  In this paper, we propose a new representation of the minimal form factors in
integrable quantum field theories. These are solutions of the two-particle form
factor equations, which have no poles on the physical sheet. Their expression
constitutes the starting point for deriving higher particle form factors and,
from these, the correlation functions of the theory. As such, minimal form
factors are essential elements in the analysis of integrable quantum field
theories. The proposed new representation arises from our recent study of form
factors in $\mathrm{T}\overline{\mathrm{T}}$-perturbed theories, where we
showed that the minimal form factors decompose into elementary building blocks.
Here, focusing on the paradigmatic sinh-Gordon model, we explicitly express the
standard integral representation of the minimal form factor as a combination of
infinitely many elementary terms, each representing the minimal form factor of
a generalised $\mathrm{T}\overline{\mathrm{T}}$ perturbation of the free
fermion. Our results can be readily extended to other integrable quantum field
theories and open various relevant questions and discussions, from the
efficiency of numerical methods in evaluating correlation functions to the
foundational question of what constitutes a "reasonable" choice for the minimal
form factor.
