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This post is a sequel of: Where does the majority of the mass of the usual matter come from?

The following answer of @hft:

Your question asks why the "current quark masses" [see http://pdg.lbl.gov/2011/download/rpp-2010-booklet.pdf at page 21] of the quarks that make up a proton don't add up to the mass of the proton. The problem is that, for the light quarks, the "current quark masses" are very different from the "constituent quark masses" [see wikipedia]. "Constituent quark masses" basically means: What you would expect from a naive model where the proton is uud and the neutron is udd. Whereas "current quark masses" means: What you would expect in a mass-independent subtraction scheme such as $\bar {MS}$ at a scale of $\mu\sim 2GeV$, which basically is a fancy way of saying "it's complicated".

The reason these are different is because the strong nuclear force (AKA Quantum Chromodynamics) is "asymptotically free", i.e., it is only easy to understand at very high energies in terms of single particles. At low energies, such as in protons, the "bag of quarks" that make up the proton can not be thought of as single particles because there is a large and difficult to determine interaction among the quarks and gluons inside the proton. This interaction energy can be though of as "making up the mass difference" although it would be just as good to say that a proton is NOT make up of just three quarks. Rather it is made up up many many quark-anti-quark pairs and gluons on top of which sit three "extra" quarks: u, u, and d

leads us to the following question:

Question: Is the Higgs field responsible of only 1% of the proton mass (through these three "extra" quarks)?

The following comment of @hft:

You're on to something there, but it's hard to say exactly. The Higgs can be thought of as responsible for generating all the quark mass terms in the QCD Lagrangian, but it doesn't generate any mass associated with interaction energy by E/c^2. Although I think it's a little more complicated than this.

calls for a proper answer to the question above.

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Simply put, yes.

Even if we account for the mass of the three quarks that make up a proton, it would only make up (at most) 1% of the mass of what we measure for an entire proton. This is because of the fluctuations in the gluon field.

Now, I won't go into detail, but basically this gluon field is ever permeating, and is only absent in the area of quark pairs or trios. (These are called quantum flux tubes, a topic for another time). Basically it's all of this energy from the fluctuating field, constantly creating and destroying virtual particles that accounts for the majority of a mass of a proton.

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