Why we have to sum in all final states of hadrons?

Question

Correct if I am wrong. In deep inelastic scattering have to sum in all final sates hadrons because we do not want to detect the hadrons. All we want to detect is the electron. Am I right?

Answer 1

Yes or no, it depends what you precisely understand by this short sentence.

The word "inelastic" doesn't really mean that we don't "want" to detect something. Physics doesn't say what we should "want".

"Deep" means that the quarks (and perhaps the electron) penetrate very close to each other and the scattering studies the behavior of the matter at distances much shorter than the size of the nuclei.

And "inelastic" means that the final state contains particles that are different from the list of particles in the initial state – usually many more particles (many new hadrons). Those are produced when the partons are trying to escape each other etc.

We don't "have" to sum over hadrons in the final states in general. But it is "useful" to some over the final states with different hadrons – i.e. to calculate the "inclusive cross sections" etc. – because the detailed decomposition of the energy among the hadrons and their precise number depends on the process "when the partons are already trying to escape and they are far from each other", not on the "deep" processes when the partons really interact with themselves (and perhaps with the extra electron).

So to see that the reactions are due to the "deep" interactions between the partons (and the electron), it's "useful" to sum the cross section over many final states. These sums are more directly related to the underlying "deep" reaction between the elementary particles.