We relate quark confinement, as measured by the Polyakov-loop order parameter, to color confinement, as described by the Kugo-Ojima/Gribov-Zwanziger scenario. We identify a simple criterion for quark confinement based on the IR behaviour of ghost and gluon propagators, and compute the order-parameter potential from the knowledge of Landau-gauge correlation functions with the aid of the functional RG. Our approach predicts the deconfinement transition in quenched QCD to be of first order for SU(3) and second order for SU(2) -in agreement with general expectations. As an estimate for the critical temperature, we obtain Tc ≃ 284MeV for SU(3).

PrologueThis lecture course 1 is intended to fill the gap between graduate courses on quantum field theory and specialized reviews or forefront-research articles on functional renormalization group approaches to quantum field theory and gauge theories.These lecture notes are meant for advanced students who want to get acquainted with modern renormalization group (RG) methods as well as functional approaches to quantum gauge theories. In the first lecture, the functional renormalization group is introduced with a focus on the flow equation for the effective average action. The second lecture is devoted to a discussion of flow equations and symmetries in general, and flow equations and gauge symmetries in particular. The third lecture deals with the flow equation in the background formalism which is particularly convenient for analytical computations of truncated flows. The fourth lecture concentrates on the transition from microscopic to macroscopic degrees of freedom; even though this is discussed here in the language and the context of QCD, the developed formalism is much more general and will be useful also for other systems. Sections which have an asterisk * in the section title contain more advanced material and may be skipped during a first reading. This is not a review. I apologize for many omissions of further interesting and important aspects of this field (and their corresponding references). General reviews and more complete reference lists can be found in [1,2,3,4,5,6,7,8]. A guide to more specialized literature is given in the Further-Reading subsections at the end of some sections.

We study electron-positron pair creation from the Dirac vacuum induced by a strong and slowly varying electric field (Schwinger effect) which is superimposed by a weak and rapidly changing electromagnetic field (dynamical pair creation). In the sub-critical regime where both mechanisms separately are strongly suppressed, their combined impact yields a pair creation rate which is dramatically enhanced. Intuitively speaking, the strong electric field lowers the threshold for dynamical particle creation -or, alternatively, the fast electromagnetic field generates additional seeds for the Schwinger mechanism. These findings could be relevant for planned ultra-high intensity lasers.PACS numbers: 12.20. Ds, 11.15.Tk, 11.27.+d. As first realized by Dirac [1], a consistent relativistic quantum description of electrons necessarily involves negative energy levels, which -in the Dirac-sea pictureare filled up in the vacuum state. This entails the striking possibility of pulling an electron out of the vacuum by means of some external influence, such as a (classical) electromagnetic field [2], where the remaining hole in the Dirac sea is then associated with a positron. Of course, to create such an electron-positron pair out of the vacuum, one has to overcome the energy gap of 2mc 2 between the filled and the empty levels. There are basically two main mechanisms for doing so: In a strong electric field E over a sufficiently long distance L, "virtual" electron-positron pair fluctuations may gain this energy when qEL ≥ 2mc2 . This pair creation process is called the Schwinger mechanism [3,4] and can be understood as tunneling through the classically forbidden region (energy gap). Thus it is suppressed exponentially O(exp{−πE S /E}) for weak fields E, where E S = m 2 c 3 /( q) is the Schwinger critical field. For E ≃ E S , the work done by separating the electron-positron pair over a Compton wavelength is of the order of the energy gap 2mc2 . Alternatively, a classical time-dependent electromagnetic field will also create electron-positron pairs in general (dynamical pair creation). However, if the frequency ω of the external field is not large enough, ω < 2mc 2 , these non-adiabatic corrections correspond to higher-order (i.e., multi-photon) processes and are also suppressed exponentially exp{− O(1/ω)} for small ω [5]. These pair-production processes are fundamental predictions of quantum electrodynamics (QED), but only the multi-photon production process has so far been observed experimentally: the positron data taken at the SLAC E-144 experiment have convincingly been explained by nphoton production with n ≃ 5 [6]. However, a verification of the Schwinger mechanism has still remained an experimental challenge [7]. Since the Schwinger mechanism is non-perturbative in the field, its discovery would help exploring the non-perturbative realm of quantum field theory in a controlled fashion. Here, we propose a new mechanism which can help to overcome the strong exponential suppression. The basic idea is similar in spirit to ide...

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In a previous paper [1], it was shown that the worldline expression for the nonperturbative imaginary part of the QED effective action can be approximated by the contribution of a special closed classical path in Euclidean spacetime, known as a worldline instanton. Here we extend this formalism to compute also the prefactor arising from quantum fluctuations about this classical closed path. We present a direct numerical approach for determining this prefactor, and we find a simple explicit formula for the prefactor in the cases where the inhomogeneous electric field is a function of just one spacetime coordinate. We find excellent agreement between our semiclassical approximation, conventional WKB, and recent numerical results using numerical worldline loops.

We develop a method to compute the Casimir effect for arbitrary geometries. The method is based on the string-inspired worldline approach to quantum field theory and its numerical realization with Monte-Carlo techniques. Concentrating on Casimir forces between rigid bodies induced by a fluctuating scalar field, we test our method with the parallel-plate configuration. For the experimentally relevant sphere-plate configuration, we study curvature effects quantitatively and perform a comparison with the "proximity force approximation", which is the standard approximation technique. Sizable curvature effects are found for a distance-to-curvature-radius ratio of a/R 0.02. Our method is embedded in renormalizable quantum field theory with a controlled treatment of the UV divergencies. As a technical by-product, we develop various efficient algorithms for generating closed-loop ensembles with Gaußian distribution.

A renormalization group flow equation with a scale-dependent transformation of field variables gives a unified description of fundamental and composite degrees of freedom. In the context of the effective average action, we study the renormalization flow of scalar bound states which are formed out of fundamental fermions. We use the gauged Nambu-Jona-Lasinio model at weak gauge coupling as an example. Thereby, the notions of a bound state or fundamental particle become scale dependent, being classified by the fixed-point structure of the flow of effective couplings.

We investigate QCD with a large number of massless flavors with the aid of renormalization group flow equations. We determine the critical number of flavors separating the phases with and without chiral symmetry breaking in SU(Nc) gauge theory with many fermion flavors. Our analysis includes all possible fermionic interaction channels in the pointlike four-fermion limit. Constraints from gauge invariance are resolved explicitly and regulator-scheme dependencies are studied. Our findings confirm the existence of an N f window where the system is asymptotically free in the ultraviolet, but remains massless and chirally invariant on all scales, approaching a conformal fixed point in the infrared. Our prediction for the critical number of flavors of the zero-temperature chiral phase transition in SU (3) is N cr f = 10.0 ± 0.29(fermion) +1.55 −0.63 (gluon), with the errors arising from approximations in the fermionic and gluonic sectors, respectively.