- 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 thepion 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 thefermion field with which it interacts. In the case of thenuclear force , the fermions would be theproton and another proton or theneutron .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 orGreen's function of theKlein-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 theFeynman 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
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