Four-fermion field theories and the Chern-Simons field: A renormalization group study (original) (raw)
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Perturbative Gross-Neveu model coupled to a Chern-Simons field: A renormalization group study
Physical Review D, 1998
In 2+1 dimensions, for low momenta, using dimensional renormalization we study the effect of a Chern-Simons field on the perturbative expansion of fermions self interacting through a Gross Neveu coupling. For the case of just one fermion field, we verify that the dimension of operators of canonical dimension lower than three decreases as a function of the Chern-Simons coupling.
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.
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.
Theoretical and Mathematical Physics, 1996
We discuss the problem of equivalence between the standard (integer-dimensional d = 2) and the d = 2 + e dimensional renormalization schemes for the complete UN-symmetrical four-fermion interaction model. To ensure the muttiplicative renormalizability of the theory, we need three charqes in the first case; m the second, me lu;cd an infinite series of independent charges g =_ {gn,n = 0, 1 .... }. After the usual MS-renormalization. there ex'tsts a UV-finite renormalization of fields. Charges g ~ el(g) exist such that the renormatized Green'.s fu,;.ctzor~s in the limit e-+ 0 depend only on the three lower charges g~(g) unth ' n. = 0, 1.2. rather th.an ou the whole .set. Thrs ensures the possibility of establishing the equivalence of the two renormahzation schemes. The results of calculations in the MS scheme up to two loops for the 13-functions, and up to three loops for the anomalous field dimension 7¢ are presented. These are presented together with the derivation of the "'projection technique" relations, which allows one to ezpress the higher renormalized composite operators of the 4F-interaction via the lower ones in the limit e-* O.
Four-fermion interaction near four dimensions
Nuclear Physics B, 1991
A large class of models with four-fermion interaction is known to be renormalizable and asymptotically free in two dimensions. It has been noticed very early, in the example of the U(N)-invariant Gross-Neveu model and within the framework of the 1/N expansion, that then these models behave also like renormalizable models in higher dimensions. Some of them are thus natural candidates for composite models of scalar particles like for example the Higgs boson. An important question, however, has to be answered: Are these models more predictive, in four dimensions, than the effective models containing the bosons explicitly? We shall show here that, like for the non-linear o~-modelwhich has been investigated earlier, the answer, at least in some perturbative sense, is negative for a large class of models. The reason can be easily understood: These models are more short-distance sensitive than normal renormalizable models. The new parameters are hidden in the cutoff procedure. In particular in some models the fermions receive masses by spontaneous chiral symmetry breaking. The property that ratio of fermion and boson masses can be predicted is simply a consequence of the JR freedom of both type of models and the natural assumption that coupling constants have generic values at the cutoff scale. We shall consider in this article for definiteness the Gross-Neveu model but it will be clear that the arguments are rather general.
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.
The fermion-fermion effective potential in the Maxwell-Chern-Simons theory
Physics Letters B, 1992
The effective nonrelativistic potential VT describing the fermion-fermion interaction in the Maxwell-Chern-Simons theory is derived to the lowest order in perturbation theory. As expected, VT is not invariant under parity and time-reversal transformations. The quantum dynamics generated by Vr becomes exactly solvable at the limits where either the Maxwell or the Chern-Simons terms disappear; in neither case electron-electron bound states show up. However, numerical calculations indicate that fermion-fermion bound states do exist in the general case.
Nuclear Physics B, 1993
We explicitly carry Out the renormalization of non-abelian Chern-Simons field theories with matter in (2+1) dimensions. All the renormalization constants are calculated to the leading two-loop order both in terms of component fields and N = 1 superfields for the fundamental representation of the SU(n), Sp(n) and SO(n) groups. Renormalization-group fixed points are found, and their stability properties are examined. It is shown that the N 2 supersymmetry is realized as an infrared fixed-point solution, where the ultraviolet divergencies cancel.