We report on an investigation of the (pi-, pi0 pi0) reaction by means of measurements of the pi0 pi0 invariant mass distributions from pi- interactions on H, D, C, Al, and Cu targets at p(pi-) = 0.408 GeV/c. The sharp, strong peak in the pi+ pi- invariant mass near 2 m(pi) reported by the CHAOS Collaboration is not seen in our pi0 pi0 data. However, we do observe a change in the shape of the pi0 pi0 invariant mass spectrum for the different targets, indicating that the pi0 pi0 interaction diminishes in the nuclear medium as represented by nuclei D, C, Al, and Cu, compared to hydrogen.

We report the first determination of the upper limit for the branching ratio of the CP forbidden decay eta to 4 pi0. No events were observed in a sample of 3.0×10^7 eta decays. The experiment was performed with the Crystal Ball multiphoton spectrometer installed in a separated pi- beam at the AGS (Alternating Gradient Synchrotron). At the 90% confidence limit, B(eta to 4 pi0) ⇐ 6.9×10^-7.

The session on precision studies of electroweak interactions is summarized. The contributions address the bilinear, trilinear, quartic as well as heavy-quark interactions of the electroweak gauge bosons. This makes up a picture of the physics of electroweak symmetry and symmetry breaking which can be investigated with the proposed design of e+e- Linear Colliders and detectors.

It has been suggested to realize a factory for 10^9 Z^0 through a linear e^+e^- collider with polarized beams. Very likely the relevant CP studies for B mesons will already have been performed at the B factories by that time, hence GIGA-Z will be a third generation b physics experiment. Yet such a facility would provide us with unique opportunities in the domain of beauty physics that would be of essential significance even in 2010: (1) Production and decay of polarized beauty baryons; (2) searching for and probing transitions driven by b –> q \nu \bar{\nu}; (3) detailed and comprehensive studies of inclusive semileptonic B_s decays.

We construct the higher twist structure functions that describe inclusive b hadron decays in regions of phase space where the hadronic decay products carry high energy but have low invariant mass. We show that, for B meson decays, there are four new non-vanishing matrix elements of non-local operators. We show that to subleading twist these decays are parametrized in terms of four functions. We compute the tree-level matching for a general heavy-to-light current and apply it to B → X_s gamma. Using a simple model for these functions we estimate the subleading twist contributions to this decay.

We present analytical results for the leading contributions of the top quark to the electroweak rho parameter at order GF^3 Mt^6 and GF^2 Mt^4 alpha_s. The Higgs boson and the gauge bosons are taken to be massless in this limit. The correction of order GF^3 Mt^6 is found to be sizeable in comparison to the the leading two-loop GF^2 Mt^4 correction, however it is much smaller than the subleading GF^2 Mt^2 MZ^2 correction.

The non-diagonal correlators of vector and scalar currents are considered at three-loop order in QCD. The full mass dependence is computed in the case where one of the quarks is massless and the other one carries the mass M. As applications we consider the single-top-quark production via the process $q\bar{q}\to t\bar{b}$ and the decay rate of a charged Higgs into hadrons. In both cases the computed NLO corrections are shown to be numerically much less important than the leading ones.

The dimensionally regularized massless non-planar double box Feynman diagram with powers of propagators equal to one, one leg off the mass shell, i.e. with p_1^2=q^2\neq 0, and three legs on shell, p_i^2=0, i=2,3,4, is analytically calculated for general values of q^2 and the Mandelstam variables s,t and u (not necessarily restricted by the physical condition s+t+u=q^2). An explicit result is expressed through (generalized) polylogarithms, up to the fourth order, dependent on rational combinations of q^2,s,t and u, and simple finite two- and three fold Mellin–Barnes integrals of products of gamma functions which are easily numerically evaluated for arbitrary non-zero values of the arguments.

We further analyze the definition and the calculation of the heavy quark impact factor at next-to-leading (NL) $\log s$ level, and we provide its analytical expression in a previously proposed k-factorization scheme. Our results indicate that k-factorization holds at NL level with a properly chosen energy scale, and with the same gluonic Green's function previously found in the massless probe case.

Non-relativistic QCD (NRQCD) predicts colour octet contributions to be significant not only in many production processes of heavy quarkonia but also in their radiative decays. We investigate the photon energy distributions in these processes in the endpoint region. There the velocity expansion of NRQCD breaks down which requires a resummation of an infinite class of colour octet operators to so-called shape functions. We model these non-perturbative functions by the emission of a soft gluon cluster in the initial state. We found that the spectrum in the endpoint region is poorly understood if the values for the colour octet matrix elements are taken as large as indicated from NRQCD scaling rules. Therefore the endpoint region should not be taken into account for a fit of the strong coupling constant at the scale of the heavy quark mass.

We discuss the non-perturbative and the radiative corrections to inclusive B decays from the point of view known from QED corrections to high energy e^+ e^- processes. Here the leading contributions can be implemented through the so called ``radiator function'' which corresponds to the shape function known in heavy hadron decays. In this way some new insight into the origin of the shape function is obtained. As a byproduct, a parameterization of the radiatively corrected shape function is suggested which can be implemented in Monte Carlo studies of inclusive heavy hadron decays.

3 Jets: The planar topologies > The calculation of the two-loop corrections to the three jet production rate and to event shapes in electron-positron annihilation requires the computation of a number of up to now unknown two-loop four-point master integrals with one off-shell and three on-shell legs. In this paper, we compute those master integrals which correspond to planar topologies by solving differential equations in the external invariants which are fulfilled by the master integrals. We obtain the master integrals

as expansions in $\e=(4-d)/2$, where $d$ is the

space-time dimension. The results are expressed in terms of newly introduced two-dimensional harmonic polylogarithms, whose properties are shortly discussed. For all two-dimensional harmonic polylogarithms appearing in the divergent parts of the integrals, expressions in terms of Nielsen's polylogarithms are given.

Extended version of lectures given at the University of Karlsruhe and at Calc-2000 school in Dubna. Properties of HQET as a field theory, methods of calculation of HQET diagrams and some simple applications are explained in detail. These lectures can be used as an additional chapter with any modern QCD textbook.

We investigate energy (momentum) distributions in J/Psi photoproduction and J/Psi production in B meson decay. In particular the upper endpoint region of the spectrum is examined where the effect of soft gluon emission from the ccbar pair becomes important. Constructing a model which is consistent with the so-called shape function formalism we consider these fragmentation effects and show that the relevance of possible colour octet contributions in the photoproduction channel is still inconclusive.

We present numerical results for the two loop QCD corrections to the ghost gluon vertex, the quark gluon vertex, and the triple gluon vertex in the massless limit at the symmetric point. We restrict ourselves to the tensor structures existing in the tree level. The corrections are used to examine different coupling constant definitions in momentum subtraction schemes and the corresponding three-loop $\beta$ functions.

A Monte Carlo generator has been constructed to simulate the reaction e^+ e^- \to \gamma + 4 \pi, where the photon is assumed to be observed in the detector. Isospin relations between the amplitudes governing tau decays into four pions and electron positron annihilation into four pions respectively have been found which allow to determine all four modes after the amplitude for the \pi^+ \pi^- 2 \pi^0 channel has been fixed. The kinematic breaking of these isospin relations as a consequence of the \pi^- – \pi^0 mass difference has also been investigated. The program is constructed in analogy to an earlier one simulating e^+ e^- \to \gamma + 2 \pi. However, it does not include final state radiation from the charged pions. Additional collinear photon radiation has been incorporated with the technique of structure functions. Predictions are presented for cms energies of 1GeV, 3GeV and 10GeV, corresponding to the energies of DAPHNE, BEBC and of B-meson factories. Using this program it is demonstrated that, even after applying realistic cuts, the event rates are sufficiently high to allow for a precise measurement of R(Q^2) in the region of Q between approximately 1GeV and 2.5GeV. The model predictions are compared to recent data from electron positron colliders and agreement is obtained within the relatively large errors.

In this paper we present a full set of 2- and 3-point functions for massless QCD at three-loop order in the MSbar scheme. The vertex functions are evaluated at the asymmetric point with one vanishing momentum. These results are used to relate the MSbar coupling constant to that of various momentum subtraction renormalization schemes at three-loop order. With the help of the known four-loop MS-bar beta-function we then determine the four-loop coefficients of the corresponding MOM beta-functions.

As an application we consider the momentum dependence (running) of the three-gluon asymmetrical vertex recently computed within the lattice approach by Ph. Boucaud et al. (JHEP 04 006 (2000)). An account of the four-loop term in the corresponding beta-function leads to a significant (around 30\%) decrease of the value of the non-perturbative power-suppressed correction to the running.

In the recent years there has been an enormous development in the evaluation of higher order quantum corrections. An essential ingredient in the practical calculations is provided by vacuum diagrams, i.e. integrals without external momenta. In this paper a program package is described which can deal with one-, two- and three-loop vacuum integrals with one non-zero mass parameter. The principle structure is introduced and the main parts of the package are described in detail. Explicit examples demonstrate the fields of application.

We present the analytic next-to-next-to-leading QCD calculation of some higher moments of deep inelastic structure functions in the leading twist approximation. We give results for the moments $N$=1,3,5,7,9,11,13 of the structure function $F_3$. Similarly we present the moments $N$=10,12 for the flavour singlet and $N$=12,14 for the non-singlet structure functions $F_2$ and $F_L$. We have calculated both the three-loop anomalous dimensions of the corresponding operators and the three-loop coefficient functions of the moments of these structure functions.

The present paper is the first of a series of papers reporting on the parallelization of the symbolic manipulation program FORM on different parallel architectures. Part I deals with workstation clusters using dedicated network hardware and the messages passing libraries (MPI and PVM). After a short introduction to the sequential version of FORM a detailed analysis of the different platforms used is given and the structure of the parallel version of FORM is explained. The forthcoming part II will describe the parallelization of FORM on SMP (symmetrical multi-processing) architectures.

Recent results for the asymptotic behavior of fermion scattering amplitudes in the Sudakov limit are presented including next-to-leading logarithmic corrections. These are used for the analysis of the dominant electroweak corrections to the fermion-antifermion pair production in $e^+e^-$ annihilation at high energy.

The calculation of exclusive observables beyond the one-loop level requires elaborate techniques for the computation of multi-leg two-loop integrals. We discuss how the large number of different integrals appearing in actual two-loop calculations can be reduced to a small number of master integrals. An efficient method to compute these master integrals is to derive and solve differential equations in the external invariants for them. As an application of the differential equation method, we compute the ${\cal O}(\epsilon)$-term of a particular combination of on-shell massless planar double box integrals, which appears in the tensor reduction of $2 \to 2$ scattering amplitudes at two loops.

The total cross section for the production of massive quarks in electron positron annihilation can be predicted in perturbative QCD. After expansion in m^2/s the quartic terms, i.e. those proportional to m^4/s^2, are calculated up to order \alpha_s^3 for vector and axial current induced rates. Predictions relevant for charm, bottom and top quarks production are presented. The \alpha_s^3 corrections are shown to be comparable to terms of order \alpha_s and \alpha_s^2. As a consequence, the predictions exhibit a sizeable dependence on the renormalization scale. The stability of the prediction is improved and, at the same time, the relative size of the large order terms decreases by replacing the running mass \bar{m}(\mu) with the scheme independent invariant one \hat{m}. By combining these results with the prediction for massless case and the quadratic mass terms the cross section for massive quark production at electron positron colliders is put under control in order \alpha_s^3 from the high energy region down to fairly low energies.

We describe a calculation of the on-shell renormalization factors in QCD

 and QED at the three loop level. Explicit results for the fermion mass 
 renormalization factor Zm and the on-shell fermion wave function 
 renormalization constant Z2 are given. We find that at O(alpha_s^3) the
 wave function renormalization constant Z2 in QCD becomes gauge dependent 
 also in the on-shell scheme, thereby disproving the ``gauge-independence'' 
 conjecture based on an earlier two-loop result. As a byproduct, we derive 
 an O(alpha_s^3) contribution to the anomalous dimension of the heavy quark 
 field in HQET. 

The class of the two-loop massless crossed boxes, with light-like external legs, is the final unresolved issue in the program of computing the scattering amplitudes of 2 –> 2 massless particles at next-to-next-to-leading order. In this paper, we describe an algorithm for the tensor reduction of such diagrams. After connecting tensor integrals to scalar ones with arbitrary powers of propagators in higher dimensions, we derive recurrence relations from integration-by-parts and Lorentz-invariance identities, that allow us to write the scalar integrals as a combination of two master crossed boxes plus simpler-topology diagrams. We derive the system of differential equations that the two master integrals satisfy using two different methods, and we use one of these equations to express the second master integral as a function of the first one, already known in the literature. We then give the analytic expansion of the second master integral as a function of epsilon=(4-D)/2, where D is the space-time dimension, up to order O(epsilon^0).

We further analyze the definition and the calculation of the heavy quark impact factor at next-to-leading (NL) $\log s$ level, and we provide its analytical expression in a previously proposed k-factorization scheme. Our results indicate that k-factorization holds at NL level with a properly chosen energy scale, and with the same gluonic Green's function previously found in the massless probe case.

In this paper the formulae are collected which are needed for the computation of the strong coupling constant and quark masses at different energy scales and for different number of active flavours. All equations contain the state-of-the-art QCD corrections up to three- and sometimes even four-loop order. For the practical implementation Mathematica is used and a package containing useful procedures is provided.

Including the radiative perturbative corrections and the short distance tachyonic gluon mass effects which mimic the ones of UV renormalons, we re-estimate the decay amplitudes, masses and widths of light hybrid mesons from QCD spectral sum rules. We show that the effects are tiny and confirm the previous lowest order results. We discuss the phenomenological impacts of our results for the vector hybrids.

We demonstrate that the angular distribution of hadrons produced in semi-inclusive deep inelastic final states is related to the inclusive longitudinal structure function. This relation could provide a new method of accessing $F_L(x,Q^2)$ in deep inelastic scattering measurements.

The production of a hard and isolated photon accompanied by one or two jets in large-Q2 deep inelastic ep scattering is calculated at next-to-leading order. We include consistently contributions from quark-to-photon fragmentation and study various differential cross sections and their dependence on isolation cut parameters. Numerical results relevant for HERA experiments are presented.

An algorithm for calculation of three-loop propagator diagrams in HQET, based on integration-by-parts recurrence relations, is constructed and implemented as a REDUCE package Grinder, and in Axiom.