From: jos.bergervoet@xs4all.nl
On 6/27/2017 4:23 AM, Lawrence Crowell wrote:
> On Wednesday, June 21, 2017 at 8:54:47 PM UTC-5, Jos Bergervoet wrote:
..
...
>> = 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.
>
> In QCD the u and d quarks are about 10MeV at most and the remaining
> mass of the proton is due to the QCD field of gauge bosons or gluons.
> The majority of the mass of a proton is due to the mass-gap induced
> by confinement. This has some bizarre consequences, and one of those
> is that the number of virtual gauge bosons running around the
> confined QCD vacuum is divergent. In other words it is infinite.
As already indicated above, I don't think it's bizarre, I just
wouldn't expect it here. You can only have an infinitude of soft
quanta if their field has a long-range tail (like 1/r or 1/r^2
for instance). Now the transition form confined QCD vacuum to
the surrounding ordinary vacuum does not seem to have this tail.
The strong interaction range outside a baryon is determined by
the pion mass, approximately, so it falls off exponentially.
Put differently, if you sum up infinitely many long-wavelength
quanta (in occupation number space, or Fock space) you will be
constructing a long-range tail of the gluon field around the
baryon. Does such a tail really exist?!
> In effect by the anti-sceening physics of chromo-charges the IR
> contribution to physics dominates and all order diagrams contribute
> equally.
That is to be expected since we talk about a bound state. Even
for a simple hydrogen atom you cannot solve that case with
perturbation theory in finite order. The question I was trying
to answer is just what the expectation value of total photon
number in Fock space would be, regardless how we have solved
the problem.
> In orders of radiation corrections g^n if g is unity then
> the most high order diagram contributes. As a result QCD is great
> at calculating scattering amplitudes where the coupling constant g
> ~ 1/E, for E the energy of scatter or transverse momentum, but it
> is no good at calculating the mass of a proton.
You mean "perturbative QCD" is not good at that. So you need
another computational tool. Still that does not prove (nor
disprove) that the bound state must have an infinitude of
gluons (in the sense of the expectation value ).
> So what do we do? We use computers. The space is gridded up into
> cells and the gauge potentials run around the edgelinks of the
> cells. The difference in gauge potentials across nodes determines
> the chromo-electric field, just as E = -&A/&t - grad A (& = partial).
> The areas bounded by edgelinks carry chromo-magnetic fields. In
> doing this I prefer to set up a dual lattice in 3-d where the
> chromo-electric field is along the dual edgelinks. Think of Faraday's
> principles. I prefer this because it puts the chromo-electric and
> magnetic fields on a dual and more equal footing. This system in
> euclidean 4-d is set to go to equilibrium or a Monte-Carlo calculation
> determines the most probable distribution of fields. These are run
> into the stress-energy tensor to find energy.
On a lattice of finite extent you cannot have soft gluons at
all, the shortest possible wavelength is restricted. So in
that case I would expect that you would certainly *not* find
infinitely many of them!
> On supercomputers a fair amount of progress has been made. The
> masses of low energy hadrons has been computed to fair accuracy.
Yes, I heard they are within 0.5% correct. But that isn't even
accurate enough to show the difference between proton and neutron
mass.. Nevertheless, I would expect that after the calculation one
can see the statistics of the gluon field in Fock space, is that
correct? I'm still trying to zoom in on the value of (purely
to help OP with his question, of course!) and since I don't expect
it to be infinite, a reasonably sized lattice would suffice to
indicate the approximate value. (The proof it can't be infinite
would then have to be added in another way.)
> There is of course the unsatisfying sense that we still do not
> understand this. The Claymath problem on mass gap is the open problem
> to address this.
>
> Is there a way to think about this analytically? Maybe. The S-dual
> of QCD is where the chromo-electric field is very weak. This is the
> gist of the Mantenen-Olive duality
>
> qg = n\hbar,
>
> where for a gauge charge q large the dual charge g is small. I think
> the graviton is maybe an entanglement of colorless pair of these
> s-dual gluons. By this if we extend QCD to SU(4) then conformal
> gravity is SU(2,2), and this has a duality with this extended QCD.
> The advantage is that a weak field theory is not hard to understand.
Still I am afraid that computing a bound state cannot become "simple"
just by using any such trick..
> The problem is that it is gravitation now, and that is not hard to
> understand either. In fact the UV limit of quantum gravity is lots
> of Planck oscillators collapsed into a black hole. This is in a way
> dual to the IR limit in QCD. So we have converted a hard problem
> "here" into a hard problem "there."
But is it hard? (To find out whether for gluons is finite or not,
I mean?) I still do not see what is wrong with my original reasoning:
without long-range tail, no infinitude of soft quanta is possible.
--
Jos
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