Editing One Pion Exchange Potential

In 1935 Hideki Yukawa proposed that the strong nuclear force is mediated by a massive particle particle, and proposed the following form for the [https://en.wikipedia.org/wiki/Yukawa_potential potential]

<math> V \left(r\right) = -g^2 \frac{\exp \left(-r\lambda\right)}{r} </math>

where <math> \lambda = \hbar/m_m c </math> is the Compton wavelength of the mediating particle, <math> \hbar </math> is the reduced planck constant, <math> c </math> is the speed of light and <math>m_m</math> is the mass of the mediator. In the limit <math> \lambda \rightarrow \infty </math> 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 <math>m_m = m_{\pi}</math>. 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

<math> g^2 \approx \hbar c </math>

One way of understanding this coupling constant is as follows. When two nucleons are at a distance <math> r </math> they communicate by exchanging particle whose de Broglie wavelength is comparable to <math> r <math>, so their momentum is <math> \hbar/r </math>, and if they are relativistic their energy is <math> \hbar c / r </math>. 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 <math> l = 0 </math>). The next leading order in the [http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Scattering_II.htm partial wave expansion] is p-wave (with angular momentum <math> l = 1 </math>), which is suppressed by the ratio of the Compton wavelengths squared of the particles and mediators. Hence,

<math> g^2 \approx \hbar c \left(\frac{m_{\pi}}{m_n}\right)^2 </math>

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

<math> V \left(\frac{\hbar}{m_{\pi} c}\right) \approx m_{\pi} c^2 \left(\frac{m_{\pi}}{m_n}\right)^2 </math>

A more rigorous derivation can be found [https://www.physics.umd.edu/courses/Phys741/xji/chapter6.pdf here].
[[Category:Particle Physics]]