Ising field theory in a magnetic field: φ3 coupling at T > Tc

Abstract

Abstract We study the “three particle coupling” $$ {\Gamma}_{11}^1\left(\xi \right) $$ Γ 11 1 ξ , in 2d Ising Field Theory in a magnetic field, as the function of the scaling parameter ξ := h/(−m)15/8, where m ∼ Tc − T and h ∼ H are scaled deviation from the critical temperature and scaled external field, respectively. The “φ3 coupling” $$ {\Gamma}_{11}^1 $$ Γ 11 1 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”: $$ {\left({\Gamma}_{11}^1\right)}^2 $$ Γ 11 1 2 , as the function of u := ξ2, is analytic in the whole complex u-plane except for the branch cut from – ∞ to – u0 ≈ – 0.03585, the latter branching point – u0 being associated with the Yang-Lee edge singularity. Under this assumption, the values of $$ {\Gamma}_{11}^1 $$ Γ 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 $$ {\Gamma}_{11}^1 $$ Γ 11 1 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.