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We study the "three particle coupling" $\Gamma_{11}^{1}(\xi)$, in $2d$ IsingField 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 deviationfrom the critical temperature and scaled external field, respectively. The"$\varphi^3$ coupling" $\Gamma_{11}^1$ is defined in terms of the residue ofthe $2 \to 2$ elastic scattering amplitude at its pole associated with thelightest particle itself. We limit attention to the High-Temperature domain, sothat $m$ is negative. We suggest "standard analyticity": $(\Gamma_{11}^1)^2$,as the function of $u:=\xi^2$, is analytic in the whole complex $u$-planeexcept for the branch cut from $-\infty$ to $-u_0 \approx -0.03585$, the latterbranching point $-u_0$ being associated with the Yang-Lee edge singularity.Under this assumption, the values of $\Gamma_{11}^1$ at any complex $u$ areexpressed through the discontinuity across the branch cut. We suggestapproximation for this discontinuity which accounts for singular expansion of$\Gamma_{11}^1$ near the Yang-Lee branching point, as well as its knownasymptotic at $u\to +\infty$. The resulting dispersion relation agrees wellwith known exact data, and with numerics obtained via Truncated Free FermionSpace Approach. This work is part of extended project of studying the S-matrixof the Ising Field Theory in a magnetic field.

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