# Why there is a 3-point interaction in this theory?

I'm studying the Lagrangian $$\mathcal{L} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi+\lambda\phi\partial_\mu\phi\partial^\mu\phi$$ And am trying to work out the Feynman rule for the 3-point function of this interaction term. The massless free part makes sense to me, but I don't quite understand what's the physical meaning of this interaction term. How it describes a different process as $$\phi^3$$ theory?

The term $$\lambda\phi^3$$ would render a vertex in Feynman diagrams that would contribute to amplitude as $$\lambda$$ only. In contrast, $$\lambda\partial_\mu\phi\partial^\mu\phi$$ would render a vertex with the same fields as $$\lambda\phi^3$$ interaction but counting as $$\sim\lambda (p_1p_2 + p_1p_3 + p_2p_3)$$ where $$p_i$$ is the momentum of each particle as incoming to the vertex. Upon conservation of momentum in the vertex, this becomes $$\sim\lambda (p_1^2 + p_2^2 + p_3^2)$$.
In other words, $$\partial_\mu \sim p_\mu$$ when treating amplitudes in momentum space instead of position space. This rule comes from a detail study of the scattering matrix and its expansion in a series.
If you work through the derivation of the Feynman rules (for example by using path integral methods), you will find that the cubic vertex you have written down will contribute a factor of $$\sim \lambda k_{1, \mu} k_2^{\mu}$$ (up to factors of $$2$$, $$\pi$$, $$-1$$, $$i$$, etc), where $$k_1$$ and $$k_2$$ are the momenta of two of the three legs of the vertex.