# Using Feynman rules to calculate amplitude

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1. Nov 24, 2015

### Higgsy

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

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2. Nov 24, 2015

### Avodyne

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

3. Nov 24, 2015

### Staff: Mentor

Those don't look like Feynman diagrams to me.

4. Nov 24, 2015

### Higgsy

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

5. Nov 29, 2015

### 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.