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?
 A: 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.
A: 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.
