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Institute for Theoretical Physics

Elementary Particle Physics Division

## Current fields of research

### Discrete symmetries

The discrete symmetries parity (P), charge conjugation (C) and time reversal (T) are all broken individually in nature, however, their product is conserved (CPT theorem). In atomic physics P-violation effects are described by the exchange of Z-bosons between the electrons of the shell and the quarks in the nucleus. We study energy shifts of geometrical (Berry-) phases as new manifestations of this interaction.O. Nachtmann,

Non conservation of the combined symmetry CP of charge conjugation and parity has been observed experimentally in the neutral K-meson system and recently also in the neutral B-meson System. We study CP-odd effects in high energy reactions as e.g. in Z-decays and electron-positron collisions at a center of mass energy of about 100 to 800 GeV. In this way information about possible extensions of the standard model of elementary particle physics can be gained (multi-Higgs-models, supersymmetric models, etc.).

### Theory of high-energy scattering

The theoretical description of hadron scattering at high energies and small momentum transfers in the framework of Quantum Chromodynamics (QCD) is the theme of this research. Neither perturbation theory nor the numerical methods of lattice gauge theory are suited for treating this problem.H.G. Dosch, O.Nachtmann, T.Paulus, A. Shoshi, F. Steffen, V .Schatz

Also involved is the Nuclear Physics Group (H.J.Pirner).

Starting from the functional integral of QCD, we have developed a way to represent the relevant scattering amplitudes as expectation values of light-like Wegner-Wilson loops. The calculation of these expectation values can be performed using non perturbative methods, especially in the framework of the model of the stochastic vacuum.

A further area of research in high-energy scattering, which is investigated in our group, is the regime of small values of the Bjorken variable x which is measured in deep inelastic scattering at HERA.

### World-line/string inspired methods in quantum field theory (QFT)

Already since the time of foundation of QFT besides the "second quantization" approach - now usually adopted in textbooks - there exists another one based on the propagation of relativistic particles in a background. We are extending the latter method using the basic concept of quantum mechanical Green's functions on Feynman graphs, also on multi-loop graphs. The formalism is much related to multi-loop calculations in modern (super) string theory in the limit of infinite string tension.M.G. Schmidt, C. Zahlten

### Masses and Weak Decays of Quarks

The fundamental parameters of the Standard Model concerning quarks can be only extracted indirectly from hadronic properties. Several methods such as chiral symmetry, QCD sum-rules, Heavy-Quark Effective Theory, and quark models are applied in order to determine the elements of the Cabbibo-Kobayashi-Maskawa matrix, to test the Standard Model, and to obtain a detailed understanding of numerous experimental results.H.G. Dosch, M. Eidemüller, M. Jamin, B. Stech

## Consequences of the non-trivial vacuum structure |

In a non-linear theory like QCD the state of lowest energy (the vacuum) most likely has a non-trival structure. This may be the key for a phenomenological understanding of several interesting and surprising effects in QCD, both in low energy physics (spectroscopy) and in high energy scattering. For many applications the vacuum structure is modeled in a Gaussian stochastic model. It provides an analytical tool for calculating processes not (yet) treatable by other means.H.G. Dosch, M. Eidemüller, D. Gromes, M. Jamin, O. Nachtmann, A. Shoshi, F. Steffen, W.Wetzel

## Exact renormalization group equations |

The link between the microscopic laws of physics and macroscopic observations is described by an exact renormalization group equation. Looking at a physical system at different length scales the effective interactions will depend on the scale. One often encounters situations where the interactions are simple at short distance scales and quite complex at long distances. This is due to coherent fluctuations of the microscopic degrees of freedom on all scales. We aim for a non-perturbative equation which can account for the flow from simplicity to complexity.J. Berges, C. Wetterich

More precisely, we consider the effective average action or coarse grained free energy which includes the effects of all quantum and thermal fluctuations with momenta larger than some infrared cutoff scale k. For large k no fluctuations are included and the effective average action corresponds to the microscopic or classical action. For k->0 all fluctuations are included and the effective average action equals the effective action. The latter generates the one particle irreducible correlations functions. In statistical physics the effective action is the free energy. Following the flow of the effective average action as k decreases from high values to zero interpolates between the microscopic action and the quantum or thermal effective action. It may be compared with looking at the theory with a microscope of variable resolution. In our formalism the relevant degrees of freedom can be different for large and small k. This is crucial, for example, for quantum chromo-dynamics where perturbative quark-gluon-physics at short distances have to be connected with baryon-meson-physics at long distances.

The flow of the effective average action obeys an exact functional differential equation. Its solution for k->0 would amount to a complete solution for the correlation functions in statistical physics or scattering amplitudes in particle physics. Exact solutions for the functional differential equation seem therefore out of reach. Approximate non-perturbative flow equations are obtained in our formalism by suitable truncations of the effective average action. Typically these are partial differential equations which are simple enough to be solved numerically. They have wide applications in particle physics and statistical physics.

## Flow equations for mesons and chiral symmetry breaking |

J. Berges, C. WetterichFor an analytic understanding of the quark-hadron phase transition it is crucial to investigate effective theories for the relevant degrees of freedom. Due to the interplay between confinement and asymptotic freedom this is particularly difficult in QCD: The relevant degrees of freedom at short distances are quarks and gluons, whereas the long-distance physics is dominated by the interaction of baryons and mesons. This poses the problem which are the relevant degrees of freedom at and above the high temperature phase transition or the high density phase transition. For the chiral aspects of the QCD phase transition the pions and the s-meson are particularly important. They are directly connected to the spontaneous symmetry breaking of the chiral symmetry. This generalizes to Kaons, etc. once the strange quark is included. All these scalars can be described by an effective linear meson field coupled to quarks or baryons.

Non-perturbative flow equations are used for an investigation of chiral symmetry breaking in QCD. We also attempt to compute the masses and decays of the light mesons. In one approach we work within the linear chiral meson model which is assumed to be valid for momenta below 700 MeV, after integrating out the gluons. For two light quark flavours we see how chiral symmetry breaking is induced by the quark fluctuations. Realistic values for the chiral condensate and the pion decay constant can be obtained.

We have investigated the predictions of this model for a state in thermal equilibrium. This is relevant for the QCD-phase transition at high temperature, both in the context of early cosmology and current heavy ion experiments. For high temperature we find a chiral second order phase transition. We have computed the behavior of the chiral condensate as well as the pion and sigma masses near the critical temperature. It is governed by the critical equation of state in the O(4) universality class. Besides a computation of the critical equation of state and the associated critical exponents and amplitudes we have also established a direct link between the critical behavior and the physics at zero temperature.

For three light quark flavours, we have done a detailed phenomenological study of the linear meson model. In particular, we have computed the next to leading effective couplings in chiral perturbation theory. We find that an expansion in the strange quark mass works well for most quantities. Certain phenomena like the mixing between the eta- and the eta'-mesons are only poorly described, however, by this expansion. They can be described satisfactorily in the linear chiral meson model.

Recent work has concentrated on the properties of effective models at high baryon density. Within the linear quark meson model a first order high density phase transition has been found. In contrast to the high temperature transition, however, confinement effects play an important role at high density. In matters, if the degrees of freedom are quark or baryons and how the transition between effective degrees of freedom is described. A first attempt to address the chiral properties of QCD in an effective model for quarks/baryons and mesons reveals a weak first order phased transition between a gas of nucleons and nuclear matter (gas-liquid transition) as well as a transition to quark matter. For the latter, the possible effects of color superconductivity have to be included in the future.

## Spontaneous color symmetry breaking in QCD |

We propose that confinement can also be described in a dual picture as a Higgs phenomenon. Due to a quark-antiquark condensate in the color octet channel the gluons acquire a mass and integer electric charges. By gluon-meson duality they are associated with the octet of vector mesons. Similarly, the nine light quark degrees of freedom correspond to the light octet of baryons and a singlet. This is quark-baryon duality.J. Jäckel, C. Wetterich

We attempt to substantiate this picture by following the flow of the effective interactions from the domain of validity of QCD perturbation theory down to momentum scales where the quark condensates form. To do this we use bosonization to introduce bosonic fields which should be the main degrees of freedom at low momenta. Spontaneous color symmetry is then associated with a non trivial expectation value for a bosonic field which transforms as a color octet.

In this context multiquark interactions generated during the flow play an important role. Especially the instanton interactions are crucial to produce a non-trivial minimum in the effective potential for the bosonic octet field. To translate these multiquark interactions to the bosonic language we continuously redefine the bosonic fields to absorb the fermionic interactions. This leads to modified flow equations for the bosonized effective average action.

### Lattice field theory

Important properties of quantum-chromodynamics (QCD), such as quark confinement, mass spectrum, phase transitions etc, are of a non-perturbative nature. The formulation of QCD on a space-time lattice allows one to study these properties in numerical simulations. The Heidelberg lattice group is currently involved in several projects:I. Bender, D. Gromes, J. Holk, H.-J. Rothe, I. Stamatescu, W. Wetzel

- Topological properties of the QCD vacuum at T=0 and T>0,
- Structure of hadrons at non-vanishing temperature,
- Casimir effect.

Project 2 concerns itself with the structural changes of mesons in the transition from the confinement to the quark-gluon plasma phase.

In project 3 lattice methods are applied to the study of the Casimir effect, where one studies the experimental consequences arising from vacuum fluctuations of quantum fields.

### Lower-dimensional field theory and BRST symmetries

K.-H. Rothe

### Quantum field theory for thermal equilibrium and non-equilibrium

This field of research combines concepts of quantum field theory and statistical mechanics. It describes matter under extreme conditions as in the very early universe or in heavy ion collisions.J. Berges T. Prokopec G. Aarts O. Nachtmann, M.G. Schmidt, C.Wetterich

Our main interest is in problems for which interactions are not weak, and standard perturbation theory cannot be applied. Part of our activity concerns exact renormalization group equations for equilibrium states or exact equations for the evolution of correlation functions for situations out of equilibrium. We also use and develop improved perturbative expansions and effective field theory descriptions in order to reduce the problem at hand to a more tractable one.

The non-equilibrium problems studied at our institute include both systems close to and far from thermal equilibrium. In particular, we address the approach to thermal equilibrium and electroweak baryon number violation at very high temperature.

A further interesting area of research in high-energy scattering, which is investigated in our group, are the diffractive processes measured in deep inelastic scattering at HERA.

### Physics of the early universe

L. Adueva, M. Doran, M. Lilley, E. Thommes, M.G. Schmidt, J. Schwindt, C.Wetterich

Recent observations of distant supernovae indicate that the expansion rate of the Universe is increasing. This suggests that the energy density of the Universe is dominated by a component with negative pressure. Assuming that this component is a cosmological constant, or vacuum energy, requires massive fine-tuning - the value of the energy density of the cosmological constant would need to be 120 orders of magnitude lower than we would naively expect on theoretical grounds.

One proposal for a solution to the `Cosmological Constant Problem' is quintessence - an extremely light particle which interacts with other matter only via gravity and whose self-interactions cause it to condense and behave like a slowly-decaying vacuum energy. The fine-tuning problem is `solved' because the particular value of the vacuum energy today is merely a coincidence related to the age of the Universe.

There has been an enormous amount of activity in this field in the last 3 years with many formulations of quintessence models, all with the same general properties but with differing specific predictions. We are investigating ways in which the specific models and quintessence in general can be either detected, constrained or ruled out via observations of the Universe such as CMB and large-scale structure.

Some publications### Neutrino Masses and Oscillations

PublicationsB. Stech, C.Wetterich

### Non-perturbative flow equations in statistical physics

T. Baier, E. Bick, C.Wetterich

**Exact renormalization group equations for electron interactions in condensed matter**

For many systems in condensed matter physics appropriate and seemingly simple physical models are known, which have been successfully resisted any successful attempt to a full understanding for decades. One example is the Hubbard model (already introduced in the sixties), which --- in its two dimensional version --- claims to be able to qualitatively describe the features of high temperature superconductors. Since the electron interaction in this model is strong, perturbative approaches fail and other methods are needed.

One nonperturbative method to deal with this problem are exact renormalization group equations. To simplify the process of motivating truncation schemes for these equations, we reformulated the Hubbard model in a way which replaces the four fermion interaction by a Yukawa coupling between the fermions and newly introduced bosons, which describe the relevant degrees of freedom of the model (e.g. particle density, antiferromagnetism or d-wave-superconductivity). To achieve this, we subdivided the lattice into plaquettes, each containing four lattice sites. To label the sites in one plaquette, we introduce a new index, which we call color (therefore the name colored Hubbard model).

In a mean field like calculation (neglecting the bosonic fluctuations) we were able to show that at least qualitatively our model yields a phase diagram similar to that of actual high temperatur superconductors. The drawback of the mean field approximation is the loss of control of the relation between the different Yukawacouplings in our model with the original four fermion coupling. Our results therefore strongly depends on the choice of these arbitrary couplings, which is unphysical. Besides, taking into account only the fermionic fluctuations tends to overemphasize the critical temperature.

The inclusion of bosonic fluctuations is achieved by using exact renormalization group equations. We use the formalism of the average effective action, which has the form of a one loop equation and is particularly well suited for motivating truncation schemes. Since in this formalism the starting point of the flow is the classical action, it often suffices to merely add wave function renormalization constants, flow dependent couplings and a potential for the bosons to get a useful ansatz for the general effective action. Currently, our group is working on solving the flow equations for the effective potential and the Yukawa couplings.