Exploring Solutions to $\phi$ and $\ket{\overrightarrow{P}}$

[Diracobama2181]

TL;DR Summary

Currently needing insight on re normalization for $$\bra{ \overrightarrow{P'}}\phi^4\ket{ \overrightarrow{P}}$$.

I already know this quantity diverges, however I was wondering where to go from there. Any resource would be appreciated. Thank you.
Useful Information:
$$\phi=\int\frac{d^3k}{2\omega_k (2\pi)^3}(\hat{a}(\overrightarrow{k})e^{-ikx}+\hat{a}(\overrightarrow{k})^{\dagger}e^{ikx}))$$
$$\ket{\overrightarrow{P}}=\hat{a}(\overrightarrow{k})^{\dagger}\ket{0}$$

 

[azntoon]

$$\bra{\overrightarrow{P}}\phi\ket{0}=\frac{1}{2\omega_k (2\pi)^3}(\hat{a}(\overrightarrow{k})e^{-ikx}+\hat{a}(\overrightarrow{k})^{\dagger}e^{ikx})$$To show that this quantity diverges, you will need to analyze the behavior of the integrand as $k \rightarrow \infty$. To do this, consider the limit of the integrand as $k \rightarrow \infty$:$$\lim_{k\rightarrow\infty}\frac{1}{2\omega_k (2\pi)^3}(\hat{a}(\overrightarrow{k})e^{-ikx}+\hat{a}(\overrightarrow{k})^{\dagger}e^{ikx})$$Since $\omega_k \propto k$, the denominator goes to infinity faster than the numerator as $k \rightarrow \infty$, so the integrand goes to 0. Therefore, the integral diverges as $k \rightarrow \infty$.