I. Dremin

The notion of wavelets is defined. It is briefly described what wavelets are, how to use them, when we do need them, why they are preferred, and where they have been applied. Then one proceeds to the multiresolution analysis and fast wavelet transform as a standard procedure for dealing with discrete wavelets. It is shown what specific features of signals (functions) can be revealed by this analysis, but cannot be found by other methods (e.g., by the Fourier expansion). Finally, some examples of practical application are given. Rigorous proofs of mathematical statements are omitted, and the reader is referred to the corresponding literature.

We review results on hadron multiplicities in high energy particle collisions. Both theory and experiment are discussed. The general procedures used to describe particle multiplicity in Quantum Chromodynamics (QCD) are summarized. The QCD equations for the generating functions of the multiplicity distributions are presented both for fixed and running coupling strengths. The mean multiplicities of gluon and quark jets, their ratio, higher moments, and the slopes of multiplicities as a function of energy scale, are among the main global features of multiplicity for which QCD results exist. Recent data from high energy e+e- experiments, including results for separated quark and gluon jets, allow rather direct tests of these results. The theoretical predictions are generally quite successful when confronted with data. Jet and subjet multiplicities are described. Multiplicity in limited regions of phase space is discussed in the context of intermittency and fractality. The problem of singularities in the generating functions is formulated. Some special features of average multiplicities in heavy quark jets are described.

Similar to electromagnetic forces, strong interactions might induce such collective effects as Cherenkov and Mach waves. Their conical structure would be responsible for specific ring-like events. The theoretical and experimental arguments in favor of these phenomena are discussed and their most important features are described.

Colliding high energy hadrons either produce new particles or scatter elastically with their quantum numbers conserved and no other particles produced. We consider the latter case here. Although inelastic processes dominate at high energies, elastic scattering contributes considerably (18-25%) to the total cross section. Its share first decreases and then increases at higher energies. Small-angle scattering prevails at all energies. Some characteristic features are seen that provide informationon the geometrical structure of the colliding particles and the relevant dynamical mechanisms. The steep Gaussian peak at small angles is followed by the exponential (Orear) regime with some shoulders and dips, and then by a power-law drop. Results from various theoretical approaches are compared with experimental data. Phenomenological models claiming to describe this process are reviewed. The unitarity condition predicts an exponential fall for the differential cross section with an additional substructure to occur exactly between the low momentum transfer diffraction cone and a power-law, hard parton scattering regime under high momentum transfer. Data on the interference of the Coulomb and nuclear parts of amplitudes at extremely small angles provide the value of the real part of the forward scattering nuclear amplitude. The real part of the elastic scattering amplitude and the contribution of inelastic processes to the imaginary part of this amplitude (the so-called overlap function) at nonforward transferred momenta are also discussed. Problems related to the scaling behavior of the differential cross section are considered. The power-law regime at highest momentum transfer is briefly described.

Content1. Introduction 2. Main facts and relations 3. The geometry of the interaction region 4. New tendencies of inelastic interactions 5. Elastic scattering outside the diffraction cone 6.

This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to corresponding literature. The multiresolution analysis and fast wavelet transform became a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for achievement of a goal. Analysis of various functions with the help of wavelets allows to reveal fractal structures, singularities etc. Wavelet transform of operator expressions helps solve some equations. In practical applications one deals often with the discretized functions, and the problem of stability of wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves by some examples only. The authors would be grateful for any comments which improve this review paper and move us closer to the goal proclaimed in the first phrase of the abstract.

A new simple measure of multiple production processes is proposed. It takes into account an eventual inhomogeneity of particle positions on the (pseudo) rapidity scale and can be used to separate prominent spike (or ring) events.

Fluctuations of the number of particles produced within small phase space bins in high energy e+e− collisions are studied in QCD using factorial moments and double trace moments. The results are discussed in connection with ideas of intermittency and fractality widely used nowadays. We stress that the quasi-power behaviour of factorial moments in hard processes, that is in the processes with large momentum transfer, is a common feature of field theories with dimensionless coupling. Explicit expressions for intermittency indices and multifractal dimensions for asymptotically high energies are given in terms of QCD anomalous dimension taking account of first subleading corrections. Running coupling effects should lead to flattening of factorial moments at very small bin sizes. A simple relation between intermittency indices and exponents of the double trace moments is derived. A purely perturbative approach based on the notion of well-developed parton cascades is formally inapplicable to a quantitative description of experiments since its intrinsic limitations seem to be hardly satisfied at available energies.

It is shown that protons become more active at the periphery with increase of their collision energy. By computing the impact parameter distribution of the proton–proton overlap function at LHC energies and comparing it with ISR (and Spp¯S for pp¯) data, we conclude that the peripheral region of protons plays an increasing role in the rise of total cross sections through multiparticle dynamics. The size of the proton as well as its blackness increase with energy. The protons become more black both in the central region and, especially, at the periphery. This effect can be related to the ridge phenomenon and to the inelastic diffraction processes at LHC energies.

The ratio of cumulant to factorial moments of experimental multiplicity distributions has been calculated for e+e− and hh interactions in a wide range of energies. As a function of the rank it exhibits an initial steep decrease and a series of oscillations around zero. Those features cannot be reproduced by the Negative Binomial Distribution. A comparable behaviour is instead predicted in high-energy perturbative QCD. The presence of a qualitatively similar behaviour for different processes and in wide energy intervals suggests speaking of an approximate scaling of the cumulant to factorial moment ratio.

Usual. Multiperipheral schemes are briefly reviewed. Different models (and, especially, the multicluster production model) are treated within the multiperipheral approach. General qualitative and semiquantitative results as well as their possible modifications are discussed and compared with experiment. Quantitative applications of the theory to many particle production experiments are considered. Unusual. In one of Dostoevsky's novels there is a servant who splits all mankind into two parts: his master and himself, and all other scum. We shall not divide all hadron multiple production models into the multiperipheral cluster model and all others. Nevertheless, we try to show that the former now gives a framework for a unified description of high multiplicity events. Let the adherents of other approaches do the same.

The properties of the quark-gluon medium observed in high energy nucleus-nucleus collisions are discussed. The main experimental facts about these collisions are briefly described and compared with data about proton-proton collisions. Both microscopic and macroscopic approaches to their description are reviewed. The chromodynamics of the quark-gluon medium at high energies is mainly considered. The energy loss of partons moving in this medium is treated. The principal conclusion is that the medium possesses some collective properties which are crucial for understanding the experimental observations.

The quantum-field theory based on the Bethe-Salpeter equation is applied to investigate properties of the leading vacuum trajectory for the elastic amplitude and to describe high-energy inelastic processes. It is shown that it naturally explains the production of fireballs at very high energies. Parameters of the leading trajectory are related to characteristics of inelastic processes.

The energy evolution of the average multiplicities of quark and gluon jets is studied in perturbative QCD. Higher order (3NLO) terms in the perturbative expansion of equations for the generating functions are found. First and second derivatives of average multiplicities are calculated. The mean multiplicity of gluon jets is larger than that of quark jets and evolves more rapidly with energy. It is shown which quantities are most sensitive to higher order perturbative and nonperturbative corrections. We define the energy regions where the corrections to different quantities are important. The latest experimental data are discussed.

Quarkonia—bound systems of a heavy quark and an antiquark—are described successfully by nonrelativistic potential models. A review is given of the various phenomenological potentials which have been proposed for the interaction of quarks, their application to the description of all the possible characteristics of charmonium and bottomonium, and the corresponding predictions for the hypothetical toponium. The role of relativistic corrections and nonperturbative effects is discussed briefly.

We analyze the multiplicity distributions of charged particles at Tevatron ($p\overline{p}$) and LHC ($pp$) energies in the framework of the independent pair parton interactions model. It is shown that the number of soft pair parton interactions (and therefore the density of the partonic medium) is large and increases with energy. The mean multiplicity at each parton interaction grows also with energy. This growth depends on the width of the rapidity window. Similar conclusions are obtained in the multiladder exchange model.

The unitarity condition unambigously requires the Orear region to appear in between the diffraction cone at low transferred momenta and hard parton scattering regime at high transferred momenta in hadron elastic scattering. It originates from rescattering of the diffraction cone processes. It is shown that such region has been observed in the differential cross section of the elastic pp-scattering at \sqrt s=7 TeV. The Orear region is described by exponential decrease with the scattering angle and imposed on it damped oscillations. They explain the steepening at the end of the diffraction cone as well as the dip and the subsequent maximum observed in TOTEM data. The failure of several models to describe the data in this region can be understood as improper account of the unitarity condition. It is shown that the real part of the amplitude can be as large as the imaginary part in this region. The overlap function is calculated and shown to be small outside the diffraction peak. Its negative sign there indicates the important role of phases in the amplitudes of inelastic processes.

Experimental data on proton-proton interactions in high energy collisions show quite a special and unexpected behaviour of the proportion of elastic scattering compared to inelastic processes with increasing energy. It decreases at the beginning (at comparatively low energies) but then starts increasing. From Intersecting Storage Rings (ISR) energies of 23.5 - 62.5 GeV up to higher energies 7 - 13 TeV at the Large Hadron Collider (LHC) it increases by a factor more than 1.5! According to intuitive classical ideas we would expect a stable tendency with increasing proportion of the break-down of protons compared to their survival probability. One can assume that either the asymptotic freedom or the extremely short time of flight of high energy protons through each other are in charge of such a surprising effect. The unquestionable principle of unitarity combined with the available experimental data on elastic scattering is used to get new conclusions about the shape of the interaction region of colliding protons. Its evolution at present energies is considered. Some predictions about its behaviour at even higher energies are described with different assumptions on relative roles of elastic scattering and inelastic processes. The shape can transform rather drastically if the proportion of elastic processes keeps rising. This unexpected property leads to an unexpected corollary. The possible origin of the effect and its interrelation to the strong interaction dynamics are speculated.

We suggest a new alternative model of positron origin in the Galaxy. It is shown in our model that interactions of the electromagnetic fields of colliding ions (ultraperipheral ion collisions) can contribute to the total production of Galactic positrons. The corresponding cross-section is estimated by using the Born approximation and the equivalent photon method. This process of ion collisions dominates in the range of subrelativistic energies and produces positrons with energies of several MeV. Despite its low efficiency, as it requires more than 0.1 erg to produce a single positron, this process may be an effective source of positrons in the Galactic medium.

We propose to use multidimensional wavelet analysis for pattern recognition in high multiplicity events and consider some examples. Wavelet analysis reveals clustering phenomena in multiparticle production processes. Results of event-by-event studies of some nucleus–nucleus collisions at 158A GeV are reported. The clusterization topology is found to be different in different events. Typical patterns include jets and ring-like structures. They are demonstrated here for five high multiplicity events.

Quantum chromodymamics (QCD) approach to the problem of multiplicity distributions in high energy particle collisions is described. The solutions of QCD equations for generating functions of the multiplicity distributions in gluon and quark jets are presented for both fixed and running coupling constants. The newly found characteristics very sensitive to distribution shapes is discussed. The predictions are confronted to experimental data. Evolution of the multiplicity distributions with decreasing phase space windows is considered and discussed in connection to the notions of intermittency and fractality. Some other QCD effects are briefly described.

Experimental findings by CMS on properties of jets and underlying events at high multiplicities in proton–proton interactions at 7 TeV are interpreted as an indication of increasing role of central collisions with small impact parameters. The value of the average impact parameter of the pp collisions as a function of the soft hadron multiplicity is estimated. We find an indication that the rates of different hard processes observed by CMS and ALICE universally depend on the underlying event charged-particle multiplicity until it becomes four times higher than average. It is shown that the increase of the overlap area of colliding protons is not sufficient to explain the rate of jet production in events with charged-particle multiplicity that is more than three times higher than average. New mechanisms are necessary, like interaction of protons in rare configurations of higher than average gluon density. Such mechanisms are not included in the present Monte Carlo event generators. Further studies are proposed.

Analytic predictions for the energy dependence of the mean multiplicities in gluon and quark jets are presented at the next-to-next-to-next-to-leading order (3NLO) of perturbative QCD and are compared to experiment. The 3NLO correction to the gluon jet multiplicity is found to be small. The corresponding theoretical expression provides a good description of available gluon jet measurements. In contrast, the 3NLO correction to the quark jet multiplicity is large and the theoretical expression does not describe the data accurately. It is shown that the well known success of the next-to-leading order (NLO) approximation in describing the energy evolution of quark jet multiplicity can be attributed to the equivalence of the quark and gluon expressions at NLO to within a constant factor, and to almost constant contributions from higher order terms to the gluon jet result.

A model of independent pair parton interactions is proposed, according to which, hadron interactions are represented by a set of independent binary parton collisions. The final multiplicity distribution is described by a convolution of the negative binomial distributions in each of the partonic collisions. As a result, it is given by a weighted sum of negative binomial distributions with parameters multiplied by the number of active pairs. Its shape and moments are considered. Experimental data on multiplicity distributions in high energy $p\bar p$ processes are well fitted by these distributions. Predictions for the CERN Large Hadron Collider and higher energies are presented. The difference between $e^+e^-$ and $p\bar p$ processes is discussed.

It is shown that a previously derived angular dependence of the differential cross sections of large-angle elastic scattering fits very well with experimental data on pp-scattering at momenta 8.1, 9.2, 10.1, 11.1 and 12.1 GeV/c. Results of a least-square test are given. A shoulder in the pp differential cross section at 1.2 ⪅ pθ ⪅ 1.4 GeV · rad/c is predicted at these energies. It connects the diffraction cone with the intermediate-angle region. The existence of this shoulder is supported by recent experimental data at 19.2 and 21.1 GeV/c. Very large angles (near 12π) are also discussed. The general variation of pp differential cross sections as functions of both angle and momentum is drawn.

Recent experimental results about the energy behavior of the total cross sections, the share of elastic and inelastic contributions to them, the peculiar shape of the differential cross section and our guesses about the behavior of real and imaginary parts of the elastic scattering amplitude are discussed. The unitarity condition relates elastic and inelastic processes. Therefore it is used in the impact-parameter space to get some information about the shape of the interaction region of colliding protons by exploiting new experimental data. The obtained results are described.

It is advocated that geometry of the interaction region of two heavy nuclei colliding at large impact parameters is important for the relative role of light-by-light scattering and QCD-initiated processes. Exclusive production of resonances is possible by dense electromagnetic fields in the interior space between the nuclei. The cross-section of the two-photon processes is evaluated and some examples are considered. It is speculated that the exclusive production of [Formula: see text]-mesons by “two-photon” processes forbidden by the Landau–Yang rule may be allowed within strong magnetic fields due to odd number of photons becoming involved.

Multiresolution wavelet analysis of pressure variations in a gas turbine compressor reveals the existence of precursors of stall and surge processes. Signals from eight pressure sensors positioned at various places within the compressor were recorded and digitized in three different operating modes in stationary conditions with a recording interval of 1 ms during 5–6 s. It has been discovered that there exists a scale of 32 intervals over which the dispersion (variance) of the wavelet coefficients shows a remarkable drop of about 40% for more than 1 s prior to the development of the malfunction. A shuffled sample of the same values of the pressure does not show such a drop demonstrating the dynamical origin of this effect. Higher order correlation moments reveal different slopes in these two regions differing by the variance values. The log–log dependence of the moments does not show clear fractal behavior because the scales of 16 and 32 intervals are not on the straight line of monofractals. This is a clear indication of the nonlinear response of the system at this scale. These results provide a means for automatic regulation of an engine, preventing possible failures.

The collective effect of emission by the forward moving partons of high energy Cherenkov gluons in nucleus-nucleus collisions at RHIC and LHC energies is considered. It can reveal itself as peaks in the pseudorapidity distribution of jets at midrapidities or as a ring-like structure of individual events in event-by-event analysis. The pseudorapidity distribution of centers of dense isolated groups of particles in HIJING model is determined. It can be considered as the background for Cherenkov gluons. If peaks above this background were found in experiment, they would indicate new collective effects.

The shapes of the inelastic interaction region of protons with energies $$\sqrt{s}=7$$ and 13 GeV are considered. The unitarity condition with different assumptions about the elastic scattering amplitude is used. It is shown that the sign of the imaginary part of the amplitude at large transferred momenta is especially important for the conclusion about this shape at small impact parameters (central collisions).

It is shown that Cherenkov radiation can be observed at TESLA in electron collisions with optical laser pulses. The prospects for it to be observed at SLC, LEP, LHC and RHIC are discussed. The conclusions are compared with results for pair production.

The coherent hadron production analogous to Cherenkov radiation of photons gives rise to the ring-like events. Being projected on the ring diameter they produce the two-bump structure recently observed for the away-side jets at RHIC. The position of the peaks and their height determine such properties of the hadronic medium as its nuclear refractive index, the parton density, the free path length and the energy loss of Cherenkov gluons. Cherenkov gluons may be responsible for the asymmetry of dilepton mass spectra near ρ-meson observed in experiment. Beside comparatively low energy gluons observed at RHIC, there could be high energy gluons at LHC, related to the high energy region of positive real part of the forward scattering amplitude and possessing different characteristics. This would allow to scan (x, Q 2 )-plane determining the parton densities in its various regions.

It is shown that the interaction region of colliding protons appears almost totally absorbing (black) at impact parameters to 0.4–0.5 fm and LHC energy of 7 TeV. The blackness of the proton interaction region for central interactions is completely defined by the ratio of the diffraction cone slope B in elastic scattering to the total cross section. The corresponding parameter is close to unity at LHC energies. The behavior of this ratio at higher energies will determine whether the interaction region structure will remind a torus or a black disk. Recent phenomenological fittings of experimental data at 7 TeV give no way of distinguishing these cases due to inaccuracies of experiments and uncertainties of the extrapolation to unmeasured regions of transferred momenta.

Abstract It is argued that the dynamics of the elastic scattering of high-energy protons at intermediate transferred momenta changes with the energy increase. It evolves from the multiple scattering at the external layer for energies about 10 GeV to the double scattering at the two subsequent layers within the colliding protons for energies about 10 TeV. The active role of the overlap function in the unitarity condition is emphasized in this context.

The spectrum of Cherenkov gluons is calculated within the framework of in-medium QCD. It is compared with experimental data on the double-humped structure around the away-side jet obtained in the experiments STAR and PHENIX at RHIC. The values of the real and imaginary parts of the permittivity of the medium created in ultrarelativistic nuclear collisions are obtained from these fits. It is shown that accounting for an additional smearing of transverse momenta of Cherenkov gluons in the medium allows to provide a very good description of the experimental data.

The equations of in-medium gluodynamics are proposed. Their classical lowest order solution is explicitly shown for a color charge moving with constant speed. For nuclear permittivity larger than 1 it describes emission of Cherenkov gluons resembling the results of classical electrodynamics. The choice of nuclear permittivity and Lorentz invariance of the problem are discussed. Effects induced by the transversely and longitudinally moving (relative to the collision axis) partons at LHC energies are described.

Surprisingly enough, the ratio of elastic to inelastic cross sections of proton interactions increases with energy in the interval corresponding to ISR $$\rightarrow $$ LHC (i.e. from 10 to 10 $$^4$$ GeV). That leads to special features of their spatial interaction region at these and higher energies. Within the framework of some phenomenological models, we show how the particular ranges of the transferred momenta measured in elastic scattering experiments expose the spatial features of the inelastic interaction region according to the unitarity condition. The difference between their predictions at higher energies is discussed. The notion of central and peripheral collisions of hadrons is treated in terms of the impact-parameter description. It is shown that the shape of the differential cross section in the diffraction cone is mostly determined by collisions with intermediate impact parameters. Elastic scattering at very small transferred momenta is sensitive to peripheral processes with large impact parameters. The role of central collisions in the formation of the diffraction cone is less significant.

Recent experimental data on elastic scattering of high energy protons show that the critical regime has been reached at LHC energies. The approach to criticality is demonstrated by increase of the ratio of elastic to total cross sections from ISR to LHC energies. At LHC it reaches the value which can result in principal change of the character of proton interactions. The treatment of new physics of hollowed toroid-like hadrons requires usage of another branch of the unitarity condition. Its further fate is speculated and interpreted with the help of the unitarity condition in combination with present experimental data. The gedanken experiments to distinguish between different possibilities are proposed.

The goals of the physics to be studied at the Large Hadron Collider (LHC) are very impressive. Four major experimental installations are ready to compete in obtaining and analyzing the data from high-energy hadron collisions. The main hope is to answer the most intricate questions ever asked concerning the most fundamental problems of matter and its fundamental forces and space structure. The design of the LHC and its four detectors is briefly described. We then review the main facts revealed previously by experimentalists at other accelerators. The most pertinent topics and the stage-by-stage plans for LHC investigations are discussed. Further prospects for high-energy physics are outlined.

It is argued that PHENIX collaboration observed for the first time the radiation of the longitudinal wake oscillations formed behind the parton penetrating the quark–gluon medium. It shifts the maximum of a hump in two-particle correlations and changes its width in the case of some special orientation of the trigger particle.

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Several new experimental discoveries in high energy proton interactions, yet unexplained by QCD, are discussed in the paper. The increase of the cross sections with increasing energy from ISR to LHC, the correlation between it and the behavior of the slope of the elastic diffraction cone, the unexpected increase of the survival probability of protons in the same energy range, the new structure of the elastic differential cross section at rather large transferred momenta (small distances) and the peculiar ridge effect in high multiplicity inelastic processes are still waiting for QCD interpretation and deeper insight in vacuum.

The unitarity condition in the impact parameter space is used to obtain some information about the shape of the interaction region of colliding protons. It is shown that, strictly speaking, a reliable conclusion can be gained only if the behavior of the elastic scattering amplitude (especially, its imaginary part) at all transferred momenta is known. This information is currently impossible to obtain from experimentation. In practice, several assumptions and models are used. They lead to different results as shown below.

Abstract Long-distance ultraperipheral collisions of two relativistic ions are considered. Clouds of photons surrounding the ions are responsible for their distant electromagnetic interaction. The perturbative approach and the method of equivalent photons are described. It is shown that the total cross section of these collisions rapidly increases with an energy increase and is especially large for heavy ions. Some experimental data and their comparison with theoretical approaches are described. Further proposals are discussed .

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It is proposed to use the energy behavior of mean multiplicities of jets propagating in a nuclear medium as the thermometer of this medium during the collision phases. The qualitative effects are demonstrated in the framework of the fixed coupling QCD with account of jet quenching.

Cherenkov gluons may be responsible for the asymmetry of dilepton mass spectra near ρ-meson observed in experiment. They can be in charge only for dileptons in the low-mass wing of the resonance. Therefore the dilepton mass spectra are flattened there and their peak is slightly shifted to lower masses compared with the in-vacuum ρ-meson mass. This feature must be common for all resonances.

It is argued that the very first signatures of the approach to the black disk asymptotical limit in hadron collisions may be observed in the differential cross section of elastic scattering. The exponentially decreasing with the angle (or |t| ) regime beyond the diffraction peak will become replaced by an oscillatory behavior or by the power-like falloff. Some estimates of energies where this can happen are presented.

Spectra of unbound electron–positron pairs (dielectrons, in brief) and photons from decays of parapositronia produced in ultraperipheral collisions of electrically charged objects are calculated. Their shapes at energies of the NICA collider are demonstrated. Soft dielectrons and photons are abundantly produced. The relevance of these processes to the astrophysical problem of cooling electron–positron pairs and the intense emission of 511 keV photons from the Galactic center is discussed.

Abstract It is shown that production of low-mass $$e^+e^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>-</mml:mo> </mml:msup> </mml:mrow> </mml:math> -pairs in ultraperipheral nuclear collisions is enhanced due to the Sommerfeld–Gamow–Sakharov (SGS) factor. This effect is especially strong near the threshold of creation of unbound $$e^+e^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>-</mml:mo> </mml:msup> </mml:mrow> </mml:math> -pairs with low masses in the two-photon fusion. Coulomb attraction of the non-relativistic components of such pairs may lead to the increased intensity of 511 keV photons. It can be recorded at the NICA collider and has some astrophysical implications. The analogous effect can be observed at LHC in dilepton production.

The pseudorapidity distribution of centers of dense isolated groups of particles in HIJING model is determined. It can be considered as the background for Cherenkov gluons. If peaks over this background were found in experiment, they would indicate the onset of new collective effects.

The PMTs of the CMS Hadron Forward calorimeter were found to generate a large size signal when their windows were traversed by energetic charged particles. This signal, which is due to Čerenkov light production at the PMT window, could interfere with the calorimeter signal and mislead the measurements. In order to find a viable solution to this problem, the response of four different types of PMTs to muons traversing their windows at different orientations is measured at the H2 beam-line at CERN. Certain kinds of PMTs with thinner windows show significantly lower response to direct muon incidence. For the four anode PMT, a simple and powerful algorithm to identify such events and recover the PMT signal using the signals of the quadrants without window hits is also presented. For the measurement of PMT responses to Čerenkov light, the Hadron Forward calorimeter signal was mimicked by two different setups in electron beams and the PMT performances were compared with each other. Superior performance of particular PMTs was observed.

It is shown that the soft mechanism of multiparticle production by Lund hadronic strings formed by colliding constituent degrees of freedom generates a shape of angular correlations similar to the ridge structure observed in the pp collisions at 7 TeV at the LHC.

Spike production is studied in pp and 7r+p/K+p collisions in the beam-momentum range of 205-360 GeV/c. The pseudorapidity distribution of spike centers exhibits two narrow peaks in pp interactions, while having one wide bump in irp and Kp interactions. The position of the peaks is consistent with the expectation from a model of coherent gluon radiation at finite length. The interference between the quark color amplitudes obtained within this model causes two off-center peaks in pp data, but only one central peak in irp and Kp data.

It is argued that the cross sections of ultraperipheral interactions of heavy nuclei can become comparable in value to those of their ordinary hadronic interactions at high energies. Simple estimates of corresponding “preasymptotic energy thresholds” are provided. The method of equivalent photons is compared with the perturbative approach. The situation at NICA/FAIR energies is discussed.

A review of the experimental data on correlations and number fluctuations in multiparticle production at high energies. The author discusses new concepts of intermittency and fractality that can be used to describe these processes.

The event-by-event analysis of multiparticle production in high energy hadron and nuclei collisions can be performed using the discrete wavelet transformation. The ring-like and jet-like structures in two-dimensional angular histograms are well extracted by wavelet analysis both for toy models and in Monte Carlo simulated events of nucleus–nucleus collisions at LHC energies. For the first time the method is applied to the jet-like events. The jet positions are located quite well by the discrete wavelet transformation of angular particle distribution even in the presence of the underlying soft process background in nucleus–nucleus collisions at LHC energies.

The quark-gluon medium described by QCD equations is considered at high energies. Within the assumptions of the linear response theory the chromopermittivity of the medium is introduced and it is argued that it exceeds 1 at TeV energies. The dispersion equations show that the proper modes of the medium reveal instability and the parton currents traversing it induce the emission of Cherenkov gluons. Their distributions at LHC can differ from those typical for lower energies of RHIC because they are determined by the high energy dependence of the chromopermittivity while the latter ones arise due to collective resonance excitations. The distinction between Cherenkov gluons and Mach waves is discussed.

It is claimed that elliptic flow, ridge and alignment are effects of azimuthal asymmetry, which have a common origin evolving with primary energy and stemming from the general structure of field-theoretical matrix elements. It interrelates a new ridge-phenomenon, recently found at the LHC and RHIC, with known coplanarity feature observed in collider jet physics as well as in cosmic ray studies.

The event-by-event analysis of multiparticle production in high energy hadron and nuclei collisions can be performed using the discrete wavelet transformation. The ring-like and jet-like structures in two-dimensional angular histograms are well extracted by wavelet analysis. For the first time the method is applied to the jet-like events with background simulated by event generators, which are developed to describe nucleus-nucleus collisions at LHC energies. The jet positions are located quite well by the discrete wavelet transformation of angular particle distribution even in presence of strong background.

It is shown that the regime of pp-interactions at 7 TeV is a critical one. The LHC data about elastic pp-scattering at 7 and 8 TeV are used to get some information about both elastic and inelastic profiles of pp-collisions. They are discussed in the context of two phenomenological models which intend to describe the high energy pp-data with high accuracy. Some predictions following from these models for an LHC energy of 13 TeV and for an energy of 95 TeV of the newly proposed collider are discussed. It is claimed that the center of the inelastic interaction region will become less dark with an increase of energy albeit very slowly.

Using the unitarity relation in combination with experimental data about the elastic scattering in the diffraction cone, it is shown how the shape and the darkness of the inelastic interaction region of colliding protons change with increase of their energies. In particular, the collisions become fully absorptive at small impact parameters at LHC energies that results in some special features of inelastic processes. Possible evolution of this shape with the dark core at the LHC to the fully transparent one at higher energies is discussed that implies that the terminology of the black disk would be replaced by the black toroid. The approach to asymptotics is disputed. The ratio of the real to imaginary parts of the nonforward elastic scattering amplitude is briefly discussed. All the conclusions are only obtained in the framework of the indubitable unitarity condition using experimental data about the elastic scattering of protons in the diffraction cone without any reference to quantum chromodynamics (QCD) or phenomenological approaches.

It is proposed to classify multiple production processes according to the Renyi dimensions of the (pseudo) rapidity distribution of secondary particles in individual events. The method is illustrated by some examples.

The general features of high-energy collisions of elementary particles are outlined. ft is argued that multiple production occurs through the production of hadronic clusters. The history and present status of the cluster concept are surveyed.

A brief survey of the strong interaction theory is presented. The basic principles of quantum chromodynamics and phenomenological approaches to strong interaction processes are described. Their predictions and recent achievements in describing experimental data at high energies are considered.

The nuclear index of refraction, the density of partons, their free path length and energy loss in the matter created in heavy-ion collisions at RHIC are estimated within the suggestion that the emission of Cherenkov gluons is responsible for the observed two-bump structure of the angular distribution of hadrons belonging to the companion (away-side) jet.

Theoretical proposals of scaling laws for the differential elastic scattering cross sections of protons are confronted with experimental data over a wide energy range. Different combinations of the transferred momentum and energy resulting from the solution of the definite partial differential equation are attempted as scaling variables. Reasonable scaling of the differential cross sections in the diffraction cone has been shown for one of these variables. The violation of the geometrical scaling is ascribed to the increase of the proton blackness with energy. The origin of high-t region violations of scaling laws is discussed.

The s-channel unitarity condition for the imaginary part of the hadronic elastic scattering amplitude outside the diffraction peak is studied within different assumptions about the behavior of its real part. The integral equation for the imaginary part is derived with the asymptotical expression for the real part inserted in the unitarity condition. The conclusions about the asymptotical approach to the black disk limit and possible zeros of the imaginary part of the amplitude are obtained. Their relation to the present day experiments is discussed.

The properties of the quark-gluon medium observed in high energy nucleus-nucleus collisions are discussed. The main experimental facts about these collisions are briefly described and compared with data about proton-proton collisions. Both microscopic and macroscopic approaches to their description are reviewed. The chromodynamics of the quark-gluon medium at high energies is mainly considered. The energy loss of partons moving in this medium is treated. The principal conclusion is that the medium possesses some collective properties which are crucial for understanding the experimental observations.

CONTENTS 1. Introduction 381 2. A Brief Review of the Experimental Data at Accelerator Energies... 382 3. The Statistical-Hydrodynamic Approach 386 4. Fragmentation Models 388 5. The Multiperipheral Model and the Reggeon Scheme 389 6. Quark Models 393 7. Comparison with the Experimental Data 394 8. Interactions with Nuclei 397 9. The Space-Time Picture 399 10. Very High Energies 401 11. Conclusions 402 Cited Literature 402

Properties of distributions of the number of trades in different intraday time intervals for five stocks traded in MICEX are studied. The dependence of the mean number of trades on the capital turnover is analyzed. Correlation analysis using factorial and Hq moments demonstrates the multifractal nature of these distributions as well as some peculiar changes in the correlation pattern. Guided by the analogy with the analysis of particle multiplicity distributions in multiparticle production at high energies, an evolution equation relating changes in capital turnover and a number of trades is proposed. We argue that such equation can describe the observed features of the distribution of the number of trades in the stock market.

The equations of in-medium gluodynamics are proposed. Their classical lowest-order solution is explicitly shown for a color charge moving with constant speed. For chromopermittivity larger than 1 it describes emission of Cherenkov gluons resembling results of classical electrodynamics. The values of the real and imaginary parts of the chromopermittivity are obtained from the fits to experimental data on the double-humped structure around the away-side jet obtained at RHIC. The dispersion of the chromopermittivity is predicted by comparing the RHIC, SPS, and cosmic-ray data. This is important for LHC experiments. Cherenkov gluons may be responsible for the asymmetry of dilepton mass spectra near ρ meson observed in the SPS experiment with excess in the low-mass wing of the resonance. This feature is predicted to be common for all resonances. The “color rainbow” quantum effect might appear according to higher-order terms of in-medium QCD if the chromopermittivity depends on color.

Threshold behavior of the cross-sections of ultraperipheral nuclear interactions is studied. Production of [Formula: see text] and [Formula: see text] pairs as well as [Formula: see text] and parapositronium is treated. The values of corresponding energy thresholds are presented and the total cross-sections of these processes at the newly constructed NICA and FAIR facilities are estimated.

Rotation of such objects as an atomic nucleus or a chromodynamical string can result in specific effects in scattering processes and multiparticle production. Secondary fragments of the rotating nucleus or of the decaying string can move like stones thrown from a sling. That would be detected as the azimuthal asymmetry of particle distributions in individual events. Non-classical states of the created particles like the Schr"\{o}dinger cats are produced. Some classical and quantum-mechanical estimates of possible effects are given. Experimental facts which can be used for their verification are discussed.

A brief survey of theoretical approaches to description of multiplicity distributions in high energy processes is given. It is argued that the multicomponent nature of these processes leads to some peculiar characteristics observed experimentally. Predictions for LHC energies are presented. It is shown that similarity of the energy dependence of average multiplicities in different reactions is not enough alone to suggest the universal mechanism of particle production in strongly-interacting systems. Other characteristics of multiplicity distributions depend on the nature of colliding partners.

The impact of different assumptions about high energy behavior of the total cross section of proton-proton interactions on the ratio of the real to imaginary part of the forward elastic scattering amplitude is analyzed. It is shown how experimental data about this ratio at LHC energies can help in the proper choice of the asymptotic dependence of the total cross section.

A short review of the two recently analyzed collective effects in dense non-Abelian matter, the photon and dilepton production in nonequilibrium glasma and polarization properties of turbulent Abelian and non-Abelian plasmas, is given.

It is shown that the soft mechanism of multiparticle production by Lund hadronic strings formed by colliding constituent degrees of freedom generates a shape of angular correlations similar to the ridge structure observed in the pp collisions at 7 TeV at the LHC.

The partial differential equation for the imaginary part of the elastic scattering amplitude is derived. It is solved in the black disk limit. The asymptotical scaling behavior of the amplitude coinciding with the geometrical scaling is proved. Its extension to preasymptotical region and modifications of scaling laws for the differential cross section are considered.

The second and third factorial moments of multiplicity distributions for gluon and quark jets are obtained from the equations for generating functions. These equations are derived from QCD with a running coupling constant, and the solutions are calculated to order $(\sqrt{{\ensuremath{\alpha}}_{s}}{)}^{3},$ i.e., the next-to-next-to-next-to leading order. The results are confronted with experimental data. A general discussion on high order corrections revealed by such an approach is given. Other possible corrections and approaches are discussed as well.

It is argued that the experimentally observed phenomenon of asymmetric shapes of vector mesons produced in nuclear media during high-energy nucleus-nucleus collisions can be explained as Fano-Feshbach resonances. It has been observed that the mass distributions of lepton pairs created at meson decays decline from the traditional Breit-Wigner shape with some excess in the low-mass wing of the resonance. It is clear that the whole phenomenon is related to some interaction with the nuclear medium. Moreover, it can be further described in quantum mechanics as the interference of direct and continuum states in the Fano-Feshbach effect. To reveal the nature of the interaction it is proposed to use a phenomenological model of the additional contribution due to Cherenkov gluons. They can be created because of the excess of the refractivity index over 1 just in the low-mass wing as required by the classical Cherenkov treatment. In quantum mechanics, this requirement is related to the positive real part of the interaction amplitude in this wing. The corresponding parameters are found from the comparison with $ \rho$ -meson data and admit reasonable explanation.