Cutkosky Cutting Rules: Understanding Rules & Where To Add i

[thoughtgaze]

OKay, so whenever I run into explanations on the cutting rules, most of the time I see the statement to replace$\frac{1}{p^2 - m^2 + i\epsilon} \rightarrow -2i\pi \delta(p^2 - m^2)$ for each propagator that has been cut

taking note that there is no factor of i in the numerator for $\frac{1}{p^2 - m^2 + i\epsilon}$

so for example, for $\phi^3$ theory we can have a loop amplitude given by

$iM(p^2) = \frac{(i\lambda)^2}{2} \int \frac{d^4k}{(2\pi)^4}\frac{i}{(k-p)^2 - m^2 + i\epsilon}\frac{i}{k^2 - m^2 + i\epsilon}$

or
$iM(p^2) = -\frac{(i\lambda)^2}{2} \int \frac{d^4k}{(2\pi)^4}\frac{1}{(k-p)^2 - m^2 + i\epsilon}\frac{1}{k^2 - m^2 + i\epsilon}$Making the cut through the diagram, and making the above defined replacement gives

$-\frac{(i\lambda)^2}{2} \int \frac{d^4k}{(2\pi)^4}[-2i\pi \delta((k-p)^2 - m^2)][-2i\pi \delta(k^2 - m^2)]$

upon simplifying we should have...

$-\frac{(\lambda)^2}{2} \int \frac{d^4k}{(2\pi)^2}[\delta((k-p)^2 - m^2)][\delta(k^2 - m^2)]$

which is off, by a minus sign, from the right answer...

I would get the right answer if I made the replacement

$\frac{i}{p^2 - m^2 + i\epsilon} \rightarrow -2i\pi \delta(p^2 - m^2)$
WITH the factor of i in the numerator

instead of $\frac{1}{p^2 - m^2 + i\epsilon} \rightarrow -2i\pi \delta(p^2 - m^2)$
WITHOUT the factor of i in the numerator

What am I doing wrong?

 

[UVCatastrophe]

The main thing I see you're doing wrong is evaluating Feynman diagrams.

 

[thoughtgaze]

Oh? Well how so?

 

[CAF123]

@thoughtgaze
I have been using Cutkosky's cutting rules extensively in my summer project but I still regard myself as an amateur in QFT so please take everything I am saying with a pinch of salt as they say :) The replacement of the off shell propagator terms with delta functions when we take a 'cut' is given by, as far as I am aware, $(k^2-m^2+i\epsilon)^{-1} \rightarrow 2 \pi \delta(k^2-m^2)$.

The factors of $i$ come into play depending on whether we are using the normal or complex conjugated version of the Feynman rules. Given a cut diagram, there is a convention that we label the vertices of the diagram black or white, black vertices follow standard Feynman rules and are accompanied by a factor of $i$ and white ones the complex conjugated version so come with a $-i$. In this set up, there is usually a corresponding theta function in the replacement above of the propagator terms so that energy flow from black to white vertices is counted positively but I am not sure if you are perhaps using another convention.

 

[thoughtgaze]

Interesting, I have not heard of this convention to treat the vertices differently by complex conjugation. Do you have any references for this particular notion?

Also, I have not seen $(k^2-m^2+i\epsilon)^{-1} \rightarrow 2 \pi \delta(k^2-m^2)$ in any reference I have, for example, peskin&schroeder eq. 7.56

 

[CAF123]

Apologies for delay in replying,

Interesting, I have not heard of this convention to treat the vertices differently by complex conjugation. Do you have any references for this particular notion?

See for example pages 9-10 of this paper http://arxiv.org/pdf/1401.3546v2.pdf.