# Yukawa force vs Nuclear force

Yukawa force is any force that is described by the potential of the form $$V = k \frac{e^{-\lambda r}}{r}$$. The nuclear force can be approximately described by such potential (with $$\lambda \sim m_\pi)$$, so it's an example of Yukawa force.

The nucleon-nucleon interaction is very complicated, and meson-exchange potentials are simple models (not derived from QCD) of the low energy regime. However, one-pion (or multi-pion) exchange is special, because the pion is the lightest state in QCD, and the one-pion exchange therefore rigorously describes the longest range part of the interaction. This can be formalized and systematically improved using chiral effective field theory.

The one-pion exchange interaction between two nucleons is $$V_{NN}= \frac{m_\pi^2}{12\pi} \frac{g_A^2}{2f_\pi^2} (\sigma_1\cdot\sigma_2)(\tau_1\cdot\tau_2) \frac{e^{-m_\pi r}}{r} + (LS-coupling)$$ This interaction depends on the spin and isospin of the two nucleons. It is attractive in $$I=0,S=1$$ and $$I=1,S=0$$, repulsive in $$I=S=0,1$$. Two neutrons have to have $$I=1$$, so the interaction between two neutrons is attractive if the total spin is 0.

The nucleon-nucleon interaction can be related to the nucleon-anti-nucleon interaction using G-parity, $$G=C\exp(-i\pi T_2)$$, a combination of $$C$$-conjugation and isospin. The G-parity of the pion is negative, so the one-pion $$N\bar{N}$$ interaction is $$V_{N\bar{N}}= - V_{NN}$$ but the two-pion amplitude has the same sign, etc. A neutron and an anti-neutron can have both $$I=0$$ and $$I=1$$ (as opposed to $$nn$$ which is always $$I=1$$). Staying with $$I=0,S=1$$, the interaction is now repulsive, but the $$I=1,S=0$$ part is attractive.

Note that if one construct phenomenological potentials using many meson exchanges, then the strong $$N\bar{N}$$ interaction is on average more attractive then the $$NN$$ interaction, see for example Buck et al.

Yukawa's theory is one model of the nuclear force. There are many such models. For a review, see Ruprecht Machleidt (2014), Scholarpedia, 9(1):30710..