Optical theorem and renormalised perturbation theory (c.f. Peskin 10.2)

[muppet]

Hi all,

On p.327 in my second edition of Peskin and Schroeder, I have an expression for the one loop correction to the 4-point amplitude of phi^4 theory:
$$i\mathcal{M}=-i\lambda - \frac{i \lambda^2}{32 \pi^2}\text{[Complicated integral]}$$

Mathematica can do the integral for me, and all that I'm interested in for the moment- I think- is the imaginary part, which happens to be -I*Pi; I therefore find that to this order,
$$\text{Im}(\mathcal{M})=\frac{\lambda^2}{32 \pi}$$

Examining the leading contribution to the optical theorem
$$\text{Im}(\mathcal{M}(t=0))=2E_{cm}p_{cm}\sigma_{TOTAL}$$
we should compare this to the cross-section $\sigma_{TOTAL}$ we get from the tree-level amplitude (eq. 4.100 in my Peskin):
$$\sigma_{TOTAL}=\frac{\lambda^2}{32 \pi s}$$

So it looks like the optical theorem is only satisfied if $2E_{cm}p_{cm}=s$, which is only true for massless particles. This disturbs me. Not only have we not assumed that our particles are massless, but the counterterms are singular in this limit; the only way I can find to make sense of this result is that it somehow corresponds to the check on the optical theorem performed in bare perturbation theory, with a divergent bare coupling $\lambda$, but having unitarity restored in a physically meaningless limit, with an assumption about a physical quantity that is not only extraneous but incorrect, doesn't make me feel any better.

Can someone please explain to me what I'm doing wrong here? Thanks.

 

[eljose79]

it is important to carefully examine and question our results and assumptions. In this case, it seems like there may be a mistake in the calculations or an error in the assumptions made. It is important to carefully check the equations and make sure all the necessary factors are included in the calculations. It may also be helpful to consult with colleagues or experts in the field to get a fresh perspective and potentially identify any errors or misunderstandings. Additionally, it could be useful to double check the results using different methods or approaches to see if they align with the expected outcomes. Overall, it is important to remain open-minded and willing to revise our understanding if new evidence or information comes to light.