Ising field theory in a magnetic field: φ ³ coupling at T > T_c

We study the “three particle coupling” Γ111(ξ) , in 2d Ising Field Theory in a magnetic field, as the function of the scaling parameter ξ := h/(−m)^15/8, where m ∼ T_c − T and h ∼ H are scaled deviation from the critical temperature and scaled external field, respectively. The “φ ³ coupling” Γ111 is defined in terms of the residue of the 2 → 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”: (Γ111)2 , as the function of u := ξ ², is analytic in the whole complex u-plane except for the branch cut from – ∞ to – u ₀ ≈ – 0.03585, the latter branching point – u ₀ being associated with the Yang-Lee edge singularity. Under this assumption, the values of Γ111 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 Γ111 near the Yang-Lee branching point, as well as its known asymptotic at u → +∞. 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.