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\begin{document}
\title{Multiple Production of $W$ Bosons in $pp$ and $pA$ Collisions}
\author{E. Braidot,
 E. Cattaruzza, A. Taracchini, D. Treleani\protect\footnote{\ \ speaker}}
\institute{Department of Theoretical Physics, University of Trieste and INFN, Section of Trieste }
\maketitle
\begin{abstract}
The production of equal sign $W$ boson pairs, through
single and double parton collisions, are comparable in magnitude at the LHC.
As a consequence of the  strong
anti-shadowing of MPI in interactions with nuclei, the double scattering contribution is further enhanced in the case
of hadron-nucleus collisions
\end{abstract}

\section{Multiple production of $W$ bosons in proton-proton collisions}

Multiple parton interactions are a manifestation of the unitarity
problem caused by the rapid increase of the parton flux at small
$x$, which leads to a dramatic growth of all cross sections with
large momentum transfer in $pp$ collisions at the
LHC\cite{Catani:2000jh}. The critical kinematical regime may be
identified by comparing the rate of double collisions with the
rate of single collisions. When the two rates become comparable
multiple collisions are no more a small perturbation and all
multiple collisions become equally important, while the production
of large $p_t$ partons becomes a common feature of the inelastic
event\cite{Sjostrand:1987su}\cite{Ametller:1987ru}.
In its simplest implementation\cite{Paver:1982yp} the double parton scattering cross
section $\sigma_D$ is given by

\begin{equation}
\sigma_D={1\over2}{\sigma_S^2\over\sigma_{eff}}\nonumber
\end{equation}

\noindent where $\sigma_S$ is the single scattering cross section.
The problem with unitarity becomes hence critical in the kinematical domain
where $\sigma_S$ and the scale factor $\sigma_{eff}$ are of the
same order.

The experimental indication is that the value of $\sigma_{eff}$ is
close to 10 mb\cite{Abe:1997xk}. One might hence conclude that one
should worry about multiple parton collisions only when the single
scattering cross section becomes comparable with $\sigma_{eff}$.
On the contrary multiple parton collisions may represent an
important effect also in cases where the single scattering cross
section is many orders of magnitude smaller that $\sigma_{eff}$.
The consideration applies to the interesting case of the
production of equal sign $W$ boson pairs. The leptonic decay
channel of $W$ bosons, which leads to final states with isolated
leptons plus missing energy, is in fact of great interest for the
search of new physics\cite{Abulencia:2007rd}.

The production of two equal sign $W$ bosons is a higher order
process in the Standard Model and two equal sign $W$ bosons can be
produced only in association with two jets\cite{Kulesza:1999zh}.
At the lowest order there are 68 diagrams at ${\cal
O}(\alpha_W^4)$ and 16 diagrams at ${\cal
O}(\alpha_S^2\alpha_W^2)$ (some of the diagrams are shown in
Fig.1) and, even though $\alpha_S>\alpha_W$, the strong and
electroweak diagrams give comparable contributions to the cross
section, which is infrared and collinear safe and can be evaluated
without imposing any cutoff in the final state quark jets.

\begin{figure}[htp]
\centering
\includegraphics[width=11cm]{Fig1.pdf}
\vspace{0cm}
\caption{Some of the three level diagrams which contribute to equal sign $W$ pairs production}
\label{fig:diagrams}
\end{figure}

\begin{figure}[htp]
\centering
\includegraphics[width=14.5cm]{Fig2.pdf}
\vspace{0cm} \caption{{\it Upper left panel}: $W$ production cross
sections by single parton scattering in $pp$ interactions as a
function of the c.m. energy.\ {\it Upper right panel}:  $W$ and
$W$ pairs production cross sections in $pp$ interactions by double
and by single parton collisions.\ {\it Lower left panel}: $W$
pairs production cross sections by double parton collisions with
correlated and uncorrelated parton densities in the case of $pp$
interactions.\ {\it Lower right panel}: $W$ pairs densities in
transverse space in the case of single and of double parton
collisions in $pp$ interactions.} \label{fig:pp cross sections}
\end{figure}

The resulting cross sections to produce $W$ bosons and $W$ boson
pairs, by single parton scattering in $pp$ interactions, are shown
in Fig.2 as a function of the c.m. energy. As apparent in the
figure (left upper panel) the cross section to produce two equal
sign $W$ bosons is five orders of magnitude smaller with respect
to the cross section to produce a single $W$ boson. The same
reduction factor is expected for the production of two equal sign $W$ bosons
through a multiple collisions processes:

\begin{equation}
\sigma_{WW}={1\over2}\sigma_W{\sigma_W\over\sigma_{eff}},\qquad{\sigma_{W}\over\sigma_{eff}}\simeq{10^2{\rm nb}\over10{\rm mb}}=10^{-5}
\end{equation}

The argument above relies on the simplest expression of the double
scattering cross section, obtained by assuming a factorized
expression for the the double parton distributions, which is
obviously inconsistent in the case of the valence because of the
correlations induced by flavor conservation. In the actual case,
given the large mass of the $W$ bosons, one may expect important
contributions of the valence also at the LHC. One may hence
normalize the double parton distributions in such a way to satisfy
the flavor sum rules and work out the double scattering cross
section accordingly. The effect on the cross section is shown in
the left lower panel of Fig.2, which shows that, at the LHC, the
cross sections is reduced by about 20\%.

The integrated rates of equal sign $W$ boson pairs, by single and
double parton collisions, are hence comparable in $pp$ collisions
at the LHC. The distribution in phase space is however rather
different in the two cases.

In the right lower panel of Fig.2 we show the distribution of the
produced $W$s, as a function of their transverse momenta. 
The distribution in transverse momenta of the produced $W$s is obtained by following the recipe of the "Poor Man's shower model" of Barger and Phillips\cite{Barger:1987nn} and using as a smearing function at low $p_t$ the expression in Eq.15 of \cite{Kulesza:2001jc}. 
The two
contributions may be separated with a cut of 15 GeV/c in the
transverse momenta of the produced $W$s. In Fig. 3 we show how the
$W^+$ bosons (left panels) and their decay electrons (right
panels) are distributed in transverse momentum and rapidity. The
case of double parton collisions is shown in the upper panels,
while the case of single parton collisions is shown in the lower
panels. In the case of a double parton collision, the $W$ bosons
are mainly produced with small transverse momenta, while the
rapidity distribution of the $W$ boson reminds the momentum of the
originating up quarks. The distributions of the final state charged
leptons is peaked at the same rapidity of the parent $W$ boson and
at a transverse momentum corresponding to 1/2 of the $W$ boson
mass.

In the case of single parton collisions (lower panels of Fig.3)
the $W$s and the corresponding decay leptons have a much broader
distribution in $p_t$ and rapidity and the characteristic peaks of
the double scatterings are completely absent. The two
contributions are hence disentangled very easily by adopting
appropriate cuts in rapidity and transverse momenta of the finally
observed charged leptons.

\begin{figure}[htp]
\centering
\vspace{0cm}
\includegraphics[width=14.5cm]{Fig3.pdf}
\vspace{0cm} \caption{$W^+W^+$ and $e^+e^+$ pairs distribution in
transverse momentum and rapidity, in the case of single parton
collision (upper panels) and of double parton collisions (lower
panels) in proton-proton  collisions.} \label{fig:y and pt distr.
in pp coll.}
\end{figure}

\section{Multiple production of $W$ bosons in proton-nucleus collisions}

As pointed out in\cite{Strikman:2001gz}, a major feature of MPI in
hadron-nucleus collisions is the strong anti-shadowing. Double
parton collisions may in fact be amplified by a factor 2 or 3 on
heavy nuclei as compared with the corresponding cross section in
hadron-nucleon collisions multiplied by the atomic mass number
$A$. Notice that for, say, values of $x$ of the order of $10^{-3}$
and for values of $Q^2>10$ GeV$^2$, the usual nuclear shadowing
correction is a much smaller effect and corresponds to a reduction
of the cross section not larger than $10\%$ even on heavy
nuclei\cite{Kulagin:2004ie}. The effect is schematically
illustrated in Fig. 4, where non additive
corrections to the nuclear structure functions are neglected, in such a way that
each nuclear parton may be associated to a given parent nucleon.
As shown in Fig.4, in proton-nucleus interactions one may hence
distinguish two different contributions to the double parton
scattering cross section, depending wether the two nuclear partons
undergoing the interactions are originated by one or by two
different target nucleons. 

\begin{figure}[htp]
\centering
\vspace{0cm}
\includegraphics[width=12cm, height=5cm]{Fig4.pdf}
\vspace{0cm} \caption{$W$ production cross sections by single
parton scattering in pp collisions as a function of the c.m.
energy.} \label{fig:diagrams}
\end{figure}

The cross section may thus be written
as the sum of two terms

\begin{equation}
\sigma_D^A=\sigma_D^A\big|_1+\sigma_D^A\big|_2
\end{equation}

and

\begin{eqnarray}
\sigma_D^A\big|_1={1\over2}{\sigma_W^2\over\sigma_{eff}}\int d^2bT(b)\propto A,\qquad\sigma_D^A\big|_2={1\over2}\sigma_W^2\int d^2bT^2(b)\propto A^{4/3}\nonumber
\end{eqnarray}


\begin{figure}[htp]
\centering
\vspace{0cm}
\includegraphics[width=14.5cm]{Fig5.pdf}
\vspace{0cm} \caption{$W$ and $W$ pairs production in
proton-proton and proton-nucleus collisions. Integrated cross
sections as a function of the c.m. energy (upper panels) and
distributions in transverse space (lower panels).} \label{fig:pp
and pA comparison}
\end{figure}

The anti-shadowing effect is apparent in Fig.5, where the $W$
production cross sections in proton-proton collisions are compared
with the cross sections in proton-nucleus collisions (after
dividing by the atomic mass number $A$). In the upper panels one
compares the cross sections as a function of the c.m. energy,
while in the lower panels one compares the distributions in
transverse momenta of the two $W^+$ bosons. The region where
double parton collisions dominate now extends to transverse
momenta of the order of 40 $GeV/c$.


In the upper panels of Fig.6 (left and right respectively) we show
the distributions in transverse momentum and rapidity of the $W^+$
bosons and of the decay leptons in $pA$ collisions. The $W$ bosons
are produced with a small transverse momentum, while the rapidity
distribution of the $W$ boson reminds the momentum of the
originating up quark. The asymmetry in rapidity is due to the
different content of up quarks in the proton as compared with the
content of up quarks in the pairs of nucleons of the target
nucleus undergoing the process ($pp$, $pn$ and $nn$). The
distributions of the final charged leptons is peaked at the same
rapidity of the parent $W$ boson and, as in the case of
proton-proton interactions, at a transverse momentum corresponding
to 1/2 of the $W$ boson mass.

\begin{figure}[htp]
\centering
\vspace{0cm}
\includegraphics[width=14.5cm]{Fig6.pdf}
\caption{$W^+W^+$ and $e^+e^+$ pairs distribution in transverse
momentum and rapidity, in the case of single parton collision
(upper panels) and of double parton collisions (lower panels) in
proton-nucleus collisions.} \label{fig:diagrams}
\end{figure}

The distributions of equal sign $W$ bosons and of the decay
leptons generated by single parton collisions in $pA$ interactions
are shown in the lower panels of Fig.6 (left and right
respectively) as a function of rapidity and transverse momenta.
The contribution of double collisions is overwhelming when
selecting leptons with transverse momenta of the order of one half
of the $W$ mass.

\section{Concluding summary}

Equal sign $W$ boson pairs are produced by a higher order process
in the SM. As a consequence, the cross section to produce two $W$
bosons with equal sign is more than two orders of magnitude
smaller in $pp$ collisions at the LHC, as compared with the cross section to produce two $W$
bosons with opposite sign. An
outcome is that the integrated cross sections, to produce two
equal sign $W$ bosons through single and double parton collisions,
are similar in magnitude. The equal sign $W$ bosons and the
corresponding decay leptons are however distributed very
differently in phase space by the two production mechanisms, which
allows to disentangle the two contributions easily by looking at
the distribution of the decay leptons. 

As a consequence of the
strong anti-shadowing of MPI in collisions with nuclei, the 
contribution of double scattering is greatly enhanced in the case of
hadron-nucleus collisions.



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