18.10. Bound states and semi-classical limit It can be proved that the renormalized coupling constant

ξ = β2 8

1 1− β2 8π

.

of the Sine–Gordon model comes from the quantum correction of the classical action.

By the same token, it is possible to prove that the exact mass of the soliton/anti-soliton is M = m

ξ

, where m is the parameter in the Lagrangian. Keeping m fixed, the semi-classical limit

β2→0 of the Sine–Gordon gives rise to a non-trivial theory.

a. Use the expression above of the mass of the soliton to express differently the mass of the breathers Bn of the Sine–Gordon, given in eqn. (18.10.21).

b. Expand mn in powers of β2 and show, that at lowest order, mn nm1. So, all these states can be considered as loosely bound states of n ‘elementary’ bosons B1.

c. Compute at the order β4 the binding energy En ≡ nm1 −mn of these states.