text:
\section{Introduction}

Semileptonic $B$ decays
\begin{equation}
\label{firsteq}
B \, \rightarrow \, X_u \,+ \,  l \, + \, \nu,
\end{equation}
where $X_u$ is any hadron final state coming from the fragmentation of the
$up$ quark, are interesting processes for the study of strong interactions
as well as of weak interactions.
The computation of the spectra in (\ref{firsteq}) is often non trivial because of the
presence of double infrared logarithms in the perturbative expansion, which
formally diverge in the endpoints and therefore must be resummed to all orders
in the QCD coupling $\alpha_S$ \cite{altetal}.
In all generality, large logarithms come from the so-called threshold
region, defined as the one having
\begin{equation}
\label{mXllEX}
m_X \, \ll \, E_X \, \le \, m_b,
\end{equation}
where $m_X$ and $E_X$ are the invariant mass and total energy of the final
hadron state $X_u$ and $m_b$ is the beauty mass.
We find useful to summarize here the main results of \cite{me,noi}.
The infrared logarithms in process (\ref{firsteq})
can be organized in a series of the form
\begin{equation}
\label{start}
\Sigma[u;\,\alpha(Q)] \, = \, 1 \, + \, \sum_{n=1}^\infty
\sum_{k=1}^{2n}
\Sigma_{n k} \, \alpha^n(Q) \, \log^k\left( \frac{1}{u} \right),
\end{equation}
and can be factorized into the universal QCD form factor $\Sigma$.
The $\Sigma_{nk}$'s are numerical coefficients whose explicit expressions
can be obtained from \cite{noi} and $\alpha=\alpha_S$ is the QCD coupling.
$Q$ is the hard scale of the process and is determined by the final hadron
energy:
\begin{equation}
Q \, = \, 2E_X.
\end{equation}
We have defined the hadron variable
\begin{equation}
u \, = \, \frac{1 - \sqrt{1 - \left(2m_X/Q\right)^2} }{1 + \sqrt{1 - \left(2m_X/Q\right)^2} }
\, \simeq \, \left(\frac{m_X}{Q}\right)^2
~~~~~~~~~~~~~~~~~~(0\le u \le 1),
\end{equation}
involving the ratio of the invariant hadron mass to the hard
scale, where in the last member we have taken the leading term in
the threshold region (\ref{mXllEX}) only. As it is well known, the
QCD form factor in  eq.~(\ref{start}) contains at most two
logarithms for each power of $\alpha$, coming from the overlap of
the soft and the collinear region in each emission. Note that the
hard scale $Q$ enters the argument of the infrared logarithms $1/u
\simeq Q^2/m_X^2$ as well as the argument of the running coupling
$\alpha=\alpha(Q)$. One can obtain a factorized form for the
triple differential distribution
--- which is the most general distribution in process (\ref{firsteq}) --- from which
all other spectra can be obtained by integration:
\begin{equation}
\label{tripla}
\frac{1}{\Gamma} \int_0^u \frac{d^3\Gamma}{dx dw du'} \, du'
\,=\, C\left[x,w;\alpha(w\,m_b)\right]\,
\Sigma\left[u;\alpha(w\,m_b)\right] \, + \, D\left[x,u,w;\alpha(w\,m_b)\right],
\end{equation}
where
\begin{equation}
w \,=\, \frac{2 E_X}{m_b}~~~~~~~~(0\le w \le 2);~~~~~~~~~~~~~~~~~~~~~
x \,=\, \frac{2 E_l}{m_b}~~~~~~~(0\le x \le 1).
\end{equation}
$\Gamma$ is the total semileptonic width,
$C\left[x,w;\alpha\right]$ is a short-distance coefficient function
independent on $u$ and
$D\left[x,u,w;\alpha\right]$ is a remainder function not containing
infrared logarithms (i.e. short-distance dominated) and
vanishing for $u\rightarrow 0$ as well as for $\alpha\rightarrow 0$.
The explicit expressions of these functions have been given in \cite{noi}.

The properties of semileptonic decay spectra are best understood
comparing them with the simpler radiative decay
\begin{equation}
\label{raddec}
B \, \rightarrow \, X_s \, + \, \gamma.
\end{equation}
In such decay we have indeed:
\begin{equation}
~~~~~~~~~~~~~~~~~
Q \, = \, m_b \left( 1 - \frac{q^2}{m_b^2} + \frac{m_X^2}{m_b^2} \right)
\, = \,   m_b \left( 1 + \frac{m_X^2}{m_b^2} \right)
\, \simeq \, m_b,
\end{equation}
where $q^{\mu}$ is the 4-momentum of the real photon --- in
general of the probe. In the radiative decays (\ref{raddec}) the
hard scale is therefore independent on the kinematics and is fixed
by the beauty mass. In the semileptonic decay (\ref{firsteq}),
$q^{\mu}$ is the dilepton momentum and we have the more general
situation $0\le q^2 \le m_b^2$; the hard scale is given in this
case by
\begin{equation}
Q \, \simeq \, m_b \left( 1 - \frac{q^2}{m_b^2} \right)
\end{equation}
and depends on the dilepton invariant mass squared $q^2$:
it cannot be identified with the heavy flavor mass $m_b$. Kinematic configurations with
\begin{equation}
m_X \, \ll \, Q \, \approx \, m_b
\end{equation}
as well as with
\begin{equation}
\label{specificSL}
m_X \, \ll \, Q \, \ll \, m_b
\end{equation}
are possible. In fact, in the radiative decays, for example, the
average hadron energy is
\begin{equation}
\langle E_X  \rangle_{rd} \, = \, \frac{1}{2} \, m_b \, \left[ 1 \, + \, O(\alpha) \right],
\end{equation}
while in the semileptonic ones we have the smaller value \cite{noi}
\begin{equation}
\langle E_X  \rangle_{sl} \, = \, \frac{7}{20} \, m_b \, \left[ 1 \, + \, O(\alpha) \right] .
\end{equation}
In general, semileptonic spectra may or may not involve integration over the
hadron energy $E_X$ and, according to \cite{me,noi},
have a different infrared structure in the two cases.
In \cite{noi} we have studied in detail the simpler case of the distributions
not integrated over the hadron energy, i.e. over the hard scale $Q$,
which have the same infrared structure
of the hadron invariant mass distribution in the radiative decay (\ref{raddec}):
\begin{equation}
\label{specraddec}
\frac{1}{\Gamma_R}\int_0^{t_s}\frac{d\Gamma_R}{dt_s'} \, dt_s'
\, = \,
C_R(\alpha) \left( 1 \, + \,
\, \sum_{n=1}^\infty
\sum_{k=1}^{2n}
\Sigma_{n k} \, \alpha(m_b)^n\, \log^k \frac{1}{t_s} \right)
\, + \, D_R(t_s;\alpha).
\end{equation}
We have defined $t_s = m_{X_s}^2/m_b^2 = 1-x_{\gamma}$,
with $x_{\gamma}=2E_{\gamma}/m_b$,
$\Gamma_R$ is the total radiative width, $C_R(\alpha)$ is a short-distance
coefficient function and $D_R(t_s;\alpha)$ is a short-distance remainder function.

In this paper, we attack the distributions integrated over the hadron energy,
which have a more complicated infrared structure than the one in (\ref{raddec}).

In sec.~\ref{sect1} we resum to next-to-leading order (NLO) the distribution
in the hadron variable $u$ defined before.
The infrared logarithms appearing in the perturbative expansion of this spectrum,
\begin{equation}
\frac{1}{\Gamma}\int_0^u\frac{d\Gamma}{du'} \, du' \, = \, C_U(\alpha)\,
\left( 1 \, + \, \sum_{n=1}^\infty
\sum_{k=1}^{2n}
\Sigma_{U nk} \, \alpha\left(m_b\right)^n \, \log^k \frac{1}{u} \right)
\, + \, D_U(u;\alpha),
\end{equation}
coincide in first order with those in the decay (\ref{raddec}):
$\Sigma_{U12}=\Sigma_{12}$ and $\Sigma_{U11}=\Sigma_{11}$,
while they differ in higher orders.
$C_U(\alpha)$ is a short-distance coefficient function and $D_U(u;\alpha)$
is a short-distance remainder function whose explicit expressions will
be given in sec.~(\ref{sect1}).

In sec.~\ref{sect2} we compute to NLO the distribution in the variable
\begin{equation}
t \, = \, \frac{m_X^2}{m_b^2}
~~~~~~~~~~~~~~~~~~~~~~~~~~(0\le t \le 1),
\end{equation}
i.e. the distribution in the invariant hadron mass squared.
The infrared logarithms appearing in this distribution,
\begin{equation}
\frac{1}{\Gamma}\int_0^t\frac{d\Gamma}{dt'} \, dt' \, = C_T(\alpha)\,
\left( 1 \, + \, \sum_{n=1}^\infty
\sum_{k=1}^{2n}
\Sigma_{T nk} \, \alpha\left(m_b\right)^n \, \log^k \frac{1}{t}  \right)
\, + \, D_T(t;\alpha),
\end{equation}
differ also at the $O(\alpha)$ single logarithm level from the
corresponding ones in (\ref{raddec}):  $\Sigma_{T11}\ne\Sigma_{11}$.
We also define a non-minimal factorization-resummation scheme
which seems to have better convergence properties of the perturbative
series than the minimal one.

We define for both spectra effective form factors which resum
the large logarithmic corrections.
These form factors involve a convolution
of a process-dependent coefficient function with the universal
QCD form factor $\Sigma$ entering the radiative decay (\ref{raddec}) and the
triple differential distribution in the decay (\ref{firsteq}).
The definition of the effective form factors we give is, in a sense,
non perturbative, because we do not look at the explicit perturbative
expansion of $\Sigma$.

Finally, in sec.~\ref{concl}, we draw our conclusions and we consider
generalizations of our results.

\section{Distribution in the hadron mass/energy ratio}
\label{sect1}

In this section we compute the resummed distribution in the variable $u$
(defined in the introduction) to next-to-leading order (NLO).
That is accomplished by integrating the resummed double distribution
in $u$ and $w$ obtained in sec.~(4) of \cite{noi} over $w$:
\begin{equation}
\label{firstu}
\frac{1}{\Gamma}\frac{d\Gamma}{du}
\, = \, \int_0^{1+u} dw \, \frac{1}{\Gamma}\frac{d^2\Gamma}{dw du}.
\end{equation}
By replacing the resummed expression on the r.h.s. of eq.~(\ref{firstu}),
we obtain:
\begin{equation}
\label{exactU}
\frac{1}{\Gamma}\frac{d\Gamma}{du}
\, = \, \int_0^{1+u} dw \, C_H(w;\,\alpha) \, \sigma\left[u;\,\alpha(w\,m_b)\right]
\, + \, \int_0^{1+u} dw \, d_H(u,w;\,\alpha).
\end{equation}
$C_H(w;\,\alpha)$ is a short-distance coefficient function having an
expansion in powers of $\alpha$:
\begin{equation}
\label{CHw}
C_H(w;\,\alpha) \, = \, C_H^{(0)}(w) \, + \, \alpha \, C_H^{(1)}(w)
\, + \, \alpha^2 \, C_H^{(2)}(w) \, + \, O(\alpha^3),
\end{equation}
with
\begin{eqnarray}
\label{CHw1}
C_H^{(0)}(w) &=& 2w^2(3-2w);
\\
\label{CHw2}
C_H^{(1)}(w) &=& \frac{C_F}{\pi} \, 2w^2(3-2w)
\left[ {\rm Li}_2(w) \, + \,\log w\log(1-w) \, - \, \frac{35}{8}
\, - \, \frac{9-4w}{6-4w} \log w \right],
\end{eqnarray}
where $C_F=(N_c^2-1)/(2N_c)$, $N_c=3$ is the number of colors and
$\alpha=\alpha(m_b)$. $\sigma\left(u;\,\alpha\right)$ is the
differential QCD form factor:
\begin{equation}
\sigma(u;\,\alpha) \, = \, \frac{d}{du} \Sigma(u;\,\alpha),
\end{equation}
with $\Sigma(u;\,\alpha)$ being the cumulative form factor
considered in the introduction,
\begin{equation}
\label{Sigma}
\Sigma(u;\,\alpha) \, = \, 1 \, + \, \alpha \, \Sigma^{(1)}(u)
\, + \, \alpha^2 \, \Sigma^{(2)}(u) \, + \, O(\alpha^3)
\end{equation}
and
\begin{equation}
\Sigma^{(1)}(u)\, = \, - \, \frac{C_F}{\pi}
\left[ \frac{1}{2}\log^2 u \, + \, \frac{7}{4}\log u \right].
\end{equation}
Finally, $d_H(u,w;\,\alpha)$ is a short-distance remainder function whose explicit
expression is not needed here.
We are interested in the threshold region (\ref{mXllEX}), which can also be defined
as the one having
\begin{equation}
u \, \ll \, 1.
\end{equation}
Since large logarithms originate only from the first integral on the r.h.s.
of eq.~(\ref{exactU}),
in order to isolate them, let us neglect at first the contribution from the
remainder function $d_H(u,w;\,\alpha)$.
The small terms for $u\rightarrow 0$ will be included later on, by expanding
the resummed expression and comparing with fixed-order spectrum,
as discussed at the end of sec.~2 of \cite{noi}.
The integration of the first term on the r.h.s. of eq.~(\ref{exactU}) over the hadron energy
is not trivial because the coefficient function $C_H(w;\,\alpha)$ depends on $w$
as well as the QCD form factor $\sigma[u;\,\alpha(w\,m_b)]$, which depends on
$w$ via the scale of the coupling $\alpha=\alpha(w\,m_b)$.
Unlike the distributions considered in \cite{noi}, we cannot factor out in this case
the universal form factor $\sigma$.

Since it is technically simpler to deal with partially-integrated form factors
rather than with differential ones, let us define the event fraction
\begin{equation}
R_U(u) \, = \, \int_0^u  \frac{1}{\Gamma}\frac{d\Gamma}{du'} du',
\end{equation}
which has the end-point values:
\begin{equation}
R_U(0) \, = \,0;~~~~~~~~~~R_U(1) \, = \,1.
\end{equation}
The spectrum in $u$ is trivially obtained by differentiation:
\begin{equation}
\frac{1}{\Gamma}\frac{d\Gamma}{du} \, = \, \frac{d}{du} \, R_U(u).
\end{equation}
Integrating both sides of eq.~(\ref{exactU}), we obtain:
\begin{eqnarray}
R_U\left[u;\alpha(m_b)\right]
&=& \int_0^u du' \int_0^{1+u'} \, dw \, C_H(w;\alpha)\,\sigma[u';\alpha(w\,m_b)]
\, + \, O(u,\alpha)
\nonumber\\
&=& \int_0^u du' \int_0^1 \, dw \, C_H(w;\alpha)\,\sigma[u';\alpha(w\,m_b)]
 + \int_0^u du' \int_1^{1+u'} \, dw \, C_H(w;\alpha)\,\sigma[u';\alpha(w\,m_b)]+
\nonumber\\
& & + \, O(u;\alpha),
\end{eqnarray}
where by $O(u;\alpha)$ we denote terms which vanish for
$u\rightarrow 0$ as well as for $\alpha\rightarrow 0$. The second
integral in the last member of the r.h.s. extends to a kinematic
region in $w$ which is $O(u)$ and therefore can be dropped in the
threshold region; we can therefore assume tree-level kinematics:
\begin{equation}
0 \, \le \, w \, \le \, 1.
\end{equation}
By exchanging the order of the integrations in the remaining integral, we obtain:
\begin{equation}
\label{Unorem}
R_U\left[u;\alpha(m_b)\right]\,=\,\int_0^1 \, dw \, C_H(w;\alpha)\,\Sigma[u;\alpha(w\,m_b)]
\, + \, O(u;\alpha).
\end{equation}
Substituting the explicit expressions for the coefficient function given in eq.~(\ref{CHw})
and of the QCD form factor given in eq.~(\ref{Sigma}), we obtain:
\begin{equation}
\int_0^1 \, dw \, C_H(w;\alpha)\,\Sigma(u;\,\alpha) \, = \,
1 \, - \,
\frac{\alpha\,C_F}{\pi}
\left[
\frac{1}{2}\log^2 u \, + \, \frac{7}{4}\log u \, + \, \frac{335}{144}
\right]
\, + \, O(\alpha^2).
\end{equation}
The next step is to factorize the event fraction $R_U(u;\,\alpha)$
into:
\begin{itemize}
\item
a QCD form factor $\Sigma_U(u;\,\alpha)$ containing the long-distance contributions,
i.e. the $\log 1/u$ terms diverging for $u\rightarrow 0$;
\item
a coefficient function $C_U(\alpha)$ containing the constant
terms for $u\rightarrow 0$;
\item
a remainder function $D_U(u;\,\alpha)$, collecting the left-over small
contributions $O(u;\alpha)$, vanishing for $u\rightarrow 0$ and for $\alpha\rightarrow 0$.
\end{itemize}
Let us write therefore:
\begin{equation}
\label{resumU} R_U\left(u;\,\alpha\right) \, = \, C_U(\alpha) \,
\Sigma_U(u;\,\alpha) \, + \, D_U(u,\,\alpha).
\end{equation}
The coefficient function, the form factor and the remainder function can
all be expanded in powers of $\alpha$:
\begin{eqnarray}
C_U(\alpha) &=& 1 \, + \, \alpha \, C_U^{(1)} \, + \, \alpha^2 \, C_U^{(2)}
\,+ \, O(\alpha^3);
\\
\Sigma_U(u;\,\alpha) &=& 1 \, + \, \alpha \, \Sigma_U^{(1)}(u) \, + \alpha^2 \, \Sigma_U^{(2)}(u)
\, + \, O(\alpha^3);
\\
D_U(u;\,\alpha) &=& \alpha \, D_U^{(1)}(u) \, + \alpha^2 \, D_U^{(2)}(u)
\, + \, O(\alpha^3).
\end{eqnarray}
The knowledge of soft-gluon dynamics allows the resummation of the
dominant terms to all orders in $\alpha$ in $\Sigma_U$; this is
instead not possible for the coefficient function and the
remainder function, which are not long-distance dominated and for
them one has to use truncated expansions.

The above conditions do not completely specify the form of the
form factor, of the coefficient function and of the remainder
function, so we have to select a factorization scheme. Let us
choose a minimal scheme, in which the form factor contains {\it
only} logarithmic terms:
\begin{equation}
\Sigma_U(u;\,\alpha) \, = \, 1 \, + \, \sum_{n=1}^{\infty}\sum_{k=1}^{2n}
\Sigma_{Unk} \, \alpha^n \, L^k.
\end{equation}
where
\begin{equation}
L \, \equiv \, \log \frac{1}{u}.
\end{equation}
From the above definition, it follows that $\Sigma_U$ has the same normalization
as $\Sigma$:
\begin{equation}
\label{SigmaU1}
\Sigma_U(1;\,\alpha)\,=\,1,
\end{equation}
which holds to any order in $\alpha$.
We obtain in first order:
\begin{eqnarray}
\Sigma_U^{(1)}(u) &=&
- \, \frac{C_F}{\pi}
\left[
\frac{1}{2}\log^2 u \, + \, \frac{7}{4}\log u
\right];
\\
\label{CU1}
C_U^{(1)}  &=& - \, \frac{335}{144} \frac{C_F}{\pi}.
\end{eqnarray}
Note that
\begin{equation}
\label{onlyfirst}
\Sigma_U^{(1)}(u)=\Sigma^{(1)}(u).
\end{equation}
As we are going to explicitly show later in this section, this
property does not hold in higher orders of $\alpha$. For
$\alpha(m_b)=0.22$, we have a first order correction to the
coefficient function of $-\,21.7\%$.

The remainder function is obtained by imposing consistency
between the resummed expression and the fixed-order one (matching).
By expanding to first order in $\alpha$ the resummed expression
(\ref{resumU})
and imposing the equality with the full $O(\alpha)$ result,
one obtains \cite{ucg}:
\begin{equation}
\label{restoU}
D_U^{(1)}(u) \, = \, \frac{C_F}{\pi}
\Big[
\frac{ u(15624 - 2688 u - 1352 u^2 + 141 u^3) }{5040}
- \frac{21-84 u - 29 u^2 + 6 u^3}{210} u \log u
\Big].
\end{equation}
As required, the remainder function goes to zero in the elastic point $u=0$
(as $u\log u$).
Taking $u=1$ in eq.~(\ref{resumU}), we obtain the following relation
between the coefficient function and the remainder function in the upper
endpoint:
\begin{equation}
C_U(\alpha) \, = \, 1 \, - \, D_U(1;\alpha),
\end{equation}
which holds to any order in $\alpha$ and is verified in first order
(see eqs.~(\ref{CU1}) and (\ref{restoU})).

The minimal scheme has been defined above by looking at the explicit form
of the event fraction $R_U(u;\,\alpha)$ as a power series in $\alpha$:
one reorganizes the series picking up the logarithmic terms and putting them
into the effective form factor $\Sigma_U$. It is also possible to give a different
definition of the minimal scheme which does not make use
of the explicit expansion of the event fraction.
Since
\begin{equation}
\int_0^1 \, dw \, C_H(w;\alpha)\,\Sigma\left[u;\,\alpha(w\,m_b)\right]
\, = \, C_U(\alpha) \, \Sigma_U(u;\,\alpha)
\end{equation}
and
\begin{equation}
\Sigma_U(1;\,\alpha) \, = \, 1,
\end{equation}
we can define the coefficient function to all orders as:
\begin{equation}
\label{defCU}
C_U(\alpha) \, = \, \int_0^1 dw \, C_H(w;\,\alpha).
\end{equation}
The effective form factor is therefore written as:
\begin{equation}
\label{defSigU}
\Sigma_U\left(u;\,\alpha\right) \,=\,
\frac{\int_0^1 dw \, C_H(w;\,\alpha) \, \Sigma[u;\,\alpha(w\,m_b)]}
     { \int_0^1 dw \, C_H(w;\,\alpha) }.
\end{equation}
The definition (\ref{defSigU}) of the minimal scheme has the following phenomenological
advantage. One may wish to use for $\Sigma(u;\alpha)$, instead of the perturbative expression,
for example the result of a fit to some experimental data or a non-perturbative model\footnote{
For the coefficient function, one can still use
the perturbative result. That is because the coefficient function,
unlike the form factor, is short-distance dominated, and therefore its perturbative
evaluation is more reliable.}.
In these cases, $\Sigma(u;w)$ does not depend on the coupling and therefore cannot be
expanded in $\alpha$, but the effective form factor $\Sigma_U$ in the minimal
scheme can still be computed by means of eq.~(\ref{defSigU}).

The representation of the effective form factor $\Sigma_U$ given in
eq.~(\ref{defSigU}) allows us to make a few general comments:
\begin{itemize}
\item
$\Sigma_U(u;\,\alpha)$ factorizes all the
threshold logarithms in the spectrum but is, unlike $\Sigma(u)$,
process dependent. That is because it involves the convolution of the universal
form factor $\Sigma(u;\,\alpha)$ with the process-dependent coefficient function
over all the hadron energies.
$C_H(w;\,\alpha)$ has the role of a probability distribution:
it gives the probability for the hadronic subprocess with hard scale
$Q\,=\,w\,m_b$ to occur.
In the $u$ distribution hadronic subprocesses with all the possible
hard scales $Q$ from zero up to $m_b$ do contribute, while in the radiative decay
(\ref{raddec}) $Q$ is kinematically fixed to the upper value $m_b$.
The relation between these two spectra therefore is not a purely short-distance
one: to relate these two distributions one has to model in some way the variation
of the form factor $\Sigma[u;\alpha(Q)]$ with $Q$ ranging from $m_b$ down to zero.
In agreement with physical intuition, the problem in the computation of the $u$
distribution is the estimate of the contributions from small hard scales,
$Q \ll m_b$, where perturbation theory is expected to fail.
However, since $C_H \, \propto \, Q^2$, small hard scales give a small
contribution to the total.
We may say that the $u$ spectrum is ``protected'' from long-distance
effects related to large logarithms with a large coupling, i.e.
related to the region in eq.~(\ref{specificSL});
\item
since $\alpha(w\,m_b)\,=\,\alpha(m_b)\,+\,O(\alpha^2)$,
the effective form factor and the universal form factor coincide in
first order, as already found by explicit computation (see eq.~(\ref{onlyfirst}));
\item
were not for the dependence of the form factor $\Sigma\,=\,\Sigma[u;\,\alpha(w\,m_b)]$
on $w$ through the running coupling, $\alpha\,=\,\alpha(w\,m_b)$,
the effective form factor $\Sigma_U$ would coincide with the universal one $\Sigma$
to all orders.
\end{itemize}

Let us now discuss the higher orders in the perturbative expansion
of $\Sigma_U(u;\,\alpha)$.
One has to insert in eq.~({\ref{defSigU}}):
\begin{itemize}
\item the QCD form factor $\Sigma$, whose next-to-next-to-leading
order corrections (NNLO) are now well established due to the
recent revaluation of the resummation constant $D_2$ in
\cite{bsgamma3}; \item the coefficient function $C_H(w;\,\alpha)$,
which is known to $O(\alpha)$, i.e. to NLO (cfr. eq.~(\ref{CHw})),
\end{itemize}
and perform the integration over $w$. To obtain the truncated
expansion of $\Sigma_U$, one simply replaces the truncated
expansions of $\Sigma$ and of $C_H$, expands the product and
integrates term by term.

The exponential structure of the threshold logarithms in $\Sigma(u;\,\alpha)$
is partially spoiled in $\Sigma_U(u;\,\alpha)$
because of the integration over the hard scale $Q\, = \, w \, m_b$,
but it is not completely ruined.
In order to simplify the representation of the large logarithms, it is
therefore convenient to introduce the
exponent of the form factor also in the effective case,
i.e. to define $G_U$ as:
\begin{equation}
\Sigma_U \, = \, e^{ G_U }.
\end{equation}
Expanding the exponent in powers of $\alpha$ up to third order
included, we have an expansion of the same form of $G$, defined in
\cite{noi}, that is:
\begin{equation}
G_U\left(u;\,\alpha\right) \,=\, \sum_{n=1}^{\infty} \sum_{k=1}^{n+1}
G_{U n k}\, \alpha^n \, L^k
\,=\,  G_{U 12} \alpha L^2 + G_{U 11} \alpha L
+ G_{U 23} \alpha^2 L^3  + G_{U 22} \alpha^2 L^2
+ G_{U 21} \alpha^2 L + \cdots,
\end{equation}
where
\begin{equation}
L \,  \equiv \, \log\frac{1}{u} \, \ge \, 0
\end{equation}
and $\alpha \,\equiv\, \alpha(m_b)$.
In order to explicitly see the effects of the integration over the
hard scale, let us give the coefficients $G_{Uij}$ up to
the third order in terms of the corresponding
coefficients $G_{ij}$ of the radiative decay (\ref{raddec}) and of the
resummation constants $A_i,~B_i,~D_i$ and of the $\beta$-function
coefficients $\beta_i$ \cite{noi}:
\begin{eqnarray}
G_{U 12} &=& G_{12};
\\
G_{U 11} &=& G_{11};
\\
G_{U 23} &=& G_{23};
\\
G_{U 22} &=& G_{22} - \frac{5}{12} A_1\beta_0;
\\
G_{U 21} &=& G_{21} - \frac{5}{6} \beta_0 \Big( B_1 + D_1 \Big);
\\
G_{U 34} &=& G_{34};
\\
G_{U 33} &=& G_{33} - \frac{5}{6} A_1 \beta_0^2;
\\
G_{U 32} &=& G_{32}
- \frac{5}{6}\,A_2\,\beta_0
- \frac{1}{36}\,A_1\,
   \Big(
      23\,{\beta_0}^2 + 15\,\beta_1
     \Big)
-  \frac{5}{6}{\beta_0}^2\,
   \Big( B_1 + 2\,D_1 \Big)  -
  \frac{5}{6}  \,{A_1}^2\,
     \beta_0\,z(2)   +
\nonumber\\
&-&
        \left(
\frac{547}{216} - 2\,z(3)
       \right)
\frac{C_F}{\pi } \, A_1\,
     \beta_0\,,
\end{eqnarray}
where $z(a)=\sum_{n=1}^{\infty}1/n^a$ is the Riemann Zeta function
and the explicit expressions of the $G_{ij}$ have been given in
sec.~(2) of \cite{noi}. For a reference use in phenomenological
analysis, let us also give the explicit expressions for the
coefficients, which can be checked against the second and third
order computations of the $u$ spectrum as soon as the latter will
become available. We report only the coefficients differing from
the starting ones, $G_{U i j} \ne G_{i j}$:
\begin{eqnarray}
G_{U 22} &=&\frac{C_F}{{\pi }^2} \,
    \Bigg[ - \frac{n_f}
       {48} - \frac{C_F\,z(2)}{2} +
      C_A\,
       \left( - \frac{5}{96}  +
         \frac{z(2)}{4} \right)
      \Bigg]  ;
\\
G_{U 21} &=&
\frac{C_F }{\pi^2} \,
\Bigg[ n_f\,
       \left( - \frac{5}{6} +
         \frac{z(2)}{6} \right)
       + C_A\,
       \left( \frac{215}{48} -
         \frac{17\,z(2)}{12} -
         \frac{z(3)}{4}
         \right)  +
      C_F\,
       \left( \frac{3}{32} +
         z(2) +
         \frac{z(3)}{2}
         \right)  \Bigg]  ;
\\
G_{U 33} &=&
\frac{C_F }{\pi^3} \,
\Bigg[
-\frac{11}{432}\,n_f^2 + C_A\,n_f \left(\frac{95}{216}
      -\frac{z(2)}{12}\right)+
    C_A^2\left(-\frac{2471}{1728} + \frac{11\,z(2)}{24}\right) +\nonumber\\
 &+&   C_F\,n_f \left(\frac{1}{16}+\frac{z(2)}{4}\right)
  - \frac{11}{8}\,C_F\,C_A\,z(2) +
    \frac{1}{3}\,C_F^2\,z(3)
 \Bigg]  ;
\\
G_{U 32} &=&
\frac{C_F }{\pi^3} \,
\Bigg[
\,n_f^2 \left( \frac{49}{432}  - \frac{z(2)}{36} \right) +
    C_F\,C_A \left(-\frac{23177}{10368} + \frac{155\,z(2)}{96}
     - \frac{11\,z(3)}{12}+ \frac{5\,z(4)}{4} \right) + \nonumber\\
&+& C_F\,n_f \left(\frac{2971}{5184}-\frac{17\,z(2)}{48}
   - \frac{z(3)}{12}\right) +
    C_F^2\left( - \frac{7\,z(3)}{4}+ \frac{z(4)}{4} \right) +
  C_A\,n_f\left(-\frac{1709}{1728} + \frac{19 z(2)}{72}-\frac{z(3)}{24}\right) +
\nonumber \\
&+&    C_A^2\left(\frac{1541}{864} - \frac{4\,z(2)}{9}
     + \frac{77\,z(3)}{48} -\frac{11\,z(4)}{16} \right)
 \Bigg],
\end{eqnarray}
where $C_A=N_c=3$ and $n_f$ is the number of active flavors.
Let us make a few comments about the results obtained:
\begin{itemize}
\item it is remarkable that we could explicitly compute $G_{U 32}$
with the knowledge of the first two orders of the coefficient
function $C_H(w;\,\alpha)$ only (cfr. eqs.~(\ref{CHw1}) and
(\ref{CHw2})). We can compute also the coefficients $G_{U n,n-1}$
for $n>3$, i.e. the exponent $G_U$ to NNLO. As explained in detail
in \cite{cattren1, noi}, NNLO means indeed that for each power of
$\alpha$ we can compute the three principal logarithms:\footnote{
If we look instead at the form factor itself, $\Sigma_U$, NNLO
means that we can compute for any $n$ all the coefficients
$\Sigma_{Unk}$ with $n-1 \le k\le 2n$:
\begin{equation}
\Sigma_{U n,2n}\,\alpha^n \, L^{2n},~~~~~
\Sigma_{U n,2n-1}\,\alpha^n \, L^{2n-1},~~~~~
\cdots,~~~~~
\Sigma_{U n,n}\,\alpha^n \, L^n,~~~~~
\Sigma_{U n,n-1}\,\alpha^n \, L^{n-1}.~~~
\end{equation}
For instance for $n=3$ we can compute the coefficients of the logarithms
from power six down to power two included.}
\begin{equation}
G_{U n,n+1}\,\alpha^n \, L^{n+1},~~~~~
G_{U n,n}\,\alpha^n \, L^n,~~~~~
G_{U n,n-1}\,\alpha^n L^{n-1}.
\end{equation}
By general counting arguments, one would expect that the NNLO
corrections to $\Sigma_U$ also require the knowledge of the NNLO
contribution to the coefficient function, i.e. of the
$O(\alpha^2)$ term $C_H^{(2)}(w)$. That is actually not the case
because the NNLO contributions proportional to $C_H^{(2)}(w)$
cancel between the numerator and the denominator in the definition
of the effective form factor in eq.~(\ref{defSigU}). However, a
complete NNLO resummation of the $u$-spectrum also requires the
knowledge of the second-order correction to the coefficient
function $C_U^{(2)}$, and for that $C_H^{(2)}(w)$ is needed (see
eq.(\ref{defCU})); \item Since the coefficients of the threshold
logarithms in the $u$ distribution and in the photon spectrum in
the radiative decay (\ref{raddec}) differ from two loops on in
NLO, the cancellation of long-distance effects in the ratio
considered in \cite{ucg}
\begin{equation}
\label{ldonly}
R(u)\, = \, \frac{ d\Gamma_R/dt_s(t_s=u) }{ d\Gamma/du(u) }
\end{equation}
is not exact but occurs only in leading order. As previously
discussed, the $u$ distribution has additional long distance
effects with respect to the radiative decay related to small
hadron energies.
\end{itemize}

The differential spectrum in $u$ is obtained from the event fraction
$R_U(u)$ by differentiation:
\begin{equation}
\frac{1}{\Gamma}\frac{d\Gamma}{du} \, = \, C_U(\alpha)
\, \sigma_U\left(u;\,\alpha\right) \, + \, d_U(u;\,\alpha),
\end{equation}
where we have defined
\begin{equation}
\sigma_U \left(u;\alpha\right)\,\equiv
\,\frac{d}{du} \Sigma_U \left(u;\alpha\right);
~~~~~~~~~~~
d_U(u;\alpha) \, \equiv \, \frac{d}{du} \, D_U(u;\alpha).
\end{equation}
The coefficient function is clearly the same in the partially-integrated
spectrum and in the differential one, while the remainder function is obtained
by differentiation and reads:
\begin{equation}
d_U^{(1)}(u) \, = \, \frac{C_F}{\pi}
\left[
\frac{36-8u-8u^2+u^3}{12} -
\frac{ 7 - 56u - 29u^2 + 8u^3}{70} \log u
\right].
\end{equation}

\section{Hadron mass spectrum}
\label{sect2}

In this section we resum to NLO the distribution
in the invariant hadron mass squared, i.e. the distribution in the
variable
\begin{equation}
\label{deft}
~~~~~~~~~~~~~~~
t \,=\, \frac{m_X^2}{m_b^2} \, = \,
\frac{u \, w^2}{(1+u)^2}\, \simeq \, u\, w^2 ~~~~~~~~~~~~~~~~~~~~~~~ (0\le t\le 1),
\end{equation}
where in the last member we have kept the leading term for
$u\rightarrow 0$ only.
This distribution is obtained by integrating the distribution
in the hadron variables $u$ and $w$ with the previous kinematic
constraint:
\begin{equation}
\label{firstt}
\frac{1}{\Gamma}\frac{d\Gamma}{dt} \, = \,
\int_D du dw  \, \frac{1}{\Gamma}\frac{d^2\Gamma}{du dw}
\delta\left[t-\frac{u \, w^2}{(1+u)^2}\right];
\end{equation}
the integration covers the whole phase space $D$ of the hadron
variables:
\begin{equation}
  \int_D du dw
\,=\, \int_0^2 dw \int_{max[0,w-1]}^1 du
\,=\, \int_0^1 du \int_0^{1+u} dw.
\end{equation}
It is convenient to evaluate the event fraction, defined
like in the previous distribution as:
\begin{equation}
R_T(t)\,=\,\int_0^t dt' \, \frac{1}{\Gamma}\frac{d\Gamma}{dt'}\,,
\end{equation}
with the end-point values $R_T(0)=0$ and $R_T(1)=1$.
Integrating both sides of eq.~(\ref{firstt}), one obtains:
\begin{equation}
R_T(t) \,=\,  \int_D du dw  \, \frac{1}{\Gamma}\frac{d^2\Gamma}{du dw}
\theta \left[t-\frac{u \, w^2}{(1+u)^2}\right].
\end{equation}
By inserting the resummed form for the double hadron distribution
and neglecting at first the remainder function, we obtain:
\begin{equation}
\label{RT}
R_T(t;\,\alpha) \,=\,
\int_D du dw \, C_H(w;\,\alpha) \, \sigma\left[u;\,\alpha(w\,m_b)\right]\,
\theta \left[t-\frac{u \, w^2}{(1+u)^2}\right] \, + \, O(t,\alpha),
\end{equation}
where by $O(t,\alpha)$ we denote terms which are zero at the threshold
and vanish for $\alpha=0$.
In order to isolate the large logarithms, let us simplify the domain $D$
into the unit square
\begin{equation}
0 \, \le \, u, \, w \, \le \,1
\end{equation}
and  simplify the kinematic constraint as well as:
\begin{equation}
\theta \left[t - u \, w^2/(1+u)^2\right]
\,\, \rightarrow \,\,
\theta[t - u \, w^2].
\end{equation}
Let us observe that the variable $t$ keeps unitary range even after
such approximations. We then obtain:
\begin{eqnarray}
R_T(t;\alpha) &=&  \int_0^1 \int_0^1
du dw \, C_H(w;\alpha) \, \sigma\left[u;\alpha(w\,m_b)\right]\,
\theta \left[t- u w^2\right] \, + \, O(t,\alpha)
\nonumber\\
&=&
\label{goodsplit}
\int_0^{\sqrt{t}} dw \, C_H(w;\alpha) \, + \,
\int_{\sqrt{t}}^1 dw \, C_H(w;\alpha) \,
\Sigma\left[t/w^2;\,\alpha(w\,m_b)\right] \, + \, O(t,\alpha).
\end{eqnarray}
Let us remark that the present  case is more involved with respect to the one
treated in the previous section, because the hadron mass squared $t$ is a combination
of both variables used for threshold resummation, namely $u$ and $w$.
Neglecting infinitesimal terms for $t\rightarrow 0$,
the expression above can be further simplified by neglecting the first integral
and integrating the second integrand over $w$ down to zero,
\begin{equation}
R_T(t;\,\alpha) \, = \,
\int_0^1
dw \, C_H(w;\alpha) \, \Sigma\left[t/w^2;\,\alpha(w\,m_b)\right] \, + \, O(\alpha,t).
\end{equation}
By inserting the first-order expressions for the coefficient function and
the form factor given in the previous section, we obtain:
\begin{equation}
\int_0^1
dw \, C_H(w;\alpha) \, \Sigma\left[t/w^2;\,\alpha(w\,m_b)\right]  \, = \,
1\, - \, \frac{\alpha \, C_F}{\pi}
\left[
\frac{1}{2}\log^2 t
\, + \, \frac{31}{12}\log t
\, + \, \frac{637}{144}
\right]
\, + \, O(\alpha^2).
\end{equation}
As anticipated in the introduction, we have a different coefficient for the single
logarithm with respect to the hadron mass distribution in the radiative decay (\ref{raddec})
or the $u$ distribution of the previous section.

As with the $u$ distribution, we introduce a resummed form of the event fraction as:
\begin{equation}
\label{R_T}
R_T(t;\,\alpha) \,=\, C_T(\alpha) \, \Sigma_T(t;\,\alpha) \,
+ \, D_T(t;\,\alpha).
\end{equation}
All the functions above have an expansion in powers of $\alpha$:
\begin{eqnarray}
C_T(\alpha) &=& 1 \, + \, \alpha \, C_T^{(1)} \, + \, \alpha^2 \, C_T^{(2)}
\,+ \, O(\alpha^3);
\\
\Sigma_T(u;\,\alpha) &=& 1 \, + \, \alpha \, \Sigma_T^{(1)}(t) \, + \alpha^2 \, \Sigma_T^{(2)}(t)
\, + \, O(\alpha^3);
\\
D_T(t;\,\alpha) &=& \alpha \, D_T^{(1)}(t) \, + \alpha^2 \, D_T^{(2)}(t)
\, + \, O(\alpha^3).
\end{eqnarray}
Let us consider a minimal factorization scheme, where
only logarithms in $t$ are factorized in
the effective form factor $\Sigma_T$.
Since we will consider a different scheme later on in this section, let
us denote the quantities in the minimal scheme with a bar.
The first-order corrections to the form factor and the coefficient function
in the minimal scheme read:
\begin{eqnarray}
\bar{\Sigma}_T^{(1)}(t) &=&
- \, \frac{C_F}{\pi}
\left[
\, \frac{1}{2}\log^2 t
\, + \, \frac{31}{12}\log t
\right];
\nonumber\\
\bar{C}_T^{(1)} &=& - \,\frac{C_F}{\pi} \, \frac{637}{144} \, = - \, 1.87744.
\end{eqnarray}
Note that the correction to the coefficient function is very large:
for $\alpha(m_b)=0.22$ it amounts to $-\,41.3\%$.

We now expand the resummed result in powers of $\alpha$ and
compare with the fixed order result,
which is known to full order $\alpha$ \cite{mxspectrum,ndf}:
\begin{equation}
R_T(t;\alpha)\,=\, 1 \,+\, \alpha \, R_T^{(1)}(t) \, + \, \alpha^2 \, R_T^{(2)}(t)
\, + \, O(\alpha^3),
\end{equation}
with
\begin{equation}
R_T^{(1)}(t) \,=\,
- \, \frac{C_F}{\pi}
\left[
\, \frac{1}{2}\log^2 t
\, + \, \frac{31}{12}\log t
\, + \, \frac{637}{144}
\, - \, \frac{97}{18} t \, + \, \frac{25}{18}t^3 \, - \, \frac{61}{144} t^4
\, + \, \left(  \frac{5}{3} t \, - \, \frac{3}{2} t^2 \, + \, \frac{1}{6} t^4 \right)\log t
\right].
\end{equation}
We obtain for the remainder function in first order:
\begin{equation}
\bar{D}_T^{(1)}(t) \, = \,
\frac{97}{18} t - \frac{25}{18}t^3 + \frac{61}{144} t^4
- \left(  \frac{5}{3} t - \frac{3}{2} t^2 + \frac{1}{6} t^4 \right)\log t.
\end{equation}
Since
\begin{equation}
\label{normSigbar}
\bar{\Sigma}_T(1;\,\alpha)\, = \, 1,
\end{equation}
taking $t=1$ in eq.~(\ref{R_T}), we obtain a relation between the coefficient function
and the remainder function in the endpoint:
\begin{equation}
\bar{C}_T(\alpha) \,  = \, 1 \, - \, \bar{D}_T(1;\,\alpha).
\end{equation}
It is a trivial matter to verify that the above relation holds true for our first-order
expressions.

A compact definition of the minimal scheme can be given as follows.
Since
\begin{equation}
\int_0^1
dw \, C_H(w;\alpha) \, \Sigma\left[t/w^2;\,\alpha(w\,m_b)\right] \, = \,
\bar{C}_T(\alpha) \, \bar{\Sigma}_T\left(t;\,\alpha\right),
\end{equation}
the coefficient function in the minimal scheme can be defined taking
$t=1$ in the above equation and using (\ref{normSigbar}):
\begin{equation}
\bar{C}_T(\alpha) \, \equiv \,
\int_0^1 dw \, C_H(w;\alpha) \, \Sigma\left[1/w^2;\,\alpha(w\,m_b)\right].
\end{equation}
The effective form factor then reads:
\begin{equation}
\bar{\Sigma}_T\left(t;\,\alpha\right) \, \equiv \,
\frac{ \int_0^1 dw \, C_H(w;\alpha) \, \Sigma\left[t/w^2;\,\alpha(w\,m_b)\right] }
{\int_0^1 dw \, C_H(w;\alpha) \, \Sigma\left[1/w^2;\,\alpha(w\,m_b)\right]}.
\end{equation}
Let us comment the above result. The effective form factor
involves a convolution over the hadron energy $w$ of the
coefficient function and the universal form factor, which cannot
be reduced to an ordinary product by the standard moment
transform. That is because the variable $w$ enters not only in the
first argument of
$\Sigma=\Sigma\left[t/w^2;\,\alpha(w\,m_b)\right]$ but also in the
scale of the coupling $\alpha=\alpha(w\,m_b)$. Analogously to the
$u$ distribution, there are long-distance effects in the effective
form factor related to small hadron energies $w \ll 1$, which are
suppressed by the coefficient function. In the present case,
however, there is an additional mechanism suppressing the small
energy contributions: since $\Sigma$ is evaluated in $t/w^2$,
small $w$'s correspond to a large argument $u=t/w^2$ of
$\Sigma(u)$, where there are no large logarithms, $\log 1/u \sim
O(1)$, and one is inclusive at the parton level. We may say that
this spectrum is ``double protected'' from the non-perturbative
long-distance effects related to small hadron energies.

The systematic expansion of the form factor is easily obtained
by writing as usual:
\begin{equation}
\bar{\Sigma}_T \, = \, e^{\bar{G}_T},
\end{equation}
one obtains:
\begin{equation}
\bar{G}_T(t;\alpha) \,=\, \sum_{n=1}^{\infty}\sum_{k=1}^{n+1}
\bar{G}_{T n k}\, \alpha^n \, L_t^k,
\end{equation}
where
\begin{equation}
L_t \, \equiv \, \log\frac{1}{t}
\end{equation}
and
\begin{eqnarray}
\bar{G}_{T 12} &=& G_{12};
\\
\bar{G}_{T 11} &=& G_{11} + \frac{5}{6} A_1;
\\
\bar{G}_{T 23} &=& G_{23};
\\
\bar{G}_{T 22} &=& G_{22} + \frac{7}{24} A_1^2 + \frac{5}{6} A_1\beta_0;
\\
\bar{G}_{T 21} &=& G_{21} + \frac{5}{6} A_2
+  A_1^2 \left[ \frac{5}{6} z(2) - \frac{47}{54} \right]
- \frac{23}{36} A_1\beta_0 + \frac{7}{12} A_1 (B_1 + D_1) +
\nonumber\\
&+& \frac{5}{6} \beta_0 D_1
+ A_1 \frac{C_F}{\pi} \left[ \frac{547}{108} - 4 z(3) \right] ;
\\
\bar{G}_{T 34} &=& G_{34};
\\
\bar{G}_{T 33} &=& G_{33} + \frac{83}{648} A_1^3 + \frac{7}{12} A_1^2\beta_0
                     + \frac{10}{9}A_1\beta_0^2;
\\
\bar{G}_{T 32} &=& G_{32} +
\frac{5}{3} \,A_2\,
     \beta_0   +
  \frac{5}{3} \,{\beta_0}^2\,
     D_1  +
\frac{83}{216} \, {A_1}^2
   \left( B_1 \, + \,D_1 \right)
+ {A_1}^2\, \beta_0\,
      \left(
        \frac{35\,z(2)}{12} - \frac{47}
           {27}   \right)
\nonumber\\
&+&  {A_1}^2\, \frac{C_F }{\pi }\,
        \left( \frac{22747}{1728} -
          12\,z(4) \right) +
\frac{7}{12} \, A_1 \, A_2
- \frac{1}{18}
A_1\, \Big( 23\,{\beta_0}^2 - 15 \,\beta_1 \Big) +
\frac{7}{12} A_1\,\beta_0\,
      \Big( B_1 + 2 D_1 \Big) +
\nonumber\\
&+&
        A_1\,\beta_0\,\frac{C_F }{\pi }\,
           \left( \frac{547}{108} -
             4\,z(3)
             \right)
+ {A_1}^3\,
   \left( - \frac{1117}
        {1296}  +
     \frac{7\,z(2)}{12} -
     \frac{5\,z(3)}
      {6} \right).
\end{eqnarray}
For comparison with future higher-order computations, let us
give the explicit values of the coefficients
$\bar{G}_{T ij} \ne G_{ij}$:
\begin{eqnarray}
\bar{G}_{T 11} &=& \frac{31\,C_F}{12\,\pi };
\\
\bar{G}_{T 22} &=& \frac{C_F }{\pi^2}
\, \Bigg[ - \frac{11\,n_f}
       {48} + C_F\,
       \left( \frac{7}{24} -
         \frac{z(2)}{2} \right)
       + C_A\,
       \left( \frac{35}{32} +
         \frac{z(2)}{4} \right)
      \Bigg];
\\
\bar{G}_{T 21} &=&
\frac{C_F}{\pi^2}
\Bigg[ n_f
       \left( \frac{z(2)}{6}- \frac{83}
            {144} \right)
       + C_F\,
       \left( \frac{941}{288} +
         \frac{11}{6} z(2) -
         \frac{7}{2} z(3) \right)  +
      C_A\,
       \left( \frac{107}{32} -
         \frac{11}{6} z(2) -
         \frac{z(3)}{4}
         \right)  \Bigg];~~~~
\\
\bar{G}_{T 33} &=&
\frac{C_F}{\pi^3}
\, \Bigg[ \frac{37}{1296}\,n_f^2 +
    C_F\,C_A\left(\frac{77}{144} - \frac{11\,z(2)}{8}\right) +
        C_F\,n_f\left(-\frac{5}{144} + \frac{z(2)}{4}\right) +\nonumber\\
&&~~~~~  + \, C_A\,n_f \left(-\frac{25}{162} - \frac{z(2)}{12}\right) +
   C_A^2\left( \frac{1057}{5184} + \frac{11\,z(2)}{24}\right) +
    C_F^2\left(\frac{83}{648} + \frac{z(3)}{3} \right)
  \Bigg];
\\
\bar{G}_{T 32} &=&
\frac{C_F}{\pi^3}
\, \Bigg[ \left( \frac{263}{2592} - \frac{z(2)}{36} \right) \,n_f^2 +
    C_F\,C_A\left(\frac{28493}{10368} + \frac{457\,z(2)}{96}
    -\frac{77\,z(3)}{12}+ \frac{5\,z(4)}{4}\right) +
\nonumber\\
&&~~ + \, C_F\,n_f\left(-\frac{2353}{5184} - \frac{47\,z(2)}{48}
     + \frac{11\,z(3)}{12}    \right) +
        C_F^2 \left(\frac{60287}{5184} + \frac{7\,z(2)}{12}
    - \frac{31\,z(3)}{12}  - \frac{47\,z(4)}{4}    \right) +
\nonumber\\
&&~~ + \, C_A\,n_f \left(-\frac{6515}{5184} + \frac{17 z(2)}{36}
    - \frac{z(3)}{24}  \right) +
        C_A^2 \left(\frac{31841}{10368} - \frac{229\,z(2)}{144}
    + \frac{77\,z(3)}{48}  - \frac{11\,z(4)}{16}    \right)
  \Bigg].~~~~~
\end{eqnarray}
Let us make a few remarks:
\begin{itemize}
\item
The coefficient of the single logarithm at $O(\alpha)$ is different
from the previous case as well as from the radiative decay:
\begin{equation}
\bar{G}_{T 11} \, \ne \, G_{U 11} \, = \, G_{11},
\end{equation}
because $\bar{G}_{T 11}$ takes a kinematic contribution from $A_1$, i.e.
from the double logarithm at one loop in $\Sigma$.

The logarithmic structure of $\bar{G}_T$ radically differs from that
of $G$ because the integration variable $w$ enters not only the argument
of the running coupling $\alpha=\alpha(w\,m_b)$
but also the argument of the logarithm $L=\log(t/w^2)$;
\item
If the hard scale was set by the heavy flavor mass, $Q\,=\,m_b$
(a kind of frozen coupling case with respect to the real case)
we would have the following values for the coefficients:
$\bar{G}_{T 22}^{fr} = G_{22} + 7/24 \; A_1^2 + 5/4\; A_1\beta_0$ and
$\bar{G}_{T 33}^{fr} = G_{33} + 83/648 \;A_1^3 + 7/8\; A_1^2\beta_0 +
 35/18\; A_1\beta_0^2$.
\end{itemize}
As discussed in the previous section, in phenomenological studies
one may want to replace the perturbative expression of
$\Sigma(u;\,\alpha)$ with a fit to some experimental data or with
a non-perturbative model. The minimal scheme cannot be used
directly in these circumstances because the effective form factor
$\bar{\Sigma}(t;\,\alpha)$ involves the integration of
$\Sigma(u;\,\alpha)$ in the unphysical region $u>1$. There are
various ways to deal with this problem. One way could be for
example replacing $\Sigma(u;\,\alpha)$ with the non-perturbative
quantity $\Sigma_{np}(u;w)$ for $u\le 1$, while still keeping the
perturbative $\Sigma(u;\,\alpha)$ in the unphysical region $u>1$.
The perturbative form factor is indeed an analytic function of
$u$, which can be continued to any value of $u$. To avoid the
inclusion of the perturbative $\Sigma(u;\,\alpha)$ for $u>1$, let
us consider instead a non-minimal scheme with an effective form
factor defined as:
\begin{equation}
\label{defT}
\Sigma_T\left(t;\,\alpha\right) \,=\,
\frac{ \int_0^1 dw \, C_H(w;\,\alpha) \, \tilde{\Sigma}\left[t/w^2;\,\alpha(w\,m_b)\right] }
{ \int_0^1 dw \, C_H(w,\alpha) },
\end{equation}
where the standard form factor $\Sigma(u;\alpha)$ has been
extended to  arguments larger than one, $u>1$, since:
\begin{equation}
\tilde{\Sigma}(u ;\alpha) \, \equiv \, \Bigg\{
\begin{array}{ll}
\Sigma(u ;\alpha)
&  {\rm for~} u \, \le \, 1;
\\
1
& {\rm for~} u \, > \, 1.
\end{array}
\end{equation}
Because of the definition, it holds:
\begin{equation}
\Sigma_T(1;\,\alpha) \,=\, 1.
\end{equation}
Explicitly one has (cfr. the r.h.s. of eq.~(\ref{goodsplit})):
\begin{equation}
\int_0^1 dw \, C_H(w;\,\alpha) \,
\tilde{\Sigma}\left[t/w^2;\,\alpha(w\,m_b)\right]
\,=\,
\int_0^{\sqrt{t}} dw \, C_H(w;\alpha) \, + \,
\int_{\sqrt{t}}^1 dw \, C_H(w;\alpha) \,
\Sigma\left[t/w^2;\,\alpha(w\,m_b)\right].
\end{equation}
The coefficient function is given in the new scheme by:
\begin{equation}
C_T(\alpha) \, = \,  \int_0^1 dw \, C_H(w,\alpha).
\end{equation}
By inserting in eq.~(\ref{defT}) the perturbative expansions
for $C_H$ and for $\Sigma$, we obtain:
\begin{equation}
\Sigma_T^{(1)}(t) \,=\,
\frac{C_F}{\pi}\Big[
- \frac{1}{2}\log^2 t
- \frac{31}{12}\log t
- \frac{151 - 232\,  t^{3/2} + 81 \, t^2 }{72}
\Big]
\end{equation}
and
\begin{equation}
\label{CT1}
C_T^{(1)} \,=\, - \frac{C_F}{\pi} \, \frac{335}{144} \, = \, - \, 0.98735.
\end{equation}
The first of the above equations shows that $\Sigma_T$, unlike $\bar{\Sigma}_T$,
is not defined in a minimal factorization scheme, i.e. it does not contain only
logarithmic terms $\alpha^n\,L_t^k$, but also contributions of a different form.
Note that the coefficient function in (\ref{CT1}) has a much smaller value than
in the minimal scheme, giving a hint of better convergence of the
perturbative series in the modified scheme.

Matching with the first-order spectrum, one obtains for the remainder function
in the modified scheme:
\begin{equation}
D_T^{(1)}(t) \,=\, \frac{C_F}{\pi}
\left[
  \frac{97}{18} t - \frac{29}{9} t^{3/2} + \frac{9}{8}t^2
- \frac{25}{18}t^3 + \frac{61}{144} t^4
-\left( \frac{5}{3} t - \frac{3}{2} t^2 + \frac{1}{6} t^4 \right)\log t
\right].
\end{equation}
The coefficients of the logarithms in the exponent of the form factor
$G_T$ are the same in the minimal scheme and in the modified one:
\begin{equation}
G_{Tij} \, = \, \bar{G}_{T ij}~~~~~~~~{\rm for}~j\ge 1.
\end{equation}
The non logarithmic coefficients in the modified scheme
are given by ($\bar{G}_{T 10}=0$ and $\bar{G}_{T 20}=0$
by definition of minimal scheme):
\begin{eqnarray}
G_{T 10} &=&
-\,\frac{23}{36} \,{A_1} \, + \,
  \frac{5}{6}\,({B_1} + D_1);
\\
G_{T 20} &=&
-\,\frac{23}{36} \,A_2 \, + \,
  \frac{7}{24}\,(B_1 + D_1)^2 +
  \frac{5}{6}\,({B_2}+D_2) +
   \frac{23}{36} \, \beta_0 \, B_1   +
\nonumber\\
&+& {A_1}\,
   \left( {B_1} + D_1 \right)
      \left( -\,\frac{47}{54}   +
        \frac{5\,z(2)}{6}
         \right)  +
  A_1^2\,
   \left( \frac{2057}{2592} -
     \frac{23\,z(2)}{36} +
     \frac{5\,z(3)}{6} \right)  +
\nonumber\\
&+&  \frac{C_F}{\pi}\,
     \left( {B_1}\,+D_1 \right)
        \left( \frac{547}{108} - 4\,z(3)
          \right)  +
      \frac{C_F}{\pi} \, A_1 \,
        \left( -\, \frac{90121}{5184}
              + \frac{10\,z(3)}{3} +
          12\,z(4) \right).
\end{eqnarray}
Explicitly:
\begin{eqnarray}
G_{T 10} &=& - \, \frac{C_F }{\pi} \, \frac{151}{72};
\\
G_{T 20} &=&
\frac{C_F}{\pi^2}
\Bigg\{
C_A\, \left( \, - \frac{344}
        {81}  + \frac{3}{2}\,z(2) +
     \frac{5}{24}\,z(3) \right)  +
  n_f\,\left( \frac{971}{1296} -
     \frac{5}{36}\,z(2)
     \right)  \, +
\nonumber\\
&&~~~~ + \, C_F\,
   \left( - \frac{79889}{3456}  -
     \frac{53}{36}\,z(2) +
     \frac{119}{12}\,z(3) + 12\,z(4) \right)
\Bigg\}.
\end{eqnarray}
The relations between the coefficient function in the minimal scheme
and in the modified one are obtained imposing that
\begin{equation}
C_T \, e^{G_T} \, = \, \bar{C}_T \, e^{\bar{G}_T} \, + \, O(t;\alpha)
\end{equation}
and read:
\begin{eqnarray}
C_T^{(1)} &=& \bar{C}_T^{(1)} \, - \, G_{T10};
\nonumber\\
 C_T^{(2)} &=& \bar{C}_T^{(2)} \, - \,
\bar{C}_T^{(1)} \, G_{T10}
\, + \, \frac{1}{2}G_{T10}^2 \, - \, G_{T20}.
\end{eqnarray}
The first equation can be directly verified by inserting our
first-order expressions.
The second order correction to the coefficient function
is unknown at present in either scheme and its determination
requires a full two-loop calculation; the second of the above
equations simply allows us to transform the coefficient function from
one scheme to another one.

The Babar collaboration has recently presented the differential spectrum in
$t$ \cite{babarmx}, which is obtained from the previous one by differentiation:
\begin{equation}
\frac{1}{\Gamma}\frac{d\Gamma}{dt} \,=\, \frac{d}{dt} R_T(t).
\end{equation}
Due to their relevance, let us present explicit formulas. The
resummed spectrum reads:
\begin{equation}
\frac{1}{\Gamma}\frac{d\Gamma}{dt} \, = \, C_T(\alpha) \,
\sigma_T(t;\,\alpha) \, + \, d_T(t;\,\alpha)
\end{equation}
where
\begin{itemize}
\item
the coefficient function is the same as in the event fraction,
since it is independent on $t$;
\item
the effective form factor is
\begin{equation}
\sigma_T(t;\,\alpha) \,=\, \frac{d}{dt} \Sigma_T(t;\,\alpha).
\end{equation}
More explicitly:
\begin{eqnarray}
\label{sigTdef}
\sigma_T(t;\,\alpha)
&=& \frac{\int_{\sqrt{t}}^1 d w/w^2 \, C_H(w;\,\alpha) \, \sigma[t/w^2;\,\alpha(w\,m_b)]}
{\int_0^1 d w \, C_H(w;\,\alpha)}
\nonumber\\
&=& \frac{\int_t^1 du/(2\sqrt{t \, u})\,
C_H\left(\sqrt{t/u};\,\alpha(w \, m_b)\right) \,
\sigma\left[u;\,\alpha\left(m_b\sqrt{t/u} \right)\right]}
{\int_0^1 d w \, C_H(w;\,\alpha)},
\end{eqnarray}
where a double representation as an integral over $w$ or over $u$
respectively has been given;
\item
the remainder function is:
\begin{equation}
d_T(t;\,\alpha) \,=\, \frac{d}{dt} D_T(t;\,\alpha)
\,=\, \alpha \, d_T^{(1)}(t) \, + \, \alpha^2 \, d_T^{(2)}(t) \, +\, O(\alpha^3).
\end{equation}
The explicit value of the first-order correction is:
\begin{equation}
d_T^{(1)}(t) \, = \, \frac{C_F}{\pi} \left[
\frac{67}{18} - \frac{29}{6}\sqrt{t} + \frac{15}{4} t -  \frac{25}{6} t^2
+ \frac{55}{36} t^3 -\frac{5}{3} \log t + 3t\log t - \frac{2}{3} t^3 \log t
\right].
\end{equation}
\end{itemize}
Let us make a comment about the non-perturbative effects entering
the differential mass distribution. The expression for the
effective form factor $\sigma_T(t;\,\alpha)$ at the second member
of eq.~(\ref{sigTdef}) involves an integration over $w$ from
$\sqrt{t}$ up to one. Since the running coupling is evaluated in
$Q=w\,m_b$, the smallest hard scale contributing to the
distribution is
\begin{equation}
Q_{min} \, = \, \sqrt{t} \,\, m_b \, = \, m_X.
\end{equation}
In order to avoid the infrared pole in the coupling --- the well-known Landau pole
--- implying a breakdown of the perturbative scheme, one has to impose the condition
\begin{equation}
 m_X \, \gg \, \Lambda_{QCD},
\end{equation}
which is also very reasonable from the physical viewpoint.
Resummed perturbation theory therefore signals that the hadron
mass distribution cannot be computed for hadron masses of the
order of the hadron scale because of the appearance  of the Landau
pole\footnote{The universal QCD form factor
$\sigma\left[u;\,\alpha(Q)\right]$ has an infrared singularity at
$ u = \exp \left[-1/(2 \, \beta_0 \, \alpha(Q)) \right]$, related
to the Landau pole for $m_X\approx\sqrt{\Lambda_{QCD}Q}\gg
\Lambda_{QCD}$. We assume that the latter has been regulated in
some way; for a recent discussion see for example
\cite{nostrareg}.}.

\section{Conclusions}
\label{concl}

In this work we have presented next-to-leading resummed expressions
for the distribution in the final hadron mass/energy ratio
and for the distribution in the invariant hadron mass
in the semileptonic decays
\begin{equation}
\label{prima}
B \, \rightarrow \, X_u \, + \, l \, + \, \nu.
\end{equation}
By expanding our formulas, we have obtained the coefficients of
all the infrared logarithms to $O(\alpha^2)$ and of the leading
ones to $O(\alpha^3)$.
These two spectra have different logarithmic structures
from each other, which are both different also from that one in the
radiative decay (\ref{raddec}).
That occurs because these spectra involve integration over the
total hadron energy $E_X$, which sets the hard scale $Q$ of the
hadronic subprocess in (\ref{prima}):
\begin{equation}
Q \,  = \, 2 E_X.
\end{equation}
Long distance effects manifest in perturbation theory in the form of large
infrared logarithms, which are usually factorized and resummed in
QCD form factors.
Universality of long-distance effects therefore shows up
in perturbation theory as the occurrence of equal form factors
in different distributions.
That implies that the spectra we have considered in this work
have different long-distance effects from each other as well as from the
radiative decay (\ref{raddec}).
There is no simple connection between these semileptonic spectra
and the hadron mass distribution in radiative decays.
For both spectra, we have introduced effective, i.e. process
dependent, form factors, which factorize the large
logarithms to all orders in perturbation theory.
These effective form factors can also be computed in a phenomenological way
by inserting, in place of the perturbative QCD form factor
$\Sigma\left[u;\,\alpha(w\,m_b)\right]$,
the form factor $\Sigma_{np}(u;\,w)$ computed with a non-perturbative model
or a fit to experimental data.
In the case of the distribution in the hadron mass squared $t$, we have also
considered a non-minimal factorization-resummation scheme, which seems
to have better convergence properties of the perturbative series with respect
to the minimal one.

There are other important semileptonic spectra which have similar properties
to those of the distributions considered here, i.e. long distance effects
which cannot be factorized into a process independent form factor, to be extracted
for example from the radiative decay \cite{inpreparation}.

Finally, we have also shown that the cancellation of long-distance effects
in the ratio constructed in \cite{ucg} occurs only in leading order
while it is violated at the level of $\alpha^2\log^2\left(\frac{1}{u}\right)$ terms.

redpajama_set_name: RedPajamaArXiv


That is Seyi Oluyole's condition for any street kid that wants to continue dancing with her. And that is her strategy to make them stay in school and excel.
Seyi Oluyole is a writer, dance choreographer, fitness trainer and the founder of Dream Nurture Foundation, a global Christian relief, developmental organization committed to bringing life and hope to children in communities covered by poverty and despondency.
The organization offers children educational opportunities while bringing out the best in them and engaging them in diverse range of human activities like sports, drama, and dance among others.
At 10, Seyi's parents sold their house at Mile 2, which made them move around a lot. Finally, they rented a house at Oyingbo, Ebute – Metta, and there, she saw girls her age who dropped out of school before JSS3 due to pregnancy.
At 14, she started a dance group with some of the children. At that time, while she was in SS1, majority of them were still in primary school and could barely read.
Opportunity presented itself when Seyi and her dance team performed at Nollywood actress, Omotola Jalade-Ekeinde's fundraising concert of her NGO. This interested many of the kids that Seyi had to use it as a condition to joining or continuing with the group.
The kids began to take school seriously and it was a way for Seyi to start working on them.
The Dream Catchers is a part of the Dream Nurture Foundation that helps poor and street kids. The NGO provides the kids with shelter, food, clothing and education. Presently, the NGO has taken in 10 children and are sponsoring them fully.
The Dream Catchers are also empowered through dance, which gives their dreams validity and helps them know that irrespective of what they have been told by others, they stand a chance at succeeding.
Some people think I am crazy with what I do; seeing children on the street and taking care of them. I started teaching the children dance gradually and when I see that they are in love with it, I tell them that if they don't do well in school they will not dance again, but because they want to dance, they pay attention in school.
I just don't take in any child. The mother or father would have to sign an agreement on the number of days they want their child to stay with me.
When I first met the children, they couldn't even speak English, but now, when they start talking, you won't remember their story.
My mum will tell me that, being a young girl, how would I get a husband. Because living alone as a young woman is not supporte by culture, let alone living with kids that are not yours. The man would think you want spend his money with the children.
These children didn't choose to be born poor. Lets help these children to grow to be better persons and have integrity, so that the first thing that comes to their mind is how to help others and not what to steal from others.


text:
Dwight D. Eisenhower was 34th president of the United States of America (1953-1961)
Birth: October 14, 1890 at Denison, Texas as David Dwight Eisenhower
Death: March 28, 1969 at Washington, D.C.
Bringing to the Presidency his prestige as commanding general of the victorious forces in Europe during World War II, Dwight D. Eisenhower obtained a truce in Korea and worked incessantly during his two terms to ease the tensions of the Cold War. He pursued the moderate policies of "Modern Republicanism," pointing out as he left office, "America is today the strongest, most influential, and most productive nation in the world."
Born in Texas in 1890, brought up in Abilene, Kansas, Eisenhower was the third of seven sons. He excelled in sports in high school, and received an appointment to West Point. Stationed in Texas as a second lieutenant, he met Mamie Geneva Doud, whom he married in 1916.
In his early Army career, he excelled in staff assignments, serving under Generals John J. Pershing, Douglas MacArthur, and Walter Krueger. After Pearl Harbor, General George C. Marshall called him to Washington for a war plans assignment. He commanded the Allied Forces landing in North Africa in November 1942; on D-Day, 1944, he was Supreme Commander of the troops invading France.
After the war, he became President of Columbia University, then took leave to assume supreme command over the new NATO forces being assembled in 1951. Republican emissaries to his headquarters near Paris persuaded him to run for President in 1952.
"I like Ike" was an irresistible slogan; Eisenhower won a sweeping victory.
Negotiating from military strength, he tried to reduce the strains of the Cold War. In 1953, the signing of a truce brought an armed peace along the border of South Korea. The death of Stalin the same year caused shifts in relations with Russia.
New Russian leaders consented to a peace treaty neutralizing Austria. Meanwhile, both Russia and the United States had developed hydrogen bombs. With the threat of such destructive force hanging over the world, Eisenhower, with the leaders of the British, French, and Russian governments, met at Geneva in July 1955.
The President proposed that the United States and Russia exchange blueprints of each other's military establishments and "provide within our countries facilities for aerial photography to the other country." The Russians greeted the proposal with silence, but were so cordial throughout the meetings that tensions relaxed.
Suddenly, in September 1955, Eisenhower suffered a heart attack in Denver, Colorado. After seven weeks he left the hospital, and in February 1956 doctors reported his recovery. In November he was elected for his second term.
In domestic policy the President pursued a middle course, continuing most of the New Deal and Fair Deal programs, emphasizing a balanced budget. As desegregation of schools began, he sent troops into Little Rock, Arkansas, to assure compliance with the orders of a Federal court; he also ordered the complete desegregation of the Armed Forces. "There must be no second class citizens in this country," he wrote.
Eisenhower concentrated on maintaining world peace. He watched with pleasure the development of his "atoms for peace" program–the loan of American uranium to "have not" nations for peaceful purposes.
Before he left office in January 1961, for his farm in Gettysburg, he urged the necessity of maintaining an adequate military strength, but cautioned that vast, long-continued military expenditures could breed potential dangers to our way of life. He concluded with a prayer for peace "in the goodness of time." Both themes remained timely and urgent when he died, after a long illness, on March 28, 1969.
Categories United States Tags 34th president, Dwight D. Eisenhower, list president, president, president of the United States, United States Post navigation
Harry S. Truman was 33rd president of the United States of America (1945-1953)
John F. Kennedy was 35th president of the United States of America (1961-1963)

redpajama_set_name: RedPajamaCommonCrawl


text:
VDL may refer to:

 Vancouver Dodgeball League
 Van Diemen's Land, the former name of the smallest Australian state of Tasmania
 Van Diemen's Land Company in Tasmania
 VDL Groep, Dutch manufacturing company
 VDL (radio), a name used by two radio stations in the Lebanon
 Vienna Definition Language in computer programming
 VHF Data Link
 Ursula von der Leyen, President of the European Commission

redpajama_set_name: RedPajamaWikipedia


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text:
Bezmer Air Base  is an air base for the Bulgarian Air Force.  The base is situated in the eastern part of the Upper Thracian Lowland, in Yambol Oblast (Region), 10 km west of the city of Yambol and 30 km southeast of the city of Sliven, between the villages of Bezmer and Bolyarsko, and near the Sofia-Burgas railway.  The base takes its name from the nearby village, which is named after Khan Bezmer of Bulgaria (7th Century AD).

History

World War I
The strategic location and particularly favorable weather conditions of the area was appreciated already during World War I, when the Imperial German Air Service built in Yambol a base for zeppelins used for reconnaissance and bombing missions to Romania, Russia, Sudan and Malta.

Post World War II
In 1955 the Bezmer Air Base hosted the 22 Fighter Air Regiment, later transformed into 22 Fighter-Bomber Air Regiment, and eventually into 22 Attack Air Base of the Bulgarian Air Force, serving as a base for Su-25 ground attack aircraft, as well as Su-22М-4 and Su-22UM-3K reconnaissance planes.

Modern times
The Bezmer Air Base is situated in the eastern part of the Upper Thracian Lowland, in Yambol Oblast (Region), 20 km west of the city of Yambol and 30 km southeast of the city of Sliven, between the villages of Bezmer and Bolyarsko, and near the Sofia-Burgas railway. The base takes its name from the nearby village, which is named after Khan Bezmer of Bulgaria (7th Century AD).

The strategic location and particularly favorable weather conditions of the area was appreciated already during World War I, when the Imperial German Air Service built in Yambol a base for zeppelins used for reconnaissance and bombing missions to Romania, Russia, Sudan and Malta.

In 1955 the Bezmer Air Base hosted the 22 Fighter Air Regiment, later transformed into 22 Fighter-Bomber Air Regiment, and eventually into 22 Attack Air Base of the Bulgarian Air Force, serving as a base for Su-25 Frogfoot ground attack aircraft, as well as Su-22М-4 and Su-22UM-3K reconnaissance planes. Aircraft and personnel from Bezmer have recently been participating in a number of joint military exercises including the PfP 'Cooperative Key' in Turkey, Bulgaria, Romania and France, 'Immediate Response 2005' and the Bulgarian-American-Romanian 'Immediate Response 2006'.

Current use
Aircraft and personnel from Bezmer have recently been participating in a number of joint military exercises including the PfP "Cooperative Key" in Turkey, Bulgaria, Romania and France, 'Immediate Response 2005' and the Bulgarian-American-Romanian "Immediate Response 2006".

The base has a modern communication, information and navigation system.  A second phase of modernization and infrastructure development is underway, including a runway extension, which would expand the range of aircraft the base can support.

The Bezmer Air Base is among the joint US-Bulgarian military bases established according to the 2006 Defense Cooperation Agreement between the United States and Bulgaria.  Some experts rank Bezmer among the six most important American military bases outside mainland USA.

United States Secretary of Defense Lloyd Austin visited the base on March 18, 2022.

Images

See also
Dobrich Air Base
Graf Ignatievo Air Base
Gabrovnitsa Air Base
Cheshnegirovo Air Base
Dobroslavtsi Air Base
Uzundzhovo Air Base
Ravnets Air Base
Vrazhdebna Air Base
List of Bulgarian Air Force bases
List of Bulgarian military bases
28th Air Detachment
Bulgaria
Military of Bulgaria
Bulgarian cosmonaut program
List of joint US-Bulgarian military bases
Bulgarian-American Joint Military Facilities

References

Citations

Bibliography

 L. Ivanov and P. Pantev eds., The Joint Bulgarian-American Military Facilities: Public opinion and strategic, political, economic, and environmental aspects, NI Plus Publishing House, Sofia, 2006 (in Bulgarian)
 L. Ivanov ed., Bulgaria: Bezmer and adjacent regions — Guide for American military, Multiprint Ltd., Sofia, 2007,

Airports in Bulgaria
Military installations of Bulgaria

de:Amerikanische Militärbasen in Bulgarien


I read this article and found it very interesting, thought it might be something for you. The article is called The Long Road To Basic Amenities In India's Schools and is located at https://archive.indiaspend.com/sectors/the-long-road-to-basic-amenities-in-indias-schools.
The Supreme Court recently directed the federal government and the state governments to provide basic amenities, including drinking water and toilet facilities, in all government and private schools within six months.
IndiaSpend's Sourjya Bhowmick looks at the 2010-11 data of the District Information System for Education (DISE), which covers government & private schools, and finds that the task is arduous with Chhattisgarh, Bihar, Jharkhand and Orissa having the least educational infrastructure. Surprisingly, 83% of Indian schools have functional computers while only 43% have electricity connection.
The Supreme Court judgment is based on a writ petition filed in 2004 and several interim orders (Oct 18, 2011 and Dec 12, 2011) given to the states to provide toilet facilities, drinking water and other necessary infrastructure for schools.
From the graph, you can see that the total does not reach 100%. The reason, as stated by DISE, is because of missing values and rounding of figures. While the number of government schools increased by around 16,558 in 2010-11 (an increase of 1.5%), the percentage share of government schools has gone down by 2% because of the addition of over 10,000 private schools.
The indicators that will be taken for this article are schools having drinking water facilities, functional girls' toilets, functional computers, electricity connection and the average dropout rate. The best 5 performing states and the worst 5 states are being taken into account.
Table 1 shows that states have been doing well in providing drinking water in schools. Even states in the bottom 5 have improved in the past two years. While over 93% of Indian schools have access to drinking water, 7% schools still do not have access to this basic infrastructure.
Table 2 shows that Jharkhand has been quickly building girls' toilets in schools. While Chhattisgarh is the worst performer now (with only 35% schools with girls' toilets), Rajasthan has the best number (92%). It is highly likely that it will be a daunting task for many states to meet the Supreme Court order of building toilets in all schools within six months.
A surprising observation here is that access to toilet facilities does not have any direct relation with girls' enrolment in schools. While over 60% schools across the country have girls' toilets, enrolment of girls continue to be low at 48%.
Let us now look at another important infrastructure – electricity to run basic facilities like lights and fans and also computer-based teaching aids.
So, over 95% schools in Gujarat have electricity connection while only 4% schools in Bihar have access to power. The all-India average of electricity connection in schools is only 43%.
Another key indicator to look at now is functional computers in schools.
The DISE document does not mention the number of computers that should be functional in every school. While Punjab is doing the best with 98% schools having functional computers, Orissa has the worst percentage (39%) of schools having functional computers. Surprisingly, 83% of schools have functional computers in India while 57% schools have no electricity connection!
Rajasthan has the worst dropout rate in the country and the state figures only once in the infrastructure list – the state is among the top 5 when it comes to schools with girls' toilets. Other states like Orissa and West Bengal, which do not have good infrastructure, however are able to retain students in schools.
So, apart from the Supreme Court-mandated facilities like drinking water and toilets, schools across the country are lacking in other education infrastructure as well. While there have been many court directives in the past on such issues, a lot still needs to be done before India can claim 100% availability of education infrastructure.
Providing basic infrastructure is tedious but is an imperative to ensure that children who are guaranteed a right to education also have the basic amenities – water, toilet, electricity – available to them so that they can learn and succeed.


text:
Q: Python: readlines() with tar-file gives StreamError: seeking backwards is not allowed. What is wrong? Ok, I saw similar questions but not exactly the same. And I can't figure out what is really wrong with this python code:
import tarfile

tar_file = tarfile.open('something.tgz', mode="r|gz")
txt_file = tar_file.extractfile('inner.txt')
lines = txt_file.readlines()
txt_file.close()
tar_file.close()

It gives StreamError: seeking backwards is not allowed due to readlines().
But this fact looks strange to me and I try to understand what I miss here.

A: The problem is with this line:
tar_file = tarfile.open('something.tgz', mode="r|gz")

According to the tarfile.open() docs, the correct mode should be either "r" - Open for reading with transparent compression (recommended) or "r:gz" - Open for reading with gzip compression. Using the pipe | character creates a stream:

Use this variant in combination with e.g. sys.stdin, a socket file object or a tape device. However, such a TarFile object is limited in that it does not allow random access

which is where you ran into problems with readlines() and seek(). When I changed that pipe | to a colon :, your code worked fine.

redpajama_set_name: RedPajamaStackExchange


Sophomore, Junior, Senior and Graduate students have 15 residence halls to choose from to create their best living environment.
These choices include Suite Style Housing for sophomores, juniors, seniors and graduate students residing in singles, doubles and triple room configurations.
While juniors, seniors and graduate students who wish to reside in our Townhouse Apartment Style housing select between double and triple room configurations. Sophomores are not eligible to reside within townhouses.
Michael E. Ageno Hall, commonly known as Ageno A, is located in the canyon. The hall is named in memory of Michael E. Ageno, a member of the class of 1963 and son of college patron, Edward S. Ageno. The second-year residence hall is suite style in design, accommodating six students per suite. Each half-suite has a double room and a single room with a shared vanity, sink, small refrigerator, and microwave. Please be aware that small refrigerators and microwaves are not provided, however you might get lucky if the students previous to you left their small refrigerators or microwave! The suites are connected by a toilet and shower and each room contains one built-in dresser and a lockable closet. Each room has independent heating controls.
Marjorie David Ageno Hall, commonly known as Ageno B, is named in memory of Marjorie David Ageno, wife of college patron Edward S. Ageno. The second-year residence hall is suite style in design, accommodating six students per suite. Each half-suite has a double room and a single room with a shared vanity, sink, small refrigerator, and microwave. Please be aware that small refrigerators and microwaves are not provided, however you might get lucky if the students previous to you left their small refrigerators or microwave! The suites are connected by a toilet and shower and each room contains one built-in dresser and a lockable closet. Each room has independent heating controls.
Ferdinand & Camille Ageno Hall, commonly known as Ageno C, was named after Edward S. Ageno's parents. The second-year residence hall is suite style in design, accommodating six students per suite. Each half-suite has a double room and a single room with a shared vanity, sink, small refrigerator, and microwave. Please be aware that small refrigerators and microwaves are not provided, however you might get lucky if the students previous to you left their small refrigerators or microwave! The suites are connected by a toilet and shower and each room contains one built-in dresser and a lockable closet. Each room has independent heating controls.
The second-year residence hall is suite style in design, accommodating six students per suite. Each half-suite has a double room and a single room with a shared vanity, sink, small refrigerator, and microwave.Please be aware that small refrigerators and microwaves are not provided, however you might get lucky if the students previous to you left their small refrigerators or microwave! The suites are connected by a toilet and shower and each room contains one built-in dresser and a lockable closet. Each room has independent heating controls. Claeys North has its own basketball court and backyard, with a barbeque.
The second-year residence hall is suite style in design, accommodating six students per suite. Each half-suite has a double room and a single room with a shared vanity, sink, small refrigerator, and microwave. Please be aware that small refrigerators and microwaves are not provided, however you might get lucky if the students previous to you left their small refrigerators or microwave! The suites are connected by a toilet and shower and each room contains one built-in dresser and a lockable closet. Each room has independent heating controls. The Legacy Garden is located just outside Claeys South.
Becket Hall is on the hill above Justin Hall, and currently houses second-year students and the Lasallian Living Learning Community. Each suite is comprised of three double bedrooms, one bathroom, and a common living space. Each bedroom contains one large built-in dresser for two users and two lockable closets. Please be aware that small refrigerators and microwaves are not provided, however you might get lucky if the students previous to you left their small refrigerators or microwave! DTA boxes are provided in the bedrooms only. Each suite has one central heating control. Becket shares a laundry room with More Hall. At the time of its construction, Becket Hall was the College's first residence to feature apartment-style living.
More Hall was built in 1967 on the hill above Justin Hall, overlooking the central campus, and currently houses second-year students. More features suites comprised of three double bedrooms, one bathroom, and a common living space. Each bedroom contains one large built-in dresser for two users and two lockable closets. Please be aware that small refrigerators and microwaves are not provided, however you might get lucky if the students previous to you left their small refrigerators or microwave! DTA boxes are provided in the bedrooms only. Each suite has one central heating control. The hall shares its laundry room with Becket Hall.
The two upper townhouses, located at the top of the hill, each have their own beautiful courtyard area, lounge and laundry room.
Edward S. Ageno Hall East, better known by its short name Ageno East, is found at the top of the hill with a breathtaking view of the campus. Each suite in this hall for juniors and seniors accomodates five to six students in double rooms and triple rooms. Apartments are equipped with a full bathroom, living room/common area, and basic furniture. A full sized fridge is provided but microwaves are not. The hall has its own courtyard space, suitable for barbeques and other gatherings. Ageno West is also home to the Santiago Living Learning Community.
Edward S. Ageno Hall West, better known by Ageno West, houses juniors and seniors in townhouses not far from the cross on the hill. This upperclassmen hall accomodates five to six students in double rooms and triple rooms. Apartments are equipped with a full bathroom, living room/common area, and basic furniture. A full sized fridge is provided but microwaves are not. The hall has its own courtyard space, suitable for barbeques and other gatherings.
Six lower townhouses, located near the Joseph Alioto Recreation Center, each have access to a beautiful courtyard area, barbeque area, and laundry room.
Freitas Hall was built in 1973, and is named for Edward W. Freitas, the brother of townhouses benefactor Judge Carlos Freitas '27. Apartments are equipped with a full bathroom, living room/common area, and basic furniture. A full sized fridge is provided but microwaves are not.
Guerrieri East is one of the two halls built in 1981, partly through the generosity of the estate of Lewis Guerrieri. The hall shares a basketball court and barbeque pits with Guerrieri West. Apartments are equipped with a full bathroom, living room/common area, and basic furniture. A full sized fridge is provided but microwaves are not.
Guerrieri West is one of the two halls built in 1981, partly through the generosity of the estate of Lewis Guerrieri. The hall shares a basketball court and barbeque pits with Guerrieri East. Apartments are equipped with a full bathroom, living room/common area, and basic furniture. A full sized fridge is provided but microwaves are not.
The Sabatte townhouse was built through the generosity of Fran Sabatte '36 and Raymond Sabatte '45, who named the building for their parents. Apartments are equipped with a full bathroom, living room/common area, and basic furniture. A full sized fridge is provided but microwaves are not.
Raymond Syufy '40, founder of Century Theaters and original benefactor of the townhouses, named Syufy Hall for Pauline Syufy, his mother. Apartments are equipped with a full bathroom, living room/common area, and basic furniture. A full sized fridge is provided but microwaves are not.
Thille Hall was named for Albert J. Thille, brother of Mary Thille. Mary was one of the four original benefactors of the Saint Mary's townhouses and the benefactress of the reconstrcted De La Salle quad. Apartments are equipped with a full bathroom, living room/common area, and basic furniture. A full sized fridge is provided but microwaves are not.


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Baker-Polito Administration Announces New Housing Choice Designations and Highlights Housing Choice Bill in Haverhill
10 additional Housing Choice Communities recognized for adopting housing best practices
HAVERHILL – Today, Governor Charlie Baker joined Haverhill Mayor James J. Fiorentini, Representative Linda Dean Campbell, Representative Christina Minicucci, Representative Lenny Mirra, Representative Andres Vargas and other local leaders to highlight An Act to Promote Housing Choices, legislation filed by Governor Baker in February that calls for targeted zoning reform to advance new housing production in Massachusetts and support the administration's goal to produce 135,000 new housing units by 2025.
Representative Campbell, Representative Minicucci, Representative Mirra, Representative Vargas, Karen Sawyer Conard, Executive Director of the Merrimack Valley Planning Commission, Dave Traggorth, President of Traggorth Companies, and Eric Chinburg, President of Chinburg Properties, joined Mayor Fiorentini in endorsing the legislation in the lead up to the May 14th Joint Committee on Housing hearing.
During today's event, Governor Baker also announced the designation of Haverhill and nine other municipalities as Housing Choice Communities. These ten communities join the 69 Housing Choice Communities designated in 2018.Launched in December 2017 as part of the Baker-Polito Administration's Housing Choice Initiative, this designation recognizes communities that adopt best practices for boosting housing production by making them eligible for competitive capital grants. The Administration has awarded $4 million to 19 communities through the Housing Choice Capital Grant Program.
"Housing Choice grants reward and encourage municipalities' efforts to increase housing production and exemplify the support we believe is critical to empowering community development," said Governor Charlie Baker. "We congratulate today's designees for their commitment to delivering important housing for residents and look forward to building on this progress by working together with the Legislature to pass our Housing Choice legislation."
"Governor Baker and I recognize that far too many families, workers and employers are limited by the lack affordable, adequate housing in their communities," said Lt. Governor Karyn Polito. "We are proud of the $1 billion our administration has invested into affordable housing programs like the Housing Development Incentive Program to advance thousands of market-rate units, and we look forward to continuing to prioritize housing at this crucial juncture."
Haverhill earned this designation for producing more than 800 new housing units over the last five years, successfully leveraging state resources such as the Housing Development Incentive Program to drive the conversion of former mills into transit-oriented, mixed-use developments and promote a variety of housing and commercial activity. The pending Housing Choice bill would make it easier for the city to continue pursuing such projects.
In addition to Haverhill, today's newly designated communities include: Ayer, Burlington, Dracut, Fall River, Franklin, Hingham, Newton, Westwood, and Wrentham. Since 2014, these municipalities have collectively produced more than 5,800 new units of housing.
The legislative proposal will enable cities and towns to adopt certain zoning best practices related to housing production by a simple majority vote, rather than the current two-thirds supermajority. While this legislation will lower the voting threshold to change zoning for all communities in the Commonwealth, it does not require cities and towns to make any of these changes. With the proposed simple majority threshold, municipalities that pursue rezoning efforts including those enabling transit-oriented or downtown-oriented new housing, would gain approval if they achieve more than 50% of the vote, as opposed to the current super majority of more than 66%. Massachusetts is currently one of only a few states to require a supermajority to change local zoning.
Zoning changes that promote best practices for housing growth that would qualify for the simple majority threshold include:
Building mixed-use, multi-family, and starter homes, and adopting 40R "Smart Growth" zoning in town centers and near transit.
Allowing the development of accessory dwelling units, or "in-law" apartments.
Approving Smart Growth or Starter Homes districts that put housing near existing activity centers.
Granting increased density through a special permit process.
Allowing for the transfer of development rights and enacting natural resource protection zoning.
Reducing parking requirements and dimensional requirements, such as minimum lot sizes.
This legislation also includes a provision, added by the Joint Committee on Housing last session, that would reduce the voting threshold for a special permit issued by a local permit granting authority to a simple majority vote, for certain multi-family or mixed-use projects with at least 10% affordable units in locations near transit or, in centers of commercial activity within a municipality.
"Massachusetts is home to the country's best-educated workforce, but also its most expensive housing," said Housing and Economic Development Secretary Mike Kennealy. "Our Housing Choice legislation would empower the production necessary to reduce housing costs, delivering relief to families while helping retain talent and securing economic growth. We look forward to working with our partners in the Legislature to make sure it passes this session."
The legislation is part of the administration's Housing Choice Initiative, which provides incentives, technical assistance, and capital grant funding to encourage new housing production that meets the long-term needs of the Commonwealth's growing, and aging, population.
"Our families, seniors, workforce, individuals with disabilities, and vulnerable populations have diverse housing needs, and today's market is not meeting them. Housing that our residents can afford is a crucial resource and An Act to Promote Housing Choices will create the tools we need to promote new housing development for those across incomes," said Acting Undersecretary of Housing and Community Development Jennifer Maddox. "We are proud of the coalition we've built in support of this legislation and the investments we've made in affordable housing, community development, and public housing to benefit the many communities which contribute to our strong, inclusive Commonwealth."
"I'm grateful that Governor Baker is coming to Haverhill to showcase his key housing legislation, which I strongly endorse," said Mayor of Haverhill James J. Fiorentini. "This legislation, aptly called the Housing Choice Initiative, will give our city options to control our own destiny. It fits perfectly with our ongoing Master Plan process and will help us as we decide where we want residential growth and what kind of residential growth we want. Most importantly, it will give us new tools to develop housing that middle class families can actually afford."
"Governor Baker's Housing Choice legislation provides a much clearer path to create dense urban housing and to preserve existing structures like the Ellis Factory Lofts. It will increase housing options and density, reduce dimensional requirements, and decreases costs, which all helps to make more affordable projects," said Eric Chinburg, President of Chinburg Properties. "We are really pleased to have completed this renovation and to breathe new life into an iconic, historic factory building. Attracting new residents and businesses to downtown Haverhill supports a vibrant place to live, work and play. Haverhill has been very supportive during this complex project and we congratulate Mayor Fiorentini and his staff staff on their approach to encourage varied housing options in the downtown core, where residents can walk to restaurant, shops and public transportation."
"The Housing Choice Initiative legislation is an important step in the development of more affordable housing across the Commonwealth," said Lisa B. Alberghini, President of the Planning Office for Urban Affairs. "We are in the midst of an affordable housing crisis, and we need to increase affordable housing opportunities to ensure the continued creation of strong, diverse and healthy communities. The Housing Choice Initiative is a significant tool in the effort to address this need."
"We have been successful in Haverhill because Mayor Fiorentini has proven that housing is key to economic development, and because Governor Baker has given us the tools like the Housing Development Incentive Tax Credit and the State Historic Tax Credit to help make it happen," said Dave Traggorth, President of Traggorth Companies. "We support the Housing Choice legislation so that we can scale up the success we've had both within Haverhill, and in other cities and towns eager to see the same incredible results."
"As one of 13 Regional Planning Agencies in the Commonwealth, the Merrimack Valley Planning Commission is proud to support the Baker-Polito Administration's efforts with the Housing Choice Initiative," said Karen Sawyer Conard, Executive Director of the Merrimack Valley Planning Commission. "Home to seven Housing Choice Community Designations and three Community Grant and Small Town Grant Awards to enhance water infrastructure needs and improve open space amenities for residents, our region is taking full advantage of this innovative program. Our communities will look to coordinate the benefits of the Housing Choice Initiative with our newly developed Regional Housing Plan to help plan for the development and preservation of housing units to meet the needs of all residents today and in the future."
"Its designation as a Housing Choice Community is a testament to the City of Haverhill's commitment to bolstering housing production and supporting a growing workforce that can proudly call this great city home," said Senator Diana DiZoglio. "I am pleased to see these efforts recognized and look forward to working alongside the City of Haverhill and communities across the Merrimack Valley to ensure all working families have sufficient access to affordable housing."
"Zoning reform is key to addressing our housing crisis and ensuring that every community does their best to meet the housing demand in Massachusetts,"said Representative Andres Vargas. "I appreciate the Governor taking the time to visit Haverhill, as we continue to make progress on housing affordability."
"We must continue to find innovative solutions to address the housing crisis, especially given our aging population," said Representative Linda Dean Campbell. "Haverhill's designation as a Housing Choice Community will give the City greater access to funding to further its innovative efforts on this issue. I look forward to working with Governor Baker and my colleagues on the state level to make it easier for our communities to adopt the housing solutions that work best for them."
"Massachusetts has enjoyed a very robust economy with strong job growth,"said Representative Lenny Mirra. "In order to see this continue, the Commonwealth will need a large increase in housing production and the Governor's plans will go a long way in making this happen."
The Baker-Polito Administration has shown a deep commitment to increasing the production of housing across income levels. Since 2015, the administration has invested more than $1 billion in affordable housing, resulting in the production and preservation of more than 17,000 housing units, including 15,000 affordable units. In 2018, Governor Baker signed the largest housing bond bill in Massachusetts history, committing more than $1.8 billion to the future of affordable housing production and preservation. The Baker-Polito Administration has also advanced the development of more than 11,000 mixed-income housing units through the successful MassWorks Infrastructure Program, reformed the Housing Development Incentive Program, and worked with communities to implement smart-growth development and planning efforts.
2019 Housing Choice Community Designations:
· Town of Ayer
· Town of Burlington
· Town of Dracut
· City of Fall River
· Town of Franklin
· City of Haverhill
· Town of Hingham
· City of Newton
· Town of Westwood
· Town of Wrentham


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The High Wycombe Furniture Archive (HWFA) was developed at Buckinghamshire New University in High Wycombe. It contains photographic and print material relating to furniture companies in the High Wycombe region, primarily from Ercol Limited and E. Gomme Limited (G-Plan).
The digitisation of our collections and the creation of the HWeFA website was funded by a Resource Enhancement Grant from the Arts and Humanities Research Council (AHRC) in 2006. The website was first launched in August 2009, with additional material added on a regular basis.


We created a Rothamsted Manor Winter wedding styled shoot with a bold red berry & navy theme with some other local wedding vendors in order to build my portfolio, try out new techniques and network with other amazing suppliers from the local area.
I worked alongside the amazing venue stylist, Sara Carlin (Venue Styling by Sara) on this wonderful red berry and navy themed shoot. It has a luxury winter wedding feel about it and it was so much fun to work with Sara and the other amazing suppliers on this project.
Please see the bottom of this post for a list of suppliers and their contact details if you want to contact any of them.
And if you have a few extra spare minutes, please take a look at the behind the scenes video that was created for us by Louise Shrigley Wedding FIlms.
Below you can find a list of all the suppliers involved in this shoot and their social media details if you want to get in touch with them.


Why Should I add a Waste Vegetable oil fuel system to my diesel?
This kit can be home built with materials from your local auto parts and hardware store. All you need is an understanding of the system, some mechanical skill and a diesel vehicle.