# 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..