# Example of truncated expectation which gives a disconnected diagram. Where am I wrong?

My notes introduce the truncated expectation in the following way: given $$S_0$$ a quadratic form, consider a generating function $$e^{W(J)} = \int \prod_x d\varphi_x e^{-S_0+ JO} = \int P(d\varphi) e^{JO}.$$ Let define the truncated expectation of $$O$$ as $$E^T(O;n) = \frac{\partial^n}{\partial J^n} W(J) \bigg|_{J=0}= \frac{\partial^n}{\partial J^n} \log \int P(d\varphi) e^{JO} \bigg|_{J=0}$$ And so for example $$E^T(O;1)=\frac{\partial}{\partial J} \log \int P(d\varphi) e^{JO} \bigg|_{J=0} =\frac{\int P(d\varphi) O}{\int P(d\varphi) }=\langle O\rangle$$.

Now, using the Wick rule, I calculate $$E^T(\varphi_1 \varphi_2 \varphi_3 \varphi_4;1)=\langle \varphi_1 \varphi_2 \varphi_3 \varphi_4\rangle = \langle \varphi_1 \varphi_2\rangle \langle \varphi_3 \varphi_4\rangle +\langle \varphi_1 \varphi_4\rangle \langle \varphi_2 \varphi_3\rangle +\langle \varphi_1 \varphi_3\rangle \langle \varphi_2 \varphi_4\rangle.$$

The rules for Feynman diagrams are: construct a Feynman diagram drawing a segment to each field and then join the different segments if there is a contraction between the field, that is if I have the expectation between the field. Then, following this rule I have that $$\langle \varphi_1 \varphi_2 \varphi_3 \varphi_4\rangle$$ is composed by disconnected diagram, because, for example for the first term that comes from the Wick rule, $$\langle \varphi_1 \varphi_2\rangle \langle \varphi_3 \varphi_4\rangle$$ I have one segment that connects the vertices $$1-2$$ and one segment for $$3-4$$.

Where am I wrong?

• I don't think you're wrong. The $O$ vertex should be treated as a connected object and the three terms you wrote are connected vacuum graphs made of a single quartic vertex attached to two tadpoles. Jan 20, 2021 at 15:44
• Ok thank you, even if I don't really understand what you mean for quartic vertex and tadpoles... I am mathematician and maybe I am not in feeling with these argument... Jan 20, 2021 at 16:53
• See my answer below. Jan 20, 2021 at 20:50
• Are your notes available? Jan 20, 2021 at 21:48

$$Z=\int P(d\varphi)e^{JO}$$ is the partition function of $$\varphi^4$$ type model. You can rigorously prove in the sense of power series in $$\mathbb{C}[[J]]$$ that $$Z$$ is a sum over all graphs with degree 4 vertices (this is math i.e. graph theory terminology) corresponding to the $$O$$'s, connected or not. When you take the log $$W$$ you get a similar sum where you only keep connected graphs. In the example above, graphs look like the number 8. You have one central vertex with 4 half edges emanating from it, with two loops (math terminology), i.e., tadpoles (physics terminology).