Calculating Terms in $\bra{P'}\phi^4\ket{P}$

[Diracobama2181]

Homework Statement

I am trying to show that $\bra{P'}\phi^4 \ket{P}$is divergent, where $\ket{P}$ is a free spin 0 boson.

Relevant Equations

$\ket{P}=a^{\dagger}(k)\ket{0}$
$\phi=\int \frac{d^3k(\hat{a}e^{-ikx}+\hat{a}^{\dagger}e^{ikx})}{2\omega_{k}(2\pi)^3}$
$\omega=\sqrt{k^2+m^2}$

From this, I find
$$\bra{P'} \phi^4 \ket{P} = \int \frac {d^3 k_1 d^3 k_2 d^3 k_3 d^3 k_4} {16 \omega_{k_1}\omega_{k_2}\omega_{k_3}\omega_{k_4} (2\pi)^{12} }\bra{0} a_{P'}(a_{k_1} a_{k_2} a_{k_3}a_{k_4}e^{i(k_1+k_2+k_3+k_4)x}+...)a^{\dagger}_{P}\ket{0}$$ (a total of 16 different terms)
Right now, I am trying to figure out how to calculate terms like
$\bra{0}a_{P'}a_{k_1}a^{\dagger}_{k_2}a_{k_3}a^{\dagger}_{k_4}a^{\dagger}_{P}\ket{0}$. Any examples would be greatly appreciated.

 

[HomogenousCow]

Commute an annihilation or creation operator to the right or left of the product to render the expectation value zero, repeat this with the new terms generated by the commutator. Although to show that this quantity is divergent you don't have to do this, just look at how the integrand scales with $k$. After doing all the commutators you will find that the integrand scales as $\sim k^{-2}$, which means that even with one integration measure $d^3 k$ the integral will still be divergent since $\int \frac{d^3 k}{k^2} \sim \Lambda$.