The phenomenon of CP violation in the $B$ system and strategies for extracting CKM phases are reviewed. We focus both on general aspects and on some recent developments including CP-violating asymmetries in $B_d$ decays, the $B_s$ system in light of a possible width difference $\Delta\Gamma_s$, charged $B$ decays, and $SU(3)$ relations among certain transition amplitudes. In order to describe the relevant non-leptonic $B$ decays, low energy effective Hamiltonians calculated beyond the leading logarithmic approximation are used. Special emphasis is given to the role of electroweak penguin operators in such transitions. These effects are analyzed both within a general framework and more specifically in view of the theoretical cleanliness of methods to determine CKM phases. Strategies for obtaining insights into the world of electroweak penguins are discussed.

Predictions for decay rates and distributions for $\tau$ decays into final states with kaons are discussed and compared with recent measurements. Special emphasis is put on new constraints for the vector current contribution in the $KK\pi$ decay modes. For the $K\pi\pi$ modes, disagreements with the experimental results can be traced back to the $K_1$ widths.

A short review of a few selected topics in Heavy Quark Effective Theory is given. Applications to exclusive decays are discussed.

We discuss the calculation of two-point three-loop functions with an arbitrary number of massive propagators and one large external momentum. The relevant subdiagrams are generated automatically. The resulting massless two-point integrals and massive tadpoles are transformed on-line to FORM-expressions ready to be used by existing FORM packages which calculate them analytically. As an example we compute the quartic mass corrections to the photon polarization function.

The effective chiral Lagrangian of the strong and electromagnetic interactions of the pseudoscalar mesons at low energies depends on a set of low energy constants. We determine the contributions to the electromagnetic coupling constants at order $O(e^2 p^2)$, which arise from resonances within a photon loop. We give some implications of our results, in particular we discuss in detail the effects on the corrections to Dashen's theorem.

Recent theoretical results on heavy flavor production and decay in the framework of perturbative QCD are reviewed. This includes calculations for top production at hadron colliders, inclusive charmonium production and the comparison between the singlet and octet mechanisms. Predictions for heavy flavor production in $e^+e^-$ annihilation will be discussed in some detail, covering both the threshold and the high energy region. The first results in NLO for heavy flavor decays will also be reviewed.

Complete two-loop electroweak corrections to leptonic anomalous magnetic moments have been calculated recently using asymptotic expansions of Feynman diagrams. Techniques developed in that context can also be applied to the electric dipole moments (EDM) of fermions. In the Standard Model the EDM was proved to vanish at two loops. In this talk we discuss a simplification of that proof.

Two-loop electroweak corrections to the electron, muon, and tau lepton anomalous magnetic moments have been computed recently. Effects of hadronic contributions to the photon propagator in two-loop QED corrections have also been reanalysed. The common technique used in both calculations was asymptotic expansion of Feynman diagrams in the ratio of lepton mass to some heavy mass scale. In this talk we present some details of this method paying particular attention to the expansion of multi-scale diagrams.

It is shown that the potential for lepton-antilepton bound states (leptonium) is the Fourier transform of the first Born approximation to the QED scattering amplitude in an 8-component equation, while 16-component equations are excluded. The Fourier transform is exact at all {\sl cms} energies $ - \infty < E < \infty$; the resulting atomic spectrum is explicitly CPT-invariant.

We reanalyze the decay $B_d\to\pi^+\pi^-$ without assuming dominance of QCD penguins with internal top-quark exchanges. In that case the weak phase of the $\bar b \to\bar d$ QCD penguin amplitude is not related in a simple way to the CKM angle $\beta$. Nevertheless it is still possible to extract the CKM angle $\alpha$ in a transparent and almost model-independent way from the time-dependent CP-violating asymmetry of the transition $B_d\to\pi^+\pi^-$. Aside from that CP asymmetry this approach needs as an input only amplitudes of decays with branching ratios of order $10^{-5}$ and will thus be well within reach at future $B$-factories.

3 pi nu_tau > Isospin violating signals in the tau → 3 pi nu_tau decay mode are discussed. For the tau → pi^- pi^- pi^+ nu_tau decay mode, isospin violation arises from the vector current contribution in the tau → omega pi nu_tau decay with the subsequent isospin violating omega decay into pi^+ pi^-. We demonstrate that such effects may be observed in presently available data through the measurement of the interference effects of these vector current contributions with the dominating axial vector current, i.e. through a measurement of the structure functions W_F,W_G,W_H and W_I. In the case of the tau → pi^0 pi^0 pi^- nu_tau decay mode, a vector current contribution is generated by eta pi mixing in the decay chain tau → eta rho nu_tau → pi^0 pi^0 pi^- nu_tau. We find that this effect is rather small, the magnitude of the associated interference terms being too low for present statistics.

In this article we provide a manifestly gauge-invariant approach to charged particles. It involves (1) Green functions of gauge-invariant operators and (2) Feynman rules which do not depend on any kind of gauge-fixing condition. First, we provide a thorough analysis of QED. We propose a specific set of gauge-invariant Green functions and describe a manifestly gauge-invariant technique to evaluate them. Furthermore we show how Green's functions and Feynman rules of the manifestly gauge-invariant approach and of the Faddeev- Popov ansatz are related to each other. Finally, we extend this manifestly gauge-invariant approach to the Standard Model of electroweak interactions.

Motivation for such an approach is abundant. A gauge-dependent framework does obstruct not only theoretical insight but also phenomenological analyses of precision experiments at LEP. Unresolved issues include the analysis of finite width effects of unstable particles and of oblique corrections parametrised by, e.g., S, T, and U. Quite another example of considerable complications caused by a gauge-dependent framework is the matching of a full and an effective theory, relevant for effective field theory analyses of the symmetry-breaking

For large values of the Higgs mass the low energy structure of the gauged linear sigma model in the spontaneously broken phase can adequately be described by an effective field theory. We present a manifestly gauge-invariant functional technique to explicitly evaluate the corresponding effective Lagrangian from the underlying theory.

CP violating signals in semileptonic $\tau$ decays are studied. 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.

The $B_s$ system is analyzed in light of a possible width difference $\Delta\Gamma_s$ between its mass eigenstates. If $\Delta\Gamma_s$ is sizable, untagged $B_s$-meson decays may allow a probe of CP violation and moreover the extraction both of the Wolfenstein parameter $\eta$ and of the notoriously difficult to measure angle $\gamma$ of the unitarity triangle. To accomplish this ambitious task, time-dependent angular distributions for untagged $B_s$ decays into admixtures of CP eigenstates and channels that are caused by $\bar b\to\bar c u\bar s$ quark-level transitions play a key role. The work described here was done in collaboration with Isard Dunietz.

NLO corrections to jet cross sections in DIS at HERA are studied, with particular emphasis on the two jet final state. High jet transverse momenta are a good criterion for the applicability of fixed order perturbation theory. A ``natural'' scale choice is the average $k_T^B$ of the jets in the Breit frame, which suggest analyzing the data in different $<k_T^B>$ intervalls.

The production of forward jets of transverse momentum $p_T(j)\approx Q$ and large momentum fraction $x_{jet}\gg x$ probes the onset of BFKL dynamics at HERA. A full ${\cal O}(\alpha_s^2)$ calculation of the inclusive forward jet cross section is presented and compared to the expected BFKL cross section.

The production of forward jets of transverse momentum $p_T(j)\approx Q$ and large momentum fraction $x_{jet}\gg x$ probes the onset of BFKL dynamics at HERA. A full ${\cal O}(\alpha_s^2)$ calculation of the inclusive forward jet cross section is presented and compared to the expected BFKL cross section. The kinematical region populated by these events and the scale dependence of the fixed order perturbative QCD cross sections are discussed.

The static potential between an infinitely heavy quark and antiquark is derived in the framework of perturbative QCD to three loops by performing a full calculation of the two-loop diagrams and using the renormalization group. The contribution of massless fermions is included.

Top quark pair production close to threshold at a next linear collider is discussed, with special focus on the influence of rescattering corrections. Numerical results are shown for the differential cross-section, i.e. the momentum distribution and the forward backward asymmetry, and for moments of the angular distribution of leptons arising from the semileptonic top quark decay.

The total cross section for electron-positron annihilation into hadrons is calculated in the region just below the B meson threshold. QCD corrections up to third order, quark mass effects, initial state radiation, the running of alpha_QED and the tails of the Upsilon resonances are included in the prediction. For given alpha_s the prediction is accurate to 0.5 percent. An experimental measurement with the corresponding precision would allow to determine alpha_s with high accuracy at intermediate energies.

Z –> W u d </EM> > We present an estimate of perturbative QCD corrections to the decay $Z \to Wu\bar d$. A simple approximate approach is described in detail. The difference of masses of $M_Z$ and $M_W$ is used as an expansion parameter. A complete analytical formula for a part of the corrections is also presented.

An asymptotic expansion of the two-loop two-point ``master'' diagram with two masses $m$ and $M$, on the mass shell $Q^2=M^2$, is presented. The treatment of the non-analytical terms arising in the expansion around the branching point is discussed. Some details of the calculation of a new class of two-loop integrals are given.

Next-to-leading order corrections to jet cross sections in deep inelastic scattering at HERA are studied. The predicted jet rates allow for a precise determination of $\alpha_s(\mu_R)$ and the gluon density at HERA. We argue, that the ``natural'' renormalization and factorization scale is in general set by the average $k_T^B$ of the jets in the Breit frame. Some implications for the associated forward jet production in the low $x$ regime at HERA are discussed.

The measurement of the polarized gluon distribution

function $\Delta g(x)$  from photon gluon fusion processes in 
electron proton deep inelastic scattering producing
two jets has been investigated. The study is based on the MEPJET
simulation program.
The size of the expected  spin asymmetry
and corresponding statistical uncertainties
for a possible measurement
with polarized beams of electrons and protons  at HERA 
have been estimated.
The results show that the asymmetry can reach a few percents.  

We present an analysis of two-loop mixed QCD and electroweak corrections to the decay of the Z boson into light quarks. We find that the naive factorization of QCD and electroweak corrections does not describe correctly the two-loop effects. The nonfactorizable corrections shift the width of the Z boson by approximately -0.55(3) MeV and increase the central value of the strong coupling constant determined at LEP by 0.001.

An effective theory approach to heavy quarkonia decays based on the $1/m_Q$ expansion is introduced. Its application to decays in which the two heavy

Higher-order hadronic contributions to the anomalous magnetic moment of the electron, muon, and $\tau$ lepton are considered in detail. As a main result we find a reduction by $-18 \times 10^{-11}$ for the g-2 of the muon as compared to previous calculations. Analytical expressions for the kernel functions of higher-order hadronic effects are presented. We employ the method of asymptotic expansions to calculate kernel functions analytically in terms of a series expansion in the lepton mass.

Strategies for extracting CKM phases from non-leptonic $B$ decays are reviewed briefly. Both general aspects and some recent developments including CP-violating asymmetries, the $B_s$ system in light of a possible width difference $\Delta\Gamma_s$, and triangle relations among $B$ decay amplitudes are discussed. The role of electroweak penguins

Next-to-leading order corrections to jet cross sections in deep inelastic scattering at HERA are studied. The predicted jet rates allow for a precise determination of $\alpha_s(\mu_R)$ at HERA over a wide range of $\mu_R$. We argue, that the ``natural'' renormalization and factorization scale is set by the average $k_T^B$ of the jets in the Breit frame and suggest to divide the data in corresponding $<k_T^B>$ intervals. Some implications for the determination of the gluon density and the associated forward jet production in the low $x$ regime at HERA are briefly discussed.

In this series of lectures the basic ideas of the $1/m_Q$ expansion in QCD ($m_Q$ is the mass of a heavy quark) are outlined. Applications to exclusive and inclusive decays

The two-loop electroweak radiative corrections to the muon's anomalous magnetic moment, $a_\mu\equiv (g_\mu-2)/2$, are presented. We obtain an overall 22.6\% reduction in the electroweak contribution $a_\mu^{\rm EW}$ from $195\times 10^{-11}$ to $151(4)\times 10^{-11}$. Implications for the full standard model prediction and an upcoming high precision measurement of $a_\mu$ are briefly discussed. Some aspects of the calculations are discussed in detail.

We review a variety of topics related to hadronic structure functions in exclusive semihadronic tau decays. We introduce the concept of structure functions and summarize the most important concepts. We then calculate the decay $\tau \to 3 \pi \nu_\tau$ for very small hadronic invariant mass to one loop in Chiral Perturbation Theory. New interesting features emerge with respect to the known results at tree level, in particular for the structure functions $w_D$ and $w_E$. We discuss the prospects for experimental verification of our predictions. Finally, we discuss various issues at higher $Q^2$, related to hadronic resonance physics. Here we consider $2\pi$, $\pi K$, $3 \pi$, $K \pi K$ and $\pi K \pi$ hadronic final states.

Recent results on the entirely analytic calculation of second order corrections to massive quark pair production vertices induced by light quark flavours are reviewed. Based on the method presented in this talk the second order effects of the insertion of any massless one-loop vacuum polarization into the gluon line of the first order diagrams can be determined. The behaviour of the corrections at threshold for vector and axial-vector current induced massive quark production is discussed.

The real and imaginary part of the vacuum polarization function $\Pi(q^2)$ induced by a massive quark is calculated in perturbative QCD up to order $\alpha_s^2$. We combine the information from small and large momentum region and from the threshold using conformal mapping and Pad\'e approximation. This leads us to formulae for $\Pi(q^2)$ valid for arbitrary $m^2/q^2$. Taking subsequently the imaginary part we get the ${\cal O}(\alpha_s^2)$ to $R \equiv \sigma(e^+ e^- \to \mbox{hadrons})/

          \sigma(e^+ e^- \to \mu^+ \mu^-)$.

This extends the calculation by K\“all\'en and Sabry from two to three loops.

The relativistic Breit Hamiltonian between electrons is transformed into an effective vector potential ${\bf A}_i$ for the $i.$th electron, ${\bf A}_i$ having the structure of a recoil–corrected hyperfine operator.

Apart from a small three--body operator, the Dirac--Breit equation

The three-loop vacuum polarization function $\Pi(q^2)$ induced by a massive quark is calculated. A comprehensive description of the method is presented. From the imaginary part the ${\cal O}(\alpha_s^2)$ result for the production of heavy quarks $R(s)=\sigma(e^+e^-\to \mbox{hadrons})/\sigma(e^+e^-\to \mu^+\mu^-)$ can be extracted. Explicit formulae separated into the different colour factors are given.

The relativistic motion of an isolated two–body system (bound or unbound) of given lab energy $K^{0}$ in QED is separated into cms motion and relative motion. The relative motion equation ${\cal K}_{L}} \psi_{L} ({\bf r}_{L} ) =0$ contains the momentum eigenvalue ${\bf K}$ of the cms motion. It is greatly simplified by a binary boost to the atomic rest frame, where $K^{0}$ and ${\bf K}$ appear only in a Lorentz–invariant combination. This boost is not a product of single–particle boosts, which are useful only for perturbative interactions. CPT–invariance is demonstrated, and orthogonality

The cross-section for the production of $t\bar{t}$ pairs in $e^+e^-$-annihilation via a virtual photon is determined up to order $\alpha_s^2$. For the cm-energies of interest the effect of the top mass may not be considered as a small parameter and the full mass dependence must be included.

Two-jet cross sections in deep inelastic scattering at HERA are calculated in next-to-leading order. The importance of higher order corrections and recombination scheme dependencies is studied for various jetalgorithms. Some implications for the determination of $\alpha_s(\mu_R^2)$, the determination of the gluon density and the associated forward jet production in the low $x$ regime at HERA are briefly discussed.

We demonstrate that time-dependent studies of angular distributions for $B_s$ decays caused by $\bar b\to\bar cu\bar s$ quark-level transitions extract {\it cleanly} and {\it model-independently} the CKM angle $\gamma$. This CKM angle could be cleanly determined from {\it untagged} $B_s$ decays alone, if the lifetime difference between the $B_s$ mass eigenstates $B_s^L$ and $B_s^H$ is sizable. The time-dependences for the relevant {\it tagged} and {\it untagged} observables are given both in a general notation and in terms of linear polarization states and should exhibit large CP-violating effects. These observables may furthermore provide insights into the hadronization dynamics of the corresponding exclusive $B_s$ decays thereby allowing tests of the factorization hypothesis.

We investigate the time-evolutions of angular distributions for $B_s$ decays into final states that are admixtures of CP-even and CP-odd configurations. A sizable lifetime difference between the $B_s$ mass eigenstates allows a probe of CP violation in time-dependent untagged angular distributions. Interference effects between different final state configurations of $B_s\rightarrow D^{*+}_s\, D^{*-}_s$, $J/\psi\,\phi$ determine the Wolfenstein parameter $\eta$ from {\it untagged} data samples, or – if one uses $|V_{ub}|/|V_{cb}|$ as an additional input – the notoriously difficult to measure CKM angle $\gamma$. Another determination of $\gamma$ is possible by using isospin symmetry of strong interactions to relate {\it untagged} data samples of $B_s\to K^{\ast+}\,K^{\ast-}$ and $B_s\to K^{\ast0}\, \overline{K^{\ast0}}$. We note that the {\it untagged} angular distribution for $B_s\to\rho^0\phi$ provides interesting information about electroweak penguins.

QCD corrections to the electron positron annihilation cross section into hadrons and to the hadronic $Z$ boson decay rate are reviewed. Formal developments are introduced in a form particularly suited for practical applications. These include the operator product expansion, the heavy mass expansion, the decoupling of heavy quarks and matching conditions. Exact results for the quark mass dependence are presented whenever available, and formulae valid in the limit of small bottom mass ($m_{\rm b}^2\ll s$) or of large top mass ($m_{\rm t }^2 \gg s$) are presented. The differences between vector and axial vector induced rates as well the classification of singlet and nonsinglet rates are discussed. Handy formulae for all contributions are collected and their numerical relevance is investigated. Prescriptions for the separation of the total rate into partial rates are formulated. The applicability of the results in the low energy region, relevant for measurements around 10 GeV and below, is investigated and numerical predictions are collected for this energy region.

Complete two-loop QCD corrections to $b\to c$ transitions are presented in the limit of zero recoil. Vector and axial-vector coefficients $\eta_{A,V}$ are calculated analytically in the limit of equal beauty and charm masses, and a series appoximation is obtained for the general mass case. $\eta_A$ is crucial for the determination of the absolute value of the Cabibbo-Kobayashi-Maskawa matrix element $V_{cb}$. The two-loop effects enhance the one-loop corrections by 21\%, removing a major theoretical uncertainty in the value of $|V_{cb}|$. Including two-loop QCD effects and previously neglected electroweak and Coulomb corrections we find $|V_{cb}| = 0.0379 \pm 0.0020({\rm stat}) \pm 0.0025({\rm syst}) \pm 0.0011({\rm theory})$.

Second order ${\cal O}(\alpha_s^2)$ corrections to the heavy quark production cross-section due to massless quarks and coloured scalars are calculated for all energies above threshold. Based on the method introduced in this letter also the gauge non-invariant second order corrections due to the pure gluonic self-energy insertion and a certain class of ${\cal{O}}(\alpha_s^3)$ and ${\cal{O}}(\alpha_s^4)$ corrections are determined. For the special choice of the gauge parameter, $\xi=4$, the leading threshold and high energy behaviour of the pure second order gluonic corrections to the cross-section are governed by the gluonic self-energy insertion.

We study logarithms of the form $\ln(m_q/m_b)$ which arise in the inclusive semileptonic decay of a bottom quark to a quark of mass $m_q$. We use the renormalization group to resum the leading radiative corrections to these terms, of the form $m_q^2\alpha_s^n\ln^n(m_q/m_b)$, $m_q^3\alpha_s^{n+1}\ln^n(m_q/m_b)$ and $m_q^4\alpha_s^n\ln^{n+1}(m_q/m_b)$. The first two resummations are trivial, while the latter involves a non-trivial mixing of four-fermi operators in the $1/m_b$ expansion. We illustrate this technique in a toy model in which the semileptonic decay is mediated by a vector interaction, before treating the more complicated case of left-handed decay.

In a small window of phase space, chiral perturbation theory can be used to make standard model predictions for tau decays into two and three pions. For $\tau\to2\pi\nu_\tau$, we give the analytical result for the relevant form factor $F_V$ up to two loops, then calculate the differential spectrum and compare with available data. For $\tau\to 3 \pi \nu_\tau$, we have calculated the hadronic matrix element to one loop. We discuss the decomposition of the three pion states into partition states and we give detailed predictions for the decay in terms of structure functions. We also compare with low energy predictions of meson dominance models. Overall, we find good agreement, but also some interesting discrepancies, which might have consequences beyond the limit of validity of chiral perturbation theory.