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

Question

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 \begin{equation} A:\mathrel{B^{n}}:~=~:\mathrel{AB^{n}}:+nA^{\bullet}B^{\bullet}:\mathrel{B^{n-1}}:\tag{1} \end{equation} 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: $$A_{1}\cdots A_{n}:\mathrel{B_{1}\cdots B_{m}}:~ ?\tag{2}$$

Answer 1

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 ${\cal R}$.

2. Then eqs. (1) and (2) become examples of a nested Wick's theorem discussed in my Phys.SE answer here.

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.