Decay and scattering terms in a field theory Lagrangian

Consider two genetic terms in a generic Poincare invariant quantum field theory:

1. A trilinear term of the form $\phi_1\phi_2\phi_3$, and

2. a quartic term of the form $\phi_1\phi_2\phi_3\phi_4$

where $\phi_i'$s can (i) all be different, (i) some can be same and some are different or (iii) all may be different. Let us also not declare whether $\phi_i$'s are all scalar, vector or fermionic.

Is it correct to say that the first term always contributes only to decays and the second term always contributes only to scattering?

• You have to be careful about whether you’re asking about the three(four)-point function in the free or fully interacting theory. For instance, the four point function of an interacting theory can be written as an expansion (a la Feynman), some of whose components are products of three-point functions (i.e. s-channel diagrams). Jun 14 '18 at 14:29

1 Answer

It is not correct.

A 3 field operator contributes to scattering amplitude (example: $\bar\psi \gamma^\mu \psi A_\mu$ in QED contributes to the electron scattering) and a 4 field operator can contribute to decays (example for three-body decays like $\mu^-\to \nu_\mu e^- \bar\nu_e$ given by a 4 fermion operator in Fermi theory)