# Why doesn't a renormalizable $\phi^4$ theory have odd diagrams?

I've been reading Zee's QFT textbook and trying to follow some lecture notes online whenever I can't grasp something. I really don't understand one thing regarding the renormalization of theories, though.

The two theories commonly used as examples of renormalizable theories are $\phi^3$ in 6 dimensions and $\phi^4$ in 4 dimensions. Apparently, $\phi^3$ can have diagrams with an odd number of external legs, but $\phi^4$ only has even terms. I don't understand where this comes from. I have assumed that even-powered theories would have even terms only, but then I started reading these 2014 lecture notes (section 2.1, p. 24) by John McGreevy and the given example for a $\phi^6$ theory is a three-legged diagram:

which made me a bit confused. How do you actually determine which diagrams are allowed in a theory?

In the lecture notes, the Feynman diagrams for the two examples of $\phi^3$ in $D=6$ and $\phi^6$ in $D=3$ are mixed up. You can easily identify this by noticing that in a scalar $\phi^n$ theory $n$ lines meet in all vertices. The example graph for $\phi^6$ theory in the lecture notes (which is wrongly listed as an example of $\phi^3$ in $D=6$) has six external legs, consistent with the statment that there are only diagrams with an even number of external legs in $\phi^n$ theory, when $n$ is even.
The reason for this statement is simple: The number of external legs $E$ has to be $$E=nV-2P$$ where $V$ is the number of vertices and $P$ the number of internal propagators. This is always even for $n$ even.
• Thanks for the answer. Regarding the point that in a $\phi^n$ theory $n$ lines meet in all vertices, I often see diagrams in textbooks/notes where the centre is filled in, like so. Is this a convention to avoid drawing all the additional lines that connect the vertices?