Composite operator approach to dynamical mass generation in the (2 + 1)-dimensional Gross–Neveu model

Using a nonperturbative approach based on the Cornwall–Jackiw–Tomboulis (CJT) effective action [Formula: see text] for composite operators, the phase structure of the simplest massless [Formula: see text]-dimensional Gross–Neveu model is investigated. We have calculated [Formula: see text] in the first-order of the bare coupling constant [Formula: see text] and have shown that there exist three different specific dependences of [Formula: see text] on the cutoff parameter [Formula: see text], and in each case, the effective action and its stationarity equations have been obtained. The solutions of these equations correspond to the fact that three different masses of fermions can arise dynamically and, respectively, three different nontrivial phases can be observed in the model.

Four Loop Scalar ϕ4 Theory Using the Functional Renormalization Group

We work with a symmetric scalar theory with quartic coupling in 4-dimensions. Using a 2PI effective theory and working at 4 loop order, we renormalize with a renormalization group method. All divergences are absorbed by one bare coupling constant and one bare mass which are introduced at the level of the Lagrangian. The method is much simpler than counterterm renormalization, and can be generalized to higher order nPI effective theories.

About renormalized effective action for the Yang-Mills theory in four-dimensional space-time

This work is related to the asymptotic approach in the renormalization theory and its problems. As the main example, the Yang-Mills theory in four-dimensional space-time is considered. It has been shown earlier [16] that using the asymptotic of the bare coupling constant one can find an expression for the renormalized effective action, however, this formula has problems (divergence ln " and infinite series). This work shows the relation of these values and provides an answer for the renormalized effective action.

NONPERTURBATIVE SOLUTION OF THE NONCONFINING SCHWINGER MODEL WITH A GENERALIZED REGULARIZATION

The nonconfining Schwinger model4 is studied with a one-parameter class of kinetic energy-like regularization. It may be thought of as a generalization over the regularization considered in Ref. 4. The phase space structure has been determined in this new situation. The mass of the gauge boson acquires a generalized expression with the bare coupling constant and the parameters involved in the regularization. The confinement and deconfinement scenario has been studied at the quark–antiquark potential level.

STRONG-COUPLING REGIME IN $g\phi^4_2$ THEORY

The problem of the strong-coupling regime is considered in the scalar superrenormalizable field theory [Formula: see text]. By using the Gaussian transform, we have found an optimal representation within which the exact strong-coupling behavior of the free energy is already obtained in the leading-order approximation. Within this representation, the interaction becomes slower as the bare coupling constant grows, so the higher-order corrections can be systematically estimated by using a modified perturbation scheme. The next-to-leading terms give rise in insignificant corrections for finite coupling. The regularization procedure regroups the initial counterterms so that the divergencies are exactly removed in final expressions. The main idea is demonstrated in the simplest examples of a plain quartic integral and the anharmonic oscillator.

INTEGRABLE PERTURBATIONS OF CFT WITH COMPLEX PARAMETER: THE M3/5 MODEL AND ITS GENERALIZATIONS

By using the thermodynamic Bethe ansatz approach, we give evidence of the existence of both massive and massless behaviors for the ϕ2,1 perturbation of the M3,5 nonunitary minimal model, thus resolving apparent contradictions in the previous literature. The two behaviors correspond to changing the perturbing bare coupling constant from real values to imaginary ones. Generalizations of this picture to the whole class of nonunitary minimal models Mp,2p±1, perturbed by their least relevant operator, lead to a cascade of flows similar to that of unitary minimal models perturbed by ϕ1,3. Various aspects and generalizations of this phenomenon and the links with the Izergin–Korepin model are discussed.

EFFECTIVE POTENTIALS FOR ϕ4-THEORY IN 2+1 DIMENSIONS

Spontaneous symmetry breaking in ϕ4-theory in 2+1 dimensions is investigated using the Gaussian approximation. The theory stays in the symmetric phase at zero temperature as long as the bare coupling constant is below a critical value λc. When λ>λc the symmetric phase is again stable when the temperature is above a transition temperature T(λ). The obtained results are compared with the predictions of the standard one-loop effective potential.

DYSON-SCHWINGER EQUATIONS IN TERMS OF RANDOM WALKS: FOUR-FERMION INTERACTIONS

Quantum field theories of interacting fermions have been recently formulated in terms of directed random walks. Using this formulation, we derive a hierarchy of equations for the correlation functions of scalar N-component four-fermion theories. These follow from an analysis of the underlying random process, and from geometric considerations. Our equations are, as we show, equivalent to the standard Dyson-Schwinger equations of motion, and are a convenient starting point for nonperturbative investigations of four-fermion theories. In particular, we discuss the physical interpretation of the gap equation in the language of random walks, and show that, in both the N→0 and N→∞ limits, an interacting theory can be obtained only for a finely tuned negative bare coupling constant.

RECOVERY OF GAUGE INVARIANCE IN STRONG-COUPLING QED

Under a novel ansatz for the vertex function, the Schwinger- Dyson equation for the fermion propagator in the cutoff QED is solved in the arbitrary gauge, taking account of the vacuum polarization in the photon propagator. For any ultraviolet cutoff Λ, there exists a bifurcation point ec(Λ) of the bare coupling constant above which the trivial fermion-mass function for massless QED bifurcates to another, nontrivial massive solution. With a proper choice of the transverse vertex function and the longitudinal vertex that respects the Ward-Takahashi identity, the critical point ec(∞) and the critical scaling behavior in the vicinity of the critical point are shown to be gauge-independent. In the arbitrary gauge, it is shown that the quenched, planar QED obeys Miransky’s scaling of the essential-singularity type and that the unquenched QED exhibits the mean-field critical behavior with classical critical exponents.

THE NAMBU-JONA-LASINIO MODEL AND GAUGE THEORIES IN 2+1 DIMENSIONS

The weakly coupled globally invariant Nambu-Jona-Lasino (NJL) model in 2+1 dimensions is shown to be equivalent to a strongly coupled gauge theory. This equivalence is demonstrated for the renormalized theories in the 1/N expansion utilizing an unconventional, cutoff-dependent bare coupling constant to take the limit of weak or strong bare couplings. The weakly coupled Abelian NJL model is renormalized to order 1/N and compared to a renormalized strongly coupled QED3. Next, the U(2) globally invariant NJL model is studied in the broken phase and renormalized to leading order. The resulting U(1)×U(1) gauge-invariant theory is shown to be equivalent to a spontaneously broken U(2) gauge theory analyzed in the 1/N expansion.