# Mechanism of chiral symmetry breaking

## Summary

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.

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. It seems that there are special structures in the gauge field which deflect quarks and lead to a dynamical coupling of right and left-handed quarks. Our investigations clearly demonstrate that center vortices could apply for this effect, low-lying Dirac eigenmodes required for chiral symmetry breaking are highly concentrated at vortex structures.

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 fulfil this theorem, i.e. the gluonic definition of the topological charge is different from the index. The reason for that lies in the special (singular) construction of these vortex configurations, which is not recognized by the gluonic definition of topological charge. The Dirac operator however measures the global winding of the gauge field and gives the correct topological charge.

### Contributors

- Manfried Faber
- Roman Höllwieser

### Collaborators

- Jeff Greensite, Physics and Astronomy Dept., San Francisco State University
- Urs M. Heller, American Physical Society
- Ŝtefan Olejník, Institute of Physics, Slovac Academy of Sciences

### Publications

1) ** Spherical vortices, fractional topological charge and lattice index theorem in SU(2) LGT.** Roman Höllwieser, Manfried Faber, Urs M. Heller. 2010. 7pp.

Prepared for 28th International Symposium on Lattice Field Theory (Lattice 2010), Villasimius, Sardinia, Italy, 14-19 Jun 2010.

Published in **PoS LATTICE2010:276,2010**.

Proceedings of Science Server

2) ** Lattice Index Theorem and Fractional Topological Charge.**

Roman Höllwieser, Manfried Faber, Urs M. Heller. May 2010. 9pp.

e-Print: **arXiv:1005.1015** [hep-lat], Abstract, PDF

3) ** Correlations between center vortices and low-lying Dirac eigenmodes.**

Roman Höllwieser, Manfried Faber, Jeff Greensite, Urs M. Heller, Ŝtefan Olejník. 2008. 5pp. Prepared for 8th Conference on Quark Confinement and the Hadron Spectrum: Confinement8, Mainz, Germany, 1-6 Sep 2008.

Published in **PoS CONFINEMENT8:036,2008**.

Proceedings of Science Server

4) **Center vortex influence on the Dirac spectrum.**

Urs M. Heller, Roman Höllwieser, Manfried Faber, Jeff Greensite, Ŝtefan Olejník. Nov 2008. 7pp. Presented at 26th International Symposium on Lattice Field Theory (Lattice 2008), Williamsburg, Virginia, 14-20 Jul 2008.

Published in **PoS LATTICE2008:258,2008**.

e-Print: arXiv:0811.4408 [hep-lat], Abstract, PDF

5) ** Center Vortices and the Dirac Spectrum.**

Roman Höllwieser, Manfried Faber, Jeff Greensite, Urs M. Heller, Ŝtefan Olejník. May 2008. (Published May 2008). 13pp. Published in **Phys.Rev.D78:054508,2008**.

e-Print: **arXiv:0805.1846** [hep-lat], Abstract, PDF

6) ** Surprises with the lattice index theorem.**

Gerald Jordan, Roman Höllwieser, Manfried Faber, Urs M. Heller. 2006. 7pp.

Contributed to 25th International Symposium on Lattice Field Theory, Regensburg, Germany, 30 Jul - 4 Aug 2007. Published in **PoS LAT2007:076,2007**.

Proceedings of Science Server

7) ** Tests of the lattice index theorem.**

Gerald Jordan, Roman Höllwieser, Manfried Faber, Urs M. Heller. Oct 2007. (Published Oct 2007). 9pp. Published in **Phys.Rev.D77:014515,2008**.

e-Print: **arXiv:0710.5445** [hep-lat], Abstract, PDF