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I understand that differential cross-sections such as $$\frac{\partial \sigma}{\partial \Omega}\left(\theta,\,\phi\right)$$ are useful observables. But if we only know $\sigma_{\text{total}}$, the total cross section for a process, is that something we can experimentally verify? Or are total cross sections only useful in ratio with other total cross sections, such as (for instance)$$ \frac{\sigma \left(e^+e^- ~ \to ~ \text{hadrons}\right)}{\sigma\left(e^+e^- ~ \to ~~~ \mu^+ \mu^- ~~\right)} \,?$$

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  • $\begingroup$ Are you asking whether total cross sections are "useful observables" (as in the title), or whether they are experimentally verifiable? If the former, I'd suggest using more precise wording in the title. $\endgroup$ – David Z May 14 '18 at 1:48
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Merriam-Webster's definition of "useful":

Definition of USEFUL

  1. capable of being put to use; especially : serviceable for an end or purpose $\bullet$ useful tools

  2. of a valuable or productive kind $\bullet$ do something useful with your life

For "cross section":

  1. a measure of the probability of an encounter between particles such as will result in a specified effect (such as scattering or capture)

So a total crossection will answer on how probable an interaction between two specific particles is. The size will define whether the interaction is weak, strong or electromagnetic, so even if only the total cross section is known or can be calculated there is useful information for subsequent studies.

Edit after edit of question, defining usefull as "experimentally verifiable".

Yes, they are. Look at this table in the particle data group listing total crossection for particles scattering off each other,as an example, proton proton scattering:

ppscatterin

As the definition of crossection says, it is proportional to the probability of scattering off each other.

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  • $\begingroup$ Thanks. I've edited my title to include "experimentally verifiable." So are total cross-sections experimentally verifiable? Or only the ratio of different cross-sections? $\endgroup$ – Dwagg May 14 '18 at 21:15
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Yes, they are experimentally verifiable.

If you have a beam of one of the particle species, and a thin plate of thickness $dx$ and number density $n$ of the other, and you're looking at a process of total cross-section $\sigma$, then the probability the particle interacts is given by:

$$ dp=n\sigma\,dx$$

or, in other words, the characteristic length scale of the interaction is:

$$ \lambda=n\sigma $$

Many experiments these days collide two beams rather than a beam and a fixed target, but using relativity you can relate the two- essentially by considering the interaction from the reference frame in which one beam is fixed.

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