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

Abstract

Abstract We study the “three particle coupling” $$ {\Gamma}_{11}^1\left(\xi \right) $$ Γ111ξ , in 2dIsing Field Theory in a magnetic field, as the function of the scaling parameterξ:=h/(−m)15/8, wherem∼Tc−Tandh∼Hare scaled deviation from the critical temperature and scaled external field, respectively. The “φ3coupling” $$ {\Gamma}_{11}^1 $$ 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 thatmis negative. We suggest “standard analyticity”: $$ {\left({\Gamma}_{11}^1\right)}^2 $$ Γ1112 , as the function ofu:=ξ2, is analytic in the whole complexu-plane except for the branch cut from – ∞ to –u0≈ – 0.03585, the latter branching point –u0being associated with the Yang-Lee edge singularity. Under this assumption, the values of $$ {\Gamma}_{11}^1 $$ Γ111 at any complexuare expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of $$ {\Gamma}_{11}^1 $$ Γ111 near the Yang-Lee branching point, as well as its known asymptotic atu →+∞. 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.