I've been learning about Wick's theorem from a variety of sources when I came across this example from "Ultracold Quantum Fields" by Stoof et al. on page 148 (Here I have simplified the notation slightly)
$$\langle \phi^*_{\alpha}\phi^*_{\alpha'}\phi_{\alpha''}\phi_{\alpha'''}\rangle= \langle\phi^*_{\alpha}\phi_{\alpha'''}\rangle\langle\phi^*_{\alpha'}\phi_{\alpha''}\rangle \pm \langle\phi^*_{\alpha}\phi_{\alpha''}\rangle\langle\phi^*_{\alpha'}\phi_{\alpha'''}\rangle$$
where the top sign refers to bosons and the lower to fermions. Why is there no $(\alpha,\alpha')$ term? Why is $\langle\phi^*_{\alpha}\phi^*_{\alpha'}\rangle\langle\phi_{\alpha''}\phi_{\alpha'''}\rangle$ excluded if Wick's theorem considers "all possible products of time-ordered expectation values of two operators"?
My guess is it has something to do with the fact that each expectation value contains only one conjugate field, but why would that be the case?
