Difference between 2-point and 4-point function in QFT

[Higgsy]

As I understand it, the 2-point fnuction is for 1 particle incoming, 1 particle outgoing. The 4-point function is for 2 particles incoming, 2 particles outgoing. Is this correct? So an N-point function describes N/2 incoming particles and N/2 outgoing particles?

Thanks!

 

[king vitamin]

The N-point function corresponds to N "external" particles, any of which can be incoming or outgoing. So the four-point function could be one incoming and three outgoing, or three incoming and one outgoing. Of course, if you're eventually interested in all external particles being on-shell to calculate amplitudes (which I assume you do here since you're trying to speak of particles incoming/outgoing), then you should conserve energy/momentum between initial and final states, which constrains certain configurations - e.g. you can't have two incoming and zero outgoing.

Sometimes we're interested in the N-point function off-shell, in which case it's not that meaningful to think of particles as incoming/outgoing.

 

[Higgsy]

Oh that makes sense. When would we be interested in the off-shell N-point function?

 

[king vitamin]

It is often the easiest way to renormalize the theory. You renormalize by setting some N-point functions evaluated at a specific value of "external" momenta to a renormalized coupling, and then replace all bare couplings with renormalized couplings. This is usually simplest with an off-shell choice, though you could choose them on-shell.

It's also useful because you can put your N-point function as a sub-diagram of a larger diagram, where your original N-point function is now integrated over some of its momentum values. There are also some advanced applications which use the analytic structure of N-point functions as a function of the external momenta.