\subsection{Joint resummation in electroweak
  production\protect\footnote{Contributing authors: A. Kulesza, G.
    Sterman, W. Vogelsang}}


\subsubsection{Introduction}
The hadronic annihilation cross sections for electroweak boson
production ($\gamma^*$, W, Z, H) provide important test cases for
resummation techniques in QCD.  Soft-gluon emission leads to large
logarithmic contributions to perturbative cross sections, both near
threshold, when the incoming partons have just enough energy to
produce the observed boson, and near measured boson transverse
momentum $Q_T\ll Q$. All-order resummations of the leading and
next-to-leading terms in such contributions have been achieved
separately for the threshold and small-$Q_T$ cases. Very recently, a
formalism has been developed that encompasses both.  For electroweak
annihilation as well as QCD cross sections, the necessary analysis for
this combination, which we refer to as joint resummation, has been
carried out in \cite{Laenen:2000ij}. In this brief report, we develop
the application of the joint resummation formalism to the
phenomenology of electroweak annihilation.

The effects of resummation are closely bound to momentum conservation.
The singular corrections associated with soft gluon emission
exponentiate in the corresponding spaces of moments, impact parameter
for transverse momentum, and Mellin (or Laplace) moments in energy for
threshold resummation.  The transforms relax momentum and energy
conservation, while their inverses reimpose it.  In joint resummation,
both transverse momentum and energy conservation are respected. This
leads to two separate transforms in the calculation, and we will
discuss below how we invert these.  This is a non-trivial issue,
since, taking into account nonleading logarithms and the running of
the coupling, resummation leads in each case to a perturbative
expression in which the scale of the coupling reflects the value of
the transform variable. Because of the singularity of the perturbative
effective coupling at $\Lambda_{\rm QCD}$, the resulting expressions
are, strictly speaking, undefined.  A closer look, however, shows that
singular contributions appear only at nonleading powers of momentum
transfer. This is an example of how perturbative resummation can
suggest the way nonperturbative dynamics is expressed in infrared safe
hard scattering functions.  In effect, perturbation theory is
ambiguous, and the resolution of its ambiguities is, by definition,
nonperturbative
\cite{Contopanagos:1994yq,Webber:1994cp,Korchemsky:1995is}.  Each
scheme for dealing with these ambiguities constitutes a specification
of perturbation theory, and implies a parameterization of
nonperturbative effects.  We hope that a joint resummation affords a
more general approach to this problem.

\subsubsection{Joint resummation method}

In the framework of joint resummation~\cite{Laenen:2000ij}, the cross
section for electroweak annihilation is given as \begin{eqnarray}
\label{resum:crsec}
\frac{d\sigma_{AB}^{\rm res}}{dQ^2\,dQ_T^2} &=& \sum_a
\sigma_{a}^{(0)}\, \int_{C_N}\, \frac{dN}{2\pi i} \,\tau^{-N}\; \int
\frac{d^2b}{(2\pi )^2} \,
e^{i{\vec{Q}_T}\cdot {\vec{b}}}\, \nonumber \\
&\times& {\cal C}_{a/A}(Q,b,N,\mu,\mu_F )\; \exp\left[ \,E_{a\bar
    a}^{\rm PT} (N,b,Q,\mu)\,\right] \; {\cal
  C}_{\bar{a}/B}(Q,b,N,\mu,\mu_F) \; , \end{eqnarray} where $\tau =Q^2/S$, $Q$
denotes the invariant mass of produced boson, and $\sigma_{a}^{(0)}$
is the Born cross section.  The flavour-diagonal Sudakov exponent
$E_{a\bar a}^{\rm PT}$ is given to next-to-leading logarithmic (NLL)
accuracy by \begin{eqnarray}
\label{resum:jointsud}
\tilde{E}_{a\bar a}^{\rm PT} (N,b,Q,\mu,\mu_F) &=&
-\int_{Q^2/\chi^2}^{Q^2} \frac{d k_T^2}{k_T^2} \; \left[
  A_a(\as(k_T))\, \ln\left( \frac{Q^2}{k_T^2} \right) +
  B_a(\as(k_T))\right] \,.  \end{eqnarray} It has the classic form of the Sudakov
exponent in the recoil-resummed $Q_T$ distribution for electroweak
annihilation, with the same functions $A$ and $B$. The latter are
perturbative series in $\as$~\cite{Dokshitzer:1980hw,Collins:1981uk};
at NLL, one only needs the second (first) order expansion of $A$
($B$)~\cite{Kodaira:1982nh}.
%\be
%A_c(\as) = \sum_{n=1}^\infty \left( \frac{\as}{2\pi} \right)^n A_c^{(n)}
%\qquad \qquad
%B_c(\as) = \sum_{n=1}^\infty \left( \frac{\as}{2\pi} \right)^n B_c^{(n)}
%\;\;.  
%\ee
% Here refer to expanded E?
The only difference between the standard Sudakov exponent for $Q_T$
resummation and Eq.~\ref{resum:jointsud} comes from the lower limit of
the integrand, i.e. the function $\chi$ that organizes the logarithms
of $N$ and $b$ in joint resummation: \begin{equation}
\label{resum:chinew}
\chi(\bar{N},\bar{b})=\bar{b} + \frac{\bar{N}}{1+\eta\,\bar{b}/
  \bar{N}}\; \qquad \qquad \bar{b}\equiv b Q {\rm e^{\gamma_E}}/2 \; .
\end{equation} This choice for $\chi(\bar{N},\bar{b})$ fulfills the requirement
of reproducing the LL and NLL terms in $N$ and $b$ when $N \rightarrow
\infty$ (at fixed $b$) and $b \rightarrow \infty$ (at fixed $N$).
Moreover, it also guarantees that in the limit $b\gg N$ the quantity
$E_{a\bar a}^{\rm PT}$ expanded up to ${\cal O}(\as)$ does not have
any subleading terms with singular behavior different from the one
present in the fixed-order result.

The functions ${\cal C}(Q,b,N,\mu,\mu_F )$ in Eq.~(\ref{resum:crsec})
\begin{equation}
\label{resum:cpdf}
{\mathcal C}_{a/H}(Q,b,N,\mu,\mu_F ) = \sum_{j,k} C_{a/j}\left(N,
  \alpha_s(\mu) \right)\, {\cal E}_{jk} \left(N,Q/\chi,\mu_F\right) \,
f_{k/H}(N ,\mu_F) \; , \end{equation} contain products of parton distribution
functions $f_{k/H}$ at scale $\mu_F$ with an evolution operator ${\cal
  E}_{jk}$ between scales $\mu_F$ and $Q/\chi$. Furthermore, as
familiar from both threshold and $Q_T$ resummation, they contain hard
coefficients $C_{a/j}(N,\as)$ that are again perturbative series in
$\as$ and have the first-order expansions \begin{eqnarray} C_{q/q}^{(1)}\left(
  N,\as \right) &=& 1+ \frac{\as}{4\pi} C_F \left(-8+\pi^2
  +\frac{2}{N(N+1)} \right) \; =\;
C_{\bar{q}/\bar{q}}^{(1)}\left( N,\as \right)\; ,\\
C_{q/g}^{(1)}\left( N,\as\right)
&=&\frac{\as}{2\pi}\frac{1}{(N+1)(N+2)}\; =\;
C_{\bar{q}/g}^{(1)}\left( N,\as\right)\; .  \end{eqnarray} To be consistent with
the NLL approximation for the $E_{a\bar a}^{\rm PT}$ part of the cross
section, the evolution matrix ${\cal E}$ is derived from the NLO
solutions of standard evolution equations~\cite{Furmanski:1982cw}.
Inclusion of the full evolution of parton densities (as opposed to
keeping only the $\sim \ln \bar{N}$ part of the anomalous dimension
that dominates near threshold, see Ref.~\cite{Kulesza:2002rh}) 
extends the joint formalism and leads to resummation of collinear logarithms not all of
which are associated with threshold corrections.

Owing to the presence of the Landau pole in the strong coupling, the
jointly resummed cross section~(\ref{resum:crsec}) requires
definitions for the inverse Mellin and Fourier transforms. In pure
threshold resummation, it was proposed to parameterize the inverse
Mellin contour as~\cite{Catani:1996yz} \begin{equation}
\label{resum:cont}
N = C + z {\rm e}^{\pm i \phi} \; , \end{equation} where $C$ lies to the right of
the singularities of the parton densities, but to the left of the
Landau pole. This is a natural definition of the contour since it
decouples the Landau pole from any finite-order expansion of the
resummed cross section.  For the jointly resummed cross section in
Eq.~(\ref{resum:crsec}), if the $b$ integration is carried out in the
standard way over the real axis, one will inevitably reach the Landau
pole. This is well-known also from pure $Q_T$ resummation and led to
the introduction of a (non-perturbative) upper value $b_{\rm max}$ for
the $b$ integral, along with a redefinition $b\to b_{\ast}=b/
\sqrt{1+b^2/b_{\rm max}^2}$. To avoid introducing a new parameter, we
treat the $b$ integral in a manner analogous to the $N$ integral
above~\cite{Laenen:2000de}: were the Landau pole not present we could,
instead of performing the $b$ integral along the real axis, use
Cauchy's theorem and divert it into complex $b$ space. Our
prescription will be to use the diverted contour also in the presence
of the Landau pole. Technically, this can be achieved by writing \begin{equation} 2
\pi\, \int_0^{\infty} \, db\,b\, \,J_0(bQ_T) \,f(b) = \pi\,
\int_0^\infty db\, b\, \left[\, h_1(bQ_T,v) + h_2(bQ_T,v)\,
\right]\,f(b) \, ,
\label{resum:J0split}
\end{equation} where the $h_i$ are related to Hankel functions and distinguish
positive and negative phases in Eq.\ (\ref{resum:J0split}). We then
can treat the $b$ integral as the sum of two contours, the one
associated with $h_1$ ($h_2$) corresponding to closing the contour in
the upper (lower) half $b$-plane~\cite{Laenen:2000de}.

\subsubsection{Transverse momentum distribution for $Z$ production}

A phenomenological application of the joint resummation formalism
requires matching the resummed distribution~(\ref{resum:crsec}) to the
fixed-order result $d \sigma^{\rm fixed} $, which we do as follows:
\begin{equation}
\frac{d \sigma}{d Q^2 d Q_T^2} = \frac{d \sigma^{\rm res}}{d Q^2 d Q_T^2}
-  \frac{d\sigma^{\rm exp(k)}}{d Q^2 d Q_T^2} +
\frac{d \sigma^{\rm fixed(k)}}{d Q^2 d Q_T^2} \,,
\label{resum:joint:match}
\end{equation}
with $d\sigma^{\rm exp(k)} $ denoting the $k$-th order expansion of
the resummed cross section. This way of matching in the conjugate
$(N,\,b)$ space evidently avoids any type of double counting. Given
the above prescription we calculate the $Q_T$ distribution for Z
production at the Tevatron and compare it with the latest CDF
data~\cite{Affolder:1999jh} in Fig.~\ref{resum:fig:cdf} (dashed
lines). We emphasize that the dashed lines are obtained without any
additional nonperturbative parameter.  Towards low $Q_T$, one expects
perturbation theory (as defined in our formalism) to fail without
nonperturbative input.  The expected form of such effects can be
derived from resummed perturbation theory
itself~\cite{Kulesza:2002rh,Korchemsky:1995is,Tafat:2001in} by examining the
full NLL jointly resummed exponent in the limit of small transverse
momentum of soft radiation.  The gross structure one finds is a simple
Gaussian function $-gb^2$ added to the Sudakov exponent
in~(\ref{resum:crsec}). A fit to the data gives a modest $g= 0.8$
GeV$^2$, similar to~\cite{Qiu:2000ga}.  We finally note that, unlike
standard $Q_T$ resummation, our resummed cross
section~(\ref{resum:joint:match}) stays positive even at very large
$Q_T$ so that an extra switching between a matched distribution and a
fixed order result is not required here.  We obtain a very good
agreement between data and the jointly resummed theoretical
distribution even out to large $Q_T$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
  \includegraphics[width=6cm,height=7.5cm,angle=90]{cdf_lesh.epsi}
  \includegraphics[width=6cm,,height=7.5cm,angle=90]{cdf_lesh_largeqt.epsi}
\end{center}
\caption{CDF data \protect \cite{Affolder:1999jh}
  on Z production, compared to joint resummation predictions (matched
  to the ${\cal O} (\as)$ result according to
  Eq.~(\ref{resum:joint:match})) without \nprt smearing (dashed) and
  with Gaussian smearing using the nonperturbative parameter $g=0.8$
  GeV$^2$ (solid). The normalizations of the curves (factor of 1.069)
  have been adjusted in order to give an optimal description.  We use
  CTEQ5M parton distribution functions with renormalization and
  factorization scales $\mu=\mu_F=Q$.
\label{resum:fig:cdf}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

