A Question about Abelian Dualities in 2+1 Dimensions

Question

I have a question from the paper "More Abelian Dualities in 2+1 Dimensions".

https://arxiv.org/pdf/1609.04012.pdf

On  page 3, it says that we flow towards IR by sending $\alpha\rightarrow\infty$ and tuning the mass to zero.

$$S[\phi ;A]=\int d^{3}x |(\partial_{\mu}-iA_{\mu})\phi|^{2}-\alpha|\phi|^{4}.$$

My question is that why this limit takes the theory to IR.

Answer 1

$\alpha \to \infty$ in the IR because the $|\phi|^4$ term has dimension 1. Then there is a comma missing, the mass is tuned to zero to achieve criticality. The mass term is relevant like always though. It has to be tuned by hand.