- #1
thoughtgaze
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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?
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?