Kei-Ichi Kondo

Renormalization in the Gauged Nambu-Jona-Lasinio Model

Based on the Cornwall-Jackiw-Tomboulis effective potential, we extensively study nonperturbative renormalization of the gauged Nambu-Jona-Lasinio model in the ladder approximation with standing gauge coupling. Although the pure Nambu-Jona-Lasinio model is not renormalizable, presence of the gauge interaction makes it possible that the theory is renormalized as an interacting continuum theory at the critical line in the ladder approximation. Extra higher dimensional operators («counter terms») are not needed for the theory to be renormalized. By virtue of the effective potential approach, the renormalization («symmetric renormalization») is performed in a phase-independent manner both for the symmetric and the spontaneously broken phase of the chiral symmetry

First-Order Phase Transition in Three-Dimensional QED with Chern-Simons Term

We have studied the chiral phase transition in three-dimensional QED in the presence of a Chern-Simons term for the gauge field. There exists a phase where the chiral symmetry is broken dynamically, and we have determined the critical line for this symmetry breaking as a function of the effective coupling and the strength of the additional Chern-Simons term. In the presence of a Chern-Simons term, the chiral phase transition turns out to be first order, in sharp contrast to the phase transition in pure three-dimensional QED.

Thirring model as a gauge theory

Abstract We give another reformulation of the Thirring model (with four-fern-don interaction of the current-current type) as a gauge theory and identify it with a gauge-fixed version of the corresponding gauge theory according to the Batalin-Fradkin formalism. Based on this formalism, we study the chiral symmetry breaking of the D -dimensional Thirring model (2 D N flavors of 4-component fermions. By constructing the gauge covariant effective potential for the chiral order parameter, up to the leading order of 1/ N expansion, we show the existence of the second order chiral phase transition and obtain explicitly the critical number of flavors N c (respectively critical four-fermion coupling G c ) as a function of the four-fermion coupling G (respectively N ), below (respectively above) which the chiral symmetry is spontaneously broken.

Dynamical breakdown of chirality and parity in (2+1)-dimensional QED

In the (2+1)-dimensional QED with and without the Chern-Simons term, we find the non-local gauge in which there is no wavefunction renormalization for the fermion in the framework of the Schwinger-Dyson equation. By solving the Schwinger-Dyson equation in the non-local gauge, we find a finite critical value N c f for the number of flavors Nf of four-component Dirac fermions, above which the chiral symmetry restores irrespective of presence or absence of the Chern-Simons term. In the same framework, we study the possibility of dynamical breakdown of the parity. It is shown that the parity is not dynamically broken. We discuss this reason from the viewpoint of the Schwinger-Dyson equation.

Spontaneous chiral-symmetry breaking in three-dimensional QED with a Chern-Simons term

In three-dimensional QED with a Chern-Simons term we study the phase structure associated with chiral-symmetry breaking in the framework of the Schwinger-Dyson equation. We give detailed analyses on the analytical and numerical solutions for the Schwinger-Dyson equation of the fermion propagator, where the nonlocal gauge-fixing procedure is adopted to avoid wave-function renormalization for the fermion. In the absence of the Chern-Simons term, there exists a finite critical number of four-component fermion flavors, at which a continuous (infinite-order) chiral phase transition takes place and below which the chiral symmetry is spontaneously broken. In the presence of the Chern-Simons term, we find that the spontaneous chiral-symmetry-breaking transition continues to exist, but the type of phase transition turns into a discontinuous first-order transition. A simple stability argument is given based on the effective potential, whose stationary point gives the solution of the Schwinger-Dyson equation.

RENORMALIZING THE GAUGED NAMBU-JONA-LASINIO MODEL

Non-perturbative renormalization is given for both symmetric and spontaneously broken phases of the chiral symmetry in the gauged Nambu-Jona-Lasinio model. We explicitly obtain β-function having non-trivial uv fixed line for the renormalized coupling as well as the bare one. In both phases the anomalous dimension is very large (> 1) near the fixed line without discontinuity.

FINITE-TEMPERATURE AND FINITE-DENSITY QED: THE SCHWINGER-DYSON EQUATION IN THE REAL-TIME FORMALISM

We derive, based on the real-time formalism (especially thermo-field-dynamics), the Schwinger-Dyson gap equation for the fermion propagator in QED and the four-fermion model at finite temperature and density. We discuss some advantages of the real-time formalism in solving the self-consistent gap equation, in comparison with the ordinary imaginary-time formalism. Once we specify the vertex function, we can write down the SD equation with only continuous variables without performing the discrete sum over Matsubara frequencies which cannot be performed in advance without further approximation in the imaginary-time formalism. By solving the SD equation obtained in this way, we find the chiral-symmetry-restoring transition at finite temperature and present the associated phase diagram of strong-coupling QED. In solving the SD equation, we consider two approximations: instantaneous-exchange and p0-independent ones. The former has a direct correspondence in the imaginary-time formalism; the latter is a new approximation beyond the former, since it is able to incorporate new thermal effects which have been overlooked in the ordinary imaginary-time solution. However, the two approximations are shown to give qualitatively the same results on the finite-temperature phase transition.

Flavor dependence and higher orders of gauge independent solutions in strong coupling gauge theory

Abstract The fermion flavor Nf dependence of non-perturbative solutions in the strong coupling phase of the gauge theory is reexamined based on the interrelation between the inversion method and the Schwinger-Dyson equation approach. Especially we point out that the apparent discrepancy on the value of the critical coupling in QED will be resolved by taking into account the higher order corrections which inevitably lead to the flavor-dependence. In the quenched QED, we conclude that the gauge-independent critical point α c = 2π 3 obtained by the inversion method to the lowest order will be reduced to the result α c = π 3 of the Schwinger-Dyson equation in the infinite order limit, but its convergence is quite slow. This is shown by adding the chiral-invariant four-fermion interaction.

Gauge independent phase structure of gauged Nambu-Jona-Lasinio and Yukawa models

We investigate the critical behavior of the gauged NJL model (QED plus four-fermion interaction) and the gauged Yukawa model by the use of the inversion method. By calculating the gauge-invariant chiral condensate in the inversion method to the lowest order, we derive the critical line which separates the spontaneous chiral-symmetry breaking phase from the chiral symmetric one. The critical exponent for the chiral order parameter associated with the second order chiral phase transition is shown to take the mean field value together with possible logarithmic correction to the mean-field prediction. All the above results are gauge-parameter independent and are compared with the previous results obtained from the Schwinger-Dyson equation for the fermion propagator.