= Solution
At lowest derivative order, a general invariant effective Lagrangian through fourth order in the fields has the schematic form
$$
\begin{aligned}
\mathcal L={}&-\frac12\operatorname{Tr}F_{\mu\nu}F^{\mu\nu}
+\operatorname{Tr}(D_\mu\phi D^\mu\phi)
-m_\phi^2\operatorname{Tr}\phi^2
-\lambda_{abcd}\phi^a\phi^b\phi^c\phi^d\\
&+\bar\psi(i\not D-m_\psi)\psi
-h_A\,\mathcal I_A(\bar\psi,\psi,\phi,\phi)
-G_B\,\mathcal J_B(\bar\psi,\psi,\bar\psi,\psi).
\end{aligned}
$$
Here $\lambda_{abcd}$ is any invariant symmetric rank-four tensor, the $\mathcal I_A$ run over invariant contractions such as $(\operatorname{Tr}\phi^2)\bar\psi\psi$ and $\bar\psi\phi^2\psi$, and the $\mathcal J_B$ run over gauge- and Lorentz-invariant four-fermion contractions. The two sign symmetries forbid scalar cubic terms and Yukawa terms $\bar\psi\phi\psi$. The covariant kinetic terms automatically contain the allowed cubic and quartic interactions involving $A_\mu$.
The canonical dimensions are
$$
[A_\mu]=[\phi]=\frac{d-2}{2},
\qquad
[\psi]=\frac{d-1}{2},
$$
and hence
$$
[g]=\frac{4-d}{2},
\quad
[\lambda]=4-d,
\quad
[h_A]=3-d,
\quad
[G_B]=2-d,
\quad
[m_\phi^2]=2,
\quad
[m_\psi]=1.
$$
A coupling is <relevant coupling>[relevant], <marginal coupling>[marginal], or <irrelevant coupling>[irrelevant] when its dimension is positive, zero, or negative at the Gaussian fixed point. Thus gauge and scalar-quartic interactions are marginal in $d=4$, the mixed two-scalar fermion bilinear is marginal in $d=3$, and four-fermion interactions are marginal in $d=2$. Quantum corrections replace this engineering classification near an interacting fixed point by the eigenvalues of its RG stability matrix.
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