace*{-8mm}

The dashed line in Fig.~\ref{fig1} indicates a typical cut, and the 
corrections in this case are of the order of 15\% and hence are sizable. Since
this result is obtained using a model, the conservative estimate is 
to assign an uncertainty of this size to the determination of
$V_{ub}$. However, it has been argued \cite{Neubert} that the situation
may be not as bad, since the first moments of the subleading functions
are known and that the situation may be improved including this
information.

\begin{figure}[htb]
\vspace*{-9mm}
\includegraphics[scale=0.8]{deltaplot.eps}
\vspace*{-11.5mm}
\caption{$\delta (E_c)$ as a function of $E_c$.}
\label{fig1}
\end{figure}
\vspace*{-11mm}
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\end{document}


