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I can model this interaction as Zee does in "Quantum field theory in a nutshell". In chapter I.4 section "from particle to force" he uses two delta functions for the source. The integral gives $E=-\frac{1}{4\pi r}e^{-mr}$

I consider $J=\delta^{(3)}(x-x_1)-\delta^{(3)}(x-x_2)$. Im assuming that this represents a particle and an antiparticle. I have done the calculations quantum and classically and I obtain $E=\frac{1}{4\pi r}e^{-mr}$

So as this simple model is used to describe the long range nucleon interactions I think I could conclude that this is the potential interaction between a neutron and an antineutron, a repulsive Yukawa potential.

I want to know If I'm right. Textbooks don't talk about this and Zee seems to say that the force is always attractive.

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1 Answer 1

In electromagnetism, electric charges of the same sign repel; opposite charges attract. That's related to the messenger's spin $J=1$.

However, your case is $J=0$ because the messenger field is a scalar pion. This situation behaves much like $J=2$ or any other even $J$ of the messenger particle: like charges attract (e.g. positive masses gravitationally attract) while opposite charges repel.

It follows that a neutron and an antineutron Yukawa-repel, too.

Anthony may have said the incorrect statement that the force is always attractive in analogy with $J=2$ gravity; while the signs are analogous, gravity differs from the Yukawa force in one additional respect: its charges (masses) are always positive, so the "universally attractive" property (between isolated objects) holds for gravity (but not the Yukawa force).

The only thing I would dispute is your term "long-range force". The Yukawa force is a textbook example of a short-range force, as guaranteed by the exponentially decreasing factor. This more properly means that the total cross section is finite; for the Coulomb interaction, it is infinite which is why the electrostatic repulsion is a long-range force (like gravity).

One should emphasize that at long distances, there are forces that are parametrically stronger than the Yukawa force you mentioned. In particular, neutrons and antineutrons are small magnets (nonzero dipole moments) so there's a magnetostatic spin-dependent force in between them, decreasing like $1/r^4$, which is still much larger because it's not suppressed by any exponential.

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