# 2-body interaction energy of 3 particles

If only two-body interaction is considered then what's the energy if I put three particles on one site?

Assume delta-interaction and interaction strength is proportional to scattering length $a_F$ which depends on the total spin of the two-particle state.

For two particle case, the interaction is simply $\hat U |F,M\rangle = U_F |F,M\rangle$

Suppose the three particles are fermions, then I will take a completely anti-symmetrized spin wave function. But now, how to express the interaction energy in terms of two-body scattering lengths?

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