  We consider irrelevant deformations of massless RSOS scattering theories by
an infinite number of higher TTbar_{s+1} operators which introduce extra
non-trivial CDD factors between left-movers and right-movers. It is shown that
the resulting theories can be UV complete after bypassing typical Hagedorn-like
singularities if the coefficients of the deformations are fine-tuned. In this
way, we have discovered that only two new UV complete QFTs are associated with
a M_p (p=3,4,...) minimal CFT based on the integrable structure of the RSOS
scattering theory. One is the massless Z_{p-1} parafermionic sinh-Gordon models
(PShG) with a self-dual coupling constant. This correspondence is confirmed by
showing that the scale-dependent vacuum energies computed by the thermodynamic
Bethe ansatz based on the S-matrices match those from the quantization
conditions for the PShG models using the reflection amplitudes. The other UV
QFT is reached from M_p by following the roaming trajectory of the Z_{p-1}
parafermionic minimal series.
