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- TL;DR
- 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.
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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