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Understanding the $Z$ production cross section in $e^{+}e^{-}rightarrow Zrightarrow mu^{+}mu^{-}$

Physics Asked by user248824 on April 25, 2021

In "Modern Particle Physics", Mark Thomson derives the total cross section for $sigmaleft(e^{+}e^{-}rightarrow Z^{0}rightarrowmu^{+}mu^{-}right)$, cf. subchapter 16.1.1 on page 431 ff. He assumes that $sqrt{s}sim m_{Z}$, s.t. the QED Feynman diagram is in first approximation negligible.

Now, in the first Eq. on page 431 that does not have a number, i.e. $$mathcal M_{fi} = – frac{g_{Z}^{2}g_{munu}}{left( s-m_{Z}^{2} + im_{Z}Gamma_{Z}right)}dots,$$ I do not understand the term that Mark Thomson wrote down, which is supposed to be the propagagtor, I assume. Usually, we would write the $W^{pm}/Z^{0}$ propagator as

$$frac{-ileft( g_{munu} – q_{mu}q_{nu}/m_{Z}^{2}right)}{left(q^{2}-m_{Z}^{2}+im_{Z}Gamma_{Z}right)}.$$ In an $s$-channel Feynman diagram, $s = q^{2}$, but what happened to the term $q_{mu}q_{nu}/m_{Z}^{2}$? After all, in the following derivation, M. Thomson does not ignore the term $s-m_{Z}^{2}$ in the denominator..

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