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\begin{document} \title{Multiparton interactions of hadrons and photons with nuclei - revealing transverse structure of nuclei and strong gluon field dynamics } 
\author{Mark Strikman}
\institute{Penn State University, University Park, PA 16802, U.S.A.}
\maketitle
\begin{abstract}
We argue that multiparton interactions in proton - nucleus collisions at the LHC should be strongly enhanced as compared to naive expectation of cross section been proportional to atomic number  - the  antishadowing phenomenon. Study of the such processes will allow to measure in a model independent way double parton distributions in nuclei and, in combination  with the $pp$ measurements - transverse correlations of partons in nucleons. It is also emphasized that ultraperipheral collisions (UPC) of nuclei will allow to study multiparton interactions 
of photons with nuclei well before the $pA$ collisions will be available at the LHC.
UPC will also
provide a quick and  effective way to test onset of a novel perturbative QCD regime of 
strong absorption for the interaction of small 
dipoles at the collider energies
in the process $\gamma +A \to J/\psi + "{\rm \,  gap"} + X$ at large momentum transfer $t$.
\end{abstract}

\section{Multiparton collisions and generalized parton distributions}
It was recognized already more than two decades ago \cite{Goebel:1979mi,*Paver:1982yp, *Humpert:1983fy,*Humpert:1984ay,*Sjostrand:1987su} that 
the increase of parton densities at small $x$ leads to a strong increase of 
the probability of nucleon-nucleon collisions where two or more partons of 
each projectile experience pair-vice independent hard interactions.
As a result at the LHC the multiparton interactions will be a generic feature of the $pp$ and $pA$ collisions. 
Although the production of multijets through the double parton
scattering mechanism was investigated in several experiments
\cite{Akesson:1986iv,Alitti:1991rd,Abe:1993rv,Abe:1997bp,Abe:1997xk,Sonnenschein:2009nh} at $pp,p\bar p$ colliders, the
interpretation of the data is somewhat  hampered by the need to model both
the longitudinal and the transverse partonic correlations at the
same time. The studies  of  proton-nucleus collisions at LHC will provide a
feasible opportunity to study separately the longitudinal and transverse 
 correlations of partons  in the nucleon as well as to check the validity of 
the underlying picture of multiple collisions.

It is worth mentioning also that understanding of multiparton interactions is important  for proper modeling of central $pp$ collisions which dominate in the production of new particles and where such multijet interactions are enhanced. Such modeling should be done in a 
 way consistent with the information about the structure of nucleons/nuclei available from hard processes which were studied at HERA. So far this is not the case (see below).

The simplest case of a multiparton process is the double parton collision. 
Since the momentum scale $p_t$ of a hard interaction corresponds 
to much smaller  transverse distances $\sim 1/p_t$ in the
coordinate space than the hadronic radius, in a double parton
collision the two interaction regions are well separated in the
transverse space. Also in the c.m. frame pairs of
partons from the colliding hadrons are located in pancakes of
thickness $\leq (1/x_1 +1 /x_2)/p_{c.m.}$. Thus two hard collisions
occur practically simultaneously as soon as $x_1, x_2$ are not too
small and hence a  cross talk between two hard collisions is not possible.
A consequence is that  the different parton
processes add incoherently in the cross section. The double parton
scattering cross section, being proportional
to the square of the elementary parton-parton cross section, is
therefore characterized by a scale factor with dimension of the
inverse of a length squared. The dimensional quantity is provided
by the nonperturbative input to the process, namely by the
multiparton distributions. In fact, because of the localization of
the interactions in transverse space, the two pairs of colliding
partons are aligned, in such a way that the transverse distance
between the interacting partons of the target hadron is
practically the same as the transverse distance between
the partons of the projectile. The double parton distribution is
therefore a function of two momentum fractions
and of their transverse distance, and it can be written as
$\Gamma(x,x',\rho, \rho')$. It  depends also on the
virtualities of the partons, $Q^2,Q'^2
 $, though to make the expressions more compact we will
not write explicitly this $Q^2$ dependence. Hence the double
parton scattering cross section for the two ``two $\to$ two''
parton processes $\alpha$
 and  $\beta$ in an inelastic interaction between hadrons
$a$ and $b$ can be written as:
\begin{eqnarray}
 \sigma_D(\alpha,\beta)&=&\frac{m}{2}\int\Gamma_a(x_1,x_2;\rho_1,\rho_2)\hat{\sigma}_{\alpha}
(x_1,x_1') \cdot 
\hat{\sigma}_{\beta}(x_2,x_2')\Gamma_b(x_1',x_2';\rho_1,\rho_2)  \,\,\,\,\,\,\,\,  \nonumber \\  & & \, \, \, \,\, \,\, \, \, \, \, \,\, \,\, \,\, \, \, \,\, \,\, \, dx_1dx_1'dx_2dx_2'd^2\rho_1 d^2\rho_2, 
\label{1}
\end{eqnarray}
\par\noindent
where $m=1$ for indistinguishable parton processes and $m=2$ for
distinguishable parton processes. We also took into account that transverse distances in the binary collisions are small as compared to the hadron size scale. Note that though the factorization 
approximation of Eq.(\ref{1}) is generally accepted in the analyses of the 
multijet processes and appears natural based on the geometry of the process 
no formal proof exists in the literature. 



The QCD factorization theorems for exclusive hard processes: $\gamma^*_L + p \to "vector \, meson\, + p$, $\gamma^*_L + p \to \gamma  + p$ give a unique tool for  determining   transverse distributions of partons in nucleons as a function of $x$ and resolution scale - the generalized parton distribution (GPD). The discussed processes are proportional to the GPDs in non-diagonal kinematics at finite longitudinal momentum transfer. However corrections for this effect are small and one can extract diagonal GPDs from the analysis of the data.They could be written as $f_j(x, Q^2, \rho)= f_j(x,Q^2) F_j(x, Q^2, \rho)$, where 
$f_j(x,Q^2)$ is the parton density and the probability to find a parton with given $x$ at transverse distance $\rho$ from the nucleon center $\int d^2 \rho F_j(x, Q^2, \rho)=1$.

Currently, the best information about the gluon transverse distributions is provided by the data on $J/\psi$ exclusive production: in the scaling limit  $d\sigma/dt \propto F_g^2(x,t)$. The analysis of the experimental data indicates that  dipole with 
$F_g=1/(1-t/m_g(x)^2) $ with the x-dependent $m_g(x)$ gives a reasonable description of the data:   $m_g^2(x=0.05) \sim 1 GeV^2, m_g^2(x=0.001) \sim 0.6 GeV^2. $

The transverse distribution of partons is expressed through $F_g(x,t)$ as 
\begin{equation}
F_g (x, \rho; Q^2) \; \equiv \; \int \frac{d^2 \Delta_\perp}{(2 \pi)^2} \; e^{i (\bf{\Delta}_\perp \bf{\rho})} \; F_g (x, t = -\bf{\Delta}_\perp^2; Q^2).
\end{equation}
In the case of the dipole parametrization one find 
\begin{equation}
F_g (x, \rho ) = \frac{m_{g}^2}{2\pi} \; \left(\frac{m_{g} \rho}{2}\right) \; K_1 (m_{g} \rho ),
\label{fg}
\end{equation}
where $K_1$ is the modified Bessel function.

 Our analysis of the data 
the transverse distribution of gluons indicates that it  is significantly  more narrow than the one which would  follow from the naive assumption that it should be the same as given by the e.m. nucleon form factors. A likely reason for the difference of sizes  is that pion field which contributes significantly to the e.m. nucleon radius gives non-negligible contribution to the gluon GPD only for   $x \le 0.1$.


The distribution over $\rho$  also somewhat broadens with decrease of $x$
with a initial broadening at $x\sim 0.05$ due to the pion field effects. Also, there are indications that transverse distribution of quarks is somewhat broader than that for gluons, for the recent analysis  and references see \cite{Frankfurt:2005mc,*Strikman:2009bd}.





Distribution over the impact parameters in $pp$ collisions with production of jets 
is given by the convolution of $F_j's$ (for simplicity we assume in the following that only gluons contribute to the jet production:
\begin{equation} P_2(b) = \int d^2 \rho_1 \int d^2 \rho_2 \delta^{(2)} (\rho_1+\rho_2 -b)F_g(x_1,Q^2,\rho_1)\cdot F_g(x_2,Q^2,\rho_2).
\end{equation}
Using parametrization of Eq.\ref{fg} one finds
\begin{equation}
P_2 (b) \;\;= \frac{m_g^2}{12 \pi} \left(\frac{m_g b}{ 2}\right)^3K_3(m_g b)
\end{equation}

If partons "i" and "j" are not correlated in the transverse plane\begin{equation}
 \Gamma_{ij}(x_1,x_2;\rho,\rho')= F_i(x_1, \rho)\cdot F_j(x_2, \rho'),
\end{equation}
one can use $P_2(b) $ to calculate the rate of the production of four jets in two binary collisions. This cross section is usually written as (we give here expression for the process studied by CDF \cite{Abe:1997xk} and D0  \cite{Sonnenschein:2009nh} of production three jets and a photon where combinatoric effect of identical collisions  is absent)
\begin{equation}
{{d\sigma(p+\bar p\to jet_1+jet_2 +jet_3+\gamma)\over d\Omega_{1,2,3,4}} \over {{d\sigma(p+\bar p\to jet_1+jet_2 )\over d\Omega_{1,2}} \cdot {d\sigma(p+\bar p\to jet_3+\gamma)\over d\Omega_{3,4}}}} = {f(x_1,x_3)f(x_2,x_4)\over \sigma_{eff} f(x_1)f(x_2)f(x_3)f(x_4)},
\end{equation}
where $f(x_1,x_3), f(x_2,x_4)$ are longitudinal light-cone double parton densities and $\sigma_{eff}$ which may depend on $x_i,p_t$ is the 
``transverse correlation area''. The CDF reported 
$\sigma_{eff}=14.5 \pm 1.7^{+ \ 1.7}_{- \ 2.3} $\ mb \cite{Abe:1997xk}.
The recent D0 analysis  \cite{Sonnenschein:2009nh} reports $\sigma_{eff} = 15.1 \pm 1.9 $ mb which is very close to the CDF result. However there is a difference in the analyses - the D0 treatment is completely inclusive, while CDF was removing the events with extra jets. The correction for this extra selection may reduce the CDF result by about 35\% \cite{Treleani:2007gi}. Hence, a more detailed comparison of two data analyses is necessary. In the following we will use the value of $\sigma_{eff}=14$ mb for numerical  estimates.

   


One can express $\sigma_{eff}$ through $P_2(b) $ as
\begin{equation}
 \sigma_{\rm eff} \;\; = \;\; 
\left[ \int d^2 b \; P_2^2(b) \right]^{-1}=\frac{28\pi }{m_g^2}
\;\; \approx \;\; 34\; \mbox{mb}.
\label{uncorrelated}
\end{equation}

This number is substantially larger than experimental result though it is smaller than a naive estimate based on the e.m. form factor of the nucleon ($\sim$ 60 \, mb) \footnote{The PYTHIA Monte Carlo  reproduces the observed rate of multijet production  assuming much more narrow distribution of partons in $\rho$ than the one allowed by the measurements of the GPDs. }. 
A more than a factor two   discrepancy between the data and Eq.\ref{uncorrelated} implies {\it presence of a strong transverse correlation between 
partons in the nucleon.}  Global fluctuations of  the transverse size of nucleons  may reduce $\sigma_{eff}$ by about $\sim 20\%$ \cite{Frankfurt:2008vi} as compared to Eq.\ref{uncorrelated}.  Larger effects may arise from 
 concentration of gluons near quarks (constituent quarks) - possible reduction of 
 $\sigma_{eff}$ by a factor of about two \cite{Frankfurt:2005mc}. 
 Together these two effects may explain magnitude of $\sigma_{eff}$ observed by CDF and D0. Additional effect results from the process of the QCD evolution since the emitted partons are localized in a small transverse area near the parton involved in the dijet process. However this effect is relevant   mostly for small enough x which were practically not covered by the CDF and D0 measurements.

Though the data are consistent with the double parton distribution been a product of two single parton distributions it would be preferable to avoid need for making this  assumption.
Studies of proton (deuteron) - nucleus collisions  would be very valuable for this purpose.

\section{Multijet production in proton  - nucleus collisions}

In the case of scattering of a hadron off a nucleus the parton density of the nucleus does not change noticeably on the scale of transverse size of the projectile hadron.  Non-additive effects in the parton densities are known to be less than few \% for $0.02\leq x \leq 0.5$. Hence they could be neglected for production of jets in this x interval (correction for these effects could be easily introduced). Therefore in this kinematics  we have to take into account only transverse  correlations of
partons in individual nucleons of the nucleus.

Thus there are  two different contributions to the double parton 
scattering cross section: $\sigma_D=\sigma^1_D+\sigma^2_D$. The first one,
$\sigma_1^D$, interaction with two partons of the same nucleon in the nucleus,
is the same as for the nucleon target (the only
difference being the enhancement of the parton flux) and the
corresponding cross section is \cite{Strikman:2001gz}
\begin{equation}
\sigma^1_D=\sigma_D\int d^2BT(B)= A\sigma_D,
\label{sigma1}
\end{equation}
\par\noindent
where
\begin{equation}
T(B)=\int_{-\infty}^{\infty} dz \rho_A(r), \int T(B)d^2B=A,
\end{equation}
\label{TB}
\par\noindent
is the nuclear thickness,
as a function of the impact parameter of the
 hadron-nucleus collision $B$.



The contribution to the term in $\Gamma_A(x_1',x_2',\rho,)$ due to the
partons originated from different nucleons of the target, $\sigma^2_D$,
can be calculated {\it solely} from the geometry of the process by
observing that the nuclear density does not change within a
transverse scale  $\left<b\right> \ll R_A$. It rapidly increases with A   
$\propto \int T^2(B)d^2B$. Taking $\sigma_{eff}$ reported by  the CDF double scattering 
experiment\cite{Abe:1997xk} we finds that   the contribution of the
second term should dominate in the case of proton - heavy nucleus collisions  \cite{Strikman:2001gz}:
\begin{equation}
R\equiv {\sigma_2 \over \sigma_1 \cdot A}
  \approx  {(A -1) \over A^2} \cdot \sigma_{eff} \int T^2(b)~d^2b 
 \approx  \left.0.68\cdot  \left({A \over 12}\right)^{0.39}\right._{| A 
\geq 12, \sigma_{eff}\sim 14 mb}.
\label{t2}
\end{equation}
 Hence we predict the �Antishadowing effect�:  for A=200, and $\sigma_{eff}$=14 mb: $\sigma_{pA}/ \sigma_{pp} \approx 4$. The effect is linear in $\sigma_{eff}$.
 Measurements with a set of nuclei
would  allow to measure the double parton distributions in nucleons and also to
check  the validity of the QCD factorization for such processes which 
appears natural but which so far was not derived in pQCD.  

Recently an event generator for the configurations in nuclei including short-range correlations was developed \cite{Alvioli:2009ab}. It allows to check the accuracy of Eq.\ref{t2} for the number of collisions where partons from two different nucleons of the nucleus are involved. It was found that for $A\sim 200$ the ratio $R$ is reduced by $\sim 5\%$.




An important 
application of the discussed process would be to investigate  transverse
correlations between the nuclear partons in the shadowing  region. This
would require a selection of   both partons of the nucleus in the 
shadowing region, $x_A\leq  x_{sh} \sim 10^{-2}$. \footnote{The A-dependence
of the ratio of $\sigma_2/\sigma_1$ in the kinematics where only one of
the nuclear partons has $ x_A\leq x_{sh}$ is practically the same as for the 
case when both nuclear partons have $x\geq x_{sh}$.} Since the shadowing effect is larger at small B and since four  jet events select smaller B than two jet events the antishadowing effect should be somewhat smaller in this case (for the same  $\sigma_{eff}$). 


It is possible to extend this analysis to the case of production of six jets. We find \cite{Strikman:2001gz}:
\begin{eqnarray}
\sigma_1^T&=&\sigma_T\int d^2BT(B)=A\sigma_T,\nonumber\\
\sigma_2^T&=&\frac{1}{3!}\int G(x_1,x_2,x_3)\hat{\sigma}(x_1,x_1')
\hat{\sigma}(x_2,x_2')\hat{\sigma}(x_3,x_3')dx_1dx_1'dx_2dx_2'dx_3dx_3'
\nonumber\\
&&\times\Bigl[ G(x_1',x_2')G(x_3')+ G(x_2',x_3')G(x_1')+
G(x_1',x_3')G(x_2')\Bigr]\nonumber\\
&&\times\int d^2BT^2(B)\frac{1}{\sigma_{eff}'},\nonumber\\
\sigma_3^T&=&\frac{1}{3!}\int G(x_1,x_2,x_3)\hat{\sigma}(x_1,x_1')G(x_1')G(x_2')G(x_3')\nonumber\\
&&\times\hat{\sigma}(x_2,x_2')\hat{\sigma}(x_3,x_3')dx_1dx_1'dx_2dx_2'dx_3dx_3'.
\int d^2BT^3(B).
\end{eqnarray}
The estimate using assumption that $\sigma_1\propto 1/\sigma_{eff}^2$ leads to prediction of a factor $\sim 12$ large antishadowing for the scattering off heavy nuclei:
\begin{equation}
\sigma_1 : \sigma_2 : \sigma_3 = 1: 1.45 \cdot (A/10)^{0.5} : 0.25 (A/10) \rightarrow 1 : 6.5 : 5.
\end{equation}

It is worth noting that studying  associated hadron production in central region, nuclear fragmentation in the multijet events would  provide additional interesting information. Indeed, four (six) jet events are due to much more central collisions 
than minimal bias $pA$ collisions. As a result one expects for moderate $x_{1p},x_{2p} \le 0.3 $ an increase of the central multiplicity, larger rate of forward neutron production, etc. At the same time a new physics is possible  
for $x_{1p} + x_{2p} \ge 0.7 $ since such a trigger may  start to select configurations in the proton with fewer gluons and also of probably of a smaller transverse size?  Another interesting limit is when one x's is moderate, while a leading hadron with moderate $p_t ~ $ few GeV/c is  detected. In this case one pair of jets serves as a trigger for centrality, while the presence / suppression of the leading  hadron measures effect of fractional energy losses in the black disk limit \cite{Frankfurt:2007rn}.

\section{Multijet production in photon  - nucleus collisions}

The $pA$ collisions at the LHC are probably rather far in the future. At the same time there appears to be another opportunity to study multiple collisions with nuclei which will be available as soon as the heavy ion program will start. It comes from the possibility to study ultraperipheral collisions of nuclei where two nuclei pass each other at large impact parameters. In this case direct strong interactions are not possible though interaction via emission of the photon by one of the nucleus (which is left practically intact) is possible,   has a large cross section and can be experimentally separated from the ordinary heavy ion collisions,
see review in \cite{Baltz:2007kq}. This will allow to measure multiparton photon wave function without need to model nucleon wave function via study of the A-dependence of the multijet production. Using information about similar collisions in $\gamma p$ collisions available at HERA it will be possible to measure reliably  $\sigma_{eff}$ for different configurations of partons in the photon wave function.
For example, for the photon  component containing heavy quarks the transverse size is $\propto 1/m_Q$ is much smaller than the nucleon size, leading to $\sigma_{eff} $ determined solely by the nucleon structure. Though $\sigma_{eff}$ in this case is significantly  smaller than for $pp$ collisions, the antishadowing effect is likely to   be large enough to perform the analysis of the correlations of partons in the photon and allow a more reliable determination of $\sigma_{eff}$ for $\gamma - p $ collisions.. 


It would be interesting also to study the gap survival probability for $\gamma A$ scattering with production of one or two pairs of jets with one of the jets of each pair in the photon fragmentation region and another  one (two) across  the gap.
This would probe both the multiparton structure of the photon and the probability of the dipole to pass through the nucleus without inelastic interactions. An important advantage of the photon is that there are several handles to regulate the transverse size of the components in the photon wave function involved in the process. For example, one can select events with different $x_{\gamma}$, with leading D-mesons, etc.

 
The simplest process which allows to  track propagation of a small dipole through the strong gluon fields in the nuclei is the process $\gamma + A \to \mbox{vector meson  + rapidity ~gap + X}$ in the kinematics where $t= (p_{\gamma}- p_{VM})^2$ is large \cite{Frankfurt:2008et}. In the rest frame of the nucleus the process corresponds to a transformation of $\gamma$ to a a $q\bar q $ pair of a small transverse size $\propto 1/\sqrt{-t}$ which interacts with a target through a two gluon ladder. 
 If the gluon fields are strong enough the interaction would approach the black disk regime of complete absorption. In this limit it is  impossible for a dipole to pass though the nucleus at small impact parameters without additional inelastic interactions. This  would reduce the A-dependence of the process from $\propto A$ to $\propto A^{1/3}$. Since the gluon fields increase with increase of energy one expects a significant deviation of the A-dependence from $\propto A$ in the LHC kinematics.  The rate of the process is sufficiently high to observe it during the first heavy ion run \cite{Frankfurt:2008et}. 
Note also that this process has several practical advantages as compared to  
 he case of coherent $J/\psi$ production. Production of hadrons in  wide range of rapidities make it easier to  trigger on these events. Also, location of the gap allows to  determine on the event by event basis which of the nuclei emitted a photon. As a result it will be feasible to study the dipole - nucleus interactions up to $\sqrt{s_{\gamma p}}\sim $ 1 TeV as compared to $\sqrt{s_{\gamma p}}\sim $ 0.2 TeV for the coherent case.
 
\section{Conclusions}
Theoretical analysis of the exclusive hard phenomena studied at HERA produced a unique information about the  transverse structure of nucleon. When combined   with the  information from the experimental studies of multiparton interactions at Tevatron, it  leads to  the unambigous conclusion that   large  transverse correlations between partons are present in the nucleon. Study of multiparton interactions with nuclei  will allow to separate longitudinal and transverse correlations of partons in  nucleons and photons. In the near future such studies will be possible in the ultraperipheral photon - lead collisions at the LHC.
Similar studies can be done at RHIC  in the deuteron - gold collisions if acceptance of detectors is increased. It appears that the
 fastest way to establish how black are interactions of small dipoles at ultra high energies  will be a study of  the  rapidity gap events with large $t$ in UPC heavy ion collisions. Studies of the leading jet production in the UPC  will also allow to investigate the regime of fractional energy losses in the proximity of the black disk regime.

 
\section*{Acknowledgments}

This work was supported by the DOE grant DE-FGO2 93ER40771.  Many thanks  to my collaborators in the research reported  in this contribution: L.Frankfurt, D.Treleani, C.Weiss, M.Zhalov. I also thank L.~Sonnenschein for the information concerning D0 multiparton interaction measurements.
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