How is the mass of hadrons calculated from constituent quarks? It seems classically it should come from three parts: (p^{2}+m^{2})^{1/2} for each constituent quark with rest mass m, plus interactions between the quarks, which would be electromagnetic Coloumb 1/R potential between any two of the three quarks, and the linear R potential between any two quarks which is responsible for confinement at strong coupling. Quantum mechanically, if you could measure the size of the hadron, and assume the quarks are confined to a box of that size, then you would only need to add the rest mass of the quarks to the particles-in-a-box energies to get the energy of the hadron, and wouldn't have to know the momentum p of each constituent quark or the coefficient of the string potential R. Is this right? I seem to recall that the u and d quarks are taken to have almost zero rest mass. So is all the energy of hadrons made from those quarks due to E&M and confinement? What about hadrons involving heavier quarks?
Those potentials are effective models, describing the complex QCD interactions. "2 quarks"/"3 quarks" are a bad model for mesons/baryons. Those are the valence quarks, and their high expectation values for the kinetic energy are an important contribution to the total energy, but you also have sea quarks and gluons with significant energy. They make things a bit easier. There, the quark masses are the dominant contribution to the total mass of the hadrons.
Still, there is an intriguing coincidence in the Standard Model: that the QCD mass is not far of the yukawian-higgs mass. Nothing in the standard model requires it. But you have then the mass of the proton nicely between the mass of, say, muon and tau. There is not a SU(3) horizontal of flavour, but still a triad of electron tau and muon should have a mass -not counting electromagnetism" exactly the double of a QCR triad (a proton, say).
Well, (maybe apart from neutrinos) all known particle masses and all energy scales are close together if you consider a log scale from the electron to the planck scale. That just means the planck scale is far away. I have no idea what that is supposed to mean.
Neither me. It is just an example to show that they are really near, yukawa-higgs on one side, QCD in the other. If something, it counterweights the argument that it is just the GUT or Planck scale which is far away. They are closer than one should expect for two completely independent parameters.
But you wrote it . What is a "a SU(3) horizontal of flavour", what do you mean with "[not] counting electromagnetism", why "should" the sum of the masses of electron+muon+tau be exactly the double of something made out of quarks? (It is not) Given enough numbers (and we have tons of hadrons), you can always find things that are close together. That does not need any special reason.
No really. We have a lot of Hadrons, but all of them are a few mass multiples away. Had the electroweak vacuum been one or two orders of magnitude higher, you would not be able to find any. The point was this, that there is no reason for the EW vacuum to break so near of QCD scale, but it does. And then when it does, the yukawas make a lot of magic to land the final mass values just in the middle of QCD world. Muon at a few percent of the pion could be other example.
I've been trying to follow the discussion on scales, but I lack basic knowledge. It's my understanding that the only dimensionful parameter in the Standard Model is the coefficient of the quadratic term in the Higgs (and possibly a mass term of a sterile neutrino). That would seem to imply there is only one scale (let's ignore the existence of the sterile neutrino scale which hasn't been proven) and as a consequence, all dimensionful parameters should either blow up or go to zero when the Higgs symmetry is restored at high temperature (although doesn't the temperature at which that happens provide a scale?). Beyond the Standard Model there seems to be one more scale, the Planck scale. So really there are only three scales, the EW scale, the Planck scale, and a sterile neutrino mass. So what's the QCD scale? Also, would pure QED have a scale? In pure QED you no longer have the EW scale because you're only considering E&M. You would have the mass terms of your charged particles, so would the pure QED scale(s) be the mass(es) of your charges particles?
The QCD scale is the scale at which the strong coupling constant blows up, i.e. the confinement scale. This article has a nice little description http://physics.stackexchange.com/questions/8926/what-is-the-significance-of-the-qcd-scale-parameter-lambda. You're right though that this scale is not present in the classical Lagrangian. It essentially appears during renormalisation. As for QED, well, it does have a Landau pole at some high energy, which I guess is kind of similar. The QED coupling constant does blow up there, but I think this is considered a problem with perturbation theory rather than something physically important like the confinement scale.
If the scale is due to renormalization then for every coupling you should have a scale. Looking at the standard model Lagrangian in the EW sector, there are two couplings. If you include QCD there are 3 couplings. All couplings will depend on scale, so you can define three scales: the scales where each coupling=1? Also, there seems to be a lot of couplings involving fermions and Higgs. Each of those presumably will depend on scale. So there should be a lot of scales?
I'm not really that familiar with how the dimensional transmutation works unfortunately. All those couplings do depend on scale, but I think only scales where strong dynamics come into play are actually considered meaningful. I think most of the couplings do not develop any poles below the Planck scale anyway.
Sorry, but now you switched the topic completely. Yes, it is unknown why the QCD scale, the particle masses and the electroweak scale just span a few orders of magnitude. But if we take this observation, finding some random "just 1% apart!" is not surprising.
I really do not understand how do you consider muon and pión "random particles". And yes still we could be looking for random coincidences with QCD but you really know that it is not the case, do you? Remenber that here in this same forum we were not looking for qcd when we added electrón muon and tau and divided by 6. We were calculating the mass MO in Koide formula for leptons, and it happened to be 313 Mev, the same mass that a consituent quark. And years later, I used Koide to calculate charm mass from bottom and top -not my idea, but Rodejohann and Zhong- and then charm and bottom to calculate strange. And it happened that the M0 of the triplet of s c and b was ,940 MeV, three times the one of leptons. if you call this random, you have a peculiar concept of randomness. Is this remark on-topic? I think yes, because the OP asked how quark and hadron mass are related. The obvious answer, vía the chiral expansión, involves constants which come from QCD, and the same happens with lattice calculations. So it is on topic to remark that the QCD scale itself could be related to quark "yukawa-higgs" masses.
Don't put words in my mouth please. I don't have an explanation why the Koide formulas work, but I would be surprised if you had. Why do you claim those masses "should" have a relation in post #4? And you still did not explain what a "a SU(3) horizontal of flavour" is. It has exactly 1 google hit: your post.