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This manuscript is the outcome of the subgroup ``PDFs, shadowing and $pA$ collisions'' from the CERN workshop ``Hard Probes in Heavy Ion Collisions at the LHC''. In addition to the experimental parameters for $pA$ collisions at the LHC, the issues discussed are factorization in nuclear collisions, nuclear parton distributions (nPDFs), hard probes as the benchmark tests of factorization in $pA$ collisions at the LHC, and semi-hard probes as observables with potentially large nuclear effects. Also, novel QCD phenomena in $pA$ collisions at the LHC are considered. The importance of the $pA$ program at the LHC is emphasized., Comment: The writeup of the working group "PDFs, shadowing and $pA$ collisions" for the CERN Yellow Report on Hard Probes in Heavy Ion Collisions at the LHC, 121 pages. Subgroup convenors: K.J. Eskola, J.w. Qiu (theory) and W. Geist (experiment). Editor: K.J. Eskola

A model for shadowing in nuclear structure functions at small $x$ and small or moderate $Q^2$ is presented using the relation with diffraction that arises from Gribov's reggeon calculus. A reasonable description of the experimental data is obtained with no fitted parameters. A comparison with predictions from other models is performed., Comment: 4 pages, LaTeX, uses enclosed moriond.sty, 2 eps figs included using epsfig; contribution to Rencontres de Moriond: QCD and High Energy Hadronic Interactions (Les Arcs, France, March 22nd-29th 2003)

The non-linear evolution of dense partonic systems has been suggested as one of the novel physics mechanisms relevant to the dynamics of hadron-nucleus and nucleus-nucleus collisions at collider energies. Here we study to what extent the description of Cronin enhancement in the framework of this non-linear evolution is consistent with the recent observation in 200 AGeV d--Au collisions at the Relativistic Heavy Ion Collider. We solve the Balitsky-Kovchegov evolution equation numerically for several initial conditions encoding Cronin enhancement. We find that the properly normalized nuclear gluon distribution is suppressed at all momenta relative to that of a single nucleon. Calculating the resulting spectrum of produced gluons in p-A and A-A collisions, we establish that the nonlinear QCD evolution is unable to generate a Cronin type enhancement, and that it quickly erases any such enhancement which may be present at lower energies., Comment: 4 pages, Latex, 3 eps-figures

A simple model for nuclear structure functions in the region of small $x$ and small and moderate $Q^2$, is presented. It is a parameter-free extension, in the Glauber-Gribov approach to nuclear collisions, of a saturation model for the nucleon. A reasonable agreement with experimental data on ratios of nuclear structure functions is obtained. The unintegrated gluon distribution and the behavior of the saturation scale which result from this model are discussed., Comment: LaTeX2e, uses enclosed moriond.sty, 4 pages, 2 eps figures included using epsfig, contribution to the XXXVIIIth Rencontres de Moriond: QCD and Hadronic Interactions, Les Arcs, France, March 22nd-29th 2003

Nuclear structure functions at small x and small or moderate $Q^2$ are studied using the relation with diffraction on nucleons which arises from Gribov's Reggeon Calculus. A reasonable description of experimental data is obtained with no fitted parameters. A comparison with other models and predictions for future lepton-ion colliders are provided. Consequences for the reduction of multiplicities in nucleus-nucleus collisions at energies of RHIC and LHC are examined., Comment: LaTeX, 32 pages, 14 eps figures included using epsfig; 1 reference updated

Different approaches to gluon shadowing at small x are reviewed. Some available results relevant for RHIC and LHC are compared., Comment: 6 pages, LaTeX2e, uses enclosed cernrep.cls, one eps figure enclosed using graphicx, contribution to the Yellow Report on Hard Probes in Heavy Ion Collisions at the LHC

Jet and dijet rates in AB collisions at LHC energies are computed using a NLO Monte Carlo code. Uncertainties due to nuclear modifications of nucleon parton densities are analyzed. Predictions for different AB collisions at LHC energies are presented., Comment: LaTeX2e, 6 pages, uses enclosed cernrep.cls, 2 figures included using graphicx; contribution to the Yellow Report on Hard Probes in Heavy Ion Collisions at the LHC

Jet and dijet rates in pA collisions at LHC energies are computed using a NLO Monte Carlo code. Uncertainties due to the choice of scale, jet-finding algorithm, nucleon parton densities and nuclear modifications of nucleon parton densities are analyzed. Predictions for different pA collisions at LHC energies are presented., Comment: LaTeX2e, 11 pages, uses enclosed cernrep.cls, 6 figures included using graphicx; contribution to the Yellow Report on Hard Probes in Heavy Ion Collisions at the LHC

A simple model for nuclear structure functions in the region of small x and small and moderate $Q^2$, is presented. It is a parameter-free extension, in the Glauber-Gribov approach to nuclear collisions, of a saturation model for the nucleon. A reasonable agreement with experimental data on ratios of nuclear structure functions is obtained. Nuclear effects in the longitudinal-to-transverse cross section ratios are found to be small. Predictions of the model for values of $x$ smaller than those available to present experiments are given. The unintegrated gluon distribution and the behaviour of the saturation scale which result from this model are shown and discussed., Comment: LaTeX, 27 pages, 9 postscript figures included using epsfig; final version: discussion on geometrical scaling clarified, comparison of gluon densities with other approaches added, Fig. 8 redone, references added and updated, some misprints corrected, results and conclusions unchanged

We study the enhancement of $c \bar c$ pair production that takes place in central heavy ion collisions due to the formation of clusters of strings. These clusters produce heavy flavors more efficiently due to their higher color. We discuss the competition between this mechanism and the well-known screening of color charge, which, above a critical string density, reduces strongly the probability of binding these $c \bar c$ pairs to form $J/\Psi$ particles. The dependence of $J/\Psi$ suppression on the centrality shows a peak at both RHIC and LHC energies corresponding to the percolation critical density., Comment: 12 pages, revtex, 3 postscript figures

Nuclear structure functions and cross-sections for heavy flavour production in lepton-nucleus collisions are investigated in the low $x$ region accessible now or in the near future. The scattering on a heavy nucleus is described by the sum of fan diagrams of BFKL pomerons, which is exact in the high-colour limit. The initial condition for the evolution at $x=0.01$ is taken from a saturation model, which reproduces the experimental data on the proton. The $A$ dependence of the structure functions is well described by a power factor $A^\alpha$, with $\alpha$ reaching values as low as 1/2 at extremely low $x$. The total cross-sections for heavy flavour production reach values of the order of mb, and the corresponding transverse momentum distributions are sizeable up to transverse momenta larger than the initial large scale $\sqrt{Q^2+4m_f^2}$., Comment: LaTeX2e, 16 pages, 6 eps figures included using epsfig; final version, some comments added, results and conclusions unchanged

First results from RHIC on charged multiplicities, evolution of multiplicities with centrality, particle ratios and transverse momentum distributions in central and minimum bias collisions, are analyzed in a string model which includes hard collisions, collectivity in the initial state considered as string fusion, and rescattering of the produced secondaries. Multiplicities and their evolution with centrality are successfully reproduced. Transverse momentum distributions in the model show a larger $p_T$-tail than experimental data, disagreement which grows with increasing centrality. Discrepancies with particle ratios appear and are examined comparing with previous features of the model at SPS., Comment: 17 pages, LaTeX, 1 table, 4 eps figures included using epsfig; final version, comments and explanations extended, references added and updated, Fig. 1 enlarged, results and conclusions unchanged

Unintegrated gluon densities in nuclei, dipole-nucleus cross-sections and quark densities are numerically investigated in the high-colour limit, with the scattering on a heavy nucleus exactly described by the sum of fan diagrams of BFKL pomerons. The initial condition for the evolution in rapidity is quickly forgotten, and the gluon density presents a "supersaturation" pattern, as previous studies indicated. Both dipole-nucleus cross-sections and quark densities present the expected saturation features. Identifying the position in transverse momentum $l$ of the maximum of the gluon distribution with the saturation momentum $Q_s(y,b)$, at large rapidities all distributions depend only on one variable $[l/Q_s(y,b)]$ or $[rQ_s(y,b)]$., Comment: LaTeX, 13 pages, 6 figures in encapsulated postscript included using epsfig; final version, with two added references plus comments, some further clarification, results and conclusions unchanged

The NA50 collaboration data on the $J/\psi$ suppression are compared with the results obtained in a comovers approach based on the Dual Parton Model (DPM). Predictions for the $J/\psi$ suppression versus the charged multiplicity - measured in the rapidity region of the dimuon trigger - are given for SPS and RHIC energies., Comment: 4 pages, contribution to QM2001

A Monte Carlo model to simulate nuclear collisions in the energy range going from SPS to LHC, is presented. The model includes in its initial stage both soft and semihard components, which lead to the formation of color strings. Collectivity is taken into account considering the possibility of strings in color representations higher than triplet or antitriplet, by means of string fusion. String breaking leads to the production of secondaries. At this point, the model can be used as initial condition for further evolution by a transport model. In order to tune the parameters and see the results in nucleus-nucleus collisions, a naif model for rescattering of secondaries is introduced. Results of the model are compared with experimental data, and predictions for RHIC and LHC are shown., Comment: LaTeX, 48 pages, 6 tables, 15 postscript figures included using epsfig

Strangeness and baryon enhancement in heavy ion collisions are discussed in the framework of the String Fusion Model. The Monte Carlo version of this model is shown to reasonably reproduce three of the features that have been pointed out as evidences of the finding of the Quark Gluon Plasma. Namely hyperon/antihyperon enhancement, dependence of the slope of the $p_{T}$ distributions on the mass of the produced particles and on the centrality of the collision, are described. Predictions for RHIC and LHC energies are presented., Comment: RevTeX, 13 pages, 5 eps figures included using epsfig; the code, with instructions for installation and use, can be taken by anonymous ftp ftp://ftp.uco.es/pub/fa1arpen/psm-1.0.uu

We propose a new criterium for saturation of the density of partons both in nucleons and nuclei. It is applicable to any multiple scattering model which would be used to compute the number of strings exchanged in $ep$ and $eA$ collisions. The criterium is based on percolation of strings, and the onset of percolation is estimated from expectations coming from the study of heavy ion collisions at high energies. We interpret this onset as an indication of saturation of the density of partons in the wave function of the hadron. In order to produce quantitative results, a particular model fitted to describe present HERA data and generalized to the nuclear case is used. Nevertheless, with the number of scatterings controlled by the relation between inclusive and diffractive processes, conclusions are weakly model-dependent as long as different models are tuned to describe the experimental data. This constitutes a new approach, based on the eikonal description of soft hadronic collisions, and different from others which employ either perturbative QCD ideas or semiclassical methods. It offers an alternative picture for saturation in the small $Q^2$ region., Comment: LaTeX, 15 pages, 2 eps figures included using epsfig; final version, abstract and discussions enlarged, references added and updated, results unchanged; more references added

Predictions on central rapidity densities of charged particles at energies of the Relativistic Heavy Ion Collider and the Large Hadron Collider, for central collisions between the largest nuclei that will be available at these accelerators, are reviewed. Differences among the results of the existing models are discussed in relation with their underlying physical basis and with the possibilities to discriminate them., Comment: 42 pages, LaTeX, 12 postscript figures included using psfig, epsf and epsfig; review article, submitted to Int. J. Mod. Phys. A/Mod. Phys. Lett. A

It is shown that solutions to the 2nd order BFKL eigenvalue equation exist for arbitrary large real values of the complex angular momentum $j$. This corresponds to a cut in the complex $j$ plane along the whole real axis, and it makes the use of the complex angular momentum variable for the calculation of the high-energy behavior somewhat questionable. The eigenfunctions contain non-perturbative pieces which behave as $\exp(-1/\alpha_s b)$ and have no counterpart in the leading-log BFKL equation. The high-energy behavior of the 2nd order BFKL Green function as found by other authors, is reproduced by excluding these non-perturbative pieces of the eigenfunctions., Comment: 19 pages, 2 figures

In the framework of a model based on nuclear absorption plus comover interaction, we compute the $E_{T}$ distribution of the $J/\psi$ in $PbPb$ collisions at SPS and compare it with available NA50 data. Our analysis suggests that the existence of new physics (deconfinement phase transition) in the region $E_{T} \lsim 100$ GeV is unlikely and that signals of new physics should rather be searched in the region $E_T \gsim 100$ GeV. The $E_{T}$ dependence of the $J/\psi$ transverse momentum has been computed. At large $E_T$ it turns out to be much flatter in the comover approach than in a phase transition framework. Estimates of the $J/\psi$ suppression at RHIC and LHC energies are also given., Comment: 31 pages, LaTeX, 9 postscript figures included using epsfig; references updated and comments added; final version to appear in Phys. Rev. C

Production of Drell-Yan pairs and open and hidden heavy flavor on nuclear targets is examined within perturbative QCD. The effects of modifications of nucleon structure functions inside the nuclear medium are considered. Besides, nuclear dependence of charmonium and bottonium absorption is studied in the framework of the Glauber-Gribov model. The low energy limit of this approach recovers the probabilistic formula usually employed for charmonium and bottonium suppression in nuclear collisions., Comment: 21 pages, LaTeX, 12 postscript figures included using epsfig; talk given at the 5th International Workshop on Hard Probes in Nuclear Collisions (Lisbon, Portugal, September 7th-18th 1997); one reference and position of the text in page corrected

The odderon singularity is studied in perturbative QCD in the framework of the Bartels-Kwieci\'nski-Praszalowicz (BKP) equation. Arguments for the odderon intercept being exactly equal to unity are given. Besides, a variational method based on a complete system of one-gluon functions is presented. For the odderon, the highest intercept calculated by this method is $1-(N_c\alpha_{s}/\pi) 0.45$. Comparison to other calculations is shown., Comment: 9 pages in LaTeX; presented by N. Armesto at the Madrid Workshop on low x Physics (Miraflores de la Sierra, Spain, June 18th-21st 1997)

The absorptive corrections resulting from the rescattering of a heavy flavour (the so-called nuclear absorption) are usually calculated with a probabilistic formula valid only in the low energy limit. We extend this formula to all energies using a quantum field theoretical approach. For charmonium and bottonium we find that the absorptive corrections in the rigorous treatment are very similar to the ones in the probabilistic approach. On the contrary, at sufficiently high energy, open charm and bottom are absorbed as much as charmonium and bottonium - in spite of the fact that their absorptive cross-sections are zero, and therefore they are not absorbed in the probabilistic model. At high enough energies there are also absorptive corrections due to the shadowing of the nucleus structure function, which are present for all systems including Drell-Yan pair production. These shadowing corrections cancel in the low energy limit., Comment: 31 pages, 5 Postscript figures; some errors in numerical calculations for EMC effect corrected

We review the two main interpretations of $J/\psi$ suppression proposed in the literature. The phase transition (or deconfining) scenario assumes that below some critical value of the local energy density (or of some other geometrical quantity which depends both on the colliding systems and on the centrality of the collision), there is only nuclear absorption. Above this critical value the absorptive cross-section is taken to be infinite, i.e. no $J/\psi$ can survive in this hot region. In the hadronic scenario the $J/\psi$ dissociates due both to nuclear absorption and to its interactions with co-moving hadrons produced in the collision. No discontinuity exists in physical observables. We show that an equally good description of the present data is possible in either scenario., Comment: 12 pages, LaTeX, uses epsfig and ioplppt; review talk given by A. Capella at the International Symposium on Strangness in Quark Matter, Santorini (Greece), April 1997; Figs. 1 and 2 not available but can be found in Refs. 13 and 6 respectively

We present a quantitative description of $J/\psi$ and $\psi^\prime$ suppression in proton-nucleus and nucleus-nucleus collisions at CERN energies. We use a conventional hadronic framework based on nuclear absorption plus final state interaction of the $J/\psi$ or $\psi^\prime$ with co-moving hadrons., Comment: 20 pages, LaTeX, 4 postscript figures included using epsfig; calculations redone, discussions extended, references updated and added, conclusions unchanged; final version, accepted for publication in Phys. Lett. B

We calculate the A-dependence of open charm and beauty as well as Drell-Yan pair production cross sections on nuclear targets taking into account the difference of quark and gluon distributions in free and bound nucleons. The results are presented for both proton-nucleus and nucleus-nucleus collisions. For the case of heavy flavour production at comparatively low energies, if $\sigma \sim A^{\alpha}$, the values of $\alpha$ are slightly higher than unity. With increasing energy $\alpha$ decreases in all cases and becomes smaller than unity. We also show $\alpha$ as a function of Feynman-$x$ of the produced $Q\overline{Q}$ or $l^+l^-$ pair and obtain that it decreases significantly in the beam fragmentation region. The sensitivity of our results to the used set of parton distributions is also discussed., Comment: 11 pages in LaTeX plus 9 postscript figures; one short comment at the end of section 3 and one reference added

It is shown that the critical threshold for percolation of the overlapping strings exchanged in heavy ion collisions can naturally explain the sharp strong suppresion of $J/\psi$ shown by the experimental data on central Pb--Pb collisions, which does not occur in central O--U and S--U collisions., Comment: 11 pages in LaTeX plus 1 postscript figure

It is shown that the fusion of strings is a source of particle production in nucleus--nucleus collisions outside the kinematical limits of nucleon--nucleon collisions. The spectrum of different particles is compared with the high energy data on p--A collisions obtaining a reasonable agreement. Results for A--B collisions at $\sqrt{s}=19.4$ AGeV and $\sqrt{s}=200$ AGeV are given. It is shown that the fusion of strings can accelerate particles up to the highest energy detected in cosmic rays without help of any additional cosmic accelerator. Also the rise of the average shower depth of maximun for cosmic rays in the energy range between $10^{16}$ eV and $10^{19}$ eV can be explained by the same mechanism without requiring any change in the chemical composition., Comment: LaTeX, 17 pages plus 7 figures in a separate postcript file

The pomeron-pomeron interaction is studied in the perturbative approach of BFKL-Bartels. The total pomeron-pomeron cross-section is proportional to $\alpha_{s}^{4}s^{\Delta}/\sqrt {t_{1}t_{2}}$ where $\sqrt{s}$ is the c.m. energy and $t_{1,2}$ are the virtualities of the colliding pomerons. Upon calculating the coefficient the cross-section is found to be of the order 2.2 mb at $\sqrt{s}= 6$ TeV and $\sqrt{-t_{1}}\sim\sqrt{-t_{2}}\sim 1$ GeV/c., Comment: 12 pages in LaTeX plus 2 postscript files

The EMC effect is studied in the perturbative QCD hard pomeron approach. In the limit $x\rightarrow 0$ and for a nucleus with a constant density the effect is found to be a function of a single variable which combines its $x$-, $Q$- and $A$-dependence. At large $Q^{2}$ the effect dies out as $\ln Q/Q$. At small but finite $x$ a change in the anomalous dimension leads to a weaker $Q$-dependence, which almost disappears as $x$ grows and the anomalous dimension becomes small. In this approach the nuclear structure function can be expressed directly in terms of the proton one. Calculations of the EMC effect on these lines give results which agree reasonably well to the existing experimental data for $x<0.05$., Comment: 18 pages in a LaTeX file plus 8 postscript figures in 8 separate files

Arguments are presented for the odderon intercept being exactly equal to unity. A variational method is presented based on a complete system of one-gluon functions. For the odderon, the highest intercept calculated by this method is $1-(3\alpha_{s}/\pi)\,0.45$., Comment: LaTeX, 19 pages plus 1 postscript figure; some small change in the introduction

It is shown that the fusion of strings is a source of particle production in nucleus--nucleus collisions outside the kinematical limits of nucleon--nucleon collisions. This fact, together with another effect of string fusion, the reduction of multiplicities, sheds some light on two of the main problems of ultrahigh energy cosmic rays, the chemical composition and the energy of the most energetic detected cosmic rays., Comment: LaTeX, 11 pages plus 3 figures in a separate postcript file

The inclusion of string fusion in Dual Parton Model results in the appearance of diquark-antidiquark pairs in the sea with comparatively large Feynman-$x$. Such antidiquarks can fragment into fast antibaryons, thus increasing several times the corresponding yields in high energy $pp$ collisions for $x_F \sim 0.8 \div 0.9$. String fusion also results in an unusual dependence of inclusive spectra on the multiplicity of secondaries. Some numerical estimations are presented., Comment: 9 pages, LaTeX, 5 postcript figures

We calculate the A-dependence of charm and beauty production cross sections on nuclear targets taking into account the difference of quark and gluon distributions in free nucleons and in nucleus. At comparatively low energies, if $\sigma \sim A^{\alpha}$, the value of $\alpha$ is slightly higher than unity. With the growth of the initial energy the value of $\alpha$ decreases and becomes smaller than unity. We also calculate the values of $\alpha$ for different Feynman-$x$ of the produced $Q\overline{Q}$ pair and obtain that they decrease significantly in the beam fragmentation region., Comment: 7 pages (LaTeX) + 3 uuencoded-compressed-tar postcript figures in a separate file

Production of charmed and beauty mesons and baryons in hadron-nucleon collisions at not very high energy is considered in the framework of the Quark-Gluon String Model. We take into account the possible corrections to the model which disappear at asymptotically high energies but could be important at comparatively low energies, essentially in the case of heavy flavour production. The case of nuclear targets is also considered and the $A$-dependence of charmed and beauty hadron yields is calculated. Predictions for the case of beauty hadron production at $\sqrt{s}$ = 39 GeV are presented., Comment: LaTeX, 17 pages plus 6 Postscript figures available in a separate tar-compressed-uuencoded file

To calculate the intercept of the multigluon system in a symmetric spatial configuration a variational method is developed based on a complete system of one-gluon functions. The method is applied to two- and three- gluon cases to compare with the known results. The convergence turns out rather slow. Ways to improve results are discussed., Comment: 18 pages (1 table, no figures), LaTeX, Santiago preprint US-FT/9-94