author subject

## Chapter 1 Nonperturbative quark-gluon dynamics

Christian S. Fischer^{1}^{1}1E-mail address:
christian.fischer@physik.tu-darmstadt.de ,
Reinhard Alkofer,
Felipe J. Llanes-Estrada,
Kai Schwenzer

Institut für Kernphysik, Technische Universität Darmstadt,
64289 Darmstadt, Germany

Institut für Physik der Karl-Franzens Universität, A-8010 Graz,
Austria

Departamento de Física Teórica I de la Universidad
Complutense, 28040 Madrid Spain

Abstract

We summarize recent results on the nonperturbative quark-gluon interaction in Landau gauge QCD. Our analytical analysis of the infrared behaviour of the quark-gluon vertex reveals infrared singularities, which lead to an infrared divergent running coupling and a linear rising quark-antiquark potential when chiral symmetry is broken. In the chirally symmetric case we find an infrared fixed point of the coupling and, correspondingly, a Coulomb potential. These findings provide a new link betwen dynamical chiral symmetry breaking and confinement.

### 1 Introduction

The relation between the two fundamental properties of QCD, confinement and dynamical chiral symmetry breaking (DSB), is surely a matter of utmost interest. Lattice calculations provide evidence that field configurations with nontrivial topological content may be at the heart of both phenomena [1, 2], but the fine details still remain elusive. Complementary to the strategy of identifying individual confining field configurations is the investigation of the correlation functions of the theory. Certainly, both confinement and DSB manifest themenselves in strong, nonperturbative correlations at small momenta. In this talk we discuss these effects and present a novel link between confinement and DSB.

### 2 Infrared behaviour of Yang-Mills theory

The infrared behaviour of Landau gauge Yang-Mills theory has been investigated in the past in a number of works in both the Dyson-Schwinger equations (DSE) framework [3, 4, 5, 6, 7, 8, 9, 10] and also within the functional renormalisation group (FRG) [11, 12, 13, 14]; for reviews see [15, 16, 17]. In the deep infrared, i.e. for external momentum scales , a general power law behaviour of one-particle irreducible Green functions with external ghost legs and external gluon legs has been derived[9, 10]:

(\theequation) |

Here, is the space-time dimension. One can show that (2) is the only infrared solution in terms of power laws of both the complete hierarchy of DSEs and FRGs [13]. The anomalous dimension is known to be positive [4, 5] and is bounded by from below [5]. With the (well justified) approximation of a bare ghost-gluon vertex in the infrared one obtains [5, 6]. This value corresponds to an infrared vanishing gluon propagator and a strongly infrared enhanced ghost,

(\theequation) |

with dressing functions and . Such a behavior of the gluon propagator implies positivity violations and therefore may be interpreted as a signal for gluon confinement [3, 8].

An important consequence of (2) is the presence of a nontrivial infrared fixed point in the running couplings related to the primitively divergent vertex functions of Yang-Mills theory:

(\theequation) |

for . The infrared value of the coupling related to the ghost-gluon vertex can be computed [5, 7] and yields for .

### 3 Infrared behavior of quenched QCD

Based on the infrared solutions (2), one can also derive the analytical infrared behavior of the quark-gluon vertex [18]. To this end one has to carefully distinguish the cases of broken or unbroken chiral symmetry. Whereas in the broken case the full quark-gluon vertex can consist of up to twelve linearly independent Dirac tensors, these reduce to a maximum of six when chiral symmetry is realized in the Wigner-Weyl mode. Correspondingly, a broken symmetry induces two tensor structures in the quark propagator, whereas only one is left when chiral symmetry is restored. In a similar way, chiral symmetry breaking reflects itself in every Green’s function with quark content.

The presence or absence of the additional tensor structures turns out to be crucial for the infrared behavior of the quark-gluon vertex. When chiral symmetry is broken (either explicitly or dynamically with a valence quark mass ) one obtains a selfconsistent solution of the vertex-DSE which behaves like

(\theequation) |

Here denotes generically any dressing of the twelve tensor structures. If, however, the Wigner-Weyl mode is realized one obtains the weaker singularity

(\theequation) |

As a consequence the running coupling from the quark-gluon vertex either is infrared divergent (’infrared slavery’) or develops a fixed point similar to the Yang-Mills couplings of eq.(\theequation):

(\theequation) |

(Here we use that the quark propagator is constant in the infrared, i.e. [19].) Note that in all couplings the irrational anomalous dimensions () of the individual dressing functions cancel in the RG-invariant products.

Finally, one can analyze the behavior of the quark four-point function which includes the (static) quark potential. With (\theequation) and (\theequation), one obtains in the Nambu-Goldstone and in the Wigner-Weyl realization of chiral symmetry. This leads to a quark-antiquark potential of

(\theequation) |

which establishes the before mentioned link between dynamical chiral symmetry breaking and confinement.

### Acknowledgments

CF thanks the organizers of Menu07 for all their efforts which made this extraordinary conference possible. This work was supported by the DFG under grant no. Al 279/5-2, by the Helmholtz-University Young Investigator Grant VH-NG-332, by the FWF under contract M979-N16, and by MEC travel grant PR2007-0110, Spain.

## References

- [1] J. Greensite, Prog. Part. Nucl. Phys. 51 (2003) 1 [hep-lat/0301023].
- [2] C. Gattringer, Phys. Rev. Lett. 97, 032003 (2006) [hep-lat/0605018];
- [3] L. von Smekal, A. Hauck and R. Alkofer, Annals Phys. 267 (1998) 1 [Erratum-ibid. 269, 182 (1998)] [hep-ph/9707327].
- [4] P. Watson and R. Alkofer, PRL 86 (2001) 5239 [hep-ph/0102332].
- [5] C. Lerche and L. von Smekal, PRD 65 (2002) 125006 [hep-ph/0202194].
- [6] D. Zwanziger, Phys. Rev. D 67 (2003) 105001 [arXiv:hep-th/0206053]; Phys. Rev. D 69 (2004) 016002 [arXiv:hep-ph/0303028].
- [7] C. S. Fischer and R. Alkofer, PLB 536 (2002) 177 [hep-ph/0202202];
- [8] R. Alkofer, W. Detmold, C. S. Fischer and P. Maris, Phys. Rev. D 70 (2004) 014014 [hep-ph/0309077];
- [9] R. Alkofer, C. S. Fischer and F. J. Llanes-Estrada, Phys. Lett. B 611 (2005) 279 [arXiv:hep-th/0412330];
- [10] M. Huber, R. Alkofer, C. S. Fischer, K. Schwenzer, 0705.3809[hep-ph].
- [11] J. M. Pawlowski, D. F. Litim, S. Nedelko and L. von Smekal, Phys. Rev. Lett. 93, 152002 (2004) [arXiv:hep-th/0312324].
- [12] C. S. Fischer and H. Gies, JHEP 0410, 048 (2004) [hep-ph/0408089].
- [13] C. S. Fischer, J. M. Pawlowski, PRD 75 (2007) 025012 [hep-th/0609009].
- [14] J. Braun, H. Gies and J. M. Pawlowski, 0708.2413[hep-th].
- [15] R. Alkofer and L. von Smekal, Phys. Rept. 353 (2001) 281 [hep-ph/0007355].
- [16] C. S. Fischer, J. Phys. G32 (2006) R253 [hep-ph/0605173].
- [17] R. Alkofer and J. Greensite, J. Phys. G34 (2007) S3 [hep-ph/0610365].
- [18] R. Alkofer, C. S. Fischer and F. J. Llanes-Estrada, arXiv:hep-ph/0607293; C. S. Fischer, F. J. Llanes-Estrada, R. Alkofer and K. Schwenzer, in preparation.
- [19] C. S. Fischer and R. Alkofer, PRD 67 (2003) 094020 [hep-ph/0301094].