Yukawa potential Information & Yukawa potential Links at HealthHaven.com

A Yukawa potential (also called a screened Coulomb potential) is a potential of the form

$V(r)= -g^2 \;\frac{e^{-mr}}{r}.$

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 $m$ . Since the field mediator is massive the corresponding force has a certain range due to its decay, which range is inversely proportional to the mass. If the mass is zero, then the Yukawa potential becomes equivalent to a Coulomb potential, and the range is said to be infinite.

In the above equation, the potential is negative, denoting that the force is attractive. The constant g is a real number; it is equal to the coupling constant between the meson field and the fermion field with which it interacts. In the case of the nuclear force, the fermions would be a proton and another proton or a neutron.

[edit] Fourier transform

The easiest way to understand that the Yukawa potential is associated with a massive field is by examining its Fourier transform. One has

$V(\mathbf{r})=\frac{-g^2}{(2\pi)^3} \int e^{i\mathbf{k \cdot r}} \frac {4\pi}{k^2+m^2} \;d^3k$

where the integral is performed over all possible values of the 3-vector momentum k. In this form, the fraction $4π / (k 2 + m 2)$ is seen to be the propagator or Green's function of the Klein-Gordon equation.

[edit] Feynman amplitude

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 $ψ(x)$ to the meson field $φ(x)$ with the coupling term

$\mathcal{L}_\mathrm{int}(x) = g\overline{\psi}(x)\phi(x) \psi(x).$

The scattering amplitude for two fermions, one with initial momentum $p 1$ and the other with momentum $p 2$ , exchanging a meson with momentum k, is given by the Feynman diagram on the right.

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 $g 2$ . 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 $- 4π / (k 2 + m 2)$ . Thus, we see that the Feynman amplitude for this graph is nothing more than

$V(\mathbf{k})=-g^2\frac{4\pi}{k^2+m^2}.$

From the previous section, this is seen to be the Fourier transform of the Yukawa potential.

[edit] References

Gerald Edward Brown and A. D. Jackson, The Nucleon-Nucleon Interaction, (1976) North-Holland Publishing, Amsterdam ISBN 0-7204-0335-9