Complex scalar field propagator

[ofer]

i am trying to understand how to express contractions of field operators via propagators.
we are talking about an interacting theory of 2 complex scalar fields,
lets call them ψ1 and ψ2.

the interaction term is: Lint=λ(ψ2)^3(ψ1)

i have found the free propagator defined as:
Df-i(x-y)=<0|T(ψi(x) ψi*(y))|0> i=1,2.

what i am strugling with is that when considering the free parts
this propagator is 0 for <0|T(ψi ψi)|0> (no complex conjugate=no anti-particle).
and also it is 0 for <0|T(ψi ψj)|0> ,i≠j.
meaning that a field can only contract with it's conjugate counter part in the free theory.

so when i try to calculate an interaction corelator in 1st order of different forms i get 0, for instantce:

<Ω|T(ψ1 ψ2*)|Ω>
this term turns out 0 because there is no one to contract with all the none-conjugate fields
in the interaction hamiltonian.
am i miss understanding things here,
or is the first corelator to be none-zero in this picture of the form:
<Ω|T(ψ2* ψ2* ψ2* ψ1* )|Ω> ??

help will be much apriciated.
thank you.

 

[NateD]

Yes, you are correct that the free propagator is only non-zero when contracting a field with its conjugate counter part. In order to calculate an interaction correlator in first order, you need to consider a term of the form <Ω|T(ψ2* ψ2* ψ2* ψ1* )|Ω>. This term corresponds to a three-point vertex involving three fields of type ψ2 and one field of type ψ1.