In 1935 Hideki Yukawa proposed that the strong nuclear force is mediated by a massive particle particle, and proposed the following form for the potential

where is the Compton wavelength of the mediating particle, is the reduced planck constant, is the speed of light and is the mass of the mediator. In the limit this expression reproduces the Coulomb potential, and the exponential term can be interpreted as the tunnelling probability for the mediating particle. A suitable mediating particle - the pion - was detected in 1947, and so . It was later found that nucleons have internal structure and are made up of quarks, and so the Yukawa theory cannot be a complete description of the strong force, but it is still a useful approximation.

What's left is to constrain the coupling constant $g$. If the mediating particle was a scalar particle (zero spin and positive parity), then the coupling constant would have been

One way of understanding this coupling constant is as follows. When two nucleons are at a distance they communicate by exchanging particle whose de Broglie wavelength is comparable to , so their momentum is , and if they are relativistic their energy is . However, the pion is not a scalar, but a rather a pseudo scalar, meaning its spin is zero but its parity is negative. This pairity precludes spherically symmetric s-wave scattering (angular momentum ). The next leading order in the partial wave expansion is p-wave (with angular momentum ), which is suppressed by the ratio of the Compton wavelengths squared of the particles and mediators. Hence,

where is the mass of a nucleon. When the nucleons are at a distance comparable to the pion Compton wavelength, the binding energy is

A more rigorous derivation can be found here.

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