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Introduction

The research of the group deals with various aspects of the fundamental interactions of nature and of the inner structure of elementary particles. It ranges from perturbative quantum field theory, as applied to hadron physics, within the framework of quantum chromodynamics, and to electroweak interactions of quarks and leptons, to extensions of Yang-Mills gauge theories within non-commutative geometry including gravity. Research in phenomenological particle physics centers on perturbative quantum field theory at the level of two loops, on the physics of hadrons containing heavy quarks, and on the physics of charged leptons and their neutrinos. On a more theoretical side, it covers algebraic Yang-Mills-Higgs theories, and their quantization, as well as their application to particle physics.

Further activities concern three-body problems in quantum physics, formal aspects of quantum perturbation theory, the inverse scattering problem, and impedance tomography. A book project on Theoretische Physik 2: "Nichtrelativistische Quantentheorie: Vom Wasserstoffatom zu den Vielteilchensystemen" was completed (F. Scheck, Springer-Verlag, 2000), revised editions of Theoretische Physik 1: "Mechanik - Von den Newtonschen Gesetzen zum deterministischen Chaos" and its English translation "Mechanics - From Newton's Laws to Deterministic Chaos", (F. Scheck, 3rd. ed. Springer-Verlag, 1999) were published.

Research Projects

Algebraic and Geometrical Methods in Quantum Field Theories

1. Algebraic Yang-Mills-Higgs Theories and Applications (R. Häußling, F. Scheck et al.)

Extensions of Yang-Mills theories to algebraic Yang-Mills-Higgs theories based on Non-commutative Geometry are by now well established. These theories provide a geometric interpretation of spontaneous symmetry breaking and incorporate chiral fermions as dynamic building blocks. They achieve a novel unified description of electroweak, strong and gravitational interactions. Our group was the first in Germany who took up this field of research about a decade ago and had an active share in its development. We have continued work on the model developed in Mainz and Marseille and its applications to particle physics.

2. Non-Commutative Geometry, Hopf Algebras and Fundamental Interactions (M. Paschke, F. Scheck, A. Sitarz et al.)

Connes' non-commutative geometry as applied to the physics of the standard model and to gravity raises important conceptual questions and gives strong hints at a better understanding of the role of quantum groups in physics. Specific issues dealt with in this context concern generalized differential calculi and connections the classification of finite Spectral Triples and their relation to Hopf algebras. Work is done also on finding and elaborating constructive examples within the spectral triple approach and in view of quantization [MZ-TH/99-53].

3. Algebraic Structures of Quantum Field Theories (D. Kreimer et al.)

The research is focused on the investigation of algebraic structures in renormalization theory and perturbative quantum field theory. In collaboration with Alain Connes (IHES and Collège de France, Paris) we achieved a reformulation of renormalization theory as a Riemann-Hilbert problem, with the splitting of an unrenormalized Green function into counterterms and finite renormalized quantities appearing as a corresponding Birkhoff decomposition, exhibiting sources of the rich analytic structure of QFT-relations to iterated integrals and shuffle algebras familiar in the theory of polylogarithms and zeta functions. Applications of these results led to high-order loop calculations in collaboration with David Broadhurst, allowing for tests of non-perturbative Borel resummation techniques, as well as to manifold connections to number theory. The detailed results obtained in 1998-1999 include a complete resolution of overlapping divergences in terms of the Hopf algebra of the classifying space of rooted trees and Feynman graphs, a classification of these Hopf algebras and an identification of the dual Lie algebras, including the relation to Hochschild cohomology. Finally, in a collaboration with E. Mielke (Mexico), chiral anomalies in space-times with torsion were studied, a topic which gave rise to controversial results in the literature due to an often erroneous use of renormalization prescriptions. [MZ-TH/98-19, 20, 31, 35, 39, 42, 43, 50, 99-14, 15, 38, 54, 55, 57].

4. Quantum Gravity (O. Lauscher, H.-J. Matschull, U. Müller, M. Reuter, F. Saueressig et al.)

One of the research topics in this field is the investigation of quantum gravitational effects within an effective field theory approach. Quantum gravity is described by a scale dependent Wilson-type effective action (coarse grained free energy or "effective average action") whose scale dependence is governed by a functional renormalization group equation. By projecting the renormalization group flow from the infinite dimensional space of all actions onto a certain particularly relevant finite dimensional subspace one can obtain approximate yet non-perturbative solutions to this equation. In particular, the scale dependence of Newton's constant and of the cosmological constant has been derived. Recent work along these lines focused on the impact which the running of Newton's constant has on the structure of black holes. In particular, it has been argued [MZ-TH/00-04] that the Hawking evaporation of a Schwarzschild black hole stops once its mass is of the order of the Planck mass, and that the final state of the evaporation process is a cold, soliton-like remnant of Planckian dimensions. The structure of the Cauchy horizon singularity formed in a realistic gravitational collapse has also been studied [MZ-TH/98-45].

One of the notorious problems of Euclidean gravity is the conformal factor instability which means that the Einstein-Hilbert action is not positive definite and that flat space is not its global minimum. It is plausible to speculate that quantum gravity effects lead to a dynamical stabilization of the conformal factor. In order to get a first understanding of how this might happen a scalar toy model with a nonstandard kinetic term has been analyzed in detail [MZ-TH/00-25]. Its (in)stability properties are similar to those of gravity theories with higher derivative actions.

From the calculational point of view the derivation of gravitational renormalization group equations is extremely involved and one might hope that, as in flat space, world-line path integrals facilitate such calculations to some extent. A first step in this direction has been taken in [MZ-TH/97-38], and in [MZ-TH/99-35] an application of this approach to the theory of Euler-Zagier sums has been described.

A further project concentrated on Witten's two-dimensional black hole which can be obtained by gauging a U(1) subgroup in the SL(2,R) WZNW theory. It was possible to show that, at the classical level the gauged SL(2,R)/U(1) WZNW theory is a completely integrable system [MZ-TH/98-21, 69, 99-26, 73]. A Lax pair and a Bäcklund transformation were constructed explicitly, and some of the results were generalized for arbitrary gauged WZNW models.

5. Quantum Gravity in Three Dimensions (H.-J. Matschull et al.)

Black holes can also be studied in three space-time dimensions, where Einstein gravity takes a particularly simple form. At the classical level, it is possible to describe the creation of a black hole within a very simple framework of just two particles passing each other and collapsing [MZ-TH/98-55, 99-17]. When quantizing such a system, one can hope to learn something about the evaporation process using simply toy models. Another aspect of Einstein gravity in three space-time dimensions is that it allows the definition of very realistic toy models, namely the so called particle models. The universe consists of only a finite number of point-like particles. Although some features of such models are more complicated than originally believed [MZ-TH/99-04], one can analyze the classical phase space and derive from this some general features of a quantum theory of gravity [MZ-TH/00-44]. For the simplest possible such model, the Kepler system of two particles coupled to the gravitational field, one can even perform the quantization exactly and make some interesting predictions about the general nature of quantum gravity [MZ-TH/99-31, 00-45].

6. Geometry of Quantum Phase-Space (M. Reuter et al.)

The aim of this program is the investigation of new geometrical structures on phase-space which are related to classical symplectic geometry in a similar way quantum mechanics is related to classical Hamiltonian dynamics. Recently the notion of a "symplectic Dirac-Kähler field" was introduced. Rather than on space-time, symplectic Dirac-Kähler fields "live" on the phase-space of an arbitrary Hamiltonian system; they are equivalent to an infinite family of metaplectic spinor fields, i.e. spinors of Sp(2N), in the same way an ordinary Dirac-Kähler field is equivalent to a (finite) multiplet of Dirac spinors [MZ-TH/99-43]. This construction has a natural interpretation within the gauge theory formulation of quantum mechanics which was proposed earlier [MZ-TH/98-51]. One finds an intriguing analogy between the lattice fermion problem (species doubling) and the problem of quantization in general.

7. Non-perturbative Properties of Yang-Mills Theories (M. Reuter et al.)

The stochastic vacuum model which describes the infrared sector of non-Abelian gauge theories in terms of a Gaussian random process has been used in order to compute the non-perturbative contributions to the 3-dimensional effective potential which arises from the electroweak model in the high temperature limit. This potential is crucial for an understanding of the electroweak phase transition in the early universe. A fading-away of the first-order phase transition with increasing Higgs mass was found, which is consistent with recent lattice results [MZ-TH/98-15]. Similar methods were used to calculate the effective gluon mass in 4 dimensions [MZ-TH/99-22].

In a different project, a novel nonlinear gauge has been introduced which leads to a hidden supersymmetry of pure Yang-Mills theory; it was argued that confinement can be understood as a consequence of the resulting Parisi-Sourlas dimensional reduction [MZ-TH/99-72].

8. Geometrical Methods in Classical and Elementary Particle Physics (M. Debert, A. Holfter, N. Papadopoulos et al.)

The connection between classical physics and quantum physics is investigated. The most remarkable difference between quantum and classical physics is the fact that in quantum physics the interpretation is part of its formulation. This has important implications in many fundamental problems in elementary particle physics.

It leads to investigations on quantization in various ways: within the framework of non-commutative geometry, deformation quantization, BRS formalism, quantum groups and Hopf algebras (see diploma theses by A. Holfter and M. Debert).

Mixing of Weak Interaction States

9. Mass Matrices and Neutrino Oscillations (G. Barenboim, R. Häußling,F. Scheck et al.)

The Cabibbo-Kobayashi-Maskawa mixing matrix for quarks and the analogous mixing matrix for neutrinos are manifestations of the mismatch between the mass eigenstates and the weak interaction states. After suitable redefinition of phases of the base states, they constitute an additional set of observables within the standard model whose dynamic origin is encoded in the mass matrices of quarks and leptons, respectively, with two charge sectors each. The latter, however, are ambiguous. Their reconstruction from the observed mixing has to cope with high-dimensional manifolds of choices and needs further physics input.

We have developed a simple, model-independent scheme that allows to reconstruct an effective mass matrix from the observable mixing matrix in a unique way. The latter is easily transformed, by means of correlated bi-unitary transformations in the two charge sectors, to any basis of chiral states, in view of making contact with individual model building. Our scheme also allows to introduce physical hypotheses about the origin of mass differences and of mixing in a transparent manner.

We showed that a consistent interpretation of all neutrino anomalies (including the LSND experiment) is achievable with three generations only, no sterile neutrino is needed [MZ-TH/98-29]. State mixing is found to be strong and to involve all three generations simultaneously. As a consequence, leptonic CP violation, if it is present, should be observable in neutrino beams from muon storage rings [MZ-TH/99-56]. Predictions for long-baseline oscillation experiments are worked out showing that a complete determination of the mixing matrix seems possible [MZ-TH/98-52]. It is also shown that the two squared mass differences of neutrinos can be obtained consistently from loop effects in left-right symmetric models, assuming primordial masses of neutrinos to vanish [MZ-TH/99-09]. Finally, investigation of hadronic CP violation within left-right symmetric models and of implications for baryogenesis and other phenomena was part of an ongoing research program with various scientists in Europe and the USA.

10. CP violation (G. Barenboim et al.)

CP violation has become the focus of increasing experimental and theoretical activity. The experimental results have pinned down the weak mixing angles to such an extent that the old problem of explaining the smallness of CP violation in the context of the three-generation Standard Model has been completely inverted to the problem of maximal CP violation. According to the majority view, the standard six-quark model is still compatible with experiment. It takes little imagination, however, to anticipate future problems for the Standard Model both in the leptonic (SuperKamiokande already known data) and in the quark sector (once the B-factories will begin to operate).

The experimental situation supplies ample motivation to investigate alternative or at least additional mechanisms for CP violation. There are in addition rather convincing theoretical arguments. The Standard Model mechanism of breaking CP explicitly in terms of complex Yukawa couplings provides a possible parameterization, but not a fundamental understanding of CP violation. This shortcoming is removed in models with spontaneous CP violation where both the gauge symmetry breaking and CP violation have a common origin. Although the mechanism of CP breaking needs to be clarified, such a unified treatment turns out to be very predictive.

In collaboration with colleagues from the University of Valencia, CERN, DESY and the Weizmann Institute, Israel) we have been working on CP violation in several extensions of the Standard Model finding possible signatures of its different formulations. In particular, as the next few years will be an exciting era for B physics, with the detailed investigation of B-hadrons at B-factories, we have concentrated ourselves in the potential for studying CP violation in the B-system [MZ-TH/98-01, 30, 40, 58, 99-03, 19].

Few-Particle Scattering

11. Theoretical Aspects of Charged-Composite Particle Scattering (E. O. Alt et al.)

Modern analyses of few-particle reactions are successfully based on momentum space integral equations of the Faddeev type which, however, are valid only if the interparticle interactions are of short range. In the presence of long-range Coulomb interactions they loose their compactness property. These problems have now been partly resolved. Indeed, if all Coulomb interactions are purely repulsive, suitably modified integral equations for the reaction amplitudes for all binary processes have been proven to possess compact kernels for all energies, thus rendering them amenable to standard solution methods [MZ-TH/98-05, 99-62]. This result supports the previously developed "screening and renormalization approach" (for a review see E. O. Alt and W. Sandhas, Collision theory for two- and three-particle systems interacting via short-range and Coulomb forces: in Coulomb Interactions in Nuclear and Atomic Few-Body Collisions, eds.: F. S. Levin and D. Micha, Plenum, New York 1996). This work is continuing.

Another investigation which is based on the successful derivation of the missing part of the boundary condition for the three-charged particle wave function [E.O. Alt and A.M. Mukhamedzhanov, Phys. Rev. A47 (1993) 2004; A.M. Mukhamedzhanov and M. Lieber, Phys. Rev. A54 (1997) 3078], concerns the derivation of NLO terms in the asymptotic expansion of the three-charged particle wave function in coordinate space [MZ-TH/98-03]. Work is in progress.

A review of various aspects of charged composite particle scattering is available in [MZ-TH/98-33].

12. Application of Three-Body Methods to Atomic Collisions (E. O. Alt et al.)

Many atomic processes are of three-body type, an important example being direct scattering and charge transfer of electrons off hydrogenic atoms. Standard atomic methods are unable to describe simultaneously the direct and exchange processes, partly due to their neglect of unitarity and channel coupling. Based on three-body integral equations we have developed an approach, to be applied at higher energies, which takes into account the constraints of two- and (partly) of three-body unitarity and treats all the open channels on equal footing. This leads to a very satisfactory description, not only of total cross sections for the electron exchange reaction but also of differential cross sections for electron direct scattering and exchange in reactions with hydrogen atoms [MZ-TH/98-07, 32, 99-41, 58].

13. Coulomb Effects in Proton-Deuteron Scattering (E. O. Alt et al.)

The investigation of nucleon-deuteron scattering is of crucial importance since it is the simplest reaction which is sensitive to off-shell effects of the fundamental nucleon-nucleon interaction and to the equally fundamental three-nucleon forces. Experimentally most extensively studied is this reaction with protons as projectiles. Above the deuteron breakup threshold, the only approach mathematically proven to be rigorously valid is the "screening and renormalization approach" (see project 11). Based on it we have succeeded to perform for the first time calculations of pd elastic scattering observables at positive energies with the realistic "Paris" NN potential [MZ-TH/98-04, 28, 99-23, 59, 60]; further publications are in progress.

14. Post-Acceleration Effects in the Coulomb Breakup of Light Nuclei (E. O. Alt et al.)

Knowledge of the astrophysical S-factors for certain nuclear fusion reactions at extremely low energies is important as ingredient in stellar evolution models, for the estimation of the relative abundance of elements and of neutrino fluxes, etc. Besides direct measurements which are greatly impeded by the required low energy of the fusing nuclei, experiments on the inverse process, namely the Coulomb breakup in the field of a heavy, fully stripped nucleus, are being performed. However, in order to extract the S-factor from such reactions one must take into account that the nucleus not only serves as a source of photons to induce the breakup of the projectile, but also influences the ejectiles afterwards (Post Coulomb Acceleration effect, PCA). Theoretical treatments of PCA effects require knowledge of the three-charged particle wave function in the appropriate asymptotic regions (see project 11). Investigations of the influence of PCA on the value of the extracted S-factors for the 7Be(p,gamma)8B reaction which determines the high energy neutrino flux from the sun are continuing [MZ-TH/98-06].

15. Coulomb Corrections in High-Energy Heavy Ion Collisions (E. O. Alt et al.)

High-energy heavy ion collisions provide an important tool for proving the existence, and studying the properties, of the quark-gluon plasma. In particular Hanbury-Brown Twiss (HBT) interferometry allows to draw conclusions on the evolving geometry of the interaction region. For the interpretation of the HBT results the Coulomb interaction between the detected particles must be taken into account in a detailed manner. We have derived a systematic procedure for dealing with Coulomb effects in three-body [MZ-TH/98-53] and n-body [MZ-TH/99-40] correlation functions which greatly improves on the standard (but only phenomenological) "Gamov factor approximation".

16. Mesomolecules (E. O. Alt et al.)

Calculation of the energy levels of antiprotonic (or mesonic) atoms represents a great challenge as masses, as well as length scales, differing by several orders of magnitudes are involved. In addition, in the case of, e.g., antiprotonic atoms the absorption of the antiprotons occurs from states with large values of the orbital angular momentum so that only transitions between states with large can be observed. Standard adiabatic approaches treat the rotational motion perturbatively and hence are not so well suited to treat such systems. We have developed a new hyperradial adiabatic approach which is applicable to Coulombic three-body systems irrespective of the masses of the constituents and in which systems with large are not more difficult to treat than those with low [MZ-TH/99-24, 42, 61]. Numerical applications are under way.

Strong Interactions at Low Energies

17. Determination of the Strange-Quark Mass (K. Schilcher et al.)

A precise determination of the strange-quark mass is of crucial importance for a better understanding of low energy phenomenology of the standard model, e.g. for a precise prediction of the CP-violating parameter '. A new determination of the strange-quark mass was carried out [MZ-TH/98-25, 65], based on the two-point function involving the axial-vector current divergences. This Green function is known in perturbative QCD up to order O(_s³), and up to dimension-six in the non-perturbative domain. The hadronic spectral function is parametrized in terms of the kaon pole, followed by its two radial excitations, and normalized at threshold according to conventional chiral symmetry. The result of a Laplace transform QCD sum rule analysis of this two-point function is: m_s(1 GeV²) = 155 ± 25 MeV. This result is in agreement with direct determinations from -decay, but in strong disagreement with the very small value of recent lattice QCD calculations.

18. Chiral Sum Rules and Duality in QCD (K. Schilcher et al.)

The ALEPH data on the vector and axial-vector spectral functions, extracted from tau-lepton decays is used in order to test local and global duality, as well as a set of four QCD chiral sum rules [MZ-TH/98-60, 99-20], These are the Das-Mathur-Okubo sum rule, the first and second Weinberg sum rules, and a relation for the electromagnetic pion mass difference. We find these sum rules to be poorly saturated, even when the upper limit in the dispersion integrals is as high as 3 GeV². Since perturbative QCD, plus condensates, is expected to be valid for |q²| O(1 GeV²) in the whole complex energy plane, except in the vicinity of the right-hand cut, we propose a modified set of sum rules with weight factors that vanish at the end of the integration range on the real axis. These sum rules are found to be precociously saturated by the data to a remarkable extent. As a byproduct, we extract for the low energy renormalization constant L₁₀ the value -4 L₁₀ = 2.43 × 10^-2, to be compared with the standard value -4 L₁₀ = (2.73 ± 0.12) × 10^-2. This in turn implies a pion polarizability _E = 3.7 × 10^-4 fm³.

19. Delta I=1/2 Enhancement and the Glashow-Schnitzer-Weinberg Sum Rule (K. Schilcher et al.)

In 1967 Glashow, Schnitzer and Weinberg derived a sum rule in the soft-pion and soft-kaon limit relating the Delta I=1/2 non-leptonic K 2 amplitude to integrals over strange and non-strange spectral functions. Using the recent ALEPH data from tau-decay, we show that the sum rule, slightly modified to reduce contributions near the cut, yields the correct magnitude of the decay amplitude corresponding to the Delta I=1/2 rule [MZ-TH/00-02].

20. Is There Evidence for Dimension-Two Corrections in QCD Two-Point Functions? (K. Schilcher et al.)

The ALEPH data on the (non-strange) vector and axial-vector spectral functions, extracted from tau-lepton decays, is used in order to search for evidence for a dimension-two contribution, C_{2 V,A}, to the operator product expansion (other than d=2 quark mass terms) [MZ-TH/99-08]. This is done by means of a dimension-two finite energy sum rule, which relates QCD to the experimental hadronic information. The average C₂ = (C_2V + C_2A)/2 is remarkably stable against variations in the continuum threshold, but depends rather strongly on _QCD. Given the current wide spread in the values of _QCD, as extracted from different experiments, we would conservatively conclude from our analysis that C₂ is consistent with zero.

21. Two-Loop Calculations in Chiral Perturbation Theory (P. Post, K. Schilcher et al.)

Recent progress in the calculation of massive Feynman integrals allows phenomenological studies of physical amplitudes to an accuracy in unison with the high precision of existing or planned experiments. Results of the two-loop calculation in full SU(3) × SU(3) chiral perturbation theory are obtained for a specific linear combination of weak and electromagnetic form factors (the one appearing in Sirlin's theorem) which does not receive contributions from O(p⁶) operators with unknown constants. For the charge radii, the corrections to the previous one-loop result turn out to be significant. To clearly identify the two-loop effects, more accurate measurements of the kaon electromagnetic and weak charge radii would be desirable. The question of convergence of the chiral expansion is addressed [MZ-TH/97-02]. The electromagnetic form factor of the neutral kaon is calculated in O(p⁶) chiral perturbation theory. At O(p⁶), a new operator enters whose arbitrary renormalization constant, however, results only in a parallel shift in the form factor F(t)/t. The two-loop effects are non-negligible, but comparison with experiment would require higher precision data [MZ-TH/00-31]. Finally all electromagnetic and semi-leptonic form factors of pions and kaons were calculated in the same order. The predictive power turns out to be limited, due to the large number of arbitrary renormalization constants (see the doctoral thesis by P. Post).

Phenomenology of Strong and Electroweak Interactions: Higher-Order Corrections

22. Radiative Corrections to the HQET Lagrangian and to Heavy Baryon Current-Current Correlators (A. Davydychev, S. Groote, J. Körner, A.A. Pivovarov et al.)

Heavy Quark Effective Theory (HQET) consists in an expansion of the QCD Lagrangian in terms of inverse powers of the heavy quark mass not unlike the case of the Foldy-Wouthuysen transformation in QED. In this way the heavy mass scale is explicitly factored out and one remains with effective heavy quark fields which no longer depend on the heavy quark mass scale. In the simplest tree level version the requisite Foldy-Wouthuysen type transformation has been carried out to a very high order [MZ-TH/93-13]. The mass independent coefficient functions become renormalized through quantum loop effects which may also induce new operators involving heavy quark fields not present at the tree level. We continued previous work on the renormalization of the coefficient functions pushing them to higher orders. In [MZ-TH/98-38], in collaboration with A.G. Grozin, we improved on the existing two-loop renormalization of the O(1/m_Q) chromomagnetic HQET operator by taking into account the possible existence of a second quark mass scale in the inner quark loops contributing to the two-loop renormalization. This is relevant for the renormalization of the chromomagnetic operator in the b-quark system where the c-quark contributes in the inner quark loop. We presented a general algorithm for the calculation of two-loop diagrams with two mass scales. In an ongoing project we have calculated the two-loop corrections and the two-loop QCD matching coefficients for the O(1/m_Q²) HQET spin-orbit and Darwin operators. The calculations were completed but need further attention and interpretation since the results show some non-trivial gauge dependence. We calculated the O(_s) radiative corrections to the current-current correlator of two heavy baryon currents where the currents are composed of two light quarks and a heavy quark with a finite mass value [MZ-TH/99-49]. This is a three-loop two-mass-scale problem which had never been attempted before. Its successful completion constitutes a significant technical advance in the technology of three-loop calculations. In the static limit we recovered the known leading result from HQET. Our results on the structure of HQET as well as various applications of HQET to heavy baryon transitions were reported at two international meetings [MZ-TH/98-63, MZ-TH/00-08]. In a number of papers we considered various phenomenological applications of HQET to heavy baryon decays involving current, pion and photon transitions (see following projects 23 - 26). These phenomenological applications involve the modeling of the light side dynamics to various degrees of sophistication.

23. Exclusive Semi-Leptonic b c and c s Decays (J. Franzkowski, J. Körner et al.)

In [MZ-TH/96-32] we developed a fully relativistic three-quark model which allows one to calculate the exclusive semi-leptonic decays of bottom and charm ground state baryons. Taking into account only the leading order of HQET the bottom and charm quarks were treated as static. We calculated the Isgur-Wise functions of these decays and, from these, determined the decay rates as well as various polarization asymmetry parameters and compared our results to experimental results where available. In subsequent papers we took the same relativistic three-quark model to study finite mass effects [MZ-TH/99-45] and the dependence of our results on the choice of the heavy baryon currents [MZ-TH/99-51]. Starting from the Bethe-Salpeter description of heavy baryons as bound states of light and heavy quarks we showed in [MZ-TH/98-13] how one can obtain the spectator quark model description of semi-leptonic heavy baryon decays from using certain reasonable approximations for the Bethe-Salpeter kernel. We made some exemplary ansätze for the Bethe-Salpeter wave functions and investigated some first phenomenological applications. These studies lead to a very useful lower bound for the Isgur-Wise function of _{b c} transitions [MZ-TH/99-12]. Together with an upper bound which results from the inclusive saturation limit we could exclude some theoretical models for the Isgur-Wise function. A further result was that the exclusive/inclusive ratio of semi-leptonic _b exceeds that of mesonic B decays. When attempting to extract the value of the KM matrix element V_bc from experimental data on semi-leptonic exclusive b c decays the zero recoil point plays a prominent role and thereby also the QCD radiative corrections at this point. In [MZ-TH/97-37, 98-68] we calculated the full two-loop radiative QCD corrections to b c transitions at the zero recoil point confirming results of Czarnecki and Melnikov on the same issue which appeared slightly earlier than our results.

24. Exclusive Non-Leptonic Decays of Heavy Baryons (J. Körner et al.)

The calculation of exclusive non-leptonic decays of heavy baryons is more difficult than the corresponding calculation for heavy meson decays because there are non-factorizing contributions in the former case which had not been evaluated up till now. The contributions of the non-factorizing graphs were reliably calculated for the first time in the framework of the relativistic three-quark model which had already been used before to calculate the semi-leptonic decays of heavy baryons [MZ-TH/97-15, 21]. The calculation shows that the non-factorizing contributions cannot be neglected as has been erroneously assumed in some existing calculations. Including the non-factorizing contributions we have calculated the rates and the polarization asymmetry parameters for various non-leptonic decays of charm and bottom baryons and compared them to existing data and to the results of other model calculations.

25. Inclusive Semi-Leptonic b c and b u Transitions (M. Fischer, S. Groote, J. Körner, M.C. Mauser et al.)

The DELPHI and OPAL collaborations at LEP have recently provided first results on the polarization of _b baryons produced on the Z-Peak. The polarization was determined from the shape of the lepton and neutrino spectrum in its semi-leptonic decay. We suggested some further observables which can be used to improve on the measurement of the polarization of the _b [MZ-TH/95-21, 32]. These are spin-momentum correlations involving the spin of the _b and the momenta of its decay products. Besides the leading contribution of free quark decay we determined the non-perturbative O(1/m²) contributions to the spin-momentum correlations which arise at next-to-leading order in the heavy quark expansion. The calculation included finite lepton mass effects relevant for decays involving the lepton. Using these results and results on the O(_s) corrections to semi-leptonic b c and b u transitions [MZ-TH/99-10] we have performed a detailed spin analysis of spin observables in the semi-inclusive decays B D_s, D_s^* + X and _b D_s, D_s^* + X at O(_s) and O(1/m²) [MZ-TH/99-27, 67].

26. One-Pion and One-Photon Transitions Between Heavy Baryons (J. Körner et al.)

To leading order in the heavy quark expansion the one-pion and one-photon transitions between heavy baryons are described through the corresponding transitions between the light diquark systems of the heavy baryons. In the simplest approach the light-side diquark transitions can be modeled using the constituent quark model which is known to work very well phenomenologically in the light quark sector. We worked out the constituent quark model relations between the various reduced HQET form factors for one-pion and one-photon transitions between the excited P-wave and the ground states of heavy baryons [MZ-TH/96-10, 38]. In a further step we determined the relevant overlap integrals using either the previously mentioned relativistic three quark model and or a light-front quark model [MZ-TH/98-14, 46, 99-06, 98-08, 67].

27. QCD Sum Rule Analysis of (M_Z) and Decays (S. Groote, J. Körner, F. Krajewski, K. Schilcher, A.A. Pivovarov et al.)

The exact value of the electromagnetic coupling constant at the mass scale of the Z boson is an important ingredient to constrain the mass of the Higgs boson through its virtual loop effects. At present the accuracy of determining (M_Z) is limited by poor e⁺e^- annihilation data at lower energies. Exploiting nonlocal duality and known higher-order QCD results we showed how the influence of data on the determination of (M_Z) could be minimized leading to a reduction of the error on (M_Z) [MZ-TH/98-02, 99-18]. In [MZ-TH/98-59] we advocated the use of Euclidean space methods for the evaluation of (M_Z). In a series of papers on decays we studied the asymptotic structure of the QCD perturbation series for some observables in decays [MZ-TH/99-37], paying special attention to strange-quark mass effects [MZ-TH/99-46], and improving this analysis by a partial resummation of the perturbation series in an effective scheme [MZ-TH/99-66]. The results of these papers show that very likely the accuracy of the present perturbative description of observables in decays is limited by the fact that the perturbative series is already now starting to show its asymptotic nature.

28. Massive Two-Loop Integrals (L. Brücher, A.I. Davydychev, H.S. Do, J. Franzkowski, A. Frink, S. Groote, J. Körner, R. Kreckel, D. Kreimer, K. Schilcher, S. Wiermann, et al.)

Massive multi-loop Feynman diagrams are becoming increasingly important in the precision analysis of experimental data in the framework of the Standard Model. In a long-term project our group is involved in the automatic evaluation of massive multi-loop Feynman diagrams (called XLOOPS). A general review of the XLOOPS program and the integration techniques that go into XLOOPS was given previously [MZ-TH/97-32] (see also [MZ-TH/98-18]). The capacity and reliability of the released version 1.0 of XLOOPS [MZ-TH/98-10] was tested in an extensive application run where one-loop contributions to the three-gauge boson coupling were studied [MZ-TH/99-69]. Recent work concentrated on the special case of self-energy and vertex diagrams with two factorizing loops [MZ-TH/98-17] and on first steps towards the inclusion of two-loop box diagrams [MZ-TH/98-16]. XLOOPS aims to do as many of the necessary multi-loop integrations in closed form as possible and, if necessary, to provide efficient numerical codes to do the remaining integrations numerically. The development of the adaptive Monte-Carlo integration code pvegas was continued to allow for further optimization [MZ-TH/98-54]. In a study of the problem of infrared and collinear divergences, results and calculational techniques of two groups have been compared. Two-loop vertex Feynman diagrams with infrared and collinear divergences are studied, on the one hand, by a small-momentum expansion augmented by Padé techniques and, on the other hand, by a numerical method using light-cone coordinates in the parallel space. The results of the two methods agree [MZ-TH/97-20]. The threshold expansion of the sunset diagram with arbitrary masses has been investigated (in collaboration with V.A. Smirnov) in [MZ-TH/98-48].

29. Vertex Functions in Quantum Chromodynamics (A.I. Davydychev et al.)

In collaboration with P. Osland, L. Saks and O.V. Tarasov, the investigation of one- and two-loop corrections to vertex functions in QCD has been continued. In Ref. [MZ-TH/98-22], two-loop contributions to the three-gluon vertex have been calculated in the case of two on-shell gluons, in an arbitrary covariant gauge. A status report on available results for the two- and three-point functions in QCD, including Ward-Slavnov-Taylor identities and the renormalization procedure, was given in [MZ-TH/98-49]. Analysis of the one-loop contributions to the quark-gluon vertex, together with the corresponding Ward-Slavnov-Taylor identities, has been performed in [MZ-TH/99-63], in an arbitrary (covariant) gauge and dimension.

30. Geometrical Approach to Feynman Diagrams and the -Expansion (A.I. Davydychev et al.)

In collaboration with R. Delbourgo, a connection between one-loop Feynman digrams with several external lines and certain quantities in non-Euclidean geometry has been investigated [MZ-TH/98-26]. Using these ideas, explicit results for arbitrary orders of the -expansion of dimensionally-regulated one- and two-loop diagrams have been obtained [MZ-TH/99-30, 44].

31. Multi-Loop Integrals for Heavy Particles and Heavy Quarks (S. Groote, J. Körner, A.A. Pivovarov et al.)

XLOOPS and the integration techniques developed for XLOOPS were also used to calculate two-loop zero recoil corrections to the b c transitions. The experience gained in these applications will prove very important for the general treatment of mass and infrared singularities to be implemented in XLOOPS at a later stage. While preparing to calculate the O(_s) corrections to the massive baryonic two-point correlator (which involve genuine massive three-loop diagrams) [MZ-TH/99-49] we discovered some very interesting general results about the class of the so-called water melon diagrams. The water melon diagrams are n-loop diagrams with two outer legs in which the inner lines all start at the left vertex and end up in the right vertex. In the two-loop case this topology is referred to as the "sunset" topology whose properties have been extensively studied in the literature. Using coordinate space methods we were able to find a one-dimensional integral representation for the class of n-loop water melon diagrams which allows for a very efficient numerical evaluation of this class of diagrams [MZ-TH/98-09, 12, 99-07]. In [MZ-TH/99-13] we investigated massive three-loop diagrams with zero external momenta ("bubble diagrams") and found some very interesting connections of these diagrams with transcendental numbers.

32. Radiative Corrections to Spin Observables for Light and Heavy Quarks (M. Fischer, S. Groote, J. Körner, J.A. Leyva, M.C. Mauser et al.)

In this complex of papers we calculated one-loop corrections to polarization observables in fundamental quark interactions involving heavy (Q) and light (q) quarks. For the process e⁺e^- QQ(g) we determined the polarization of the gluon [MZ-TH/96-34, 97-17], calculated spin-spin correlations for the two heavy quarks in the final state [MZ-TH/97-27] and determined the effect of gluon bremsstrahlung on the anti-collinearity of the heavy quark pair [MZ-TH/97-39]. For the process t b + W⁺ we calculated the O(_s) radiative corrections to the eight spin observables that can be measured in this process [MZ-TH/98-47, 99-10]. The results of these papers will become very useful for the analysis of the wealth of data on top decays expected to become available from the future hadron colliders TEVATRON (Run II) and LHC. In [MZ-TH/99-11] we determined the imaginary parts of the one-loop contributions to the four-point functions which are needed for an assessment of the magnitude of T-odd variables (also involving spin) in deep inelastic scattering and in the Drell-Yan process.

33. Phenomenology of QCD, Many-Loop Diagrams and High-Precision Determination of Standard Model Parameters (A.A. Pivovarov et al.)

The vector-current correlator of heavy quarks is computed analytically near the production threshold in next-to-next-to-leading order in perturbative and relativistic expansion in the coupling constant and velocity of the heavy quark within the non-relativistic Coulomb approximation. This allowed us to precisely determine the b-quark mass and the strong coupling constant using sum rules for the Upsilon system. The next-to-next-to-leading-order corrections are found to be of the same order as the next-to-leading-order ones and numerically important [MZ-TH/98-11].

In [MZ-TH/98-61, 99-65] we presented reviews of analytical results for e⁺e^- and cross sections near the top-production threshold up to the next-to-next-to-leading order of NRQCD.

Radiative corrections in the total hadronic decay rate of the -lepton and some moments of its differential distributions are studied employing perturbative QCD and the operator product expansion. It was found that quadratic quark-mass corrections in the strange-quark mass to the decay rate ratio R_tau to the order O(_s³ m²) contribute appreciably to the Cabibbo-suppressed decay modes of the -lepton. The numerical value for the strange-quark mass is found to be m_s(1 GeV) = 200 ± 40_exp ± 30_th MeV [MZ-TH/98-66].

Two-loop corrections to the correlation function for tensor currents in gluodynamics were calculated. In a drastic contrast to corrections for scalar and pseudo-scalar gluonic correlators where corrections are huge, the tensor correlator correction has a standard magnitude comparable to the usual QCD corrections in the quark world [MZ-TH/99-21].

A new (resummed) solution to the Schrödinger equation for the non-relativistic Green function which is used for describing the heavy quark-antiquark pair production near the threshold in e⁺e^- annihilation is found [MZ-TH/99-52].

34. Radiative Effects in Deep Inelastic Electron Proton Scattering (H. Spiesberger et al.)

The increasing precision of deep inelastic scattering experiments at HERA asks for Standard Model predictions with a correspondingly high accuracy. Recently the description one-loop electroweak radiative corrections for charged current interactions was improved [MZ-TH/99-29] (see also [MZ-TH/99-02]). In addition, with the higher luminosity at HERA, one will be able to study processes where a radiated photon is identified in the final state [MZ-TH/99-32]. One example is the production of an isolated photon accompanied by a number of jets. With QCD corrections at order O(_s) available [MZ-TH/98-57, 99-05, 25] this process can now be used for meaningful tests of QCD.

35. Deep Inelastic Electron Proton and Electron Photon Scattering at High Energies (H. Spiesberger et al.)

The HERA experiments at very high energies allow to access the new field of QCD dynamics at very low Bjorken x. A study of hitherto ignored higher-twist effects due to gluonic operators, expected to become important in this new kinematic range, was started in [MZ-TH/99-36].

Deep inelastic e scattering as measured at a high-energy linear e⁺e^- collider (TESLA) would allow to test predictions of QCD in an unambiguous way since at the very high momentum transfers reachable at TESLA, model-dependent hadronic components of the photon structure functions would play no role [MZ-TH/99-33].

Phenomenology of New Physics

36. Physics Beyond the Standard Model in Electron Proton Scattering and e⁺e^- Annihilation (H. Spiesberger et al.)

In 1997, experiments at the electron-proton collider HERA have found an excess in the number of deep inelastic scattering events at highest momentum transfers. In addition, more events with high-energetic muons in the final state than expected were observed. This raised speculations about physics beyond the Standard Model. In a series of papers we explored possible explanations within and beyond the Standard Model for these experimental observations [MZ-TH/98-56, 62, 64, 99-01, 39]. One of the most exciting possible explanations involves leptoquarks, new states which carry the quantum numbers of both leptons and quarks, or supersymmetric partners of quarks with R-parity violating couplings. If leptoquarks or squarks are responsible for the observations made at HERA, one should be able to find deviations from Standard Model predictions also in e⁺e^- annihilation [MZ-TH/99-34].

One-Loop Corrections in Lattice Gauge Theory

37. Symanzik Improvement to One-Loop Order on Anisotropic Lattices (S. Groote et al.)

In [MZ-TH/99-64] we worked on the Symanzik improvement for an anisotropic lattice within lattice gauge theory. Coefficients of the Wilson action and the so-called D234 action were improved by calculating the one-loop radiative corrections to the rest mass, the wave function renormalization and a relative factor between the spatial and temporal components, known as speed of light renormalization factor. This calculation was accomplished by considering the quark self energy diagram, taking into account both the massive and the massless quark case. The paper contains tables which are of help for lattice calculations, as recent citations show.

Electrical Impedance Tomography (EIT)

38. Electrical Impedance Tomography (EIT) (C. Lehmann, C. Mrakawa, K. Schilcher et al.)

EIT is an imaging technique of medical diagnostics, in which one tries to obtain information on the conductivity distribution in the interior of a body by measuring currents and voltages at the surface of the body. If successfully implemented, EIT would provide a non-invasive complementary alternative to more conventional imaging techniques. We have built an electrical impedance tomograph which makes it possible to reconstruct impedance distributions in the interior of a cylinder by means of measurements of currents and potentials on its boundary. The reconstruction algorithm is based on Newton's algorithm for the solution of non-linear integral equations, where the electric fields are calculated by the methods of finite elements and finite integrals. The results can be represented graphically on a PC. The resolution is fine enough to envisage applications in medical diagnostics [MZ-TH/00-32]. Most previous EIT algorithms use two-dimensional models for the conductivity. Since the electric current does not flow in straight lines, this approximation seems to be unwarranted. We discuss a new approach to three-dimensional electrical impedance imaging based on a reduction of the information to be demanded from a reconstruction algorithm. Images are obtained from a single measurement by suitably simplifying the geometry of the measuring chamber and by restricting the nature of the object to be imaged and the information required from the image. In particular we seek to establish the existence or non-existence of a single object (or a small number of objects) in a homogeneous background and the location of the former in the (x,y)-plane defined by the measuring electrodes . Given in addition the conductivity of the object rough estimates of its position along the z-axis may be obtained. The approach may have practical applications [MZ-TH/00-54].

Scientific Papers, Publications and Conference Contributions

Theses

Doctoral Theses

98-T1

"Elektrische Impedanztomographie mit realistischen Daten", [ps files],
I. Savelsberg (November 1998),
supervisor: K. Schilcher

Diploma Theses

99-D1

"Modelle für fermionische Massenmatrizen und Zustandsmischungen"
("Models for fermionic mass matrices and state mixtures"), [ps files],
Christian Pöselt (February 1999),
supervisor: F. Scheck

99-D2

"Spezielle Aspekte der nichtkommutativen Geometrie"
("Aspects of non-commutative geometry"), [ps files],
Alexander Holfter (March 1999),
supervisor: N. Papadopoulos

99-D3

"Hopf-Algebren und Quantengruppen in Anwendung auf universell-einhüllende Lie-Algebren und diskrete Eichtheorien"
("Hopf algebras and quantum groups and their application to universal covering Lie algebras and discrete gauge theories"), [ps files],
Markus Debert (Oktober 1999),
supervisor: N. Papadopoulos