# Yukawa interaction between Dirac particles is universally attractive?

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?

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• The Yukawa interaction between a particle and its antiparticle is repulsive. See physics.stackexchange.com/a/20652/26129. – higgsss Dec 2 '15 at 22:57
• This should be discussed in any QFT text book. See for example Peskin & Schroeder section 4.7 – Olof Dec 3 '15 at 9:23

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.

• After further thought, I have concluded that the interaction between a fermion and an anti-fermion is also attractive for this model (scalar boson exchange between Dirac particles). The argument is rather convoluted and too long to confine to a comment, but it boils down to an attractive interaction between an anti-neutron and a spherically symmetric nucleus. It also relies on the use of the Dirac hole theory of anti-particles applied to a relativistic mean field model of finite nuclei (sometimes called the sigma-omega model). – Lewis Miller Dec 6 '15 at 14:37

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 $m_\text{photon}\to0$.

• The ρ and ω mesons are multi-pion resonances (relatively sharp so we speak of them as real particles). There are less sharp multi-pion resonances that behave as scalar (spin 0) "pseudo-mesons" that contribute Yukawa-like attractive interactions between nucleons. These are equally important for nuclear binding. We know this because the pion (being pseudo-scalar rather than scalar) contributes nothing to nuclear binding at the mean field level for doubly magic nuclei. See: physics.stackexchange.com/questions/126533/… – Lewis Miller Dec 5 '15 at 3:42
• I'm aware of the interpretation of the $\sigma/f_0$ as a wide two-pion resonance, but your comment is the first I've heard of a similar interpretation for the $\rho$ or $\omega$. I'm intrigued but skeptical. If several isovector pions can interfere to make an isoscalar $\omega$, I'd expect the same sort of process to generate an "isotensor" meson, but that's not consistent with the very successful $q\bar q$ model for mesons. – rob Dec 6 '15 at 23:24
• When I was in grad school (1960s) it was common to refer to the ρ and ω mesons as multi-pion resonances. Here is a link to a paper on that subject: authors.library.caltech.edu/10327/1/ZACpr62.pdf I believe the theory in vogue at that time was called boot strapping. That was just after these particles were discovered and before the quark model was accepted so the point you raise would not have been addressed. I was able to find this: sciencedirect.com/science/article/pii/S0370269305012803 but I don't know whether isotensor mesons are accepted or not. – Lewis Miller Dec 8 '15 at 4:06