# 2-loop correction to exact 3-point vertex in a complex scalar field theory with cubed interaction

I am a graduate student with 1 quarter of relativistic QFT at the level of Srednicki (covered up to Chapter 30 this Fall). This question is not in any book that I know off and it wasn't assigned as homework but I tagged it under homework and exercises. The theory in question is similar to one in Srendicki but with a different interaction term (refer to Problem 9.3). Given this Lagrangian density: $$\mathcal{L} = -Z_\phi \partial^\mu \phi\partial_\mu\phi^\dagger -Z_m m^2\phi^\dagger\phi+\frac{1}{6}Z_g g(\phi^3 +(\phi^\dagger)^3)$$

where the scalar field $$\phi$$ is a complex field, are there any two-loop 1 particle irreducible corrections to the exact 3-point vertex function(s). I attempted to draw one but I am still unsure if it follows the Feynman rules for this theory.

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The reasoning for my attempt is the following:

1. Adopting the ("charged arrows") convention that lines are given arrows pointing away from $$J$$ sources and towards $$J^\dagger$$ sources, the vertices allowed by the theory are lines with all 3 arrows pointing either towards or away from the vertex. In case you don't know what I mean by the $$J$$'s, they come from adding a source term to the Lagrangian $$J\phi^\dagger+J^\dagger\phi$$ to take functional derivatives and then they are set to $$J = J^\dagger = 0$$. This is part of the typical argument used to obtain Feynman diagrams as done in Chapter 9 of Srednicki. I left out details regarding momentum labels here.
2. A correction to the 3-point vertex has to have 3 external lines.
3. I attempted to draw more "traditional" looking 2 loop vertex corrections taking diagrams from $$\phi^3$$ theory but it was not possible to assign arrows in a way that agrees with the vertices allowed by this theory. This is an example of such a diagram

After attempting to assign the charge arrows to many of these unsuccessfully, I went for a diagram where one of the external lines went over an internal line (but didn't touch). Since I have never seen one like this before and it doesn't seem like it breaks any rule, I thought this might be a valid two-loop correction.

Question: Does my attempted diagram (hand-drawn above) break any rules and is it generally a rule that an external line can go over an internal one?

1. Yes, Feynman diagrams do not have to be planar.

2. For $$L=2$$ loops and $$E=3$$ external legs, it follows

• that a connected diagram has $$L=I-(V-1)$$, cf. e.g. eq. (8) in my Phys.SE answer here,

• and that a 3-valent diagram has $$3V=2I+E$$,

which leads to $$V=5$$ vertices and $$I=6$$ internal propagators.

3. Let us consider possible incoming 3-vertex 2-loop 1PI diagrams.

• Such a diagram would consists of 3 incoming vertices $$Y_i,Y_i,Y_i$$ and 2 outgoing vertices $$Y_o,Y_o$$.

• Each incoming external leg must be attached to a $$Y_i$$ vertex. Two external legs cannot be attached to the same $$Y_i$$ vertex since this would lead to a 1PR diagram. So each external leg is attached to a different $$Y_i$$.

• Two vertices of same kind cannot be directly connected with a propagator as it would violate arrow direction.

• The two internal lines of a $$Y_i$$ vertex cannot be attacted to the same $$Y_o$$ vertex since this would lead to a 1PR diagram. So each internal line of a $$Y_i$$ vertex is attacted to a different $$Y_o$$ vertex.

It is not hard to see that OP's 1st diagram is the only possibility.

4. Similarly, OP's 2nd diagram is the only possible outgoing 3-vertex 2-loop 1PI diagram.

• Thanks! Lingering question: the relationships between $L, I, V$ and $E$ are not what I have seen in this text. Only similar relation I have encountered is between the external (sources), propagators and vertices $E = 2P -3V$ in $\phi^3$ theory. Are these results taken from graph theory, and are these something typically discussed in this context? Also, would you define 3-trivalent? (edited) Dec 20, 2023 at 18:26
• I updated the answer. Dec 20, 2023 at 18:42