Admissibility and uniqueness of renormalization functions h_i

Determine whether different combinations of singular parts can be employed in the renormalization functions h_i(Λ, s, β) specified in Definition 34-d-5 so that the renormalized limit exists upon removing the regularization Λ → +∞, and clarify any uniqueness conditions for such choices.

Background

The paper defines a general renormalization framework (Definition 34-d-5) in which divergent contributions arising from a spectrum deformation β → β(Λ) are removed by shifting internal parameters and applying overall multipliers, encoded via a family of functions h_i(Λ, s, β). These functions are required to diverge as Λ → +∞ for fixed s and to vanish as s → 0 for fixed Λ.

Throughout the analysis, concrete renormalizations subtract the singular part r(Λ) from phase factors to yield finite limits for the regularized functionals. However, the authors emphasize that this construction does not render the h_i unique and raise the question of whether alternative combinations of singular parts could also achieve valid renormalized limits. This ambiguity motivates a precise investigation into admissible choices and potential uniqueness conditions.