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If we try to separate two quarks bound into a meson or a hadron, the energy in the gluon field eventually will be large enough to spawn a quark-antiquark pair.

How far can we stretch that gluon field before it "snaps"? What's the average distance? I can't seem to find it anywhere.

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From lattice calculations (see String Tension of Quark-Anti-Quark Pairs in Lattice QCD) it has been found that the string tension of the quarks, in the case of pions, is given by $$ \sqrt{\sigma}\sim460\ \mathrm{MeV} $$ which is equivalent to a length of $\sim 2.7\ \mathrm{fermi}$.

In the case of the charmonium ($\bar c c$), the tension (see Charmonium potential from full lattice QCD) is close to $$ \sqrt{\sigma}\sim394\ \mathrm{MeV} $$ that is, a lenght of $3.1\ \mathrm{fermi}$.

For a bottomonium state ($\bar b b$), the best estimate I've found (see Bottomonium states versus recent experimental observations in the QCD-inspired potential model) is $$ \sqrt\sigma\sim 410\ \mathrm{MeV} $$ i.e., $3\ \mathrm{fermi}$.

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  • $\begingroup$ This number is too large, see my answer here: physics.stackexchange.com/questions/238218/… $\endgroup$ – Thomas May 27 '16 at 15:46
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    $\begingroup$ @Thomas Where did you get the $1\ \mathrm {GeV/fm}$ value? You can see in my links (which are more recent than yours) that the LQCD results are closer to $400\ \mathrm{MeV/fm}$... $\endgroup$ – AccidentalFourierTransform May 27 '16 at 15:50
  • $\begingroup$ (460 MeV)^2=1 GeV/fm $\endgroup$ – Thomas May 27 '16 at 15:52
  • $\begingroup$ @Thomas welp, according to wolfram alpha its more like $2.7\ \mathrm{fm}$. $\endgroup$ – AccidentalFourierTransform May 27 '16 at 15:54
  • $\begingroup$ (460 MeV)^2 = (460 MeV)/(200 MeV*fm) = 460 MeV * 2.3/fm = 1 GeV/fm $\endgroup$ – Thomas May 27 '16 at 16:01
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From the HyperPhysics site on Quarks:

It is postulated that it may actually increase with distance at the rate of about 1 GeV per fermi. A free quark is not observed because by the time the separation is on an observable scale, the energy is far above the pair production energy for quark-antiquark pairs. For the U and D quarks the masses are 10s of MeV so pair production would occur for distances much less than a fermi. You would expect a lot of mesons (quark-antiquark pairs) in very high energy collision experiments and that is what is observed.

From what I can gather, the actual increase in energy per fermi is not known to a high accuracy because the range is so incredibly small. Considering that the strong nuclear force is only able to participate in interactions at distances of less than $10^{-15}$ m it is understandable that no exact numbers exist for the maximum separation two quarks can have.

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    $\begingroup$ Check. But you pop hadrons out of the vacuum, so, let's say a π⁰, so then very crudely 0.135 fermis.... more than a tenth of a fermi, if needs be. $\endgroup$ – Cosmas Zachos May 27 '16 at 18:55

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