# Divergent verticies in mesonic scalar theory [closed]

Considering the following Lagrangian density:

$$\mathcal{L} = - \frac{1}{2} ( \partial_{\mu} \phi \partial^{\mu} \phi + m^2 \phi^2) + \bar{\psi} (i \gamma^{\mu} \partial_{\mu} - m) \psi + g \bar{\psi} \psi \phi.$$

it's superficial degree of divergence, for $$d$$ dimensions, can be written as:

$$D = d + V \left( \frac{d-4}{2} \right) - N_f \left( \frac{d-2}{2} \right) - N_{b} \left( \frac{d-1}{2} \right)$$

and with $$d=4$$:

$$D = 4- \frac{3}{2}N_f - N_{b}$$

where $$N_f$$ is the number of external fermion lines, $$N_b$$ is the number of external boson lines and $$V$$ the number of verticies.

My question is, if we add auto-interaction terms to the third and fourth powers in the scalar theory, ($$\frac{1}{3!} \lambda_3 \phi^3$$ , $$\frac{1}{4!} \lambda_4 \phi^4$$) from the degree of divergence how can we conclude that the potentially divergent verticies happen for the values:

$$(N_{b},N_f) = (2,0),(0,2),(1,2),(3,0),(4,0) \hspace{0,5cm} ?$$