Non Relativistic Limit of a Model of Fermions interacting through a Chern-Simons Field

We study the non relativistic limit of a Model of Fermions interacting through a Chern-Simons Field, from a perspective that resembles the Wilson’s Renormalization Group approach, instead of the more usual approach found in most texts of Field Theory. The solution of some difficulties, and a new understanding of non relativistic models is given. Models of a Chern-Simons [1] field interacting with non relativistic bosons [2] or fermions [3] have being studied in the literature both for its interest in general understanding of field theory by itself as for its application to Condensed Matter Physics [4]. The use of these models face in general, the difficulties of their non renormalizability. This fact is perhaps, the main reason for the interest, on the results of Bergman and Lozano [2], in one loop, later extended to three loops [5]. Their model consists of a non-relativistic boson field φ with a ∗e-mails:[email protected] 1 quartic self interaction and minimally interacting with a Chern-Simons field A : L = φ ( i d dt + eA ) φ− 1 2m |(~ ∇− ie ~ A)φ| −0 4 (φφ) + θ 2 ǫA∂A. (1) The only primitively divergent Green Function is the boson four point function. Up to one loop, the model can be made finite by the choice of a renormalized coupling constant λ through the equation; λ0 = λ + m 4π ( λ − 4e 2 mθ ln ( Λ M ) ) (2) where Λ is an ultraviolet (UV) cut-off and M an arbitrary constant (the renormalization constant) with dimension of mass. Their main observation is that at the critical value, λ = | 2 e mθ |, the one loop contribution vanishes and no renormalization of λ is needed. At this choice of λ the model regains the scale invariance that it has at classical level, and the relative wave function of the two bosons reproduces the Aharonov-Bohm scattering amplitude [8] up two the second Bohr order. The model of non relativistic fermions interacting with the Chern-Simons field was also discussed in [2] and studied in more details in [3]. In this last paper it is shown that the one loop scattering of two fermions with spins of the same sign (in 2+1 dimension the spin is a pseudo-scalar) is finite in one loop, due to the contact interaction represented by the Pauli interaction, that is already present in the minimal interaction of the fermions with the gauge field. As for the scattering of two fermions of opposite spins the Pauli interaction does not have any role and the amplitude is divergent unless a quartic fermionic interaction of the form c(Λ)ψψ φφ where ψ and φ represent respectively fermions with spin plus and minus 1/2, and c is a constant that depends logarithmically on the UV cut off Λ. This last fact poses a new problem. If the non relativistic model is thought to be the low energy limit of a more fundamental model of relativistic Dirac fermions interacting with a Chern-Simons field, in the way that this limit is generally taken in most texts [9], it would come from a similar quartic interaction in the relativistic fermions. As is well known one such interaction is non renormalizable! We will show that this is, in fact, a false 2 problem. No quartic non renormalizable self interaction is needed in the “parent” relativistic model if a new perspective on the non relativistic limit in field theory is taken. Before going to the description of this new limit, lets us briefly resume, in an example, the “Classical Non Relativistic Limit”, and discuss why it is not always correct. Let us consider, in 2+1 dimension, a 2 component Dirac fermion field Ψ, that represents a spin plus fermion and its anti-fermion, interacting with an external electromagnetic field A, as described by the Lagrangian density (the gamma matrices are γ . = σ ,γ . = iσ and γ . = iσ where σ are the Pauli matrices) Lrel = Ψ̄ {