# Cross sections

nrqed
Homework Helper
Gold Member
thanks for the response! yes, I realized I miss-used the expression cross section and replaced it with amplitude. I thought the two processes were related due to their similarity when drawing Feynman diagrams but wasnt sure, thanks for clarifying.

Just for further understanding... can we say that the cross section is proportional to $$G^2 E_{cm}^2$$ since $$\bar{M^2}= 64G^2 (p \cdot k')(p' \cdot k)$$? If so, then we can say that $$\frac{d \sigma}{dE_{cm}^2} =const$$ and then could we say that it is safe to assume that the total cross section approaches infinity as $$E_{cm}$$ increases?
I don't have time to check now but how did you get that the cross section is proportional to G^2 E_{cm}^2 (it might be obvious but I don't see). Did you include the phase space factors (which tend to suppress the cross sections)

I don't have time to check now but how did you get that the cross section is proportional to G^2 E_{cm}^2 (it might be obvious but I don't see). Did you include the phase space factors (which tend to suppress the cross sections)
I thought the phase space factor comes into play when calculating the differential decay rate:

$$d \Gamma = \frac{1}{2M} |M|^2 dQ$$

where as the differential cross section is given by:

$$\frac{d \sigma}{d \Omega} = \frac{1}{64 \pi^2 E_{cm}^2 } \frac{p_f}{p_i} |M|^2$$

nrqed
Homework Helper
Gold Member
I thought the phase space factor comes into play when calculating the differential decay rate:

$$d \Gamma = \frac{1}{2M} |M|^2 dQ$$

where as the differential cross section is given by:

$$\frac{d \sigma}{d \Omega} = \frac{1}{64 \pi^2 E_{cm}^2 } \frac{p_f}{p_i} |M|^2$$
That's what I meant: there is a suppressing 1/E_cm^2 factor from phase space (phase space is everything that multiplies the square of the amplitude).
What is your M^2 expressed in terms of CM quantities?

$$\frac{d \sigma}{d \Omega} = \frac{1}{64 \pi^2 s } \frac{p_f}{p_i} |M|^2$$

$$|M|^2= 64G^2 (p \cdot k')(p' \cdot k)$$

$$u=-2 k \cdot p' = -2 p \cdot k'$$

Seems to me that the cross section is proportional to $$G^2 \frac{u^2}{s}$$

I had thought $$s=p \cdot k'=p' \cdot k$$

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by the way, how does the calculation for the amplitude look?

also, would the calculation for the amplitude $$|M|^2$$ look the same for IVB theory except there would be a $$\frac{g^2}{M_W^2 - q^2}$$ term in front of the amplitude? I'm not sure how this "changes" anything compared to the 4 fermi result since they are only constants out in front.