Regular Series

Vol. 28 (1997), No. 12, pp. 2557 – 3078

Small-x Physics and BFKL Dynamics

Acta Phys. Pol. B 28, 2557 (1997)

page 2557 •


After a brief review of the parton model in deep inelastic lepton–proton scattering DGLAP evolution and its relationship to BFKL evolution is discussed. The dipole picture of BFKL evolution is developed in the context of heavy onium-heavy onium scattering. The BFKL evolution equation is derived and solved. At very high energies BFKL behavior leads to scattering cross sections that exceed unitarity limits. A simple picture of unitarity is described in dipole language. Finally, the present state of the experimental search for BFKL evolution is given.

A Unified BFKL and DGLAP Evolution Equations for Quarks and Gluons

Acta Phys. Pol. B 28, 2577 (1997)

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We discuss the partonic description of the HERA data for \(F_2\), paying particular attention to a recent \(k_{\rm T}\) factorization approach which unifies DGLAP and BFKL effects.

Deep Inelastic Scattering at HERA

Acta Phys. Pol. B 28, 2587 (1997)

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Precise \(\mathcal {F}_2\) measurements have been performed in deep inelastic (DIS) \(e^+p\) scattering by H1 and ZEUS over a wide range in \(x\) from 10\(^{-5}\) to 0.3 and in \(Q^2\) from 0.16 to 5000 GeV\(^2\). \(\mathcal {F}_2\), as well as the gluon momentum density \(xg(x,Q^2)\), extracted from \(d\mathcal {F}_2/d\) ln\(Q^2\) exhibit a fast rise as \(x \to 0\) which shows that the proton’s quark and gluon densities at small \(x\) are large. The data on \(\mathcal {F}_2\) as well as the first measurements of the longitudinal structure function \(\mathcal {F}_{\rm L}\) and the charm contribution to \(\mathcal {F}_2\) are in agreement with DGLAP evolution. Comparison of results obtained for very small values of \(Q^2\) with theoretical models suggests that for \(x \lt 10^{-2}\) the transition from soft to hard scattering occurs at \(Q^2\) values below \(\approx 1\) GeV\(^2\). The DIS NC and CC measurements have been extended to large values of \(Q^2 \approx M^2_W\,, M^2_W\). For \(Q^2\) above \(M^2_W\), the weak force is found to have similar strength as the electromagnetic one. Combining \(e^+p\) data from 1994–97 running for a total integrated luminosity of 58 pb\(^{-1}\) H1 and ZEUS have observed an excess of NC events over the predictions of the Standard Model (SM) for \(Q^2 \gt 15000\) GeV\(^2\). The excess occurs at large \(x\) corresponding to large masses \(M\) of the \(e\) — \(hadron\) system, \(M\) around 200–230 GeV. The statistical probability that the data are compatible with the SM is at the 0.2–1% level. The data are compared with expectations for contact interactions, leptoquarks and squarks.

Diffractive Production of C-Even Mesons in \(\gamma p\) and \(ep\) Collisions

Acta Phys. Pol. B 28, 2647 (1997)

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We consider the processes of diffractive charge-parity \(C=+1\) neutral meson production in virtual photon proton collision at high energies. Due to the opposite \(C\)-parities of photon and meson \(M^+(M+ = \eta _C\,, \pi ^0\,, a_2)\) this process probes the \(t\)-channel \(C=-1\) odderon exchange which is described here as noninteracting three-gluon exchange. The possibility to study this reactions at \(ep\) collider HERA is discussed briefly.

The Structure of the Photon in Hard Hadronic Processes

Acta Phys. Pol. B 28, 2659 (1997)

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The concept of the structure of the photon is discussed and the progress in the measurement of various structure functions of the photon as well of parton distributions in the photon is shortly reviewed.

Virtual Photon Collisions at Very High Energies

Acta Phys. Pol. B 28, 2673 (1997)

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In the framework of the dipole picture of the BFKL pomeron we calculate the total \(\gamma ^* \gamma ^*\) cross section of the virtual photons. We show that the dipole model reproduces the results obtained earlier from \(k_T\)-factorization up to the selection of the scale determining the length of the QCD cascade. The choice of scale turns out to be important for the numerical outcome of the calculations.

Single Particle Density and Density Matrix from the QCD Cascade in DLA Approximation

Acta Phys. Pol. B 28, 2683 (1997)

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The structure of the QCD gluonic cascade in configuration space is investigated. The explicit form of the inclusive single particle density in configuration space transverse coordinates is derived in the double logarithmic approximation (DLA) of QCD. The possible simplification of the multiparton density matrix formalism for DLA approach is found and discussed.

Charm Production in Deep Inelastic Lepton–Hadron Scattering

Acta Phys. Pol. B 28, 2715 (1997)

page 2715 •


A review is given of the QCD corrections to charm quark production in deep inelastic electron–proton scattering. An outline of the computation of the virtual photon–parton subprocesses, from which one obtains the heavy quark coefficient. functions, is given. The dominant production mechanisms are discussed. Further we show that the asymptotic heavy quark coefficient functions, computed in the limit \(Q^2 \gg m^2\), can be derived using the operator product expansion technique. Further we present the various schemes proposed in the literature to describe the charm component of the structure function and compare them with the most recent data from the experiments carried out at HERA.

Nucleon Structure Functions from the Instanton Vacuum: Leading and Non-Leading Twists

Acta Phys. Pol. B 28, 2751 (1997)

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We review the description of nucleon structure functions in the instanton vacuum. This includes the calculation of the twist-2 parton distributions at a low normalization point as well as higher-twist matrix elements. The instanton vacuum with its inherent small parameter, the packing fraction of the instanton medium, \(\bar \rho /R\), provides a consistent picture of the non-perturbative gluon degrees of freedom at the scale \(\bar \rho ^{-1} \simeq \) 600 MeV. The twist-2 quark and antiquark distribution are of order unity, while the twist-2 gluon distribution is of order \((\bar \rho /R)^4\). Twist-4 matrix elements determining power corrections to the Bjorken, Ells–Jaffe and Gross–Llewellyn-Smith sum rules are found to be of order \((\bar \rho /R)^4\). We present numerical estimates for the parametrically large quantities.

Quantum Field Degrees of Freedom and Asymptotics of the Skyrmion Field

Acta Phys. Pol. B 28, 2765 (1997)

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A brief introduction to the Skyrme model is given. An approximate method is proposed for constructing stationary soliton solutions of the Skyrme model with nonzero spin and isospin values. Tue vicinity of the skyrmion center is represented as a rigidly rotating field configuration and is quantized in terms of collective coordinates. Nonzero modes are taken into account at the periphery by means of perturbation theory. In this approach, the asymptotic behavior of the soliton field is consistent with the Yukawa law, and the transition to the chiral limit is smooth.

Disoriented Chiral Condensate: Theory and Phenomenology

Acta Phys. Pol. B 28, 2773 (1997)

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The basic ideas underlying the production dynamics and search techniques for disoriented chiral condensate are described.

Fixed Target Experiments with Heavy Ions at CERN

Acta Phys. Pol. B 28, 2793 (1997)

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Recent data on fixed target experiments with relativistic heavy ions at CERN SPS are reviewed.

On Space-Time Evolution of Nuclear Collisions at CERN-SPS Energies

Acta Phys. Pol. B 28, 2817 (1997)

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We make an attempt to combine information from different CERN–SPS data on nuclear interactions to get a rough picture of the space-time evolution of these collisions. Topics discussed include: formation time of hadrons, energy densities reached in heavy-ion interactions, dependence of longitudinal radii \(R_{\rm L}\) on nucleon numbers of colliding nuclei, \(e^+e^-\) and strangeness production and \(J/{\mit \Psi }\) suppression. We argue that the CERN–SPS data on heavy-ion collisions might be consistent with the picture of the space-time evolution consisting of a stage of intermediate gluons, with duration of 0.5–2.0 fm/\(c\) depending on the size of the system, followed by the stage of interacting hadron gas. The behaviour of hadron gas is supposed to be strongly influenced by the formation time of hadrons and its Lorentz dilation.

Hadronic Signatures of Deconfinement in Relativistic Nuclear Collisions

Acta Phys. Pol. B 28, 2841 (1997)

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We describe the remarkable accomplishments of the current heavy ion Pb–Pb collision experiments involving strange particle production, carried out at 158A GeV at CERN–SPS. Together with earlier 200A GeV S-induced reactions, these results imply that, at central rapidity, a novel mechanism of strangeness production arises, accompanied by excess entropy formation. We argue that: (i) these results are consistent with the formation of a space-time localized, highly excited, dense state of matter; (ii) the freeze-out properties of strange hadrons are suggestive of the formation of a color-deconfined, thermally and nearly chemically equilibrated phase, which provides at present the only comprehensive framework to describe all experimental data; (iii) the matter fireball is undergoing a transverse expansion with nearly the velocity of sound of relativistic matter; longitudinal expansion is not in the scaling regime. We present a first analysis of the recent Pb–Pb results and discuss several alternative reaction scenarios. We evaluate quantitatively strangeness production in the deconfined quark–gluon phase and obtain yields in agreement with the experimental observations made in 200A GeV S–W and 158A GeV Pb–Pb interactions. We also present a qualitative discussion of \(J/{\mit \Psi }\) results consistent with our understanding of strange particle results.

Photon and Dilepton Production from Hot Out-Off-Equilibrium Media

Acta Phys. Pol. B 28, 2873 (1997)

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The electromagnetic emissivity from QCD media away from equilibrium is studied in the framework of closed time path thermal field theory. For the dilepton rate a nonequilibrium mesonic medium is considered applying finite temperature perturbation theory for \(\pi \)–\(\rho \) interactions. The dilepton rate is derived up to the order \(g^2_{\rho }\). For hard photon production a quark gluon plasma is assumed and calculations are performed in leading order in the strong coupling constant. These examples are chosen in order to investigate the role of possible pinch terms in boson and in fermion propagators, respectively. The implications of the results for phenomenology are also discussed.

Electron Scattering and Nuclear Structure

Acta Phys. Pol. B 28, 2897 (1997)

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These lectures are divided into two parts. First, an elementary introduction to electron scattering is presented, starting with the scattering of a non-relativistic lepton from a charge distribution, the extending to a relativistic Dirac electron, and finally including the quantum dynamics of the target. The relation to gamma decay is discussed. The analysis is repeated in a covariant manner and target structure functions defined. The parity violating asymmetry in the scattering of longitudinally polarized electrons arising from the exchange of a \(Z\) is calculated. The structure functions are evaluated for deep-inelastic scattering in the quark–parton model where they exhibit Bjorken scaling. The asymmetry arising from scattering polarized nucleons on polarized nucleons is calculated in this model, as is the parity-violating asymmetry. The second part of the lecture series presents an overview of the current status of electron scattering, including a description of CEBAF. This latter material already appears in the published literature.

Skyrmions in Nuclei

Acta Phys. Pol. B 28, 2919 (1997)

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The applications of skyrmions to the derivation of the nucleon–nucleon force are now over a dozen years old, and this occasion is used to assess the degree of success of the endeavor. A very brief review is given of the use of skyrmions for determining single-baryon properties. Then their use for two-nucleon system is described, with attention to the use of the product ansatz, the full structure of the lagrangian, baryon resonance admixtures, dilatons, and exact solutions for the \(B=2\) system in order to find the sources of attraction in the central potential. We briefly address possible insights into the behavior of the nucleon in nuclei achieved from the skyrmion approach.

The Gluon Condensate in QCD at Finite Temperature

Acta Phys. Pol. B 28, 2937 (1997)

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We begin with the discussion of the relationship between the trace of the energy momentum tensor and the gluon condensate at finite temperatures. Using the recent numerical data from the simulations of lattice gauge theory for quantum chromodynamics (QCD) we present the computational evaluations for the gluon condensate. A short discussion of the properties of deconfinement and the implications on the high temperature limit are included. We also mention the case of the massive quarks where some of the properties of the condensate appear to change. We put together these results with some ideas related to the dilatation current. We draw the conclusion that the nature of the strong interactions implies that the thermodynamics of quarks and gluons never approach even at very high temperatures that of an ideal ultrarelativistic gas.

Various Shades of Blue’s Functions

Acta Phys. Pol. B 28, 2949 (1997)

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We discuss random matrix models in terms of elementary operations on Blue’s functions (functional inverse of Green’s functions). We show that such a description encapsulates the essence of a number of physical phenomena whether at/or away from the critical points and provides an elegant way for discussing them. We illustrate these assertions by borrowing on a number of recent results in effective QCD in vacuum and matter, as well as effective quantum systems of disorder with strong or weak dissipation (Hatano–Nelson localization).

Confinement of Colour by Dual Superconductivity

Acta Phys. Pol. B 28, 2995 (1997)

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The evidence from lattice that colour is confined by dual superconductivity of QCD vacuum is reviewed. Open problems are discussed.

Dual Vortices, Spontaneous Gauge Symmetry Breaking and Confinement

Acta Phys. Pol. B 28, 3009 (1997)

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We apply lattice methods to the physics of quark confinement. We exploit the close correspondence between the confinement of color flux and vortices in superconductivity.

Standard Model with Duality: Theoretical Basis

Acta Phys. Pol. B 28, 3027 (1997)

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The Dualized Standard Model which has a number of very interesting physical consequences is itself based on the concept of a nonabelian generalization to electric-magnetic duality. This paper explains first the reasons why the ordinary (Hodge) * does not give duality for the nonabelian theory and then reviews the steps by which these difficulties are surmounted, leading to a generalized duality transform formulated in loop spare. The significance of this in relation to the Dualized Standard Model is explained, and possibly also to some other areas.

Standard Model with Duality: Physical Consequences

Acta Phys. Pol. B 28, 3041 (1997)

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The Dualized Standard Model offers a natural place both to Higgs fields and to fermion generations with Higgs fields appearing as frame vectors in internal symmetry space and generation appearing as dual colour. If they are assigned those niches, it follows that there are exactly 3 generations of fermions, and that at the tree-level, only one generation has a mass (fermion mass hierarchy) while the CKM matrix is the identity. However, loop corrections lift this degeneracy giving nonzero CKM mixing and masses to fermions of the two lower generations. A recent calculation to 1-loop level, with just a few parameters, yields a very good fit. to the empirical CKM matrix and sensible values also to the lower generation masses. In addition, predictions are obtained, in areas ranging from low energy flavour-changing neutral current decays to extremely high energy cosmic rays, which are testable in experiments now being planned.

Visualization of the Topological Structure of the Vacuum in Lattice QCD

Acta Phys. Pol. B 28, 3057 (1997)

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We analyze the topological structure of the pure gluonic SU(2) vacuum at finite temperature in both phases of the theory by computing correlation functions between the topological charge density and the monopole density in maximum abelian projection. On gauge average we find a nontrivial spatial correlation between both topological objects. We show that the coexistence of monopoles and instantons also holds per gauge configuration.

Resummation of Perturbative QCD by Padé Approximants

Acta Phys. Pol. B 28, 3067 (1997)

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In this lecture I present some of the new developments concerning the use of Padé Approximants (PA’s) for resuming perturbative series in QCD. It is shown that PA’s tend to reduce the renormalization scale and scheme dependence as compared to truncated series. In particular it is proven that in the limit where the \(\beta \) function is dominated by the 1-loop contribution, there is an exact symmetry that guarantees invariance of diagonal PA’s under changing the renormalization scale. In addition it is shown that in the large fib approximation diagonal PA’s can be interpreted as a systematic method for approximating the flow of momentum in Feynman diagrams. This corresponds to a new multiple scale generalization of the Brodsky– Lepage–Mackenzie (BLM) method to higher orders. I illustrate the method with the Bjorken sum rule and the vacuum polarization function.


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