So I've come to the understanding that a hadrons mass does not come from the constituent quarks but instead comes from a combination of things like binding energy and the mass energy of virtual gluons and quarks inside the hadron. But this doesn't make any sense from an energy stand point. If you have for example; 3 quarks that aren't bound together. And then you bind them together into a hadron, suddenly they gain mass. Wait a second! Doesn't this violate the conservation of energy? You could say that the mass is the result of the binding energy. But for a bound system, the binding energy must be negative. And so a bound system of quarks should be less massive than an unbound system. Why would quark ever want to bind together if the resulting system has much more energy? To summarize it all up, where does this mass come from?
2 Answers
When you say
"binding energy must be negative"
you have a built-in assumption that the strength of an interaction goes to zero at large distance. That's true for gravitation and for electrostatics, for the weak interaction and even for the effective strong-force (the nuclear one), but it is not true of the underlying QCD strong force whose potential is grows linearly with distance.
Pulling quarks apart doesn't get any easier as they separate, it stays just as hard (for a while, then a pair production event occurs), which means that you can't measure the energy of the bound system relative a zero set at infinite remove.
Given a rule like $$V_{QCD} \propto r \,,$$ you have to conclude that the binding energy is positive unless the quarks are found at zero distance with respect to one another. Measurements of hadron sizes are non-vanishing, so you expect positive binding energy.
Where does a hadrons mass come from?
Well, according to this guy, it comes from its energy-content. Some will tell you it comes from something else. But take a look at what CERN physicist Gian Guidice says on page 174 of A Zeptospace Odyssey:
He's saying that 99% of a hadron's mass is down to E=mc². As to why E=mc² somehow doesn't apply to a body such as a quark, I don't know.
Edit:
So I've come to the understanding that a hadrons mass does not come from the constituent quarks but instead comes from a combination of things like binding energy and the mass energy of virtual gluons and quarks inside the hadron. But this doesn't make any sense from an energy stand point.
Correct. Because virtual particles are virtual, not real. See anna v's answer here where she says "virtual particles exist only in the mathematics of the model". And binding energy is negative. It doesn't represent extra mass, it represents less. Check out the mass deficit.
If you have for example; 3 quarks that aren't bound together.
When you do, make sure you make a call to the guys in Stockholm. Because we've never ever seen a free quark. Even though the quark concept has been around for fifty years. You might want to google on quark confinement to find out what some people think about that.
And then you bind them together into a hadron, suddenly they gain mass. Wait a second! Doesn't this violate the conservation of energy? You could say that the mass is the result of the binding energy. But for a bound system, the binding energy must be negative.
Well spotted that man!
And so a bound system of quarks should be less massive than an unbound system. Why would quark ever want to bind together if the resulting system has much more energy? To summarize it all up, where does this mass come from?
It comes from the energy-content. E=mc² is not wrong. A radiating body loses mass. And spookily enough, proton-antiproton annihilation is rather like two radiating bodies losing mass. All of it:
Image credit CSIRO, see The Big Bang & the Standard Model of the Universe
And then all those fundamental quarks and gluons, and all that fundamental strong force, have totally utterly gone.
