Yukawa interaction between Dirac particles is universally attractive?

Question

Can anyone provide me a specific reference to (or supply themselves) the derivation of the fact that the Yukawa interaction$$\mathcal{L}_{\text{int}} = -g\overline{\psi} \psi \phi$$between Dirac particles is universally attractive, i.e. one finds an attractive interaction between particles-particles, particles-antiparticles, and antiparticles-antiparticles?

Answer 1

I derived the Yukawa interaction between fermions as part of my dissertation research in the latter-1960s. You are correct in stating that this interaction is attractive between fermions and also between anti-fermions, but my derivation does not cover the fermion anti-fermion interaction so I can't dispute @higgsss' comment above. My intuition, nevertheless, supports your assertion.

It is important to note that these statements apply only when the exchanged boson transforms like a Lorentz scalar (spin 0). For other transformation properties the results differ. For example, if the boson is a vector particle (spin 1) the interaction is repulsive (as is the case with the electrostatic interaction between two electrons (here the boson (photon) mass is 0).

Here is a brief sketch of my derivation. Start with the full Lagrangian, including the non-interacting fermion and boson terms. Take the variational derivative with respect to the boson field. You get the inhomogeneous Klein-Gordon equation with the fermion field source terms on the rhs. Use the Green's function for the homogeneous KG equation to obtain the solution to the inhomogeneous equation as an integral over the fermion density weighted by the Green's function. If I remember correctly, the integral can be obtained by performing a line integration continued into the complex plane at +/- infinity (thereby inclosing a pole). The Yukawa function drops out as the residue of the pole. I'm remembering this from having done it over 45 years ago, and I doubt this is how its done today, but this is probably how Yukawa did it.

Answer 2

It's not generally true that like-particle interations are attractive; it depends on the transformation of the mediating particle under rotations. Here's a related question which mentions the relationship between mediator spin and whether attraction is between like charges or between unlike charges.

The strong interaction is mediated by not one, but a whole forest of mesons. The lightest of these is the pion, which has a mass $$ \frac{\hbar c}{m_\pi} \approx \rm 1.4\,fm. $$ Since the Yukawa potential for the pion is $$ V \propto - \frac1r \exp {\frac{-m_\pi r}{\hbar c}}, $$ we find that nucleons more than a couple of femtometers from each other don't have any interaction energy, because of the exponential in the pion mass, and so we call the Yukawa force a "contact interaction" and say things like "the radius of a nucleon is a little more than a femtometer."

The next mesons that come in are the $\rho$ and $\omega$, which have mass around $\rm 800\,MeV \to 0.25\,fm$, and which both have unit spin. The Yukawa potential for these mesons has the same form as for the pion, but a different length scale. This gives us the second major feature of the nuclear interaction: nucleons don't like to "touch," because a large repulsive interaction kicks in when they come close to overlapping each other.

You can think of the Coulomb potential $V = \pm\frac{\alpha\hbar c}{r}$ between two unit charges as a Yukawa potential in the limit $m_\text{photon}\to0$.