We present the O(alpha_s) corrections to the cross section for the reaction e+e- –> gamma^* –> Q \bar Q in the energy region close to the threshold. We assume that the energy of the reaction is such that both the perturbative expansion in the strong coupling constant and expansion in the relative velocity of the heavy quarks can be used. We explicitly obtain terms O(alpha_s^2/beta^2, alpha_s^2/beta, alpha_s^2) in the relative correction to the threshold cross section. Using the ideas of asymptotic expansions, we demonstrate how an expansion of Feynman diagrams in the threshold region is constructed. From this analysis we obtain a matching relation between the vector current in full QCD and the quark–antiquark current in NRQCD at leading order in 1/m and the second order in the strong coupling constant.

Predictions based on the chirally normalized vector meson dominance model for decay rates and distributions of $\tau$ decays into $K\pi\pi\nu_\tau$ final states are discussed. Disagreements with experimental results can be traced back to the $K_1$ widths.

For the vertex corrections to the partial decay rate $\Gamma(Z\to b\bar{b})$ involving the top quark only the leading terms of order $\alpha\alpha_s$ in the $1/M_t$ expansion are known. In this work we compute the missing next-to-leading corrections. They turn out to be equally important as the leading term and therefore have to be included into an ${\cal O}(\alpha\alpha_s)$ analysis to arrive at reliable predictions.

Next-to-leading order QCD predictions for 1-jet and 2-jet cross sections in polarized deep inelastic scattering at HERA energies are presented. Whereas the QCD corrections to the total polarized cross section are very large, only moderate corrections are found for the dijet cross sections.

This talk gives a review of recent two- and three-loop calculations to electroweak precision observables. Emphasis is put on the calculation of low energy observables, in particular the muon anomalous magnetic moment and the electric dipole moment of the neutron.

The asymmetry distributions for several kinematic variables are considered for finding a systematic way to maximize the signal for the extraction of the polarized gluon density. The relevance of mass effects for the corresponding dijet cross section is discussed and the different approximations for including mass effects are compared. We also compare via the programs PEPSI and MEPJET two different Monte Carlo (MC) approaches for simulating the expected signal in the dijet asymmetry at a polarized HERA.

CP violating signals in semileptonic $\tau$ decays induced by an exotic scalar exchange are studied in a completely model-independent way. These can be observed in decays of unpolarized single $\tau$'s even if their rest frame cannot be reconstructed. No beam polarization is required. The importance of the two-meson channel, in particular the $K\pi$ final state is emphasized.

Next-to-leading order QCD predictions for 1-jet and 2-jet cross sections in deep inelastic scattering with complete neutral current ($\gamma^*$ and $Z$) and charged current ($W^\pm$) exchange are presented.

We review the physics involved in the production and decay of top quarks in e^+e^- → ttbar near threshold, with special emphasis on the recent theoretical study on the decay process of top quarks in the threshold region. The energy-angular distribution of l^+ in semileptonic top decays is calculated including the full O(alpha_s) corrections. Various effects of the final-state interactions are elucidated. A new observable is defined near threshold, which depends only on the decay of free polarized top quarks, and thus it can be calculated without bound-state effects or the final-state interactions.

We consider |\Delta F| = 1 (F = S,C or B) nonleptonic effective hamiltonian in a renormalization scheme which allows to consistently use fully anticommuting \gamma_5 at any number of loops, but at the leading order in the Fermi coupling G_F. We calculate two-loop anomalous dimensions and one-loop matching conditions for the effective operators in this scheme. Finally, we transform our results to one of the previously used renormalization schemes, and find agreement with the original calculations.

We derive an algorithm for automatic calculation of perturbative \beta-functions and anomalous dimensions in any local quantum field theory with canonical kinetic terms. The infrared rearrangement is performed by introducing a common mass parameter in all the propagator denominators. We provide a set of explicit formulae for all the necessary scalar integrals up to three loops.

Recent analytical and numerical results for the three-loop polarization function allow to present a phenomenological analysis of the cross section for massive quark production in electron positron annihilation to order $\alpha_s^2$. Numerical predictions based on fixed order perturbation theory are presented for charm and bottom production above 5 and 11.5 GeV, respectively. The contribution from these energy regions to $\alpha(M_Z^2)$, the running QED coupling constant at scale M_Z, are given. The dominant terms close to threshold, i.e. in an expansion for small quark velocity $\beta$, are presented.

In the recent literature, constraints on the CKM angle $\gamma$ arising from the branching ratios for $B^\pm\to\pi^\pm K$ and $B_d\to\pi^\mp K^\pm$ decays received a lot of attention. An important theoretical limitation of the accuracy of these bounds is due to rescattering effects, such as $B^+\to\{\pi^0K^+\}\to\pi^+K^0$. We point out that these processes are related to penguin topologies with internal up quark exchanges and derive $SU(2)$ isospin relations among the $B^+\to\pi^+K^0$ and $B_d^0\to\pi^-K^+$ decay amplitudes by defining ``tree and ``penguin amplitudes in a proper way, allowing the derivation of generalized bounds on the CKM angle $\gamma$. We propose strategies to obtain insights into the dynamics of penguin processes with the help of the decays $B_{u,d}\to K\overline{K}$ and $B^\pm\to\pi^\pm K$, derive a relation among the direct CP-violating asymmetries arising in these modes, and emphasize that rescattering effects can be included in the generalized bounds on $\gamma$ completely this way. Moreover, we have a brief look at the impact of new physics.

In this article we evaluate mass corrections up to $O((m^2/q^2)^6)$ to the three-loop polarization function induced by an axial-vector current. Special emphasis is put on the evaluation of the singlet diagram which is absent in the vector case. As a physical application $O(\alpha_s^2)$ corrections to the production of top quarks at future $e^+e^-$ colliders is considered. It is demonstrated that for center of mass energies $\sqrt{s} >~ 500$ GeV the inclusion of the first seven terms into the cross section leads to a reliable description.

The large center of mass energy and increasing statistical precision for a wide range of hadronic final state observables at the HERA lepton-proton collider has provided a detailed testing ground for QCD dynamics. Fully flexible next-to-leading order calculations are mandatory on the theoretical side for such tests and will be discussed in detail. Next-to-leading order QCD predictions for one- and two-jet cross sections in deep inelastic scattering with complete neutral current ($\gamma^\ast$ and/or $Z$) and charged current ($W^\pm$) exchange together with leading order results for three- and four-jet final states are presented. The theoretical framework, based on the phase space slicing method and the use of universal crossing functions, is described in detail. All analytical formulae necessary for the next-to-leading order calculations are provided. The numerical results are based on the fully differential $ep \rightarrow n$ jets event generator \docuname\ which allows to analyze any infrared and collinear safe observable and general cuts in terms of parton 4-momenta. The importance of higher order corrections is studied for various jet algorithms. Implications and comparisons with (ongoing) experimental analyses for jet cross sections at high $Q^2$, the determination of $\alpha_s(\mu_R)$, the gluon density, power corrections in event shapes and the associated forward jet production in the low $x$ regime at HERA are discussed. A study of jet cross sections in polarized electron and polarized proton collisions shows that dijet events provide a good measurement of the polarized gluon distribution $\Delta g(x_g)$, in a region, where $x_g \Delta g(x_g)$ is expected to show a maximum.

We investigate the rare decay $\Lambda_b \to \Lambda \gamma$ which receives both short and long distance contributions. We estimate the long distance contributions and find them very small. The form factors are obtained from $\Lambda_c \to \Lambda \ell \bar{\nu}_\ell$ using heavy quark symmetry and a pole model. The short distance piece opens a window to new physics and we discuss the sensitivity of $\Lambda_b \to \Lambda \gamma$ to such effects.

Some of the recent developments in the theory and applications of the heavy quark expansion for inclusive processes are reviewed. The topics concern semileptonic and nonleptonic decays, lifetimes and the semileptonic branching fraction.

Using impact-factor representation, we consider the lepton pair production by an incident high-energy photon in a strong electromagnetic field of a nucleus. By summing leading terms of perturbation series, we obtain a simple formula for the amplitude, valid to all orders in ${\cal O}(\alpha Z)$ and arbitrary field of the nucleus. Using these results, we derive, in a simple manner, the results for the lepton pair production by a virtual incident photon in a Coulomb field. For real incident photon our results coincide with the known ones. Also, a particular example of a non-Coulomb potential is discussed in some detail.

Next-to-leading order QCD corrections to three jet heavy quark production in $e^+ e^-$ collisions, including quark mass effects, are presented. The extraction of the b-quark mass form LEP data is considered and the first experimental evidence for the running of a quark mass is discussed.

Dipole radiation in QCD is derived to the second order in $\alpha_s$. A power-like evolution of the spin-singlet heavy quark operators is obtained to the same accuracy. We give an ${\cal O}(\alpha_s^2)$ relation between the short-distance low-scale running heavy quark mass and the $\overline{\rm MS}$ mass. We discuss the properties of an effective QCD coupling $\alpha_s^{(d)}(E)$ which governs the dipole radiation. We argue that this coupling is advantageous for describing heavy quark physics.

A brief review of CP violation in $K$ and $B$ decays is given. While the observables $\varepsilon$ and Re$(\varepsilon'/\varepsilon)$ describing CP violation in neutral $K$ decays do not allow a powerful test of the CKM mechanism of CP violation, rare $K\to\pi\nu\overline{\nu}$ decays and in particular the $B$-meson system are much more promising in this respect. After a brief look at the $K$ system, selected aspects of CP-violating effects in $B$ decays are discussed. It is pointed out that combined branching ratios for $B\to\pi K$ modes, which have been observed recently by the CLEO collaboration, may allow to derive stringent constraints on the CKM angle $\gamma$ that could open a window to new physics.

We extend the approach based on the light-cone expansion and the heavy quark effective theory to the inclusive semileptonic decay of an unpolarized $b$-flavored hadron. It is applied to calculate the semileptonic decay width of the $\Lambda_b$ baryon and its ratio to that of the $B$ meson. We obtain $\Gamma_{\rm SL}(\Lambda_b)= 51\pm 9 \ |V_{cb}|^2\mbox{ps}^{-1}$ and $\Gamma_{\rm SL}(\Lambda_b)/\Gamma_{\rm SL}(B)= 1.052\pm 0.006$. From the latter, the semileptonic branching fraction for $\Lambda_b$ is predicted to be $(8.7\pm 0.7)\%$.

A Monte Carlo study of the azimuthal angular distribution around the virtual boson-proton beam axis for dijet events in DIS at HERA is presented. In the presence of typical acceptance cuts on the jets in the laboratory frame, the azimuthal distribution is dominated by kinematic effects rather than the typical $\cos\phi$ and $\cos 2\phi$ dependence predicted by the QCD matrix elements. This implies that the $\phi$ dependent part of the QCD matrix elements contributes even to the dijet production cross section. Neglecting this $\phi$ dependence leads to an error of about 5-8\% in the production cross section for typical acceptance cuts in the laboratory frame. We also present first NLO results on the $\phi$-decorrelation of the jets through NLO effects.

Dijet production in DIS is an important laboratory for testing our understanding of perturbative QCD. Flexible NLO Monte Carlo programs allow to investigate general jet definition schemes. For forward jet production at $p_{Tj}\approx Q$ and $x_{jet}\gg x$ the NLO predictions fall well below HERA data, thus providing evidence for BFKL dynamics.

ttbar → b l+ nu bbar W- Near Top Quark Threshold > We calculate final-state interaction corrections to the energy-angular distribution of l+ in semi-leptonic top quark decay, where the parent top quark is produced via e+e- → t\bar{t} near threshold. These are the corrections due to gluon exchange between t and \bar{b} (\bar{t} and b) and between b and \bar{b}. Combining with previously known other corrections, we explicitly write down the l+ energy-angular distribution including the full O(alpha_s)=O(beta) corrections near t\bar{t} threshold. Numerical analyses of the final-state interaction corrections are given. We find that they deform the l+ distribution typically at the 10% level. We also find that all qualitative features of the numerical results can be understood from intuitive pictures. The mechanisms of various effects of the final-state interactions are elucidated. Finally we define an observable which is proper to the decay process of the top quark (dependent only on d\Gamma_{t → b l+ \nu}/ dE_l d\Omega_l of a free polarized top quark) near t\bar{t} threshold. Such a quantity will be useful in extracting the decay property of the top quark using the highly polarized top quark samples.

Using the recent four-loop calculations of the QCD beta-function and the three-loop matching coefficients we study the induced error in $\alpha_s(m_Z)$ obtained from $\alpha_s(m_tau)$ due to the evolution procedure. We show that, when consistent matching and running is used at this order, these errors are pushed below 0.0005 in $\alpha_s(m_Z)$.

Searches for new physics using muons are reviewed. Particular attention is paid to muon number non-conserving processes, like the decay $\mu\to e\gamma$ and muon–electron conversion in muonic atoms. Also, experimental determinations and theoretical predictions for the muon anomalous magnetic moment are reviewed.

Recent progress in calculating ${\cal O}(\alpha _s ^2)$ corrections to inclusive and exclusive semileptonic $b \to c$ transitions is reviewed. The impact of these corrections on the $|V_{cb}|$\ determination from both inclusive and exclusive $B$ decays is discussed.

We calculate final-state interaction corrections to the energy-angular distribution of l^+ in semi-leptonic top quark decay, where the parent top quark is produced via e^+e^- \to t\bar{t} near threshold. These are the corrections due to gluon exchange between t and \bar{b} (\bar{t} and b) and between b and \bar{b}. Combining with previously known other corrections, we explicitly write down the l^+ energy-angular distribution including full O(\alpha_s)=O(\beta) corrections near threshold. Numerical analyses of the final-state interaction corrections are presented. We find that they deform the l^+ distribution typically at 10% level. Also we elucidate mechanisms of the various effects of the final-state interactions. We find that all qualitative features of the numerical results can be understood from intuitive pictures. The natures that \bar{b} is emitted preferrably in the parent \bar{t}'s spin direction and that there is attractive force between \bar{b} and \bar{t} generate: shift of |p_t|-distribution, forward-backward asymmetry of t, modification of the top-quark polarization vector, and energy-angle-correlated deformation of l^+ distributions.

Measurements of the combined branching ratios for $B^\pm \to \pi^\pm K$ and $B_d \to \pi^\mp K^\pm$ allow interesting constraints on the CKM angle $\gamma$, the ratio $r \equiv |T'|/|P'|$ of the current-current and penguin operator contributions to $B_d \to \pi^\mp K^\pm$, and the CP asymmetry in that decay. Present CLEO results for these branching ratios indicate where problems with consistency of the Standard Model may arise in the future. In this paper we discuss scenarios of new physics in these decays and investigate their implications for the above constraints.

CP-violating effects in non-leptonic $B_s$-meson decays are reviewed. Special emphasis is given to implications arising from the width difference $\Delta\Gamma_s$ between the $B_s$ mass eigenstates. If $\Delta\Gamma_s$ is found to be sizable, certain untagged $B_s$-meson decays may allow interesting probes of CP violation, extractions of the CKM angle $\gamma$ and the Wolfenstein parameter $\eta$, and may indicate physics beyond the Standard Model.

c transitions > We have recently presented the complete O(alpha_s^2) corrections to the semileptonic decay width of the b quark at maximal recoil. Here we discuss various technical aspects of that calculation and further applications of similar methods. In particular, we describe an expansion which facilitates the phase space integrations and the treatment of the mixed real–virtual corrections, for which Taylor expansion does not work and the so-called eikonal expansion must be employed. Several terms of the expansion are given for the O(alpha_s^2) QCD corrections to the differential semileptonic decay width of the b –quark at maximal recoil. We also demonstrate how the light quark loop corrections to the top quark decay rate can be obtained using the same methods. We briefly discuss the application of these techniques to the calculation of the O(alpha_s^2) correction to zero recoil sum rules for heavy flavor transitions.

Transition rates for coherent muon-electron conversion in muonic atoms, mu+N –> e+N, are computed for various types of muon number violating interactions. Attention is paid to relativistic atomic effects, Coulomb distortion, finite nuclear size, and nucleon distributions. Discrepancies with previously published results are pointed out and explained. Results are presented for several elements of current and future experimental interest.

In this paper the three-loop polarization functions \Pi(q^2) are calculated for the cases of an external vector, axial-vector, scalar or pseudo-scalar current. Results are presented for the imaginary part which directly leads to the cross section $\sigma(e^+e^- \to Z \to hadrons)$ and to the Higgs decay rates, respectively.

Motivated by recent CLEO measurements of $B\to\pi K$ modes, we investigate their implications for the CKM angle $\gamma$ and a consistent description of these decays within the Standard Model. Interestingly it turns out that already the measurement of the combined branching ratios $B^\pm\to\pi^\pm K$ and $B_d\to\pi^\mp K^\pm$ allows to derive stringent constraints on $\gamma$ which are complementary to the presently allowed range for that angle. This range, arising from the usual fits of the unitarity triangle, is typically symmetric around $\gamma = 90^\circ$, while our method can in principle {\it exclude} a range of this kind. Consistency within the Standard Model implies furthermore bounds on the ratio $r\equiv|T'|/|\tilde P|$ of the current-current and penguin operator contributions to $B_d\to\pi^\mp K^\pm$, and upper limits for the CP-violating asymmetry arising in that decay. Commonly accepted means to estimate $r$ yield values at the edge of compatibility with the present CLEO measurements.

The heavy quark expansion in QCD became a widely used tool in heavy quark physics. The theoretical aspects of the $1/m_Q$ expansion ($m_Q$ being the mass of the heavy quark) and some techniques as they appear in practical calculations are reviewed. Some applications to processes involving heavy quark decays are discussed.

We review the highlights of quark mixing, particle-antiparticle mixing, CP violation and rare K- and B-decays in the Standard Model. The top quark discovery, the precise measurement of its mass, the improved knowledge of the couplings $V_{cb}$ and $V_{ub}$, and the calculations of NLO short distance QCD corrections improved considerably the predictions for various decay rates, the determination of the couplings $V_{td}$ and $V_{ts}$ and of the complex phase in the Cabibbo-Kobayashi-Maskawa matrix. After presenting the general theoretical framework for weak decays, we discuss the following topics in detail: i) the CKM matrix, its most convenient parametrizations and the unitarity triangle, ii) the CP-violating parameter $\epsilon_K$ and $B^0_{d,s}-\bar B^0_{d,s}$ mixings, iii) the ratio $\epsilon'/\epsilon$, iv) the rare K-decays $K_L\to\pi^0e^+e^-$, $K^+\to\pi^+\nu\bar\nu$, $K_L\to\pi^0\nu\bar\nu$ and $K_L\to\mu^+\mu^-$, v) the radiative decays $B\to X_s\gamma$ and $B\to X_s l^+l^-$, vi) the rare B-decays $B\to X_{s,d}\nu\bar\nu$ and $B_{d,s}\to l^+l^-$, vii) CP violation in neutral and charged B-decays putting emphasis on clean determinations of the angles of the unitarity triangle, and viii) the role of electroweak penguins in B-decays. We present several future visions demonstrating very clearly the great potential of CP asymmetries in B-decays and of clean K-decays such as $K^+\to\pi^+\nu \bar\nu$ and $K_L\to\pi^0\nu\bar\nu$ in the determination of the CKM parameters and in decisive testing of the Standard Model. An outlook for the coming years ends our review.

D* Transitions > We present the complete O(alpha_s^2) corrections to the Wilson coefficient of the unit operator in the zero-recoil sum rule for the B–>D* transition. We include both perturbative and power-suppressed nonperturbative effects in a manner consistent with the operator product expansion. The impact of these corrections on |V_cb| extracted from semileptonic B–>D* decays near zero recoil is discussed. The mixing of the heavy quark kinetic operator with the unit operator at the two loop level is obtained. O(alpha_s) corrections to a number of power-suppressed operators are calculated.

Three-loop corrections to the scalar and pseudo-scalar current correlator are calculated. By applying the large momentum expansion mass terms up to order (m^2/q^2)^4 are evaluated analytically. As an application O(\alpha_s^2) corrections to the decay of a scalar and pseudo-scalar Higgs boson into top quarks are considered. It is shown that for a Higgs mass not far above the $t\bar{t}$ threshold these higher order mass corrections are necessary to get reliable results.

Three-loop QCD corrections to the vector current correlator are considered. The large momentum procedure is applied in order to evaluate mass corrections up to order $(m^2/q^2)^6$. The inclusion of the first seven terms to the ratio $R=\sigma(e^+e^-\to\mbox{hadrons})/\sigma(e^+e^-\to\mu^+\mu^-)$ leads to reliable predictions from the high energy region down to relatively close to threshold.

We present a complete three-loop calculation of the electric dipole moment of the $u$ and $d$ quarks in the Standard Model. For the $d$ quark, more relevant for the experimentally important neutron electric dipole moment, we find cancellations which lead to an order of magnitude suppression compared with previous estimates.

We present analytical results for the $O(\alpha_s^2)$ contributions to the functions $\eta _A$ and $\eta _V$ which parameterize QCD corrections to semileptonic $b \to c$ transitions at zero recoil. Previously obtained approximate results are confirmed. The methods of computing the relevant two-loop diagrams with two mass scales are discussed in some detail.

An eight-component formalism is proposed for the relativistic two-fermion problem. In QED, it extends the applicability of the Dirac equation with hyperfine interaction to the positronium case. The use of exact relativistic two-body kinematics entails a CP-invariant spectrum which is symmetric in the total cms energy. It allows the extension of recent $\alpha^6$ recoil corrections to the positronium case, and implies new recoil corrections to the fine and hyperfine structures and to the Bethe logarithm.

We investigate the rare decay $\Lambda_b \to \Lambda \gamma$ which receives both short and long distance contributions. We estimate the long distance contributions and find them very small. The form factors are obtained from $\Lambda_c \to \Lambda \ell \bar{\nu}_\ell$ using heavy quark symmetry and a pole model. The short distance piece opens a window to new physics and we discuss the sensitivity of $\Lambda_b \to \Lambda \gamma$ to such effects.

We present a complete ${\cal O}(\alpha_s^2)$ correction to the differential width of the inclusive semileptonic decay $b\to cl\nu_l$ at the kinematical point of vanishing invariant mass of the leptons, $q^2=0$. Together with the recently computed ${\cal O}(\alpha_s^2)$ correction at the upper boundary of the lepton invariant mass spectrum, this new information permits an estimate of the ${\cal O}(\alpha_s^2)$ effect in the total inclusive semileptonic decay width $b\to cl\nu_l$. We argue that the non-BLM part of the ${\cal O}(\alpha_s^2)$ correction gives at most $1\%$ correction to the inclusive semileptonic decay width $b\to cl\nu_l$. This significantly improves the credibility of extracting $|V_{cb}|$ from the inclusive semileptonic decays of the $b$-hadrons.

We present the coefficient of the chromomagnetic interaction operator, the only unknown coefficient in the Heavy Quark Effective Theory (HQET) lagrangian up to the $1/m$ level, with the two–loop accuracy by matching scattering amplitudes of an on–shell heavy quark in an external field in full QCD and HQET, and obtain the two–loop anomalous dimension of this operator in HQET.

A full analytic calculation of the two-loop diagrams contributing to the static potential in QCD is presented in detail. Using a renormalization group improvement, the ``three-loop'' potential in momentum space is thus derived and the third coefficient of the $\beta$-function for the V-scheme is given. The Fourier transformation to position space is then performed, and the result is briefly discussed.

A QCD based calculation of the photon spectrum in semi-inclusive \eta_Q decays is performed. The method applied is an effective theory based on a 1/m_Q expansion of QCD.