This paper presents an exhaustive, step‑by‑step derivation of the bare two‑loop potential contributions to the sine–Gordon kink mass, using the rigorously reduced meson Hamiltonian. Every algebraic manipulation is shown in full detail. The exact bare cubic vertex is derived in closed form. The normalisation of the Green’s functions is carefully treated, leading to a consistent numerical framework. The sausage (cubic cubic) diagram is shown to diverge logarithmically in the ultraviolet, while the figure‑eight (quartic tadpole) diagram is finite after normal‑ordering. Numerical evaluation with a smooth Gaussian regulator confirms that the two bare diagrams do not cancel; renormalisation is required. The exact two‑loop correction from integrability is \ (+^2/ (16) \). We outline the counterterm diagrams (mass insertion, cubic tadpole, vacuum energy subtraction) that will cancel the divergence and yield this value. The paper establishes the correct bare Feynman rules and provides a solid foundation for the complete first‑principles derivation.
Kalmykov et al. (Sun,) studied this question.