Ising Field Theory in a magnetic field: $\varphi^3$ coupling at $T > T_c$

We study the "three particle coupling" $\Gamma_{11}^{1}(\xi)$, in $2d$ Ising Field Theory in a magnetic field, as the function of the scaling parameter $\xi:=h/(-m)^{15/8}$, where $m \sim T_c-T$ and $h \sim H$ are scaled deviation from the critical temperature and scaled external field, respectively. The "$\varphi^3$ coupling" $\Gamma_{11}^1$ is defined in terms of the residue of the $2 \to 2$ elastic scattering amplitude at its pole associated with the lightest particle itself. We limit attention to the High-Temperature domain, so that $m$ is negative. We suggest "standard analyticity": $(\Gamma_{11}^1)^2$, as the function of $u:=\xi^2$, is analytic in the whole complex $u$-plane except for the branch cut from $-\infty$ to $-u_0 \approx -0.03585$, the latter branching point $-u_0$ being associated with the Yang-Lee edge singularity. Under this assumption, the values of $\Gamma_{11}^1$ at any complex $u$ are expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of $\Gamma_{11}^1$ near the Yang-Lee branching point, as well as its known asymptotic at $u\to +\infty$. The resulting dispersion relation agrees well with known exact data, and with numerics obtained via Truncated Free Fermion Space Approach. This work is part of extended project of studying the S-matrix of the Ising Field Theory in a magnetic field.