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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:

enter image description here

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

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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.

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  • $\begingroup$ 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? $\endgroup$ – TKG May 2 '15 at 14:27
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    $\begingroup$ I don't think that there is a universally accepted interpretation of such graphs, so it depends on the context! I have seen this notation particularly frequently to denote the collection of (or even the sum of) all diagrams with the given number of external legs, which is sometimes called the exact n-point vertex. But this might be simply because of special books/recources that I have consulted. $\endgroup$ – physicus May 2 '15 at 14:36

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