Igor M. Dremin

Wavelets and their uses

This review paper is intended to give a useful guide for those who want to apply the discrete wavelet transform in 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 the corresponding literature. The multiresolution analysis and fast wavelet transform have become a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for the achievement of a goal. Analysis of various functions with the help of wavelets allows one to reveal fractal structures, singularities etc. The wavelet transform of operator expressions helps solve some equations. In practical applications one often deals with the discretized functions, and the problem of stability of the 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 to a few examples only. The authors would be grateful for any comments which would move us closer to the goal proclaimed in the first phrase of the abstract.

Elastic scattering of hadrons

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 can be seen that provide information on 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 decrease. 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 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) 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.

Quantum chromodynamics and multiplicity distributions

The quantum chromodynamics (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 both for fixed and for running coupling constants. Characteristics have been found which are very sensitive to distributions shapes. The predictions are compared with experimental data. Evolution of the multiplicity distributions with decreasing phase space windows is considered and discussed in relation to the notions of intermittency and fractality. Some other QCD effects are briefly described.

The quark?gluon medium

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 Interaction Region of High Energy Protons

New proton-proton collision data from the LHC have considerably extended the energy range over which the structure of the proton-proton interaction region can be studied. In this paper, we combine the unitarity relation with experimental data on elastic scattering in the diffraction cone to show how the shape and the darkness of the inelastic interaction region of colliding protons change with increasing the proton energy. In particular, at LHC energies, small-impact-parameter collisions become fully absorptive, with some implications for inelastic processes as well. The possibility of changing from the black-core scenario at LHC energies to the fully transparent scenario at higher energies is discussed — a phenomenon that implies changing from the black disk to black toroid terminology. As the asymptotic behavior is approached, a different regime may arise. The parameter determining the opacity of central collisions also crucially affects the differential cross section of elastic scattering outside the diffraction cone, where all phenomenological models fail for the LHC energies. It is in this region where the ratio of the real to imaginary part of the elastic scattering amplitude in nonforward scattering becomes a decisive factor, as indeed it should according to the unitarity condition. Our results make it possible for the first time to estimate this ratio outside the diffraction cone by comparing with data for LHC energies, and it turns out to be drastically different from the values measured at forward scattering. Moreover, both real and imaginary parts are found to behave differently in different phenomenological models and in the approach based on the unitarity condition. This problem is still to be resolved. All the conclusions are made solely in the framework of the indubitable unitarity condition using experimental data on elastic proton scattering in the diffraction cone, without resorting to other theoretical methods, such as quantum chromodynamics or phenomenological models.

Critical regime of proton elastic scattering at the LHC

It is shown that the darkness of the interaction region of protons is governed by the ratio of the slope of the diffraction cone to the total cross section. At LHC energies, it becomes completely absorptive at small impact parameters. The lower limit of the ratio is determined. That imposes some restrictions on its energy behavior. It is argued that the black disk terminology should be replaced by the black torus.

Torus or black disk

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.

Multiparticle production and quantum chromodynamics

The theory of strong interactions, quantum chromodynamics (QCD), is quite successful in predicting and describing many features of multiparticle production processes at high energies. In this paper, the general perturbative QCD approach to these processes (primarily e+e−-annihilation) is briefly formulated, and associated problems are discussed. It is shown that analytical calculations at the parton level using a low-momentum cutoff are surprisingly adequate in describing experimental data on the final hadronic state in multiparticle production processes at high energies — even though the perturbative expansion parameter is not very small. More importantly, the perturbative QCD not only describes existing data but also predicts many qualitatively new and intriguing phenomena.

Towards a common origin of the elliptic flow, ridge, and alignment

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.

Cherenkov radiation and pair production by particles traversing laser beams

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.