Perturbative Gross-Neveu model coupled to a Chern-Simons field: A renormalization group study (original) (raw)

Four-fermion field theories and the Chern-Simons field: A renormalization group study

Physical Review D, 1999

In (2+1) dimensions, we consider the model of a N flavor, two-component fermionic field interacting through a Chern-Simons field besides a four fermion self-interaction which consists of a linear combination of the Gross-Neveu and Thirring like terms. The four fermion interaction is not perturbatively renormalizable and the model is taken as an effective field theory in the region of low momenta. Using Zimmerman procedure for reducing coupling constants, it is verified that, for small values of the Chern-Simons parameter, the origin is an infrared stable fixed point but changes to ultraviolet stable as α becomes bigger than a critical α c . Composite operators are also analyzed and it is shown that a specific four fermion interaction has an improved ultraviolet behavior as N increases. Fermionic quartic interactions have been very important for the clarification of conceptual aspects as well as for the applications of Quantum Field Theory. Illustrative examples of such dual role are provided by the Thirring and Nambu-Jona Lasinio models. However, perturbative studies of the models have been hampered by the fact that only in two dimensions they are renormalizable. If the number of flavors is high enough, a better ultraviolet behavior is achieved in the context of the 1/N expansion which turns out to be renormalizable up

Fermionic matter coupled to higher derivative Chern–Simons theories. II

Journal of Mathematical Physics, 1996

The diagrammatic and the Feynman rules for the higher derivative Chern-Simons theories in ͑2ϩ1͒ dimensions coupled to fermionic matter are constructed. This is done by starting from the path-integral quantization. Once the diagrammatic and the Feynman rules are given, the regularization and renormalization problem of this higher derivative model is analysed in the framework of the perturbation theory. The unitarity problem related with the possible appearance of ghost states with negative norm is also discussed. Finally, the BRST formalism for the model is constructed and some interesting differences with respect to the formalism applied to usual Chern-Simons models are presented.

Low energy limit of the Chern-Simons theory coupled to fermions

Physical Review D

We study the nonrelativistic limit of the theory of a quantum Chern-Simons field minimally coupled to Dirac fermions. To get the nonrelativistic effective Lagrangian one has to incorporate vacuum polarization and anomalous magnetic moment effects. Besides that, an unsuspected quartic fermionic interaction may also be induced. As a by product, the method we use to calculate loop diagrams, separating low and high loop momenta contributions, allows to identify how a quantum nonrelativistic theory nests in a relativistic one.

Chern–Simons theory with vector fermion matter

The European Physical Journal C, 2012

We study three dimensional conformal field theories described by U (N) Chern-Simons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger-Dyson equation in lightcone gauge, we compute the exact planar free energy of the theory at finite temperature on R 2 as a function of the 't Hooft coupling λ = N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |λ| = 1; the conformal theory does not exist for |λ| > 1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three point functions up to two loops. We also discuss a lightcone Hamiltonian formulation of this theory where a W ∞ algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U (1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.

Chern-Simons perturbation theory

Translations of mathematical monographs, 2002

We study the perturbation theory for three dimensional Chern-Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the action obtained by BRS gauge fixing in the Lorentz gauge has a superspace formulation. The basic properties of the propagator and the Feynman rules are written in a precise manner in the language of differential forms. Using the explicit description of the propagator singularities, we prove that the theory is finite. Finally the anomalous metric dependence of the 2-loop partition function on the Riemannian metric (which was introduced to define the gauge fixing) can be cancelled by a local counterterm as in the 1-loop case [28]. In fact, the counterterm is equal to the Chern-Simons action of the metric connection, normalized precisely as one would expect based on the framing dependence of Witten's exact solution.

θ effects in Chern-Simons (2+1)-dimensional QED with a four-Fermi interaction

Physical Review D, 1993

We investigate the effects of the Chern-Simons coupling on the high energy behavior in the (2 + 1)-dimensional Chern-Simons QED with a four-Fermi interaction. Using the 1/N expansion we discuss the Chern-Simons effects on the critical four-Fermi coupling at O(1/N ) and the β function around it. Highenergy behavior of Green's functions is also discussed. By explicit calculation, we find that the radiative correction to the Chern-Simons coupling vanishes at O(1/N ) in the broken phase of the dynamical parity symmetry. We argue that no radiative corrections to the Chern-Simons term arise at higher orders in the 1/N expansion.

Matter Chern Simons theories in a background magnetic field

Journal of High Energy Physics, 2019

We study large N 2+1 dimensional fermions in the fundamental representation of an SU(N)k Chern Simons gauge group in the presence of a uniform background magnetic field for the U (1) global symmetry of this theory. The magnetic field modifies the Schwinger Dyson equation for the propagator in an interesting way; the product between the self energy and the Greens function is replaced by a Moyal star product. Employing a basis of functions previously used in the study of non-commutative solitons, we are able to exactly solve the Schwinger Dyson equation and so determine the fermion propagator. The propagator has a series of poles (and no other singularities) whose locations yield a spectrum of single particle energies at arbitrary t’ Hooft coupling and chemical potential. The usual free fermion Landau levels spectrum is shifted and broadened out; we compute the shifts and widths of these levels at arbitrary t’Hooft coupling. As a check on our results we independently solve for the pro...

The dynamical structure of higher dimensional Chern-Simons theory

Nuclear Physics B, 1996

Higher dimensional Chern-Simons theories, even though constructed along the same topological pattern as in 2+1 dimensions, have been shown recently to have generically a non-vanishing number of degrees of freedom. In this paper, we carry out the complete Dirac Hamiltonian analysis (separation of first and second class constraints and calculation of the Dirac bracket) for a group G × U (1). We also study the algebra of surface charges that arise in the presence of boundaries and show that it is isomorphic to the WZW 4 discussed in the literature. Some applications are then considered. It is shown, in particular, that Chern-Simons gravity in dimensions greater than or equal to five has a propagating torsion.

Spin-1 massive particles coupled to a Chern-Simons field

Physical Review D, 1999

We study spin one particles interacting through a Chern-Simons field. In the Born approximation, we calculate the two body scattering amplitude considering three possible ways to introduce the interaction: (a) a Proca like model minimally coupled to a Chern-Simons field, (b) the model obtained from (a) by replacing the Proca's mass by a Chern-Simons term and (c) a complex Maxwell-Chern-Simons model minimally coupled to a Chern-Simons field. In the low energy regime the results show similarities with the Aharonov-Bohm scattering for spin 1/2 particles. We discuss the one loop renormalization program for the Proca's model. In spite of the bad ultraviolet behavior of the matter field propagator, we show that, up to one loop the model is power counting renormalizable thanks to the Ward identities satisfied by the interaction vertices.