Using Feynman rules to calculate amplitude

[Higgsy]

Given a diagram, how is one supposed to apply the feynman rules to calculate the feynman amplitude?

 

[Avodyne]

This is explained in any QFT textbook. What book/article did you get your picture from?

 

[jtbell]

Those don't look like Feynman diagrams to me.

 

[Higgsy]

Srednicki. But these are not calculated in srednicki. To clarify, they are the vacuum feynman diagrams for $$\phi ^{4}$$ scalar theory

 

[vanhees71]

These are vacuum "bubble" diagrams, i.e., they contribute to the vacuum->vacuum transition amplitude in perturbation theory. To evaluate S-matrix elements you don't need them, because they cancel in the LSZ reduction formula via the correct normalization of the scattering amplitude.

To formally evaluate them you just use the Feynman rules and use any regularization procedure you like. Dimensional regularization is pretty convenient also in $\phi^4$ theory. Take the "8 diagram". The vertex stands for $-\mathrm{i} \lambda/4!$. Then you have 3 ways to connect the first leg at the vertex with another line and then only 1 to connect the remaining legs. Thus you have a symmetry factor $3$. The final dim-reg expression is.
$$\mathrm{i} V=\frac{\mathrm{i} \lambda \mu^{2 \epsilon}}{8} \int_{\mathbb{R}^d} \frac{\mathrm{d}^d l_1}{(2 \pi)^d} \int_{\mathbb{R}^d} \frac{\mathrm{d}^d l_2}{(2 \pi)^d} \frac{1}{(m^2-l_1^2)(m^2-l_2^2)},$$
where $d=4-2 \epsilon$ is the dimension of space-time.