# Feynman diagrams for $\varphi^3 + \varphi^4$

Given $$L= -\frac{1}{2}\partial^\mu \varphi \partial_\mu \varphi - \frac{1}{2}m^2\varphi^2 + \frac{\lambda_3}{3!}\varphi^3 - \frac{\lambda_4}{4!}\varphi^4$$

I am trying to find the Feynman diagrams. I have understood how to do it for $$\varphi^3$$ and $$\varphi^4$$ separately (that is, when one of the two $$\lambda$$ is zero), but I have troubles with the combined case. In total there should be 13 additional, connected, diagrams, as compared to the two separate cases (excluding tadpoles), but I only manage to find 7, making me suspect that I have misunderstood something basic.

Anyway, I have the two following conditions: $$1\leq E\leq4 ; 0\leq V_1+V_2 \leq 4$$ where $$V_1$$ is the number of three-leg vertices, and $$V_2$$ the number of four-leg vertices.

I deduce the following condition for the number of external sources: $$E = 2P - 3V_1 - 4V_2$$

For the four different possible number of external sources I then have:

$$E=1;V_1=1;V_2=1;P=4$$ - one diagram, which however is a tadpole

$$E=2;V_1=2;V_2=1;P=6$$ - two diagrams

$$E=3;V_1=1;V_2=1;P=5$$ - two diagrams

$$E=4;V_1=2;V_2=1;P=7$$ - two diagrams

However, there should be a total of 13 diagrams to find. Can someone see something wrong in what I have written above, or provide some hints on how to find the remaining diagrams?

(note: I already have all the diagrams where one of either $$V_1$$ or $$V_2$$ is zero)

• I don't understand your question. I can draw an infinite number of connected diagrams for this theory. Feb 24 '20 at 16:44
• I am still new to this, so I am sorry if I have stated something in a confusing or incorrect way. It is an exercise problem, and it states explicitly "Draw all diagrams that are connected and without tadpoles, for 0<=V1+2V2<=4 and 1<=E<=4", and mentions that there are 13 in total (with V1, V2>0). I deduce the number of propagators from the condition E=2P-3V1-4V2, which then effectively limits the number of diagrams one could construct, no? It is similar to problem 9.2 here: hep.ucsb.edu/people/cag/qft/QFT_Notes_9.pdf
– a20
Feb 24 '20 at 18:23
• I see now. Your question was not very clear. Is seemed like you were imposing these conditions yourself for some reason unbeknownst to me. Feb 24 '20 at 19:01
• I am not sure your formula $E=2P - 3V_1-4V_2$ is correct. For the one loop correction to the two point function of the $\varphi^3$ theory I find $2P - 3V_1 = 4-6 =-2$. Also do you need all connected diagrams or only the one particle irreducible? Feb 24 '20 at 19:14
• I have not heard about "one particle irreducible" yet in the book I am following (Srednicki). The formula is the same as Srednicki derives in chapter 9, if you put $V_2$ to zero, but there he only considers an interaction lagrangian with $\varphi^3$.
– a20
Feb 24 '20 at 19:45

The diagrams you are searching for have coefficients $$\hat{g}^{V_1}\hat{\lambda}^{V_2}$$, see also (2.1), and in particular the thirteen tadpole-free diagrams are: The best way I know of of producing complete sets of diagrams and their coefficients at any given order (while being confident nothing is missing and that all factors are correct) is to compute the path integral directly.