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So as I understand it the order of $\phi$ in a scalar Quantum field theory is indicative of the number of lines entering a given vertex. For example for $\phi^3$ this leads to vertices like the one shown with 3 lines entering it. My question is why do we talk about $n$-point vertices $iV_{n}(k_{1},...,k_{n})$ in $\phi^3$ theory when $\phi^3$ means that vertices have 3 lines entering them?

vertex in $\phi^3$

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    $\begingroup$ When you say "n-point vertices", do you mean n-point functions? $\endgroup$
    – ACuriousMind
    Dec 14 '15 at 0:16
  • $\begingroup$ For example a three-point vertex function $iV_{3}(k_{1},...,k_{n})$ means the sum of all one-particle irreducible diagrams with three external lines, with the external propagators removed. This concept is then extended (for example srednicki pg. 115 printed version) to higher order 1PI vertices. $\endgroup$
    – boson
    Dec 14 '15 at 0:24
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A $\phi^n$ scalar field theory usually refers to a theory which contains in its Lagrangian a term proportional to $\phi^n$. This means that in a Feynman diagram the vertex where $n$ fields/lines meet is allowed. This does not mean that only $n$-point functions are nonzero.

In fact, for the $\phi^3$ example you gave, I can draw the following graph with two lines "entering" the diagram, and where the vertices are "cubic":

which is a contribution to the two-point function (at one-loop order) of $\phi^3$ theory.


EDIT (after comment):

Similarly, although one cannot directly join four lines at a point since there is no $\phi^4$ term in the Lagrangian of $\phi^3$ theory, one can find contributions to an effective 4-point vertex, such as:

at one loop.

So, while you cannot draw a diagram like this:

You can draw one like this (at higher loop order):

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  • $\begingroup$ Right but the vertices are still cubic. I'm talking about the vertices being quartic in $\phi^3$ $\endgroup$
    – boson
    Dec 14 '15 at 0:25
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    $\begingroup$ @boson: You're right that you cannot draw a quartic vertex directly in the diagram since it is not in the original Lagrangian, i.e. there is no "tree-level" contribution, as Srednicki points out in the beginning of chapter 17. Nevertheless, you can draw a loop diagram with 4 $\phi$s going in, as in his Figure 17.1. This diagram works effectively as a 4-vertex, since you can include it as part of a bigger diagram (which generically will have more loops). $\endgroup$
    – J-T
    Dec 14 '15 at 0:45

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