We give a theoretical framework to obtain a low-energy effective theory of quantum chromodynamics (QCD) towards a first-principle derivation of confinement/deconfinement and chiral-symmetry breaking/restoration crossover transitions. In fact, we demonstrate that an effective theory obtained using simple but nontrivial approximations within this framework enables us to treat both transitions simultaneously on equal footing. A resulting effective theory is regarded as a modified and improved version of nonlocal Polyakov-loop extended Nambu-Jona-Lasinio (nonlocal PNJL) models proposed recently by Hell, R\"ossner, Cristoforetti, and Weise, and Sasaki, Friman, and Redlich, extending the original (local) PNJL model by Fukushima and others. A novel feature is that the nonlocal NJL coupling depends explicitly on the temperature and Polyakov loop, which affects the entanglement between confinement and chiral-symmetry breaking, together with the cross term introduced through the covariant derivative in the quark sector considered in the conventional PNJL model. The chiral-symmetry breaking/restoration transition is controlled by the nonlocal NJL interaction, while the confinement/deconfinement transition in the pure gluon sector is specified by the nonperturbative effective potential for the Polyakov loop obtained recently by Braun, Gies, Marhauser, and Pawlowski. The basic ingredients are a reformulation of QCD based on new variables and the flow equation of the Wetterich type in the Wilsonian renormalization group. This framework can be applied to investigate the QCD phase diagram at finite temperature and density.

We give an argument for deriving analytically the infrared Abelian dominance in a gauge-invariant way for the Wilson loop average in SU(2) Yang-Mills theory. In other words, we propose a possible mechanism for realizing the dynamical Abelian projection in the SU(2) gauge-invariant manner without breaking color symmetry. This supports validity of the dual superconductivity picture for quark confinement. We also discuss the stability of the vacuum with magnetic condensation as a by-product of this result.

We derive general relationships between the number of complex poles of a propagator and the sign of the spectral function originating from the branch cut in the Minkowski region under some assumptions on the asymptotic behaviors of the propagator. We apply this relation to the mass-deformed Yang-Mills model with one-loop quantum corrections, which is identified with a low-energy effective theory of the Yang-Mills theory, to show that the gluon propagator in this model has a pair of complex conjugate poles or “tachyonic” poles of multiplicity two, in accordance with the fact that the gluon field has a negative spectral function, while the ghost propagator has at most one “unphysical” pole. Finally, we discuss implications of these results for gluon confinement and other nonperturbative aspects of the Yang-Mills theory.

We give a new way of looking at the Cho-Faddeev-Niemi (CFN) decomposition of the Yang-Mills theory to answer how the enlarged local gauge symmetry re

Complex singularities have been suggested in propagators of confined particles, e.g., the Landau-gauge gluon propagator. We rigorously reconstruct Minkowski propagators from Euclidean propagators with complex singularities. As a result, the analytically continued Wightman function is holomorphic in the tube, and the Lorentz symmetry and locality are kept intact, whereas the reconstructed Wightman function violates the temperedness and the positivity condition. Moreover, we argue that complex singularities correspond to confined zero-norm states in an indefinite metric state space.