Wicks Contraction without contractions at the same spacetime point

[QFT1995]

Homework Statement

Compute the following

$$\mathcal{T} \langle {0}| \prod_i^Ne^{\imath \beta_i \phi(x_i)} |0 \rangle$$ without contractions at the same point

Relevant Equations

Wick's theorem

If I'm computing

$$\mathcal{T} \langle 0 | \prod_i^Ne^{\imath \beta_i \phi(x_i)} | 0\rangle $$
where the contractions at the same spacetime point are ignored, can I simply insert a complete set of states (product now outside of expression) between each exponential to give

$$\mathcal{T} \prod_i^N \langle 0 |e^{\imath \beta_i \phi(x_i)} | 0\rangle$$
and then the only terms not contributing to contractions at the same spacetime point is the term 1 in the exponential which gives

$$\mathcal{T} \prod_i^N\langle 0 |1 | 0\rangle = 1$$
since
$$ \langle 0 | 0\rangle=1$$
or is this wrong?

 

[Asim Riaz]

Yes, this is correct. The only terms that would contribute to contraction at the same spacetime points are the ones where $\beta_i$ and $\beta_j$ differ by an integer, and since the $\beta_i$'s are chosen independent from each other, these terms do not exist. Therefore, the only contribution to the expression is 1.