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%% subsection 2.1 Hadron Energy Spectrum in the ACCMM Approach [slac-pub-7245-0-0-2-1 in slac-pub-7245-0-0-2: ^slac-pub-7245-0-0-2 >slac-pub-7245-0-0-2-2]
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\subsection{\usemenu{slac-pub-7245::context::slac-pub-7245-0-0-2-1}{Hadron Energy Spectrum in the ACCMM Approach}}\label{subsection::slac-pub-7245-0-0-2-1}
In the ACCMM model the $\bar{B}$-meson consists of a $b$-quark and a spectator
antiquark $\bar q$, flying back-to-back in the $\bar{B}$-rest frame
with 3-momentum vectors $\vec{p}$ and $-\vec{p}$, respectively;
the momentum distribution $\Phi(p)$ of the spectator is assumed to be
of Gaussian form
\be
\label{wave}
\Phi(p) = \frac{4}{\sqrt{\pi} p_F^3} \, \exp
\left(- \frac{p^2}{p_F^2}\right) \quad ; \quad p =|\vec{p}| \quad ,
\ee
normalized according to
\be
\int_0^\infty \, dp \, p^2 \, \Phi(p) =1 \quad .
\ee
The basic feature of this model is the requirement that in the $\bar B$
rest frame the energies of the two constituents have to add up to the
$\bar B$-meson mass $m_B$. This is possible only when at least one of
the masses of the constituents is allowed to be 3-momentum dependent.
Usually, the mass of the spectator $m_{sp}$ is considered to be
fixed to a constituent quark mass
value, while the $b$-quark mass becomes
momentum dependent,
\be
\label{mbp}
m_b(p)^2 = m_B^2 + m_{sp}^2 - 2 m_B \,
\sqrt{p^2+m_{sp}^2} \quad .
\ee
We now consider the semileptonic decay
$ \bar B =(b,{\overline q}) \rightarrow u (+g) \ell \overline \nu +
\overline{q}$. The symbols
$p_b$, $p_u$, $p_g$, $p_\ell$, $p_{\overline \nu}$, $p_q$, $p_B$, and
$p_H$ denote the four-momenta of the $b$-quark,
the $u$-quark, the gluon,
the charged lepton, the anti-neutrino,
the spectator, the $B$-meson,
and the hadronic matter of the final state, respectively.
In addition, we define the four vector $q^\mu$
\be
q = p_\ell + p_{\overline {\nu}} \quad .
\ee
Due to the fact that $p_b^\mu + p_q^\mu = p_B^\mu$ in the ACCMM model,
the vector
$q^\mu$ is the same with or without the spectator four-momentum, viz.,
\be
q= p_B - p_H = p_b +p_q - p_u (- p_g) -p_q = p_b -p_u (-p_g) \quad .
\ee
For this reason, it is technically easier
to concentrate first on the double differential distribution
$d^2 \Gamma/dq^2dq^0$,
because the spectator does not appear explicitly.
To achieve this,
we first derive the double differential decay rate (including the
$O(\a_s)$ radiative corrections)
for $b \to u (+g) \ell {\overline \nu}$, where
the $b$-quark decays at rest.
For simplicity, the $u$-quark is assumed to be massless.
The kinematical variables $q^2$ and $q^0$ vary in the region
\bea
\label{rangeq2q0}
0 \le q^2 \le m_b^2 \quad , \nonumber \\
\left( q^2 \right)^{1/2} \le q^0 \le {m_b^2 + q^2 \over 2 m_b} \quad .
\eea
For each $q^2$, the tree-level process
(as well as the virtual gluon corrections)
are concentrated at the upper
endpoint of the $q^0$ range in Eq.(\docLink{slac-pub-7245-0-0-2.tcx}[rangeq2q0]{6}).
In the following it is more convenient to introduce
dimensionless variables $x$ and $y$ which vary
independently in the range
$[0,1]$:
\bea
q^2 = x^2 m_b^2, \nonumber \\
q^0 = [x + {1 \over 2} (1-x)^2 y] m_b.
\eea
% The Jacobian associated with this change of variables is given by
% \be
% dq^2 d q^0 = m_b^3 x (1-x)^2 dx dy.
% \ee
With the above change of variables, we now proceed to calculate
the double differential distribution $d^2 \Gamma / dx dy$.
In terms of the new variables $(x,y)$, the tree-level contribution
and the virtual corrections are concentrated at $y=1$, while the
bremsstrahlung corrections give rise to a continuous distribution in
the whole $(x,y)$ domain. Individually, both the
virtual corrections and bremsstrahlung contributions
suffer from infrared and collinear divergences, which
occur at $y=1$ for a given value of $x$.
However, if the variable $y$ is integrated over a range
$s_0 \le y \le 1$ ($s_0 < 1$), these singularities cancel,
such that the quantity
\be
\label{pert}
{d \Gamma \over d x} (s_0) = \int_{s_0}^1 {d^2 \Gamma \over d x dy} d y
\nonumber \\
= \int_0^1 {d^2 \Gamma \over dx dy} dy - \int_0^{s_0} {d^2 \Gamma_{
brems} \over dx dy} dy
\ee
remains finite. The first term on the right hand side (RHS)
in Eq.(\docLink{slac-pub-7245-0-0-2.tcx}[pert]{8}) has been calculated in Ref.~\cite{17};
the result is
\be
\label{contr1}
{d \Gamma \over d x} \equiv \int_0^1 {d^2 \Gamma \over dx dy} dy
= 4 x (1-x^2)^2 (1 + 2 x^2) \, [ 1 -{2 \alpha_s \over 3 \pi} G(x)]
\, \Gamma_0
\quad ,
\ee
where $\Gamma_0$ and the function $G(x)$
which containes the radiative corrections are given by
\bea
\label{gamma0}
\Gamma_0 &=& \frac{G_F^2 \, m_b^5 \, |V_{ub}|^2}{192 \pi^3}
\quad ,
\\
G(x) &=&
\frac{[8x^2(1-x^2 -2x^4) \log x
+ 2(1-x^2)^2 (5 + 4 x^2) \log(1-x^2) -
(1-x^2)(5 + 9x^2 - 6 x^4) ]}{2 (1-x^2)^2 (1 + 2x^2)}
\nonumber \\
&& + \pi^2 + 2 \mbox{Li}(x^2) - 2 \mbox{Li}(1-x^2) \quad.
\eea
Here $\mbox{Li}(x)$ is the Spence function defined as
\be
\mbox{Li}(x) = - \int_0^x {dt \over t} \log(1-t).
\ee
Because the tree-level contribution and virtual
corrections are concentrated
at $y=1$, the second term on the RHS of Eq.~(\docLink{slac-pub-7245-0-0-2.tcx}[pert]{8})
contains the gluon bremsstrahlung contribution
only (as indicated by the notation).
As the endpoint region is cut off by $s_0 < 1$,
this term is finite; consequently, the infrared and collinear
regularization is not necessary from the very beginning.
We have calculated this term; the result is
\bea
\label{contr2}
\int_0^{s_0} {d^2 \Gamma_{brems} \over d x d y} d y =
4 x (1 - x^2)^2 (1 + 2x^2) \, {2 \alpha_s \over 3 \pi}
\, \left( \log^2(1-s_0) + H(s_0) \right) \,
\Gamma_0 \quad ,
\eea
where
\bea
\label{h0}
H(s_0) = && \int_0^{s_0} dy \Big( {4 \over 1 - y}
\log{2 - y(1-x) + \kappa
\over 2} \nonumber \\
&&- {(1-x)(3 + x + xy - y) \over (1+x)^2}
\Big[\log(1-y) - 2 \log{2 - y(1-x) + \kappa \over 2} \Big]
\nonumber \\
&&- { \kappa \over 2 (1 + x)^2 (1 + 2x^2) }
\Big[{7 (1+x) (1 + 2 x^2) \over 1 - y} + (1-x)(3 - 2 x^2) \Big]\Big).
\eea
The quantity $\kappa$ in Eq. (\docLink{slac-pub-7245-0-0-2.tcx}[h0]{14}) is defined as
$\kappa = \sqrt {y^2 (1-x)^2 + 4 xy}$.
Combining the two contributions, Eqs.(\docLink{slac-pub-7245-0-0-2.tcx}[contr1]{9}) and (\docLink{slac-pub-7245-0-0-2.tcx}[contr2]{13}),
we obtain
\be
{1 \over \Gamma_0} {d \Gamma \over d x} (s_0)
= 4 x (1-x^2)^2 (1 + 2x^2) \,
\left[ 1 - {2 \alpha_s \over 3 \pi} \log^2 (1 - s_0)
- {2 \alpha_s \over 3 \pi}
\left( G(x) + H(s_0) \right) \right] \, \Theta(1 - s_0).
\ee
The double logarithms arise from the soft and
collinear gluons and become
important as $s_0 \rightarrow 1$. Resumming these double logarithmic
terms to all orders, we get
\be
{1 \over \Gamma_0} {d \Gamma \over d x}(s_0)
= 4 x (1 - x^2)^2 (1 + 2 x^2) \,
\exp\left(-{2 \alpha_s \over 3 \pi} \log^2 (1 - s_0) \right) \,
\left[ 1 - {2 \alpha_s \over 3 \pi} \left( G(x)
+ H(s_0) \right) \right] \,
\Theta (1 - s_0).
\ee
This expression enables us to reproduce the Sudakov exponentiated
double-differential decay rate by differentiating with respect to $s_0$.
\bea
\label{doubleexpon}
{1 \over \Gamma_0} {d^2 \Gamma \over d x d y} &=&
- {d \over d s_0} \Big( {1 \over \Gamma_0}
{d \Gamma \over d x}(s_0) \Big)_{s_0 = y}
=\nonumber \\
&& - 4 x (1 - x^2)^2 (1 + 2 x^2) \,
\exp\Big( - {2 \alpha_s \over 3 \pi} \log^2 (1 - y) \Big)
\nonumber \\
&\times & \left\{
{4 \alpha_s \over 3 \pi} {\log(1-y) \over (1-y)}
\Big[ 1 - {2 \alpha_s \over 3 \pi} \big( G(x) + H(y) \big) \Big]
-{2 \alpha_s \over 3 \pi} {d H \over d y}(y) \right\}
.
\label{doublediff}
\eea
To get the parton level hadron energy spectrum for a $b$ quark
decaying at rest, we first re-express
Eq.(\docLink{slac-pub-7245-0-0-2.tcx}[doubleexpon]{17}) in terms of the variables $(q^2,q^0)$ and then
integrate over $q^2$; this leads to the
distribution $d\Gamma/dq^0$. As the
hadronic energy $E_{had}$ is related to $q^0$ by $E_{had}=m_b - q^0$,
the spectrum $d\Gamma/dE_{had}$ for a $b$--quark decaying at rest
is readily obtained.
To get rid of the $b$--quark mass dependence $m_b^5$ in the decay width
(as seen e.g. in Eq.(\docLink{slac-pub-7245-0-0-2.tcx}[gamma0]{10})) and errors thereof,
we present in the following the
differential branching ratio $dBR/dE_{had}$
that is obtained by dividing $d\Gamma/dE_{had}$
by the theoretical semileptonic $b$--quark decay width $\Gamma_{sl}$
and multiplying this ratio by the experimentally measured semileptonic
branching ratio $BR_{sl}=(10.4 \pm 0.4) \%$ \cite{18}:
\be
\label{brdiff}
\frac{dBR(B \to X_u \ell \nu)}{dE_{had}} = \left(
\frac{1}{\Gamma_{sl}} \frac{d\Gamma}{dE_{had}} \right) \, BR_{sl} .
\ee
The semileptonic decay width,
neglecting the small $B \to X_u \ell \nu$
contribution, reads
\be
\label{semileptonic}
\G_{sl} = \frac{G_F^2 \, m_b^5 \, |V_{cb}|^2}{192 \p^3} \,
g(m_c/m_b) \, \left( 1 -
\frac{2 \a_s(m_b)}{3 \p} f(m_c/m_b) \right) ,
\ee
where the phase space function $g(u)$ is defined as
\be
\label{gu}
g(u) = 1 - 8 u^2 + 8 u^6 - u^8 - 24 u^4 \log u \quad ,
\ee
and the radiative correction function in an approximate analytic form
is given by~\cite{11}:
\be
\label{fu}
f(u) = \left( \p^2 - \frac{31}{4} \right) \, (1-u)^2 + \frac{3}{2}
.
\ee
The result (based on the the
Sudakov improved differential branching ratio
as just derived) is shown by the dash-dotted line in Fig. 1 (using
$m_b=4.85$ GeV and $m_c=1.45$ GeV). To emphasize the effects of
the Sudakov resummation, we have also plotted
the result obtained in pure $O(\a_s)$ perturbation theory \cite{19},
i.e., without
exponentiation (short-dashed line).
This curve exhibits a (integrable) double logarithmic divergence
when $E_{had}=m_b/2$ is approached from above.
The effects of the exponentiation
are therefore most prominent in the region
around $m_b/2$ as illustrated in Fig.~1, because
Sudakov-exponentiation suppresses the
singularity just mentioned.
We stress, however,
that the effect of exponentiation is
negligibly small in the whole region
below the
charmed hadron threshold we are mostly interested in.
This should be contrasted with the case of
the energy spectrum of the charged lepton near
the kinematical endpoint where these effect are most pronounced.
Another interesting aspect concerning
the radiative corrections to the hadron
energy spectrum is that the kinematical boundary of the hadron energy
depends on whether the final state contains bremsstrahlung gluons
or not.
In the absence of bremsstrahlung gluons,
the kinematical endpoint is at $m_b/2=2.425$ GeV as can be seen
form the long-dahed curve representing
the result without QCD corrections.
Fig.~1 clearly illustrates that, above $E_{had} = 2.425$ GeV, a
significant tail due to gluon bremsstrahlung, hence of
order $\alpha_s(m_b)$ in strength, is present in the spectrum.
\begin{figure}[htb]
\vspace{0.10in}
\centerline{
\epsfig{file=fig1.ps,height=3in,angle=0,clip=}
}
\vspace{0.08in}
\caption[]{The long-dashed line shows the hadron energy spectrum for a
$b$-quark decaying at rest without taking into account any QCD corrections.
The short-dashed line is the corresponding spectrum including
$O(\a_s)$ virtual and bremsstrahlung corrections. The result after
exponentiating the Sudakov double logarithms
(discussed in the present section) is shown by the dash-dotted line.
The solid line shows
the hadron energy spectrum for a $B$-meson decaying at rest.
The bound state effects are calculated with the ACCMM model
with $p_F=344$ MeV and $m_{sp}=150$ MeV
(corresponding to $\langle m_b \rangle =4.85$ GeV).
The open circles show the hadron energy spectrum due to a $B$-meson
decaying at flight with a momentum $|p_B|=330$ MeV.
\label{fig:1}}
\end{figure}
We now turn to implement the bound--state effects using the ACCMM model.
We start from the double differential distribution
$d \Gamma/dq^2dq^0$ in Eq.~(\docLink{slac-pub-7245-0-0-2.tcx}[doubleexpon]{17}). As the $b$-quark moves
with momentum $\vec{p}$ inside the $B$-meson,
we first replace the mass $m_b$ by $m_b(p)$ as given in Eq.~(\docLink{slac-pub-7245-0-0-2.tcx}[mbp]{3}).
We then Lorentz boost the double
differential distribution and get the spectrum for a $b$-quark
decaying at flight (momentum $\vec{p}$).
Finally we convolute the spectrum with the ACCMM distribution
function given in Eq.~(\docLink{slac-pub-7245-0-0-2.tcx}[wave]{1}). This leads to the
ACCMM averaged quantity
$d \Gamma/dq^2dq^0$ for a $B$-meson decaying at rest.
Again, this distribution is straightforwardly
converted to $d\Gamma/dE_{had}$.
To get the differential branching ratio, we replace
the $b$--quark mass $m_b$ in the expression
for the semileptonic decay width in Eq. (\docLink{slac-pub-7245-0-0-2.tcx}[semileptonic]{19})
by the ACCMM average value $\langle m_b \rangle$ derived by applying
the ACCMM convolution to the total semileptonic decay rate.
This ACCMM averaged $b$-quark mass is given by
\be
\label{mbeff}
\langle m_b^5 \rangle = \int dp \, p^2 \,
\big(m_b(p)\big)^5 \, \Phi(p) \quad . \ee
The $m_b - m_c$ mass difference is reliably calculated to be 3.40 GeV
by HQET~\cite{20}. Therefore, we set
$m_c= \langle m_b \rangle - \, 3.40 \, \mbox{GeV}$ in the $\Gamma_{sl}$.
The result is shown in Fig. 1 by the solid line; here we have used
$p_F=344$ MeV and $m_{sp}=150$ MeV,
which yields $\langle m_b \rangle =4.85$ GeV
according to Eq. (\docLink{slac-pub-7245-0-0-2.tcx}[mbeff]{22}).
As shown in Fig. 1, the main effect of the Fermi motion
of the $b$-quark inside $B$-meson is to
shift the perturbative hadron energy spectrum
uniformly to higher energies by about
300 MeV. We will elaborate this point in more
detail when discussing the HQET
approach to the bound-state effects, where the same result
can be understood in a more transparent way.
There is one more source of Doppler shift to the hadron energy spectrum.
At a symmetric $B$-factory the $B$-meson is produced from the decay of the
$\Upsilon(4S)$ resonance. Due to the released binding energy
the $B$-mesons are produced with a momentum
$|\vec{p}_B| \approx 330$ MeV in the $\Upsilon(4S)$ rest frame.
We have also worked out this effect to the spectrum; the corresponding
result is shown in Fig. 1 by open circles.
As this effect is very small, we will not consider it any more
in foregoing discussions.
Although the ACCMM model has two input parameters, viz.
$p_F$ and $m_{sp}$,
it turns out that the hadron energy spectrum is sensitive only to
one parameter, namely, the average $b$--quark
mass $\langle m_b \rangle$,
which is a function of $p_F$ and $m_{sp}$
(see Eq.~(\docLink{slac-pub-7245-0-0-2.tcx}[mbeff]{22}))~\footnote{
It has been pointed out that transitions from heavy quarks
to light quarks with masses $m_q \le (\bar{\Lambda} m_b)^{1/2}$ always
give rise effectively to a one-parameter dependence~\cite{16}.}.
To illustrate this point, we have chosen two different pairs of
$(p_F,m_{sp})$ values, which both correspond
to the same value of $\langle
m_b \rangle$
(=4.85 GeV in the present case).
The correspondig result plotted in Fig.~2 indicates little dependence
on $p_F$ and $m_{sp}$, once $\langle m_b \rangle$ is hold fixed.
In Fig.~3, we have varied $\langle m_b \rangle$ over the
range indicated in the plot. We have found that the hadron energy
spectrum depends on $\langle m_b \rangle$ rather strongly,
especially in the
region below the charmed hadron threshold.
\begin{figure}[htb]
\vspace{0.10in}
\centerline{
\epsfig{file=fig2.ps,height=3in,angle=0,clip=}
}
\vspace{0.08in}
\caption[]{The hadron energy spectrum
(based on the ACCMM model) is shown for
two different pairs of model paramters $(p_F,m_{sp})$.
The solid line corresponds to
$(374 \ \mbox{MeV}, 0)$ while the dashed line represents the result for
$(252 \ \mbox{MeV}, 300 \, \mbox{MeV})$.
Both set correspond to the same value of the $b$-quark mass $\langle m_b
\rangle =4.85$ GeV.
\label{fig:2}}
\end{figure}
\begin{figure}[htb]
\vspace{0.10in}
\centerline{
\epsfig{file=fig3.ps,height=3in,angle=0,clip=}
}
\vspace{0.08in}
\caption[]{The hadron energy spectrum
based on the ACCMM approach is shown
for different values of the $b$-quark mass $\langle m_b \rangle$.
\label{fig:3}}
\end{figure}
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