How does the exchange of gluons between quarks "bind" them together?
How does the exchange of photons bind an electron and proton together?
How does exchange of words bind a group of people together?
Your question can be understood at several levels. The way I understand it, it may be re-phrased
in a non-relativistic picture of quark-quark interaction via gluon exchange.
All the way up to
there are many other possible interpretations.
As in, why does the fact that virtual gluons pass between quarks constantly mean that they stay together as neutrons and protons?
In other words, how does the virtual gluon transmit an attractive force?
It does not. Gluon exchange can result in both attractive and repulsive interactions.
You indeed may want to review how the photon transmit electromagnetic interaction.
Though I can't say I fully understand it myself, Richard Feynman compares gluons and chromodynamics with photons and electrodynamics, talking about a coupling constant referred to as j for photons and g for gluons, which seems relative to amplitude, in the book 'QED: The Strange Theory of Light and Matter', g being much larger and more complex to calculate than j.
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