# Why four-point vertex function in $\phi^3$ theory?

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?

• When you say "n-point vertices", do you mean n-point functions? Dec 14 '15 at 0:16
• 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. Dec 14 '15 at 0:24

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):

• Right but the vertices are still cubic. I'm talking about the vertices being quartic in $\phi^3$ Dec 14 '15 at 0:25
• @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).
– J-T
Dec 14 '15 at 0:45