The sine-Gordon model is the model with the same Cartan matrix but an imaginary β. This Cartan matrix corresponds to the Lie algebra s u ( 2 ) . This has a single simple root, α α 1 = 1 and Coxeter label n 1 = 1 , but the Lagrangian is modified for the affine theory: there is also an affine root α α 0 = − − 1 and Coxeter label n 0 = 1 . One can expand ϕ ϕ as ϕ ϕ 0 α α 0 + ϕ ϕ 1 α α 1 , but for the affine root ⟨ ⟨ α α 0 , α α 0 ⟩ ⟩ = 0 , so the ϕ ϕ 0 component decouples.