# Feynman rules for this real scalar field in 2d

## Homework Statement

Consider the following real scalar field in two dimensions:

$S = \int d^2 x ( \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2 - g \phi^3)$

What are the Feynman rules for calculating $< \Omega | T(\phi_1 ... \phi_n ) | \Omega >$
2. Homework Equations

For a phi-4 theory in 4d:

Each propagator contributes a Feynman propagator $D_F (x-y)$
Each vertex z (4 lines to a point) contributes $\frac{-i g}{4!} \int d^4 z$
3. The Attempt at a Solution

I just wanted to check my understanding is okay. Adapting the rules for a phi-4 theory (this is phi-3 theory, yes?):

Each propagator contributes a Feynman propagator $D_F (x-y)$ (same as before)
Each vertex (now only 3 lines to a point because it is a phi-3 theory) contributes $\frac{-i g}{3!} \int d^3 z$ (because phi-3 not phi-4) or $\frac{-i g}{2!} \int d^2 z$ (because 2d not 4d)

In fourier space, one would integrate over momentum as $\int \frac{d^3 p}{(2 \pi)^3}$

Is this correct for adapting a phi-4 theory to phi-3?
Does going from 4d to 2d change anything here I'm missing?

Thanks