How one can use Wick's theorem for the product $A:\mathrel{B^{n}}:$?

I try to use Wick's theorem in the case that some products we deal with are already normal ordered.

My guess is that it could be something like $$$$A:\mathrel{B^{n}}:~=~:\mathrel{AB^{n}}:+nA^{\bullet}B^{\bullet}:\mathrel{B^{n-1}}:\tag{1}$$$$ I tried to prove that by induction but I failed, maybe the formula is similar and I am somehow close, or maybe my intuition totally fails. How one could approach such a problem?

Also, how would that Wick's expansion look in the general case: $$A_{1}\cdots A_{n}:\mathrel{B_{1}\cdots B_{m}}:~ ?\tag{2}$$

• Which context? Which references? Which pages? Nov 26, 2022 at 12:23
• I don't know if the context is important I try to calculate OPE of the current and vertex operator for the free boson conformal field theory and this is the problem I met by the way. But I think the problem is interesting on its own. Nov 26, 2022 at 12:30

1. There is usually a second implicitly written operator ordering besides the normal order. [This plays a role in e.g. eq. (2).] E.g. in the context of 2D conformal field theory, there is typically an implicitly written radial ordering $${\cal R}$$.
3. Eq. (1) is correct, because the only possible terms are a term with no contraction and $$n$$ terms with a single $$AB$$ contraction.
4. Eq. (2) becomes a sum of all possible $$AA$$ and $$AB$$ (but not $$BB$$) contractions.