Yukawa potential

In particle physics, a Yukawa potential (also called a screened Coulomb potential) is a potential of the form

$V_\text{Yukawa}(r)= -g^2 \;\frac{e^{-mr}}{r}$,

where g is a magnitude scaling constant, m is the mass of the affected particle and r is the radial distance to the particle. It is worth noting that 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.

History

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, which is inversely proportional to the mass.

Relation to Coulomb potential

Figure 1: A comparison of Yukawa potentials where g=1 and with various values for m.

Figure 2: A "long-range" comparison of Yukawa and Coulomb potentials' strengths where g=1.

If the mass is zero, then the Yukawa potential becomes equivalent to a Coulomb potential, and the range is said to be infinite.

As the mass approaches 0, the exponential term goes to 1,

$m \rightarrow 0: e^{-m r}\Rightarrow e^0 \Rightarrow 1$.

So in this limit the equation,

$V_{\text{Yukawa}}(r)= -g^2 \;\frac{e^{-mr}}{r}$,

becomes a form of the Coulomb potential,

$V_{\text{Coulomb}}(r)= -g^2 \;\frac{1}{r}$.

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 goes to zero rather quickly.

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 / (k² + m²) 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 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 p₁ and the other with momentum p₂, 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 g². 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 / (k² + m²). 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.

See also

• Yukawa interaction

References

Texts

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