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It is suggested that the Wick rotation angle is a genuine dynamical degree of freedom, rather than just a technical device needed to improve the convergence of functional integrals. The one-loop effective potential \(V(\theta )\) of the Wick angle \(\theta \) is calculated, using heat-kernel regularisation. It is found that when the number of fermionic degrees of freedom exceeds the number of bosonic degrees of freedom, the real part of \(V(\theta )\) is minimized, and the imaginary part is stationary, uniquely in \(D = 4\) dimensions, at the value \(\theta = \pi \), corresponding to Lorentzian signature.

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The lecture is an introduction to understanding pion production in semiclassical terms.

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We analyse the QCD chiral phase transition in the nonlinear and linear \(\sigma \)-model. The strategy is the same in both cases. We fix the parameters of the effective meson theory at temperature \(T = 0\) and extrapolate the models to temperatures in the vicinity of the phase transition. The linear \(\sigma \)-model in SU(3)\(\times \)SU(3) gives a first order phase transition at \(T_c=164\) MeV. At this temperature chiral SU(2)\(\times \)SU(2) is restored. The discontinuity in the energy density is small \(\Delta \epsilon /\epsilon \approx 10\%\). We also calculate meson masses as a function of temperature.

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The spatial dependence of the finite-temperature meson correlation function in the scalar and pseudoscalar channel is studied in the Nambu–Jona–Lasinio model. The screening masses, obtained from the asymptotic behaviour of the static correlation function, are found to differ substantially from the dynamic masses, defined by a pole of the meson propagator. In particular, at high temperatures, the meson screening masses axe large, although there are no well defined meson modes. In the high-temperature limit, the screening masses approach \(2\pi T\), which corresponds to a gas of non-interacting, massless quarks. However interaction effects remain substantial well beyond the chiral transition temperature. The overall temperature dependence of the screening masses is in agreement with lattice results.

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We argue that different Z\(_{\rm N}\) thermal vacua of hot pure Yang–Mills theory distinguished in the standard approach by different values of Polyakov loop average \(\langle {\rm P} \rangle _{\rm T}\) corresponds actually to one and the same physical state. A critical discussion of the argument which are usually put forward in favor of the opposite conclusion (that, in pure continuum Yang–Mills theory, distinct Z\(_{\rm N}\)-phases may coexist in the physical space being separated by the domain walls with finite surface energy) is given. In particular, we note that the same arguments can be applied with an equal ease to abelian theories and would lead to the existence of the walls in the high-T 4-dim QED and to appearance of the queer high-T solitons with the mass \(\propto \) T\(^2\)/e in the Schwinger model. We emphasize that these configurations may be relevant for the Euclidean path integral but whether they correspond to real Minkowski space objects is unclear.

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The thermodynamics of a quark plasma is studied within the two-flavour Nambu–Jona–Lasinio model. We review the results that have been obtained in the mean-field (Hartree) approximation, including chiral and quark number susceptibilities, in which we treat contributions to the thermodynamic potential which arise to order 1 in the number of colours \(N_c\).

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I discuss a comprehensive approach to the space-like physics in high temperature QCD in three dimensions. The approach makes use of dimensional reduction. I suggest that this approach is useful for high temperature QCD in four dimensions.

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QCD point-to-point correlation functions for various mesonic and baryonic channels are calculated using the so called Random Instanton Liquid Model. The results are compared to experimental data and the results of recent lattice calculations. We also briefly discuss the present status of the theory of interacting instantons.

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We study the spectrum of the QCD Dirac operator near zero virtuality. We argue that it can be described by a random matrix theory with the chiral structure of QCD. In the large \(N\) limit, this model reduces to the low energy limit of the QCD partition function put forward by Leutwyler and Smilga. We conjecture that the microscopic limit of its spectral density Is universal and reproduces that of QCD. Using random matrix methods we obtain its exact analytical expression. This result is compared to numerically calculated spectra for a liquid of instantons, and we find a very satisfactory agreement.

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I discuss the general aspects of heavy hadron spectroscopy. The interplay between heavy–light hadrons and light mesons is discussed using a double expansion in the number of colors and the heavy quark mass. For a large number of colors heavy hadrons emerge as solitons.

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Recent data on the Gottfried sum and less recent ones on the pion–nucleon sigma term seem to disagree with naive parton-based expectations on the light (i.e., up, down and strange) quark content of the nucleon. We show that these discrepancies are resolved if nonperturbative contributions are included in the analysis of the data. These appear both in the computation of matrix elements of operators, and in their QCD scale dependence, and depend strongly on the quantum numbers of the given state.

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Summarization of our investigations of the stability of the three dimensional, classical spin field with Hopf index is presented. Example of a variational approach to functionals with topological terms is given. A nonnumerical method of analysis of a nonlinear evolution equations is presented.

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In two dimensional SU\((N)\) theories confinement can be understood as a topological property of the vacuum. In the bosonized version of two dimensional theories non trivial boundary conditions (topology) play a crucial role. They are inevitable if one wants to describe non singlet states. In abelian bosonization, color is the charge of a topological current in terms of a non-linear meson field. We show that confinement appears as the dynamical collapse of the topology associated with its non trivial boundary conditions.

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A critical survey is made of the mechanism of color confinement by “dual superconductivity” of QCD vacuum. Tests by Monte Carlo simulations on a lattice are reviewed.

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Critical slowing down of local algorithms in lattice gauge theories can be used to extract physics at different scales of length. We show that renormalization constants of the topological susceptibility of QCD vacuum can be determined numerically, without any use of perturbation theory, by a heating and cooling technique based on the above mentioned principle. The comparison with perturbative computations is discussed.

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We present the recent developments and applications in the Yennie–Frautschi–Suura (YFS) Monte approach to higher order radiative corrections to the SU(2)\(_{\rm L} \times \) U\(_1 \times \) SU(3)\(^{\rm c}\) model of elementary particle interactions. We focus on the high precision Z\(^0\) physics and on recent results relevant to the physics issues for the SSC/LHC type colliding beam device. lii both areas of investigation, we conclude that the YFS Monte Carlo approach permits a complete assessment of the role of the higher order radiative effects in the comparison between theory and experiment at the required level of precision.

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We review recent developments concerning the partonic structure of the proton. We describe a recent parton distribution analysis which incorporates the new precision deep inelastic scattering and related data. We discuss the application of perturbative QCD in the small \(x\) regime which is currently being probed, for the first time, by the experiments at HERA. We introduce, and consider applications of, the BFKL equation. We summarise the status of parton shadowing calculations. We briefly discuss implications of the new polarized neutron structure function measurements.

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The two versions of the electroweak models are investigated: the left–right (LR) model (both the symmetric and asymmetric cases) and the composite one (CM). The total cross sections of the W-pair production are found in both models. The analysis of LR model results show that already at LEP II energies we can either establish or limit such parameters of LR model as \({\mit \Delta }\rho _{\rm M}\), \(m_{{\nu _{\rm R}}}\) and \(g_{\rm R}\). It is also found that the investigation of the single production of Z\(_2\)-boson in pp collisions will be the good tool for the \(g_{\rm R}\) definition. For the CM the deviations value from the SM is mainly defined by the multiple moments values.

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I discuss the problem of computing \(B + L\) violation at high energies as that of solving classical differential equations. These equations involve boundary conditions at initial and a final times which are anti-Feynman, and involve solving non-linear differential equations which are complex, even if the original Hamiltonian was real.

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This note presents the results obtained by the H\(_1\) collaboration at HERA from the analysis of deep inelastic electron–proton scattering data collected in 1992 — the first year of HERA operation. Measurements of the structure function \(F_2(x, Q^2)\), particle energy flow, jet rates and results of direct searches for leptoquarks are discussed.