Quantum Field Theory
- World-line/string inspired methods in quantum field theory
- Exact renormalization group equations
- Non-perturbative flow equations in statistical physics
- Flow equations for mesons and chiral symmetry breaking
- Spontaneous color symmetry breaking in QCD
- Lattice field theory
As a new project, we investigate the origin and effects of quantum forces in three systems of different length scales:
- in nanomechanical systems (10 nano meter - 1 micro meter), we determine quantitatively the geometry-dependent Casimir forces between component parts;
- in laser physics, we compute the interactions between strong laser fields and quantum fluctuations inducing vacuum polarization (100 femto meter);
- in systems governed by the strong interaction (<1 femto meter), we evaluate the induced forces between topological degrees of freedom (instantons, vortices).
J. Berges, C. WetterichThe 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.
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.
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.
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.
J. Jäckel, C. WetterichWe 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.
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.
- 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.