Quantum corrections significantly influence the quantities observed in modern particle physics. The corresponding theoretical computations are usually quite lengthy which makes their automation mandatory. This review reports on the current status of automatic calculation of Feynman diagrams in particle physics. The most important theoretical techniques are introduced and their usefulness is demonstrated with the help of simple examples. A survey over frequently used programs and packages is provided, discussing their abilities and fields of applications. Subsequently, some powerful packages which have already been applied to important physical problems are described in more detail. The review closes with the discussion of a few typical applications for the automated computation of Feynman diagrams, addressing current physical questions like properties of the $Z$ and Higgs boson, four-loop corrections to renormalization group functions and two-loop electroweak corrections.

Recent theoretical results in tau lepton physics and their implications for the experimental analysis are presented. The paper is split into the following chapters: 1. Introduction, 2. Weak Couplings, 3. Inclusive Decays, Perturbative QCD and Sum Rules, 4. Exclusive Decays, 5. Beyond the Standard Model.

We consider the assumption that a tachyonic gluon mass imitates short-distance nonperturbative physics of QCD. The phenomenological implications include modifications of the QCD sum rules for correlators of currents with various quantum numbers. The new 1/Q^2 terms allow to resolve in a natural way old puzzles in the pion and scalar-gluonium channels. They lead to a slight reduction of the values of the running light quark masses from the (pseudo)scalar sum rules and of alpha_s(M_\tau) from tau decay data. Further tests can be provided by precision measurements of the correlators on the lattice and by the e^+e^- –> hadrons data.

The production of final state photons in hadronic $Z$-boson decays can be used to study the quark-to-photon fragmentation function $D_{q\to \gamma}(z,\mu_{F})$. Currently, two different observables are used at LEP to probe this function: the `photon' +~1 jet rate and the inclusive photon energy distribution. We outline the results of a calculation of the `photon' +~1 jet rate at fixed ${\cal O}(\alpha \alpha_{s})$, which yield a next-to-leading order determination of the quark-to-photon fragmentation function $D_{q\to \gamma}(z,\mu_{F})$. The resulting predictions for the isolated photon rate and the inclusive photon spectrum at the same, fixed order, are found to be in good agreement with experimental data. Furthermore, we outline the main features of conventional approaches using parameterizations of the resummed solutions of the evolution equation and point out deficiencies of these currently available parameterizations in the large $z$-region. We finally demonstrate that the ALEPH data on the `photon' +~1 jet rate are able to discriminate between different parameterizations of the quark-to-photon fragmentation function, which are equally allowed by the OPAL photon energy distribution data.

Recent results for electroweak radiative corrections will be presented with emphasis on those connecting QCD and electroweak interactions. This includes purely elctroweak corrections at two-loop order and beyond, hadronic effects specific to the $b\bar b$ final state in $Z$ decays, the impact of perturbative QCD on the $\rho$ parameter and related observables, and mixed QCD and electroweak vertex corrections. Estimates for the theoretical uncertainties are given.

Matching of light-light quark currents in QCD with a heavy flavour and in the low-energy effective QCD is calculated in MSbar at two loops. Heavy-light HQET currents are similarly considered.

The recoil corrections to the energy levels of leptonium are calculated to the order $\alpha^6$ and $\alpha^6 \log\alpha$ for all states except those having hyperfine mixing. A correction to the hyperfine operator is derived in which the potential appears in the denominator.

We compute the renormalization of the complete CKM matrix in the MSbar scheme and perform a renormalization group analysis of the CKM parameters. The calculation is simplified by studying only the Higgs sector, which for the \beta-function of the CKM matrix is at one loop the same as in the full Standard Model. The renormalization group flow including QCD corrections can be computed analytically using the hierarchy of the CKM parameters and the large mass differences between the quarks. While the evolution of the Cabibbo angle is tiny V_{ub} and V_{cb} increase sizably. We compare our results with the ones in the full Standard Model.

We present an analytic calculation of the m*alpha^6 recoil corrections to the hyperfine splitting (HFS) of the ground state energy levels in positronium. We find \Delta E_{\rm rec} = m alpha^6 (-1/6 ln(alpha) + 331/432 -ln(2)/4 - 17 Zeta(3)/(8pi^2) + 5/(12 pi^2)) \approx m alpha^6 (- 1/6 ln(alpha) + 0.37632), confirming Pachucki's numerical result \cite{Ph}. We present a complete analytic formula for the m*alpha^6 HFS of the positronium ground state and, including m*alpha^7*ln^2(alpha) effects, find E(1^3S_1)-E(1^1S_0))_{theory} = 203 392(1) MHz. This differs from the experimental results by about 3 standard deviations.

We include the full second-order corrections to the static QCD potential in the analysis of the $t\bar{t}$ threshold cross section. There is an unexpectedly large difference between the QCD potential improved by the renormalization-group equation in momentum space and the potential improved by the renormalization-group equation in coordinate space. This difference remains even at a fairly short distance $1/r \simeq 100$~GeV and its origin can be understood within perturbative QCD. We scrutinize the theoretical uncertainties of the QCD potential in relation to the $t\bar{t}$ threshold cross section. In particular there exists a theoretical uncertainty which limits our present theoretical accuracy of the $t\bar{t}$ threshold cross section at the peak to be $\delta \sigma_peak/\sigma_peak \simgt 6\%$ within perturbative QCD.

A sizeable difference in the differential production cross section of top and antitop quarks, respectively, is predicted for hadronically produced heavy quarks. It is of order $\alpha_s$ and arises from the interference between charge odd and even amplitudes respectively. For the TEVATRON it amounts up to 15\% for the differential distribution in suitable chosen kinematical regions. The resulting integrated forward-backward asymmetry of 4–5\% could be measured in the next round of experiments. At the LHC the asymmetry can be studied by selecting appropriately chosen kinematical regions. Furthermore, a slight preference at LHC for centrally produced antitop is predicted, with top quarks more abundant at large positive and negative rapidities.

We extend the investigation of the reparametrization invariance of Heavy Quark Effective Theory to O(1/m_Q^3). We show that to this order reparametrization invariance can only be maintained if the the reparametrization transformation is renormalized properly.

We propose a method for a QCD based calculation of one-particle inclusive decays of the form B \to \bar D X or B \to \bar D^* X. It is based on the heavy mass limit and a short distance expansion of the amplitudes, which yield a power series in the parameter 1/M^2_X for the spectra and in \Lambda_QCD m_b/(m_b - m_c)^2 for the rates. We study the leading term of this expansion for the case of the semi–leptonic decays B \to \bar D X l^+ \nu.

Models of neutrino masses are discussed capable of explaining in a natural way the maximal mixing between $\nu_{\mu}$ and $\nu_{\tau}$ observed by the Super-Kamiokande collaboration. For three generations of leptons two classes of such models are found implying: a) $\Delta {m_{23}}^2 \ll \Delta {m_{12}}^2 \approx \Delta {m_{13}}^2 $ and a small mixing between $\nu_{e}$ and the other two neutrinos, b) $\Delta {m_{12}}^2 \ll \Delta {m_{13}}^2 \approx \Delta {m_{23}}^2$ and a nearly maximal mixing for solar neutrino oscillations in vacuum.

Recent developments in the field of automatic computation of Feynman diagrams using asymptotic expansions are reviewed. The hadronic decay rate of the Z-boson is taken as an example for their physical application.

We complete the one loop renormalization of the HQET lagrangian at O(1/m_Q^3) including four fermion operators with two heavy and two light quark fields in the operator basis. It is shown that as a consequence the short distance coefficients of the operators bilinear in the heavy quark field receive nontrivial corrections.

We review recent calculations of second order corrections to heavy quark decays. Techniques developed in this context are applicable to many processes involving heavy charged particles, for example the nonabelian dipole radiation. Calculations involving relativistic radiating particles are also discussed.

We present O(alpha_s^2) corrections to the decay t–>bW in the limit of a very large top quark mass, m_t » m_W. We find that the O(alpha_s^2) effects decrease the top quark decay width Gamma_t by about 2%: Gamma_t = Gamma_0(1-0.8alpha_s(m_t)-1.7alpha_s^2). The complete corrections are smaller by about 24% than their estimate based on the BLM effects O(beta_0 alpha_s^2). We explain how to compute a new type of diagrams which contribute to Gamma(t –> bW) at the O(alpha_s^2) level.

In the virtual presence of a heavy quark t, the interactions of a CP-odd scalar boson A, with mass M_A « 2M_t, with gluons and light quarks can be described by an effective Lagrangian. We analytically derive the coefficient functions of the respective physical operators to three loops in quantum chromodynamics (QCD), adopting the modified minimal-subtraction (MS-bar) scheme of dimensional regularization. Special attention is paid to the proper treatment of the gamma_5 matrix and the Levi-Civita epsilon tensor in D dimensions. In the case of the effective ggA coupling, we find agreement with an all-order prediction based on a low-energy theorem in connection with the Adler-Bardeen non-renormalization theorem. This effective Lagrangian allows us to analytically evaluate the next-to-leading QCD correction to the A → gg partial decay width by considering massless diagrams. For M_A = 100GeV, the resulting correction factor reads 1+(221/12)alpha_s^(5)(M_A)/pi +165.9(alpha_s^(5)(M_A)/pi)^2 approx 1+0.68+0.23. We compare this result with predictions based on various scale-optimization methods.

We obtain model independent bounds for the form factors which arise in semileptonic B → Pi decays. To this end we derive a theoretical restriction for possible combinations of the value of the form factor and its derivatives at the endpoint region. These restrictions are then used as an input in a - slightly modified - general formalism, which has already been deduced and applied in earlier publications. Our results can be used to set constraints on the form factors which have to be obeyed by all model dependent parametrizations.

A sizeable difference in the differential production cross section of top and antitop quarks, respectively, is predicted for hadronically produced heavy quarks. It is of order $\alpha_s$ and arises from the interference between charge odd and even amplitudes respectively. For the TEVATRON it amounts up to 15\% for the differential distribution in suitable chosen kinematical regions. The resulting integrated forward-backward asymmetry of 4–5\% could be measured in the next round of experiments. At the LHC the asymmetry can be studied by selecting appropriately chosen kinematical regions.

In this work radiative corrections in the total hadronic decay rate of the tau-lepton and some moments of its differential distributions are studied employing perturbative QCD and the operator product expansion. We calculate quadratic quark mass corrections in the strange mass to the decay rate ratio R_{\tau} to the order O(\alpha_s^3 m^2) and find that they contribute appreciably to the Cabibbo suppressed decay modes of the $\tau$-lepton. Using the results of a recent experimental analysis, we obtain m_s(1 GeV ) = 200 \pm 40_{exp} \pm 30_{th} MeV.

The b quark low-scale running mass m_kin is determined from an analysis of the Upsilon sum rules in the next-to-next-to-leading order (NNLO). It is demonstrated that using this mass one can significantly improve the convergence of the perturbation series for the spectral density moments. We obtain m_kin( 1 GeV ) = 4.56 \pm 0.06 GeV. Using this result we derive the value of the MS-bar mass m: m ) = 4.20 \pm 0.1 GeV. Contrary to the low-scale running mass, the pole mass of the b quark cannot be reliably determined from the sum rules. As a byproduct of our study we find the NNLO analytical expression for the cross section e+e- –> Q\bar Q of the quark antiquark pair production in the threshold region, as well as the energy levels and the wave functions at the origin for the ^1S_3 bound states of Q\bar Q.

We compute the one-loop anomalous dimension for the light cone distribution function of a heavy quark and solve the corresponding evolution equation analytically. Some implications of the results for inclusive $B$ decays are discussed.

A sizeable difference in the differential production cross section of top and antitop quarks, respectively, is predicted for hadronically produced heavy quarks. It is of order $\alpha_s$ and arises from the interference between charge odd and even amplitudes respectively. For the TEVATRON it amounts up to 15\% for the differential distribution in suitable chosen kinematical regions. The resulting integrated forward-backward asymmetry of 4–5\% could be measured in the next round of experiments. At the LHC the asymmetry can be studied by selecting appropriately chosen kinematical regions. Furthermore, a slight preference at LHC for centrally produced antitop is predicted, with top quarks more abundant at large positive and negative rapidities.

We present complete O(alpha_s^2) corrections to the decay b–>c+l+nu at the point where the invariant mass of the leptons sqrt{q^2} equals the c quark mass. We use this result, together with previously obtained corrections at the ends of the q^2 spectrum, to estimate the total width of the semileptonic b–>c decay with O(alpha_s^2) accuracy, essential for the |V_{cb}| determination.

A correlator of the vector current of a heavy quark is computed analytically near threshold in the next-to-next-to-leading order in perturbative and relativistic expansion that includes $\al_s^2$, $\al_sv$ and $v^2$ corrections in the coupling constant and velocity of the heavy quark to the nonrelativistic Coulomb approximation. Based on this result, the numerical values of the $b$-quark pole mass and the strong coupling constant are determined from the analysis of sum rules for the $\Upsilon$ system.

Tau lepton production and decay to hadronic final states offer the unique possibility for studying low energy hadron dynamics and the weak couplings of the tau at the same time. Eventually one may even be sensitive to signals of physics beyond the Standard Model, like deviations from the V-A structure of the charged currents, or even discover CP violation. Tau polarization and the polarization of the decay products may be crucial for these investigations.

In the infinite mass limit for a heavy quark its spin decouples from the QCD dynamics, which leads to the heavy-quark spin symmetry. After a short discussion of spin symmetry some applications are considered.

We calculate the next-to-next-to-leading order correction to the cross section for top quark pair production in e+e- annihilation in the threshold region, resumming all O [ ( alpha_s/beta )^n ( alpha_s^2, beta^2, alpha_s beta ) ] terms of perturbation series. We find that the magnitude of the NNLO correction is comparable to the size of the NLO corrections.

We derive a formula which relates the QCD potentials in momentum space and in position space in terms of the beta-function of the renormalization-group equation for the potential. This formula is used to study the theoretical uncertainties in the potential and in particular in its application to the determination of the pole mass m_b when we use perturbative expansions. We demonstrate the existence of these uncertainties for the Richardson potential explicitly and then discuss the limited theoretical accuracy in the perturbative QCD potential. We conclude that a theoretical uncertainty of m_b much below 100 MeV would not be achievable within perturbative QCD.

We include the full second-order corrections to the static QCD potential in the analysis of the ttbar threshold cross section. There is an unexpectedly large difference between the QCD potential improved by the renormalization-group equation in momentum space and the potential improved by the renormalization-group equation in coordinate space. This difference remains even at a fairly short distance 1/r ~ 100 GeV and its origin can be understood within perturbative QCD. We scrutinize the theoretical uncertainties of the QCD potential in relation to the ttbar threshold cross section. In particular there exists a theoretical uncertainty which limits our present theoretical accuracy of the ttbar threshold cross section at the peak to be not better than 6% within perturbative QCD.

A sizeable difference in the differential production cross section of top and antitop quarks, respectively, is predicted for hadronically produced heavy quarks. It is of order $\alpha_s$ and arises from the interference between charge odd and even amplitudes respectively. For the TEVATRON it amounts to approximately $5-10\%$ in the region where the cross section is large and could therefore be measured in the next round of experiments. At the LHC the asymmetry can be studied by selecting appropriately chosen kinematical regions.

An evaluation of the effective QED coupling at the scale M_Z is presented. It employs the predictions of perturbative QCD for the cross section of electron positron annihilation into hadrons up to order \alpha_s^2, including the full quark mass dependence, and of order \alpha_s^3 in the high energy region. This allows to predict the input for the dispersion relations over a large part of the integration region. The perturbative piece is combined with data for the lower energies and the heavy quark thresholds. The result for the hadronic contribution to the running of the coupling \Delta\alpha^{(5)}_{\rm had}(M_Z^2)= (277.5 \pm 1.7)\times 10^{-4} leads to (\alpha(M_Z^2))^{-1}=128.927 \pm 0.023. Compared to previous analyses the uncertainty is thus significantly reduced.

We consider the three-loop singlet diagrams induced by axial-vector, scalar and pseudo-scalar currents. Expansions for small and large external momentum $q$ are presented. They are used in combination with conformal mapping and Pad\'e approximations in order to arrive at results for the polarization functions valid for all $q^2$. Results are presented for the imaginary parts which are directly related to physical quantities like the production of top quarks or the decay of scalar or pseudo-scalar Higgs bosons.

We include the full second-order corrections to the static QCD potential in the analysis of the ttbar threshold cross section. Then we examine the difference between the results obtained in the momentum-space approach and in the coordinate-space approach, which was found recently. Contrary to our expectation, the reduction of the difference by the inclusion of the second-order corrections is very small. There still remains a non-negligible deviation, which originates from the difference in the construction of the potentials in the two spaces. We scrutinize this problem. In particular, we estimate our present theoretical uncertainty of the ttbar threshold cross section at the peak to be at least 6% within perturbative QCD.

We present a calculation of the renormalized HQET Lagrangian at order O(1/m_Q^3) in the one particle sector. The anomalous dimensions of local operators and time ordered products of dimension 7 contributing at this order are calculated in the one loop approximation. We show that a careful treatment of the time ordered products is necessary to arrive at a gauge independent renormalized lagrangian. Our result sets the stage for an investigation of reparametrization invariance at O(1/m_Q^3).

We analyze sum rules for the \Upsilon system with resummation of threshold effects on the basis of the nonrelativistic Coulomb approximation. We found the pole mass of the bottom quark m_b=4.75 \pm 0.04 GeV and the strong coupling constant \alpha_s(M_Z)=0.118 \pm 0.006. The origin of the contradiction between two recent estimates obtained within the same formal framework is clarified.