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

1. Aug 2, 2012

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.