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I give an overview of recent calculations of the renormalization group \(\beta \)-function and the quark mass anomalous dimension at the 4-loop order of perturbative Quantum Chromodynamics. In addition I discuss the order \(\alpha _s^3\) contribution to the Ellis-Jaffe sum rule for the structure function \(g_1\) of polarized deep inelastic lepton-nucleon scattering. The calculations discussed in this talk were performed in collaborations with S.A Larin and J.A.M. Vermaseren.

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We analyze the second order QCD corrections to the fragmentation functions \(F_k^{\rm H}(x,Q^2)\) (\(k=T,L,A\)) which are measured in \(e^+~e^-\) annihilation From these fragmentation functions one can derive the integrated transverse (\(\sigma _T\)), longitudinal (\(\sigma _L\)) and asymmetric (\(\sigma _A\)) cross sections. The sum \(\sigma _{\rm tot}=\sigma _T+\sigma _L\) corrected up to order \(\alpha _s^2\) agrees with the well known result in the literature. It turns out that the order \(\alpha _s^2\) corrections to the transverse and asymmetric quantities are small in contrast to our findings for \(F_L^{\rm H}(x,Q^2)\) and \(\sigma _L\) where they turn out to be large. Therefore in the latter case one gets a better agreement between the theoretical predictions and the data obtained from he LEP experiments.

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The role of various symmetries in the evaluation of splitting functions and coefficient functions is discussed. The scale invariance in hard processes is known to be a guiding tool to understand the dynamics. We discuss the constraints on splitting functions coming from various symmetries such as scale, conformal and supersymmetry. We also discuss the Drell–Levy–Yan relation among splitting and coefficient functions in various schemes. The relations coming from conformal symmetry are also presented.

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I summarize the calculation of \(O(\alpha _s)\) corrections to the production of a hard and isolated photon accompanied by one or two jets in deep inelastic lepton nucleon scattering at HERA.

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We discuss some of the problems that may occur in the calculation of complicated Feynman diagrams. These include the group independent evaluation of color factors, and the summation techniques that are needed for the expansion of diagrams into their Mellin moments.

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The method of the large mass expansion (LME) is investigated for selfenergy and vertex functions in two-loop order. It has the technical advantage that in many cases the expansion coefficients can be expressed analytically. As long as only one non-zero external momentum squared, \(q^2\), is involved also the Taylor expansion (TE) w.r.t. small \(q^2\) yields high precision results in a domain sufficient for most applications. In the case of only one non-zero mass \(M\) and only one external momentum squared, the expansion w.r.t. \(q^2/M^2\) is identical for the TE and the LME. In this case the combined techniques yield analytic expressions for many diagrams, which are quite easy to handle numerically.

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The general lines of the derivation and the main properties of the master equations for the master amplitudes associated to a given Feynman graph are recalled. Some results for the 2-loop self-mass graph with 4 propagators are presented.

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An \(osp(1,2)\)–covariant Lagrangian quantization of general gauge theories is introduced which also applies to massive fields. It generalizes the Batalin-Vilkovisky and the \(Sp(2)\)–covariant field–antifield approach and guarantees symplectic invariance of the quantized action. Massive gauge theories with closed algebra are considered as an example.

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The algebraic method of renormalization is applied to the standard model of electroweak interactions. We present the most important modifications compared to theories with simple groups.

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An algorithm for the reduction of Feynman integrals with any number of loops and external momenta to a minimal set of basic integrals is proposed. The method is based on the new algorithms for evaluating tensor integrals, representation of generalized recurrence relations for a given kind of integrals as a linear system of PDEs and the reduction of this system to a standard form. Basic integrals reveal as parametric derivatives of the system in the standard form and the number of basic integrals in the minimal set is determined by the dimension of the solution space of the system of PDEs.

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The precise electroweak measurements obtained from LEP and SLD Z data are presented. Since 1996 LEP has been running above the W pair threshold, and results on the mass and couplings of the W boson are also given. The LEP results are combined with the SLD, Tevatron and \(\nu \)-nucleon results to infer the mass of the Higgs boson. This is especially interesting in light of recent improvements in the calculation of the radiative corrections.

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I discuss the importance of the \(O(g^4 m_t^2\,/\,M_w^2)\) corrections to the effective electroweak angle and \(M_w\) in the indirect determination of the Higgs mass. I emphasize the rôle of a very precise \(M_w\) measurement on the \(M_H\) estimate.

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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.

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To avoid loss of sensitivity in the search for new physics in single- and multi-photon final states with large missing energy at LEP, precise predictions for the Standard Model irreducible background are required. At LEP1 the theoretical situation is satisfactory. Going to LEP2, some improvements are necessary. To this aim, the matrix elements for the processes \(e^+ e^- \to \nu \bar \nu n\gamma \), with \(n=1,2,3\), are exactly computed in the Standard Model, including the possibility of anomalous couplings for single-photon production. Due to the presence of observed photons in the final state, particular attention is paid to the treatment of higher-order QED corrections. Comparisons with existing calculations are shown and commented. An improved version of the event generator NUNUGPV is presented.

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We present the results of the calculation of the ‘photon’ + 1 jet rate at \({\cal O}(\alpha \alpha _{s})\) for LEP energies. By comparing these results with the data from the ALEPH Collaboration we make a next-to-leading order determination of the quark-to-photon fragmentation function \(D_{q\to \gamma }(z,\mu _{F})\) at \({\cal O}(\alpha \alpha _{s})\). The predictions obtained using this fragmentation function for the isolated rate, defined as the ‘photon’ + 1 jet rate for \(z\gt 0.95\), are found in good agreement with the ALEPH data. The next-to-leading order corrections are moderate demonstrating the perturbative stability of this particular isolated photon definition. We have also computed the inclusive photon energy distribution and found good agreement with the OPAL data.

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The calculation of the two-loop corrections to the electroweak gauge boson quartic couplings, growing quadratically with the Higgs boson mass, is reviewed. The potential of the CERN Large Hadron Collider and \(e^+e^-\) linear collider to study such anomalous interactions is discussed.

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Diagrammatic two-loop results are presented for the leading QCD corrections to the masses of the neutral \({\cal CP}\)-even Higgs bosons in the Minimal Supersymmetric Standard Model (MSSM). The results are valid for arbitrary values of the parameters of the Higgs and scalar top sector of the MSSM. Their impact on a precise prediction for the mass of the lightest Higgs boson is briefly discussed.

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Higher-order contributions to the precision observables \(\Delta r\), \(\sin ^2 \theta _{\rm eff}\) and \({\mit \Gamma }_l\) are discussed. The Higgs-mass dependence of the observables is investigated at the two-loop level, and exact results are derived for the Higgs-dependent two-loop corrections associated with fermions. The top-quark corrections are compared with the results obtained by an expansion in the top-quark mass up to next-to-leading order. For the pure fermion-loop contributions to \(\Delta r\) results up to four-loop order are derived. They allow to investigate the validity of the commonly used resummation of the leading fermionic contributions to \(\Delta r\).

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Some recent results on the effective interactions of a light gravitino with ordinary particles are reviewed. In particular, the low-energy behaviour of electron-positron and photon-photon annihilation into gravitinos are carefully discussed, and a new "low-energy theorem" is established in the electron-positron case. These results are applied to derive model independent bounds on the supersymmetry breaking scale and to organize the search for a superlight gravitino at high-energy colliders.

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The one-loop MSSM contributions to the weak dipole moments of the \(\tau \) lepton and the \(b\) quark (at the \(Z\) peak) as well as the electromagnetic and weak dipole form factors of the \(t\) quark (at arbitrary \(s\gt 4m^2_t\)) are reviewed. Emphasis is given to the relevance of the \(t\)-quark CP-violating weak and electric dipole form factors as a part of the full one-loop correction to the process \(e^+e^-\to t\bar t\) in the MSSM.

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The main physics items to be studied in the \(\phi \)-factory DA\(\Phi \)NE under commissioning in Frascati are discussed. Experimental searches for CP violation, study of chiral structure of pseudoscalar and vector mesons, pion–pion phase shifts, hadronic \(e^+e^-\) cross-sections will be studied with the detector KLOE, nuclear physics with kaons will be investigated with the detector FINUDA, and formation of Kaonic atoms with the detector DEAR.

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Recently a rephasing-invariant definition of the CP-mixing parameter for Indirect CP Violation has been introduced. This is made possible by the explicit use of the CP operator in the analysis. The problem is the determination of the CP operator for a CP violating scenario. We discuss it and provide a definite solution.

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The role of higher Fock-state contributions to exclusive charmonium decays will be discussed. It will be argued that the \(J\,/\,\psi \) (\(\psi '\)) decays into baryon-antibaryon pairs are dominated by the valence Fock-state contributions. \(P\)-wave charmonium decays, on the other hand, receive strong contributions from the \(c\overline {c} g\) Fock states since the valence Fock-state contributions are suppressed in these reactions. Numerical results for the decay widths of \(J\,/\,\psi (\psi ') \to B\overline {B}\) and \(\chi _{c J}\to P\overline {P}\) will be also presented and compared to data.

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The next-to-leading-order (NLO) analysis of the decay \(B \to X_s \gamma \) is reviewed in this talk. The theoretical framework and the various ingredients for a theoretical prediction of the branching ratio Br\((B\rightarrow X_s\gamma )\) are briefly outlined. The numerical analysis focuses on an estimate of the theoretical uncertainties. It is pointed out that the theoretical error is presently dominated by parametric uncertainties that may be reduced in the future. In view of oncoming measurements at future B factories this underlines the importance of the decay \(B \to X_s \gamma \) for the search for new physics.

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We review some aspects of our calculation of two-loop QCD corrections to \(b \to c\) decay, which confirmed the results of Czarnecki and Melnikov.

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We review some features and results of the calculations performed with the program SIXPHACT for six fermion final states at Linear Collider.

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We give an overview of the problems and developments associated with the calculation of radiative corrections to off-shell gauge-boson pair production in \(e^+e^-\) collisions.

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I present a short overview over \(W\) pair production and studies of angular differential cross-sections with and without initial state radiation applying semi-analytical methods and using the Fortran program GENTLE. The influence of anomalous couplings to this process is also discussed.

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The status of the calculation of the \({\cal O}(\alpha )\) 1-loop radiative corrections to \(e^+e^-\to 4f\) and some technical progresses are presented.

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The scope of constrained differential renormalization is to provide renormalized expressions for Feynman graphs, preserving at the same time the Ward identities of the theory. It has been shown recently that this can be done consistently at least to one loop for abelian and non-abelian gauge theories. We briefly review these results, evaluate as an example the gluon selfenergy in both coordinate and momentum space, and comment on anomalies.

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We summarize recent results connecting multiloop Feynman diagram calculations to different parts of mathematics, with special attention given to the Hopf algebra structure of renormalization.

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The calculation of the next-to-leading order QCD corrections to the production of a neutral Higgs-boson pair is briefly summarized. The dominant production mechanism is gluon fusion via top-quark loops, so that the QCD corrections can be considered, to good approximation, in the limit of a heavy top-quark mass.

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The decoupling relations for the strong coupling constant, \(\alpha _s\), and the light quark masses are considered in the framework of perturbative QCD. A low-energy theorem is derived which connects the decoupling constants to the coefficient functions of the effective Lagrangian responsible for the decay of a Higgs boson in the intermediate-mass range. The evaluation of the imaginary part of the correlators formed by the corresponding operators completes the calculation of the decay of the Higgs boson into hadrons.

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A connection between one-loop \(N\)-point Feynman diagrams and certain geometrical quantities in non-Euclidean geometry is discussed. A geometrical way to calculate the corresponding Feynman integrals is considered.

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It is shown that conventional mass renormalization, when applied to photonic or gluonic corrections to unstable particle propagators, leads to non-convergent series in the resonance region. A solution of this problem, based on the concepts of pole mass and width, is presented. In contrast with the \(Z\) case, the conventional on-shell definition of mass for \(W\) bosons and unstable quarks contains an unbounded gauge dependence in next-to-leading order. The on-shell and pole definitions of width are shown to coincide if terms of \(O({\mit \Gamma }^2)\) and higher are neglected, but not otherwise.

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High statistics data have been presented by ZEUS for hard photoproduction of \(D^*\) mesons. The measured cross sections for large \(p^{D^*}_{\perp }\), \(\eta ^{D^*}\) lie above those predicted by QCD-NLO calculations. There is a substantial contribution from resolved photons indicating the existence of charm excitation in the photon. Photoproduction of beauty quarks has been observed for the first time by H1. The measured cross section lies above the theoretical expectations calculated in QCD-LO by a substantial factor. First results on quasielastic photoproduction of Upsilons have been presented by ZEUS. The observed cross section lies above the theoretical predictions. QCD-NLO fits to data for the proton structure function \(F_2\) from H1 and ZEUS have provided rather precise determinations of the density of gluons in the proton. The resulting predictions for the charm contribution \(F^c_2\) to \(F_2\) are consistent with \(F^c_2\) obtained directly from charm production by DIS. An analysis of \(F_2\) data at small \(Q^2\) indicates that the transition from soft hadron-like scattering to DIS occurs somewhere between 0.5 and 3 GeV\(^2\). A QCD-NLO analysis by ZEUS with \(F_2\) measurements starting at \(Q^2 = 1\) GeV\(^2\) shows that a good description of the data can be obtained. Surprisingly, while at \(Q^2 = 7, 20\) GeV\(^2\) the gluon density (\(g\))at small \(x\) is much larger than that for the singlet quarks (\({\mit \Sigma }\)), the situation appears to be reversed at \(Q^2 = 1\) GeV\(^2\) where \(g \lt {\mit \Sigma }\). Diffraction dissociation of the virtual photon into high mass hadrons represents a substantial part of the DIS cross section. Its energy dependence is similar to that of the sum of all DIS channels and leads to a pomeron trajectory which lies above that observed in hadron-hadron scattering. The behaviour of the diffractive structure function suggests a substantial contribution from partonic interactions. H1 and ZEUS previously had reported an excess of events above the Standard Model predictions at large \(Q^2\), high \(x\) seen in the 1994-96 data. Analysis of the 1997 data, corresponding to about the same integrated luminosity, has not added to this excess.