Editing Yukawa interaction (section)

==Classical potential==
{{main|Yukawa potential}}

If two [[fermion]]s interact through a Yukawa interaction mediated by a '''Yukawa particle''' of mass <math>\mu</math>, the potential between the two particles, known as the ''Yukawa potential'', will be:

<math display="block">V(r) = -\frac{g^2}{\,4\pi\,} \, \frac{1}{\,r\,} \, e^{-\mu r}</math>

which is the same as a [[Coulomb potential]] except for the sign and the exponential factor. The sign will make the interaction attractive between all particles (the electromagnetic interaction is repulsive for same electrical charge sign particles). This is explained by the fact that the Yukawa particle has spin zero and even spin always results in an attractive potential. (It is a non-trivial result of [[quantum field theory]]<ref>{{cite book |author=A. Zee |title=Quantum Field Theory in a Nutshell |edition=2nd |publisher=World Scientific |year=2010 |chapter=I.5 |isbn=978-0691140346}}</ref> that the exchange of even-spin [[bosons]] like the [[pion]] (spin 0, Yukawa force) or the [[graviton]] (spin 2, [[gravity]]) results in forces always attractive, while odd-spin bosons like the [[gluons]] (spin 1, [[strong interaction]]), the [[photon]] (spin 1, [[electromagnetic force]]) or the [[rho meson]] (spin 1, Yukawa-like interaction) yields a force that is attractive between opposite charge and repulsive between like-charge.) The negative sign in the exponential gives the interaction a finite effective range, so that particles at great distances will hardly interact any longer (interaction forces fall off exponentially with increasing separation).

As for other forces, the form of the Yukawa potential has a geometrical interpretation in term of the [[field line]] picture introduced by [[Faraday]]: The {{Math|{{sfrac|1|''r''}}}} part results from the dilution of the field line flux in space. The force is proportional to the number of field lines crossing an elementary surface. Since the field lines are emitted isotropically from the force source and since the distance {{Mvar|r}} between the elementary surface and the source varies the apparent size of the surface (the [[solid angle]]) as {{Math|{{sfrac|1|''r''<sup>2</sup>}}}} the force also follows the {{Math|{{sfrac|1|''r''<sup>2</sup>}}}}&nbsp;dependence. This is equivalent to the {{Math|{{sfrac|1|''r''}}}} part of the potential. In addition, the exchanged mesons are unstable and have a finite lifetime. The disappearance ([[radioactive decay]]) of the mesons causes a reduction of the flux through the surface that results in the additional exponential factor <math>~e^{-\mu r}~</math> of the Yukawa potential. Massless particles such as [[photons]] are stable and thus yield only {{Math|{{sfrac|1|''r''}}}} potentials. (Note however that other massless particles such as [[gluons]] or [[gravitons]] do not generally yield {{Math|{{sfrac|1|''r''}}}} potentials because they interact with each other, distorting their field pattern. When this self-interaction is negligible, such as in weak-field gravity ([[Newtonian gravitation]]) or for very short distances for the [[strong interaction]] ([[asymptotic freedom]]), the {{Math|{{sfrac|1|''r''}}}} potential is restored.)