Editing Appendix/Ramblings/StrongNuclearForce (section)

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
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From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
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<math>~V(r)</math>
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<math>~=</math>
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<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
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where, the constant <math>~k</math> determines the field energy per unit length and is called string tension.  For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement).  It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances.  For example charmonium:  <math>~r \approx 0.4~\mathrm{fm}</math>.
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Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
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This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
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At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
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<math>~V(r)</math>
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<math>~=</math>
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<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
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where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant.  To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
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The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves.  This more powerful force is mediated by particles called gluons.  Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons.  There are charges, called colored charges on the quarks, but protons and neutrons are color neutral.  Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons.  That also cannot be simply described.  Various potentials are used to calculate nuclear interactions.
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From the arXiv preprint of a review article by [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force A. Deur, S. J. Brodsky, &amp; G. F. de T&eacute;ramond (2020)] titled, ''The QCD Running Coupling'':
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(middle of p. 10) "We illustrate this behavior" &#8212; that is, "&hellip; the scale dependence of the coupling" &#8212; "for the coupling that arises in the static case of heavy sources and which provides a simple physical picture.  Historically, and in the case of linear theories with massless force carriers, a force coupling constant is a universal coefficient that links the force to the 'charges' of two bodies (''e.g.,'' the electric charge for electricity or the mass for gravity) divided by the distance dependence <math>1/r^2</math>."
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(middle of p. 10, continued) "In QFT (quantum field theory) &hellip; for weak enough forces, the first Born approximation dominates higher order contributions and the <math>1/q^2</math> propagator in momentum yields the familiar <math>1/r^2</math> factor in coordinate space.  However, higher orders do contribute and deviations from the <math>1/r^2</math> law thus occur.  This extra r-dependence is folded in the coupling which then acquires a scale dependence."
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