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Hints: The starting point is the 2-point relation $$T(\phi(x)\phi(y)) ~-~:\phi(x)\phi(y): ~=~ C(x,y)~{\bf 1}, \qquad C(x,y)~\equiv~\langle 0 | T(\phi(x)\phi(y))|0\rangle,\tag{1} $$ cf. this Phys.SE post. The relevant Wick's theorem is a nested Wick's theorem $$ T(:\phi(x)^n::\phi(y)^m:)~=~\exp\left( ...


This is explained in my Phys.SE answer here. In a nutshell, under appropriate assumptions, one may show that $$ \text{Contract}[\phi(x)\phi(y)]~=~D_F(x-y) ~{\bf 1},$$ where ${\bf 1}$ is the identity operator.

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