In the paper "More Abelian Dualities in 2+1 Dimensions",
https://arxiv.org/abs/1609.04012
on page 4, it is said that the theory
S[ϕ;a]=∫d3x{−14e2fμνfμν+|(∂μ−iaμ)ϕ|2−α|ϕ|4},
where fμν=∂μaν−∂νaμ, in the infrared limit (e→∞ α→∞), has no Maxwell term. i.e. 14e2da∧⋆da=0.
My understanding is the following.
1. Under a change of variables: x=x′/b, where 0<b<1, one has
S[ϕ;a]=∫d3x{−14e2fμνfμν+|(∂μ−iaμ)ϕ|2−α|ϕ|4}=∫d3x′{−14e2b−1(∂′μaν−∂′νaμ)(∂′μaν−∂′νaμ)+b−1(∂′μϕ∗)(∂′μϕ)+ib−2aμϕ∗(∂′μϕ)−ib−2aμ(∂′μϕ∗)ϕ+b−3aμaμϕ∗ϕ−b−3α|ϕ|4},
where ∂′μ=∂/∂x′μ.
2. Defining new fields ϕ′=b−1/2ϕ, a′μ=b−1aμ and f′μν=∂′μa′ν−∂′νa′μ, one has
S[ϕ;a]=∫d3x′{−14e2bf′μνf′μν+(∂′μϕ′∗)(∂′μϕ′)+ia′μϕ′∗(∂′μϕ′)−ia′μ(∂′μϕ′∗)ϕ′+a′μa′μϕ′∗ϕ′−b−1α|ϕ′|4}
3. Defining new couplings 1/e′2=b/e2, and α′=b−1α, one has
S[ϕ;a]=S[ϕ′;a′]=∫d3x′{−14e′2f′μνf′μν+|(∂′μ−ia′μ)ϕ′|2−α′|ϕ′|4}
Under the change of variables x=x′/b, the infrared limit is b→0+. This is equivalent to e′→∞ and α′→∞.
Is that correct? Do I need to compute the corresponding beta functions?