On the Coleman-Hill Theorem

The Coleman-Hill theorem prohibits the appearance of radiative corrections to the topological mass (more precisely, to the parity-odd part of the vacuum polarization tensor at zero momentum) in a wide class of abelian gauge theories in 2+1 dimensions. We re-express the theorem in terms of the effective action rather than in terms of the vacuum polarization tensor. The theorem so restated becomes somewhat stronger: a known exception to the theorem, spontaneously broken scalar Chern-Simons electrodynamics, obeys the new non-renormalization theorem. Whereas the vacuum polarization does receive a one-loop, parity-odd correction, this does not translate to a radiative contribution to the Chern-Simons term in the effective action. We also point out a new situation, involving scalar fields and parity-odd couplings, which was overlooked in the original analysis, where the conditions of the theorem are satisfied and where the topological mass does, in fact, get a radiative correction. The existence of the Chern-Simons (CS) term in 2+1 dimensional gauge theories [1] has fueled a large body of research over the last several years, in fields varying from condensed 1 matter physics to pure mathematics. The term leads to fractional-statistics excitations (relevant to the fractional quantum Hall effect) [2], while its topological nature in the nonabelian case has yielded information on the classification of lower-dimensional manifolds and knot invariants [3]. It is odd under parity, and provides for a gauge-invariant mass for the relevant vector bosons. The coefficient of the non-abelian CS term must be quantized for the theory to be consistent [4–6]. This quantization must be respected by radiative corrections, and, indeed, in pure SU(N) gauge theory, it has been found that the coefficient (appropriately normalized so that the quantization is to integer values) receives a one-loop correction which changes its value by the integer N [7]. If the gauge field is coupled to matter fields which spontaneously break the symmetry, the situation is much more delicate in the non-abelian case. With complete breaking of the symmetry, the topological mass itself receives a correction which is a complicated function of the parameters of the theory, and certainly no quantization condition is satisfied, in general [8]. However, the quantization of the coefficient of the CS term itself might be salvaged, since there exist other terms which are not of a topological nature and therefore whose coefficients need not be quantized, yet which contribute to the topological mass. The nonquantization of the radiative correction to the topological mass might thus be a combination of a quantized correction to the CS term along the lines of [7] along with a non-quantized correction to the other terms, as was suggested [8]. More alarming is the case of a non-abelian theory spontaneously broken to a non-abelian subgroup. There, the topological mass is found not to be quantized (similar to the situation in [8]), yet there are no terms other than the CS term which make a contribution to the topological mass [9]. It would appear, therefore, that in such theories the CS term does receive a non-quantized radiative correction, and thus that they are not consistent at a quantum level, following the reasoning of [4–6]. A parallel but quite different situation arises in the abelian case, where no quantization condition is required. There is thus no a prioro restriction on radiative corrections, yet 2 such corrections are in fact few and far between. Coupling the photon to a fermion yields a correction to the linear term in a momentum expansion of the parity-odd part of the photon vacuum polarization tensor π μν (whether or not the CS term is there initially) at one loop [1,4,10], but not at two loops [11]. Inspired by this unexpected result, Coleman and Hill [12] devised a proof that, under very general conditions, the only correction to the linear term in a momentum expansion of π μν comes from fermions at one loop. In particular, they have emphasized that the result is valid even for nonrenormalizable interactions in the presence of gaugeand Lorentz-invariant regularization. Situations exist where the conditions given by Coleman and Hill are satisfied, yet where radiative corrections to the topological mass do nonetheless arise. Namely, this can occur if there are new parity-violating interaction terms in the initial Lagrangian, a possibility which was overlooked in [12]. For instance, if the photon is coupled to massive vector particles which themselves violate parity (a possibility in 2+1 dimensions) there is a correction to the topological mass [13]. Furthermore, even scalar fields can have such parity-violating interactions. Indeed, the following interaction Lagrangian Lint = jμ (