Yukawa potential

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 the proton and another proton or the neutron.

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

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

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

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 $psi(x)$ to the meson field $phi(x)$ with the coupling term

:$mathcal\{L\}\_mathrm\{int\}(x)\; =\; goverline\{psi\}(x)phi(x)\; psi(x)$

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

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

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

References

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