In this Seiberg's SUSY lecture

the professor said that the following theory with Kinetic and W potential terms:

$$

K=|\phi|^2

$$

$$

W=m\phi^2+g \phi^3

$$

1. "It is not a valid theory in $4d$, but it is infrared (IR) free, thus it is not an interacting QFT."

2. "In $2d$ and $3d$, they are valid interacting QFTs."

Could someone explain the logic?

- what do free (quadratic lagrangian?) and interacting (higher order non-quadratic lagrangian?) mean respect to UV or respect to IR?

- I thought the $K=|\phi|^2$ requires derivative to be a kinetic term? Is he incorrect?

- I thought the $W=m\phi^2+g \phi^3$ are both relevant operators in the IR in 4d. Thus $g \phi^3$ changes the IR dynamics? Should this lead to an interacting QFT in 4d at IR?

- I thought the $m\phi^2$ is relevant and $g \phi^3$ is a marginal operator in the IR in 3d. Thus $g \phi^3$ again changes the IR dynamics? Should this lead to an interacting QFT in 3d?

- I thought the $m\phi^2$ is marginal and $g \phi^3$ is an irrelevant operator in the IR in 2d. Should this lead to a free QFT in 2d?

;

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