How do they measure quark masses? Quark charges?

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The measurement of quark masses, specifically the up (u) and down (d) quarks, is derived from the mass of hadrons, such as protons and neutrons, which are composed of these quarks. The majority of the proton's mass arises from the strong interactions between quarks rather than their individual masses. Researchers utilize the relationship between hadron masses and quark masses to compute the latter. For detailed methodologies, refer to the technical document available at PDG.

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Hello,

So, I know quarks are confined in baryons. In a proton, there are "3" quarks, but the sum of their masses is not the mass of the proton. This implies a major fraction of the proton mass comes from interactions. My question is, how then do they measure quark u and d masses? And simmilarily, their charges?
 
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It is possible to compute how the masses of quark-containing particles (such as the proton, neutron, and pion, collectively known as hadrons) depend on the quark masses. Then, the observed values of the hadron masses can be used to determine the quark masses.

For complete details, see

http://pdg.lbl.gov/2012/reviews/rpp2012-rev-quark-masses.pdf

This is a technical document, intended for experts, but I think it's pretty readable nonetheless, and gives a good picture of what physicists actually do to figure these things out. One key point is that, because quarks are confined, their "mass" does not have as clear and simple a definition as it does for unconfined particles such as the electron.
 
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