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## Summary:

- I'm trying to work out theamplitudes for the various Feynman diagrams from the Dyson series expansion of the S-matrix. Sometimes I'm left with constant integrals over all space and other anomalies.

## Main Question or Discussion Point

This is QFT for the gifted amateur, chapter 19, which is generating the various Feynman diagrams and rules. Some calculations are given but I encounter various problems when trying to work them all out.

The starting point is that we want to calculate:

$$\langle q| \hat S | p \rangle = (2\pi)^3 \sqrt{2E_q}\sqrt{2E_p}\langle 0|\hat a_{\vec q} \hat S a_{\vec p}^{\dagger}| 0 \rangle$$

Where:

$$\hat S = T[1 + (\frac{-i\lambda}{4!})\int d^4x \ \hat \phi(x)^4 + (\frac{-i\lambda}{4!})^2(\frac 1 {2!})\int d^4xd^4y \ \hat \phi(x)^4 \hat \phi(y)^4 + \dots]$$

First, if we take the term in ##\lambda##, we have the term:

$$\langle 0|T[\hat a_{\vec q} \hat \phi(x)^4 a_{\vec p}^{\dagger}]| 0 \rangle$$

And, using Wick's theorem I get two non-zero terms coming out of this. The first is covered in the book:

$$12\langle 0|\ [\hat a_{\vec q} \hat \phi(x)] \ [\hat \phi(x) \hat \phi(x)] \ [\hat \phi(x) a_{\vec p}^{\dagger}]| 0 \rangle$$

Where I've used ##[ \ \ ]## to indicate a contraction.

But, I was also looking at the term:

$$3\langle 0|\ [\hat a_{\vec q} a_{\vec p}^{\dagger} ] \ [\hat \phi(x) \hat \phi(x)] \ [\hat \phi(x) \hat \phi(x)]| 0 \rangle$$

Is this term valid? In any case, it leads to an infinity:

$$\delta^4(q-p) \int d^4x \bigg ( \int \frac{d^4k}{(2\pi)^4} \big ( \frac{i}{k^2 - m^2 + i\epsilon} \big ) \bigg )^2$$

A similar thing happens for the ##\lambda^2## term. I have an extra term in the integral that does not correspond to any diagram:

$$[\hat a_{\vec q} a_{\vec p}^{\dagger} ] \ [\hat \phi(x) \hat \phi(y)]^4$$

I can see why the diagram would not make sense, but I can't see why that term vanishes from the integral.

Finally, similar terms crop up in trying to calculate the integrals for other diagrams. I get the right answer except I still have an integral of the form ##\int d^4 x## in front of everything. Mathematically, it all comes back to the same issue, as above.

Any help would be very welcome.

Thanks.

The starting point is that we want to calculate:

$$\langle q| \hat S | p \rangle = (2\pi)^3 \sqrt{2E_q}\sqrt{2E_p}\langle 0|\hat a_{\vec q} \hat S a_{\vec p}^{\dagger}| 0 \rangle$$

Where:

$$\hat S = T[1 + (\frac{-i\lambda}{4!})\int d^4x \ \hat \phi(x)^4 + (\frac{-i\lambda}{4!})^2(\frac 1 {2!})\int d^4xd^4y \ \hat \phi(x)^4 \hat \phi(y)^4 + \dots]$$

First, if we take the term in ##\lambda##, we have the term:

$$\langle 0|T[\hat a_{\vec q} \hat \phi(x)^4 a_{\vec p}^{\dagger}]| 0 \rangle$$

And, using Wick's theorem I get two non-zero terms coming out of this. The first is covered in the book:

$$12\langle 0|\ [\hat a_{\vec q} \hat \phi(x)] \ [\hat \phi(x) \hat \phi(x)] \ [\hat \phi(x) a_{\vec p}^{\dagger}]| 0 \rangle$$

Where I've used ##[ \ \ ]## to indicate a contraction.

But, I was also looking at the term:

$$3\langle 0|\ [\hat a_{\vec q} a_{\vec p}^{\dagger} ] \ [\hat \phi(x) \hat \phi(x)] \ [\hat \phi(x) \hat \phi(x)]| 0 \rangle$$

Is this term valid? In any case, it leads to an infinity:

$$\delta^4(q-p) \int d^4x \bigg ( \int \frac{d^4k}{(2\pi)^4} \big ( \frac{i}{k^2 - m^2 + i\epsilon} \big ) \bigg )^2$$

A similar thing happens for the ##\lambda^2## term. I have an extra term in the integral that does not correspond to any diagram:

$$[\hat a_{\vec q} a_{\vec p}^{\dagger} ] \ [\hat \phi(x) \hat \phi(y)]^4$$

I can see why the diagram would not make sense, but I can't see why that term vanishes from the integral.

Finally, similar terms crop up in trying to calculate the integrals for other diagrams. I get the right answer except I still have an integral of the form ##\int d^4 x## in front of everything. Mathematically, it all comes back to the same issue, as above.

Any help would be very welcome.

Thanks.

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