# $\phi\phi\to\phi\phi$ scattering in $\phi^4$-theory and Feynman diagrams at the tree-level

What are the tree-level Feynman diagrams for $\phi\phi\to\phi\phi$ scattering in $\phi^4$-theory? I know that there are four diagrams at tree-level (zero loop). But each diagram is considered only once. Why is that? There are one-loop diagrams in which external leg receives one-loop correction. I agree that they are disconnected diagrams but if we remove the disconnected part wouldn't that contribute to a tree level process? Sorry if the question is unclear because I could not attach relevant diagrams.

• There is only one diagram at tree level. Third one in this image – FrodCube May 29 '18 at 10:31
• @FrodCube But why? Why is this diagram taken only once? If we consider you diagrams with one-loop radiative correction at one of the external legs, and chop off the one-loop as disconnected portion I reproduce your diagram. So why consider your diagram only one? – mithusengupta123 May 29 '18 at 11:38

## 1 Answer

Consideration of correlation functions in an interacting QFT (i.e one in which you wish to describe a scattering event) means you can neglect disconnected diagrams order by order in the perturbative expansion. Formally, this amounts to a generating functional with connected diagrams 'exponentiated' but that is a detail. So, there is only one diagram contributing at tree level for the given process, namely the four point vertex. The three other diagrams are non-interactive and just represent propagation of fields between two source points.

A systematic way to obtain the contributing diagrams is to label your asymptotic in and out fields as originating from source points $x_1,x_2,x_3$ and $x_4$. Then you make all possible attachments of these lines resulting in inequivalent diagrams at a particular desired loop/coupling order. At tree level, for example, you would have four lines originating from the $x_i$ and all meeting at a vertex.

Therefore, at one loop, you will indeed have four diagrams corresponding to radiative corrections you can make to the four external legs. These are not $1\text{PI}$ but by amputating them in the manner in which you describe will separate a source point from the rest of the connected diagram so it will not yield the tree level result. Instead, these external leg corrections are usually dealt with in a consistent programme of renormalisation, so typically one does not include them.

There are, however, three other non trivial one loop diagrams contributing to $\phi \phi \rightarrow \phi \phi$.