# SUSY Kinetic and $W$ potential terms: RG flow — free or interacting

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?

The terms $$K$$ and $$W$$ are not kinetic and potential terms, rather $$K$$ is a K"ahler potential and $$W$$ is a superpotential. Neither term enters the Lagrangian directly, but they are used to construct it. For details see, for example, Cyril Closset's lecture notes on supersymmetry.