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\section{QCD RESUMMATION\protect\footnote{Section coordinator:
    E. Laenen}$^{,}$~\protect\footnote{Contributing authors: C.
    Bal\'azs, R. Ball, M. Cacciari, S. Catani, D. de Florian, S.
    Forte, E. Gardi, M. Grazzini, N. Kidonakis, E. Laenen, S. Moch, P.
    Nadolsky, P.Nason, A. Kulesza, L. Magnea, F. Olness, R. Scalise,
    G. Sterman, W. Vogelsang, R. Vogt, C.-P. Yuan}}\label{sec:resum,qcdsm}

\subsection{Introduction}

In our working group we investigated a variety of issues concerning
the relevance of resummation for observables at TeV colliders.
Resummation is a catch-all name for efforts to extend the predictive
power of QCD by summing large (logarithmic) corrections to all orders
in perturbation theory. In practice, the resummation formalism depends
on the observable at issue, through the type of logarithm to be
resummed, and the resummation methods.

A number of resummation formalisms (threshold resummation, $Q_T$ or
recoil resummation and any renormalization-group resummation) have now
matured to the point where one can employ them for precision physics.
It is known how to organize the associated logarithms to all orders
and to any accuracy, at least in principle.  Such resummation
formalisms therefore constitute a systematically improvable
calculational scheme, like standard perturbation theory.  It is also
known how to consistently match these resummations to finite order
perturbation theory. In our working group, the full
next-to-next-to-leading logarithmic threshold and $Q_T$ resummation
were performed for the inclusive Higgs production cross section and
its transverse momentum respectively. Further studies addressing the
value of resummation for precision physics were done for heavy quark
production cross sections and fragmentation functions. The
applicability of joint resummation, the combination of threshold and
recoil resummation, was examined in detail for electroweak
annihilation. The relation between small-$x$ resummation, the DGLAP
evolution equation, and precision analysis of the deep-inelastic
structure function at HERA was further explored. Detailed application
of these results to TeV colliders is still to come.

Resummed cross sections are inherently ambiguous because they require
a prescription to handle singularities due to very soft radiation.
These ambiguities take the form of power corrections, about which
there is still much to learn. Our working group has studied the
characteristics of various prescriptions, and the power corrections
that they imply.

Both within our working group, and in joint sessions with others,
numerous fruitful discussions took place based on short 
presentations by participants. These presentations and discussions 
addressed both the topics mentioned and reported on below, as well 
as the issues of resummation and Monte Carlo programs (by V. Ilyin)
and next-to-next-leading logarithmic threshold resummation for 
Drell-Yan and deep-inelastic scattering (by A. Vogt).

In general, the studies performed in the QCD resummation working
group, whose reports now follow, as well as the discussions held
strengthen the view that QCD resummation does, must, and will play an 
important part in the quantitative study of observables at TeV colliders.


\subsection{Higgs boson transverse momentum at the
  LHC\protect\footnote{Contributing authors: C. Bal\'azs, D. de
    Florian, A. Kulesza}}


 
 
\subsubsection{Introduction} 
 
The underlying mechanism of the electroweak symmetry breaking (EWSB)
is an uncovered sector of the Standard Model (SM), thus the physical
remnant of the spontaneous EWSB, the Higgs boson, is the primary
object of search at present and future colliders.
% 
At the CERN Large Hadron Collider (LHC), a Standard Model (like) Higgs
boson can mainly be produced in 14 TeV center of mass energy
proton--proton collisions by the partonic subprocess $g g$ (via heavy
quark loop) $\to H X$ (see section A.1 of ref. \cite{Cavalli:2002vs}). 
The extraction
of the signal requires the accurate prediction of its production rate,
as well as the transverse momentum ($Q_T$) distribution of the Higgs
boson and its decay products and backgrounds, since the shape of these
distributions can dictate the analysis strategies for the search
\cite{Balazs:2000sz}.
% 
To reliably predict these distributions, especially in the low to
medium $Q_T$ region where the bulk of the rate is, the effects of the
multiple soft--gluon emission have to be included. This, and the need
for the systematic inclusion of the higher order QCD corrections
require the extension of the standard hadronic factorization theorem
to the low $Q_T$ region. With a smooth matching to the usual
factorization formalism, it is possible to obtain a sound prediction
in the full $Q_T$ range.
% In the next section this extended factorization formalism is summarized. 
 
 
\subsubsection{Low $Q_T$ Factorization}\label{sec:low-qt,resum,qcdsm} 
 
In this section the low transverse momentum factorization formalism is
summarized briefly.  We consider the case of the inclusive
hard-scattering process where the final-state system $F$ is produced
by the collision of the two hadrons $h_1$ and $h_2$.  The final state
$F$ is a generic system of non-strongly interacting particles, such as
{\em one} or {\em more} vector bosons $(\gamma^*, W, Z, \dots)$, Higgs
particles ($H$) and so forth.
 
When calculating fixed order QCD corrections to the cross section
%${d\sigma}/{dQ dQ_T^2 d\phi}$, 
the hadronic factorization theorem is invoked. While the transverse
momentum $Q_T$ of the produced system is of the order of its invariant
mass $Q$, this calculation is reliable. But the standard factorization
fails when the $Q_T \ll Q$, as a result of multiple soft and collinear
emission of gluons from the initial state. As an indication of this
problem, the ratio of the two very different scales appear in
logarithmic corrections of the form $\alpha_S^n/Q_T^2 \ln^{m}
Q^2/Q_T^2$ ($1<m<2n-1$), which spoil the convergence of fixed-order
calculation in the low $Q_T$ region.
% These logs are not absorbed by the parton  
%distribution functions, unlike the ones originating from purely collinear parton  
%emission, so the factorization has to be changed to resum them. 
% 
These logarithmically-enhanced terms, not absorbed by the parton
distribution functions, have to be evaluated at higher perturbative
orders, and possibly resummed to all orders in the QCD coupling
constant $\as$.
% 
%As a result, the $Q_T$  
%distribution calculated using the conventional hadronic factorization theorem is  
%unphysical in the low $Q_T$ region. 
 
% The solution 
 
To resolve the problem, the differential cross section is split into a
part which contains all the contribution from the logarithmic terms
(res.), and into a regular term (fin.):
\begin{eqnarray} 
\label{resum:eq1} 
       \frac{d\sigma_{F}}{dQ dQ_T^2 d\phi} =  
\left[ \frac{d\sigma_{F}}{dQ dQ_T^2 d\phi} \right]_{\rm res.} +  
\left[ \frac{d\sigma_{F}}{dQ dQ_T^2 d\phi} \right]_{\rm fin.} \, , 
\end{eqnarray} 
where $\phi$ denotes the remaining independent kinematical variables
of the final system.  Since the second term on the right hand side in
Eq.(\ref{resum:eq1}) does not contain potentially large logs, it can
be calculated using the usual factorization. The first term has to be
evaluated differently, keeping in mind that failure of the standard
factorization occurs because it neglects the transverse motion of the
incoming partons in the hard scattering.
 
In the Fourier conjugate transverse position (${\vec b}$) space the
resummed component of the cross section
\cite{Dokshitzer:1980hw}--\cite{Davies:1984hs}
can be written as
\begin{eqnarray} 
\label{resum:resgen} 
 \frac{Q^2\, d\sigma_{F}}{dQ^2 dQ_T^2d \phi} = \sum_{a,b} 
\int dx_1 \, dx_2\, db  \frac{b}{2} J_0(b Q_T) \; 
f_{a/h_1}\left(x_1,\frac{b_0^2}{b^2}\right)   
f_{b/h_2}\left(x_2,\frac{b_0^2}{b^2}\right) 
 s\, W_{ab}^{F}(x_1 x_2 s; Q, b, \phi) \,. 
\end{eqnarray} 
Here, the resummed partonic cross section $W_{ab}^{F}$ is
\begin{eqnarray} 
\label{resum:unw} 
W_{ab}^{F}(s; Q, b, \phi) &= \sum_c \displaystyle \int dz_1\,  
dz_2  
\; C_{ca}^{F}\left(\as(b_0^2/b^2), z_1\right)  
\; C_{{\bar c}b}^{F}\left(\as(b_0^2/b^2), z_2\right) 
\; \delta\left(Q^2 - z_1 z_2 s\right) \nonumber \\ 
&\cdot\, 
\displaystyle \frac{d\sigma_{c{\bar c}}^{(LO) \,F}(Q^2,\phi)}{d\phi} \; 
H_c^{F}\left(\as(Q^2), \phi\right) 
%\f{d\sigma_{c{\bar c}}^{F}(Q^2, \as(Q^2),\phi)}{d\phi} 
\;S_c(Q,b) \;\;, 
%\end{align} 
\end{eqnarray} 
where $\sigma_{c{\bar c}}^{(LO)}$ is the lowest order partonic
cross-section and $S_c(Q,b)$ the Sudakov form factor (with $c=q,g$)
\begin{equation} 
\label{resum:sudakov} 
S_c(Q,b)=\exp \left\{ -\int_{b_0^2/b^2}^{Q^2} \frac{dq^2}{q^2}  
\left[ A_c\left(\as(q^2)\right) \;\ln \left(\frac{Q^2}{q^2}\right)  
+ B_c\left(\as(q^2)\right) \right] \right\}\, . 
\end{equation} 
In the usual CSS approach \cite{Collins:1985kg}, the coefficient
function $H_c^{F}$ does not appear (i.e. $H_c^{F}\equiv 1$), with the
consequence that both the coefficient functions
$C_{ab}(\as(b_0^2/b^2),z)$ and the Sudakov form factor $S_c(Q,b)$
become process dependent, a certainly unpleasent feature.  As
discussed in Ref.~\cite{Catani:2000vq}, the inclusion of $H_c^F$ is
sufficient to make the Sudakov form factor $S_c(Q,b)$ and the
coefficient functions $C_{ab}(\as(b_0^2/b^2),z)$ process-independent.
In a similar way as it happens for parton densities, which have to be
defined by fixing a factorization scheme (e.g. the ${\overline {\rm
    MS}}$ scheme or the DIS scheme), there is an ambiguity in the
factors on the right-hand side of Eq.~(\ref{resum:unw}), which have to
be defined by choosing a `resummation scheme'. Note that the choice of
a `resummation scheme' amounts to defining $H_c^{F}$ (or $C_{ab}$) for
a {\em single} process. Having done that, the process-dependent factor
$H_c^{F}$ and the universal factors $S_c$ and $C_{ab}$ in
Eq.~(\ref{resum:unw}) are unambiguously determined for any other
process.
 
The resummation formula in Eq.~(\ref{resum:unw}) has a simple physical
origin.  When the final-state system $F$ is kinematically constrained
to have a small transverse momentum, the emission of accompanying
radiation is strongly inhibited, so that
%only % CsB: part of the low Q_T effect comes from hard partons balancing each other 
mostly soft and collinear partons (i.e. partons with low transverse
momenta $Q_T$) can be radiated in the final state. The
process-dependent factor $H_c^{F}(\as(Q^2))$ embodies hard
contributions produced by virtual corrections at transverse-momentum
scales $Q_T \sim Q$. The form factor $S_c(Q,b)$ contains real and
virtual contributions due to soft (the function $A_c(\as)$ in
Eq.~(\ref{resum:sudakov})) and flavour-conserving collinear (the
function $B_c(\as)$ in Eq.~(\ref{resum:sudakov})) radiation at scales
$Q \gtap Q_T \gtap 1/b$. At very low momentum scales, $Q_T \ltap 1/b$,
only real and virtual contributions due to collinear radiation remain
(the coefficient functions $C_{ab}(\as(b_0^2/b^2),z)$).
 
The $A_c$ and $B_c$ functions, as well as the coefficients $C_{ab}$
are free of large logarithmic corrections and safely calculable
perturbatively as expansions in the strong coupling $\as$
\begin{equation} 
A_c(\alpha_S) = 
\sum_{n=1}^\infty  
\left( \frac{\alpha _S}\pi \right)^n A_c^{(n)}, 
 ~~~  
B_c(\alpha_S) = 
\sum_{n=1}^\infty  
\left( \frac{\alpha _S}\pi \right)^n B_c^{(n)}, 
 ~~~  
C_{ab}^F(\alpha_s,z) = 
\sum_{n=0}^\infty  
\left( \frac{\alpha _S}\pi \right)^n C^{F(n)}_{ab}(\alpha_S,z)\, , 
\end{equation} 
with a similar expansion for the coefficient function $H_c^F$.  More
detailed expression for the coefficients can be found in
\cite{Kodaira:1982nh,Catani:1988vd,Davies:1984hs,deFlorian:2000pr,deFlorian:2001zd
}.  The coefficients of the perturbative expansions $A_c^{(n)}$,
$B_c^{(n)}$ and $C_{ab}^{F(n)}(z)$ are the key of the resummation
procedure since their knowledge allows to perform the resummation to a
given {\it logarithmic} order: $A^{(1)}$ leads to the resummation of
leading logarithmic (LL) contributions, $\{ A^{(2)}, B^{(1)} \}$ give
the next-to-leading logarithmic (NLL) terms, $\{ A^{(3)}, B^{(2)},
C^{(1)}\}$ give the next-to-next-to-leading logarithmic (NNLL) terms,
and so forth\protect\footnote{A concensus has not been reached regarding 
the classification of the so-called LL, NLL, NNLL etc. terms, and their
corresponding $B$ contents. The above classification of subsect. 
\ref{sec:low-qt,resum,qcdsm}  is used in subsects.\ref{higgs-prod-nnll} and
\ref{softglue-higgs}.  Another classification is used in subsect. \ref{qt-css}.
The motivation for the latter is presented in
sect.\ref{sec:resumps,mc,qcdsm}.}\label{ftnote-classif}.

In particular, the coefficient $B^{(2)}$ has been recently computed
\cite{deFlorian:2000pr, deFlorian:2001zd} for both $q\bar{q}$ and $gg$
channels allowing to extend the analysis to NNLL accuracy.  Even
though there is no analytical result available for it, the coefficient
$A_{q,g}^{(3)}$ has been extracted numerically with a very good
precision in Ref.~\cite{Vogt:2000ci}.
% 
After matching the resummed and fixed order cross sections, it is
expected that the normalization of the resummed cross section
reproduces the fixed order total rate (at which the $A$, $B$ and $C$
functions are evaluated), since when expanded and integrated over
$Q_T$ they deviates only in higher order terms.~\cite{Balazs:1997xd}
 
 
\subsubsection{Higgs $Q_T$ at the LHC using the CSS formalism}\label{qt-css}
 
The low $Q_T$ factorization formalism, described in the previous
section, is utilized to calculate the QCD corrections to the
production of Higgs
bosons at the LHC. %~\cite{BalazsYuanH}  
In the low $Q_T$ region this calculation takes into account the
effects of the multiple--soft gluon emission including the Sudakov
exponent $S$, and the non--perturbative contributions ${\cal
  F}_{a/h}$. At the next-to-leading-logarithmic order (NLL) the
$A^{(1,2)}$, and $B^{(1,2)}$ coefficients are included in the Sudakov
exponent.  The normalization changing effect of the ${\cal
  O}(\alpha_S^3)$ virtual corrections are also taken into account by
including the coefficient ${C}_{gg}^{(1)}$, which ensures agreement
with the ${\cal O}(\alpha_S^3)$ total rate.  At the
leading-logarithmic order (LL) the coefficients $A^{(1)}$, and
$B^{(1)}$ and ${C}_{gg}^{(0)}$ are included. By matching to the ${\cal
  O}(\alpha_S^3)$ fixed order distributions a prediction is obtained
for the Higgs production cross section in the full $Q_T$ range which
is valid up to ${\cal O}(\alpha_S^3)$. The expressions for the
included $A$, $B$ and $C$ coefficients and further details of this
calculation are given in an earlier work.~\cite{Balazs:1997xd} The
analytic results are coded in the ResBos Monte Carlo event generator.
 
 
\begin{figure}[htbp] 
\begin{center} 
  \includegraphics[width=\textwidth]{LHC_mH125_GeV.epsi}
\end{center} 
\caption{ 
  Higgs boson transverse momentum distributions at the LHC calculated
  by ResBos (curves) and PYTHIA (histograms). The solid curve was
  calculated at NLL \protect\footnote{Notice that this is not the
    conventional definition of LL and NLL accuracy.} (including
  $A^{(1,2)}$, $B^{(1,2)}$, and $C^{(0,1)}$).  The dashed curve is LL
  (includes $A^{(1)}$, $B^{(1)}$, and $C^{(0)}$).  For PYTHIA, the
  original output with default input parameters rescaled by a factor
  of $K = 2$ (solid), and one calculated by the altered input
  parameter value $Q_{max}^2 = s$ (dashed) are shown.  The lower
  portion, with a logarithmic scale, also shows the high $Q_T$ region.
  In the last frame all are normalized to the solid curve.}
\label{resum:Fig:PYTHIA} 
\end{figure} 
 
Fig.~\ref{resum:Fig:PYTHIA} compares the Higgs boson transverse
momentum distributions at the LHC calculated by ResBos (curves) and by
PYTHIA (histograms from version 6.136). We use a Higgs mass of 125 GeV
and CTEQ5M parton distributions. The solid curve was calculated at NLL
(including $A^{(1,2)}$, $B^{(1,2)}$, and $C^{(0,1)}$).  The dashed
curve is LL (includes $A^{(1)}$, $B^{(1)}$, and $C^{(0)}$).  The shape
of these curves are quite similar reflecting the small uncertainty in
the shape of the resummed prediction. The normalization changes
considerably after including the sub-leading tower of logs, since both
the $C^{(1)}$ and the $B^{(2)}$ coefficients receive contributions
from the ${\cal O}(\alpha_S^3)$ virtual corrections, which are known
to be large.
 
In the low and intermediate $Q_T$ ($\lesssim 100$ GeV) region the
shape of the default PYTHIA histogram agrees better with the LL ResBos
curve, justifying the similar physics that goes into PYTHIA and the LL
calculation.  For high $Q_T$, the PYTHIA prediction falls under the
ResBos curve, since ResBos mostly uses the exact fixed order ${\cal
  O}(\alpha_S^3)$ matrix elements in that region, while PYTHIA still
relies on the multi--parton radiation ansatz. PYTHIA can be tuned to
agree with ResBos in the high $Q_T$ region, by changing the maximal
virtuality a parton can acquire in the course of the shower (dashed
histogram).  Matrix element corrections to this process in PYTHIA are
expected to fix this discrepancy in the near future.  Since showering
is attached to a process after the hard scattering takes place, and
the parton shower occurs with unit probability, it does not change the
total cross section for Higgs boson production given by the hard
scattering rate.  Thus, the total rate is given by PYTHIA at ${\cal
  O}(\alpha_S^2)$. Thus in Fig.~\ref{resum:Fig:PYTHIA} the dashed
PYTHIA histograms are plotted with their rate normalized to the NLL
ResBos curve.
 
\subsubsection{Higgs production to NNLL accuracy}\label{higgs-prod-nnll} 
 
As has been discussed in the introduction, the coefficients appearing
in Eq.~(\ref{resum:unw}) (with the exception of $A^{(n)}$) are
dependent on the resummation scheme.  Under a change from scheme $S$
to $S'$, the coefficients are modified as
\begin{eqnarray} 
H_{c\, S}^{(1)F} & \to & H_{c\, S'}^{(1)F} \;, \nonumber \\ 
C_{ab\, S}^{(1)}(z) & \to & C_{ab\, S'}^{(1)}(z)= 
C_{ab\, S}^{(1)}(z) +\frac{1}{2} \delta_{ab} \delta(1-z) \left( 
H_{a\, S}^{(1)F} - H_{a\, S'}^{(1)F} \right) \;, \nonumber \\ 
B_{c\, S}^{(2)} & \to & B_{c\, S'}^{(2)}= B_{c\, S}^{(2)} + \beta_0 \left( 
H_{c\, S}^{(1)F} - H_{c\, S'}^{(1)F} \right) 
\label{transf} 
\end{eqnarray} 
One possible scheme is the scheme-$H$\cite{Catani:2000vq}, where
$H_{g\, H}\equiv 1$ and the coefficients agree with the corresponding
to the CSS formulation for this process.  Another posibility consist
in the $\overline{MS}$ scheme, where $B^{(2)}$ is defined to be
proportional to the coefficient of the $\delta(1-z)$ term in the
two-loop $gg$ splitting function \cite{Catani:2000vq}.  Based on the
physical interpretation of the different pieces in the resummation
formulae in Eq.~(\ref{resum:unw}), it is possible to define the
`collinear resummation scheme', where only terms of collinear origin
(i.e. the ones originated from the $n-4$ component of the splitting
functions, see \cite{deFlorian:2000pr, deFlorian:2001zd} ) remain in
the coefficient function $C$.
 
 
Even though the cross section for Higgs production is a physical
observable and therefore independent on the chosen scheme, the
truncation of the perturbative expansion at the level of the resummed
coefficients introduces an explicit dependence on the scheme. We will
use this scheme dependence as a way to quantify the perturbative
stability of the resummed result, estimating the size of the
unaccounted higher order terms as well as of the non-perturbative
contributions.  It follows from~(\ref{transf}) that the scheme
dependence first enters at the NNLL level.  In Fig.
\ref{resum:Fig:higgsnnll} we show preliminary results for the $Q_T$
distribution for Higgs production at LHC ($M_H=150$ GeV) at LL, NLL
and NNLL, in the last case using the three schemes discussed above.

The results are obtained using the code developed
in~\cite{Kulesza:2001jc}, adapted for the case of inclusive Higgs
production in the $gg$ channel (in the limit of infinite top quark
mass) and to genuine NNLL accuracy, i.e. including up to $A^{(3)}$,
$B^{(2)}$, $C^{(1)}$ and $H^{(1)}$.

Matching to the perturbative result has been performed only at LO, but
this will not affect our conclusions since we are mainly interested in
the low $Q_T$ region where the resummed contribution completely
dominates the cross section.  The matching at ${\cal O}(\alpha_S^4)$
can be performed by using the NLO calculation of
\cite{deFlorian:1999zd}. Furthermore we have not included any
non-perturbative contribution and the parton distributions correspond
to the CTEQ5M set.
\begin{figure}[htbp] 
\begin{center} 
  \includegraphics[width=0.8\textwidth]{hqt.eps}
\end{center} 
\caption{\label{resum:Fig:higgsnnll}{$Q_T$ distribution for Higgs
    production at LHC to LL, NLL and NNLL accuracy. The three NNLL
    lines correspond to the choice of different resummation schemes.
  }}
\end{figure} 
As can be observed, the NNLL corrections are rather large. The main
reason for that is the appearence of the $C^{(1)}$ and $H^{(1)}$
coefficients, which contain the large virtual (and soft) corrections
for Higgs production, which are known to considerably increase (up to
almost a factor of two) the inclusive cross section. Higher order
terms are expected to be smaller, at least from the observation of the
rather good perturbative stability concerning the dependence on the
resummation scheme at NNLL.\\
{\bf Acknowledgements:} C.B. and D.de F. thank the organizers of the
Les Houches conference for their financial support.  C.B. also thanks
I. Puljak for the PYTHIA curves. A.K. would like to thank W.~Vogelsang
for many useful discussions. D.de F. thanks M. Grazzini for
discussions.
 
 

\subsection{Soft-gluon effects in Higgs boson production at hadron
  colliders\protect\footnote{Contributing authors: S. Catani, D. de
    Florian, M. Grazzini and P. Nason}}\label{softglue-higgs}


The most important mechanism for SM Higgs boson production at hadron
colliders is gluon--gluon fusion through a heavy (top) quark loop.
NLO QCD corrections to this process are known to be large
\cite{Dawson:1991zj,Djouadi:1991tk,Spira:1995rr}: their effect
increases the LO inclusive cross section by about 80-100$\%$.  The NLO
corrections are weakly dependent on the mass $M_t$ of the top quark,
thus justifying the evaluation of higher-order terms within the
large-$M_t$ approximation.  Recently, the calculation of the NNLO
corrections in the large-$M_t$ limit has been completed
\cite{Harlander:2000mg,Catani:2001ic,Harlander:2001is,Catani:2001cr,Harlander:2002wh}.
Their effect is moderate at the LHC: in the case of a light Higgs, the
$K$-factor is about $2.3$--$2.4$, corresponding to an increase of
about 25$\%$ with respect to NLO.  The NNLO effect is more sizeable at
the Tevatron where $K\sim 3$, the increase being of about $50 \%$ with
respect to NLO.  The bulk of the NNLO contributions is due to soft and
collinear radiation
\cite{Catani:2001ic,Harlander:2001is,Catani:2001cr}, the hard
radiation \cite{Harlander:2002wh} giving only a small (typically
5-7$\%$) correction.  Multiple soft-gluon emission beyond NNLO can
thus be important, particularly at the Tevatron.  Here we investigate
the effects of resummation of soft (Sudakov) emission to all orders.
More details will be given elsewhere.

The cross section ${\hat \sigma}_{gg}$ for the partonic subprocess
$gg\to H+X$ at the centre--of--mass energy ${\hat s}=M_H^2/z$ can be
written as
\begin{equation}
\label{resum:spart}
{\hat \sigma}_{gg}({\hat s},M_H^2) = \sigma_0 \;z \;G_{gg}(z) \;,
\end{equation}
where $M_H$ is the Higgs mass, $\sigma_0$ is the Born-level cross
section and $G_{gg}$ is the perturbatively computable coefficient
function.  Soft-gluon resummation has to be performed (see
\cite{Catani:1996yz} and references therein) in the Mellin (or
$N$-moment) space ($N$ is the moment variable conjugate to $z$).  The
{\em all-order} resummation formula for the coefficient function
${G}_{gg}$ is \cite{Catani:2001ic,Kramer:1996iq}:
\begin{equation}
\label{resum:resformula} 
{G}_{gg, \, N}^{{\rm (res)}} =\as^2(\mu_R^2)\,
C_{gg}(\as(\mu^2_R),M_H^2/\mu^2_R;M_H^2/\mu_F^2) \, 
\Delta_{N}^{H}(\as(\mu^2_R),M_H^2/\mu^2_R;M_H^2/\mu_F^2)\; ,
\end{equation}
where $\mu_R$ and $\mu_F$ denote the renormalization and factorization
scales, respectively.  The function $C_{gg}(\as)$ contains all the
terms that are constant in the large-$N$ limit,
% : they are 
produced by hard virtual contributions and non-logarithmic soft
corrections.  It can be computed as a power series expansion in $\as$
as
\begin{equation}
\label{resum:coef}
C_{gg}(\as(\mu^2_R),M_H^2/\mu^2_R;M_H^2/\mu_F^2) =  
1 + \sum_{n=1}^{+\infty} \;  
\left( \frac{\as(\mu^2_R)}{\pi} \right)^n \; 
C_{gg}^{(n)}(M_H^2/\mu^2_R;M_H^2/\mu_F^2) \;\;,
\end{equation}
where the perturbative coefficients $C_{gg}^{(n)}$ are closely related
to those of the $\delta(1-z)$ contribution to $G_{gg}(z)$.  The
radiative factor $\Delta_{N}^{H}$ embodies the large logarithmic terms
due to soft-gluon radiation.  To implement resummation, the radiative
factor is expanded to a given logarithmic accuracy as
\begin{equation} 
\label{resum:deltannll} 
\Delta_N^{H} = 
\exp \Big\{ \frac{}{} \ln N \; g^{(1)}(\lambda) + 
g^{(2)}\!\left(\lambda,\frac{M_H^2}{\mu^2_R};\frac{M_H^2}{\mu_F^2}\right)  
+ \as(\mu^2_R) 
g^{(3)}\!\left(\lambda,\frac{M_H^2}{\mu^2_R};\frac{M_H^2}{\mu_F^2}\right)  
+ {\cal O}(\as^2(\as \ln N)^k) \Big\} , 
\end{equation} 
such that the functions $g^{(1)}, g^{(2)}$ and $g^{(3)}$ respectively
collect the leading logarithmic (LL), next-to-leading logarithmic
(NLL) and next-to-next-to-leading logarithmic (NNLL) terms with
respect to the expansion parameter $\lambda=\as(\mu^2_R) \ln N$.

NLL resummation \cite{Catani:2001ic} is controlled by three
perturbative coefficients, $A_g^{(1)}, A_g^{(2)}$ and $C^{(1)}_{gg}$.
The coefficients $A_g^{(1)}$ and $A_g^{(2)}$, which appear in the
functions $g^{(1)}$ and $g^{(2)}$, are well known (see
\cite{Catani:1996yz} and references therein).  The coefficient
$C^{(1)}_{gg}$ in Eq.~(\ref{resum:coef}) is extracted from the NLO
result.

At NNLL accuracy three new coefficients are needed
\cite{Catani:2001ic}: the coefficient $C_{gg}^{(2)}$ in
Eq.~(\ref{resum:coef}) and two coefficients, $D^{(2)}$ and
$A_g^{(3)}$, which appear in the NNLL function $g^{(3)}$.  The
functional form of $g^{(3)}$ was computed in Ref.~\cite{Vogt:2000ci}.
The coefficients $D^{(2)}$ and $C_{gg}^{(2)}$ are obtained
\cite{Catani:2001ic} from the NNLO result. The coefficient $A_g^{(3)}$
is not yet fully known: we use its exact $N_f^2$-dependence
\cite{Bennett:1998ch} and the approximate numerical estimate of
Ref.~\cite{vanNeerven:2000wp}.

Finally, to take into account the dominant collinear logarithmic
terms, the coefficient $C_{gg}^{(1)}$ in the resummation formula can
be modified as \cite{Catani:2001ic}
\begin{equation}
\label{resum:colterm}
C_{gg}^{(1)} \rightarrow C_{gg}^{(1)} + 2 A_g^{(1)} \; \frac{\ln N}{N} \;.
\end{equation}

In the following we present a preliminary study of the resummation
effect at the Tevatron and the LHC.  The hadron-level cross section is
obtained by convoluting the partonic cross section in
Eq.~(\ref{resum:spart}) with the parton distributions of the colliding
hadrons. We use the MRST2000 set \cite{Martin:2000gq}, which includes
approximated NNLO parton distributions, with $\as$ consistently
evaluated at one-loop, two-loop, three-loop order for the LO, NLO
(NLL), NNLO (NNLL) results, respectively.  All the results correspond
to the choice $\mu_F=\mu_R=M_H$.  The $K$-factors in
Figs.~\ref{resum:fig:lhc} and \ref{resum:fig:tev} are defined with
respect to the LO cross section. The resummed calculations are always
matched to the corresponding fixed-order calculations, i.e. we
consider the full fixed-order result and include higher-order
resummation effects.
%%====================================
\begin{figure}[htb]
\begin{center}
  \includegraphics[width=12cm,height=7.5cm,angle=0]{lhc2000.epsi}
\end{center}
\caption{\label{resum:fig:lhc}{Resummed K-factors at the LHC}.}
\end{figure}
%%====================================
In Fig.~\ref{resum:fig:lhc} we plot the K-factors at the LHC, as a
function of the Higgs mass.  On the left-hand side we show the NLL
result, matched to NLO, with and without the inclusion of the
collinear term in Eq.~(\ref{resum:colterm}).  The NLO result is
reported for comparison.  On the right-hand side we show the NNLL
results, again with and without the inclusion of the collinear term.
The matching is done to the NNLO result\footnote{More precisely, we
  include all the soft and virtual contributions and the hard terms in
  the form $(1-z)^n$ up to $n=1$. Higher powers of $(1-z)$ give very
  small effect \cite{Harlander:2002wh}.}  \cite{Harlander:2002wh}.  We
also plot the resummed NLL result obtained by including $C_{gg}^{(2)}$
and $D^{(2)}$ only.  We see that the inclusion of the collinear term
is numerically not very relevant.  Soft-gluon resummation at NLL
accuracy increases the NLO cross section by 13-20 $\%$, the effect
being more sizeable at high $M_H$.  Going from NNLO to NNLL the effect
is smaller, with an increase of $\sim 6$-9$\%$ in the full range of
$M_H$.  Note also that NLL resummation gives a good approximation of
the complete NNLO calculation.

%%====================================
\begin{figure}[htb]
\begin{center}
  \includegraphics[width=12cm,height=7.5cm,angle=0]{tev2000.epsi}
\end{center}
\caption{\label{resum:fig:tev}{Resummed K-factors at the Tevatron Run II}.}
\end{figure}
%%====================================
In Fig.~\ref{resum:fig:tev} we report the analogous results at the
Tevatron.  Here the resummation effects are larger. Going from NLO to
NLL accuracy, the cross section increases by 25-30$\%$.  NNLL
resummation increases the NNLO cross section by $\sim 12$-$16\%$ when
$M_H$ varies in the range 100-200~GeV.  These results are not
unexpected \cite{Catani:2001cr}, since at the Tevatron the Higgs boson
is produced closer to threshold and the effect of multiple soft-gluon
emission is more important.

\input{modif-JR}

\subsection{Transverse momentum and threshold resummations in heavy
  quark production\protect\footnote{Contributing authors: N.
    Kidonakis, P.M. Nadolsky, F. Olness, R. Scalise, C.-P. Yuan}}


Theoretical analysis of QCD problems involving a single scale
parameter is highly developed; however, the problem becomes complex
when more than one distinct scale is involved. Unfortunately, most of
the interesting experimental measurements fall in the multi-scale
category.  In recent years, powerful theoretical tools have been
developed to describe such complex reactions.  The fundamental problem
with the analysis of multi-scale regimes in the context of
perturbation theory is that the truncated series in the strong
coupling $\alpha_s$ may converge poorly due to the presence of large
logarithms of dimensionless quantities.
The ``big bad logs'' can appear in a number of guises.  Logarithms of
the form ($\alpha_s^n/\qt^2) \ln^m(\qt^2/Q^2)$, $m \le 2n-1$ appear
when calculating differential distributions in transverse momentum.
Separately, we encounter logarithms of the form $\alpha_s^n
[\ln^m(1-z)/1-z]_+$, $m \le 2n-1$, with $z=Q^2/s$, in the production
of heavy quarks near threshold.  The solution to the problem is to
find all-order sums of large logarithmic terms from first principles
of the theory, such as gauge and renormalization group invariance.

The soft-gluon resummation formalism of Collins, Soper, Sterman (CSS)
\cite{Collins:1988ig,Collins:1985kg} sums large logs of the form
$\ln^m(\qt^2/Q^2)$; this formalism has enjoyed a number of theoretical
advances in recent years
%, including the improvement of the
%nonperturbative sector and the introduction of new
%resummation schemes
\cite{Berger:2002mt}.  Logarithms of the second type, $\ln^m(1-z)$,
arise in the production of heavy quarks near threshold
\cite{Kidonakis:1997gm,Kidonakis:1996aq,Laenen:1998kp,Kidonakis:1999ze,Kidonakis:2000ui}.
More recently, new techniques have been developed to perform joint
resummation to handle both types of logs simultaneously
\cite{Laenen:2000ij}.

In this study, we address the impact of $q_T$ and threshold logarithms
on the heavy quark production. Our approach is to first examine the
$\qt$ and threshold resummation results separately, compare the
contributions in various kinematic regions, and finally consider the
joint resummation.  The dependence of results on the mass $M$ of the
heavy quarks gives an additional dimension to this problem.  Our work
on $q_T$ resummation is motivated, in part, by recently calculated
${\cal O}(\alpha_s)$ cross sections for heavy quark production in
semi-inclusive deep-inelastic scattering (SIDIS)
\cite{Kretzer:2001tc}, and by newly developed methods for resummation
of $q_T$ logarithms in massless SIDIS
\cite{Meng:1996yn,Nadolsky:2000ky,Nadolsky:1999kb}. These results can
be combined to obtain improved description of differential heavy-quark
distributions in a large range of collision energies.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\sec{Outline of the Problem}

Our calculations use the ACOT variable flavor number factorization
scheme \cite{Aivazis:1994pi,Collins:1998rz}, which itself is a
resummation of logarithms involving the heavy quark mass via the DGLAP
evolution equation. This scheme is successfully applied to obtain
accurate predictions for inclusive quantities (e.g. the charm
structure function $F_2^c(x,Q^2)$) both at asymptotically high
energies $s \gg M^2$ and near the threshold $s \approx M^2$.
 
For less inclusive observables, application of the ACOT scheme in a
fixed-order calculation is often not sufficient due to the presence of
logarithms different from $\ln(Q^2/M^2)$. Fixed-order calculations,
such as the calculation in \cite{Kretzer:2001tc}, have to combine
subprocesses with different numbers of final-state particles. This
combination leads to unphysical ``plus'' and ``delta-function''
distributions; it also leaves large logarithmic terms in the
coefficient functions.  The dynamical origin of this deficiency lies
in the intensive soft QCD radiation, which accompanies particle
reactions near boundaries between $n$- and $(n+1)$-particle
kinematics. In SIDIS, unphysical distributions appear in the current
fragmentation region, which in our notations corresponds to the limit
$q_T^2 \ll Q^2$. Note that this phenomenon occurs when the hard scale
of the reaction is much larger than the heavy-quark mass, {\it i.e.},
$Q^2 \gg M^2$.


The key result of the CSS resummation formalism is that all large
logarithms due to the soft radiation in the high-energy limit can be
resummed into a Sudakov exponent.  This result can be summarized in
the following master equation:\footnote{In such a short report, the
  equations presented can only be schematic; complete notations will
  be defined in our upcoming publication; see also
  \cite{Meng:1996yn,Nadolsky:2000ky,Nadolsky:1999kb}.}
%
\begin{equation}
\frac{d\sigma}{dQ^2\, dq_T^2}=
\frac{\sigma_0}{s} \ 
\int \frac{d^2 b}{(2\pi)^2} \ e^{i \vec q_T \cdot \vec b} \  
\left( C^{\mathrm{in}} \otimes f \right)
\left( C^{\mathrm{out}} \otimes d \right)
e^{-S} \ 
+ \sigma_{\mathrm{FO}} - \sigma_{\mathrm{ASY}} \, .
\label{resum:CSS}
\end{equation}
%
Here $b$ is the impact parameter, $f$ and $d$ are parton distribution
functions and fragmentation functions, respectively.  $C^{\mathrm{in}}$,
$C^{\mathrm{out}}$ contain perturbative corrections to contributions from the
incoming and outgoing hadronic jets, respectively.  The factor
$e^{-S}$ is the Sudakov exponential, which includes an all-order sum
of perturbative logarithms $\ln^m{q_T^2/Q^2}$ at $b\lesssim 1\mbox {
  GeV}^{-1}$ and nonperturbative contributions at $b\gtrsim 1\mbox{
  GeV}^{-1}$.  Finally, $\sigma_{\mathrm{FO}}$ is the fixed-order expression
for the considered cross section, while $\sigma_{\mathrm{ASY}}$ ({\it
  asymptotic piece}) is the perturbative expansion of the $b$-space
integral up to the same order of $\alpha_s$ as in $\sigma_{FO}$.  At
small $q_T$, where terms $\ln^m(q_T^2/Q^2)$ are large, $\sigma_{\mathrm{FO}}$
cancels well with $\sigma_{\mathrm{ASY}}$, so that the cross section
(\ref{resum:CSS}) is approximated well by the $b$-space integral.  At
$q_T \gtrsim Q$, where the logarithms are no longer dominant, the
$b$-space integral cancels with $\sigma_{\mathrm{ASY}}$, so that the cross
section (\ref{resum:CSS}) is equal to $\sigma_{\mathrm{FO}}$ up to higher order
corrections.

\begin{figure}[thbp]
  \includegraphics[width=0.8\textwidth]{./resum1.eps}
\caption{ Balance of various terms in the ACOT scheme and resummed cross
  section. Graphs show $q_T$ on the $x$-axis and $d\sigma/dq_T^2$ on
  the $y$-axis.}
\label{resum:fig2}
\end{figure}
In extending the CSS resummation from the massless to the massive
case, we observe that the proof of the resummation formula in the
original papers \cite{Collins:1988ig,Collins:1985kg} does not require
quarks to be massless; nor does it rely on the usage of the
$\overline{MS}$ scheme for regularization of soft and collinear
singularities. Hence the reorganization of perturbative calculations
as in Eq.~(\ref{resum:CSS}) is also justified for massive quarks.
Note, however, that nonvanishing quark masses regulate collinear
singularities, so that at least some of the logarithms are not
dominant in the threshold region.
On the other hand, the discussion in Ref.~\cite{Collins:1998rz}
suggests that masses can be set to zero for those heavy quarks that
enter the hard scattering subprocess directly from the proton.

This approximation, which differs from the complete mass-dependent
cross section by higher-order terms, significantly simplifies the
analysis. We therefore drop heavy-quark masses in the hard scattering
subdiagrams with incoming heavy quark lines.  This approximation leads
to massless expressions for the perturbative Sudakov factor and
$C$-functions for quark-initiated subprocesses, and mass-dependent
$C$-functions for gluon-initiated subprocesses.

Figure~\ref{resum:fig2} qualitatively illustrates the balance of
various terms in the resummation formula in various regions of phase
space for the lowest-order contributions. First, consider a
fixed-order calculation in the simplified ACOT scheme near the
threshold. In this region, the cross section $d\sigma/dq_T^2$ is well
approximated by the ${\cal O}(\alpha_s)$ photon-gluon diagram
(Fig.~\ref{resum:fig2}a).  To this diagram, we add the lowest-order
$\gamma^* q$ term, which resums powers of $\ln(Q^2/M^2)$ and
contributes at $q_T = 0$ (Fig.~\ref{resum:fig2}b).  We also subtract
the overlap between the two diagrams (Fig.~\ref{resum:fig2}c), which
is approximately equal to, but not the same as, the $\gamma^* q$
contribution. The resulting distribution (Fig.~\ref{resum:fig2}d) is
close to the fixed-order result, but has discontinuities in the small
$q_T$ region.  These discontinuities are amplified when $Q$ increases.

In the resummed cross section, the fusion diagram still dominates near
the threshold, but now the resummed cross section and subtracted
asymptotic piece are smooth functions, which cancel well at all values
of $q_T$ (Fig.~\ref{resum:fig2}e-h). Hence, the distribution is
physical in the whole range of $q_T$. Finally, at $Q^2 \gg M^2$ the
small-$q_T$ region the $\gamma^* g$ fusion contribution is dominated
by the $1/q_T^2$ term (Fig.~\ref{resum:fig2}i).  Such singular terms
are summed through all orders in the $b$-space integral corresponding
to Fig.~\ref{resum:fig2}j and are canceled in the fusion contribution
by subtracting the asymptotic piece (Fig.~\ref{resum:fig2}k).

As mentioned above, threshold resummation organizes logs of the form
$\alpha_s^n [\ln^m(1-z)/1-z]_+$, which can yield a large contribution
when $z \rightarrow 1$
\cite{Kidonakis:1997gm,Kidonakis:1996aq,Laenen:1998kp,Kidonakis:1999ze,Kidonakis:2000ui}.
When combining the results of both the $\qt$ and threshold
resummation, an important consideration is whether there is an overlap
in resummed terms, which would result in double counting. While in a
general context this delicate problem makes the double resummation
non-trivial, this is not an issue for the low order case we are
examining here; for this reason, we can compute the contributions
separately and compare them without any complications.

Our analysis in this study will continue well past the time scale of
the present workshop, and detailed results will be described in a
separate publication. New experimental measurements will contain a
wealth of information of high precision -- if only we can extract and
analyze it. Theoretical tools must be refined to keep up with this
progress.  The resummation formalisms that we are studying here
provide an efficient way to go beyond low order perturbative
calculations and describe the heavy quark data accurately in a wide
range of kinematical variables. With these new methods at hand, we
will be well prepared for examination of new detailed information from
HERA, TEVATRON Run II, and the LHC.


\subsection{Soft gluon resummation for fragmentation
  functions\protect\footnote{Contributing author: M. Cacciari}}


When one is considering the issue of performing tests of perturbative
QCD (pQCD) predictions and of calculational techniques, production
processes involving heavy quarks (i.e. of mass $m \gg \Lambda_{QCD}$)
can be thought of as an ideal choice for at least two distinct
reasons.
\begin{itemize}
\item The large mass of the quark acts as a cutoff for collinear
  singularities, allowing perturbative calculations to give finite
  results without resorting to factorization into phenomenological
  functions, hence allowing, at least in principle, direct comparisons
  to experimental data.
\item When the quarks undergo a non-perturbative
  hadronization\footnote{This is the case for charm and bottom, while
    the top quarks decay weakly before hadronizing.} before being
  observed, say, as a heavy-light meson like $D$ or $B$, the extent to
  which this process modifies the perturbative predictions can be
  expected to be of the order of magnitude of (powers of) the ratio
  $\overline{\Lambda}/m$, where $\overline{\Lambda}$ is a hadronic
  scale of the order of a few hundred MeV . One can therefore hope to
  be less dependent on the inclusion of non-perturbative
  parametrizations (for instance parton distribution and fragmentation
  function) which, by introducing additional phenomenological degrees
  of freedom, can obscure the real predictive power of the underlying
  theoretical framework.
\end{itemize}

Such nice features do of course come at a price. Perturbative
calculations, while finite, will however contain in higher order
logarithms of the various physical scales entering the problem. For
instance, heavy quark hadroproduction at transverse momentum $p_T$ or
heavy quark fragmentation in $e^+e^-$ collisions at a center-of-mass
energy $Q$ will display $\alpha_s^2(\alpha_s^n\log^k(p_T/m))$ or
$\alpha_s^n\log^k(Q/m)$ (with $0\le k \le n$) terms respectively. When
$Q,\;p_T \gg m$ such logarithms, remnants of the screening of the
collinear singularities, grow large, eventually spoiling the
convergence of perturbation theory. Analogously, observing the heavy
quark close to a phase space boundary will constrain soft radiation
and result in Sudakov logarithms which can also grow very large.

Therefore, in order to really exploit the aforementioned possibly
superior qualities of heavy quark processes, one must first take care
of properly and accurately evaluating the perturbative result. In
particular, this means performing an all-order resummation of the
large logarithms previously mentioned, and matching to an exact fixed
order result for recovering the non-leading terms.

Resummation of ``collinear'' large logarithms to next-to-leading log
(NLL) accuracy was first performed in
Ref.~\cite{Mele:1990yq,Mele:1991cw} for $e^+e^-$ collisions. Sudakov
logarithms were also resummed to leading logarithmic order (LL).
Reference~\cite{Dokshitzer:1996ev} extended the resummation of Sudakov
logs to NLL level.

The papers previously mentioned only directly addressed heavy quark
production in $e^+e^-$ collisions.
Reference~\cite{Mele:1990yq,Mele:1991cw} did however put forward
convincing arguments about the existence of a process-independent {\sl
  heavy quark fragmentation function}. Making use of this
universality, NLL resummation of collinear logs was first performed
for hadron-hadron collisions in \cite{Cacciari:1994mq}, and then
extended to other processes as well
\cite{Cacciari:1996fs,Cacciari:1996ej}. These resummations where then
matched to complete next-to-leading (NLO) perturbative calculations
\cite{Cacciari:1998it,Cacciari:2001td} to provide reliable results in
the whole range of transverse momenta.

The universality has recently been fully exploited in
Ref.~\cite{Cacciari:2001cw}, where the heavy quark fragmentation
function has been extracted by making no reference to any explicit
production process.  The production cross section for a heavy quark of
mass $m$ at a scale $Q$ (or $p_T$) can therefore always be written as
a convolution of this process-independent function $D^{\rm
  ini}(\mu_F,m)$ and a process dependent coefficient function
$C(Q,\mu_F)$:\footnote{For simplicity we will not be showing parton
  indices and summation over them.}
\begin{equation}
\sigma(Q,m) = C(Q,\mu_F) \otimes D^{\rm ini}(\mu_F,m) + {\cal
O}((m/Q)^p) \; .
\end{equation}
The factorization scale $\mu_F$ separates the two functions, and
Altarelli-Parisi evolution can be used to resum the collinear
logarithms in $D^{\rm ini}(\mu_F,m)$ by evolving from an initial scale
$\mu_{0F} \simeq m$ up to the hard scale $\mu_F \simeq Q$, so that
$D^{\rm ini}(\mu_F,m) = E(\mu_F,\mu_{0F}) \otimes D^{\rm
  ini}(\mu_{0F},m)$.

Making use of this factorization it has been possible to resum the
Sudakov logarithms to NLL accuracy for $D^{\rm ini}(\mu_{0F},m)$ and
$C(Q,\mu_F)$ separately. The resummed result for $D^{\rm
  ini}(\mu_{0F},m)$ is, of course, also process independent (though
factorization-scheme dependent), and its Mellin moments can be written
(we use the ${\overline {\rm MS}}$ factorization-scheme) in the form
\cite{Cacciari:2001cw}:
\begin{eqnarray}
D_N^{{\rm ini}, \,S}(\alpha_s(\mu_0^2);\mu_0^2,\mu_{0F}^2,m^2) &\!\!\!=& \!\!\! 
\left\{ 1 + \frac{\alpha_s(\mu_0^2) C_F}{\pi}\left[
-\frac{\pi^2}{6} + 1 - \gamma_E^2 + \gamma_E + \left(\frac{3}{4} -
\gamma_E\right) \ln\frac{\mu_{0F}^2}{m^2}\right]\right\}\nonumber\\
\label{resum:sudHQini}
&\!\!\!\times&\!\!\! \exp \Bigl[ \ln N \;g^{(1)}_{\rm ini}(\lambda_0) + 
g^{(2)}_{\rm ini}(\lambda_0,m^2/\mu_0^2;m^2/\mu_{0F}^2) 
\Bigr]\, .
\label{resum:Sudres}
\end{eqnarray}
Explicit results for the $g_{\rm ini}^{(1)}$ and $g_{\rm ini}^{(2)}$
functions can be found in Ref.~\cite{Cacciari:2001cw}.

Contrary to that of the ``collinear'' $\log(p_T/m)$ terms, resummation
of the Sudakov logs is not of direct phenomenological importance for
studying inclusive $p_T$ distributions in heavy quark production at
the Tevatron or the LHC.  Indeed, unless one gets very close to phase
space limits, only low-$N$ moments of $D_N^{{\rm ini}}$, around $N
\sim 5$, are relevant\cite{Nason:1999ta}. Sudakov resummation, on the
other hand, starts producing visible effects around $N \sim 10$
\cite{Cacciari:2001cw}.

Having a reliable theoretical prediction for $D^{{\rm
    ini}}(\mu_{0F},m)$ (and, with the analogous NLL Sudakov
resummation of $C(Q,\mu_F)$ also performed in
Ref.~\cite{Cacciari:2001cw}, for the measurable single-inclusive
heavy-quark cross section in $e^+e^-$ collisions) is however useful
when extracting from experimental data the non-perturbative
contributions which must complement the perturbative calculation in
order to get a good description of the data. A generic ``hadron
level'' cross section for $B$ or $D$ meson production in $e^+e^-$ or
hadron collisions can be described by\footnote{The coefficient
  function $C(Q,\mu_F)$ and the heavy quark function $D^{\rm
    ini}(\mu_F,m)$ can of course be evaluated to different levels of
  theoretical accuracy, for instance including NLL Sudakov resummation
  and matching to complete NLO calculations, as in
  Ref.~\cite{Cacciari:2001cw}. Sudakov resummation for the coefficient
  function $C(Q,\mu_F)$ is at present only available for $e^+e^-$
  collisions. It is however less important for hadronic collisions,
  where phase space edges (and hence large-$N$ moments), are rarely
  experimentally probed (an exception to this might be $D$ meson
  photoproduction data at HERA).}
\begin{equation}
\sigma_{\rm had}(Q,m) = C(Q,\mu_F) \otimes D^{\rm ini}(\mu_F,m) 
\otimes D^{\rm
np}(\epsilon_1,...,\epsilon_n)\; .
\label{resum:hadlev}
\end{equation}
The function $D^{\rm np}(\epsilon_1,...,\epsilon_n)$ can represent
either a phenomenological parametrization or an attempt to establish
the form of such a contribution through the analysis of power
corrections.  It can be argued~\cite{Cacciari:1997wr,Cacciari:1997du}
that it will be, at least to some reasonable accuracy level, as
process independent as its perturbative counterpart $D^{\rm
  ini}(\mu_F,m)$. In any case, it will depend on a set of parameters
$(\epsilon_1,...,\epsilon_n)$ which, pending a full solution of
non-perturbative QCD, can only be determined by comparison with the
data.

\begin{figure}[thbp]
  \includegraphics[width=\textwidth]{./aleph2001.eps}
\caption{ALEPH data for weakly-decaying $B$ mesons are fitted to a
  convolution of the perturbative prediction and a non-perturbative
  function (see text).  Solid line: $\alpha = 27.45$, fitted in $N$
  space, fixed by the $\langle x_E\rangle$ (i.e.  $N=2$) point only.
  Dashed line: $\alpha = 29.1$, fitted as before, but no Sudakov
  resummation in the pQCD part. Dotted line: $\alpha = 17.7$, fitted
  with resummation and using all points in $x_E$ space. The dot-dashed
  line is the purely perturbative contribution, as defined in
  Ref.~\cite{Cacciari:2001cw}.}
\label{resum:fig1}
\end{figure}

An example of such a determination is given in Fig.~\ref{resum:fig1}.
ALEPH $e^+e^-$ data~\cite{Heister:2001jg} for weakly-decaying $B$
meson production are fitted to the Kartvelishvili et
al.~\cite{Kartvelishvili:1978pi} non-perturbative, one-parameter
function\footnote{Despite its simplicity, this function is
  known~\cite{Heister:2001jg} to provide one of the best fits to the
  experimental data. Alternatively, one could use the form
\begin{equation}
D_N^{\rm np}(\overline{\Lambda})
 = \exp\left\{\sum_{k=1}^\infty \frac{(-1)^k}{k!\; k
}\left(\frac{\overline{\Lambda} (N-1)}{m}\right)^k\right\}\;,
\end{equation}
suggested by the structure of the Sudakov resummation for $D^{\rm
  ini}(\mu_F,m)$. Upon identyifing $\alpha$ with
$2m/\overline{\Lambda}$, the results of the two expressions are
virtually indistinguishable, even when including only a few terms of
the series.}  $D^{\rm np}(x;\alpha) = (\alpha+1)(\alpha+2) x^\alpha
(1-x)$.  The solid line is obtained by fitting Eq.~\ref{resum:hadlev}
to the $\langle x_E\rangle$ point only (the average energy fraction),
in $N$ space. The dashed line is obtained by performing the same kind
of fit but excluding the Sudakov resummation of
Eq.~\ref{resum:Sudres}. One can see this degrades the result when
comparing to $x_E$ data in the right panel.

While giving in this particular example a worse result, in $x_E$
space, than the one obtained by fitting the $x_E$ distribution itself
(dotted line in Fig.~\ref{resum:fig1}), the determination of the
non-perturbative parameter(s) from $N$ space data is however
theoretically more self-consistent. Non-perturbative effects increase
in size as $x_E$ gets closer to one. Hence when fitting in $x_E$ space
one is not only using many experimental points to fix a single (or a
few) parameter(s) while however not having the functional form on firm
theoretical grounds, but one is also adjusting the non-perturbative
contribution to fit points where its weight greatly differs. In $N$
space, on the other hand, the non-perturbative contribution increases
with $N$. Fitting only $\langle x_E\rangle$ means using the point
where it is smallest, and then one can still check if the shape of the
curve in $x_E$ is correctly predicted.

A theoretically sensible program for evaluating heavy meson cross
sections at the Tevatron or the LHC would therefore entail:
\begin{itemize}
\item[a)] Analysing the structure of the process-independent function
  $D^{\rm ini}(\mu_{0F},m)$, and trying to extract information about
  its power corrections, which build the non-perturbative function
  $D^{\rm np}$. Full results for this step are at present not yet
  available in the literature, and work is in progress.  Some of the
  characteristics of the non-perturbative contribution are however
  well established. It is for instance well known
  \cite{Jaffe:1994ie,Nason:1997pk} that the leading power correction
  is linear in $\overline{\Lambda}/m$, where $m$ is the heavy quark
  mass and $\overline{\Lambda}$ a hadronic scale of the order of a few
  hundred MeV.
\item[b)] Fitting the first few $N$ moments of $e^+e^-$ data, either
  with a theoretically motivated non-perturbative function (see
  previous item) or simply with a given functional form, and
  extracting the phenomenological parameters. Ideally, one would fit
  only as many moments as the number of free parameters.
\item[c)] Checking the resulting shape in $x_E$ space against $e^+e^-$
  data. Good agreement might not be necessary if one is only
  interested in predicting hadronic cross sections\footnote{We pointed
    out before that $N$ moments around $N\sim 5$ are more important to
    this aim, and we can see from Fig.~\ref{resum:fig1} that fitting
    in $x_E$ space actually leads to a {\sl poorer} description of
    low-$N$ moment data. The magnitude of the discrepancy is in this
    case of the order of 10\%, in agreement with the observations made
    in Ref.~\cite{Nason:1999ta}.}, but it provides a good guidance as
  to how sensible the chosen non-perturbative parametrization is.
\item[d)] Evaluating the hadronic cross sections at the Tevatron or
  the LHC using the fitted non-perturbative form, together with the
  proper\footnote{It is of course important that the perturbative
    terms included at this stage in $D^{\rm ini}(\mu_{0F},m)$ match
    the ones employed when fitting the non-perturbative contribution.}
  $D^{\rm ini}(\mu_{0F},m)$, and a specific coefficient function.  A
  matched approach like the one presented in
  Ref.~\cite{Cacciari:1998it} can also be used at this stage, to
  improve the result at transverse momentum values of the order of the
  heavy quark mass or below.
\end{itemize}




\subsection{Analytic resummation and power
  corrections\protect\footnote{Contributing author: L. Magnea}}

A characteristic feature of resummations is the fact that they
highlight the inherent limitations of perturbation theory. In QCD,
these limitations are particularly severe, reflecting the complex
nonperturbative structure of the theory: the perturbative expansion
for any IR safe observable must be divergent, and furthermore not
summable even \`a la Borel, as a consequence of the presence of the
Landau pole in the infrared evolution of the running coupling. In the
case of IR safe quantities, this ambiguity in the perturbative
prediction can be shown to be suppressed by powers of the hard scale,
and it is interpreted as a signal of the occurence of nonperturbative
corrections of the same parametric size.

Soft gluon resummations always lead to expressions in which the Landau
pole appears explicitly. On the other hand, it should be kept in mind
that resummations at a fixed logarithmic accuracy do not in general
yield correct estimates for the size of the relevant power correction.
This was first shown in Ref.~\cite{Beneke:1995pq} in the case of the
inclusive Drell--Yan cross section. In that case, threshold
resummation carried out at NLL level suggests a power correction
proportional to $N \Lambda/Q$, where $N$ is the Mellin variable
conjugate to the soft gluon energy fraction in the CM frame; it turns
out, however, that this power correction is cancelled by the inclusion
of nonlogarithmic terms, which can be exactly evaluated in the large
$n_F$ limit. This observation lead the authors of
Ref.~\cite{Catani:1996yz} to introduce the ``minimal prescription''
for the evaluation of the partonic cross section in momentum space;
with this prescription, the inverse Mellin transform of the partonic
cross section is evaluated picking a contour of integration located to
the left of the Landau pole: as a consequence, the presence of the
pole does not affect the result, and information about power
corrections must be supplied independently.

It should be emphasized, however, that a more precise relationship
between threshold resummations and power corrections can still be
established, by fully exploiting the factorization properties of soft
and collinear radiation. The refactorization of the partonic cross
section proposed in Ref.~\cite{Sterman:1987aj}, in fact, is valid up
to corrections that are suppressed by powers of $N$: thus if the
functions involved are evaluated maintaining this accuracy, one can
expect that they should encode correctly at least the information
concerning power corrections of the form $N^p (\Lambda/Q)^m$ for the
leading values of $p$ and $m$, while corrections suppressed by extra
powers of $N$ might still be missed. This fact has been verified in
Ref.~\cite{Sterman:1999gz}, where the cancellation of the leading
power correction for the Drell--Yan cross section, established in
Ref.~\cite{Beneke:1995pq} in the large $n_F$ limit, was reproduced. 
In the more general setting of joint resummation, the same result was
obtained in Ref.~\cite{Laenen:2000ij}, and yet a different approach to
reproduce it is discussed in the next section~\cite{Gardi:2001di}.

One may wonder to what extent phenomenological results are affected by
the inclusion of power suppressed effects, or, in other words, by the
choice of different methods to regulate the Landau singularity. The
minimal prescription can be seen as one possible regulator, operating
at the level of the inverse Mellin transform.
Recently~\cite{Magnea:2000ss,Magnea:2001ge} it was observed that
another gauge invariant regularization procedure is given simply by
dimensional regularization: using a dimensionally continued version of
the running coupling, one may show that the Landau singularity moves
away from the integration contour for sufficiently large values of $d
= 4 - 2 \epsilon$. Resummed expression are then analytic functions of
the coupling and of $\epsilon$, and the Landau singularity appears as
a cut. To see how this comes about, recall that in $d = 4 - 2
\epsilon$, the $\beta$ function acquires $\epsilon$ dependence, so
that
\begin{equation}
\beta(\epsilon, \alpha_s) \equiv \mu \frac{\partial \alpha_s}{\partial \mu} = 
- 2 \epsilon \alpha_s + \hat{\beta} (\alpha_s)~, 
\label{resum:beta}
\end{equation}
where $\hat{\beta} (\alpha_s) = - b_0 \alpha_s^2/(2 \pi) + {\cal
  O}(\alpha_s^3)$.  As a consequence, the running coupling also
becomes dimension dependent.  At one loop,
\begin{equation}
\overline{\alpha} \left(\frac{\mu^2}{\mu_0^2}, \alpha_s(\mu_0^2), 
\epsilon \right) = 
\alpha_s(\mu_0^2) \left[\left(\frac{\mu^2}{\mu_0^2}\right)^\epsilon - 
\frac{1}{\epsilon} \left(1 - \left(\frac{\mu^2}{\mu_0^2}\right)^\epsilon
\right) \frac{b_0}{4 \pi}\alpha_s(\mu_0^2) \right]^{-1}~.
\label{resum:loalpha}
\end{equation}
It is easy to see that the running coupling in
Eq.~(\ref{resum:loalpha}) has a qualitatively different behavior with
respect to its four dimensional counterpart. First of all, it vanishes
as $\mu^2 \to 0$ for $\epsilon < 0$, as appropriate for infrared
regularization. This is a consequence of the fact that the one loop
$\beta$ function, for $\epsilon < 0$, has two distinct fixed points:
the one at the origin in coupling space is now a Wilson--Fisher fixed
point, whereas the asymptotically free fixed point is located at
$\alpha_s = - 4 \pi \epsilon/b_0$. Furthermore, the location of the
Landau pole becomes $\epsilon$ dependent, and it is given by
\begin{equation}
\mu^2 = \Lambda^2 \equiv Q^2 \left(1 + \frac{4 \pi \epsilon}{b_0 
\alpha_s(Q^2)} \right)^{-1/\epsilon}~.
\label{resum:lapo}
\end{equation}
The pole is not on the real axis in the $\mu^2$ plane, {\it i.e.} not
on the integration contour of typical resummed formulas, provided
$\epsilon < - b_0 \alpha_s(Q^2)/(4 \pi)$. One may then perform the
scale integrals, and get analytic expressions in which the ambiguity
due to nonperturbative phenomena shows up as a cut originated by the
Landau pole in the integrand. This idea was applied in
Ref.~\cite{Magnea:2000ss} to the resummed quark form factor, which can
be evaluated in terms of simple analytic functions since it is a
function of a single scale.  Here we will briefly illustrate the
application of the formalism to the simplest resummed partonic cross
sections (DIS and Drell--Yan), at the leading-log
level~\cite{Magnea:2001ge}.

\subsubsection{An example: LL threshold resummation for DIS and Drell--Yan}
\label{resum:disdy}

Resummation of threshold ($x \to 1$) logarithms, both for DIS and
Drell-Yan, was performed at NNL level
in~\cite{Sterman:1987aj,Catani:1989ne}.  A formulation closer to the
present approach was later given in~\cite{Contopanagos:1997nh}.
Applying the latter formalism, one may express the Mellin transform of
the DIS structure function $F_2(x, Q^2/\mu^2,
\alpha_s(\mu^2),\epsilon)$, at the LL level, as a simple exponential
\begin{equation}
F_2 \left(N, \frac{Q^2}{\mu^2}, \alpha_s(\mu^2), \epsilon \right) = 
F_2 \left(1 \right) ~\exp \left[ \frac{C_F}{\pi} \int_0^1 d z 
\frac{z^{N - 1} - 1}{1 - z} 
\int_0^{(1 - z) Q^2} \frac{d \xi^2}{\xi^2}~
\bar{\alpha} \left(\frac{\xi^2}{\mu^2},\alpha_s(\mu^2),\epsilon
\right) \right]~.
\label{resum:dis1} \nonumber
\end{equation}
Integration of the running coupling around $\xi^2 = 0$ generates the
leading collinear divergences, which can be factorized by subtracting
the resummed parton distribution
\begin{equation}
\psi \left(N, \frac{Q^2}{\mu^2}, \alpha_s(\mu^2), 
\epsilon \right) = \exp \left[ \frac{C_F}{\pi} \int_0^1 d z 
\frac{z^{N - 1} - 1}{1 - z} \int_0^{Q^2} \frac{d \xi^2}{\xi^2}~
\bar{\alpha} \left(\frac{\xi^2}{\mu^2},\alpha_s(\mu^2),\epsilon
\right) \right]~. \label{resum:psims} \nonumber
\end{equation}
The IR and collinear finite resummed partonic DIS cross section is
then defined by taking the ratio of Eqs.~(\ref{resum:dis1}) and
(\ref{resum:psims}), as $\widehat{F}_2 = F_2/\psi$.

The integration over the renormalization scale $\xi$ is easily
performed by means of the change of variables $d \xi/\xi = d
\alpha/\beta (\epsilon, \alpha)$, obtaining the compact expression
\begin{equation}
\widehat{F}_2 \left(N, \frac{Q^2}{\mu^2}, \alpha_s(\mu^2), 
\epsilon \right) = \widehat{F}_2 \left(1\right) \exp \left[
- \frac{4 \pi C_F}{b_0} \int_0^1 \frac{z^{N - 1} - 1}{1 - z}
\log \left( \frac{\epsilon + a((1 - z) 
Q^2)}{\epsilon + a(Q^2)} \right) \right]~, 
\label{resum:dis2} \nonumber
\end{equation}
where $a(\mu^2) = b_0 \alpha_s (\mu^2)/(4 \pi)$.
Eq.~(\ref{resum:dis2}) is manifestly finite, though ambiguous due to
the cut, as $\epsilon \to 0$.

The expected power correction can now be evaluated by taking the limit
$\epsilon \to 0$ with $\alpha_s(Q^2)$ fixed. This limit depends on how
the cut is approached, and the size of the ambiguity is easily
evaluated.  One finds
\begin{equation}
\delta \widehat{F}_2 \left( N, \alpha_s (Q^2) \right) \propto N 
\frac{\Lambda^2}{Q^2}~ \left( 1 + {\cal O} \left(\frac{1}{N}\right) + 
{\cal O} \left(\frac{\Lambda^2}{Q^2} \right) \right)~,
\end{equation}
as expected in DIS.

The resummed expression for the Drell-Yan partonic cross section, at
the leading $\log N$ level, is very similar. One finds
\begin{equation}
\widehat{\sigma}_{DY} \left(N, \frac{Q^2}{\mu^2}, \alpha_s(\mu^2), 
\epsilon \right) = \frac{\sigma_{DY} \left(N, \frac{Q^2}{\mu^2}, 
\alpha_s(\mu^2), \epsilon \right)}{\psi^2 \left(N, \frac{Q^2}{\mu^2}, 
\alpha_s(\mu^2), \epsilon \right)}~,
\end{equation}
where $\sigma_{DY}$ differs from $F_2$ because of a factor of two in
the exponent (due to the presence of two radiating quarks in the
initial state for the DY process), and because phase space dictates
that the upper limit of the scale integration should be $(1 - z)^2
Q^2$ instead of $(1 - z) Q^2$. Thus one finds
\begin{equation}
\delta \widehat{\sigma}_{DY} \left( N, \alpha_s (Q^2) \right) 
\propto N \frac{\Lambda}{Q}~
\left( 1 + {\cal O} \left(\frac{1}{N}\right) + {\cal O} \left(
\frac{\Lambda}{Q} \right) \right)~.
\label{resum:dypc}
\end{equation}
Eq.~(\ref{resum:dypc}) is the result that must be expected from a LL
resummation, in agreement with~\cite{Contopanagos:1994yq}. On the
other hand, one may note that the cancellation of this leading power
ambiguity, in the context of the refactorization formalism of
Ref.~\cite{Sterman:1987aj}, is of a purely kinematical nature: the
non--logarithmic terms needed for the cancellation arise by matching
the kinematic variable used in computing $\sigma_{DY}$ (energy
fraction in the CM frame) with the one used in defining the parton
distribution $\psi$ (light--cone momentum fraction). The cancellation
then clearly survives dimensional continuation.

\subsubsection{Directions for further work}
\label{resum:wor}

It is interesting to notice that Eq.~(\ref{resum:dis2}) and its
generalizations to the Drell--Yan process and to subleading logarithms
can be seen as the first ingredients of a prescription to handle the
Landau pole in phenomenological applications, similar but not
identical to the minimal prescription. Performing the $z$ integration
for a fixed negative value of $\epsilon < - a(Q^2)$ one finds an
analytic function whose inverse Mellin transform can be computed
without having to worry about the Landau pole at all. To illustrate
this fact, note that if one introduces the well--known
substitution~\cite{Catani:1989ne}
\begin{equation}
\int_0^1 d z \frac{z^{N-1} - 1}{1 - z} f(z) \rightarrow 
- \int_0^{1 - 1/N} d z \frac{1}{1 - z} f(z)~,
\label{resum:ca32}
\end{equation}
valid at LL level, the integration in the exponent can be performed
analytically, even for finite $\epsilon$. At LL level and in the
$\overline{MS}$ scheme, for example, one finds that the exponent of
the resummed partonic Drell--Yan cross section reads
\begin{eqnarray}
E_{DY}^{(\overline{MS})} \left( N, \alpha_s (Q^2) \right) & = & 
- \frac{8 C_F}{b_0} \left[\log N \log \left( -\frac{a(Q^2)}{\epsilon} 
N^{2 \epsilon} \right) - \epsilon \log^2 N \right. \label{resum:edyms} \\ 
& - & \left. \frac{1}{2 \epsilon} \left({\rm Li}_2 \left(1 + 
\frac{\epsilon}{a(Q^2)} \right) - {\rm Li}_2 \left(\frac{a(Q^2) + 
\epsilon}{a(Q^2) N^{2 \epsilon}} \right) \right) \right]~. \nonumber
\end{eqnarray}
Eq.~(\ref{resum:edyms}) is essentially the dimensional continuation of
the function $g_1$ of Ref.~\cite{Catani:1989ne}, in the
$\overline{MS}$ scheme. A similar expression can be derived in the DIS
scheme, and slightly more cumbersome ones can be written for the NLL
function $g_2$ in either scheme. It would be interesting to develop
such a `dimensional prescritpion' in an actual phenomenological
application.

To conclude, we have observed that dimensional continuation can be
used to regulate in a gauge invariant way the Landau singularity,
which characterizes resummed expressions for QCD amplitudes and cross
sections.  Applying the formalism to the DIS and Drell--Yan cross
sections reproduces the known results for the expected power
corrections, and the method may be turned into a practical
prescription to evaluate inverse Mellin transforms bypassing the
Landau pole. Possible interesting generalizations include applications
to existing resummations for event shapes in $e^+ e^-$ annihilation
and for the production of coloured final states in hadronic
collisions. One may safely say that the relationship between
resummations and power corrections has not yet been fully explored.
New approaches, such as the one of Ref.~\cite{Gardi:2001di}, or a
sharpening of old tools, may yet yield a deeper insight both in the
theory and in the phenomenology of perturbative QCD.


\subsection{Threshold Resummation and Power Corrections 
  in the Drell-Yan Process by Dressed Gluon
  Exponentiation\protect\footnote{Contributing author: E. Gardi}}


The Drell-Yan (DY) process, where a lepton pair is produced in a
hadronic collision $h_a + h_b\longrightarrow l^+l^- + X$, is a
classical example where initial-state radiation determines the cross
section. At the perturbative level, \beq \frac{d\sigma}{d
  Q^2}=\left(\frac{4\pi\alpha_{\QED}^2}{9Q^2\, s}\right)
\sum_{i,j}\frac{dx_i}{x_i}\,\frac{dx_j}{x_j}\,f_a(x_i,Q^2)\,f_b(x_j,Q^2)
\, w_{ij}(z,Q^2).  \end{equation} Here $f_a(x_i,Q^2)$ are the twist-two parton
distribution functions, $w_{ij}(z,Q^2)$ is the partonic cross section
and $z=Q^2/(p+\bar{p})^2$ where $p$ and $\bar{p}$ are the momenta of
the incoming quark and antiquark.

We consider the threshold region where the invariant mass of the
produced lepton pair, $Q^2$, approaches the total center of mass
energy $s$, so at the partonic level $z\longrightarrow 1$.
Conseqently, the total energy of QCD radiation in the final state is
strongly constrained, the physical scale for gluon emission $(1-z)Q$
is low and that multiple emission has a significant r\^ole.
Perturbatively, this implies that fixed-order calculations are
insufficient, and that large Sudakov logs, $\ln (1-z)$, must be
resummed to all
orders~\cite{Sterman:1987aj,Collins:1988ig,Catani:1989ne,Contopanagos:1997nh,Catani:1991rr,Vogt:2000ci}.
It is clear that close enough to the threshold non-perturbative
corrections are enhanced as
well~\cite{Qiu:1991xx,Contopanagos:1994yq,Korchemsky:1995is,Beneke:1995pq,Akhoury:1997pb}:
they appear as powers of~$1/(Q(1-z))$ rather than as powers of~$1/Q$.
Contrary to structure functions in deep inelastic scattering, the DY
cross section does not have an operator product expansion. The
analysis of power correction must therefore rely on perturbative
tools, namely on renormalons. Infrared renormalons reflect the
sensitivity of Feynman diargams to the behaviour of the coupling in
the infrared. This sensitivity manifests itself in the large order
behaviour of the perturbative expansion.  Assuming that power
corrections which are associated with renormalons dominate, the form
of the power corrections can be deduce from the lowers order Feynman
diagrams with an off-shell
gluon~\cite{Beneke:1995qe,Ball:1995ni,Dokshitzer:1996qm}.  In the DY
case the lowest order Feynman diagram with an off-shell
gluon~\cite{Beneke:1995pq} (or the leading term in the flavour
expansion) reflects sensitivity of the form $1/(Q^2(1-z)^2)$. The
absence of a correction $1/(Q(1-z))$ is very intriguing, particularly
because such corrections do appear in other QCD observables, e.g. in
event-shape variables~\cite{Dokshitzer:1996qm} which share with the DY
case the structure of leading and next-to-leading Sudakov logs. (In
fact, the similarity is superficial~\cite{Gardi:2001di}. The
difference between the pattern of power corrections, as well as
sub-leading Sudakov logs, originates in the fact that in DY the
radiation is from initial-state partons and the constrain is on the
total energy, whereas in event shapes the radiation is from
final-state partons and the constrain is on the invariant mass of the
jet.).  The main problem with the renormalon method, however, is that
higher order Feynman diagrams, that are sub-leading in the flavour
expansion may generate stronger corrections~$\sim 1/(Q(1-z))$. The
diagrams which might be relevant are purely
non-Abelian~\cite{Korchemsky:1996iq}: the emission of a (dressed)
gluon off a virtual gluon, which is exchanged between the initial
partons.  Currently, this question remains open and we do not address
it here.

Since non-perturbative corrections, and particularly those associated
with the running coupling, cannot be unambiguously separated from the
resummed perturbation theory, whenever power corrections are
non-negligible, renormalon resummation must be employed. Perturbative
terms are alway parametrically larger than the ambiguous power
correction and they admit a different functional form. Therefore, the
resummation cannot be replaced by parametrization of power
corrections, but only supplemented by it. A most striking
demonstration of this fact is
provided~\cite{Gardi:1999dq,Gardi:2001ny,Gardi:2002bg} by the analysis
of event shape variables, where not only the magnitude of the power
correction but also the extracted value of $\alpha_s$ crucially depend
on the resummation.  In particular, in the calculation of differential
cross sections, such as the DY, in the threshold region, a fixed
logarithmic accuracy is insufficient. Power accuracy can be reached
only if the Sudakov exponent itself is evaluated to such accuracy.
This is the aim of DGE~\cite{Gardi:2001ny,Gardi:2002bg,Gardi:2001di}.

Let us now briefly describe the application of DGE in the case of
DY~\cite{Gardi:2001di}. The first stage is to evaluate the cross
section at the level of a single gluon emission. The gluon is assumed
to be off-shell. The gluon virtuality $k^2\equiv \lambda Q^2$ provides
the argument of the running coupling in the renormalon calculation.
Being interested in logarithmically enhanced terms, and since such
terms originate in the case under cosideration from the phase-space
region where {\em all} components of the gluon momentum are small, the
soft approximation can be used.  The partonic cross section is,
\begin{eqnarray}
\label{resum:DY_cs1}
\left.w(z,Q^2)\right\vert_{\osg}\simeq
\frac{C_F \alpha_s}{2\pi}
\,\int_{\beta_1}^{\beta_2}
\,d\beta\,\left[
\frac{2}{\left(1-z-\beta\right)\beta}
\right],
\end{eqnarray}
where $\beta=k_{+}/p_{+}=2k\bar{p}/2p\bar{p}$ is the longitudinal
momentum fraction of the gluon in the direction of the emitting quark
($p=(p_{+},0,0)$) in the gauge $A_{+}=0$, and the phase-space limits
(see~\cite{Gardi:2001di}) are $\beta_{1,2}=(1-z+\lambda\pm\Delta)/2$,
with $\Delta\equiv \sqrt{(1-z)^2-2\lambda(1+z)+\lambda^2}$.

The leading order result in Eq.~(\ref{resum:DY_cs1}) is promoted to a
resummed one by integrating over the running coupling.  The Borel
representation of the Single Dressed Gluon (SDG) partonic cross
section is \begin{equation} \left.w(z,Q^2)\right\vert_{\SDG}=\frac{C_F
  \alpha_s}{2\pi}\,\int_0^{\infty}d{u}\,\exp\left(-{{u} \ln
    Q^2/\bar{\Lambda}^2}\right)\,B_{\SDG}(z,u)\,
\frac{\sin\pi{u}}{\pi{u}}\,A_B(u), \end{equation} where $A_B(u)=1$ for the
1-loop running coupling, which is used below as an example, and
\begin{eqnarray}
\label{resum:B_SDG_DY}
B_{\SDG}(z,u) &=&
\,\frac{4}{1-z}\,\int_{1}^{(1-\sqrt{z})^2} \,d\lambda\,\lambda^{-u-1} \,\\
& &+\,\int_{0}^{(1-\sqrt{z})^2} {d\lambda} \,\lambda^{-u-1} \,
\,\left[\frac{4}{\sqrt{(1-z)^2-2\lambda(1+z)+\lambda^2}}-\frac{4}{1-z}
\right]. \nonumber
\end{eqnarray}
The modification of the lower integration limit over $\lambda$ for the
singular~$1/({1-z})$ term corresponds to factorization of gluons with
virtuality smaller than $Q^2$ into the parton distribution factors
$f_{a,b}(x_i,Q^2)$.

In Eq.~(\ref{resum:B_SDG_DY}) the phase-space limits are exact: the
upper integration limit was deduced from the condition $\beta_{2}\leq
\beta_1$, yielding~\hbox{$\Delta({\lambda_{\max}})=0$}, and therefore
$\lambda_{\max}=(1-\sqrt{z})^2$.  Since the integration over $\lambda$
is restricted to $\lambda \ll 1-z$, one can replace the integration
limit by $\lambda_{\max}\simeq (1-z)^2/4$ and approximate the
$\lambda$-dependent denominator in Eq.~(\ref{resum:B_SDG_DY}) by
$\sqrt{(1-z)^2-4\lambda}$. We thus find
\begin{eqnarray}
\label{resum:B_x_DY}
B_{\SDG}(z,u) &=&\frac{4}{1-z}\, \frac1u\,\left[
1-\,\frac{\sqrt{\pi}\,\Gamma(1-u)}{\Gamma(\frac12
-u)}\,\left(\frac{1-z}{2}\right)^{-2u}\right].
\end{eqnarray}
As was stressed in~\cite{Beneke:1995pq}, a further approximation where
also the term $4\lambda$ is neglected does not influence the leading
and next-to-leading Sudakov logs, however, since the integral in
Eq.~(\ref{resum:B_SDG_DY}) extends to $\lambda_{\max}\simeq
(1-z)^2/4$, it is not legitimate for power correction analysis.

At the second step of the calculation, the SDG result is
exponentiated: under the assumption of independent emission, the cross
section with any number of gluons can be written, in Mellin space,
$w(N,Q^2)\equiv \int_0^1 dz\, z^{N-1}\, w(z,Q^2) $, as
\begin{eqnarray}
\label{resum:DY_cs_exp}
\left.\ln w(N,Q^2)\right\vert_{\DGE}&=&\int_0^1 dz \left(z^{N-1}-1\right)\left. w(z,Q^2)\right\vert_{\SDG}\nonumber \\
&=&\frac{C_F \alpha_s}{2\pi}\,\int_0^{\infty}d{u}\,\exp\left(-{{u} \ln
Q^2/\bar{\Lambda}^2}\right)\,B_{N}(u)\,
 \frac{\sin\pi{u}}{\pi{u}}\,A_B(u),
\end{eqnarray}
where the Borel function is \begin{equation}
\label{resum:BN_DY} 
B_N(u)=\int_0^1 dz \left(z^{N-1}-1\right)B_{\SDG}(z,u)=2\left(e^{2u\ln
    N}-1\right)\Gamma(-u)^2-\frac4u\ln N.  \end{equation}

Eqs.~(\ref{resum:DY_cs_exp}) and (\ref{resum:BN_DY}) summarize our
final result for the DGE of the DY cross section. It contains both
perturbative and non-perturbative information. At the perturbative
level the exponent can be written in the standard way, in terms of
functions which sum all the contributions at a fixed logarithmic
accuracy, \begin{equation} \ln
\left.w(N,Q^2)\right\vert_{\DGE}=\frac{C_F}{2\beta_0}
\sum_{k=1}^{\infty}{{\bar{A}(Q^2)}^{k-2}}\,\,
f_{k}\!\left({\bar{A}}(Q^2)\ln N\right),
\label{resum:log_expansion}
\end{equation} where $\bar{A}(Q^2)\equiv \bar{\alpha}_s(Q^2)\beta_0/\pi$ and
1-loop running coupling was assumed.  The first two~functions
\begin{eqnarray}
\label{resum:f_k_12_DY}
f_{1}(\xi)&=& 2\,(1-2\xi)\,\ln(1 - 2\,\xi ) + 4\,\xi    \nonumber \\
f_{2}(\xi)&=& - 4\,\gamma \,\ln(1 - 2\,\xi )
\end{eqnarray}
are well
known~\cite{Sterman:1987aj,Collins:1988ig,Catani:1989ne,Contopanagos:1997nh,Catani:1991rr,Vogt:2000ci}.
We stress that in spite of the fact that the actual calculation is
done in the large $\beta_0$ limit, the DGE reuslt is {\em exact} to
next-to-leading logarithmic accuracy, provided that the 1-loop
coupling is replaced by the 2-loop coupling in the ``gluon
bremsstrahlung''
scheme~\cite{Catani:1991rr,Gardi:2001ny,Gardi:2002bg,Gardi:2001di}.
The large $\beta_0$ contribution to the higher order functions is
given by
\begin{eqnarray*}
\begin{array}{ll}
f_{3}(\xi)={1.33}/{(1-2\,\xi )} +6.58\,\xi \\
f_{4}(\xi)={2.12}/{(1-2\,\xi)^{2}}   \\
f_{5}(\xi)={4.00}/{(1 -2\,\xi)^{3}} - 19.48\,\xi  \\
f_{6}(\xi)={11.59}/{(1-2\,\xi)^{4}}  \\
f_{7}(\xi)={48.42}/{(1-2\,\xi)^{5}} +91.56\,\xi  \\
f_{8}(\xi)={238.80}/{(1-2\,\xi)^{6}}  \\
f_{9}(\xi)={1438.66}/{(1-2\,\xi)^{7}} - 527.14\,\xi  \\
f_{10}(\xi)={10078.0}/{(1-2\,\xi)^{8}}.
\end{array}
\end{eqnarray*}
We see that sub-leading logs are enhanced by factorially increasing
coefficients as well as an increasing singularity at $\xi=1/2$.
Therefore, truncation of this expansion has a significant impact on
the result. Such truncation would also induce renormalization scale
dependence. When power accuracy is required the sum, up to the minimal
term, or alternatively the Borel integral in
Eq.~(\ref{resum:DY_cs_exp}) must be computed.

Of course, due to the renormalon singularities of the Borel function
(\ref{resum:BN_DY}) the Borel integral (\ref{resum:DY_cs_exp}), as
written, is ill-defined. A prescription must be given.  Since the
full, non-perturbative result is unambiguous, power corrections can be
deduced from this ambiguity.  The first, crucial conclusion is that
the power corrections in the threshold region should exponentiate
together with the perturbative sum, so that the correction factorizes
in Mellin space: $w(N,Q^2) \longrightarrow w(N,Q^2) \, w_{\rm
  NP}(N,Q^2)$.  This is fully consistent with the formulation of the
resummation in terms of an evolution equation for a Wilson-line
operator~\cite{Korchemsky:1995is}, where the non-perturbative
correction appears the initial condition for the evolution.  The
singularities of the Borel integrand are located at {\em integer}
values of $u$.  Equation~(\ref{resum:BN_DY}) has double poles at all
integers, but the $\sin(\pi u)/\pi u$ factor of in
Eq.~(\ref{resum:DY_cs_exp}) leaves only simple poles. As observed in
Ref.~\cite{Beneke:1995pq}, this singularity structure implies that the
leading power correction at large $Q^2$ (and not too large $z$) is
$1/Q^2(1-z)^2$. Closer to $z=1$ sub-leading power corrections of the
form $1/(Q^2(1-z)^2)^n$, where $n$ is an integer, become important.
The non-perturbative correction factor in Mellin space
is~\cite{Gardi:2001di}, \begin{equation} w_{\rm
  NP}(N,Q^2)=\exp\left\{\sum_{n=1}^{\infty}\frac{C_F\,(-1)^n}{{\beta_0}\,n\,
    (n!)^2}\,\omega_n
  \,\left(\frac{{\bar{\Lambda}}^2N^2}{Q^2}\right)^n\right\}, \end{equation}
where $\omega_n$ are non-perturbative parameters.\\
{\bf Acknowledgements:} It a pleasure to thank Yuri Dokshitzer,
Gregory Korchemsky and Douglas Ross for very useful discussions.



\subsection{NNLO expansions of threshold-resummed heavy quark cross
  sections\protect\footnote{Contributing authors: N. Kidonakis, E.
    Laenen, S. Moch, R. Vogt}}


Long- and short-distance dynamics in QCD for inclusive hadronic
hard-scattering cross sections are factorized into universal,
non-perturbative parton distribution functions and fragmentation
functions, and perturbatively calculable hard scattering functions.
Remnants of long-distance dynamics in a hard scattering function can
become large in regions of phase space near partonic threshold and
dominate higher order corrections.  Such Sudakov corrections assume
the form of distributions that are singular at partonic threshold.
Threshold resummation organizes these double-logarithmic corrections
to all orders, thereby presumably extending the predictive power of
QCD to these phase space regions.  One use for resummed cross sections
is to provide, upon expansion in $\alpha_s$, estimates of finite
higher order corrections that are not yet known exactly. Here we
discuss next-to-next-to-leading order (NNLO) estimates for
double-differential heavy quark hadroproduction cross sections.  These
are based on expansions of their next-to-leading logarithmic resummed
versions
\cite{Kidonakis:1996aq,Kidonakis:1997gm,Laenen:1998qw,Kidonakis:2000ui,Kidonakis:2001nj}.

\subsubsection{Kinematics and threshold-singular functions}

The definition of the threshold depends on the observable to be
resummed. For double-differential cross sections various choices are
possible. In one-particle inclusive (1PI) kinematics, in our case
defined by the partonic kinematics ($ij =q\Bar{q},gg$),
\begin{equation}
\label{resum:eq:4}
 i(k_1) + j(k_2) \rightarrow {\rm{Q}}(p_1) +
X'[\Bar{\rm{Q}}](p_2')\,,
\end{equation}
and the corresponding invariants $s=2k_1\cdot k_2,\, t_1 = -2k_2\cdot
p_1,\, u_1 = -2k_1\cdot p_1,\; s_4 \equiv s+t_1+u_1$, the threshold
condition is
\begin{equation}
  \label{resum:eq:1}
s_4 =  0\,.
\end{equation}
The singular functions organized by threshold resummation are the
plus-distributions
\begin{equation}
  \label{resum:eq:2}
  \left[{\ln^{l}(s_4/m^2)\over s_4}\right]_+\,.
\end{equation}
Pair-invariant mass (PIM) kinematics is defined by
\begin{equation}
  \label{resum:eq:3}
   i(k_1) +j(k_2) \rightarrow {\rm{Q}}{\Bar{\rm{Q}}}(p') + X'(k)\,,
\end{equation}
% with the variables $ p'^2 = M^2$ and $\cos\theta$, where $\theta$ is
with the variables $ p^{\prime \, 2} = M^2$ and $\cos\theta$, where $\theta$ is
the polar scattering angle in the partonic c.m. frame. The threshold
is set by
\begin{equation}
  \label{resum:eq:5}
   1-\frac{M^2}{s} \equiv 1-z = 0
\end{equation}
with the corresponding singular functions
\begin{equation}
  \label{resum:eq:6}
  \left[{\ln^{l}(1-z)\over 1-z}\right]_+\,.
\end{equation}
At threshold,
\begin{equation}
  \label{resum:eq:11}
t_1 = - \frac{M^2}{2} \left( 1 - \beta_M\, {\rm{cos}} \theta \right)\,,\qquad
u_1 = - \frac{M^2}{2} \left( 1 + \beta_M\, {\rm{cos}} \theta \right)\, ,
\end{equation}
with $\beta_M=\sqrt{1-4m^2/M^2}$.  We denote corrections as leading
logarithmic (LL) if $l=2i+1$ in Eqs.~(\ref{resum:eq:2}) and
(\ref{resum:eq:6}) at order $O(\alpha^{i+3}_s),\; i=0,1,\ldots$, as
next-to-leading logarithmic (NLL) if $l=2i$, etc.  Threshold
resummation is best performed in moment space, defined by the Laplace
transform with respect to $w_K$,
\begin{equation} {\tilde f}(N) = \int\limits_0^\infty dw_K
\,e^{-N w_K} f(w_K)\,,
\label{resum:laplacetfm}
\end{equation} 
where $w_{\opi} = s_4/m^2$ and $w_{\pim} = 1-z$.  In moment space the
singular functions become linear combinations of $\ln^k(\tilde{N})$
with $k\leq l+1$ and $\tilde{N}=N\exp(\gamma_E)$.

The NNLO double-differential partonic cross sections for which we
derived approximate results are
\begin{equation}
  \label{resum:eq:7}
 (\mathrm{\opi}):\; s^2 \frac{d^2 \sigma_{ij}(t_1,u_1,s_4)}
{dt_1\,du_1},\quad
 (\mathrm{\pim}):\;
  s \frac{d^2 \sigma_{ij}(s,M^2,\cos\theta)}{dM^2\,d\cos\theta}\,.
\end{equation}
We can only sketch the derivation here and refer to
\cite{Kidonakis:2001nj} for all details. The resummed
double-differential partonic cross section has the following
functional form in moment space (here in 1PI kinematics, for the PIM
result use Eq.~(\ref{resum:eq:11}) and multiply the right hand side by
$\beta_M/2$):
\begin{eqnarray}
\label{resum:eq:9}
&& \hspace{-5mm}
s^2 \frac{d^2 \tilde\sigma^{\rm res}_{ij}(t_1,u_1,N)}
{dt_1\,du_1}
\,=\,{\rm Tr}\Bigg\{
H_{ij}(m^2,m^2) \\ 
\nonumber & &\hspace{0mm} \times
{\rm \bar{P}} \exp\left[\int_m^{m/N} {d\mu'\over\mu'} 
(\Gamma^{ij}_S)^{\dagger}\left(\alpha_s(\mu^{\prime})\right)\right]
{\tilde S}_{ij}\!\left(1\right)
{\rm P} \exp\left[\int_m^{m/N} {d\mu'\over\mu'} 
\Gamma^{ij}_S\left(\alpha_s(\mu^{\prime})\right)\right] \Bigg\} \\
\nonumber & & \times
\exp\left(\tilde{E}_{i}(N_u,\mu,\mu_R)\right)\, 
\exp\left(\tilde{E}_{j}(N_t,\mu,\mu_R)\right)\,\, \exp\Bigg\{ 2\,
\int\limits_{\mu_R}^{m}{d\mu'\over\mu'}\,\, \Bigl(
\gamma_i\left(\alpha_s(\mu^{\prime})\right) +
\gamma_j\left(\alpha_s(\mu^{\prime})\right) \Bigr) \Bigg\}\, ,
\end{eqnarray}
where any $t_1$ andf $u_1$ dependence is suppressed.  The indicated
trace is in the space spanned by tensors that can couple the $SU(3)$
representations of the partons in (\ref{resum:eq:4}) to singlets:
$2$-dimensional for the $q\Bar{q}$ and $3$-dimensional for the $gg$
channel.  ($\rm \bar{P}$) $\rm P$ refers to (anti)-path-ordering in
$\mu'$.  The two-loop expansion of $\exp(\tilde{E}_{i})$ may be
written schematically as
\begin{eqnarray}
\label{resum:eq:8}
\exp(\tilde{E}_{i}(N_u,\mu,m))
\simeq 1 + \frac{\alpha_s}{\pi}\left(\sum_{k=0}^2 C^{i,(1)}_{k}\ln^k(N_u)  \right)
+ \left(\frac{\alpha_s}{\pi}\right)^2\left(
\sum_{k=0}^4 C^{i,(2)}_{k}\ln^k(N_u)
\right)
+\ldots
\end{eqnarray}
The coefficients $C^{i,(n)}_k$ are given in
Ref.~\cite{Kidonakis:2001nj}.  Similar expansions can be given for the
other $N$-dependent factors in Eq.~(\ref{resum:eq:9}). Momentum space
expressions to NNLL-NNLO are obtained by gathering together all terms
at ${\cal O}(\alpha_s^3)$ and ${\cal O}(\alpha_s^4)$, performing an
inverse Laplace transform, and matching the $N$-independent $H_{ij}$
and $\tilde{S}_{ij}(1)$ coefficients to known exact ${\cal
  O}(\alpha_s^3)$ results. The resulting, approximate, NNLO cross
sections in Eq.~(\ref{resum:eq:7}) have NNLL accuracy in the sense
stated below Eq.~(\ref{resum:eq:11}).  They are rather long and are
given explicitly in Ref.~\cite{Kidonakis:2001nj}.


\subsubsection{Numerical results}

We have so far performed numerical studies of these results for the
\textit{inclusive} partonic cross sections $\sigma_{ij}(s,m^2,\mu^2)$.
We must then attribute any differences in integrating either the PIM
or 1PI results to an ambiguity of our estimates. It is convenient to
express these inclusive partonic cross sections in terms of
dimensionless scaling functions $f^{(k,l)}_{ij}$ that depend only on
$\eta = s/4m^2 -1$, as follows:
\begin{eqnarray}
\label{resum:scalingfunctions}
\sigma_{ij}(s,m^2,\mu^2) &=& \frac{\alpha_s^2(\mu)}{m^2}
\sum\limits_{k=0}^{\infty} \,\, \left( 4 \pi \alpha_s(\mu) \right)^k
\sum\limits_{l=0}^k \,\, f^{(k,l)}_{ij}(\eta) \,\,
\ln^l\left(\frac{\mu^2}{m^2}\right) \, .
\end{eqnarray} 
From our results for the double-differential cross section we have
constructed LL, NLL, and NNLL approximations to $f^{(k,l)}_{ij}(\eta)$
for $k\leq 2,\, l\leq k$, and for both the $q{\Bar{q}}$ and $gg$
channel. For $k=1$ exact results are
known~\cite{Nason:1988xz,Beenakker:1989bq,Beenakker:1991ma}.  For
$k=2$ and $l=1,2$ we have derived exact results using renormalization
group methods. Our best NNLO estimate consists of all exactly known
scaling functions, together with the NNLL estimate of
$f^{(2,0)}_{ij}$.

We now show a few representative results. To exhibit the relevance of
threshold approximations we write the inclusive hadronic cross section
as a pointwise product in $\eta$ of the partonic cross section and the
parton flux $\Phi_{ij}$
\begin{eqnarray}
\label{resum:eq:10}
\sigma_{h_1h_2}(S,m^2) &=&  \sum\limits_{i,j = q,{\Bar{q}},g} \,\,
\int_{-\infty}^{\log_{10}(S/4m^2-1)} d\log_{10}\eta \, \frac{\eta}{1+\eta} 
\ln(10) \, \Phi_{ij}(\eta,\mu^2)\,\,
\sigma_{ij}(\eta,m^2,\mu^2)\, .
\end{eqnarray}
We use the two-loop expression of $\alpha_s$ and the CTEQ5M parton
distributions \cite{Lai:1999wy} at LO, NLO, and NNLO.

As an example, in Fig.~\ref{resum:fig:nnlo-part-flux} we show the
$q\Bar{q}$ NLO scaling function to various accuracies and the
corresponding partonic flux for $t\Bar{t}$ production at the Tevatron,
showing the values of $\eta$ where the integral in
Eq.~(\ref{resum:eq:10}) gets the most weight.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIGURE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htp]
\begin{center}
  \includegraphics[width=5cm,height=7.5cm,angle=-90]{./fqq-1PI.epsi}
  \includegraphics[width=5cm,,height=7.5cm,angle=-90]{./phiqq.epsi}
\end{center}
\caption{(a) The $\eta$-dependence of the scaling functions 
  $f^{(k,0)}_{q{\Bar{q}}}(\eta),\;k=0,1$ in the
  ${\Bar{\rm{MS}}}$-scheme and 1PI kinematics.  We show the exact
  results for $f^{(k,0)}_{q{\Bar{q}}},\;k=0,1$ (solid lines), the LL
  approximation to $f^{(1,0)}_{q{\Bar{q}}}$ (dotted line), the NLL
  approximation to $f^{(1,0)}_{q{\Bar{q}}}$ (dashed line) and the NNLL
  approximation to $f^{(1,0)}_{q{\Bar{q}}}$ (dashed-dotted line).  (b)
  The $q\Bar{q}$ CTEQ5M parton flux for $t\Bar{t}$ production at the
  Tevatron (upper three curves, $\sqrt{S}=1.8$ TeV and $m=175$ GeV)
  and for $b\Bar{b}$ production at HERA-B (lower three curves
  $\sqrt{S}=41.6$ GeV and $m=4.75$ GeV).  The curves correspond to
  $\mu=m$ (solid curves), $\mu=m/2$ (dotted curves), and $\mu=2m$
  (dashed curves). }
\label{resum:fig:nnlo-part-flux}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The partonic results show that the NNLL approximation at NLO
approximates the exact result very well. In
Ref.~\cite{Kidonakis:2001nj} we have studied the quality of the NNLL
approximation more extensively and find this conclusion to hold to
NNLO and for both channels. The other plot in
Fig.~\ref{resum:fig:nnlo-part-flux} shows that the flux selects
partonic processes that are reasonably close to threshold, making our
approximations phenomenologically relevant.

While QCD corrections to the top quark inclusive cross section at the
Tevatron are fairly modest, our NNLO threshold estimates help to
substantially reduce factorization scale dependence, as we show in
Fig.~\ref{resum:fig:nnlo-ttbar-scale}.  The latter is expected on
general grounds \cite{Oderda:1999im,Sterman:2000pu}.
\begin{figure}[htp]
\begin{center}
  \includegraphics[width=6cm,height=7.5cm,angle=-90]{mu-1PI.epsi}
  \includegraphics[width=6cm,height=7.5cm,angle=-90]{mu-PIM.epsi}
\end{center}
\caption{The $\mu$-dependence of the top quark cross section 
  at the Tevatron with $\sqrt{S}=1.8$ TeV and $m=175$ GeV for the sum
  of the $q \bar{q}$ and $gg$ channels in the $\overline{\rm MS}$
  scheme. We show the Born (upper solid line at small $\mu/m$) and the
  exact NLO (lower solid line at small $\mu/m$) cross sections, the
  1PI approximate NNLL-NNLO cross section (dashed line) and the 1PI
  NNLO estimate with only $f^{(2,0)}_{q{\Bar{q}}}$ and
  $f^{(2,0)}_{gg}$ NNLL approximate (dashed-dotted line).  (b) The
  same as (a) in PIM kinematics.}
\label{resum:fig:nnlo-ttbar-scale}
\end{figure}
We have checked that using presently available, almost NNLO densities
\cite{Martin:2000gq} lead to very similar results. Based on our
approximations we provide, by averaging the results from PIM and 1PI
kinematics, NNLL-NNLO estimates for the following inclusive top cross
sections
\begin{equation}
  \label{resum:eq:34}
  \sigma_{t\bar{t}}(1.8\,\mathrm{TeV})
 = 5.8
\pm 0.4 
\pm 0.1 \;\;\mathrm{pb}\,,
\end{equation}
\begin{equation}
\label{resum:eq:35}
  \sigma_{t\bar{t}}(2.0\,\mathrm{TeV})
 = 8.0  
\pm 0.6
\pm 0.1 \;\;\mathrm{pb}\,.
\end{equation}
For the HERA-B bottom cross section, we find
\begin{equation}
\label{resum:eq:36}
  \sigma_{b\bar{b}}(41.6\,\mathrm{GeV})
 = 30 
\pm  8 
\pm 10 \;\;\mathrm{nb}\,.
\end{equation}
The first set of errors indicates the kinematics ambiguity while the
second is an estimate of the scale uncertainty.  Note that the scale
uncertainty for top production is now considerably smaller than the
kinematics uncertainty.


\subsection{High energy resummation\protect\footnote{Contributing
    authors: R. Ball, S. Forte}}



\subsubsection{Perturbative QCD at HERA}

QCD has been tested at HERA~\cite{Forte:1999kh,Chekelian:2001pi} over
the last several years to an accuracy which is now comparable to that
of tests of the electroweak sector at LEP: perturbative QCD turns out
to provide an embarrassingly successful description of the HERA data,
even in kinematic regions where simple fixed--order perturbative
predictions should fail. This success is most strikingly demonstrated
by the comparison with the data of the scaling violations of structure
functions predicted by the QCD evolution
equations~\cite{Chekanov:2001qu,Adloff:2000qk}: the data agree with
the theory over five orders of magnitude in both $x$ and $Q^2$.

The significance of this sort of result is somewhat obscured by the
need to fit the shape of parton distributions at a reference scale,
which might suggest that deviations from the predicted behaviour could
be accommodated by changing the shape of the parton distribution.
However, this is not true because of the predictive nature of the QCD
result: given the shape of partons at one scale, there is no freedom
left to fit the data at other scales. This predictivity is
particularly transparent in the small $x$ region, where the
fixed--order QCD result actually becomes asymptotically independent of
the parton distribution, apart from an overall normalization. Indeed,
the data for $\ln F_2$ plotted versus the variable
$\sigma\equiv\ln{x_0\over x}\ln {\as(Q^2_0)\over \as(Q^2)}$ are
predicted to lie on a straight line, with universal slope
$2\gamma=12/\sqrt{33-2 n_f}$ (double asymptotic
scaling~\cite{Ball:1994du,DeRujula:1974rf}).  The predicted scaling is
spectacularly borne out by the data, as shown in
Fig.~\ref{resum:fig:bf1}: in fact, the data are now so accurate that
one can see the change in slope when passing the $b$ threshold, and
indeed double scaling is only manifest if one separates data in the
regions where $\alpha_s$ runs with $N_f=4$ from those with
$N_f=5$.\footnote{The fact that the observed slope is somewhat smaller
  than the predicted one, especially at low $Q^2$, is due to NLO
  corrections~\cite{Forte:1995vs} as well as corrections due to the
  ``small'' eigenvalue of perturbative
  evolution~\cite{Mankiewicz:1997sd}.}  Equally good agreement with
fixed--order perturbation theory is seen when considering less
inclusive observables.
\begin{figure}
%\begin{center}
  \includegraphics[width=0.48\linewidth,clip]{scal4.ps}
  \includegraphics[width=0.45\linewidth,clip]{scal5.ps}
\caption{Double asymptotic scaling of the H1 data~\cite{Adloff:2000qk}. The
  scaling variable $\sigma\equiv\ln(x_0/ x)\ln (\as(Q_0^2)/\as(Q^2))$
  is defined with $x_0=0.1$, $Q_0=1~{\rm GeV}$; the rescaling factor
  $R_F$ is as in Ref.~\cite{Ball:1994du}. Only data with $\rho\ge 1$,
  $\sigma\ge 1$, $Q^2\ge4~{\rm GeV}^2$; $x\le0.03$ are plotted. Left:
  $Q^2\le m_b^2$; right: $Q^2> m_b^2$. The straight line is the
  asymptotic prediction.}
\label{resum:fig:bf1}
%\end{center}
\end{figure}

This agreement of the data with fixed--order perturbative QCD
computations is very surprising, in that the perturbative expansion
receives contributions of order $\as\ln{1\over x}$ so one would expect
higher--order corrections to be non--negligible whenever
$\as\ln{1\over x}\gsim 1$, i.e. in most of the HERA region. As is well
known, the resummation of leading $\ln{1\over x}$ (LLx) contributions
to gluon--gluon scattering, and thus to a wide class of hard
processes, including small $x$ scaling violations of structure
functions, is accomplished by means of the BFKL evolution
equation~\cite{Lipatov:1976zz,Fadin:1975cb,Kuraev:1976ge}. Matching
the BFKL approach to standard perturbative computation, however, is
nontrivial~\cite{Ball:1995vc,Ellis:1995gv}, while the BFKL equation
itself seems to be unstable towards the inclusion of higher order
corrections~\cite{Fadin:1998py}. Hence, the main problem in
understanding HERA physics, i.e. perturbative QCD at small $x$ is that
of establishing ``consistency of the BFKL approach with the more
standard DGLAP~\cite{Altarelli:1977zs,Gribov:1972ri} evolution
equations''~\cite{McLerran:2001sr}, which embody the leading $\ln Q^2$
(LLQ$^2$) resummation on which perturbative QCD is based.  This
problem is now
solved~\cite{Ball:1999sh,Altarelli:1999vw,Ciafaloni:1999yw}, and on
the basis of this solution it is possible to combine the available
information on perturbation theory at small $x$, and use it to explain
the unexpected success of fixed--order calculations.

\subsubsection{Duality}

Let us for definiteness consider the prototype problem of the
description of small $x$ scaling violations of parton distributions.
For simplicity, consider the case of a single parton distribution
$G(x,Q^2)$, which can be thought of as the dominant eigenvector of
perturbative evolution. Scaling violations are then described by the
Altarelli-Parisi equation satisfied by $G(x,Q^2)$, and thus summarized
by the Altarelli--Parisi splitting function
$P(x,\as)$~\cite{Altarelli:1977zs}.

The basic result which allows the study of scaling violations at small
$x$ is {\it duality} of perturbative
evolution~\cite{Ball:1997vf,Altarelli:1999vw,Altarelli:2000mh},
namely, the fact that, because the Altarelli-Parisi equation is an
integro--differential equation in the two variables $t\equiv\ln
Q^2/\Lambda^2$ and $\xi\equiv 1/x$, it can be equivalently cast in the
form of a differential equation in $t$ satisfied by the $x$--Mellin
transform
\begin{equation}
G(N,t)=\int^{\infty}_{0}\! d\xi\, e^{-N\xi}~G(\xi,t),
\label{resum:nmel}
\end{equation}
or a differential equation in $\xi$ satisfied by the $Q^2$--Mellin
transform
\begin{equation} 
G(\xi,M)=\int^{\infty}_{-\infty}\! dt\, e^{-Mt}~G(\xi,t)
\label{resum:mmel} 
\end{equation} 
of the parton distribution.  The pair of dual evolution equations are
\begin{eqnarray}
\frac {d}{dt}G(N,t)&=&\gamma(N,\as)~G(N,t)
\label{resum:tevol}\\
\frac {d}{d\xi}G(\xi,M)&=&\chi(M,\as)~G(\xi,M),
\label{resum:xevol}
\end{eqnarray}
where Eq.~(\ref{resum:tevol}) is the standard renormalization--group
equation, with anomalous dimension $\gamma(N,t)$, and
Eq.~(\ref{resum:xevol}) is essentially the BFKL equation. Duality is
the statement that the solutions of these two equations coincide to
all perturbative orders, up to power suppressed corrections, provided
their kernels are related by
\begin{equation}
\chi(\gamma(N,\as),\as)=N.\label{resum:dual}
\end{equation}
This means that the BFKL and Altarelli-Parisi equations describe the
same physics: it is the choice of the kernel to be used in the
evolution equation which determines which is the large scale which is
resummed.  We can then discuss the construction and resummation of the
kernel irrespective of the specific evolution equation where it is
used, with the understanding that the kernel can be equivalently
viewed as a $\gamma(N,\as)$ or a $\chi(M,\as)$, the two being related
by Eq.~(\ref{resum:dual}). Before doing this, we sketch how duality
can be proven order by order in perturbation theory.

\subsubsection{Fixed coupling}

Perturbative duality is most easy to prove when the coupling does not
run, since in this case the two scales $t$ and $\xi$ appear in the
Altarelli--Parisi equation in a completely symmetric way. It is
convenient to introduce the double--Mellin transform $G(N,M)$ of the
parton distribution. The solution to the Altarelli--Parisi equation in
$M,N$ space has the form (which can be e.g. obtained by performing an
$M$--mellin transform Eq.~(\ref{resum:mmel}) of the solution to the
renormalization--group Eq.~(\ref{resum:tevol}))
\begin{equation}
G(N,M)=\frac{G_0(N)}{M-\gamma(N,\as)},
\label{resum:tsoln}
\end{equation}
where $G_0(N)$ is a boundary condition at a reference scale $\mu^2$.

The inverse Mellin transform of Eq.~(\ref{resum:tsoln}) coincides with
the residue of the simple pole in the $M$ plane of $e^{t M} G(N,M)$,
and thus its scale dependence is entirely determined by the location
of the simple pole of $G(N,M)$~(\ref{resum:tsoln}) , namely, the
solution to the equation
\begin{equation}
M=\gamma(N,\as). 
\label{resum:pole}
\end{equation} 
The pole condition Eq.~(\ref{resum:pole}) can be equivalently viewed
as an implicit equation for $N$: $N=\chi(M,\as)$, where $\chi$ is
related to $\gamma$ by Eq.~(\ref{resum:dual}). Hence, the function
\begin{equation}
G(N,M)=\frac{F_0(N)}{N-\chi(M,\as)},
\label{resum:xisoln}
\end{equation}
corresponds to the same $G(t,x)$ as Eq.(\ref{resum:tsoln}), because
the location of the respective poles in the $M$ plane are the same,
while the residues are also the same, provided the boundary conditions
are matched by
\begin{equation}
 G_0(N)=-
{ F_0(\gamma(\as,N))\over\chi^\prime(\gamma(\as,N))}.
\label{resum:bcmatch}\end{equation}

Eq.~(\ref{resum:xisoln}) is immediately recognized as the $N$-Mellin
of the solution to the evolution equation~(\ref{resum:xevol}) with
boundary condition $F_0(M)$ (at some reference $x=x_0$), which is what
we set out to prove.  In general, the analytic continuation of the
function $\chi$ defined by Eq.~(\ref{resum:dual}) will be such that
Eq.~(\ref{resum:pole}) has more than one solution (i.e. $\gamma$ is
multivalued). In this case, poles further to the left in the $M$ plane
correspond to power--suppressed contributions, while poles to the
right correspond to contributions beyond perturbation theory (they do
not contribute when the inverse $M$--Mellin integral is computed along
the integration path which corresponds to the perturbative region).

It is easy to see that upon duality the leading--order
$\chi=\as\chi_0$ is mapped onto the leading singular
$\gamma=\gamma_s(\as/N)$, and conversely the leading--order
$\gamma=\as\gamma_0$ is mapped onto the leading singular
$\chi=\chi_s(\as/M)$. In general, the expansion of $\chi$ in powers of
$\as$ at fixed $M$ is mapped onto the expansion of $\gamma$ in powers
of $\as$ at fixed $\as/N$, and conversely.  So in particular at
LLQ$^2$ it is enough to consider $\gamma_0$ or $\chi_s$, and at LLx it
is enough to consider $\gamma_s$ or $\chi_0$. The running of the
coupling is a LLQ$^2$ but NLLx effect, so beyond LLx the discussion
given so far is insufficient.

\subsubsection{Running coupling}

The generalization of duality to the running coupling case is
nontrivial because the running of the coupling breaks the symmetry of
the two scales $\xi$ and $t$ in the Altarelli--Parisi equation.
Indeed, upon $M$--Mellin transform~(\ref{resum:mmel}) the usual
one--loop running coupling becomes the differential operator
\begin{equation}
\ah = \frac{\as}{1-\beta_0 \as \smallfrac{d}{dM}}+\cdots,
\label{resum:ahdef}
\end{equation}
where $d\as/dt= -\beta_0\as^2$.

Consider for simplicity the LLx $x$--evolution equation, i.e.
Eq.~(\ref{resum:xevol}) with $\chi=\as\chi_0(M)$, and include running
coupling effects by replacing $\as$ with the differential operator
Eq.~(\ref{resum:ahdef}). We can solve the equation perturbatively by
expanding the solution in powers of $\as$ at fixed $\as/N$: the
leading--order solution is given by Eq.~(\ref{resum:xisoln}), the
next--to--leading order is obtained by substituting this back into the
equation and retaining terms up to order $\beta_0\as$, and so
on~\cite{Ball:1999sh}. We can then determine the associate $G(N,t)$ by
inverting the $M$--Mellin, and try to see whether this $G(N,t)$ could
be obtained as the solution of a renormalization group (RG)
equation~(\ref{resum:tevol}).

The inverse Mellin is again given by the residue of the pole of $e^{t
  M} G(N,M)$ in the M--plane, where $G(N,M)$ is now the perturbative
solution. When trying to identify this with a solution to
Eq.~(\ref{resum:tevol}) there are two potential sources of trouble.
The first is that now the perturbative solution at order
$(\as\beta_0)^n$ has a $(2n+1)$--st order pole. Therefore, the
scale--dependence of the inverse Mellin is now a function of both
$\as$ and $t$, whereas the solution of a RG equation depends on $t$
only through the running of $\as$. Hence it is not obvious that a dual
anomalous dimension will exist at all. The second is that even if a
dual $\gamma$ does exist, it is not obvious that it will depend only
on $\chi$ and not also on the boundary condition $F_0(M)$
Eq.~(\ref{resum:xisoln}): in such case, the running of the coupling in
the $\xi$--evolution equation would entail a breaking of
factorization.

However, explicit calculation shows that it is possible to match the
anomalous dimension and the boundary condition order by order in
perturbation theory in such a way that both duality and factorization
are respected. Namely, the solution to the leading--twist running
coupling $x$--evolution Eq.~(\ref{resum:xevol}) with kernel $\ah
\chi_0$ and boundary condition $G_0(M)$ is the same as that of the
renormalization group Eq.~(\ref{resum:tevol}) with boundary conditions
and anomalous dimension given by
\begin{eqnarray}
&&\gamma(\as(t),\as(t)/N)=\gamma_s(\as(t)/N)+\as(t)\beta_0\Delta\gamma_{ss}(\as(t)/N)+\nonumber\\
&&\quad\qquad+
(\as(t)\beta_0)^2 \Delta\gamma_{sss}(\as(t)/N) +O(\as(t)\beta_0)^3\\
\label{resum:efgam}
&&G_0(\as,N)=G_0(N)+\as\beta_0 \Delta^{(1)}G_0(N)+
(\as\beta_0)^2\Delta^{(2)} G_0(N)
+O(\as\beta_0)^3,
\label{resum:efbc}
\end{eqnarray}
where the leading terms $\gamma_s$ and $G_0(N)$ are given by
Eqs.~(\ref{resum:dual}) and (\ref{resum:bcmatch}) respectively. The
subleading corrections are
\begin{eqnarray}
\Delta\gamma_{ss}&=&-
\frac{\chi_0''\chi_0}
% {2\chi_0'^2}\\
{2\chi_0^{\prime \, 2}}\\
\Delta^{(1)}G_0(N)&=&{2{\chi_0'}^2
F_0-\chi_0\left(F_0'\chi_0''-\chi_0'F_0''\right)\over 2{\chi_0'}^3},
\label{resum:nlocorr}
\end{eqnarray}
where all derivatives are with respect to the arguments of $\chi_0(M)$
and $F_0(M)$, which are then evaluated as functions of $\gamma_s(\as/N
)$.  The sub--subleading correction to the anomalous dimension is
\begin{equation}
\Delta\gamma_{sss}=-\chi_0^2{15{\chi_0''}^3-16
\chi_0'\chi_0''\chi_0'''+3 {\chi_0'}^2\chi_0''''\over 24 {\chi_0'}^5},
\label{resum:nnlocorr}
\end{equation}
and we omit the very lengthy expression for $\Delta^{(2)}G_0(N)$.  The
fact that duality and factorization hold up to NNLLx is nontrivial,
and suggests that they should hold to all orders. An all--order proof
can be in fact constructed~\cite{Altarelli:2001ji}.


Once the corrections to duality Eq.~(\ref{resum:efgam}) are
determined, they can be formally re-interpreted as additional
contributions to $\chi$: namely, one can impose that the duality
Eq.~(\ref{resum:dual}) be respected, in which case the kernel to be
used in it is an ``effective'' $\chi$, obtained from the kernel of the
$x$--evolution Eq.~(\ref{resum:xevol}) by adding to it running
coupling corrections order by order in perturbation theory: $\chi_0$
will be free of such correction, $\chi_1$ will receive a correction
\begin{equation}
\Delta\chi_1=\beta_0{1\over
2}{\chi_0(M)\chi_0''(M)\over{\chi_0'}^2(M)},
\label{resum:efchi}
\end{equation}
and so forth. Applying duality to the known one--loop anomalous
dimensions $\gamma_0$ thus gives us the resummation of the all--order
singular contributions $\chi(\as/M)$ to this effective $\chi$, which
include the running coupling correction Eq.~(\ref{resum:efgam}) and
its higher--order generalizations.

\subsubsection{Resummation}
Because the first two orders of the expansion of $\chi$ in powers of
$\as$ at fixed $M$ and of the expansion of $\gamma$ in powers of $\as$
at fixed $N$ are known, it is possible to exploit duality of
perturbative evolution to combine this information into anomalous
dimension which accomplish the simultaneous resummation of leading and
next--to--leading logs of $x$ and $Q^2$.  In fact, it turns out that
both a small $M$ and a small $N$ resummation of anomalous dimensions
are necessary in order to obtain a stable perturbative expansion,
while unresummed anomalous dimensions lead to instabilities. Both
sources of instability are generic consequences of the structure of
the perturbative expansion, and could have been predicted before the
actual explicit computation~\cite{Fadin:1998py} of subleading
small-$x$ corrections.

\subsubsection{Small M}
The perturbative expansion of $\chi$ at fixed $M$ is very badly
behaved in the vicinity of $M\sim0$: at $M=0$, $\chi_0$ has a simple
pole, $\chi_1$ has a double pole and so on.  In practice, this spoils
the behaviour of $\chi$ in most of the physical region $0< M< 1$.
Because $1/M^k$ is the Mellin transform of ${\Lambda^2\over Q^2}
{1\over k!}\ln^{k-1}(Q^2/\Lambda^2)$, these singularities correspond
to logs of $Q^2$ which are left unresummed in a LLx or NLLx
approach~\cite{Salam:1998tj}.

The resummation of these contributions may be understood in terms of
momentum conservation, which implies that $\gamma(1,\as)=0$ (note our
definition of the $N$--Mellin transform~(\ref{resum:nmel}), and also
that $\gamma$ is to be identified with the large eigenvector of the
anomalous dimension matrix).  The duality Eq.~(\ref{resum:dual}) then
implies that a momentum--conserving $\chi$ must satisfy
$\chi(0,\as)=1$. This, together with the requirement that $\chi$
admits a perturbative expansion in powers of $\as$, implies that in
the vicinity of $M=0$, the generic behaviour of the kernel is
\begin{equation}
\chi_s\tozero M {\as\over\as+\kappa M}={\as\over\kappa M}-{\as^2\over
(\kappa M)^2}+{\as^3\over(\kappa M)^3}+\dots\quad,
\label{resum:momcons}
\end{equation} 
where $\kappa$ is a numerical constant which turns out to be
$\kappa=\pi/C_A$.  Hence we understand that there must be an
alternating--sign series of poles at $M=0$, which sums up to a regular
behaviour.  In fact, we can systematically resum singular
contributions to $\chi$ to all orders in $\as$ by including in $\chi$
the terms $\chi_s(\as/M)$ derived from the leading order
$\gamma_0(N)$, and similarly at next--to--leading order, and so on.
Because the usual anomalous dimension automatically respects momentum
conservation order by order in $\as$, in order to remove the small $M$
instability of the expansion of $\chi$ at fixed $M$, it is sufficient
to improve the expansion by promoting it to a ``double leading''
expansion which combines the expansions in powers of $\as$ at fixed
$M$ and at fixed $\as/M$~\cite{Altarelli:1999vw}.  For example, at
leading order $\chi=\as\chi_0(M)+\chi_s(\as/M)-{\rm d.c.}$, where the
subtraction refers to the double--counting of the $\as/M$ term which
is present both in $\as\chi_0$ and in $\chi_s(\as/M)$. This expansion
of $\chi$ is dual Eq.~(\ref{resum:dual}) to an analogous expansion of
$\gamma$, where at leading order
$\gamma=\as\gamma_0(M)+\gamma_s(\as/M)-{\rm d.c.}$, and so forth. Both
expansions are well behaved at small $M$, i.e. large $N$.  At this
level, it is already clear that the impact of the inclusion of
small-$x$ corrections is moderate: indeed, it turns out that the
double--leading kernel is quite close to the usual two--loop kernel,
except at the smallest values of $N$, i.e. in the neighbourhood of the
minimum of $\chi(M)$~\cite{Altarelli:1999vw}.

\subsubsection{Small N}

The improved double--leading expansion of the anomalous dimension
still requires resummation at small $N$. This is because, even though
the next--to--leading correction to the double--leading evolution
kernel is small for all fixed $M$, it is actually large if $N$ is
fixed and small.  This in turn follow from the fact that the leading
$\chi$ kernel has a minimum, so the small $N=\chi$ region corresponds
by duality Eq.~(\ref{resum:dual}) to the vicinity of the minimum where
the kernel is almost parallel to the $\gamma=M$ axis.

At small $N$, unlike at small $M$, there is no principle like momentum
conservation which may provide a fixed point of the expansion and thus
fix the all--order behaviour.  The only way out is thus to treat this
all--order behaviour as a free parameter. Namely, we introduce a
parameter $\lambda$ which is equal to the value of the all--order
kernel $\chi$ at its minimum, and then we expand about this all--order
minimum. In practice, this means that we reorganize the expansion of
$\chi$ according to~\cite{Ball:1999sh}
\begin{eqnarray}
\chi(M,\as)&=&\as \chi_0(M)+\as^2\chi_1(M)+\dots\nonumber\\
&=&\as \tilde \chi_0(M) +\as^2\tilde\chi_1(M)+\dots,
\label{resum:clamsub}
\end{eqnarray}
where
\begin{equation}
\as\tilde \chi_0(M,\as)\equiv \alpha_s
\chi_0(M)+\sum_{n=1}^\infty \alpha_s^{n+1} c_n,\qquad 
\tilde\chi_i(M)\equiv\chi_i(M)-c_i,
\label{resum:ctil}
\end{equation}
and the constants $c_i$ are chosen in such a way that
\begin{equation}
\lambda \equiv\as\tilde\chi_0(\half)=\as\chi_0(\half)+\Delta \lambda.
\label{resum:lamdef}
\end{equation}
is the all--order minimum of $\chi$.  Of course, in practice
phenomenological predictions will only be sensitive to the value of
$\lambda$ in the region where very small values of $N$ are probed,
i.e. at very small $x$.


\subsubsection{Phenomenology}

\begin{figure}
%\begin{center}
  \includegraphics[width=0.4\linewidth,clip]{chi2.ps}
  \includegraphics[width=0.4\linewidth,clip]{lamg.ps}
\caption{$\chi^2$ (left) and starting gluon slope $G(x, \hbox{4 GeV}^2)\sim
  x^{-\lambda_g}$ (right) for the fit~\cite{Altarelli:2000mh} to the
  95 H1 data~\cite{Adloff:2000qk} as a function of the resummation
  parameter $\lambda$ Eq.~(\ref{resum:lamdef}), for the two
  resummation prescriptions discussed in text. The fits are performed
  with $\as(M_z)=0.119$.}
%\end{center
\label{resum:fig:bf2}
\end{figure}

Using duality and the resummation discussed above, one can construct
resummed expressions for anomalous dimensions and coefficient
functions, and wind up with resummed expressions for physical
observables which may be directly compared to the data.  The need to
resum the small $N$ behaviour entails that phenomenological
predictions will necessarily depend on the parameter $\lambda$
Eq.~(\ref{resum:lamdef}). When the resummed double--leading expansion
is constructed, a further ambiguity arises in the treatment of
double--counting terms. This ambiguity is related to the nature of the
small $N$ singularities of the anomalous dimension, which control the
asymptotic small $x$ behaviour.  Specifically, according to the way
the double--counting is treated, the $N=0$ poles of the one-- and
two--loop result may survive in the resummed result (`S--resummation')
or not (`R--resummation').  Both alternatives are compatible with the
known low--order information on the evolution kernel, and can be taken
as two extreme resummation schemes which parametrize our ignorance of
higher order perturbative terms.  Since the resummed terms also have a
cut starting at $N=\lambda$, whether or not these low--$N$ poles are
present only makes a difference if $\lambda$ turns out to be small,
$\lambda\lsim0.3$.

The $\chi^2$ and starting gluon slope for a
fit~\cite{Altarelli:2000mh} to the recent H1 data~\cite{Adloff:2000qk}
for the deep--inelastic cross section are shown in
Fig.~\ref{resum:fig:bf2}, as a function of $\lambda$ and for the two
different resummation prescriptions. It is clear that if the
perturbative $N=0$ poles do not survive the resummation (R
resummation) then only a fine--tuned value of $\lambda\approx 0.2$ is
acceptable, whereas if they do survive (S resummation) essentially any
$\lambda\lsim 0$ gives a good fit.

Figure~\ref{resum:fig:bf2} demonstrates that it is possible to
accommodate the success of simple fixed--order approach within a fully
resummed scheme, and in fact the resummed calculation is in somewhat
better agreement with the data than the fixed order one. Even though
the effects of the resummation are necessarily small (otherwise the
success of the fixed order prediction could not be explained) they do
have a significant impact in the extraction of the parton
distribution: the gluon comes out to be significantly more
valence--like than in an unresummed fit.  Hence, the use of resummed
perturbation theory is crucial for the extraction of reliable parton
distributions at small $x$.

From a theoretical point of view, we see that current data already
pose very stringent constraints on the unknown high--orders of the
perturbative expansion: only a rather soft high--energy behaviour of
the deep-inelastic cross--section is compatible with the data. The
resummed parton distributions may in principle now be used to compute
hadronic cross-sections at the Tevatron and LHC: the resummation of
hadronic heavy quark production is discussed in some
detail in Ref.~\cite{Ball:2001pq}.\\
{\bf Acknowledgments:} A sizable part of this contribution is based on
work done in collaboration with G.~Altarelli. This work was supported
in part by EU TMR contract FMRX-CT98-0194 (DG 12 - MIHT).




%\bibliography{resumbib}



