The quark Schwinger–Dyson equation in temporal Euclidean space

Abstract

We present an elementary nonperturbative method to obtain Green's functions (GFs) for timelike momenta. We assume there are no singularities in the second and the fourth quadrants of the complex plane of space momentum components and perform a 3d analogue of Wick rotation. This procedure defines Greens functions in a timelike Euclidean space. As an example we consider the quark propagator in QCD. While for weak coupling, this method is obviously equivalent to perturbation theory, for a realistic QCD coupling a complex part of the quark mass and renormalization wave function has been spontaneously generated even below the standard perturbative threshold. Therefore, our method favors a confinement mechanism based on the lack of real poles.

Confinement, Chiral Symmetry Breaking and complex mass in QED2+1 in Minkowski and Euclidean spaces

“…In the paper [19] it was proposed that N-dimensional analog of Wick rotations performed for space components of Minkowski N + 1-vector can be partially useful for the study of a strong coupling quantum field theory. In even dimensional 3+1QCD it is the nonperturbative mechanism responsible for complex mass generation which is responsible for the absence of real pole type singularities in Greens functions evaluated at real their arguments which thus makes the nonperturbative calculations feasible there.…”

“…Furthermore, we derive the SDE in the Temporal Euclidean (E T ) space in the Section IV and prove that this exactly leads to the original Minkowski formulation for timelike momenta. Recall here, E T space metric is obtained from Minkowski one by the multidimensional Wick rotations, but now instead for the time component, it is made for all the space coordinates of the Lorentz three vector [19].…”

Chiral Symmetry Breaking and confinement in Minkowski space QED2+1

“…In the paper [25] it was proposed that N-dimensional analog of Wick rotations performed for space components of Minkowski N + 1-vector can be partially useful for the study of a strong coupling quantum field theory. In even dimensional 3+1QCD it is the nonperturbative mechanism (there is no imaginary unit i presented in the measure) which is fully responsible for the complex mass generation phenomena.…”

“…Recall here, TE space metric is obtained from Minkowski one by the multidimensional Wick rotations, but now instead for the time component, it is made for all the space coordinates of the Lorentz three vector. Usage of TE space instead of more standard spacelike Euclidean one was suggested first time in [25,35] as a powerful method when approximating above pronounced nonperturbative phenomena in true Minkowski space.…”

Multiple hits in wire chambers and other particle detectors

“…It is also possible to use a more symmetrical setup where the 3 wire planes (two cathode planes and one anode plane in the case of a MWPC, or 3 different readout directions on other types of detectors) make angles of 60 degrees with each other, giving rise to some hexagonal 'honeycomb' structure. This idea has been implemented using multi-GEM detectors for X-ray imaging [6], and the same idea has also been proposed and experimented for Cherenkov photon detectors (where there are typically simultaneous hits), in [7]. In order to analyze the principle however, which we set out to do in this paper, the exact geometry doesn't matter, as long as the topological relationships between the coordinates are the same.…”