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arestes
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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 [tex]\Delta[/tex]-pole,
[tex]q M_1 = \frac{-M_{\Delta}\Gamma_{\Delta}}{s-M^2_{\Delta}+iM_{\Delta}\Gamma_{\Delta}}[/tex]
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[tex](x, \theta) = -8\pi \sum_l (2l +1)M_l(s) P_l(cos\theta)[/tex]
where [tex]P_l[/tex] 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 [tex] \Gamma [/tex] 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.
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