Dirac Eigenmodes and Center Vortices in SU(2) Lattice-QCD

The vortex model was first proposed as an explanation of quark confinement in Quantum Chromodynamics (QCD). Vortices are closed color magnetic flux lines, leading to narrowing of the color electric flux lines, a small gluonic flux tube - the gluon string, and therefore a linear potential between quarks and antiquarks. During the last years numerical results showed that the vortex model is able to explain the strength of the gluonic flux tube, the string tension. Furthermore, the relevance of vortices for topological charge and chiral symmetry breaking was emphasised. In calculations of the topological charge on center vortex configurations we found a discrepancy with the so called index theorem. The index of a Dirac operator, i.e. the difference between positive and negative zero modes, is according to this theorem equal to the topological charge of a gauge configuration. However, some vortex configurations (spherical vortices) do not validate this theorem, the reason for that is still obscure. Furthermore we found special configurations with half-integer topological charge, their relevance and physical meaning is of great interest. To clarify these issues we plan to investigate different fermion representations, with regard to half-integer topological charge especially adjoint representations. Next to confinement, chiral symmetry breaking is the most important low-energy phenomenon in QCD, and a full understanding of this effect is of vital importance. For massless quarks the Lagrangian of QCD does not lead to an interaction of right- and left-handed quarks, of quarks with spins directed in and against the direction of momentum. This is the chiral symmetry of the QCD Lagrangian. The numerical investigations show that confinement is connected with a dynamical coupling of right and left-handed quarks, a dynamical breaking of chiral symmetry. This suggests that the topological excitations which are responsible for confinement are also the origin of the dynamical symmetry breaking. We plan to investigate if there are special structures in the gauge field which deflect quarks and lead to a dynamical coupling of right and left-handed quarks. Therefore, we want to analyse the color structure of center vortices.

Lattice QCD (LQCD) is the main tool for probing quantum chromodynamics (QCD), the theory of quarks and gluons, in the non-perturbative regime. It is a gauge theory formulated on a grid or lattice of points in space and time, where gauge degrees of freedom are put on the edges of the lattice, so-called links. Fermions, i.e. the quarks, sit on the sites of the lattice and their interactions are described by the Dirac equation. We use two different lattice formulations of the Dirac operator, i.e. overlap and asqtad staggered fermions, and calculate their eigenmodes and eigenvalues with respect to the underlying gauge field. With these methods we analyse the QCD vacuum and want to understand the important degrees of freedom and physical mechanisms leading to its fundamental properties, i.e.: Quark Confinement: There are no isolated quarks observed in nature, the binding energy between quarks and anti-quarks seems to be so high that instead of breaking up a quark- anti-quark pair, a new quark-anti-quark pair pops out of the QCD vacuum. Topological Charge: Gauge fields are characterized by topological quasi-particle excitations, their number is a topological invariant and can be measured with (lattice) gauge field definitions or by the number of Dirac zero modes via the Atiyah-Singer index theorem. Chiral Symmetry Breaking: For massless quarks the Lagrangian of QCD, the function describing the dynamics of QCD, does not lead to an interaction between right- and left- handed quarks, of quarks with spins directed in and against the direction of momentum. Numerical investigations show that confinement is connected with a dynamical coupling of right and left-handed quarks, a dynamical breaking of chiral symmetry. This suggests that the topological excitations which are responsible for confinement are also the origin of the dynamical symmetry breaking. In lattice QCD it was shown that magnetic fluxes (vortices) condense in the vacuum and compress the electric flux between quark and anti-quark to a string leading to confinement. Due to the center symmetry of the action the flux of such center vortices is quantised. Wilson loops around center vortices lead to center elements corresponding to an Aharonov-Bohm like effect in QCD. These properties are the key ingredients in the vortex model of confinement, which is theoretically appealing and was confirmed by a multitude of numerical calculations, both, in lattice Yang-Mills theory and within a corresponding infrared effective model. Lattice simulations further indicate that vortices may be responsible for topological charge and chiral symmetry breaking as well. Our investigations clearly demonstrate that center vortices do apply for these effects, they attract Dirac zero modes not only through intersection and writhing points, but also through their colour structure, and the interplay of these topological charge contributions from center vortices leads to near-zero modes, which by the Banks-Casher relation are responsible for a finite chiral condensate, the order parameter of chiral symmetry breaking. Thus, center vortices unify all non-perturbative phenomena in a common framework.