Scattering Problem - Unstable particles and a version of the optical theorem

[arestes]

Homework Statement

Problem: Show that unitarity (of the S-Matrix which has implications to the amplitud of scattering of 2-2 bodies scattering of unstable particles) fixes the numerator of the Partial wave amplitude near the $$\Delta$$-pole,
$$q M_1 = \frac{-M_{\Delta}\Gamma_{\Delta}}{s-M^2_{\Delta}+iM_{\Delta}\Gamma_{\Delta}}$$
where we are working in the center of mass reference and q is the magnitude of the momentum which is common for a 2-2 scattering (of unstable particles) and the M_1 on the LHS is part of the partial wave expansion of the amplitude (see below) and M_Delta is the mass of the resonance on the RHS

Homework Equations

Optical theorem. Partial wave expansion and everything in the first sections of chapter 11 to 11.5 of these lecture notes: http://www.nat.vu.nl/~mulders/QFT-0E.pdf
especially the partial wave expansion in 11.48 there

M$$(x, \theta) = -8\pi \sum_l (2l +1)M_l(s) P_l(cos\theta)$$

where $$P_l$$ are the legendre polynomials and I assume that the functions M_l are defined through this relation

Actually, I'm trying to solve exercise 11.2 of those lecture notes which looks like an especial case of equation 11.59

The Attempt at a Solution

Seriously, even though the lecture notes says it is straightforward (using the unitarity condition mentioned there) I cannot see how to proceed. I looked at eq. 11.49 since it is one version of the unitarity condition. then I have to somehow relate it to the width of unstable particles. Then they say that we need a modified propagator that includes the parameter $$\Gamma$$ so that we can get a time evolution which is decaying exponentially as it should. Then I swear I cannot relate it to the definition of the width.

 

[Avodyne]

The lecture notes won't download for me, but I think I can guess what's required. Unitarity says that $q M_1$ takes the form

$$q M_1 = {e^{2i\delta_1}-1\over 2i}$$

where $\delta_1$ is real. So, first show that $e^{2i\delta_1}$ can be written in the form

[tex]e^{2i\delta_1}={a-ib\over a+ib}[/itex]<br /> <br /> where $a$ and $b$ are real. Then work out $q M_1$ in terms of $a$ and $b$, and compare with the expression you were given.[/tex]

 

[arestes]

Thanks for the hint. I uploaded the pdf (hopefully it will be approved soon) here since for some reason their server was down.Their server must come online anytime soon because it's from a university. I can see that using the form of the exponential of the phase shift would indeed help for the problem, I don't see it would help for the general case in equation 11.59

$$q (M_l)_{ij}(s) = \frac{-M\sqrt{\Gamma_i \Gamma_j}}{s-M^2+iM\Gamma}$$
where $$s=(E_1 +E_2)^2 = E^2$$ and $$\Gamma = \sum_n \Gamma_n$$

because of the square roots there.

I'm trying to show that the exponential of the phase shift is of that form but that also seems to appear out of nowhere (I'm following those lecture notes) so I'll be looking at other books about scattering.

Could you please check those lecture notes?
thanks

 

[Avodyne]

Sorry, the notes do not define what the subscripts i and j mean in this context, and I'm not able to guess.

 

[arestes]

Yeah well... the subscripts i and j are inherited from the amplitude after we make the Partial wave expansion on equation 11.48, (they relate states). Actually I need to show only the version in problem 11.2 which has only one Gamma and the mass of the resonance M_Delta... It's so frustrating...