Kurt Langfeld

Deconfinement in SU(2) Yang-Mills theory as a center vortex percolation transition

By fixing lattice Yang-Mills configurations to the maximal center gauge and subsequently applying the technique of center projection, one can identify center vortices in these configurations. Recently, center vortices have been shown to determine the string tension between static quarks at finite temperatures (center dominance); also, they correctly reproduce the deconfining transition to a phase with vanishing string tension. After verifying center dominance also for the so-called spatial string tension, the present analysis focuses on the global topology of vortex networks. General arguments are given supporting the notion that the deconfinement transition in the center vortex picture takes the guise of a percolation transition. This transition is detected in Monte Carlo experiments by concentrating on various slices through the closed vortex surfaces; these slices, representing loops in lattice universes reduced by one dimension, clearly exhibit the expected transition from a percolating to a non-percolating, deconfined, phase. The latter phase contains a large proportion of vortex loops winding around the lattice in the Euclidean time direction. At the same time, an intuitive picture clarifying the persistence of the spatial string tension in the deconfined phase emerges.

Gluon propagator and quark confinement

Abstract The gluon propagator is investigated in Landau and in maximal center gauge for the gauge group SU (2) by means of lattice gauge simulations. We find that Gribov ambiguities arising from the implementation of Landau gauge have a small influence on the propagator. In agreement with previous findings, we obtain that the small momentum behavior is dominated by a mass, which is of order (1.48±0.05) σ , where σ is the string tension. By removing the confining vortices from the full Yang–Mills ensemble, we convert full YM-theory into a theory which does not confine quarks. We find that in the latter case the strength of the gluon propagator in the intermediate momentum range is strongly reduced. The spectral functions which reproduce the numerical data for the propagators are analyzed using a generalized Maximum Entropy Method.

Confinement and scaling of the vortex vacuum of SU(2) lattice gauge theory

Abstract The magnetic vortices which arise in SU(2) lattice gauge theory in center projection are visualized for a given time slice. We establish that the number of vortices piercing a given 2-dimensional sheet is a renormalization group invariant and therefore physical quantity. We find that roughly 2 vortices pierce an area of 1 fm 2 .

Propagators and running coupling from SU(2) lattice gauge theory

We perform numerical studies of the running coupling constant αR(p2) and of the gluon and ghost propagators for pure SU(2) lattice gauge theory in the minimal Landau gauge. Different definitions of the gauge fields and different gauge-fixing procedures are used, respectively, for gaining better control over the approach to the continuum limit and for a better understanding of Gribov-copy effects. We find that the ghost–ghost–gluon vertex renormalization constant is finite in the continuum limit, confirming earlier results by all-order perturbation theory. In the low momentum regime, the gluon form factor is suppressed while the ghost form factor is divergent. Correspondingly, the ghost propagator diverges faster than 1/p2 and the gluon propagator appears to be finite. Precision data for the running coupling αR(p2) are obtained. These data are consistent with an IR fixed point given by limp→0αR(p2)=5(1).

Color screening, Casimir scaling, and domain structure in G(2) and SU(N) gauge theories

We argue that screening of higher-representation color charges by gluons implies a domain structure in the vacuum state of non-Abelian gauge theories, with the color magnetic flux in each domain quantized in units corresponding to the gauge group center. Casimir scaling of string tensions at intermediate distances results from random spatial variations in the color magnetic flux within each domain. The exceptional G(2) gauge group is an example rather than an exception to this picture, although for G(2) there is only one type of vacuum domain, corresponding to the single element of the gauge group center. We present some numerical results for G(2) intermediate string tensions and Polyakov lines, as well as results for certain gauge-dependent projected quantities. In this context, we discuss critically the idea of projecting link variables to a subgroup of the gauge group. It is argued that such projections are useful only when the representation-dependence of the string tension, at some distance scale, is given by the representation of the subgroup.

Center vortices of Yang-Mills theory at finite temperatures

Abstract Recent lattice calculations performed at zero temperature and in the maximal center gauge indicate that quark confinement can be understood in this gauge as due to fluctuations in the number of magnetic vortices piercing a given Wilson loop. This development has led to a revival of the vortex condensation theory of confinement. For a SU(2) gauge group, we show that also at finite temperatures, center vortices are the relevant collective infrared degrees of freedom determining the long-range static quark potential; in particular, their dynamics reflect the transition to the deconfining phase.Recent lattice calculations performed at zero temperature and in the maximal center gauge indicate that quark confinement can be understood in this gauge as due to fluctuations in the number of magnetic vortices piercing a given Wilson loop. This development has led to a revival of the vortex condensation theory of confinement. For a SU(2) gauge group, we show that also at finite temperatures, center vortices are the relevant collective infrared degrees of freedom determining the long-range static quark potential; in particular, their dynamics reflect the transition to the deconfining phase.

Loops and loop clouds: A Numerical approach to the worldline formalism in QED

A numerical technique for calculating effective actions of electromagnetic backgrounds is proposed, which is based on the string-inspired worldline formalism. As examples, we consider scalar electrodynamics in three and four dimensions to one-loop order. Beyond the constant-magnetic-field case, we analyze a step-function-like magnetic field exhibiting a nonlocal and nonperturbative phenomenon: “magnetic-field diffusion”. Finally, generalizations to fermionic loops and systems at finite temperature are discussed. The computation of effective energies and effective actions of a quantum system is a general and frequently occurring problem in quantum field theory. In a given classical background, virtual quantum processes such as vacuum polarization modify the properties of the vacuum and are responsible for new nonlinear and nonlocal phenomena. These phenomena can often be described by an effective action that takes the virtual quantum processes into account. A prominent example is given by the Heisenberg-Euler action 1,2 , describing the nonlinear corrections to the Maxwell action which are induced by virtual loops of electrons and positrons. The Heisenberg-Euler action is the one-loop QED effective action, spinor eff = lndet(−i∂ / + A / − im), scalar eff = −lndet(−(∂ + iA) 2 + m 2 ), (1) (here in Euclidean formulation), evaluated for a constant electromagnetic background field, and thereby represents a valid approximation of low-energy QED for all backgrounds that vary slowly with respect to the Compton wavelength 1/m. Technical difficulties increase markedly if one wants to go beyond this approximation. The standard means is the derivative expansion 3 ; its application is however limited to backgrounds whose spacetime variation does not exceed the scale of the electron mass or the scale of the field strength. Moreover, the enormous proliferation of terms restricts the actual computations to low orders. Only for very special (and very rare) field configurations can the derivative expansion be summed up 4 . Numerical methods developed so far aim at the integration of the corresponding differential equation of the operators in Eq. (1); therefore, they have been applied only � Talk given by H. Gies at the Fifth Workshop on Quantum Field Theory under the Influence of

Density of States in Gauge Theories

The density of states is calculated for the SU(2), SU(3), and a compact U(1) lattice gauge theories using a modified version of the Wang-Landau algorithm. We find that the density of states of the SU(2) gauge theory can be reliably calculated over a range of 120,000 orders of magnitude for lattice sizes as big as 20(4). We demonstrate the potential of the algorithm by reproducing the SU(2) average action, its specific heat, and the critical couplings of the weak first order transition in U(1).

Center vortices and Dirac eigenmodes in SU(2) lattice gauge theory

Abstract We study the interplay between Dirac eigenmodes and center vortices in SU(2) lattice gauge theory. In particular, we focus on vortex-removed configurations and compare them to an ensemble of configurations with random changes of the link variables. We show that removing the vortices destroys all zero modes and the near zero modes are no longer coupled to topological structures. The Dirac spectrum for vortex-removed configurations in many respects resembles a free spectrum thus leading to a vanishing chiral condensate. Configurations with random changes leave the topological features of the Dirac eigensystem intact. We finally show that smooth center vortex configurations give rise to zero modes and topological near zero modes.

Signals of confinement in Green functions of SU(2) Yang-Mills theory.

It has been well established that the removal of center vortices from SU(2) lattice configurations results in the loss of confinement. The running coupling constant, gluon form factor, and ghost form factor are studied in the Landau gauge for the full and the vortex removed theory. In the latter case, a strong suppression of the running coupling constant and the gluon form factor at low momenta is observed, and the IR singularity of the ghost form factor disappears. Hence, the removal of the vortices generates a theory for which Zwanzigers horizon condition for confinement is no longer satisfied.