  We study the scaling functions associated with the lightest particle mass
$M_1$ in 2d Ising field theory in external magnetic field. The scaling
functions depend on the scaling parameter $\xi = h/|m|^{\frac{15}{8}}$, or
related parameter $\eta = m / h^{\frac{8}{15}}$. Analytic properties of $M_1$
in the high-T domain were discussed in arXiv:2203.11262. In this work, we study
analyticity of $M_1$ in the low-T domain. Important feature of this analytic
structure is represented by the Fisher-Langer's branch cut. The discontinuity
across this branch cut determines the behavior of $M_1$ at all complex $\xi$
via associated low-T dispersion relation. Also, we put forward the "extended
analyticity" conjecture for $M_1$ in the complex $\eta$-plane, similar to the
analyticity of the free energy density previously proposed in
arXiv:hep-th/0112167. The extended analyticity implies the "extended dispersion
relation", which we verify against the numerics from the Truncated Free Fermion
Approach (TFFSA), giving strong support to the conjecture.
