= Solution
Use $\hbar=c=1$, the <Minkowski metric> $\eta=\operatorname{diag}(1,-1)$, and $p^2=(p^0)^2-(p^1)^2$. The momentum-space <Feynman rules> for the <real scalar field> are
$$
\boxed{\text{internal line: }\frac{i}{p^2-m^2+i\epsilon},\qquad
\text{four-leg vertex: }-ig_4.}
$$
At each <interaction vertex> impose two-dimensional <momentum conservation>; before removing the overall conservation factor this is $(2\pi)^2\delta^{(2)}(\sum_jp_j)$. Integrate each independent <loop momentum> with $d^2k/(2\pi)^2$, and divide a <Feynman diagram> by its <symmetry factor>. External lines carry the same <scalar propagator> in an unamputated <two-point correlation function>; they are removed in an amputated <scattering amplitude>.
The quadratic <action> is inverted with the <Feynman i-epsilon prescription> to obtain the <scalar propagator>. Expanding the <Dyson series> gives $-ig_4/4!$ per insertion; the $4!$ attachments in the <Wick theorem> cancel the factorial. Fourier integration over the insertion point supplies the <momentum conservation> factor. Automorphisms that leave the labelled external attachments fixed account for the remaining <symmetry factor>. In two spacetime dimensions the <mass dimensions> are $[\phi]=0$ and $[g_4]=2$.
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