A Massive Renormalizable Abelian Gauge Theory in 2+1 Dimensions

The standard formulation of a massive Abelian vector field in $2+1$ dimensions involves a Maxwell kinetic term plus a Chern-Simons mass term; in its place we consider a Chern-Simons kinetic term plus a Stuekelberg mass term. In this latter model, we still have a massive vector field, but now the interaction with a charged spinor field is renormalizable (as opposed to super renormalizable). By choosing an appropriate gauge fixing term, the Stuekelberg auxiliary scalar field decouples from the vector field. The one-loop spinor self energy is computed using operator regularization, a technique which respects the three dimensional character of the antisymmetric tensor $\epsilon_{\alpha\beta\gamma}$. This method is used to evaluate the vector self energy to two-loop order; it is found to vanish showing that the beta function is zero to two-loop order. The canonical structure of the model is examined using the Dirac constraint formalism.Comment: LaTeX, 17 pages, expanded reference list and discussion of relationship to previous wor

Off-Diagonal Elements of the DeWitt Expansion from the Quantum Mechanical Path Integral

The DeWitt expansion of the matrix element M_{xy} = \left\langle x \right| \exp -[\case{1}{2} (p-A)^2 + V]t \left| y \right\rangle, $(p=-i\partial)$ in powers of $t$ can be made in a number of ways. For $x=y$ (the case of interest when doing one-loop calculations) numerous approaches have been employed to determine this expansion to very high order; when $x \neq y$ (relevant for doing calculations beyond one-loop) there appear to be but two examples of performing the DeWitt expansion. In this paper we compute the off-diagonal elements of the DeWitt expansion coefficients using the Fock-Schwinger gauge. Our technique is based on representing $M_{xy}$ by a quantum mechanical path integral. We also generalize our method to the case of curved space, allowing us to determine the DeWitt expansion of \tilde M_{xy} = \langle x| \exp \case{1}{2} [\case{1}{\sqrt {g}} (\partial_\mu - i A_\mu)g^{\mu\nu}{\sqrt{g}}(\partial_\nu - i A_\nu) ] t| y \rangle by use of normal coordinates. By comparison with results for the DeWitt expansion of this matrix element obtained by the iterative solution of the diffusion equation, the relative merit of different approaches to the representation of $\tilde M_{xy}$ as a quantum mechanical path integral can be assessed. Furthermore, the exact dependence of $\tilde M_{xy}$ on some geometric scalars can be determined. In two appendices, we discuss boundary effects in the one-dimensional quantum mechanical path integral, and the curved space generalization of the Fock-Schwinger gauge.Comment: 16pp, REVTeX. One additional appendix concerning end-point effects for finite proper-time intervals; inclusion of these effects seem to make our results consistent with those from explicit heat-kernel method

Structure of the Effective Potential in Nonrelativistic Chern-Simons Field Theory

We present the scalar field effective potential for nonrelativistic self-interacting scalar and fermion fields coupled to an Abelian Chern-Simons gauge field. Fermions are non-minimally coupled to the gauge field via a Pauli interaction. Gauss's law linearly relates the magnetic field to the matter field densities; hence, we also include radiative effects from the background gauge field. However, the scalar field effective potential is transparent to the presence of the background gauge field to leading order in the perturbative expansion. We compute the scalar field effective potential in two gauge families. We perform the calculation in a gauge reminiscent of the $R_\xi$-gauge in the limit $\xi\rightarrow 0$ and in the Coulomb family gauges. The scalar field effective potential is the same in both gauge-fixings and is independent of the gauge-fixing parameter in the Coulomb family gauge. The conformal symmetry is spontaneously broken except for two values of the coupling constant, one of which is the self-dual value. To leading order in the perturbative expansion, the structure of the classical potential is deeply distorted by radiative corrections and shows a stable minimum around the origin, which could be of interest when searching for vortex solutions. We regularize the theory with operator regularization and a cutoff to demonstrate that the results are independent of the regularization scheme.Comment: 24 pages, UdeM-LPN-TH-93-185, CRM-192

Operator regularization and the chiral anomaly

Operator regularization is used to compute anomalies in axial gauge theories to one- and two-loop order. To one-loop order, we consider the decay of a U(1) axial current into two, three, and four SU(N) vector currents. This is done first by directly computing the relevant Green functions, and then by a functional approach. The one-loop anomaly is found to be proportional to trF*F. We note that the quadrilinear term in this expression is zero. The Schwinger expansion, employed in operator regularization to generate Green functions, is then used to derive the Schwinger–de Witt WKB (Wentzel–Kramers–Brillouin) expansion of the heat kernel, recovering the known diagonal elements and also giving the previously unknown off-diagonal elements. For the case of a constant background electromagnetic field, these off-diagonal terms can also be obtained from an exact expression for the heat kernel given by Schwinger. We then use this result, in conjunction with the functional technique originally introduced in one-loop calculations, to compute exactly the amplitude for the decay of the U(1) axial current into both U(1) and O(3) constant gauge fields to two-loop order. With the appropriate choice of mass scale parameters, a vanishing result is obtained, in agreement with the Adler–Bardeen result. </jats:p

Operator regularization and massive Yang–Mills theory

Operator regularization has proved to be a practical method for computing finite, symmetry preserving Green's functions to arbitrary order in perturbation theory. This finiteness has been shown to persist in such nonrenormalizable theories as [Formula: see text] and quantum gravity; in this paper we apply the technique to massive gauge theories. The mass for the vector particle is inserted by hand using the Stueckleberg–Kunimasa–Goto mechanism; no degree of freedom associated with the Higgs scalar is present in the initial Lagrangian. Several one-loop calculations serve to illustrate that no divergences ever arise in this theory if one uses operator regularization. The one-loop effective action is examined in both 4 and n = (4 − ε) dimensions. The result is finite in 4 dimensions and in (4 − ε) dimensions has the pole structure at ε = 0 found by Kafiev. In four dimensions, we find that, in contrast to the case of renormalizable theories, radiatively induced interactions occur with coefficients depending on log (p2/μ2). Consequently μ2 is no longer an arbitrary parameter but must be fixed experimentally. </jats:p