where g is a magnitude scaling constant, i.e. is the amplitude of potential, m is the mass of the affected particle, r is the radial distance to the particle, and k is another scaling constant, which finally the product of km is the inverse scope. The potential is monotone increasing, implying that the force is always attractive.
In interactions between a meson field and a fermion field, the constant g is equal to the coupling constant between those fields. In the case of the nuclear force, the fermions would be a proton and another proton or a neutron.
Hideki Yukawa showed in the 1930s that such a potential arises from the exchange of a massive scalar field such as the field of the pion whose mass is . Since the field mediator is massive the corresponding force has a certain range, which is inversely proportional to the mass.
Relation to Coulomb potential
If the mass is zero (i.e., m=0), then the Yukawa potential equals a Coulomb potential, and the range is said to be infinite. In fact, we have:
Consequently, the equation
simplifies to the form of the Coulomb potential
A comparison of the long range potential strength for Yukawa and Coulomb is shown in Figure 2. It can be seen that the Coulomb potential has effect over a greater distance whereas the Yukawa potential approaches zero rather quickly. However, any Yukawa potential or Coulomb potential are non-zero for any large r.
The easiest way to understand that the Yukawa potential is associated with a massive field is by examining its Fourier transform. One has
The Yukawa potential can be derived as the lowest order amplitude of the interaction of a pair of fermions. The Yukawa interaction couples the fermion field to the meson field with the coupling term
The Feynman rules for each vertex associate a factor of g with the amplitude; since this diagram has two vertices, the total amplitude will have a factor of . The line in the middle, connecting the two fermion lines, represents the exchange of a meson. The Feynman rule for a particle exchange is to use the propagator; the propagator for a massive meson is . Thus, we see that the Feynman amplitude for this graph is nothing more than
From the previous section, this is seen to be the Fourier transform of the Yukawa potential.
- Brian Robert Martin; Graham Shaw (2008). Particle Physics. p. 18.
- H. Yukawa, On the interaction of elementary particles. (1935) Proc. Phys. Math. Soc. Japan. 17 48