From: jos.bergervoet@xs4all.nl   
      
   On 6/18/2017 6:40 PM, Lawrence Crowell wrote:   
   > On Sunday, June 18, 2017 at 8:17:13 AM UTC-5, James Goetz wrote:   
   >> Gluons bond quarks into baryons (i.e., protons and neutrons). For   
   >> example, two up quarks and one down quark form a proton while one up   
   >> quark and two down quarks form a neutron. Is there one gluon per one   
   >> baryon or two gluons per one baryon or what is the ratio of gluons to   
   >> baryons?   
   >   
   > If you or I could definitively answer this question a Nobel prize would   
   > be next. In a perturbation series there are various orders. The lowest   
   > order has each quark in a baryon connected by a gluon. So that is one   
   > gluon per quark, which are sometimes called baryons, or for say the   
   > proton there are 3 gluons per proton. However, to higher order there are   
   > loops and higher hbar corrections and these are amplitudes with more   
   > gluons. The higher order diagrams are a tangle or mesh of gluons. Also   
   > at higher orders quark-antiquark pairs come into play.   
   >   
   > QCD has a vast solution space, and perturbative QCD lets us look at some   
   > tiny slices or are like small paths in it. The solution space in its   
   > entirety is a vast unknown.   
      
   But wouldn't it be possible, despite not having the full solution,   
   to have a number based on the field shape and strength?   
      
   For instance take the similar question about the number of photons   
   in a Hydrogen atom. We can say that the field in coordinate space   
   has an extent of the atomic size, so in momentum space the modes   
   of the photon fields we need are of energy in the order of 1 keV   
   (the inverse of the Bohr radius or van der Waals radius). And the   
   energy in the photon field is in the order of the binding energy,   
   i.e. about 10eV, so the number of photons is:   
       = 10eV / 1keV = 0.01   
   Or to put it differently: the photon field will be a superposition   
   consisting mainly of the vacuum state (99%) with about 1% admixture   
   of 1-photon states (an integral over the required Fourier components   
   of course). And most likely it also contains in the order of about   
   0.01% of 2-photon state terms, and so on (if the 'harmonic state'   
   for the field is any guideline, that is..)   
      
   Now we can play this game for a baryon (especially if we use a   
   simple description of bound quarks, for instance the old Isgur-Karl   
   model, but that's an exercise for the reader). Since quarks are   
   quite light to begin with, half of the total energy might be in the   
   gluon field now. And the radius of the baryon is of course 1 fm,   
   so requiring a field shape with momentum space modes of 200 MeV,   
   on average. Therefore:   
      = 0.5GeV / 200MeV = 2.5   
   So a baryon contains about 2.5 gluon quanta, it seems.   
      
   One might wonder (in both cases) whether the Fourier components for   
   the shape of the fields do contain an infra-red tail with infinitely   
   many soft photons (or gluons) so that simply dividing field energy   
   by energy of the average photon would not be correct. After all   
   this is exactly the case for scattering problems, where we do indeed   
   have infinitely many photons, and most of them extremely low energy.   
   But for these bound states there is no such effect as far as I can   
   see (the tails in k-space do not give divergent integrals). The   
   above estimates should be more or less right.   
      
   --   
   Jos   
      
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